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\begin{document}
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\begin{flushright}
hepph/9709356\\
version 7, January 2016
\end{flushright}
\begin{center}
{\Large\bf A Supersymmetry Primer}\\
\vspace{0.14in}
{\sc Stephen P.~Martin} \\
Department of Physics, Northern Illinois University, DeKalb IL 60115
\end{center}
\begin{center}
\begin{minipage}[]{0.86\linewidth}
I provide a pedagogical introduction to supersymmetry. The level of
discussion is aimed at readers who are familiar with the Standard Model
and quantum field theory, but who have had little or no prior exposure to
supersymmetry. Topics covered include: motivations for supersymmetry, the
construction of supersymmetric Lagrangians, superspace and superfields,
soft supersymmetrybreaking interactions, the Minimal Supersymmetric
Standard Model (MSSM), $R$parity and its consequences, the origins of
supersymmetry breaking, the mass spectrum of the MSSM, decays of
supersymmetric particles, experimental signals for supersymmetry, and
some extensions of the minimal framework.
\end{minipage}
\end{center}
\tableofcontents
\newpage
\setlength{\baselineskip}{1.05\baselineskip}
%\renewcommand{\baselinestretch}{1.15}
\begin{quotation}
\noindent
\mbox{``We are, I think, in the right Road of Improvement, for we are making Experiments."}\\
Benjamin Franklin
\end{quotation}
\section{Introduction}\label{sec:intro}
\setcounter{equation}{0}
\setcounter{figure}{0}
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\setcounter{footnote}{1}
The Standard Model of highenergy physics, augmented by neutrino masses,
provides a remarkably successful description of presently known phenomena.
The experimental frontier has advanced into the TeV range with no
unambiguous hints of additional structure. Still, it seems clear that the
Standard Model is a work in progress and will have to be extended to
describe physics at higher energies. Certainly, a new framework will be
required at the reduced Planck scale
$\MPlanck = (8 \pi G_{\rm Newton})^{1/2} = 2.4 \times 10^{18}$ GeV,
where quantum gravitational effects become important. Based only on a
proper respect for the power of Nature to surprise us, it seems nearly as
obvious that new physics exists in the 16 orders of magnitude in energy
between the presently explored territory near the electroweak scale,
$M_W$, and the Planck scale.
The mere fact that the ratio $\MPlanck/M_W$ is so huge is already a
powerful clue to the character of physics beyond the Standard Model,
because of the infamous ``hierarchy problem" \cite{hierarchyproblem}. This
is not really a difficulty with the Standard Model itself, but rather a
disturbing sensitivity of the Higgs potential to new physics in almost any
imaginable extension of the Standard Model. The electrically neutral part
of the Standard Model Higgs field is a complex scalar $H$ with a classical
potential
\beq
V = m_H^2 H^2 + {\lambda} H^4\> .
\label{higgspotential}
\eeq
The Standard Model requires a nonvanishing vacuum expectation value (VEV)
for $H$ at the minimum of the potential. This occurs if $\lambda > 0$
and $m_H^2 < 0$, resulting in $\langle H \rangle =
\sqrt{m_H^2/2\lambda}$.
We know experimentally that $\langle H \rangle$ is approximately 174 GeV
from measurements of the properties of
the weak interactions.
The 2012 discovery
%\cite{Higgsdiscovery,Higgsproperties,Higgsmass}
\cite{Higgsdiscovery}\cite{Higgsmass}
of the Higgs boson with a mass
near 125 GeV implies that, assuming the Standard Model is correct as an
effective field theory, $\lambda = 0.126$ and $m_H^2 = (\mbox{92.9 GeV})^2$.
(These are running $\msbar$ parameters evaluated at a renormalization scale
equal to the topquark mass, and include the effects of 2loop corrections.)
The problem is that $m_H^2$ receives enormous quantum
corrections from the virtual effects of every particle or other phenomenon
that couples, directly or indirectly, to the Higgs field.
For example, in Figure \ref{fig:higgscorr1}a we have a correction to $m_H^2$
from a loop containing a Dirac fermion $f$ with mass $m_f$.%
\begin{figure}[b]
\begin{center}
\begin{picture}(108,63)(54,27)
\SetWidth{0.9}
\DashLine(55,0)(22,0){4}
\DashLine(55,0)(22,0){4}
\CArc(0,0)(22,0,360)
\Text(50,7)[c]{$H$}
\Text(0,32)[c]{$f$}
\Text(0,34.5)[c]{(a)}
\end{picture}
\hspace{1.4cm}
\begin{picture}(108,63)(54,27)
\SetWidth{0.9}
\Text(0,37)[c]{$S$}
\Text(41,5)[c]{$H$}
\DashLine(46,12)(46,12){4}
\SetWidth{1.5}
\DashCArc(0,8)(20,90,270){5}
\Text(0,34.5)[c]{(b)}
\end{picture}
\end{center}
\vspace{0.3cm}
\caption{Oneloop quantum corrections to the Higgs squared mass parameter
$m_H^2$, due to (a) a Dirac fermion $f$, and (b) a scalar $S$.
\label{fig:higgscorr1}}
\end{figure}
If the Higgs
field couples to $f$ with a term in the Lagrangian $\lambda_f H \sbar f
f$, then the Feynman diagram in Figure \ref{fig:higgscorr1}a yields a
correction
\beq
\Delta m_H^2 \>=\>
{\lambda_f^2\over 8 \pi^2} \Lambda_{\rm UV}^2 + \ldots .
\label{quaddiv1}
\eeq
Here $\Lambda_{\rm UV}$ is an ultraviolet momentum cutoff used to regulate
the loop integral; it should be interpreted as at least the energy scale
at which new physics enters to alter the highenergy behavior of the
theory. The ellipses represent terms proportional to $m_f^2$, which
grow at most logarithmically
with $\Lambda_{\rm UV}$ (and actually differ for the real and imaginary
parts of $H$). Each of the leptons and quarks of the Standard Model can
play the role of $f$; for quarks, eq.~(\ref{quaddiv1}) should be
multiplied by 3 to account for color. The largest correction comes when
$f$ is the top quark with $\lambda_f\approx 0.94$. The problem is that if
$\Lambda_{\rm UV}$ is of order $\MPlanck$, say, then this quantum
correction to $m_H^2$ is some 30 orders of magnitude larger than the
required value of $m_H^2 \approx (92.9$ GeV$)^2$. This is only directly a
problem for corrections to the Higgs scalar boson squared mass, because
quantum corrections to fermion and gauge boson masses do not have the
direct quadratic sensitivity to $\Lambda_{\rm UV}$ found in
eq.~(\ref{quaddiv1}). However, the quarks and leptons and the electroweak
gauge bosons $Z^0$, $W^\pm$ of the Standard Model all obtain masses from
$\langle H \rangle$, so that the entire mass spectrum of the Standard
Model is directly or indirectly sensitive to the cutoff
$\Lambda_{\rm UV}$.
One could imagine that the solution is to simply pick a $\Lambda_{\rm UV}$
that is not too large. But then one still must concoct some new physics at
the scale $\Lambda_{\rm UV}$ that not only alters the propagators in the
loop, but actually cuts off the loop integral. This is not easy to do in a
theory whose Lagrangian does not contain more than two derivatives, and
higherderivative theories generally suffer from a failure of either
unitarity or causality \cite{EliezerWoodard}. In string theories, loop
integrals are nevertheless cut off at high Euclidean momentum $p$ by
factors $e^{p^2/\Lambda^2_{\rm UV}}$. However, then $\Lambda_{\rm UV}$
is a string scale that is usually\footnote{Some attacks on the
hierarchy problem, not reviewed here,
are based on the proposition that the ultimate cutoff
scale is actually close to the electroweak scale, rather
than the apparent Planck scale.} thought to
be not very far below $\MPlanck$.
Furthermore, there are contributions similar to eq.~(\ref{quaddiv1}) from the virtual
effects of any heavy particles that might exist, and these involve
the masses of the heavy particles (or other high physical mass scales),
not just the cutoff. It cannot be overemphasized that merely choosing
a regulator with no quadratic divergences does {\em not} address the hierarchy problem.
The problem is not really the quadratic divergences, but rather the quadratic sensitivity
to high mass scales. The latter are correlated with quadratic divergences for some,
but not all, choices of ultraviolet regulator. The absence of quadratic divergences
is a necessary, but not sufficient, criterion for avoiding the hierarchy problem.
For example, suppose there exists a heavy complex scalar particle $S$ with
mass $m_S$ that couples to the Higgs with a Lagrangian term $ \lambda_S
H^2 S^2$. Then the Feynman diagram in Figure~\ref{fig:higgscorr1}b
gives a correction
\beq
\Delta m_H^2 \>=\> {\lambda_S\over 16 \pi^2}
\left [\Lambda_{\rm UV}^2  2 m_S^2
\> {\rm ln}(\Lambda_{\rm UV}/m_S) + \ldots
\right ].
\label{quaddiv2}
\eeq
If one rejects the possibility of a physical interpretation of
$\Lambda_{\rm UV}$ and uses dimensional regularization on the loop
integral instead of a momentum cutoff, then there will be no $\Lambda_{\rm
UV}^2$ piece. However, even then the term proportional to $m_S^2$ cannot
be eliminated without the physically unjustifiable tuning of a
counterterm specifically for that purpose. This illustrates that
$m_H^2$ is sensitive to the
masses of the {\it heaviest} particles that $H$ couples to; if $m_S$ is
very large, its effects on the Standard Model do not decouple, but instead
make it difficult to understand why $m_H^2$ is so small.
This problem arises even if there is no direct coupling between the
Standard Model Higgs boson and the unknown heavy particles. For example,
suppose there exists a heavy fermion $F$ that, unlike the quarks and
leptons of the Standard Model, has vectorlike quantum numbers and
therefore gets a large mass $m_F$ without coupling to the Higgs field. [In
other words, an arbitrarily large mass term of the form $m_F \overline F
F$ is not forbidden by any symmetry, including weak isospin $SU(2)_L$.] In
that case, no diagram like Figure~\ref{fig:higgscorr1}a exists for $F$.
Nevertheless there will be a correction to $m_H^2$ as long as $F$ shares
some gauge interactions with the Standard Model Higgs field; these may be
the familiar electroweak interactions, or some unknown gauge forces that
are broken at a very high energy scale inaccessible to experiment. In
either case, the twoloop Feynman diagrams in Figure~\ref{fig:higgscorr2}
yield a correction%
\begin{figure}
\begin{center}
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%\Text(0,35.5)[]{(a)}
\Text(0,28)[]{$F$}
\Text(48,15)[]{$H$}
\end{picture}
%
\hspace{1.6cm}
%
\begin{picture}(98,52)(49,17)
\SetWidth{0.9}
\DashLine(45,22)(0,22){4}
\DashLine(45,22)(0,22){4}
\CArc(0,12)(22,0,180)
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\Photon(0,22)(22,12){2}{4.5}
%\Text(0,35.5)[]{(b)}
\Text(0,28)[]{$F$}
\Text(41,15)[]{$H$}
\end{picture}
\vspace{0.25cm}
\end{center}
\caption{Twoloop corrections to the Higgs squared mass parameter
involving a heavy fermion $F$ that couples only indirectly to the Standard
Model Higgs through gauge interactions. \label{fig:higgscorr2}}
\end{figure}
%
\beq
\Delta m_H^2 \>=\> C_H T_F \left ( {g^2 \over 16 \pi^2} \right )^2
\left [ a \Lambda_{\rm UV}^2 + 24 m_F^2 \>{\rm ln} (\Lambda_{\rm UV}/m_F)
+ \ldots \right ],
\label{quaddiv3}
\eeq
where $C_H$ and $T_F$ are group theory factors\footnote{Specifically,
$C_H$ is the quadratic Casimir invariant of $H$, and $T_F$ is the Dynkin
index of $F$ in a normalization such that $T_F=1$ for a Dirac fermion (or
two Weyl fermions) in a fundamental representation of $SU(n)$.} of order
1, and $g$ is the appropriate gauge coupling. The coefficient $a$ depends
on the method used to cut off the momentum integrals. It does not arise at
all if one uses dimensional regularization, but the $m_F^2$ contribution
is always present with the given coefficient. The numerical factor
$(g^2/16 \pi^2)^2$ may be quite small (of order $10^{5}$ for electroweak
interactions), but the important point is that these contributions to
$\Delta m_H^2$ are sensitive both to the largest masses and to the
physical ultraviolet cutoff in the theory, presumably of order $\MPlanck$. The
``natural" squared mass of a fundamental Higgs scalar, including quantum
corrections, therefore seems to be more like $\MPlanck^2$ than the
experimental value. Even very indirect contributions from
Feynman diagrams with three or more loops can give unacceptably large
contributions to $\Delta m_H^2$. The argument above applies not just for
heavy particles, but for arbitrary highscale physical phenomena such as
condensates or additional compactified dimensions.
It could be that the Higgs boson field is not fundamental, but rather is the result of
a composite field or collective phenomenon. Such ideas are certainly still worth
exploring, although they typically present difficulties in their simplest
forms. In particular, so far the 125 GeV Higgs boson does appear to have properties
consistent with a fundamental scalar field. Or, it could be that the ultimate
ultraviolet cutoff scale, and therefore the mass scales of all presently undiscovered particles and condensates, are much lower than the Planck scale.
But, if the Higgs boson is a fundamental particle, and there really
is physics far above the electroweak scale, then we have two remaining
options: either we must make the rather bizarre assumption that {\it none} of the
highmass particles or condensates couple (even
indirectly or extremely weakly) to the Higgs scalar field, or else some
striking cancellation is needed between the various contributions to
$\Delta m_H^2$.
The systematic cancellation of the dangerous contributions to $\Delta
m_H^2$ can only be brought about by the type of conspiracy that is better
known to physicists as a symmetry. Comparing eqs.~(\ref{quaddiv1}) and
(\ref{quaddiv2}) strongly suggests that the new symmetry ought to relate
fermions and bosons, because of the relative minus sign between fermion
loop and boson loop contributions to $\Delta m_H^2$. (Note that
$\lambda_S$ must be positive if the scalar potential is to be bounded from
below.) If each of the quarks and leptons of the Standard Model is
accompanied by two complex scalars with $\lambda_S = \lambda_f^2$, then
the $\Lambda_{\rm UV}^2$ contributions of Figures~\ref{fig:higgscorr1}a
and \ref{fig:higgscorr1}b will neatly cancel \cite{quadscancel}. Clearly,
more restrictions on the theory will be necessary to ensure that this
success persists to higher orders, so that, for example, the contributions
in Figure~\ref{fig:higgscorr2} and eq.~(\ref{quaddiv3}) from a very heavy
fermion are canceled by the twoloop effects of some very heavy bosons.
Fortunately, the cancellation of all such contributions to scalar masses
is not only possible, but is actually unavoidable, once we merely assume
that there exists a symmetry relating fermions and bosons, called a
{\it supersymmetry}.
A supersymmetry transformation turns a bosonic state into a fermionic
state, and vice versa. The operator $Q$ that generates such
transformations must be an anticommuting spinor, with
\beq
Q {\rm Boson}\rangle = {\rm Fermion }\rangle, \qquad\qquad
Q {\rm Fermion}\rangle = {\rm Boson }\rangle .
\eeq
Spinors are intrinsically complex objects, so $Q^\dagger$ (the hermitian
conjugate of $Q$) is also a symmetry generator. Because $Q$ and
$Q^\dagger$ are fermionic operators, they carry spin angular momentum 1/2,
so it is clear that supersymmetry must be a spacetime symmetry. The
possible forms for such symmetries in an interacting quantum field theory
are highly restricted by the HaagLopuszanskiSohnius extension \cite{HLS} of the
ColemanMandula theorem \cite{ColemanMandula}. For realistic theories that, like the
Standard Model, have chiral fermions (i.e., fermions whose left and
righthanded pieces transform differently under the gauge group) and thus
the possibility of parityviolating interactions, this theorem implies
that the generators $Q$ and $Q^\dagger$ must satisfy an algebra of
anticommutation and commutation relations with the schematic form
\beq
&&\{ Q, Q^\dagger \} = P^\mu , \label{susyalgone}
\\
&&\{ Q,Q \} = \{ Q^\dagger , Q^\dagger \} = 0 , \label{susyalgtwo}
\\
&&[ P^\mu , Q ] = [P^\mu, Q^\dagger ] = 0 ,\label{susyalgthree}
\eeq
where $P^\mu$ is the fourmomentum generator of spacetime translations.
Here we have ruthlessly suppressed the spinor indices on $Q$ and
$Q^\dagger$; after developing some notation we will, in section
\ref{subsec:susylagr.freeWZ}, derive the precise version of
eqs.~(\ref{susyalgone})(\ref{susyalgthree}) with indices restored. In the
meantime, we simply note that the appearance of $P^\mu$ on the righthand
side of eq.~(\ref{susyalgone}) is unsurprising, because it transforms under
Lorentz boosts and rotations as a spin1 object while $Q$ and $Q^\dagger$
on the lefthand side each transform as spin1/2 objects.
The singleparticle states of a supersymmetric theory fall into
irreducible representations of the supersymmetry algebra, called {\it
supermultiplets}. Each supermultiplet contains both fermion and boson
states, which are commonly known as {\it superpartners} of each other. By
definition, if $\Omega\rangle$ and $\Omega^\prime \rangle$ are members
of the same supermultiplet, then $\Omega^\prime\rangle$ is proportional
to some combination of $Q$ and $\dagg{Q}$ operators acting on
$\Omega\rangle $, up to a spacetime translation or rotation. The
squaredmass operator $\BDpos P^2$ commutes with the operators $Q$,
$\dagg{Q}$, and with all spacetime rotation and translation operators, so
it follows immediately that particles inhabiting the same irreducible
supermultiplet must have equal eigenvalues of $\BDpos P^2$, and therefore
equal masses.
The supersymmetry generators $Q,Q^\dagger$ also commute with the
generators of gauge transformations. Therefore particles in the same
supermultiplet must also be in the same representation of the gauge group,
and so must have the same electric charges, weak isospin, and color
degrees of freedom.
Each supermultiplet contains an equal number of fermion and boson degrees
of freedom. To prove this, consider the operator $(1)^{2s}$ where $s$ is
the spin angular momentum. By the spinstatistics theorem, this operator
has eigenvalue $+1$ acting on a bosonic state and eigenvalue $1$ acting
on a fermionic state. Any fermionic operator will turn a bosonic state
into a fermionic state and vice versa. Therefore $(1)^{2s}$ must
anticommute with every fermionic operator in the theory, and in particular
with $Q$ and $Q^\dagger$. Now, within a given supermultiplet, consider the
subspace of states $ i \rangle$ with the same eigenvalue $p^\mu$ of the
fourmomentum operator $P^\mu$. In view of eq.~(\ref{susyalgthree}), any
combination of $Q$ or $Q^\dagger$ acting on $i\rangle$ must give another
state $i^\prime\rangle$ with the same fourmomentum eigenvalue. Therefore
one has a completeness relation $\sum_i i\rangle\langle i  = 1$ within
this subspace of states. Now one can take a trace over all such states of
the operator $(1)^{2s} P^\mu$ (including each spin helicity state
separately):
\beq
\sum_i \langle i  (1)^{2s} P^\mu  i \rangle
&=&
\sum_i \langle i  (1)^{2s} Q Q^\daggeri\rangle
+\sum_i\langle i  (1)^{2s} Q^\dagger Q  i \rangle
\nonumber\\
&=&
\sum_i \langle i  (1)^{2s} Q Q^\dagger  i \rangle
+ \sum_i \sum_j \langle i  (1)^{2s} Q^\dagger j \rangle \langle j  Q
 i \rangle\qquad{}
\nonumber\\
&=&
\sum_i \langle i  (1)^{2s} Q Q^\dagger  i \rangle +
\sum_j \langle j  Q (1)^{2s} Q^\dagger  j \rangle
\nonumber\\
&=&\sum_i \langle i  (1)^{2s} Q Q^\dagger  i \rangle 
\sum_j \langle j  (1)^{2s} Q Q^\dagger  j \rangle
\nonumber \\
&=& 0.
\eeq
The first equality follows from the supersymmetry algebra relation
eq.~(\ref{susyalgone}); the second and third from use of the completeness
relation; and the fourth from the fact that $(1)^{2s}$ must anticommute
with $Q$. Now $\sum_i \langle i  (1)^{2s} P^\mu  i \rangle = \, p^\mu$
Tr[$(1)^{2s}$] is just proportional to the number of bosonic degrees of
freedom $n_B$ minus the number of fermionic degrees of freedom $n_F$ in
the trace, so that
\beq
n_B= n_F
\label{nbnf}
\eeq
must hold for a given $p^\mu\not= 0$ in each supermultiplet.
The simplest possibility for a supermultiplet consistent with
eq.~(\ref{nbnf}) has a single Weyl fermion (with two spin helicity states,
so $n_F=2$) and two real scalars (each with $n_B=1$). It is natural to
assemble the two real scalar degrees of freedom into a complex scalar
field; as we will see below this provides for convenient formulations of
the supersymmetry algebra, Feynman rules, supersymmetryviolating effects,
etc. This combination of a twocomponent Weyl fermion and a complex scalar
field is called a {\it chiral} or {\it matter} or {\it scalar}
supermultiplet.
The nextsimplest possibility for a supermultiplet contains a spin1
vector boson. If the theory is to be renormalizable, this must be a gauge
boson that is massless, at least before the gauge symmetry is
spontaneously broken. A massless spin1 boson has two helicity states, so
the number of bosonic degrees of freedom is $n_B=2$. Its superpartner is
therefore a massless spin1/2 Weyl fermion, again with two helicity
states, so $n_F=2$. (If one tried to use a massless spin3/2 fermion
instead, the theory would not be renormalizable.) Gauge bosons must
transform as the adjoint representation of the gauge group, so their
fermionic partners, called {\it gauginos}, must also. Because the adjoint
representation of a gauge group is always its own conjugate, the gaugino
fermions must have the same gauge transformation properties for
lefthanded and for righthanded components. Such a combination of
spin1/2 gauginos and spin1 gauge bosons is called a {\it gauge} or {\it
vector} supermultiplet.
If we include gravity, then the spin2 graviton (with 2 helicity states,
so $n_B=2$) has a spin$3/2$ superpartner called the gravitino. The
gravitino would be massless if supersymmetry were unbroken, and so it has
$n_F=2$ helicity states.
There are other possible combinations of particles with spins that can
satisfy eq.~(\ref{nbnf}). However, these are always reducible to
combinations\footnote{For example, if a gauge symmetry were to
spontaneously break without breaking supersymmetry, then a massless vector
supermultiplet would ``eat'' a chiral supermultiplet, resulting in a
massive vector supermultiplet with physical degrees of freedom consisting
of a massive vector ($n_B=3$), a massive Dirac fermion formed from the
gaugino and the chiral fermion ($n_F=4$), and a real scalar ($n_B=1$).} of
chiral and gauge supermultiplets if they have renormalizable interactions,
except in certain theories with ``extended" supersymmetry. Theories with
extended supersymmetry have more than one distinct copy of the
supersymmetry generators $Q,\dagg{Q}$. Such models are mathematically
interesting, but evidently do not have any phenomenological prospects. The
reason is that extended supersymmetry in fourdimensional field theories
cannot allow for chiral fermions or parity violation as observed in the
Standard Model. So we will not discuss such possibilities further,
although extended supersymmetry in higherdimensional field theories might
describe the real world if the extra dimensions are compactified in an
appropriate way, and extended supersymmetry in four dimensions provides
interesting toy models and calculation tools.
The ordinary, nonextended, phenomenologically
viable type of supersymmetric model is sometimes called $N=1$
supersymmetry, with $N$ referring to the number of supersymmetries (the
number of distinct copies of $Q, \dagg{Q}$).
In a supersymmetric extension of the Standard Model
%\cite{FayetHsnu,FayetMSSM,Rparity},
\cite{FayetHsnu}\cite{Rparity},
each of the known fundamental
particles is therefore in either a chiral or gauge supermultiplet, and
must have a superpartner with spin differing by 1/2 unit. The first step
in understanding the exciting phenomenological consequences of this
prediction is to decide exactly how the known particles fit into
supermultiplets, and to give them appropriate names. A crucial observation
here is that only chiral supermultiplets can contain fermions whose
lefthanded parts transform differently under the gauge group than their
righthanded parts. All of the Standard Model fermions (the known quarks
and leptons) have this property, so they must be members of chiral
supermultiplets. The bosonic partners of the quarks and leptons therefore must
be spin0, and not spin1 vector bosons.\footnote{In particular, one cannot attempt to make a
spin1/2 neutrino be the superpartner of the spin1 photon; the neutrino
is in a doublet, and the photon is neutral, under weak isospin.}
The names
for the spin0 partners of the quarks and leptons are constructed by
prepending an ``s", for scalar. So, generically they are called {\it
squarks} and {\it sleptons} (short for ``scalar quark" and ``scalar
lepton"), or sometimes {\it sfermions}.
The lefthanded and righthanded pieces of the quarks and
leptons are separate twocomponent Weyl fermions with different gauge
transformation properties in the Standard Model, so each must have its own
complex scalar partner. The symbols for the squarks and sleptons are the
same as for the corresponding fermion, but with a tilde
($\phantom{.}\stilde{\phantom{.}}\phantom{.}$)
used to denote the superpartner of a Standard
Model particle. For example, the superpartners of the lefthanded and
righthanded parts of the electron Dirac field are called left and
righthanded selectrons, and are denoted $\stilde e_L$ and $\stilde e_R$.
It is important to keep in mind that the ``handedness" here does not refer
to the helicity of the selectrons (they are spin0 particles) but to that
of their superpartners. A similar nomenclature applies for smuons and
staus: $\stilde \mu_L$, $\stilde\mu_R$, $\stilde\tau_L$, $\stilde \tau_R$.
The Standard Model neutrinos (neglecting their very small masses) are
always lefthanded, so the sneutrinos are denoted generically by
$\stilde\nu$, with a possible subscript indicating which lepton flavor
they carry: $\stilde\nu_e$, $\stilde\nu_\mu$, $\stilde\nu_\tau$. Finally,
a complete list of the squarks is $\stilde q_L$, $\stilde q_R$ with
$q=u,d,s,c,b,t$. The gauge interactions of each of these squark and
slepton fields are the same as for the corresponding Standard Model
fermions; for instance, the lefthanded squarks $\stilde u_L$ and $\stilde
d_L$ couple to the $W$ boson, while $\stilde u_R$ and $\stilde d_R$ do
not.\setcounter{footnote}{1}
It seems clear that the Higgs scalar boson must reside in a chiral
supermultiplet, since it has spin 0. Actually, it turns out that just one
chiral supermultiplet is not enough. One reason for this is that
if there were only one Higgs chiral supermultiplet, the electroweak gauge
symmetry would suffer a gauge anomaly, and would be inconsistent as a
quantum theory. This is because the conditions for cancellation of gauge
anomalies include $ {\rm Tr}[T_3^2 Y] = {\rm Tr}[Y^3] = 0, $ where $T_3$
and $Y$ are the third component of weak isospin and the weak hypercharge,
respectively, in a normalization where the ordinary electric charge is
$Q_{\rm EM} = T_3 + Y$. The traces run over all of the lefthanded Weyl
fermionic degrees of freedom in the theory. In the Standard Model, these
conditions are already satisfied, somewhat miraculously, by the known
quarks and leptons. Now, a fermionic partner of a Higgs chiral
supermultiplet must be a weak isodoublet with weak hypercharge $Y=1/2$ or
$Y=1/2$. In either case alone, such a fermion will make a nonzero
contribution to the traces and spoil the anomaly cancellation. This can be
avoided if there are two Higgs supermultiplets, one with each of $Y=\pm
1/2$, so that the total contribution to the anomaly traces from the two
fermionic members of the Higgs chiral supermultiplets vanishes by
cancellation. As we will see in section \ref{subsec:mssm.superpotential},
both of these are also necessary for another completely different reason:
because of the structure of supersymmetric theories, only a $Y=1/2$ Higgs
chiral supermultiplet can have the Yukawa couplings necessary to give
masses to charge $+2/3$ uptype quarks (up, charm, top), and only a
$Y=1/2$ Higgs can have the Yukawa couplings necessary to give masses to
charge $1/3$ downtype quarks (down, strange, bottom) and to the charged
leptons.
We will call the $SU(2)_L$doublet complex scalar fields with
$Y=1/2$ and $Y=1/2$ by the names $H_u$ and $H_d$,
respectively.\footnote{Other notations in the literature have
$H_1, H_2$ or $H,\sbar H$ instead of $H_u, H_d$. The notation used here
has the virtue of making it easy to remember which Higgs VEVs
gives masses to which type of quarks.} The weak isospin components of
$H_u$ with $T_3=(1/2$, $1/2$) have electric charges $1$, $0$
respectively, and are denoted ($H_u^+$, $H_u^0$). Similarly, the
$SU(2)_L$doublet complex scalar $H_d$ has $T_3=(1/2$, $1/2$) components
($H_d^0$, $H_d^$). The neutral scalar that corresponds to the physical
Standard Model Higgs boson is in a linear combination of $H_u^0$ and
$H_d^0$; we will discuss this further in section
\ref{subsec:MSSMspectrum.Higgs}. The generic nomenclature for a spin1/2
superpartner is to append ``ino" to the name of the Standard Model
particle, so the fermionic partners of the Higgs scalars are called
higgsinos. They are denoted by $\stilde H_u$, $\stilde H_d$ for the
$SU(2)_L$doublet lefthanded Weyl spinor fields, with weak isospin
components $\stilde H_u^+$, $\stilde H_u^0$ and $\stilde H_d^0$, $\stilde
H_d^$.
\renewcommand{\arraystretch}{1.4}
\begin{table}[tb]
\begin{center}
\begin{tabular}{ccccc}
\hline
\multicolumn{2}{c}{Names}
& spin 0 & spin 1/2 & $SU(3)_C ,\, SU(2)_L ,\, U(1)_Y$
\\ \hline\hline
squarks, quarks & $Q$ & $({\stilde u}_L\>\>\>{\stilde d}_L )$&
$(u_L\>\>\>d_L)$ & $(\>{\bf 3},\>{\bf 2}\>,\>{1\over 6})$
\\
($\times 3$ families) & $\sbar u$
&${\stilde u}^*_R$ & $u^\dagger_R$ &
$(\>{\bf \overline 3},\> {\bf 1},\> {2\over 3})$
\\ & $\sbar d$ &${\stilde d}^*_R$ & $d^\dagger_R$ &
$(\>{\bf \overline 3},\> {\bf 1},\> {1\over 3})$
\\ \hline
sleptons, leptons & $L$ &$({\stilde \nu}\>\>{\stilde e}_L )$&
$(\nu\>\>\>e_L)$ & $(\>{\bf 1},\>{\bf 2}\>,\>{1\over 2})$
\\
($\times 3$ families) & $\sbar e$
&${\stilde e}^*_R$ & $e^\dagger_R$ & $(\>{\bf 1},\> {\bf 1},\>1)$
\\ \hline
Higgs, higgsinos &$H_u$ &$(H_u^+\>\>\>H_u^0 )$&
$(\stilde H_u^+ \>\>\> \stilde H_u^0)$&
$(\>{\bf 1},\>{\bf 2}\>,\>+{1\over 2})$
\\ &$H_d$ & $(H_d^0 \>\>\> H_d^)$ & $(\stilde H_d^0 \>\>\> \stilde H_d^)$&
$(\>{\bf 1},\>{\bf 2}\>,\>{1\over 2})$
\\ \hline
\end{tabular}
\caption{Chiral supermultiplets in the Minimal Supersymmetric Standard Model.
The spin$0$ fields are complex scalars, and the spin$1/2$ fields are
lefthanded twocomponent Weyl fermions.\label{tab:chiral}}
\vspace{0.6cm}
\end{center}
\end{table}
We have now found all of the chiral supermultiplets of a minimal
phenomenologically viable extension of the Standard Model. They are
summarized in Table \ref{tab:chiral},
classified according to their transformation
properties under the Standard Model gauge group $SU(3)_C\times SU(2)_L
\times U(1)_Y$, which combines $u_L,d_L$ and $\nu,e_L$ degrees of freedom
into $SU(2)_L$ doublets. Here we follow a standard convention, that all
chiral supermultiplets are defined in terms of lefthanded Weyl spinors,
so that the {\it conjugates} of the righthanded quarks and leptons (and
their superpartners) appear in Table \ref{tab:chiral}.
This protocol for defining chiral
supermultiplets turns out to be very useful for constructing
supersymmetric Lagrangians, as we will see in section \ref{sec:susylagr}.
It is also useful to have a symbol for each of the chiral supermultiplets
as a whole; these are indicated in the second column of
Table \ref{tab:chiral}. Thus, for
example, $Q$ stands for the $SU(2)_L$doublet chiral supermultiplet
containing $\stilde u_L,u_L$ (with weak isospin component $T_3=1/2$), and
$\stilde d_L, d_L$ (with $T_3=1/2$), while $\sbar u$ stands for the
$SU(2)_L$singlet supermultiplet containing $\stilde u_R^*, u_R^\dagger$.
There are three families for each of the quark and lepton supermultiplets,
Table \ref{tab:chiral} lists the firstfamily representatives. A family
index $i=1,2,3$ can be affixed to the chiral supermultiplet names ($Q_i$,
$\sbar u_i, \ldots$) when needed, for example
$(\sbar e_1, \sbar e_2, \sbar e_3)=
(\sbar e, \sbar \mu, \sbar \tau)$. The bar on $\sbar u$, $\sbar d$, $\sbar
e$ fields is part of the name, and does not denote any kind of
conjugation.
The Higgs chiral supermultiplet $H_d$
(containing $H_d^0$, $H_d^$, $\stilde H_d^0$, $\stilde H_d^$) has
exactly the same Standard Model gauge quantum numbers as the lefthanded
sleptons and leptons $L_i$, for example ($\stilde \nu$, $\stilde e_L$, $\nu$,
$e_L$). Naively, one might therefore suppose that we could have been more
economical in our assignment by taking a neutrino and a Higgs scalar to be
superpartners, instead of putting them in separate supermultiplets. This
would amount to the proposal that the Higgs boson and a sneutrino should
be the same particle. This attempt played a key role in some of the first
attempts to connect supersymmetry to phenomenology \cite{FayetHsnu}, but
it is now known to not work. Even ignoring the anomaly cancellation
problem mentioned above, many insoluble phenomenological problems would
result, including leptonnumber nonconservation and a mass for at least
one of the neutrinos in gross violation of experimental bounds. Therefore,
all of the superpartners of Standard Model particles are really new
particles, and cannot be identified with some other Standard Model state.
\renewcommand{\arraystretch}{1.55}
\begin{table}[t]
\begin{center}
\begin{tabular}{cccc}
\hline
Names & spin 1/2 & spin 1 & $SU(3)_C, \> SU(2)_L,\> U(1)_Y$\\
\hline\hline
gluino, gluon &$ \stilde g$& $g$ & $(\>{\bf 8},\>{\bf 1}\>,\> 0)$
\\
\hline
winos, W bosons & $ \stilde W^\pm\>\>\> \stilde W^0 $&
$W^\pm\>\>\> W^0$ & $(\>{\bf 1},\>{\bf 3}\>,\> 0)$
\\
\hline
bino, B boson &$\stilde B^0$&
$B^0$ & $(\>{\bf 1},\>{\bf 1}\>,\> 0)$
\\
\hline
\end{tabular}
\caption{Gauge supermultiplets in
the Minimal Supersymmetric Standard Model.\label{tab:gauge}}
\vspace{0.45cm}
\end{center}
\end{table}
The vector bosons of the Standard Model clearly must reside in gauge
supermultiplets. Their fermionic superpartners are generically referred to
as gauginos. The $SU(3)_C$ color gauge interactions of QCD are mediated by
the gluon, whose spin1/2 coloroctet supersymmetric partner is the
gluino. As usual, a tilde is used to denote the supersymmetric partner of
a Standard Model state, so the symbols for the gluon and gluino are $g$
and $\stilde g$ respectively. The electroweak gauge symmetry
$SU(2)_L\times U(1)_Y$ is associated with spin1 gauge bosons $W^+, W^0,
W^$ and $B^0$, with spin1/2 superpartners $\stilde W^+, \stilde W^0,
\stilde W^$ and $\stilde B^0$, called {\it winos} and {\it bino}. After
electroweak symmetry breaking, the $W^0$, $B^0$ gauge eigenstates mix to
give mass eigenstates $Z^0$ and $\gamma$. The corresponding gaugino
mixtures of $\stilde W^0$ and $\stilde B^0$ are called zino ($\stilde Z^0$)
and photino ($\stilde \gamma$); if supersymmetry were unbroken, they would
be mass eigenstates with masses $m_Z$ and 0. Table \ref{tab:gauge}
summarizes the gauge
supermultiplets of a minimal supersymmetric extension of the Standard
Model.
The chiral and gauge supermultiplets in Tables \ref{tab:chiral} and
\ref{tab:gauge} make up the
particle content of the Minimal Supersymmetric Standard Model (MSSM). The
most obvious and interesting feature of this theory is that none of the
superpartners of the Standard Model particles has been discovered as of
this writing. If supersymmetry were unbroken, then there would have to be
selectrons $\stilde e_L$ and $\stilde e_R$ with masses exactly equal to
$m_e = 0.511...$ MeV. A similar statement applies to each of the other
sleptons and squarks, and there would also have to be a massless gluino
and photino. These particles would have been extraordinarily easy to
detect long ago. Clearly, therefore, {\it supersymmetry is a broken
symmetry} in the vacuum state chosen by Nature.
An important clue as to the nature of supersymmetry breaking can be
obtained by returning to the motivation provided by the hierarchy problem.
Supersymmetry forced us to introduce two complex scalar fields for each
Standard Model Dirac fermion, which is just what is needed to enable a
cancellation of the quadratically sensitive $(\Lambda_{\rm UV}^2)$ pieces
of eqs.~(\ref{quaddiv1}) and (\ref{quaddiv2}). This sort of cancellation
also requires that the associated dimensionless couplings should be
related (for example $\lambda_S = \lambda_f^2$). The necessary relationships
between couplings indeed occur in unbroken supersymmetry, as we will see
in section \ref{sec:susylagr}. In fact, unbroken supersymmetry guarantees
that quadratic divergences in scalar squared masses, and therefore the quadratic sensitivity to high mass scales, must vanish to all
orders in perturbation theory.\footnote{A simple way to understand this is
to recall that unbroken supersymmetry requires the degeneracy of scalar
and fermion masses. Radiative corrections to fermion masses are known to
diverge at most logarithmically in any renormalizable
field theory, so the same must be
true for scalar masses in unbroken supersymmetry.} Now, if broken
supersymmetry is still to provide a solution to the hierarchy problem even
in the presence of supersymmetry breaking, then the relationships between
dimensionless couplings that hold in an unbroken supersymmetric theory
must be maintained. Otherwise, there would be quadratically divergent
radiative corrections to the Higgs scalar masses of the form
\beq
\Delta m_H^2 = {1\over 8\pi^2} (\lambda_S  \lambda_f^2)
\Lambda_{\rm UV}^2 + \ldots .
\label{eq:royalewithcheese}
\eeq
We are therefore led to consider ``soft" supersymmetry breaking. This
means that the effective Lagrangian of the MSSM can be written in the form
\beq
\lagr = \lagr_{\rm SUSY} + \lagr_{\rm soft},
\eeq
where $\lagr_{\rm SUSY}$ contains all of the gauge and Yukawa interactions
and preserves supersymmetry invariance, and $\lagr_{\rm soft}$ violates
supersymmetry but contains only mass terms and coupling parameters with {\it
positive} mass dimension. Without further justification, soft
supersymmetry breaking might seem like a rather arbitrary requirement.
Fortunately, we will see in section \ref{sec:origins} that theoretical
models for supersymmetry breaking do indeed yield effective Lagrangians
with just such terms for $\lagr_{\rm soft}$. If the largest mass scale
associated with the soft terms is denoted $m_{\rm soft}$, then the
additional nonsupersymmetric corrections to the Higgs scalar squared mass
must vanish in the $m_{\rm soft}\rightarrow 0$ limit, so by dimensional
analysis they cannot be proportional to $\Lambda_{\rm UV}^2$. More
generally, these models maintain the cancellation of quadratically
divergent terms in the radiative corrections of all scalar masses, to all
orders in perturbation theory. The corrections also cannot go like $\Delta
m_H^2 \sim m_{\rm soft} \Lambda_{\rm UV}$, because in general the loop
momentum integrals always diverge either quadratically or logarithmically,
not linearly, as $\Lambda_{\rm UV} \rightarrow \infty$. So they must be of
the form
\beq
\Delta m_{H}^2 =
m_{\rm soft}^2
\left [{\lambda\over 16 \pi^2}\> {\rm ln}(\Lambda_{\rm UV}/m_{\rm soft})
+ \ldots \right ].
\label{softy}
\eeq
Here $\lambda$ is schematic for various dimensionless couplings, and the
ellipses stand both for terms that are independent of $\Lambda_{\rm UV}$
and for higher loop corrections (which depend on $\Lambda_{\rm UV}$
through powers of logarithms).
Because the mass splittings between the known Standard Model particles and
their superpartners are just determined by the parameters $m_{\rm soft}$
appearing in $\lagr_{\rm soft}$, eq.~(\ref{softy}) tells us that the
superpartner masses should not be too huge.\footnote{This is
obviously fuzzy and subjective. Nevertheless, such subjective criteria can be useful,
at least on a personal level, for making choices about
what research directions to pursue, given finite time and money.}
Otherwise, we would lose our
successful cure for the hierarchy problem, since the $m_{\rm soft}^2$
corrections to the Higgs scalar squared mass parameter would be
unnaturally large compared to the square of the electroweak breaking scale
of 174 GeV. The top and bottom squarks and the winos and bino give
especially large contributions to $\Delta m_{H_u}^2$ and $\Delta
m_{H_d}^2$, but the gluino mass and all the other squark and slepton
masses also feed in indirectly, through radiative corrections to the top
and bottom squark masses. Furthermore, in most viable models of
supersymmetry breaking that are not unduly contrived, the superpartner
masses do not differ from each other by more than about an order of
magnitude. Using $\Lambda_{\rm UV} \sim \MPlanck$ and $\lambda \sim 1$ in
eq.~(\ref{softy}), one estimates that $m_{\rm soft}$, and therefore the masses
of at least the lightest few superpartners, should probably not be much
greater than the TeV scale, in order for the MSSM scalar potential
to provide a Higgs VEV resulting in $m_W,m_Z$ = 80.4, 91.2 GeV without
miraculous cancellations. While this is a fuzzy criterion, it
is the best reason for the continued optimism among many theorists that
supersymmetry will be discovered at the CERN
Large Hadron Collider, and can be studied at a future $e^+ e^$ linear
collider with sufficiently high energy.
However, it should be noted that the hierarchy problem was {\it not} the
historical motivation for the development of supersymmetry in the early
1970's. The supersymmetry algebra and supersymmetric field theories were
originally concocted independently in various disguises
%\cite{RNS,Golfand,WessZumino,Volkov}
\cite{RNS}\cite{Volkov}
bearing little resemblance to the MSSM. It is quite impressive that a
theory developed for quite different reasons, including purely aesthetic
ones, was later found to provide a solution for the hierarchy problem.
One might also wonder whether there is any good reason why all of the
superpartners of the Standard Model particles should be heavy enough to
have avoided discovery so far. There is. All of the particles in the MSSM
that have been found so far, except the 125 GeV Higgs boson,
have something in common; they would
necessarily be massless in the absence of electroweak symmetry breaking.
In particular, the masses of the $W^\pm, Z^0$ bosons and all quarks and
leptons are equal to dimensionless coupling constants times the Higgs VEV
$\sim 174 $ GeV, while the photon and gluon are required to be massless by
electromagnetic and QCD gauge invariance. Conversely, all of the
undiscovered particles in the MSSM have exactly the opposite property;
each of them can have a Lagrangian mass term in the absence of electroweak
symmetry breaking. For the squarks, sleptons, and Higgs scalars this
follows from a general property of complex scalar fields that a mass term
$m^2 \phi^2$ is always allowed by all gauge symmetries. For the
higgsinos and gauginos, it follows from the fact that they are fermions in
a real representation of the gauge group. So, from the point of view of
the MSSM, the discovery of the top quark in 1995 marked a quite natural
milestone; the alreadydiscovered particles are precisely those that had
to be light, based on the principle of electroweak gauge symmetry. There
is a single exception: it has long been known that at least
one neutral Higgs scalar boson had to be lighter
than about 135 GeV if the minimal version of supersymmetry is correct, for
reasons to be discussed in section \ref{subsec:MSSMspectrum.Higgs}.
The 125 GeV Higgs boson discovered in 2012 is presumably this particle, and the fact
that it was not much heavier can be counted as a successful prediction of supersymmetry.
An important feature of the MSSM is that the superpartners listed in
Tables \ref{tab:chiral} and \ref{tab:gauge}
are not necessarily the mass eigenstates of the theory.
This is because after electroweak symmetry breaking and supersymmetry
breaking effects are included, there can be mixing between the electroweak
gauginos and the higgsinos, and within the various sets of squarks and
sleptons and Higgs scalars that have the same electric charge. The lone
exception is the gluino, which is a color octet fermion and therefore does
not have the appropriate quantum numbers to mix with any other particle.
The masses and mixings of the superpartners are obviously of paramount
importance to experimentalists. It is perhaps slightly less obvious that
these phenomenological issues are all quite directly related to one
central question that is also the focus of much of the theoretical work in
supersymmetry: ``How is supersymmetry broken?" The reason for this is that
most of what we do not already know about the MSSM has to do with
$\lagr_{\rm soft}$. The structure of supersymmetric Lagrangians allows
little arbitrariness, as we will see in section \ref{sec:susylagr}.
In fact, all of the dimensionless couplings and all but one mass term in
the supersymmetric part of the MSSM Lagrangian correspond directly to
parameters in the ordinary Standard Model that have already been measured
by experiment. For example, we will find out that the supersymmetric
coupling of a gluino to a squark and a quark is determined by the QCD
coupling constant $\alpha_S$. In contrast, the supersymmetrybreaking part
of the Lagrangian contains many unknown parameters and, apparently, a
considerable amount of arbitrariness. Each of the mass splittings between
Standard Model particles and their superpartners correspond to terms in
the MSSM Lagrangian that are purely supersymmetrybreaking in their origin
and effect. These soft supersymmetrybreaking terms can also introduce a
large number of mixing angles and CPviolating phases not found in the
Standard Model. Fortunately, as we will see in section
\ref{subsec:mssm.hints}, there is already strong evidence that the
supersymmetrybreaking terms in the MSSM are actually not arbitrary at
all. Furthermore, the additional parameters will be measured and
constrained as the superpartners are detected. From a theoretical
perspective, the challenge is to explain all of these parameters with a
predictive model for supersymmetry breaking.
The rest of the discussion is organized as follows. Section
\ref{sec:notations} provides a list of important notations. In section
\ref{sec:susylagr}, we will learn how to construct Lagrangians for
supersymmetric field theories, while section \ref{sec:superfields}
reprises the same subject, but using the more elegant superspace formalism.
Soft supersymmetrybreaking couplings are
described in section \ref{sec:soft}. In section \ref{sec:mssm}, we will
apply the preceding general results to the special case of the MSSM,
introduce the concept of $R$parity, and explore the importance of the
structure of the soft terms. Section \ref{sec:origins} outlines some
considerations for understanding the origin of supersymmetry breaking, and
the consequences of various proposals. In section \ref{sec:MSSMspectrum},
we will study the mass and mixing angle patterns of the new particles
predicted by the MSSM. Their decay modes are considered in section
\ref{sec:decays}, and some of the qualitative features of experimental
signals for supersymmetry are reviewed in section \ref{sec:signals}.
Section \ref{sec:variations} describes some sample variations on the
standard MSSM picture. The discussion will be lacking in historical
accuracy or perspective; the reader is
encouraged to consult the many outstanding books
%\cite{WessBaggerbook,Rossbook,Srivastavabook,Freundbook,Westbook,
%Mohapatrabook,BailinLovebook,Buchbinder:1998qv,Ramondbook,Weinbergbook,Dreesbook,
%Baerbook,Binetruybook,Terningbook,Dinebook,Freedman:2012zz,Shifman:2012zz},
\cite{WessBaggerbook}\cite{Shifman:2012zz},
review articles
%\cite{HaberKanereview,Nillesreview,Sohnius:1985qm,GGRS,VNreview,ACNreview,
%Jonesreview,HaberTASI,Ramondreview,Baggerreview,Peskin:1997qi,
%DPFpheno,Lykkenreview,Dawsonreview,Dinereview,Gunionreview,
%Shifmanreview,Tatareview,Luty:2005sn,Aitchison:2005cf,
%Chung:2003fi,Terning:2003th,Murayama:2000dw,Bertolini:2013via}
\cite{HaberKanereview}\cite{Bertolini:2013via}
and the reprint volume
\cite{reprints}, which contain a much more consistent guide to the
original literature.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Interlude: Notations and Conventions}\label{sec:notations}
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This section specifies my notations and conventions. Fourvector indices
are represented by letters from the middle of the Greek alphabet
$\mu,\nu,\rho,\dots= 0,1,2,3$. The contravariant fourvector position and
momentum of a particle are
\beq
x^\mu = (t,\, \vec{x}), \qquad\qquad p^\mu = (E,\, \vec{p}) ,
\eeq
while the fourvector derivative is
\beq
\partial_\mu = ({\partial / \partial t},\, \vec{\nabla}) .
\eeq
The spacetime metric is
\beq
\eta_{\mu\nu} =
{\mbox{diag}}(\BDplus 1, \BDminus 1, \BDminus 1, \BDminus 1),
\eeq
so that $p^2 = \BDpos m^2$ for an onshell particle of mass $m$.
It is overwhelmingly convenient to employ twocomponent Weyl spinor
notation for fermions, rather than fourcomponent Dirac or Majorana
spinors. The Lagrangian of the Standard Model (and any supersymmetric
extension of it) violates parity; each Dirac fermion has lefthanded and
righthanded parts with completely different electroweak gauge
interactions. If one used fourcomponent spinor notation instead, then
there would be clumsy left and righthanded projection operators
\beq
P_L = (1  \gamma_5)/2, \qquad\qquad
P_R = (1 + \gamma_5)/2
\eeq
all over the place. The twocomponent Weyl fermion notation has the
advantage of treating fermionic degrees of freedom with different gauge
quantum numbers separately from the start, as Nature intended for us to
do. But an even better reason for using twocomponent notation here is
that in supersymmetric models the minimal building blocks of matter are
chiral supermultiplets, each of which contains a single twocomponent Weyl
fermion.
Because twocomponent fermion notation may be unfamiliar to some
readers, I now specify my conventions by showing how they correspond to
the fourcomponent spinor language. A fourcomponent Dirac fermion
$\Psi_{\sst D}$ with mass $M$ is described by the Lagrangian
\beq
\lagr_{\rm Dirac}
\,=\, i \overline\Psi_{\sst D} \gamma^\mu \partial_\mu \Psi_{\sst D}
 M \overline \Psi_{\sst D} \Psi_{\sst D}\> .
\label{diraclag}
\eeq
For our purposes it is convenient to use the specific representation of
the 4$\times$4 gamma matrices given in $2\times$2 blocks by
\beq
\gamma^\mu = \pmatrix{ 0 & \sigma^\mu \cr
\sigmabar^\mu & 0\cr},
\qquad\qquad
\gamma_5 = \pmatrix{1 & 0\cr 0 & 1\cr},
\eeq
where
\beq
&&\sigma^0 = \sigmabar^0 = \pmatrix{1&0\cr 0&1\cr},\qquad
\,\>\>\>\>\>\>\>\>\sigma^1 = \sigmabar^1 = \pmatrix{0&1\cr 1&0\cr},
\nonumber\\
&&\sigma^2 = \sigmabar^2 = \pmatrix{ 0&i\cr i&0\cr},\qquad
\>\>\>\sigma^3 = \sigmabar^3 = \pmatrix{1&0\cr 0&1\cr}
\> .
\label{pauli}
\eeq
In this representation, a fourcomponent Dirac spinor is written in terms of 2
twocomponent, complex,\footnote{For obscure reasons,
in much of the specialized literature on supersymmetry
a bar ($\overline{\psi}$) has been used to represent the
conjugate of a twocomponent spinor, rather than a dagger ($\psi^\dagger$).
Here, I maintain consistency with essentially all other
quantum field theory textbooks by using the dagger notation for the
conjugate of a twocomponent spinor.}
anticommuting objects $\xi_\alpha$ and
$(\chi^\dagger)^{\dot{\alpha}} \equiv \chi^{\dagger\dot\alpha}$, with two
distinct types of spinor indices $\alpha=1,2$ and $\dot{\alpha}=1,2$:
\beq
\Psi_{\sst D} =
\pmatrix{\xi_\alpha\cr {\chi^{\dagger\dot{\alpha}}}\cr} .
\label{psidirac}
\eeq
It follows that
\beq
\overline\Psi_{\sst D} =
\Psi_{\sst D}^\dagger \pmatrix{0 & 1\cr 1 & 0\cr} =
\pmatrix{\chi^\alpha &
\xi^\dagger_{\dot{\alpha}}\cr }
\> .
\label{psid}
\eeq
Undotted (dotted) indices from the beginning of the Greek alphabet are
used for the first (last) two components of a Dirac spinor. The field
$\xi$ is called a ``lefthanded Weyl spinor" and $\chi^\dagger$ is a
``righthanded Weyl spinor". The names fit, because
\beq
P_L \Psi_{\sst D} = \pmatrix{\xi_\alpha \cr 0\cr},\qquad\qquad
P_R \Psi_{\sst D} = \pmatrix{0\cr \chi^{\dagger\dot{\alpha}}\cr}
\> .
\eeq
The Hermitian conjugate of any lefthanded Weyl spinor is a righthanded
Weyl spinor:
\beq
\psi^{\dagger}_{\dot{\alpha}}
\equiv
(\psi_\alpha)^\dagger = (\psi^\dagger)_{\dot{\alpha}}
\, ,
\eeq
and vice versa:
\beq
( \psi^{\dagger\dot{\alpha}} )^\dagger =
\psi^\alpha.
\eeq
Therefore, any particular fermionic degrees of freedom can be described
equally well using a lefthanded Weyl spinor (with an undotted index) or
by a righthanded one (with a dotted index). By convention, all names of
fermion fields are chosen so that lefthanded Weyl spinors do not carry
daggers and righthanded Weyl spinors do carry daggers, as in
eq.~(\ref{psidirac}).
The heights of the dotted and undotted spinor indices are important; for
example, comparing eqs.~(\ref{diraclag})(\ref{psid}), we observe that the
matrices $(\sigma^\mu)_{\alpha\dot{\alpha}}$ and
$(\sigmabar^\mu)^{\dot{\alpha}\alpha}$ defined by eq.~(\ref{pauli}) carry
indices with the heights as indicated. The spinor indices are raised and
lowered using the antisymmetric symbol
\beq
\epsilon^{12} = \epsilon^{21} =
\epsilon_{21} = \epsilon_{12} = 1, \qquad\qquad \epsilon_{11} = \epsilon_{22} =
\epsilon^{11} = \epsilon^{22} = 0,
\label{eq:defepstwo}
\eeq
according to
\beq
\xi_\alpha = \epsilon_{\alpha\beta}
\xi^\beta,\qquad\qquad\!\!\!\!\!\!\!\!\!
\xi^\alpha = \epsilon^{\alpha\beta}
\xi_\beta,\qquad\qquad\!\!\!\!\!\!\!\!\!
\chi^\dagger_{\dot{\alpha}} = \epsilon_{\dot{\alpha}\dot{\beta}}
\chi^{\dagger\dot{\beta}},\qquad\qquad\!\!\!\!\!\!\!\!\!
\chi^{\dagger\dot{\alpha}} = \epsilon^{\dot{\alpha}\dot{\beta}}
\chi^\dagger_{\dot{\beta}}\>.\qquad{}
\eeq
This is consistent since
$\epsilon_{\alpha\beta} \epsilon^{\beta\gamma} =
\epsilon^{\gamma\beta}\epsilon_{\beta\alpha} = \delta_\alpha^\gamma$
and
$\epsilon_{\dot{\alpha}\dot{\beta}} \epsilon^{\dot{\beta}\dot{\gamma}} =
\epsilon^{\dot{\gamma}\dot{\beta}}\epsilon_{\dot{\beta}\dot{\alpha}} =
\delta_{\dot{\alpha}}^{\dot{\gamma}}$.
\vspace{0.1cm}
As a convention, repeated spinor indices contracted like
\beq
{}^\alpha\hspace{0.03cm}{}_\alpha \qquad\quad \mbox{or}
\qquad\quad
{}_{\dot{\alpha}}\hspace{0.03cm}{}^{\dot{\alpha}}\,
\eeq
can be suppressed.
In particular,
\beq
\xi\chi \equiv \xi^\alpha\chi_\alpha = \xi^\alpha \epsilon_{\alpha\beta}
\chi^\beta = \chi^\beta \epsilon_{\alpha\beta} \xi^\alpha =
\chi^\beta \epsilon_{\beta\alpha} \xi^\alpha = \chi^\beta \xi_\beta \equiv
\chi\xi
\label{xichi}
\eeq
with, conveniently, no minus sign in the end. [A minus sign appeared in
eq.~(\ref{xichi}) from exchanging the order of anticommuting spinors, but
it disappeared due to the antisymmetry of the $\epsilon$ symbol.]
Likewise, $\xi^\dagger \chi^\dagger$ and $\chi^\dagger \xi^\dagger $ are
equivalent abbreviations for $\chi^\dagger_{\dot{\alpha}} \xi^{\dagger
\dot{\alpha}} = \xi^\dagger_{\dot{\alpha}} \chi^{\dagger \dot{\alpha}}$,
and in fact this is the complex conjugate of $\xi\chi$:
\beq
(\xi\chi)^* = \chi^\dagger \xi^\dagger = \xi^\dagger \chi^\dagger.
\eeq
In a similar way, one can check that
\beq
(\chi^\dagger \sigmabar^\mu \xi)^* \,=\,
\xi^\dagger \sigmabar^\mu \chi \,=\, \chi \sigma^\mu \xi^\dagger
\,=\,
(\xi\sigma^\mu\chi^\dagger)^*
\label{yetanotheridentity}
\eeq
stands for $\xi^\dagger_{\dot{\alpha}}(\sigmabar^\mu)^{\dot{\alpha}\alpha}
\chi_\alpha$, etc. Note that when taking the complex conjugate of a spinor bilinear,
one reverses the order. The spinors here are assumed to be
classical fields; for quantum fields the complex conjugation operation in these
equations would be replaced by Hermitian conjugation in the Hilbert
space operator sense.
Some other identities that will be useful below include:
\beq
(\chi^\dagger \sigmabar^\nu \sigma^\mu \xi^\dagger)^* =
\xi\sigma^\mu \sigmabar^\nu \chi \>=\>
\chi \sigma^\nu \sigmabar^\mu \xi \>=\>
(\xi^\dagger \sigmabar^\mu \sigma^\nu \chi^\dagger)^*,
\label{eq:dei}
\eeq
and the Fierz rearrangement identity:
\beq
\chi_{\alpha}\> (\xi\eta) &=&
 \xi_{\alpha}\> (\eta\chi)  \eta_\alpha\> (\chi\xi) ,
\label{fierce}
\eeq
and the reduction identities
\beq
&&
\sigma_{\alpha\dot{\alpha}}^\mu \,
\sigmabar_\mu^{\dot{\beta}\beta}
\,\>=\>\,
\BDpos 2 \delta_\alpha^\beta
\delta_{\dot{\alpha}}^{\dot{\beta}} ,
\label{eq:feif}
\\
&&
\sigma_{\alpha\dot{\alpha}}^\mu \,
\sigma_{\mu\beta\dot{\beta}}
\,\>=\>\,
\BDpos 2 \epsilon_{\alpha\beta}
\epsilon_{\dot{\alpha}\dot{\beta}} ,
\label{eq:rickettsisok}
\\
&&
\sigmabar^{\mu\dot\alpha\alpha} \,
\sigmabar_\mu^{\dot\beta\beta}
\,\>=\>\,
\BDpos 2 \epsilon^{\alpha\beta}
\epsilon^{\dot{\alpha}\dot{\beta}} ,
\label{eq:pagehousesux}
\\
&&
\bigl[ \sigma^\mu \sigmabar^\nu + \sigma^\nu \sigmabar^\mu
\bigr ]_\alpha{}^\beta
\,=\,
\BDpos 2 \eta^{\mu\nu} \delta_\alpha^\beta
,
\label{pauliidentA}
\\
&&
\bigl[ \sigmabar^\mu \sigma^\nu + \sigmabar^\nu \sigma^\mu \bigr
]^{\dot{\beta}}{}_{\dot{\alpha}}
\,=\,
\BDpos 2 \eta^{\mu\nu} \delta_{\dot{\alpha}}^{\dot{\beta}} ,
\label{pauliidentB}
\\
&&\sigmabar^\mu \sigma^\nu \sigmabar^\rho \>=\>
\BDpos \eta^{\mu\nu} \sigmabar^\rho
\BDplus \eta^{\nu\rho} \sigmabar^\mu
\BDminus \eta^{\mu\rho} \sigmabar^\nu
\BDminus i \epsilon^{\mu\nu\rho\kappa} \sigmabar_\kappa ,
\label{eq:lloydhouserules}
\\
&&\sigma^\mu \sigmabar^\nu \sigma^\rho \>=\>
\BDpos \eta^{\mu\nu} \sigma^\rho
\BDplus \eta^{\nu\rho} \sigma^\mu
\BDminus \eta^{\mu\rho} \sigma^\nu
\BDplus i \epsilon^{\mu\nu\rho\kappa} \sigma_\kappa ,
\label{eq:pagehouseisOK}
\eeq
where $\epsilon^{\mu\nu\rho\kappa}$ is the totally antisymmetric tensor
with $\epsilon^{0123}=+1$.
With these conventions, the Dirac Lagrangian eq.~(\ref{diraclag}) can now
be rewritten:
\beq
\lagr_{\rm Dirac}
\, = \, i \xi^\dagger \sigmabar^\mu \partial_\mu \xi
+ i \chi^\dagger \sigmabar^\mu \partial_\mu \chi
 M (\xi\chi + \xi^\dagger \chi^\dagger)
\eeq
where we have dropped a total derivative piece
$i\partial_\mu(\chi^\dagger \sigmabar^\mu \chi)$,
which does not affect the action.
A fourcomponent Majorana spinor can be obtained from the Dirac spinor of
eq.~(\ref{psid}) by imposing the constraint $\chi = \xi$, so that
\beq
\Psi_{\rm M} = \pmatrix{\xi_\alpha \cr \xi^{\dagger\dot{\alpha}}\cr},
\qquad\qquad
\overline\Psi_{\rm M}
= \pmatrix{ \xi^\alpha & \xi^\dagger_{\dot{\alpha}} \cr}.
\eeq
The fourcomponent spinor form of the
Lagrangian for a Majorana fermion with mass $M$,
\beq
\lagr_{\rm Majorana} \,=\,
{i\over 2}\overline\Psi_{\rm M} \gamma^\mu \partial_\mu \Psi_{\rm M}
 {1\over 2} M \overline\Psi_{\rm M} \Psi_{\rm M}
\eeq
can therefore be rewritten as
\beq
\lagr_{\rm Majorana} \,=\,
i\xi^\dagger \sigmabar^\mu\partial_\mu \xi 
{1\over 2} M(\xi\xi + \xi^\dagger\xi^\dagger)
\eeq
in the more economical twocomponent Weyl spinor representation. Note that
even though $\xi_\alpha$ is anticommuting, $\xi\xi$ and its complex
conjugate $\xi^\dagger\xi^\dagger$ do not vanish, because of the
suppressed $\epsilon$ symbol, see eq.~(\ref{xichi}). Explicitly, $\xi\xi =
\epsilon^{\alpha\beta} \xi_\beta \xi_\alpha = \xi_2\xi_1  \xi_1 \xi_2 = 2
\xi_2 \xi_1$.
More generally, any theory involving spin1/2 fermions can always be
written in terms of a collection of lefthanded Weyl spinors $\psi_i$
with
\beq
\lagr \,=\, i \psi^{\dagger i} \sigmabar^\mu \partial_\mu\psi_i
+ \ldots
\eeq
where the ellipses represent possible mass terms, gauge interactions, and
Yukawa interactions with scalar fields. Here the index $i$ runs over the
appropriate gauge and flavor indices of the fermions; it is raised or
lowered by Hermitian conjugation. Gauge interactions are obtained
by promoting the ordinary derivative to a gaugecovariant derivative:
\beq
\lagr \,=\, i \psi^{\dagger i} \sigmabar^\mu \nabla_\mu\psi_i
+ \ldots
\eeq
with
\beq
\nabla_\mu\psi_i \,=\, \partial_\mu\psi_i \BDplus i g_a A^a_\mu {T^a_i}^j \psi_j,
\eeq
where $g_a$ is the gauge coupling corresponding to the Hermitian
Lie algebra generator matrix $T^a$ with vector field $A^a_\mu$.
There is a different $\psi_i$ for the lefthanded piece and for the
hermitian conjugate of the righthanded piece of a Dirac fermion.
Given any expression involving bilinears of fourcomponent
spinors
\beq
\Psi_i = \pmatrix{ \xi_i\cr\chi_i^\dagger\cr},
\eeq
labeled by a flavor or gaugerepresentation index $i$, one can
translate into twocomponent Weyl spinor language (or vice versa) using
the dictionary:
\beq
&&\overline\Psi_i P_L \Psi_j = \chi_i\xi_j,\qquad\qquad\qquad
\overline\Psi_i P_R \Psi_j = \xi_i^\dagger \chi_j^\dagger,\qquad\>{}\\
&&\overline\Psi_i \gamma^\mu P_L \Psi_j = \xi_i^\dagger \sigmabar^\mu
\xi_j
,\qquad\qquad
\overline\Psi_i \gamma^\mu P_R \Psi_j = \chi_i \sigma^\mu \chi^\dagger_j
\qquad\>\>\>{}
\eeq
etc.
Let us now see how the Standard Model quarks and leptons are described in
this notation. The complete list of lefthanded Weyl spinors can be given
names corresponding to the chiral supermultiplets in Table \ref{tab:chiral}:
\beq
Q_i & = &
\pmatrix{u\cr d},\>\,
\pmatrix{c\cr s},\>\,
\pmatrix{t\cr b},
\\
\sbar u_i & = &
\>\>\sbar u ,\>\,\sbar c,\>\, \sbar t,
\\
\sbar d_i & = &
\>\>\sbar d ,\>\,\sbar s,\>\, \sbar b
\\
L_i & = &
\pmatrix{\nu_e\cr e},\>\,
\pmatrix{\nu_\mu\cr \mu},\>\,
\pmatrix{\nu_\tau\cr \tau},\>\,
\\
\sbar e_i & = &
\>\>\sbar e ,\>\,\sbar \mu,\>\, \sbar \tau .
\eeq
Here $i=1,2,3$ is a family index. The bars on these fields are part of the
names of the fields, and do {\it not} denote any kind of conjugation.
Rather, the unbarred fields are the lefthanded pieces of a Dirac spinor,
while the barred fields are the names given to the conjugates of the
righthanded piece of a Dirac spinor. For example, $e$ is the same thing
as $e_L$ in Table \ref{tab:chiral},
and $\sbar e$ is the same as $e_R^\dagger$. Together
they form a Dirac spinor:
\beq
\pmatrix{e\cr {\sbar e}^\dagger} \equiv \pmatrix{e_L \cr e_R} ,
\label{espinor}
\eeq
and similarly for all of the other quark and charged lepton Dirac
spinors. (The neutrinos of the Standard Model are not part of a Dirac
spinor, at least in the approximation that they are massless.) The fields
$Q_i$ and $L_i$ are weak isodoublets, which always go together when one is
constructing interactions invariant under the full Standard Model gauge
group $SU(3)_C\times SU(2)_L \times U(1)_Y$. Suppressing all color and
weak isospin indices, the kinetic and gauge part of the Standard Model
fermion Lagrangian density is then
\beq
\lagr \,=\,
iQ^{\dagger i}\sigmabar^\mu \nabla_\mu Q_i
+ i\sbar u^{\dagger }_i\sigmabar^\mu \nabla_\mu \sbar u^i
+ i\sbar d^{\dagger }_i\sigmabar^\mu \nabla_\mu \sbar d^i
+ i L^{\dagger i}\sigmabar^\mu \nabla_\mu L_i
+ i \sbar e^{\dagger }_i\sigmabar^\mu \nabla_\mu \sbar e^i
\qquad{}
\eeq
with the family index $i$ summed over, and $\nabla_\mu$ the
appropriate Standard Model covariant derivative. For example,
\beq
\nabla_\mu \pmatrix{ \nu_e \cr e} &=&
\left [ \partial_\mu \BDplus i g W^a_\mu (\tau^a/2)
\BDplus i g' Y_L B_\mu \right ]
\pmatrix{ \nu_e \cr e}
\\
\nabla_\mu \overline e &=& \left [ \partial_\mu
\BDplus i g' Y_{\sbar e} B_\mu \right ] \sbar e
\eeq
with $\tau^a$ ($a=1,2,3$) equal to the Pauli matrices, $Y_L = 1/2$ and
$Y_{\sbar e} = +1$. The gauge eigenstate weak bosons are related to
the mass eigenstates by
\beq
W^\pm_\mu &=& (W_\mu^1 \mp i W_\mu^2)/\sqrt{2} ,
\\
\pmatrix{Z_\mu \cr A_\mu} &=&
\pmatrix{\cos\theta_W &  \sin\theta_W \cr
\sin\theta_W & \cos\theta_W \cr }
\pmatrix{W^3_\mu \cr B_\mu} .
\eeq
Similar expressions hold for the other quark and lepton gauge eigenstates,
with $Y_Q = 1/6$, $Y_{\sbar u} = 2/3$, and $Y_{\sbar d} = 1/3$. The
quarks also have a term in the covariant derivative corresponding to gluon
interactions proportional to $g_3$ (with $\alpha_S = g_3^2/4 \pi$) with
generators $T^a = \lambda^a/2$ for $Q$, and in the complex conjugate
representation $T^a = (\lambda^a)^*/2$ for $\sbar u$ and $\sbar d$, where
$\lambda^a$ are the GellMann matrices.
For a more detailed discussion of the twocomponent fermion notation, including many worked examples in which it is employed to calculate crosssections and decay rates in the Standard Model and in supersymmetry, see ref.~\cite{DHM}, or a more concise
account in \cite{Martin:2012us}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Supersymmetric Lagrangians}\label{sec:susylagr}
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In this section we will describe the construction of supersymmetric
Lagrangians. The goal is a recipe that will allow us
to write down the allowed interactions and mass terms of a general
supersymmetric theory, so that later we can apply the results to the
special case of the MSSM. In this section, we will not use the superfield
\cite{superfields} language, which is more elegant and efficient
for many purposes, but requires a more specialized machinery and
might seem rather cabalistic at first.
Section \ref{sec:superfields} will provide the
superfield version of the same material.
We begin by considering the simplest example of a supersymmetric theory in
four dimensions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The simplest supersymmetric model: a free chiral
supermultiplet}\label{subsec:susylagr.freeWZ}
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The minimum fermion content of a field theory in four dimensions consists
of a single lefthanded twocomponent Weyl fermion $\psi$. Since this is
an intrinsically complex object, it seems sensible to choose as its
superpartner a complex scalar field $\phi$. The simplest action we can
write down for these fields just consists of kinetic energy terms for
each:
\beq
&&S = \int d^4x\>
\left (\lagr_{\rm scalar} + \lagr_{\rm fermion}\right ) ,
\label{Lwz} \\
&&\lagr_{\rm scalar} =
\BDpos \partial^\mu \phi^* \partial_\mu \phi ,
\qquad\qquad
\lagr_{\rm fermion} =
i \psi^\dagger \sigmabar^\mu \partial_\mu \psi .
\eeq
This is called the massless, noninteracting {\it WessZumino model}
\cite{WessZumino}, and it corresponds to a single chiral supermultiplet as
discussed in the Introduction.
A supersymmetry transformation should turn the scalar boson field $\phi$
into something involving the fermion field $\psi_\alpha$. The simplest
possibility for the transformation of the scalar field is
\beq
\deltaeps \phi = \epsilon \psi,\qquad\qquad
\deltaeps \phi^* = \epsilon^\dagger \psi^\dagger ,
\label{phitrans}
\eeq
where $\epsilon^\alpha$ is an infinitesimal, anticommuting, twocomponent
Weyl fermion object that parameterizes the supersymmetry transformation. Until
section \ref{subsec:origins.gravitino}, we will be discussing global
supersymmetry, which means that $\epsilon^\alpha$ is a constant,
satisfying $\partial_\mu \epsilon^\alpha=0$. Since $\psi$ has dimensions
of [mass]$^{3/2}$ and $\phi$ has dimensions of [mass], it must be that
$\epsilon$ has dimensions of [mass]$^{1/2}$. Using eq.~(\ref{phitrans}),
we find that the scalar part of the Lagrangian transforms as
\beq
\deltaeps
\lagr_{\rm scalar} \,=\,
\BDpos \epsilon \partial^\mu \psi \> \partial_\mu \phi^*
\BDplus \epsilon^\dagger \partial^\mu \psi^\dagger \> \partial_\mu \phi .
\label{Lphitrans}
\eeq
We would like for this to be canceled by $\deltaeps\lagr_{\rm fermion}$,
at least up to a total derivative, so that the action will be invariant
under the supersymmetry transformation. Comparing eq.~(\ref{Lphitrans})
with $\lagr_{\rm fermion}$, we see that for this to have any chance of
happening, $\deltaeps \psi$ should be linear in $\epsilon^\dagger$ and in
$\phi$, and should contain one spacetime derivative. Up to a
multiplicative constant, there is only one possibility to try:
\beq
\deltaeps\psi_\alpha
\,=\,
 i (\sigma^\mu \epsilon^\dagger)_\alpha\> \partial_\mu \phi,
\qquad\qquad
\deltaeps\psi^\dagger_{\dot{\alpha}}
\,=\,
i (\epsilon\sigma^\mu)_{\dot{\alpha}}\> \partial_\mu \phi^* .
\label{psitrans}
\eeq
With this guess, one immediately obtains
\beq
\deltaeps \lagr_{\rm fermion} =
\epsilon \sigma^\mu \sigmabar^\nu \partial_\nu \psi\> \partial_\mu \phi^*
+\psi^\dagger \sigmabar^\nu \sigma^\mu \epsilon^\dagger \>
\partial_\mu \partial_\nu \phi
\> .
\label{preLpsitrans}
\eeq
This can be simplified by employing the
Pauli matrix identities eqs.~(\ref{pauliidentA}), (\ref{pauliidentB})
and using the fact that partial derivatives commute
$(\partial_\mu\partial_\nu = \partial_\nu\partial_\mu)$. Equation
(\ref{preLpsitrans}) then becomes
\beq
\deltaeps \lagr_{\rm fermion} & = &
\BDneg \epsilon\partial^\mu\psi\> \partial_\mu\phi^*
\BDminus \epsilon^\dagger \partial^\mu\psi^\dagger\> \partial_\mu \phi
\nonumber\\ &&
 \partial_\mu \left (
\epsilon \sigma^\nu \sigmabar^\mu \psi \> \partial_\nu \phi^*
\BDminus \epsilon \psi\> \partial^\mu \phi^*
\BDminus \epsilon^\dagger \psi^\dagger \> \partial^\mu \phi \right ).
\label{Lpsitrans}
\eeq
The first two terms here just cancel against $\deltaeps\lagr_{\rm
scalar}$, while the remaining contribution is a total derivative. So we
arrive at
\beq
\deltaeps S =
\int d^4x \>\>\, (\deltaeps \lagr_{\rm scalar} + \deltaeps
\lagr_{\rm fermion})
= 0,\>
\label{invar}
\eeq
justifying our guess of the numerical multiplicative factor made in
eq.~(\ref{psitrans}).
We are not quite finished in showing that the theory described by
eq.~(\ref{Lwz}) is supersymmetric. We must also show that the
supersymmetry algebra closes; in other words, that the commutator of two
supersymmetry transformations parameterized by two different spinors
$\epsilon_1$ and $\epsilon_2$ is another symmetry of the theory. Using
eq.~(\ref{psitrans}) in eq.~(\ref{phitrans}), one finds
\beq
(\delta_{\epsilon_2} \delta_{\epsilon_1} 
\delta_{\epsilon_1} \delta_{\epsilon_2}) \phi
\,\equiv\,
\delta_{\epsilon_2} (\delta_{\epsilon_1} \phi) 
\delta_{\epsilon_1} (\delta_{\epsilon_2} \phi)
\,=\,
i ( \epsilon_1 \sigma^\mu \epsilon_2^\dagger
+ \epsilon_2 \sigma^\mu \epsilon_1^\dagger)\> \partial_\mu \phi
. \label{coophi}
\eeq
This is a remarkable result; in words, we have found that the commutator
of two supersymmetry transformations gives us back the derivative of the
original field. In the Heisenberg picture of quantum mechanics
$\BDpos i\partial_\mu$ corresponds to the generator of
spacetime translations $P_\mu$, so eq.~(\ref{coophi}) implies the form of the
supersymmetry algebra that was foreshadowed in eq.~(\ref{susyalgone}) of
the Introduction. (We will make this statement more explicit before the
end of this section, and prove it again a different way in section \ref{sec:superfields}.)
All of this will be for nothing if we do not find the same result for the
fermion $\psi$. Using eq.~(\ref{phitrans}) in eq.~(\ref{psitrans}), we get
\beq
(\delta_{\epsilon_2} \delta_{\epsilon_1} 
\delta_{\epsilon_1} \delta_{\epsilon_2}) \psi_\alpha
\,=\,
i(\sigma^\mu\epsilon_1^\dagger)_\alpha\> \epsilon_2 \partial_\mu\psi
+i(\sigma^\mu\epsilon_2^\dagger)_\alpha \> \epsilon_1 \partial_\mu\psi
{}.
\eeq
This can be put into a more useful form by applying the Fierz identity
eq.~(\ref{fierce})
with $\chi = \sigma^\mu \epsilon_1^\dagger$,
$\xi = \epsilon_{2} $, $\eta = \partial_\mu \psi$, and again with
$\chi = \sigma^\mu \epsilon_2^\dagger$,
$\xi = \epsilon_{1} $, $\eta = \partial_\mu \psi$, followed
in each case by an application
of the identity eq.~(\ref{yetanotheridentity}). The result is
\beq
(\delta_{\epsilon_2} \delta_{\epsilon_1} 
\delta_{\epsilon_1} \delta_{\epsilon_2}) \psi_\alpha
& = &
i (\epsilon_1 \sigma^\mu \epsilon_2^\dagger
+\epsilon_2 \sigma^\mu \epsilon_1^\dagger) \> \partial_\mu \psi_\alpha
%\nonumber \\ & &
+ i \epsilon_{1\alpha} \> \epsilon_2^\dagger \sigmabar^\mu \partial_\mu\psi
 i \epsilon_{2\alpha} \> \epsilon_1^\dagger \sigmabar^\mu
\partial_\mu\psi
{}.\phantom{xxxxx}
\label{commpsi}
\eeq
The last two terms in (\ref{commpsi}) vanish onshell; that is, if the
equation of motion $\sigmabar^\mu\partial_\mu \psi = 0$ following from the
action is enforced. The remaining piece is exactly the same spacetime
translation that we found for the scalar field.
The fact that the supersymmetry algebra only closes onshell (when the
classical equations of motion are satisfied) might be somewhat worrisome,
since we would like the symmetry to hold even quantum mechanically. This
can be fixed by a trick. We invent a new complex scalar field $F$, which
does not have a kinetic term. Such fields are called {\it auxiliary}, and
they are really just bookkeeping devices that allow the symmetry algebra
to close offshell. The Lagrangian density for $F$ and its complex
conjugate is simply
\beq
\lagr_{\rm auxiliary} = F^* F \> .
\label{lagraux}
\eeq
The dimensions of $F$ are [mass]$^2$, unlike an ordinary scalar field,
which has dimensions of [mass]. Equation (\ref{lagraux}) implies the
notveryexciting equations of motion $F=F^*=0$. However, we can use the
auxiliary fields to our advantage by including them in the supersymmetry
transformation rules. In view of eq.~(\ref{commpsi}), a plausible thing to
do is to make $F$ transform into a multiple of the equation of motion for
$\psi$:
\beq
\deltaeps F =  i \epsilon^\dagger \sigmabar^\mu \partial_\mu \psi,
\qquad\qquad
\deltaeps F^* = i\partial_\mu \psi^\dagger \sigmabar^\mu \epsilon .
\label{Ftrans}
\eeq
Once again we have chosen the overall factor on the righthand sides by
virtue of foresight. Now the auxiliary part of the Lagrangian density
transforms as
\beq
\delta \lagr_{\rm auxiliary} =
i \epsilon^\dagger \sigmabar^\mu \partial_\mu \psi \> F^*
+i \partial_\mu \psi^\dagger \sigmabar^\mu \epsilon \> F ,
\eeq
which vanishes onshell, but not for arbitrary offshell field
configurations. Now, by adding an extra term to
the transformation law for $\psi$ and $\psi^\dagger$:
\beq
\delta \psi_\alpha =
 i (\sigma^\mu \epsilon^\dagger)_{\alpha}\> \partial_\mu\phi
+ \epsilon_\alpha F,
\qquad\>\>
\delta \psi_{\dot{\alpha}}^\dagger =
i (\epsilon\sigma^\mu)_{\dot{\alpha}}\> \partial_\mu \phi^*
+ \epsilon^\dagger_{\dot{\alpha}} F^* ,
\label{fermiontrans}
\eeq
one obtains an additional contribution to $\deltaeps \lagr_{\rm fermion}$,
which just cancels with $\deltaeps \lagr_{\rm auxiliary}$, up to a total
derivative term. So our ``modified" theory with $\lagr = \lagr_{\rm
scalar} +\lagr_{\rm fermion} + \lagr_{\rm auxiliary}$ is still invariant
under supersymmetry transformations. Proceeding as before, one now obtains
for each of the fields $X=\phi,\phi^*,\psi,\psi^\dagger,F,F^*$,
\beq
(\delta_{\epsilon_2} \delta_{\epsilon_1} 
\delta_{\epsilon_1} \delta_{\epsilon_2}) X &=&
i (\epsilon_1 \sigma^\mu \epsilon_2^\dagger +
\epsilon_2 \sigma^\mu \epsilon_1^\dagger) \> \partial_\mu X
\label{anytrans}
\eeq
using eqs.~(\ref{phitrans}), (\ref{Ftrans}), and (\ref{fermiontrans}), but
now without resorting to any equations of motion. So we have
succeeded in showing that supersymmetry is a valid symmetry of the
Lagrangian offshell.
In retrospect, one can see why we needed to introduce the auxiliary field
$F$ in order to get the supersymmetry algebra to work offshell. Onshell,
the complex scalar field $\phi$ has two real propagating degrees of
freedom, matching the two spin polarization states of $\psi$. Offshell,
however, the Weyl fermion $\psi$ is a complex twocomponent object, so it
has four real degrees of freedom. (Going onshell eliminates half of the
propagating degrees of freedom for $\psi$, because the Lagrangian is
linear in time derivatives, so that the canonical momenta can be
reexpressed in terms of the configuration variables without time
derivatives and are not independent phase space coordinates.) To make the
numbers of bosonic and fermionic degrees of freedom match offshell as
well as onshell, we had to introduce two more real scalar degrees of
freedom in the complex field $F$, which are eliminated when one goes
onshell. This counting is summarized in Table \ref{table:WZdofcounting}.
The auxiliary field formulation is especially useful when discussing
spontaneous supersymmetry breaking, as we will see in section
\ref{sec:origins}.
\renewcommand{\arraystretch}{1.4}
\begin{table}[tb]
\begin{center}
\begin{tabular}{cccc}
\hline
& $\phi$ & $\psi$ & $F$ \\
\hline
onshell ($n_B=n_F=2$) & 2 & 2 & 0 \\
\hline
offshell ($n_B=n_F=4$) & 2 & 4 & 2 \\
\hline
\end{tabular}
\caption{Counting of real degrees of freedom in the
WessZumino model.\label{table:WZdofcounting}}
\vspace{0.45cm}
\end{center}
\end{table}
Invariance of the action under a continuous symmetry transformation always implies
the existence of a conserved current, and supersymmetry is no exception.
The {\it supercurrent} $J^\mu_\alpha$ is an anticommuting fourvector. It
also carries a spinor index, as befits the current associated with a
symmetry with fermionic generators \cite{ref:supercurrent}. By the usual
Noether procedure, one finds for the supercurrent (and its hermitian
conjugate) in terms of the variations of the fields
$X=\phi,\phi^*,\psi,\psi^\dagger,F,F^*$:
\beq
\epsilon J^\mu + \epsilon^\dagger J^{\dagger\mu}
&\equiv &
\sum_X \, \delta X\>{\delta\lagr\over \delta(\partial_\mu X)}
 K^\mu ,
\label{Noether}
\eeq
where $K^\mu$ is an object whose divergence is the variation of the
Lagrangian density under the supersymmetry transformation, $\delta \lagr =
\partial_\mu K^\mu$. Note that $K^\mu$ is not unique; one can always
replace $K^\mu$ by $K^\mu + k^\mu$, where $k^\mu$ is any vector satisfying
$\partial_\mu k^\mu=0$, for example $k^\mu = \partial^\mu \partial_\nu
a^\nu  \partial_\nu\partial^\nu a^\mu$ for any fourvector $a^\mu$.
A little work reveals that, up to
the ambiguity just mentioned,
\beq
J^\mu_\alpha = (\sigma^\nu\sigmabar^\mu\psi)_\alpha\> \partial_\nu \phi^*
, \qquad\qquad
J^{\dagger\mu}_{\dot{\alpha}}
= (\psi^\dagger \sigmabar^\mu \sigma^\nu)_{\dot{\alpha}}
\> \partial_\nu \phi .
\label{WZsupercurrent}
\eeq
The supercurrent and its hermitian conjugate are separately conserved:
\beq
\partial_\mu J^\mu_\alpha = 0,\qquad\qquad
\partial_\mu J^{\dagger\mu}_{\dot{\alpha}} = 0 ,
\eeq
as can be verified by use of the equations of motion. From these currents
one constructs the conserved charges
\beq
Q_\alpha = {\sqrt{2}}\int d^3 \vec{x}\> J^0_\alpha,\qquad\qquad
Q^\dagger_{\dot{\alpha}} = {\sqrt{2}} \int d^3\vec{x} \>
J^{\dagger 0}_{\dot{\alpha}} ,
\eeq
which are the generators of supersymmetry transformations. (The factor of
$\sqrt{2}$ normalization is included to agree with an arbitrary historical
convention.) As quantum mechanical operators, they satisfy
\beq
\left [ \epsilon Q + \epsilon^\dagger Q^\dagger , X \right ]
= i{\sqrt{2}} \> \delta X
\label{interpolistheworstbandonearth}
\eeq
for any field $X$, up to terms that vanish onshell. This
can be verified explicitly by using the canonical equaltime
commutation and anticommutation relations
\beq
&&[ \phi(\vec{x}), \pi(\vec{y}) ] \,=\,
[ \phi^*(\vec{x}), \pi^*(\vec{y}) ] \,=\, i \delta^{(3)}(\vec{x}\vec{y}
\hspace{0.25mm}) , \qquad\>\>{}\\
&&
\{
\psi_\alpha (\vec{x}),
\psi^\dagger_{\dot{\alpha}} (\vec{y}) \} \,=\,
(\sigma^0)_{{\alpha}\dot{\alpha}}\,\delta^{(3)}(\vec{x}\vec{y}
\hspace{0.25mm} ), \qquad{}
\eeq
which follow from the free field theory Lagrangian eq.~(\ref{Lwz}). Here $\pi =
\partial_0 \phi^*$ and $\pi^* = \partial_0 \phi$ are the momenta conjugate
to $\phi$ and $\phi^*$ respectively.
Using eq.~(\ref{interpolistheworstbandonearth}), the content of
eq.~(\ref{anytrans}) can be expressed in terms of canonical commutators as
\beq
\Bigl [
\epsilon_2 Q + \epsilon_2^\dagger Q^\dagger,\,
\bigl [
\epsilon_1 Q + \epsilon_1^\dagger Q^\dagger
,\, X
\bigr ] \Bigr ]

\Bigl [
\epsilon_1 Q + \epsilon_1^\dagger Q^\dagger,\,
\bigl [
\epsilon_2 Q + \epsilon_2^\dagger Q^\dagger
,\, X
\bigr ] \Bigr ]
=\qquad\qquad &&
\nonumber
\\
2( \epsilon_1 \sigma^\mu \epsilon_2^\dagger
\epsilon_2 \sigma^\mu \epsilon_1^\dagger)\, i\partial_\mu X
,&&
\qquad\>{}
\label{epsalg}
\eeq
up to terms that vanish onshell. The spacetime momentum operator is
$P^\mu = (H, \vec{P})$, where $H$ is the Hamiltonian and $\vec{P}$ is the
threemomentum operator, given in terms of the canonical fields by
\beq
H &=&
\int d^3\vec{x} \left [
\pi^*\pi +
(\vec{\nabla} \phi^*)
\cdot (\vec{\nabla} \phi)
+ i \psi^\dagger \vec{\sigma} \cdot \vec{\nabla} \psi
\right ] ,
\\
\vec P &=&
\int d^3\vec{x} \left (
\pi \vec{\nabla} \phi
+\pi^* \vec{\nabla} \phi^*
+ i \psi^\dagger \sigmabar^0 \vec{\nabla} \psi
\right )
.
\eeq
It generates spacetime translations on the fields $X$ according to
\beq
[P^\mu, X ] = \BDneg i \partial^\mu X.
\label{supergrass}
\eeq
Rearranging the terms in eq.~(\ref{epsalg}) using the Jacobi identity,
we therefore have
\beq
\Bigl [ \bigl [
\epsilon_2 Q + \epsilon_2^\dagger Q^\dagger,\,
\epsilon_1 Q + \epsilon_1^\dagger Q^\dagger \bigr ]
,\, X
\Bigr ]
&= & 2(\BDneg\epsilon_1 \sigma_\mu \epsilon_2^\dagger
\BDplus\epsilon_2 \sigma_\mu \epsilon_1^\dagger )\, [ P^\mu , X ],
\label{epsalg2}
\eeq
for any $X$, up to terms that vanish onshell, so it must be that
\beq
\bigl [
\epsilon_2 Q + \epsilon_2^\dagger Q^\dagger,\,
\epsilon_1 Q + \epsilon_1^\dagger Q^\dagger \bigr ]
&= & 2(\BDneg\epsilon_1 \sigma_\mu \epsilon_2^\dagger
\BDplus\epsilon_2 \sigma_\mu \epsilon_1^\dagger)\, P^\mu .
\label{epsalg3}
\eeq
Now by expanding out eq.~(\ref{epsalg3}), one obtains the precise form of
the supersymmetry algebra relations
\beq
&&\{ Q_\alpha , Q^\dagger_{\dot{\alpha}} \} =
\BDpos 2\sigma^\mu_{\alpha\dot{\alpha}} P_\mu,
\label{nonschsusyalg1}\\
&&\{ Q_\alpha, Q_\beta\} = 0
, \qquad\qquad
\{ Q^\dagger_{\dot{\alpha}}, Q^\dagger_{\dot{\beta}} \} = 0
,
\phantom{XXX}
\label{nonschsusyalg2}
\eeq
as promised in the Introduction. [The commutator in eq.~(\ref{epsalg3})
turns into anticommutators in eqs.~(\ref{nonschsusyalg1}) and
(\ref{nonschsusyalg2}) when the anticommuting
spinors $\epsilon_1$ and $\epsilon_2$ are extracted.] The results
\beq
[Q_\alpha, P^\mu ] = 0, \qquad\qquad [Q^\dagger_{\dot{\alpha}},
P^\mu] = 0
\eeq
follow immediately from eq.~(\ref{supergrass}) and the fact that the
supersymmetry transformations are global (independent of position in
spacetime). This demonstration of the supersymmetry algebra in terms of
the canonical generators $Q$ and $Q^\dagger$ requires the use of the
Hamiltonian equations of motion, but the symmetry itself is valid
offshell at the level of the Lagrangian, as we have already shown.
\subsection{Interactions of chiral supermultiplets}\label{subsec:susylagr.chiral}
\setcounter{footnote}{1}
\renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{equation}}
\setcounter{equation}{0}
In a realistic theory like the MSSM, there are many chiral
supermultiplets, with both gauge and nongauge interactions. In this
subsection, our task is to construct the most general possible theory of
masses and nongauge interactions for particles that live in chiral
supermultiplets. In the MSSM these are the quarks, squarks, leptons,
sleptons, Higgs scalars and higgsino fermions. We will find that the form
of the nongauge couplings, including mass terms, is highly restricted by
the requirement that the action is invariant under supersymmetry
transformations. (Gauge interactions will be dealt with in the following
subsections.)
Our starting point is the Lagrangian density for a collection of free
chiral supermultiplets labeled by an index $i$, which runs over all gauge
and flavor degrees of freedom. Since we will want to construct an
interacting theory with supersymmetry closing offshell, each
supermultiplet contains a complex scalar $\phi_i$ and a lefthanded Weyl
fermion $\psi_i$ as physical degrees of freedom, plus a nonpropagating
complex auxiliary
field $F_i$. The results of the previous
subsection tell us that the free part of the Lagrangian is
\beq
\lagr_{\rm free} &=&
\BDpos \partial^\mu \phi^{*i} \partial_\mu \phi_i
+ i {\psi}^{\dagger i} \sigmabar^\mu \partial_\mu \psi_i
+ F^{*i} F_i ,
\label{lagrfree}
\eeq
where we sum over repeated indices $i$ (not to be confused with the
suppressed spinor indices), with the convention that fields $\phi_i$ and
$\psi_i$ always carry lowered indices, while their conjugates always carry
raised indices. It is invariant under the supersymmetry transformation
\beq
&
\delta \phi_i = \epsilon\psi_i ,
\qquad\>\>\>\>\>\qquad\qquad\qquad
\phantom{xxxi}
&
\delta \phi^{*i} = \epsilon^\dagger {\psi}^{\dagger i} ,
\label{phitran}
\\
&
\delta (\psi_i)_\alpha =
 i (\sigma^\mu {\epsilon^\dagger})_{\alpha}\, \partial_\mu
\phi_i + \epsilon_\alpha F_i ,
\qquad
&
\delta ({\psi}^{\dagger i})_{\dot{\alpha}}=
i (\epsilon\sigma^\mu)_{\dot{\alpha}}\, \partial_\mu
\phi^{*i} + \epsilon^\dagger_{\dot{\alpha}} F^{*i} ,
\phantom{xxxx}
\\
&
\delta F_i =  i \epsilon^\dagger \sigmabar^\mu\partial_\mu \psi_i ,
\qquad\qquad\qquad
\phantom{xxi}
&
\deltaeps F^{* i} =
i\partial_\mu {\psi}^{\dagger i} \sigmabar^\mu \epsilon\> .
\phantom{xxx}
\label{eq:Ftran}
\eeq
We will now find the most general set of renormalizable interactions for
these fields that is consistent with supersymmetry. We do this working in
the field theory before integrating out the auxiliary fields. To begin,
note that in order to be renormalizable by power counting, each term must
have field content with total mass dimension $\leq 4$. So, the only
candidate terms are:
\beq
\lagr_{\rm int} =
\left ({1\over 2} W^{ij} \psi_i \psi_j + W^i F_i + x^{ij} F_i F_j \right )
+ \conj  U,
\label{pretryint}
\eeq
where $W^{ij}$, $W^i$, $x^{ij}$, and $U$ are polynomials in the scalar
fields $\phi_i, \phi^{*i}$, with degrees $1$, $2$, $0$, and $4$,
respectively. [Terms of the form $F^{*i} F_j$ are already included in
eq.~(\ref{lagrfree}), with the coefficient fixed by the transformation
rules (\ref{phitran})(\ref{eq:Ftran}).]
We must now require that $\lagr_{\rm int}$ is invariant under the
supersymmetry transformations, since $\lagr_{\rm free}$ was already
invariant by itself. This immediately requires that the candidate term
$U(\phi_i, \phi^{*i})$ must vanish. If there were such a term, then under
a supersymmetry transformation eq.~(\ref{phitran}) it would transform into
another function of the scalar fields only, multiplied by $\epsilon\psi_i$
or ${\epsilon^\dagger}{\psi}^{\dagger i}$, and with no spacetime
derivatives or $F_i$, $F^{*i}$ fields. It is easy to see from
eqs.~(\ref{phitran})(\ref{pretryint}) that nothing of this form can possibly
be canceled by the supersymmetry transformation of any other term in the
Lagrangian. Similarly, the dimensionless coupling $x^{ij}$ must be zero,
because its supersymmetry transformation likewise cannot possibly be
canceled by any other term. So, we are left with
\beq
\lagr_{\rm int} =
\left (\half W^{ij} \psi_i \psi_j + W^i F_i \right ) + \conj
\label{tryint}
\eeq
as the only possibilities. At this point, we are not assuming that $W^{ij}$
and $W^i$ are related to each other in any way. However, soon we will
find out that they {\it are} related, which is why we have chosen to use
the same letter for them. Notice that eq.~(\ref{xichi}) tells us that
$W^{ij}$ is symmetric under $i\leftrightarrow j$.
It is easiest to divide the variation of $\lagr_{\rm int}$ into several
parts, which must cancel separately. First, we consider the part that
contains four spinors:
\beq
\delta \lagr_{\rm int} _{\rm 4spinor} = \left [
{1\over 2} {\delta W^{ij} \over \delta \phi_k}
(\epsilon \psi_k)(\psi_i \psi_j)
{1\over 2} {\delta W^{ij} \over \delta \phi^{*k}}
({\epsilon^\dagger}{\psi}^{\dagger k})(\psi_i \psi_j) \right ]
+ \conj
\label{deltafourferm}
\eeq
The term proportional to $(\epsilon \psi_k)(\psi_i\psi_j)$ cannot cancel
against any other term. Fortunately, however, the Fierz identity
eq.~(\ref{fierce}) implies
\beq
(\epsilon \psi_i) (\psi_j \psi_k) + (\epsilon \psi_j) (\psi_k \psi_i)
+ (\epsilon \psi_k) (\psi_i\psi_j) = 0 ,
\eeq
so this contribution to $\delta\lagr_{\rm int}$ vanishes identically if
and only if $\delta W^{ij}/\delta \phi_k$ is totally symmetric under
interchange of $i,j,k$. There is no such identity available for the term
proportional to $({\epsilon^\dagger } {\psi}^{\dagger k})(\psi_i\psi_j)$.
Since that term cannot cancel with any other, requiring it to be absent
just tells us that $W^{ij}$ cannot contain $\phi^{*k}$. In other words,
$W^{ij}$ is holomorphic (or complex analytic) in the complex fields $\phi_k$.
Combining what we have learned so far, we can write
\beq
W^{ij} = M^{ij} + y^{ijk} \phi_k
\eeq
where $M^{ij}$ is a symmetric mass matrix for the fermion fields, and
$y^{ijk}$ is a Yukawa coupling of a scalar $\phi_k$ and two fermions
$\psi_i \psi_j$ that must be totally symmetric under interchange of
$i,j,k$. It is therefore possible, and it turns out to be convenient, to
write
\beq
W^{ij} = {\delta^2 \over \delta\phi_i\delta\phi_j} W
\label{expresswij}
\eeq
where we have introduced a useful object
\beq
W =
{1\over 2} M^{ij} \phi_i \phi_j + {1\over 6} y^{ijk} \phi_i \phi_j \phi_k,
\label{superpotential}
\eeq
called the {\it superpotential}.\index{superpotential} This is not a
scalar potential in the ordinary sense; in fact, it is not even real. It
is instead a holomorphic function of the scalar fields $\phi_i$ treated as
complex variables.
Continuing on our vaunted quest, we next consider the parts of
$\delta \lagr_{\rm int}$ that contain a spacetime derivative:
\beq
\delta \lagr_{\rm int} _\partial &=& \left (
i W^{ij}\partial_\mu \phi_j \, \psi_i \sigma^\mu {\epsilon^\dagger}
+ i W^i\, \partial_\mu \psi_i\sigma^\mu {\epsilon^\dagger}
\right ) +\conj
\label{wijwi}
\eeq
Here we have used the identity eq.~(\ref{yetanotheridentity}) on the
second term, which came from $(\delta F_i)W^i$. Now we can use
eq.~(\ref{expresswij}) to observe that
\beq
W^{ij} \partial_\mu \phi_j =
\partial_\mu \left ( {\delta W\over \delta{\phi_i}}\right ) .
\label{parttwo}
\eeq
Therefore, eq.~(\ref{wijwi}) will be a total derivative if
\beq
W^i = {\delta W\over \delta \phi_i}
= M^{ij}\phi_j + {1\over 2} y^{ijk} \phi_j \phi_k\> ,
\label{wiwiwi}
\eeq
which explains why we chose its name as we did. The remaining terms in
$\delta \lagr_{\rm int}$ are all linear in $F_i$ or $F^{*i}$, and it is
easy to show that they cancel, given the results for $W^i$ and $W^{ij}$
that we have already found.
Actually, we can include a linear term in the superpotential without
disturbing the validity of the previous discussion at all:
\beq
W = L^i \phi_i +
{1\over 2} M^{ij} \phi_i \phi_j + {1\over 6} y^{ijk} \phi_i \phi_j \phi_k
.
\label{superpotentialwithlinear}
\eeq
Here $L^i$ are parameters with dimensions of [mass]$^2$, which affect only
the scalar potential part of the Lagrangian. Such
linear terms are only allowed when $\phi_i$ is a gauge singlet, and there are
no such gauge singlet chiral supermultiplets in the MSSM with minimal
field content. I will therefore omit this term from the remaining
discussion of this section. However, this type of term does play an
important role in the discussion of spontaneous supersymmetry breaking, as
we will see in section \ref{subsec:origins.general}.
To recap, we have found that the most general nongauge interactions for
chiral supermultiplets are determined by a single holomorphic function of the
complex scalar fields, the superpotential $W$. The auxiliary fields $F_i$
and $F^{*i}$ can be eliminated using their classical equations of motion.
The part of $\lagr_{\rm free} + \lagr_{\rm int}$ that contains the
auxiliary fields is $ F_i F^{*i} + W^i F_{i} + W^{*}_i F^{*i}$, leading to
the equations of motion
\beq
F_i = W_i^*,\qquad\qquad F^{*i} = W^i \> .
\label{replaceF}
\eeq
Thus the auxiliary fields are expressible algebraically (without any
derivatives) in terms of the scalar fields.
After making the replacement\footnote{Since $F_i$ and $F^{*i}$ appear only
quadratically in the action, the result of instead doing a functional
integral over them at the quantum level has precisely the same effect.}
eq.~(\ref{replaceF}) in $\lagr_{\rm free} + \lagr_{\rm int}$, we obtain
the Lagrangian density
\beq
\lagr = \BDpos\partial^\mu \phi^{*i} \partial_\mu \phi_i
+ i \psi^{\dagger i} \sigmabar^\mu \partial_\mu \psi_i
\half \left (W^{ij} \psi_i \psi_j + W^{*}_{ij} \psi^{\dagger i}
\psi^{\dagger j} \right )
 W^i W^{*}_i.
\label{noFlagr}
\eeq
Now that the nonpropagating fields $F_i, F^{*i}$ have been eliminated, it
follows from eq.~(\ref{noFlagr}) that the scalar potential for the theory
is just given in terms of the superpotential by
\beq
V(\phi,\phi^*) = W^k W_k^* = F^{*k} F_k =
\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx.}
&&
\nonumber
\\
M^*_{ik} M^{kj} \phi^{*i} \phi_{j}
+{1\over 2} M^{in} y_{jkn}^* \phi_i \phi^{*j} \phi^{*k}
+{1\over 2} M_{in}^{*} y^{jkn} \phi^{*i} \phi_j \phi_k
+{1\over 4} y^{ijn} y_{kln}^{*} \phi_i \phi_j \phi^{*k} \phi^{*l}
\, .
&&
\label{ordpot}
\eeq
This scalar potential is automatically bounded from below; in fact, since
it is a sum of squares of absolute values (of the $W^k$), it is always
nonnegative. If we substitute the general form for the superpotential
eq.~(\ref{superpotential}) into eq.~(\ref{noFlagr}), we obtain for the
full Lagrangian density
\beq
\lagr &=&
\BDpos \partial^\mu \phi^{*i} \partial_\mu \phi_i  V(\phi,\phi^*)
+ i \psi^{\dagger i} \sigmabar^\mu \partial_\mu \psi_i
 \half M^{ij} \psi_i\psi_j  \half M_{ij}^{*} \psi^{\dagger i}
\psi^{\dagger j}
\nonumber\\
&&  \half y^{ijk} \phi_i \psi_j \psi_k  \half y_{ijk}^{*} \phi^{*i}
\psi^{\dagger j} \psi^{\dagger k}.
\label{lagrchiral}
\eeq
Now we can compare the masses of the fermions and scalars by looking at
the linearized equations of motion:
\beq
\partial^\mu\partial_\mu \phi_i &=&
\BDneg
M_{ik}^{*} M^{kj} \phi_j
+ \ldots,\\
i\sigmabar^\mu\partial_\mu\psi_i &=&
M_{ij}^{*} \psi^{\dagger j}+\ldots,
\qquad\qquad
i\sigma^\mu\partial_\mu\psi^{\dagger i} \>=\>
M^{ij} \psi_j +\ldots .\qquad\>\>\>{}
\label{linfermiontwo}
\eeq
One can eliminate $\psi$ in terms of $\psi^\dagger$ and vice versa in
eq.~(\ref{linfermiontwo}), obtaining [after use of the identities
eqs.~(\ref{pauliidentA}) and (\ref{pauliidentB})]:
\beq
\partial^\mu\partial_\mu \psi_i =
\BDneg M_{ik}^{*} M^{kj} \psi_j
+ \ldots ,\qquad\qquad
\partial^\mu\partial_\mu \psi^{\dagger j} =
\BDneg \psi^{\dagger i} M_{ik}^{*} M^{kj}+\ldots
\> .
\eeq
Therefore, the fermions and the bosons satisfy the same wave equation with
exactly the same squaredmass matrix with real nonnegative eigenvalues,
namely ${(M^2)_i}^j = M_{ik}^{*} M^{kj}$. It follows that diagonalizing
this matrix by redefining the fields with a unitary matrix gives a
collection of chiral supermultiplets, each of which contains a
massdegenerate complex scalar and Weyl fermion, in agreement with the
general argument in the Introduction.
\subsection{Lagrangians for gauge
supermultiplets}\label{subsec:susylagr.gauge}
\setcounter{footnote}{1}
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The propagating degrees of freedom in a gauge supermultiplet are a
massless gauge boson field $A_\mu^a$ and a twocomponent Weyl fermion
gaugino $\lambda^a$. The index $a$ here runs over the adjoint
representation of the gauge group ($a=1,\ldots ,8$ for $SU(3)_C$ color
gluons and gluinos; $a=1,2,3$ for $SU(2)_L$ weak isospin; $a=1$ for
$U(1)_Y$ weak hypercharge). The gauge transformations of the vector
supermultiplet fields are
\beq
A^a_\mu &\rightarrow& A^a_\mu
\BDminus \partial_\mu \Lambda^a + g f^{abc} A^b_\mu \Lambda^c ,
\label{Agaugetr}
\\
\lambda^a &\rightarrow& \lambda^a + g f^{abc} \lambda^b \Lambda^c
,
\label{lamgaugetr}
\eeq
where $\Lambda^a$ is an infinitesimal gauge transformation parameter, $g$
is the gauge coupling, and $f^{abc}$ are the totally antisymmetric
structure constants that define the gauge group. The special case of an
Abelian group is obtained by just setting $f^{abc}=0$; the corresponding
gaugino is a gauge singlet in that case. The conventions are such that for
QED, $A^\mu = (V, \vec{A})$ where $V$ and $\vec{A}$ are the usual electric
potential and vector potential, with electric and magnetic fields given by
$\vec{E} = \vec{\nabla} V  \partial_0 \vec{A}$ and $\vec{B} =
\vec{\nabla} \times \vec{A}$.
The onshell degrees of freedom for $A^a_\mu$ and $\lambda^a_\alpha$
amount to two bosonic and two fermionic helicity states (for each $a$), as
required by supersymmetry. However, offshell $\lambda^a_\alpha$ consists
of two complex, or four real, fermionic degrees of freedom, while
$A^a_\mu$ only has three real bosonic degrees of freedom; one degree of
freedom is removed by the inhomogeneous gauge transformation
eq.~(\ref{Agaugetr}). So, we will need one real bosonic auxiliary field,
traditionally called $D^a$, in order for supersymmetry to be consistent
offshell. This field also transforms as an adjoint of the gauge group
[i.e., like eq.~(\ref{lamgaugetr}) with $\lambda^a$ replaced by $D^a$] and
satisfies $(D^a)^* = D^a$. Like the chiral auxiliary fields $F_i$, the
gauge auxiliary field $D^a$ has dimensions of [mass]$^2$ and no kinetic
term, so it can be eliminated onshell using its algebraic equation of
motion. The counting of degrees of freedom is summarized in Table
\ref{table:gaugedofcounting}.
\renewcommand{\arraystretch}{1.45}
\begin{table}[tb]
\begin{center}
\begin{tabular}{cccc}
\hline
& $A_\mu$ & $\lambda$ & $D$ \\
\hline
onshell ($n_B=n_F=2$) & 2 & 2 & 0 \\
\hline
offshell ($n_B=n_F=4$) & 3 & 4 & 1 \\
\hline
\end{tabular}
\caption{Counting of real degrees of freedom for each gauge
supermultiplet. \label{table:gaugedofcounting}}
\vspace{0.4cm}
\end{center}
\end{table}
Therefore, the Lagrangian density for a gauge supermultiplet ought to be
\beq
\lagr_{\rm gauge} = {1\over 4} F_{\mu\nu}^a F^{\mu\nu a}
+ i \lambda^{\dagger a} \sigmabar^\mu \nabla_\mu \lambda^a
+ {1\over 2} D^a D^a ,
\label{lagrgauge}
\eeq
where
\beq
F^a_{\mu\nu} = \partial_\mu A^a_\nu  \partial_\nu A^a_\mu
\BDminus g f^{abc} A^b_\mu A^c_\nu
\label{eq:YMfs}
\eeq
is the usual YangMills field strength, and
\beq
\nabla_\mu \lambda^a = \partial_\mu \lambda^a
\BDminus g f^{abc} A^b_\mu \lambda^c
\label{ordtocovlambda}
\eeq
is the covariant derivative of the gaugino field. To check that
eq.~(\ref{lagrgauge}) is really supersymmetric, one must specify the
supersymmetry transformations of the fields.
The forms of these follow from the requirements that they should be linear
in the infinitesimal parameters $\epsilon,\epsilon^\dagger$ with
dimensions of [mass]$^{1/2}$, that $\delta A^a_\mu$ is real, and that
$\delta D^a$ should be real and proportional to the field equations for
the gaugino, in analogy with the role of the auxiliary field $F$ in the
chiral supermultiplet case. Thus one can guess, up to multiplicative
factors, that\footnote{The supersymmetry transformations
eqs.~(\ref{Atransf})(\ref{Dtransf}) are nonlinear for nonAbelian gauge
symmetries, since there are gauge fields in the covariant
derivatives acting on the gaugino fields and in the field strength $F_{\mu
\nu}^a$. By adding even more auxiliary fields besides $D^a$, one can make
the supersymmetry transformations linear in the fields; this is easiest to
do in superfield language (see sections \ref{subsec:vectorsuperfields},
\ref{subsec:superspacelagrabelian}, and \ref{subsec:superspacelagrnonabelian}).
The version in this section, in which those extra auxiliary fields
have been eliminated, is called ``WessZumino gauge" \cite{WZgauge}.}
\beq
&& \delta A_\mu^a =
 {1\over \sqrt{2}} \left (\epsilon^\dagger \sigmabar_\mu
\lambda^a + \lambda^{\dagger a} \sigmabar_\mu \epsilon \right )
,
\label{Atransf}
\\
&& \delta \lambda^a_\alpha =
\BDneg {i\over 2\sqrt{2}} (\sigma^\mu \sigmabar^\nu \epsilon)_\alpha
\> F^a_{\mu\nu} + {1\over \sqrt{2}} \epsilon_\alpha\> D^a
,
\\
&& \delta D^a = {i\over \sqrt{2}} \left (
\epsilon^\dagger \sigmabar^\mu \nabla_\mu \lambda^a
+\nabla_\mu \lambda^{\dagger a} \sigmabar^\mu \epsilon \right ) .
\label{Dtransf}
\eeq
The factors of $\sqrt{2}$ are chosen so that the action obtained by
integrating $\lagr_{\rm gauge}$ is indeed invariant, and the phase of
$\lambda^a$ is chosen for future convenience in treating the MSSM.
It is now a little bit tedious, but straightforward, to also check that
\beq
(\delta_{\epsilon_2} \delta_{\epsilon_1} \delta_{\epsilon_1}
\delta_{\epsilon_2} ) X =
i (\epsilon_1\sigma^\mu \epsilon_2^\dagger
+\epsilon_2\sigma^\mu \epsilon_1^\dagger) \nabla_\mu X
\label{joeyramone}
\eeq
for $X$ equal to any of the gaugecovariant fields $F_{\mu\nu}^a$,
$\lambda^a$, $\lambda^{\dagger a}$, $D^a$, as well as for arbitrary
covariant derivatives acting on them. This ensures that the supersymmetry
algebra eqs.~(\ref{nonschsusyalg1})(\ref{nonschsusyalg2}) is realized on
gaugeinvariant combinations of fields in gauge supermultiplets, as they
were on the chiral supermultiplets [compare eq.~(\ref{anytrans})]. This
check requires the use of identities
eqs.~(\ref{eq:dei}), (\ref{eq:feif}) and (\ref{eq:lloydhouserules}).
If we had not included the auxiliary field
$D^a$, then the supersymmetry algebra eq.~(\ref{joeyramone}) would hold
only after using the equations of motion for $\lambda^a$ and
$\lambda^{\dagger a}$. The auxiliary fields satisfies a trivial equation
of motion $D^a=0$, but this is modified if one couples the gauge
supermultiplets to chiral supermultiplets, as we now do.
\subsection{Supersymmetric gauge
interactions}\label{subsec:susylagr.gaugeinter}
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Now we are ready to consider a general Lagrangian density for a
supersymmetric theory with both chiral and gauge supermultiplets. Suppose
that the chiral supermultiplets transform under the gauge group in a
representation with hermitian matrices ${(T^a)_i}^j$ satisfying $[T^a,T^b]
=i f^{abc} T^c$. [For example, if the gauge group is $SU(2)$, then
$f^{abc} = \epsilon^{abc}$, and
for a chiral supermultiplet transforming in the fundamental
representation the $T^a$ are $1/2$ times the Pauli
matrices.] Because supersymmetry and gauge transformations commute,
the scalar, fermion, and auxiliary fields must be in the same
representation of the gauge group, so
\beq
X_i &\rightarrow& X_i + ig \Lambda^a (T^a X)_i
\eeq
for $X_i = \phi_i,\psi_i,F_i$. To have a gaugeinvariant Lagrangian, we
now need to replace the ordinary derivatives $\partial_\mu \phi_i$,
$\partial_\mu \phi^{*i}$, and $\partial_\mu \psi_i$
in eq.~(\ref{lagrfree}) with covariant derivatives:
\beq
\nabla_\mu \phi_i &=& \partial_\mu \phi_i \BDplus i g A^a_\mu (T^a\phi)_i
\label{ordtocovphi}
\\
\nabla_\mu \phi^{*i} &=& \partial_\mu \phi^{*i} \BDminus i g A^a_\mu (\phi^* T^a)^i
\\
\nabla_\mu \psi_i &=& \partial_\mu \psi_i \BDplus i g A^a_\mu (T^a\psi)_i .
\label{ordtocovpsi}
\eeq
Naively, this simple procedure achieves the goal of coupling the vector
bosons in the gauge supermultiplet to the scalars and fermions in the
chiral supermultiplets. However, we also have to consider whether there
are any other interactions allowed by gauge invariance and involving the
gaugino and $D^a$ fields, which might have to be included to make a
supersymmetric Lagrangian. Since $A^a_\mu$ couples to $\phi_i$ and
$\psi_i$, it makes sense that $\lambda^a$ and $D^a$ should as well.
In fact, there are three such possible interaction terms that are
renormalizable (of field mass dimension $\leq 4$), namely
\beq
(\phi^* T^a \psi)\lambda^a,\qquad
\lambda^{\dagger a} (\psi^\dagger T^a
\phi),
\qquad {\rm and} \qquad
(\phi^* T^a \phi) D^a .
\label{extrater}
\eeq
Now one can add them, with unknown dimensionless coupling coefficients, to
the Lagrangians for the chiral and gauge supermultiplets, and demand that
the whole mess be real and invariant under supersymmetry,
up to a total derivative. Not surprisingly, this is possible only if the
supersymmetry transformation laws for the matter fields are modified to
include gaugecovariant rather than ordinary derivatives. Also, it is
necessary to include one strategically chosen extra term in $\delta F_i$,
so:
\beq
&&
\delta \phi_i = \epsilon\psi_i
\label{gphitran}\\
&&\delta \psi_{i\alpha} =
i (\sigma^\mu \epsilon^\dagger)_{\alpha}\> \nabla_\mu\phi_i + \epsilon_\alpha F_i
\\
&&\deltaeps F_i = i \epsilon^\dagger \sigmabar^\mu \nabla_\mu \psi_i
\> + \> \sqrt{2} g (T^a \phi)_i\> \epsilon^\dagger \lambda^{\dagger a} .
\eeq
After some algebra one can now fix the coefficients for the terms in
eq.~(\ref{extrater}), with the result that the full Lagrangian density
for a renormalizable supersymmetric theory is
\beq
\lagr & = & \lagr_{\rm chiral} + \lagr_{\rm gauge}
\nonumber\\
&&  \sqrt{2} g
(\phi^* T^a \psi)\lambda^a
 \sqrt{2} g\lambda^{\dagger a} (\psi^\dagger T^a \phi)
+ g (\phi^* T^a \phi) D^a .
\label{gensusylagr}
\eeq
Here $\lagr_{\rm chiral }$ means the chiral supermultiplet Lagrangian
found in section \ref{subsec:susylagr.chiral} [e.g., eq.~(\ref{noFlagr})
or (\ref{lagrchiral})], but with ordinary derivatives replaced everywhere
by gaugecovariant derivatives, and $\lagr_{\rm gauge}$ was given in
eq.~(\ref{lagrgauge}). To prove that eq.~(\ref{gensusylagr}) is invariant
under the supersymmetry transformations, one must use the identity
\beq
W^i (T^a \phi)_i = 0.
\label{wgaugeinvar}
\eeq
This is precisely the condition that must be satisfied anyway in order for
the superpotential, and thus $\lagr_{\rm chiral}$, to be gauge invariant.
The second line in eq.~(\ref{gensusylagr}) consists of interactions whose
strengths are fixed to be gauge couplings by the requirements of
supersymmetry, even though they are not gauge interactions from the point
of view of an ordinary field theory. The first two terms are a direct
coupling of gauginos to matter fields; this can be thought of as the
``supersymmetrization" of the usual gauge boson couplings to matter fields.
The last term combines with the $D^a D^a/2$ term in $\lagr_{\rm gauge}$ to
provide an equation of motion
\beq
D^a = g (\phi^* T^a \phi ).
\label{solveforD}
\eeq
Thus, like the auxiliary fields $F_i$ and $F^{*i}$, the $D^a$ are
expressible purely algebraically in terms of the scalar fields. Replacing
the auxiliary fields in eq.~(\ref{gensusylagr}) using
eq.~(\ref{solveforD}), one finds that the complete scalar potential is
(recall that $\lagr$ contains $V$):
\beq
V(\phi,\phi^*) = F^{*i} F_i + \half \sum_a D^a D^a = W_i^* W^i +
\half \sum_a g_a^2 (\phi^* T^a \phi)^2.
\label{fdpot}
\eeq
The two types of terms in this expression are called ``$F$term" and
``$D$term" contributions, respectively. In the second term in
eq.~(\ref{fdpot}), we have now written an explicit sum $\sum_a$ to cover
the case that the gauge group has several distinct factors with different
gauge couplings $g_a$. [For instance, in the MSSM the three factors
$SU(3)_C$, $SU(2)_L$ and $U(1)_Y$ have different gauge couplings $g_3$,
$g$ and $g^\prime$.] Since $V(\phi,\phi^*)$ is a sum of squares, it is
always greater than or equal to zero for every field configuration. It is
an interesting and unique feature of supersymmetric theories that the
scalar potential is completely determined by the {\it other} interactions
in the theory. The $F$terms are fixed by Yukawa couplings and fermion
mass terms, and the $D$terms are fixed by the gauge interactions.
By using Noether's procedure [see eq.~(\ref{Noether})], one finds the
conserved supercurrent
\beq
J_\alpha^\mu &\!\!\!=\!\!\!&
(\sigma^\nu\sigmabar^\mu \psi_i)_\alpha\, \nabla_\nu \phi^{*i}
+ i (\sigma^\mu \psi^{\dagger i})_\alpha\, W_i^*
\nonumber
\\ &&
\BDplus {1\over 2 \sqrt{2}}
(\sigma^\nu \sigmabar^\rho \sigma^\mu
\lambda^{\dagger a})_\alpha\, F^a_{\nu\rho}
+ {i\over {\sqrt{2}}} g_a \phi^* T^a \phi
\> (\sigma^\mu \lambda^{\dagger a})_\alpha , \>\>\>\>{}
\label{supercurrent}
\eeq
generalizing the expression given in eq.~(\ref{WZsupercurrent}) for the
WessZumino model. This result will be useful when we discuss certain
aspects of spontaneous supersymmetry breaking in section
\ref{subsec:origins.gravitino}.
\subsection{Summary: How to build a supersymmetric
model}\label{subsec:susylagr.summary}
\setcounter{footnote}{1}
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In a renormalizable supersymmetric field theory, the interactions and
masses of all particles are determined just by their gauge transformation
properties and by the superpotential $W$. By construction, we found that
$W$ had to be a holomorphic function of the complex scalar fields $\phi_i$,
which are always defined to transform under supersymmetry into lefthanded
Weyl fermions. In an equivalent language, to be covered
in section \ref{sec:superfields}, $W$ is
said to be a function of chiral {\it superfields} \cite{superfields}. A
superfield is a single object that contains as components all of the
bosonic, fermionic, and auxiliary fields within the corresponding
supermultiplet, for example $\Phi_i \supset (\phi_i,\psi_i,F_i)$. (This is
analogous to the way in which one often describes a weak isospin doublet
or a color triplet by a multicomponent field.) The gauge quantum numbers and
the mass dimension of a chiral superfield are the same as that of its
scalar component. In the superfield formulation, one writes instead of
eq.~(\ref{superpotentialwithlinear})
\beq
W &=&
L^i \Phi_i +
{1\over 2}M^{ij} \Phi_i\Phi_j +{1\over 6} y^{ijk} \Phi_i \Phi_j \Phi_k
,
\label{superpot}
\eeq
which implies exactly the same physics. The derivation of all of our
preceding results can be obtained somewhat more elegantly using superfield
methods, which have the advantage of making invariance under supersymmetry
transformations manifest by defining the Lagrangian in terms of integrals
over a ``superspace" with fermionic as well as ordinary commuting
coordinates. We have purposefully avoided this extra layer of notation so far, in
favor of the more pedestrian, but more familiar and accessible, component
field approach. The latter is at least more appropriate for making contact
with phenomenology in a universe with supersymmetry breaking.
The specification of the superpotential
is really just a code for the terms that it implies in the Lagrangian, so the
reader may feel free to think of the superpotential either as a function
of the scalar fields $\phi_i$ or as the same function of the superfields
$\Phi_i$.
Given the supermultiplet content of the theory, the form of the
superpotential is restricted by the requirement of gauge invariance [see
eq.~(\ref{wgaugeinvar})]. In any given theory, only a subset of the
parameters $L^i$, $M^{ij}$, and $y^{ijk}$ are allowed to be nonzero. The
parameter $L^i$ is only allowed if $\Phi_i$ is a gauge singlet. (There are
no such chiral supermultiplets in the MSSM with the minimal field
content.) The entries of the mass matrix $M^{ij}$ can only be nonzero for
$i$ and $j$ such that the supermultiplets $\Phi_i$ and $\Phi_j$ transform
under the gauge group in representations that are conjugates of each
other. (In the MSSM there is only one such term, as we will see.)
Likewise, the Yukawa couplings $y^{ijk}$ can only be nonzero when
$\Phi_i$, $\Phi_j$, and $\Phi_k$ transform in representations that can
combine to form a singlet.
The interactions implied by the superpotential eq.~(\ref{superpot}) (with
$L^i=0$) were listed in
eqs.~(\ref{ordpot}), (\ref{lagrchiral}),
and are shown\footnote{Here, the auxiliary fields have been
eliminated using their equations of motion (``integrated out").
One can instead give Feynman rules that include
the auxiliary fields, or directly in terms of superfields on superspace,
although this is usually less practical for phenomenological
applications.} in Figures~\ref{fig:dim0} and \ref{fig:dim12}.
Those in
Figure~\ref{fig:dim0} are all determined by the dimensionless parameters
$y^{ijk}$. The Yukawa interaction in Figure~\ref{fig:dim0}a corresponds to
the nexttolast term in eq.~(\ref{lagrchiral}).%
\begin{figure}
\begin{center}
\begin{picture}(66,60)(0,0)
\SetWidth{0.85}
\ArrowLine(0,0)(33,12)
\ArrowLine(66,0)(33,12)
\DashLine(33,52.5)(33,12){4}
\ArrowLine(33,32.2501)(33,32.25)
\Text(0.75,10.5)[c]{$j$}
\Text(66,10)[c]{$k$}
\Text(26,48)[c]{$i$}
\Text(33,12)[c]{(a)}
\end{picture}
%
\hspace{1.8cm}
%
\begin{picture}(66,60)(0,0)
\SetWidth{0.85}
\ArrowLine(33,12)(0,0)
\ArrowLine(33,12)(66,0)
\DashLine(33,52.5)(33,12){4}
\ArrowLine(33,32.25)(33,32.2501)
\Text(0.75,10.5)[c]{$j$}
\Text(66,10)[c]{$k$}
\Text(26,48)[c]{$i$}
\Text(33,12)[c]{(b)}
\end{picture}
%
\hspace{1.8cm}
%
\begin{picture}(60,60)(0,0)
\SetWidth{0.85}
\DashLine(0,0)(30,30){4}
\DashLine(60,0)(30,30){4}
\DashLine(0,60)(30,30){4}
\DashLine(60,60)(30,30){4}
\ArrowLine(16.5,16.5)(16.501,16.501)
\ArrowLine(43.5,16.5)(43.499,16.501)
\ArrowLine(16.5,43.5)(16.499,43.501)
\ArrowLine(43.5,43.5)(43.501,43.501)
\Text(2,9)[c]{$i$}
\Text(64,10)[c]{$j$}
\Text(2,54)[c]{$k$}
\Text(63.5,54)[c]{$l$}
\Text(30,12)[c]{(c)}
\end{picture}
\end{center}
\caption{The dimensionless nongauge interaction
vertices in a supersymmetric theory:
(a) scalarfermionfermion Yukawa
interaction $y^{ijk}$,
(b) the complex conjugate interaction $y_{ijk}$, and
(c) quartic scalar interaction $y^{ijn}y^*_{kln}$.
\label{fig:dim0}}
\end{figure}
%
For each particular
Yukawa coupling of $\phi_i \psi_j \psi_k$ with strength $y^{ijk}$, there
must be equal couplings of $\phi_j \psi_i \psi_k$ and $\phi_k \psi_i
\psi_j$, since $y^{ijk}$ is completely symmetric under interchange of any
two of its indices as shown in section \ref{subsec:susylagr.chiral}.
The arrows on the fermion and scalar lines point in the direction for
propagation of $\phi$ and $\psi$ and opposite the direction of propagation
of $\phi^*$ and $\psi^\dagger$. Thus there is also a vertex corresponding
to the one in Figure~\ref{fig:dim0}a but with all arrows reversed,
corresponding to the complex conjugate [the last term in
eq.~(\ref{lagrchiral})]. It is shown in Figure \ref{fig:dim0}b. There is
also a dimensionless coupling for $\phi_i \phi_j \phi^{*k}\phi^{*l}$, with
strength $y^{ijn} y^*_{kln}$, as required by supersymmetry [see the last
term in eq.~(\ref{ordpot})]. The relationship between the Yukawa
interactions in Figures~\ref{fig:dim0}a,b and the scalar interaction of
Figure \ref{fig:dim0}c is exactly of the special type needed to cancel the
quadratic divergences in quantum corrections to scalar masses, as
discussed in the Introduction [compare Figure~\ref{fig:higgscorr1},
and eq.~(\ref{eq:royalewithcheese})].
Figure~\ref{fig:dim12} shows the only interactions corresponding to
renormalizable and supersymmetric vertices with coupling dimensions of
[mass] and [mass]$^2$.%
\begin{figure}
\begin{center}
\begin{picture}(66,62)(0,0)
\SetWidth{0.85}
\DashLine(33,52.5)(33,12){4}
\DashLine(0,0)(33,12){4}
\DashLine(66,0)(33,12){4}
\ArrowLine(33,32.25)(33,32.2501)
\ArrowLine(16.5,6)(16.5165,6.006)
\ArrowLine(49.5,6)(49.4835,6.006)
\Text(1.75,10.5)[c]{$j$}
\Text(65,10)[c]{$k$}
\Text(26,48)[c]{$i$}
\Text(33,12)[c]{(a)}
\end{picture}
%
\hspace{0.93cm}
%
\begin{picture}(66,62)(0,0)
\SetWidth{0.85}
\DashLine(33,52.5)(33,12){4}
\DashLine(0,0)(33,12){4}
\DashLine(66,0)(33,12){4}
\ArrowLine(33,32.2501)(33,32.25)
\ArrowLine(16.5165,6.006)(16.5,6)
\ArrowLine(49.4835,6.006)(49.5,6)
\Text(1.75,10.5)[c]{$j$}
\Text(65,10)[c]{$k$}
\Text(26,48)[c]{$i$}
\Text(33,12)[c]{(b)}
\end{picture}
%
\hspace{0.93cm}
%
\begin{picture}(72,62)(0,0)
\SetWidth{0.85}
\ArrowLine(0,12)(36,12)
\ArrowLine(72,12)(36,12)
\Line(33,9)(39,15)
\Line(39,9)(33,15)
\Text(0,19)[c]{$i$}
\Text(72,20)[c]{$j$}
\Text(36,12)[c]{(c)}
\end{picture}
%
\hspace{0.93cm}
%
\begin{picture}(72,62)(0,0)
\SetWidth{0.85}
\ArrowLine(36,12)(0,12)
\ArrowLine(36,12)(72,12)
\Line(33,9)(39,15)
\Line(39,9)(33,15)
\Text(0,19)[c]{$i$}
\Text(72,20)[c]{$j$}
\Text(36,12)[c]{(d)}
\end{picture}
%
\hspace{0.94cm}
%
\begin{picture}(72,62)(0,0)
\SetWidth{0.85}
\DashLine(0,12)(36,12){4}
\DashLine(72,12)(36,12){4}
\ArrowLine(17.99,12)(18,12)
\ArrowLine(55,12)(55.01,12)
\Line(33,9)(39,15)
\Line(39,9)(33,15)
\Text(72,20)[c]{$i$}
\Text(0,20)[c]{$j$}
\Text(36,12)[c]{(e)}
\end{picture}
\end{center}
\caption{Supersymmetric dimensionful couplings:
(a) (scalar)$^3$ interaction vertex $M^*_{in} y^{jkn}$ and
(b) the conjugate interaction $M^{in} y^*_{jkn}$,
(c) fermion mass term $M^{ij}$ and
(d) conjugate fermion mass term $M^*_{ij}$,
and
(e) scalar squaredmass term $M^*_{ik}M^{kj}$.
\label{fig:dim12}}
\end{figure}
%
First, there are (scalar)$^3$ couplings in Figure
\ref{fig:dim12}a,b, which are entirely determined by the superpotential
mass parameters $M^{ij}$ and Yukawa couplings $y^{ijk}$, as indicated by
the second and third terms in eq.~(\ref{ordpot}). The propagators of the
fermions and scalars in the theory are constructed in the usual way using
the fermion mass $M^{ij}$ and scalar squared mass $M^*_{ik}M^{kj}$. The
fermion mass terms $M^{ij}$ and $M_{ij}$ each lead to a chiralitychanging
insertion in the fermion propagator; note the directions of the arrows in
Figure~\ref{fig:dim12}c,d. There is no such arrowreversal for a scalar
propagator in a theory with exact supersymmetry; as depicted in
Figure~\ref{fig:dim12}e, if one treats the scalar squaredmass term as an
insertion in the propagator, the arrow direction is preserved.
Figure~\ref{fig:gauge} shows the gauge
interactions in a supersymmetric theory. Figures \ref{fig:gauge}a,b,c
occur only when the gauge group is nonAbelian, for example for $SU(3)_C$
color and $SU(2)_L$ weak isospin in the MSSM.%
\begin{figure}
\begin{center}
%
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\hspace{1.2cm}
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\hspace{1.2cm}
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\hspace{1.2cm}
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\Text(27.5,12.1)[c]{(d)}
\end{picture}
%
\hspace{1.2cm}
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\Text(27.5,12.1)[c]{(e)}
\end{picture}
%
\vspace{1.0cm}
%
\begin{picture}(50,50)(0,0)
\SetScale{1.1}
\SetWidth{0.85}
\ArrowLine(0,0)(25,15)
\ArrowLine(25,15)(50,0)
\Photon(25,42)(25,15){2}{4}
\Text(27.5,12.1)[c]{(f)}
\end{picture}
%
\hspace{1.33cm}
%
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\SetScale{1.1}
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\DashLine(50,0)(25,15){3.6}
\Text(27.5,12.1)[c]{(g)}
\end{picture}
%
\hspace{1.33cm}
%
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\SetScale{1.1}
\Photon(25,42)(25,15){2.25}{4}
\SetWidth{0.85}
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\ArrowLine(25,15)(50,0)
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\Text(27.5,12.1)[c]{(h)}
\end{picture}
%
\hspace{1.33cm}
%
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\SetScale{1.1}
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\DashLine(0,0)(25,25){4}
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\Text(27.5,12.1)[c]{(i)}
\end{picture}
\end{center}
\caption{Supersymmetric gauge interaction vertices.
\label{fig:gauge}}
\end{figure}
Figures \ref{fig:gauge}a and
\ref{fig:gauge}b are the interactions of gauge bosons, which derive from
the first term in eq.~(\ref{lagrgauge}). In the MSSM these are exactly the
same as the wellknown QCD gluon and electroweak gauge boson vertices of
the Standard Model. (We do not show the interactions of ghost fields,
which are necessary only for consistent loop amplitudes.) Figures
\ref{fig:gauge}c,d,e,f are just the standard interactions between gauge
bosons and fermion and scalar fields that must occur in any gauge theory
because of the form of the covariant derivative; they come from
eqs.~(\ref{ordtocovlambda}) and (\ref{ordtocovphi})(\ref{ordtocovpsi})
inserted in the kinetic part of the Lagrangian. Figure \ref{fig:gauge}c
shows the coupling of a gaugino to a gauge boson; the gaugino line in a
Feynman diagram is traditionally drawn as a solid fermion line
superimposed on a wavy line. In Figure~\ref{fig:gauge}g we have the
coupling of a gaugino to a chiral fermion and a complex scalar [the first
term in the second line of eq.~(\ref{gensusylagr})]. One can think of this
as the ``supersymmetrization" of Figure \ref{fig:gauge}e or
\ref{fig:gauge}f; any of these three vertices may be obtained from any
other (up to a factor of ${\sqrt{2}}$) by replacing two of the particles
by their supersymmetric partners. There is also an interaction in
Figure~\ref{fig:gauge}h which is just like Figure~\ref{fig:gauge}g but
with all arrows reversed, corresponding to the complex conjugate term in
the Lagrangian [the second term in the second line in
eq.~(\ref{gensusylagr})]. Finally in Figure~\ref{fig:gauge}i we have a
scalar quartic interaction vertex [the last term in eq.~(\ref{fdpot})],
which is also determined by the gauge coupling.
The results of this section can be used as a recipe for constructing the
supersymmetric interactions for any model. In the case of the MSSM, we
already know the gauge group, particle content and the gauge
transformation properties, so it only remains to decide on the
superpotential. This we will do in section
\ref{subsec:mssm.superpotential}. However, first we will revisit the structure of
supersymmetric Lagrangians in section \ref{sec:superfields} using the manifestly
supersymmetric formalism of superspace and superfields, and then describe the
general form of soft supersymmetry breaking terms in section \ref{sec:soft}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Superspace and superfields}\label{sec:superfields}
\setcounter{figure}{0}
\setcounter{table}{0}
\setcounter{footnote}{2}
\renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{equation}}
In this section, the basic ideas of superspace and superfields are covered. These ideas
provide elegant tools for understanding the structure of supersymmetric theories, and
are essential for analyzing and
communicating ideas about the formal structure of supersymmetry
in the most succinct ways. However, they are also not strictly necessary;
the discussion given above shows how supersymmetry can be defined and studied
completely without the superspace and superfield notation. The reader who is mainly
interested in phenomenological aspects of supersymmetric extensions
of the Standard Model is encouraged to skip this section, especially
on a first reading. The other sections (mostly) do not depend on it.
\subsection{Supercoordinates, general superfields, and
superspace differentiation and integration\label{subsec:supercoordinates}}
\setcounter{footnote}{2}
\setcounter{equation}{0}
Supersymmetry can be given a geometric interpretation using superspace,
a manifold obtained by adding four fermionic coordinates to the usual bosonic spacetime coordinates
$t,x,y,z$. Points in superspace are labeled by coordinates:
\beq
x^\mu,\> \theta^\alpha,\> \theta^\dagger_{\dot\alpha}.
\label{eq:supercoordinates}
\eeq
Here $\theta^\alpha$ and $\theta^\dagger_{\dot\alpha}$
are constant complex anticommuting twocomponent spinors with dimension
[mass]$^{1/2}$. In the superspace formulation, the component fields of
a supermultiplet
are united into a single superfield, a function of these superspace coordinates.
We will see below that infinitesimal translations in superspace coincide with the
global supersymmetry transformations that we have already found in component field language. Superspace thus allows an elegant and
manifestly invariant definition of supersymmetric field theories.
Differentiation and integration on spaces with anticommuting coordinates are defined
by analogy with ordinary commuting variables. Consider first, as a warmup
example, a single anticommuting variable
$\eta$ (carrying no spinor indices). Because $\eta^2=0$,
a power series expansion in $\eta$
always terminates, and a general function is linear in $\eta$:
\beq
f(\eta) = f_0 + \eta f_1.
\label{eq:deffuneta}
\eeq
Here $f_0$ and $f_1$ may be functions of other commuting or
anticommuting variables, but not $\eta$. One of them will be
anticommuting (Grassmannodd), and the other is commuting (Grassmanneven).
Then define:
\beq
\frac{df}{d\eta} = f_1 .
\label{eq:defdereta}
\eeq
The differential operator $\frac{d}{d\eta}$ anticommutes
with every Grassmannodd object,
so that if $\eta'$ is distinct from $\eta$ but also anticommuting, then
\beq
\frac{d(\eta'\eta)}{d\eta} = \frac{d(\eta\eta')}{d\eta} = \eta'.
\eeq
To define an integration operation with respect to $\eta$, take
\beq
\int d\eta = 0, \qquad\quad
\int d\eta\> \eta = 1,
\label{eq:defderetabasis}
\eeq
and impose linearity.
This defines the Berezin integral \cite{Berezin}
for Grassmann variables, and gives
\beq
\int d\eta \,f(\eta)
\> = \> f_1 .
\label{eq:definteta}
\eeq
Comparing eqs.~(\ref{eq:defdereta}) and (\ref{eq:definteta}) shows
the peculiar fact that
differentiation and integration are the same thing
for an anticommuting variable.
The definition eq.~(\ref{eq:defderetabasis}) is motivated by the fact that it implies
translation invariance,
\beq
\int d\eta \> f(\eta + \eta') = \int d\eta \> f(\eta),
\eeq
and the integration by parts formula
\beq
\int d\eta \> \frac{df}{d\eta} = 0,
\label{eq:funtheoeta}
\eeq
in analogy with the fundamental theorem of the calculus
for ordinary commuting variables.
The anticommuting Dirac delta function has the defining property
\beq
\int d\eta\> \delta(\eta  \eta')\, f(\eta) &=& f(\eta'),
\eeq
which leads to
\beq
\delta (\eta  \eta') &=& \eta  \eta'.
\eeq
For superspace with coordinates $x^\mu, \theta^\alpha, \theta^\dagger_{\dot\alpha}$,
any superfield can be expanded in a power series in the anticommuting variables, with components
that are functions of $x^\mu$. Since there are two
independent components of $\theta^\alpha$ and
likewise for $\theta^\dagger_{\dot\alpha}$, the expansion always
terminates, with each term containing at
most two $\theta$'s and two $\theta^\dagger$'s.
A general superfield is therefore:
\beq
S(x, \theta, \theta^\dagger) =
a
+ \theta \xi
+ \theta^\dagger\hspace{1pt} \chi^\dagger
+ \theta\theta b
+ \thdthd c
+ \thetasigmamuthetadagger v_\mu
+ \thdthd \theta \eta
+ \theta \theta \theta^\dagger\hspace{1pt} \zeta^\dagger
+ \theta\theta\thdthd d.
\label{eq:gensuperfield}
\eeq
To see that there are no other independent contributions, note the identities
\beq
\theta_\alpha \theta_\beta \>=\> \frac{1}{2} \epsilon_{\alpha\beta} \theta\theta
,
\qquad
\theta^\dagger_{\dot \alpha}\hspace{1pt} \theta^\dagger_{\dot \beta}
\>=\> \frac{1}{2} \epsilon_{\dot\beta\dot\alpha}
\thdthd
,
\qquad
\theta_\alpha
\theta^\dagger_{\dot\beta}
\>=\> \frac{1}{2}
\sigma^\mu_{\alpha\dot\beta}
(\thetasigmamuloweredthetadagger)
,
\eeq
derived from eqs.~(\ref{eq:defepstwo}) and (\ref{eq:feif}).
These can be used to rewrite any term into the forms given
in eq.~(\ref{eq:gensuperfield}).
Some other identities involving the anticommuting coordinates that are
useful in checking results below are:
\beq
(\theta \xi)( \theta\chi)
&=& \frac{1}{2}(\theta\theta)( \xi\chi)
,
\qquad\qquad
(\theta^\dagger
\hspace{1pt}
\xi^\dagger)
\,
(\theta^\dagger
\hspace{1pt}
\chi^\dagger)
\>=\>
\frac{1}{2}(\thdthd ) (\xi^\dagger\hspace{1pt}\chi^\dagger)
,
\\
(\theta \xi )(\theta^\dagger
\hspace{1pt}
\chi^\dagger )
&=&
\frac{1}{2} (\thetasigmamuthetadagger )
(\xi \sigma_\mu \chi^\dagger ),
\\
\theta^\dagger \sigmabar^\mu \theta &=& \theta \sigma^\mu \theta^\dagger
\>=\>
(\theta^\dagger \sigmabar^\mu \theta)^*,
\\
\theta \sigma^\mu \sigmabar^\nu \theta &=&
\BDpos\eta^{\mu\nu} \theta\theta ,
\qquad\qquad\qquad
\theta^\dagger\hspace{1pt} \sigmabar^\mu \sigma^\nu \theta^\dagger \>=\>
\BDpos\eta^{\mu\nu} \thdthd.
\eeq
These follow from identities already given in section
\ref{sec:notations}.
The general superfield $S$ could be either commuting or anticommuting, and could carry
additional Lorentz vector or spinor indices. For simplicity, let us
assume for the rest of this subsection that it is Grassmanneven and
carries no other indices. Then, without further restrictions, the
components of the general superfield $S$ are 8 bosonic fields $a,b,c,d$
and $v_\mu$, and 4 twocomponent fermionic fields
$\xi,\chi^\dagger,\eta,\zeta^\dagger$. All of these are complex functions
of $x^\mu$. The numbers of bosons and fermions do agree (8 complex, or 16
real, degrees of freedom for each), but there are too many of them to
match either the chiral or vector supermultiplets encountered in the
previous section. This means that the general superfield is a reducible
representation of supersymmetry.
In sections \ref{chiralsuperfields} and \ref{subsec:vectorsuperfields}
below, we will see how chiral and vector superfields are
obtained by imposing constraints on the general case
eq.~(\ref{eq:gensuperfield}).
Derivatives with respect to the anticommuting coordinates are defined by
\beq
\frac{\partial\phantom{x}}{\partial\theta^\alpha} (\theta^\beta)
\>=\> \delta^\beta_\alpha ,
\qquad
\frac{\partial\phantom{x}}{\partial\theta^\alpha}
(\theta^\dagger_{\dot\beta}) \>=\> 0 ,
\qquad
\frac{\partial\phantom{x}}{\partial\theta^\dagger_{\dot\alpha}}
(\theta^\dagger_{\dot\beta}) \>=\>
\delta^{\dot\alpha}_{\dot\beta} ,
\qquad
\frac{\partial\phantom{x}}{\partial\theta^\dagger_{\dot\alpha}}
(\theta^{\beta}) \>=\> 0 .
\eeq
Thus, for example,
$\frac{\partial\phantom{x}}{\partial\theta^\alpha}(\psi\theta) = \psi_\alpha$
and
$\frac{\partial\phantom{x}}{\partial\theta_\alpha}(\psi\theta) = \psi^\alpha$
for an anticommuting spinor $\psi_\alpha$, and
$\frac{\partial\phantom{x}}{\partial\theta^\alpha}(\theta\theta) = 2
\theta_\alpha$ and
$\frac{\partial\phantom{x}}{\partial\theta_\alpha}(\theta\theta) = 2
\theta^\alpha$.
To integrate over superspace, define
\beq
d^2\theta
\>=\> \frac{1}{4} d\theta^\alpha d \theta^\beta \epsilon_{\alpha\beta}
,
\qquad\quad
d^2\theta^\dagger \>=\> \frac{1}{4} d\theta^\dagger_{\dot\alpha}
d \theta^\dagger_{\dot\beta}
\epsilon^{\dot\alpha\dot\beta}
,
\eeq
so that, using eq.~(\ref{eq:defderetabasis}),
\beq
\int d^2\theta \,\theta\theta = 1,\qquad\qquad
\int d^2\theta^\dagger \,\thdthd = 1.
\eeq
Integration of a general superfield therefore just picks out the
relevant coefficients of $\theta\theta$ and/or
$\thdthd$
in eq.~(\ref{eq:gensuperfield}):
\beq
\int d^2\theta\> S(x, \theta, \theta^\dagger) &=&
b(x)
+ \theta^\dagger\hspace{1pt} \zeta^\dagger (x)
+ \thdthd d(x) ,
\\
\int d^2\theta^\dagger \> S(x, \theta, \theta^\dagger) &=&
c(x)
+ \theta \eta(x)
+ \theta \theta d(x) ,
\\
\int d^2\theta d^2 \theta^\dagger \> S(x, \theta, \theta^\dagger) &=& d(x) .
\label{eq:heckuvajobtimmy}
\eeq
The Dirac delta functions with respect to integrations
$d^2\theta$ and $d^2\theta^\dagger$ are:
\beq
\delta^{(2)} (\theta  \theta') = (\theta  \theta')(\theta  \theta'),
\qquad\quad
\delta^{(2)} (\theta^\dagger  \theta^{\prime\dagger}) =
(\theta^\dagger  \theta^{\prime\dagger})
(\theta^\dagger  \theta^{\prime\dagger}) ,
\label{eq:deltathetatheta}
\eeq
so that
\beq
\int d^2\theta \> \delta^{(2)}(\theta) \, S(x,\theta,\theta^\dagger)
&=& S(x,0,\theta^\dagger)
\>=\>
a(x) + \theta^\dagger\hspace{1pt} \chi^\dagger (x) +
\thdthd c(x) ,
\phantom{xxxxx}
\\
\int d^2\theta^\dagger \> \delta^{(2)}(\theta^\dagger) \,
S(x,\theta,\theta^\dagger)
&=&
S(x,\theta,0)
\> = \>
a(x) + \theta \xi(x) + \theta\theta b(x) ,
\\
\int d^2\theta d^2\theta^\dagger \>
\delta^{(2)}(\theta)\delta^{(2)}(\theta^\dagger) \,
S(x,\theta,\theta^\dagger)
&=&
S(x,0,0)
\>=\>
a(x).
\eeq
The integrals of total derivatives with respect to the fermionic
coordinates vanish:
\beq
\int d^2\theta
\frac{\partial\phantom{x}}{\partial\theta^{\alpha}} \mbox{(anything)} = 0
,
\qquad\quad
\int d^2\theta^\dagger
\frac{\partial\phantom{x}}{\partial\theta^\dagger_{\dot\alpha}}
\mbox{(anything)} = 0,
\label{eq:totalthetaderivsvanish}
\eeq
just as in eq.~(\ref{eq:funtheoeta}). This allows for integration by parts.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Supersymmetry transformations the superspace
way\label{subsec:superspacetransformations}}
\setcounter{footnote}{2}
\setcounter{equation}{0}
To formulate supersymmetry transformations in terms of superspace,
define the following differential operators that act on superfields:
\beq
\hat Q_\alpha &=& i \frac{\partial\phantom{x}}{\partial\theta^\alpha}
 (\sigma^\mu \theta^\dagger)_\alpha \partial_\mu
,
\qquad\qquad
\hat Q^\alpha \>=\> i\frac{\partial\phantom{x}}{\partial\theta_\alpha}
+ (\theta^\dagger \sigmabar^\mu)^\alpha \partial_\mu
,
\label{eq:defQhat}
\\
\hat Q^{\dagger\dot\alpha} &=&
i\frac{\partial\phantom{x}}{\partial\theta^\dagger_{\dot\alpha}}
 (\sigmabar^\mu \theta)^{\dot\alpha} \partial_\mu
,
\qquad\qquad\>
\hat Q^{\dagger}_{\dot\alpha} \>=\>
i\frac{\partial\phantom{x}}{\partial\theta^{\dagger\dot\alpha}}
+(\theta \sigma^\mu)_{\dot\alpha} \partial_\mu
.
\label{eq:defQdaggerhat}
\eeq
These obey the usual
product rules for derivatives, but with a minus sign for anticommuting through
a Grassmannodd object. For example:
\beq
\hat Q_\alpha (S T) &=& (\hat Q_\alpha S) T + (1)^S S (\hat Q_\alpha T)
\label{eq:grassmanproductrule}
\eeq
where $S$ and $T$ are any superfields, and $(1)^S$ is equal to
$1$ if $S$ is Grassmannodd, and $+1$ if $S$ is Grassmanneven.
Then the supersymmetry transformation parameterized by infinitesimal
$\epsilon$, $\epsilon^\dagger$ for any superfield $S$ is given
by\footnote{The factor of $\sqrt{2}$ is a convention, not universally
chosen in the literature, but adopted here in order to avoid $\sqrt{2}$
factors in the supersymmetry transformations of section
\ref{subsec:susylagr.freeWZ} while maintaining consistency.}
\beq
\sqrt{2}\, \delta_{\epsilon} S &=&
i
(\epsilon \hat Q + \epsilon^\dagger \hat Q^\dagger) S
\>\,=\>\,
\Bigl (\epsilon^\alpha \frac{\partial\phantom{x}}{\partial\theta^\alpha}
+ \epsilon^\dagger_{\dot\alpha}
\frac{\partial\phantom{x}}{\partial\theta^\dagger_{\dot\alpha}}
+ i \bigl [\epsilon \sigma^\mu \theta^\dagger
+ \epsilon^\dagger \sigmabar^\mu \theta \bigr ] \partial_\mu \Bigr ) S
\label{eq:superspacedefsupertrans}
\\
&=&
S (x^\mu + i \epsilon \sigma^\mu \theta^\dagger
+ i \epsilon^\dagger \sigmabar^\mu \theta
, \> \theta\! +\! \epsilon
,\> \theta^\dagger\! +\! \epsilon^\dagger)

S (x^\mu, \, \theta,\, \theta^\dagger) ,
\label{eq:supertranslation}
\eeq
The last equality follows
by a Taylor expansion to first order in $\epsilon$ and $\epsilon^\dagger$.
Equation (\ref{eq:supertranslation}) shows that a supersymmetry
transformation can be viewed as a translation in superspace, with:
\beq
\theta^\alpha &\rightarrow& \theta^\alpha + \epsilon^\alpha,
\\
\theta^\dagger_{\dot\alpha} &\rightarrow& \theta^\dagger_{\dot\alpha}
+ \epsilon^\dagger_{\dot\alpha},
\\
x^\mu &\rightarrow& x^\mu + i \epsilon \sigma^\mu \theta^\dagger
+ i \epsilon^\dagger \sigmabar^\mu \theta .
\eeq
Since $\hat Q$, $\hat Q^\dagger$ are linear differential operators, the product or
linear combination of any superfields satisfying
eq.~(\ref{eq:superspacedefsupertrans}) is again a superfield with the
same transformation law.
It is instructive and useful to work out the supersymmetry
transformations of all of the component fields of the general superfield
eq.~(\ref{eq:gensuperfield}). They are:
\beq
\sqrt{2}\,
\delta_\epsilon a &=& \epsilon \xi
+ \epsilon^\dagger\hspace{1pt}\chi^\dagger ,
\label{eq:gensuperfieldtransa}
\\
\sqrt{2}\,
\delta_\epsilon \xi_{\alpha} &=& 2 \epsilon_\alpha b
\BDplus (\sigma^\mu \epsilon^\dagger)_\alpha (v_\mu \BDminus i \partial_\mu a)
,
\\
\sqrt{2}\,
\delta_\epsilon \chi^{\dagger{\dot\alpha}} &=&
2 \epsilon^{\dagger{\dot\alpha}} c
\BDminus (\sigmabar^\mu \epsilon)^{\dot \alpha} (v_\mu \BDplus i \partial_\mu a)
,
\\
\sqrt{2}\,
\delta_\epsilon b &=& \epsilon^\dagger\hspace{1pt} \zeta^\dagger
 \frac{i}{2} \epsilon^\dagger
\sigmabar^\mu \partial_\mu \xi
,
\\
\sqrt{2}\,
\delta_\epsilon c &=& \epsilon \eta  \frac{i}{2} \epsilon
\sigma^\mu \partial_\mu \chi^\dagger
,
\\
\sqrt{2}\,
\delta_\epsilon v^\mu &=&
\epsilon \sigma^\mu \zeta^\dagger
\epsilon^\dagger \sigmabar^\mu \eta
\frac{i}{2} \epsilon \sigma^\nu \sigmabar^\mu \partial_\nu \xi
+\frac{i}{2} \epsilon^\dagger \sigmabar^\nu \sigma^\mu \partial_\nu \chi^\dagger
,
\\
\sqrt{2}\,
\delta_\epsilon \eta_{\alpha} &=& 2 \epsilon_\alpha d
 i (\sigma^\mu \epsilon^\dagger)_\alpha \partial_\mu c
\BDplus \frac{i}{2} (\sigma^\nu \sigmabar^\mu \epsilon)_\alpha \partial_\mu v_\nu
,
\\
\sqrt{2}\,
\delta_\epsilon \zeta^{\dagger{\dot\alpha}} &=&
2 \epsilon^{\dagger{\dot \alpha}} d
 i (\sigmabar^\mu \epsilon)^{\dot\alpha} \partial_\mu b
\BDminus \frac{i}{2} (\sigmabar^\nu \sigma^\mu \epsilon^\dagger )^{\dot \alpha}
\partial_\mu v_\nu
,
\\
\sqrt{2}\,
\delta_\epsilon d &=&
\frac{i}{2}
\epsilon^\dagger \sigmabar^\mu \partial_\mu \eta
\frac{i}{2}
\epsilon \sigma^\mu \partial_\mu \zeta^\dagger .
\label{eq:gensuperfieldtransd}
\eeq
Note that since the terms on the righthand sides all have exactly one $\epsilon$
or one $\epsilon^\dagger$,
boson fields are always transformed into fermions and vice versa.
It is probably not obvious yet that the supersymmetry transformations as
just defined coincide with those found in section \ref{sec:susylagr}.
This will become clear below when we discuss the specific form of chiral
and vector superfields and the Lagrangians that govern their dynamics.
Meanwhile, however, we can compute the anticommutators of $\hat Q$, $\hat
Q^\dagger$ from eqs.~(\ref{eq:defQhat}), (\ref{eq:defQdaggerhat}), with
the results:
\beq
\Bigl \lbrace \hat Q_\alpha,\, \hat Q^\dagger_{\dot\beta} \Bigr \rbrace
&=&
2 i \sigma^\mu_{\alpha\dot\beta}\partial_\mu
\>\,=\,\> \BDpos 2 \sigma^\mu_{\alpha\dot\beta} \hat P_\mu,
\label{eq:diffopSUSYalgebra1}
\\
\Bigl \lbrace \hat Q_\alpha,\, \hat Q_{\beta} \Bigr \rbrace &=& 0,\qquad\quad
\Bigl \lbrace \hat Q^\dagger_{\dot \alpha},\, \hat Q^\dagger_{\dot\beta}
\Bigr \rbrace \>=\> 0.
\label{eq:diffopSUSYalgebra2}
\eeq
Here, the differential operator generating spacetime translations is
\beq
\hat P_\mu = \BDpos i \partial_\mu.
\eeq
Eqs.~(\ref{eq:diffopSUSYalgebra1})(\ref{eq:diffopSUSYalgebra2}) have the same
form as the supersymmetry algebra given in
eqs.~(\ref{nonschsusyalg1}), (\ref{nonschsusyalg2}).
It is important to keep in
mind the conceptual distinction between the unhatted objects
$Q_\alpha, Q^\dagger_{\dot\alpha},
P^\mu$ appearing in section \ref{subsec:susylagr.freeWZ},
which are operators acting on the Hilbert space of quantum states,
and the corresponding hatted objects
$\hat Q_\alpha, \hat Q^\dagger_{\dot\alpha},
\hat P^\mu$, which are differential operators acting on functions in
superspace. For any superfield
quantum mechanical operator $X$ in the Heisenberg picture,
the two kinds of operations are related by
\beq
\bigl [X,\, \epsilon Q + \epsilon^\dagger\hspace{1pt} Q^\dagger \bigr ]
&=&
(\epsilon \hat Q + \epsilon^\dagger\hspace{1pt} \hat Q^\dagger) X ,
\\
\bigl [X ,\, P_\mu \bigr ] &=& \hat P_\mu X.
\eeq
\subsection{Chiral covariant derivatives\label{subsec:supercovariantderivatives}}
\setcounter{equation}{0}
\setcounter{footnote}{2}
To construct Lagrangians in superspace, we will later want to use
derivatives with respect to the anticommuting coordinates, just as
ordinary Lagrangians are built using spacetime derivatives
$\partial_\mu$. We will also use such derivatives to impose constraints
on the general superfield in a way consistent with the supersymmetry
transformations. However, $
%\displaystyle\frac{
\partial
%\phantom{x}}{
/
\partial\theta^\alpha
%}
$ is not appropriate for
this purpose, because it is not supersymmetric covariant:
\beq
\delta_\epsilon \left (\frac{\partial S}{\partial \theta^\alpha} \right )
\>\not=\>
\frac{\partial\phantom{x}}{\partial\theta^\alpha} (\delta_\epsilon S),
\eeq
and similarly for
$
%\displaystyle \frac{
\partial
%\phantom{x}}{
/
\partial\theta^\dagger_{\dot\alpha}
%}
$. This means
that derivatives of a superfield with respect to
$\theta_\alpha$ or $\theta^\dagger_{\dot\alpha}$ are not superfields; they do not transform the right way.
To fix this, it is useful to define the chiral covariant derivatives:
\beq
D_\alpha &=& \frac{\partial\phantom{x}}{\partial\theta^\alpha}
i (\sigma^\mu \theta^\dagger)_\alpha \partial_\mu
,
\qquad\qquad
D^\alpha \>=\> \frac{\partial\phantom{x}}{\partial\theta_\alpha}
+i (\theta^\dagger \sigmabar^\mu)^\alpha \partial_\mu
.
\eeq
For a Grassmanneven superfield $S$, one can then define the antichiral covariant derivative to obey:
\beq
\Dcon_{\dot\alpha} S^* \equiv (D_\alpha S)^*,
\label{eq:defDcon}
\eeq
which implies
\beq
\Dcon^{\dot\alpha} &=&
\frac{\partial\phantom{x}}{\partial\theta^\dagger_{\dot\alpha}}
i (\sigmabar^\mu \theta)^{\dot\alpha} \partial_\mu
,
\qquad\qquad\>
\Dcon_{\dot\alpha} \>=\>
\frac{\partial\phantom{x}}{\partial\theta^{\dagger\dot\alpha}}
+i (\theta \sigma^\mu)_{\dot\alpha} \partial_\mu
.
\eeq
One may now check that
\beq
\Bigl \lbrace \hat Q_\alpha ,\, D_\beta \Bigr \rbrace \>=\>
\Bigl \lbrace \hat Q^\dagger_{\dot\alpha} ,\, D_\beta \Bigr \rbrace \>=\>
\Bigl \lbrace \hat Q_\alpha ,\, \Dcon_{\dot\beta} \Bigr \rbrace \>=\>
\Bigl \lbrace \hat Q^\dagger_{\dot\alpha} ,\, \Dcon_{\dot\beta}
\Bigr \rbrace \>=\> 0 .
\label{eq:QDanticommute}
\eeq
Using the supersymmetry transformation definition of eq.~(\ref{eq:superspacedefsupertrans}), it follows that
\beq
\delta_\epsilon \left ( D_\alpha S \right )
= D_\alpha\left ( \delta_\epsilon S \right ),
\qquad
\qquad
\delta_\epsilon \left ( \Dcon_{\dot\alpha} S \right )
= \Dcon_{\dot\alpha} \left ( \delta_\epsilon S \right )
.
\label{eq:Dsupercon}
\eeq
Thus the derivatives $D_\alpha$ and $\Dcon_{\dot\alpha}$ are indeed
supersymmetric covariant; acting on superfields, they return superfields.
This crucial property makes them useful both for defining constraints on superfields in a
covariant way, and for defining superspace Lagrangians involving
anticommuting spinor coordinate derivatives. These derivatives are linear differential operators, obeying product rules
exactly analogous to eq.~(\ref{eq:grassmanproductrule}).
The chiral and antichiral covariant derivatives also can be checked to
satisfy the useful
anticommutation identities:
\beq
\Bigl \lbrace D_\alpha,\, \Dcon_{\dot\beta} \Bigr \rbrace &=&
2 i \sigma^\mu_{\alpha\dot\beta}\partial_\mu,
\label{eq:diffopderivs}
\\
\Bigl \lbrace D_\alpha,\, D_{\beta} \Bigr \rbrace &=& 0,\qquad\quad
\Bigl \lbrace \Dcon_{\dot \alpha},\,
\Dcon_{\dot\beta} \Bigr \rbrace \>=\> 0.
\label{eq:diffopderivs2}
\eeq
This has exactly the same form as the supersymmetry algebra in
eqs.~(\ref{eq:diffopSUSYalgebra1}) and (\ref{eq:diffopSUSYalgebra2}),
but $D, \Dcon$ should not be
confused with
the differential operators for supersymmetry transformations,
$\hat Q, \hat Q^\dagger$. The operators $D, \Dcon$ do not represent a second supersymmetry.
The reader might be wondering why we use an overline notation for $\Dcon$, but a dagger
for $\hat Q^\dagger$. The reason is that the dagger and the overline
denote different kinds of conjugation. The dagger
on $\hat Q$ represents Hermitian conjugation in the same sense that
$\hat P = i \partial_\mu$ is an Hermitian differential operator on an
inner product space, but the overline on $\Dcon$ represents
complex conjugation in the same sense that
$\partial_\mu$ is a real differential operator, with
$(\partial_\mu \phi)^* = \partial_\mu \phi^*$. Recall that if we
define the inner product on the space of functions of $x^\mu$ by:
\beq
\langle \psi  \phi \rangle = \int \!d^4x\> \psi^*(x) \phi(x),
\eeq
then, using integration by parts,
\beq
\langle \psi \hat P \phi \rangle = \left (\langle \phi \hat P \psi \rangle\right )^*
\eeq
Similarly, the dagger on the differential operator $\hat Q^\dagger$ denotes Hermitian
conjugation with respect to the inner product defined by integration of
complex superfields over superspace. To see this, define, for any
two classical superfields
$S(x,\theta,\theta^\dagger)$ and $T(x,\theta,\theta^\dagger)$, the inner product:
\beq
\langle T  S \rangle = \int d^4x\int\! d^2\theta\! \int\! d^2\theta^\dagger\> T^* S.
\eeq
Now one finds, by integration by parts over superspace, that with the definitions
in eqs.~(\ref{eq:defQhat}) and (\ref{eq:defQdaggerhat}),
\beq
\langle T  \hat Q^\dagger_{\dot\alpha} S \rangle =
\left ( \langle S \hat Q_\alpha T \rangle \right )^* .
\label{eq:TSQcon}
\eeq
In contrast, the definition of $\Dcon$ in eq.~(\ref{eq:defDcon}) is analogous
to the equation $(\partial_\mu \phi)^* = \partial_\mu \phi^*$ for functions
on ordinary spacetime; in that sense, $\partial_\mu$ is a real differential operator,
and similarly $\Dcon_{\dot\alpha}$ is the conjugate of $D_\alpha$.
This is more than just notation; if we
defined $D^\dagger_{\dot\alpha}$ from $D_\alpha$ in a way analogous to
eq.~(\ref{eq:TSQcon}), then one can check that
it would not be equal to $\Dcon_{\dot\alpha}$ as defined above.
Note that the dagger on the quantum field theory operator $Q^\dagger_{\alpha}$
(without the hat) represents yet another sort of Hermitian conjugation,
in the quantum mechanics Hilbert space sense.
It is useful to note that, using eq.~(\ref{eq:totalthetaderivsvanish}),
\beq
\int d^2\theta \,D_\alpha \mbox{(anything)} \qquad\mbox{and}\qquad
\int d^2\theta^\dagger \,\Dcon_{\dot\alpha} \mbox{(anything)}
\eeq
are each total derivatives with respect to $x^\mu$. This enables integration by parts
in superspace
of Lagrangian terms with respect to either $D_\alpha$ or $\Dcon_{\dot\alpha}$. Another useful fact is that
acting three consecutive times with either of $D_\alpha$ or $\Dcon_{\dot\alpha}$ always produces a vanishing result:
\beq
D_\alpha D_\beta D_\gamma \mbox{(anything)} = 0 \qquad\mbox{and}\qquad
\Dcon_{\dot\alpha} \Dcon_{\dot\beta} \Dcon_{\dot\gamma} \mbox{(anything)} = 0.
\label{eq:DDDeq0}
\eeq
This follows from eq.~(\ref{eq:diffopderivs2}), and is true essentially
because the spinor indices on the anticommuting derivatives can only have two values.
\subsection{Chiral superfields\label{chiralsuperfields}}
\setcounter{equation}{0}
\setcounter{footnote}{2}
To describe a chiral supermultiplet,
consider the superfield $\Phi(x,\theta,\theta^\dagger)$
obtained by imposing the constraint
\beq
\Dcon_{\dot \alpha} \Phi &=& 0.
\label{eq:leftchiralsuperfieldconstraint}
\eeq
A field satisfying this constraint is said to be a chiral (or
leftchiral) superfield, and its complex conjugate $\Phi^*$ is called
antichiral (or rightchiral) and satisfies
\beq
D_{\alpha} \Phi^* &=& 0.
\label{eq:antichiralsuperfieldconstraint}
\eeq
These constraints are
consistent with the transformation rule for general superfields
because of eq.~(\ref{eq:Dsupercon}).
To solve the constraint eq.~(\ref{eq:leftchiralsuperfieldconstraint}) in
general, it is convenient to define
\beq
y^\mu \equiv x^\mu \BDminus i \thetasigmamuthetadagger ,\>
\label{eq:defineycoord}
\eeq
and change coordinates on superspace to the set:
\beq
y^\mu,\>\theta^\alpha,\>\theta^\dagger_{\dot\alpha}.
\eeq
In terms of these variables, the chiral covariant derivatives have the
representation:
\beq
D_\alpha &=& \frac{\partial\phantom{x}}{\partial\theta^\alpha}
2i (\sigma^\mu \theta^\dagger)_\alpha
\frac{\partial\phantom{x}}{\partial y^\mu} ,
\qquad\qquad
D^\alpha \>=\> \frac{\partial\phantom{x}}{\partial\theta_\alpha}
+2i (\theta^\dagger \sigmabar^\mu)^\alpha
\frac{\partial\phantom{x}}{\partial y^\mu} ,
\phantom{xxxx}
\label{eq:Dinyrep}
\\
\Dcon^{\dot\alpha} &=&
\frac{\partial\phantom{x}}{\partial\theta^\dagger_{\dot\alpha}} ,
\qquad\qquad\qquad\qquad\qquad\>\>\>\>\>
\Dcon_{\dot\alpha} \>=\>
\frac{\partial\phantom{x}}{\partial\theta^{\dagger\dot\alpha}} .
\label{eq:Ddaggerinyrep}
\eeq
Equation (\ref{eq:Ddaggerinyrep}) makes it clear that the chiral
superfield
constraint eq.~(\ref{eq:leftchiralsuperfieldconstraint}) is solved by any
function of $y^\mu$ and $\theta$, as long as it is not a function of $\theta^\dagger$.
Therefore, one can expand:
\beq
\Phi \,=\,
\phi(y) + \sqrt{2}\theta \psi(y) + \theta\theta F(y) ,
\label{eq:Phiinyrep}
\eeq
and similarly
\beq
\Phi^* \,=\,
\phi^*(y^*) + \sqrt{2}\theta^\dagger\hspace{1pt} \psi^\dagger(y^*)
+ \thdthd F^*(y^*) .
\label{eq:Phistarinyrep}
\eeq
The factors of $\sqrt{2}$ are conventional, and
$y^{\mu *} = x^\mu \BDplus i \thetasigmamuthetadagger $.
The chiral covariant derivatives in terms of the coordinates
$(y^*,\theta,\theta^\dagger)$ are also sometimes useful:
\beq
D_\alpha &=& \frac{\partial\phantom{x}}{\partial\theta^\alpha}
,
\qquad\qquad
\qquad\qquad\>
\qquad\quad\>
D^\alpha \>=\> \frac{\partial\phantom{x}}{\partial\theta_\alpha}
,
\\
\Dcon^{\dot\alpha} &=&
\frac{\partial\phantom{x}}{\partial\theta^\dagger_{\dot\alpha}}
2i (\sigmabar^\mu \theta)^{\dot\alpha}
\frac{\partial}{\partial y^{\mu *}}
,
\qquad\qquad\>
\Dcon_{\dot\alpha} \>=\>
\frac{\partial\phantom{x}}{\partial\theta^{\dagger\dot\alpha}}
+2i (\theta \sigma^\mu)_{\dot\alpha}
\frac{\partial}{\partial y^{\mu *}}
.
\eeq
According to eq.~(\ref{eq:Phiinyrep}),
the chiral superfield independent degrees of freedom are a complex scalar $\phi$,
a twocomponent fermion $\psi$,
and an auxiliary field $F$, just as found in
subsection \ref{subsec:susylagr.freeWZ}.
If $\Phi$ is a free fundamental chiral superfield,
then assigning it dimension [mass]$^1$
gives the canonical mass dimensions to the component fields, because
$\theta$ and $\theta^\dagger$ have dimension [mass]$^{1/2}$.
Rewriting the chiral superfields in terms of the
original coordinates $x, \theta, \theta^\dagger$, by expanding in a power series in the
anticommuting coordinates, gives
\beq
\Phi \!\!&=&\!\!
\phi(x)
\BDminus i \thetasigmamuthetadagger \partial_\mu \phi(x)
\BDminus \frac{1}{4} \theta\theta\thdthd \partial_\mu \partial^\mu
\phi(x)
+ \sqrt{2}\theta \psi(x)
\nonumber \\ &&
 \frac{i}{\sqrt{2}} \theta\theta
\theta^\dagger \sigmabar^\mu \partial_\mu \psi(x)
+ \theta\theta F(x),
\label{eq:Phiinxrep}
\\
\Phi^*
\!\!&=&\!\!
\phi^*(x)
\BDplus i \thetasigmamuthetadagger \partial_\mu \phi^*(x)
\BDminus \frac{1}{4} \theta\theta\thdthd \partial_\mu \partial^\mu
\phi^*(x)
+ \sqrt{2}\theta^\dagger\hspace{1pt} \psi^\dagger(x)
\nonumber \\ &&
 \frac{i}{\sqrt{2}} \thdthd \theta \sigma^\mu \partial_\mu
\psi^\dagger(x)
+ \thdthd F^*(x).
\label{eq:Phistarinxrep}
\eeq
Depending on the situation, eqs.~(\ref{eq:Phiinyrep})(\ref{eq:Phistarinyrep}) are sometimes
a more convenient representation than
eqs.~(\ref{eq:Phiinxrep})(\ref{eq:Phistarinxrep}).
By comparing the general superfield case
eq.~(\ref{eq:gensuperfield}) to eq.~(\ref{eq:Phiinxrep}), we see that
the latter can be
obtained from the former by identifying component fields:
\beq
&& a = \phi,
\qquad\quad
\xi_\alpha = \sqrt{2} \psi_\alpha,
\qquad\qquad
b = F,
\label{eq:specSPhione}
\\
&&
\chi^{\dagger\dot\alpha} =0,
\qquad\qquad
c = 0,
\qquad\qquad
v_\mu = \BDneg i \partial_\mu \phi,
\qquad\qquad
\eta_\alpha = 0,
\phantom{xxx}
\\
&&
\zeta^{\dagger\dot\alpha} =
\frac{i}{\sqrt{2}} (\sigmabar^\mu \partial_\mu \psi)^{\dot\alpha} ,
\qquad\qquad
d = \BDneg \frac{1}{4} \partial_\mu \partial^\mu \phi.
\label{eq:specSPhitwo}
\eeq
It is now straightforward to obtain the supersymmetry transformation
laws for the component fields of
$\Phi$, either by using
$\sqrt{2} \delta_\epsilon \Phi = i
(\epsilon \hat Q + \epsilon^\dagger \hat
Q^\dagger) \Phi$, or by plugging
eqs.~(\ref{eq:specSPhione})(\ref{eq:specSPhitwo}) into the
results for a general superfield,
eqs.~(\ref{eq:gensuperfieldtransa})(\ref{eq:gensuperfieldtransd}).
The results are
\beq
\delta_\epsilon \phi &=& \epsilon \psi,\\
\delta_\epsilon \psi_\alpha &=&
i (\sigma^\mu \epsilon^\dagger)_\alpha \partial_\mu \phi +
\epsilon_\alpha F
,
\\
\delta_\epsilon F &=& i \epsilon^\dagger \sigmabar^\mu \partial_\mu \psi,
\label{eq:runningupthathill}
\eeq
in agreement with eqs.~(\ref{phitrans}), (\ref{Ftrans}), (\ref{fermiontrans}).
One way to construct a chiral superfield (or an antichiral superfield) is
\beq
\Phi \,=\,
\Dcon\Dcon S
\,\equiv\,
\Dcon_{\dot\alpha} \Dcon^{\dot \alpha} S,
\qquad\qquad
\Phi^* \,=\, DDS^* \,\equiv\, D^\alpha D_\alpha S^*,
\label{eq:chiralfromgeneral}
\eeq
where $S$ is any
general superfield. The fact that these are chiral and antichiral, respectively,
follows immediately from eq.~(\ref{eq:DDDeq0}).
The converse is also true;
for every chiral superfield $\Phi$, one can find a superfield $S$ such that
eq.~(\ref{eq:chiralfromgeneral}) is true.
Another way to build a chiral superfield is as
a function $W(\Phi_i)$ of other chiral superfields $\Phi_i$
but not antichiral superfields; in other words, $W$ is holomorphic in
chiral superfields treated as complex variables.
This fact follows immediately from
the linearity and product rule properties of the differential operator
$\Dcon_{\dot \alpha}$ appearing in the constraint
eq.~(\ref{eq:leftchiralsuperfieldconstraint}). It will be useful below
for constructing superspace Lagrangians.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Vector superfields\label{subsec:vectorsuperfields}}
\setcounter{equation}{0}
\setcounter{footnote}{2}
A vector (or real) superfield $V$ is obtained by
imposing the constraint $V = V^*$.
This is equivalent to imposing the following constraints
on the components of the
general superfield eq.~(\ref{eq:gensuperfield}):
\beq
a \> = \> a^*,
\qquad
\chi^\dagger \>=\> \xi^\dagger,
\qquad
c \> = \> b^*,
\qquad
v_\mu \>=\> v_\mu^*,
\qquad
\zeta^\dagger \>=\> \eta^\dagger,
\qquad
d = d^*.
\label{eq:constrainvectorfromgeneral}
\eeq
It is also convenient and traditional to define:
\beq
\eta_\alpha \>=\> \lambda_\alpha  \frac{i}{2} (\sigma^\mu \partial_\mu \xi^\dagger)_\alpha ,
\qquad
v_\mu = A_\mu,
\qquad
d = \frac{1}{2} D \BDminus \frac{1}{4} \partial_\mu \partial^\mu a.
\label{eq:redefinevectorfromgeneral}
\eeq
The component expansion of the vector superfield is then
\beq
V(x,\theta,\theta^\dagger) &=&
a
+ \theta \xi
+ \theta^\dagger\hspace{1pt} \xi^\dagger
+ \theta\theta b
+ \thdthd b^*
+ \thetasigmamuthetadagger A_\mu
+ \thdthd \theta
(\lambda  \frac{i}{2} \sigma^\mu \partial_\mu \xi^\dagger)
\nonumber
\\
&&
+ \theta \theta \theta^\dagger\hspace{1pt}
(\lambda^\dagger  \frac{i}{2} \sigmabar^\mu \partial_\mu \xi)
+ \theta\theta\thdthd
( \frac{1}{2} D \BDminus \frac{1}{4} \partial_\mu \partial^\mu a).
\label{eq:vectorsuperfieldexpansion}
\eeq
The supersymmetry transformations of these components can be obtained either from
$\sqrt{2} \delta_\epsilon V = i(\epsilon \hat Q + \epsilon^\dagger \hat Q^\dagger) V$,
or by plugging
eqs.~(\ref{eq:constrainvectorfromgeneral})(\ref{eq:redefinevectorfromgeneral})
into the results for
a general superfield,
eqs.~(\ref{eq:gensuperfieldtransa})(\ref{eq:gensuperfieldtransd}).
The results are:
\beq
\sqrt{2}\,
\delta_\epsilon a &=& \epsilon \xi + \epsilon^\dagger\hspace{1pt}\xi^\dagger
\\
\sqrt{2}\,
\delta_\epsilon \xi_{\alpha} &=& 2 \epsilon_\alpha b
\BDplus (\sigma^\mu \epsilon^\dagger)_\alpha (A_\mu \BDminus i \partial_\mu a)
,
\\
\sqrt{2}\,
\delta_\epsilon b &=& \epsilon^\dagger\hspace{1pt} \lambda^\dagger
 i \epsilon^\dagger \sigmabar^\mu \partial_\mu \xi
,
\\
\sqrt{2}\,
\delta_\epsilon A^\mu &=&
\BDneg i \epsilon \partial^\mu \xi
\BDplus i \epsilon^\dagger \partial^\mu \xi^\dagger
+ \epsilon \sigma^\mu \lambda^\dagger
 \epsilon^\dagger \sigmabar^\mu \lambda ,
\\
\sqrt{2}\,
\delta_\epsilon \lambda_\alpha &=&
\epsilon_\alpha D \BDminus \frac{i}{2} (\sigma^\mu \sigmabar^\nu \epsilon)_\alpha
(\partial_\mu A_\nu  \partial_\nu A_\mu)
,
\\
\sqrt{2}\,
\delta_\epsilon D &=&
 i \epsilon \sigma^\mu \partial_\mu \lambda^\dagger
 i \epsilon^\dagger \sigmabar^\mu \partial_\mu \lambda
\label{eq:sqrttwodeltaD}
\eeq
A superfield cannot be both chiral and real at the same time,
unless it is identically constant (i.e., independent of $x^\mu$, $\theta$,
and $\theta^\dagger$). This follows from
eqs.~(\ref{eq:specSPhione})(\ref{eq:specSPhitwo}), and
(\ref{eq:constrainvectorfromgeneral}).
However, if $\Phi$ is a chiral superfield, then $\Phi+\Phi^*$ and
$i (\Phi  \Phi^*)$ and $\Phi \Phi^*$ are all real (vector) superfields.
As the notation chosen in
eq.~(\ref{eq:vectorsuperfieldexpansion}) suggests,
a vector superfield that is used to represent a gauge supermultiplet contains
gauge boson, gaugino, and gauge auxiliary fields $A^\mu$, $\lambda$, $D$
as components. (Such a vector superfield $V$ must be dimensionless in order for the component fields to have the canonical mass dimensions.)
However, there are other component fields in $V$ that did
not appear in sections \ref{subsec:susylagr.gauge} and
\ref{subsec:susylagr.gaugeinter}.
They are: a real scalar $a$, a twocomponent fermion $\xi$,
and a complex scalar $b$, with mass dimensions
respectively 0, $1/2$, and 1. These are additional auxiliary fields,
which can be ``supergauged" away.
To see this, suppose $V$ is the vector superfield for a $U(1)$
gauge symmetry, and consider the ``supergauge transformation":
\beq
V &\rightarrow & V + i (\Omega^*  \Omega),
\label{eq:u1superfieldgt}
\eeq
where $\Omega$ is a chiral superfield gauge transformation parameter,
$\Omega = \phi + \sqrt{2}\theta \psi + \theta\theta F+\ldots$.
In components, this transformation is
\beq
a &\rightarrow& a + i (\phi^*  \phi),
\\
\xi_\alpha &\rightarrow& \xi_\alpha  i\sqrt{2} \psi_\alpha ,
\\
b &\rightarrow& b  iF ,
\\
A_\mu &\rightarrow& A_\mu \BDminus \partial_\mu (\phi + \phi^*) ,
\label{eq:AgaugetransfromLambda}
\\
\lambda_\alpha &\rightarrow& \lambda_\alpha ,
\\
D &\rightarrow& D.
\label{eq:DgaugetransfromLambda}
\eeq
Equation (\ref{eq:AgaugetransfromLambda})
shows that eq.~(\ref{eq:u1superfieldgt}) provides the vector
boson field with the usual gauge transformation, with parameter 2Re$(\phi)$.
By requiring the gauge transformation to take a supersymmetric
form, it follows that
appropriate independent choices of
Im$(\phi)$, $\psi_\alpha$, and $F$ can also change $a$, $\xi_\alpha$,
and $b$ arbitrarily.
Thus the supergauge transformation
eq.~(\ref{eq:u1superfieldgt}) has ordinary gauge transformations as a
special case.
In particular, supergauge transformations can eliminate
the auxiliary fields $a$, $\xi_\alpha$,
and $b$ completely.
A superspace Lagrangian for a vector superfield
must be invariant under the
supergauge transformation
eq.~(\ref{eq:u1superfieldgt}) in the Abelian case, or a
suitable generalization given below for the nonAbelian case. After making a
supergauge transformation to eliminate $a,\xi$, and $b$,
the vector superfield is said to be in
WessZumino gauge, and is simply given by
\beq
V_{{\rm WZ}\>{\rm gauge}} \>=\>
\thetasigmamuthetadagger A_\mu
+ \thdthd \theta \lambda
+ \theta \theta \theta^\dagger\hspace{1pt} \lambda^\dagger
+ \frac{1}{2} \theta\theta\thdthd D.
\label{eq:WZgauge}
\eeq
The restriction of the vector superfield to WessZumino gauge
is not consistent with the linear superspace version of supersymmetry
transformations. This is because
$\sqrt{2} \delta_{\epsilon} (V_{{\rm WZ}\>{\rm gauge}})$
contains
$\BDneg \theta^\dagger \sigmabar^\mu \epsilon A_\mu \BDplus \theta \sigma^\mu \epsilon^\dagger A_\mu
+ \theta\theta \epsilon^\dagger \lambda^\dagger +
\theta^\dagger\theta^\dagger \epsilon \lambda$,
and so the supersymmetry transformation of the WessZumino gauge
vector superfield is not in WessZumino gauge.
However, a supergauge transformation can always restore
$\delta_{\epsilon} (V_{{\rm WZ}\>{\rm gauge}})$ to WessZumino gauge.
Adopting WessZumino gauge is equivalent to partially fixing
the supergauge, while still maintaining the full freedom
to do ordinary gauge transformations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{How to make a Lagrangian in superspace\label{superspacelagr}}
\setcounter{equation}{0}
\setcounter{footnote}{2}
So far, we have been concerned with the structural features of fields in
superspace. We now turn to the dynamical issue of how to construct
manifestly supersymmetric actions. A key observation is that the integral
of any superfield over all of superspace is automatically
invariant:
\beq
\delta_\epsilon A = 0,\quad\mbox{for}\quad
A = \int d^4x \int d^2\theta d^2 \theta^\dagger \> S(x, \theta, \theta^\dagger).
\label{eq:defsuperspaceaction}
\eeq
This follows immediately from the fact that $\hat Q$ and $\hat Q^\dagger$
as defined in eqs.~(\ref{eq:defQhat}), (\ref{eq:defQdaggerhat}) are sums
of total derivatives with respect to the superspace coordinates
$x^\mu,\theta,\theta^\dagger$, so that $(\epsilon \hat Q +
\epsilon^\dagger \hat Q^\dagger)S$ vanishes upon integration. As a check,
eq.~(\ref{eq:gensuperfieldtransd}) shows that the
$\theta\theta\thdthd$ component of a
superfield transforms into a total spacetime derivative.
Therefore, the action governing the dynamics of a theory can have
contributions of the form of eq.~(\ref{eq:defsuperspaceaction}), with
reality of the action demanding that $S$ is some real (vector) superfield
$V$. From eq.~(\ref{eq:supertranslation}), we see that the principle of
global supersymmetric invariance is embodied in the requirement that the
action should be an integral over superspace which is unchanged under
rigid translations of the superspace coordinates. To obtain the
Lagrangian density ${\cal L}(x)$, one integrates over only the fermionic
coordinates. This is often written in the notation:
\beq
[V]_D \,\equiv\, \int d^2\theta d^2\theta^\dagger\> V(x,\theta,\theta^\dagger)
\,=\,
V(x,\theta,\theta^\dagger) \Bigl _{\theta\theta\thdthd}
\,=\,
\frac{1}{2} D \BDminus \frac{1}{4} \partial_\mu \partial^\mu a
\label{eq:DtermLag}
\eeq
using eq.~(\ref{eq:heckuvajobtimmy}) and the form of $V$
in eq.~(\ref{eq:vectorsuperfieldexpansion}) for the last equality. This is
referred to as a
$D$term contribution to the Lagrangian
(note that the $\partial_\mu \partial^\mu a$ part will
vanish upon integration $\int d^4 x$).
Another type of contribution to the action can be inferred from the fact
that the $F$term of a chiral superfield also transforms into a total
derivative under a supersymmetry transformation, see
eq.~(\ref{eq:runningupthathill}).
This implies that
one can have a
contribution to the Lagrangian density of the form
\beq
[\Phi]_F \,\equiv\, \Phi \Bigl _{\theta\theta} \,=\,
\int d^2\theta\, \Phi \Bigl _{\theta^\dagger =0} \,=\,
\int d^2\theta d^2\theta^\dagger\,
\delta^{(2)}(\theta^\dagger)
\,
\Phi \>=\>F,
\label{eq:PhiF}
\eeq
using the form of $\Phi$ in eq.~(\ref{eq:Phiinxrep}) for the last equality.
This satisfies $\delta_{\epsilon} (\int d^4 x [\Phi]_F) = 0$.
The $F$term of a chiral superfield
is complex in general, but
the action must be real, which can be ensured if
this type of contribution to the Lagrangian is accompanied
by its complex conjugate:
\beq
[\Phi]_F + {\rm c.c.} \>=\>
\int d^2\theta d^2\theta^\dagger\,
\left [
\delta^{(2)}(\theta^\dagger)
\,
\Phi
+
\delta^{(2)}(\theta)
\,
\Phi^*
\right ].
\label{eq:FtermLag}
\eeq
Note that the identification of the $F$term component of a chiral superfield is
the same in the $(x^\mu,\theta,\theta^\dagger)$ and
$(y^\mu,\theta,\theta^\dagger)$ coordinates, in the sense that in both cases, one
simply isolates the $\theta\theta$ component. This follows because the
difference between $x^\mu$ and $y^\mu$ is higher order in
$\theta^\dagger$. It is a useful trick, because many
calculations involving chiral superfields are
simpler to carry out in terms of $y^\mu$.
Another possible try would be to take the $D$term of a chiral superfield. However,
this is a waste of time, because
\beq
[\Phi]_D
\,=\, \int d^2\theta d^2\theta^\dagger\> \Phi
\,=\,
\Phi \Bigl _{\theta\theta\thdthd}
\,=\,
\frac{1}{4} \partial_\mu \partial^\mu \phi,
\label{eq:tryPhiD}
\eeq
where the last equality follows from eq.~(\ref{eq:Phiinxrep}), and
$\phi$ is the scalar component of $\Phi$. Equation (\ref{eq:tryPhiD})
is a total derivative, so adding it (and its complex conjugate) to the Lagrangian
density has no effect.
Therefore, the two ways of making a supersymmetric Lagrangian are to take the
$D$term component of a real superfield, and to take the $F$term component of
a chiral superfield, plus the complex conjugate.
When building a Lagrangian, the real superfield $V$ used in
eq.~(\ref{eq:DtermLag}) and the chiral superfield $\Phi$ used in
eq.~(\ref{eq:FtermLag}) are usually composites, built out of more
fundamental superfields. However, contributions from fundamental
fields $V$ and $\Phi$ are allowed, when $V$ is the vector superfield for
an Abelian gauge symmetry and when $\Phi$ is a singlet under all
symmetries.
It is always possible to rewrite a $D$ term contribution to a Lagrangian
as an $F$ term contribution, by the trick of noticing that
\beq
\Dcon\Dcon (\theta^\dagger \theta^\dagger) &=&
DD(\theta\theta) \>\,=\,\> 4,
\label{eq:DDthetatheta}
\eeq
and using the fact that $\delta^{(2)}(\theta^\dagger) =
\theta^\dagger\theta^\dagger$
from eq.~(\ref{eq:deltathetatheta}).
Thus, by integrating by parts twice with respect to $\theta^\dagger$:
\beq
[V]_D
&=&
\frac{1}{4} \int d^2\theta d^2\theta^\dagger\, V
\,
\Dcon\Dcon (\thdthd)
\>=\>
\frac{1}{4} \int d^2\theta d^2\theta^\dagger\,
\delta^{(2)}(\theta^\dagger) \,
\Dcon\Dcon V
+ \ldots\phantom{xxxx}
\label{eq:rewriteDasFz}
\\
&=&
\frac{1}{4} \bigl [\Dcon\Dcon V \bigr ]_F
+
\ldots.
\label{eq:rewriteDasF}
\eeq
The $\ldots$ indicates total derivatives with respect to $x^\mu$,
coming from the two integrations by parts. As noted in section \ref{chiralsuperfields},
$\Dcon\Dcon V$ is always a chiral superfield.
If $V$ is real, then the imaginary part of eq.~(\ref{eq:rewriteDasF})
is a total derivative, and the result can be rewritten as
$\frac{1}{8} \bigl [\Dcon\Dcon V \bigr ]_F + {\rm c.c.}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Superspace Lagrangians for chiral supermultiplets\label{superspacelagrchiral}}
\setcounter{equation}{0}
\setcounter{footnote}{2}
In section \ref{chiralsuperfields}, we verified that the chiral
superfield components have the same supersymmetry transformations as the
WessZumino model fields. We now have the tools to complete the
demonstration of equivalence by reconstructing the Lagrangian in
superspace language. Consider the composite superfield
\beq
\Phi^{*i} \Phi_j
&=&
\phi^{*i} \phi_j
+ \sqrt{2} \theta \psi_j \phi^{*i}
+ \sqrt{2} \theta^\dagger\hspace{1pt} \psi^{\dagger i} \phi_j
+ \theta\theta \phi^{*i} F_j
+ \thdthd \phi_j F^{*i}
\nonumber \\ &&
+ \thetasigmamuthetadagger \left [
\BDneg i \phi^{*i} \partial_\mu \phi_j
\BDplus i \phi_j \partial_\mu \phi^{*i}
 \psi^{\dagger i} \sigmabar_\mu \psi_j
\right ]
\nonumber \\ &&
+ \frac{i}{\sqrt{2}} \theta\theta\theta^\dagger \sigmabar^\mu (
\psi_j \partial_\mu \phi^{*i}  \partial_\mu \psi_j \phi^{*i})
+ \sqrt{2} \theta\theta \theta^\dagger \psi^{\dagger i} F_j
\nonumber \\ &&
+ \frac{i}{\sqrt{2}} \thdthd\theta \sigma^\mu (
\psi^{\dagger i} \partial_\mu \phi_j
 \partial_\mu \psi^{\dagger i} \phi_j)
+ \sqrt{2} \thdthd \theta \psi_j F^{*i}
\nonumber \\ &&
+ \theta\theta\thdthd
\Bigl [
F^{*i} F_j
\BDplus \frac{1}{2}\partial^\mu \phi^{*i} \partial_\mu \phi_j
\BDminus \frac{1}{4}\phi^{*i} \partial^\mu \partial_\mu \phi_j
\BDminus \frac{1}{4}\phi_j \partial^\mu \partial_\mu \phi^{*i}
\nonumber \\ && \quad
+ \frac{i}{2} \psi^{\dagger i} \sigmabar^\mu \partial_\mu \psi_j
+ \frac{i}{2} \psi_j \sigma^\mu \partial_\mu \psi^{\dagger i}
\Bigr ] .
\label{eq:PhistariPhij}
\eeq
where
all fields
are evaluated as functions of $x^\mu$ (not $y^\mu$ or $y^{\mu *}$).
For $i=j$, eq.~(\ref{eq:PhistariPhij}) is a real (vector) superfield, and
the massless freefield
Lagrangian for each chiral superfield is just obtained by taking the $\theta\theta\thdthd$ component:
\beq
[\Phi^* \Phi]_D = \int d^2\theta d^2\theta^\dagger \,\Phi^* \Phi
\>=\> \BDpos \partial^\mu \phi^* \partial_\mu \phi
+ i \psi^\dagger \sigmabar^\mu \partial_\mu \psi
+ F^* F + \ldots .
\label{eq:freelagrphisuperspace}
\eeq
The $\ldots$ indicates a total derivative part, which may be
dropped since this is destined to be integrated $\int d^4 x$. Equation
(\ref{eq:freelagrphisuperspace}) is exactly the Lagrangian density
obtained in section \ref{subsec:susylagr.freeWZ} for the massless free
WessZumino model.
To obtain the superpotential interaction and mass terms, recall that products of
chiral superfields are also superfields. For example,
\beq
\Phi_i \Phi_j &=& \phi_i \phi_j
+ \sqrt{2}\theta (\psi_i \phi_j + \psi_j \phi_i)
+ \theta\theta (\phi_i F_j + \phi_j F_i  \psi_i \psi_j)
,
\label{eq:superfieldPhiPhi}
\\
\Phi_i \Phi_j \Phi_k &=& \phi_i \phi_j \phi_k
+ \sqrt{2}\theta (\psi_i \phi_j \phi_k
+ \psi_j \phi_i \phi_k + \psi_k \phi_i \phi_j)
\nonumber
\\ &&
+ \,\theta\theta (\phi_i \phi_j F_k + \phi_i \phi_k F_j +
\phi_j \phi_k F_i
 \psi_i \psi_j \phi_k  \psi_i \psi_k \phi_j
 \psi_j \psi_k \phi_i
),
\label{eq:superfieldPhiPhiPhi}
\eeq
where the presentation has been simplified by taking the component fields
on the right sides to be functions of $y^\mu$ as given in
eq.~(\ref{eq:defineycoord}). More generally, any holomorphic function of
a chiral superfields is a chiral superfield.
So, one may form a complete Lagrangian as
\beq
{\cal L}(x) &=& [\Phi^{*i} \Phi_i]_D +
\left ( \left [ W(\Phi_i) \right ]_F
+ {\rm c.c.} \right ),
\label{eq:oslo}
\eeq
where $W(\Phi_i)$
can be any holomorphic function
of the chiral superfields (but not antichiral superfields)
taken as complex variables, and
coincides with the superpotential $W(\phi_i)$ that was treated
in subsection \ref{subsec:susylagr.chiral} as a function of the scalar components.
For $W = \frac{1}{2} M^{ij} \Phi_i \Phi_j
+ \frac{1}{6} y^{ijk} \Phi_i \Phi_j \Phi_k$, the result of
eq.~(\ref{eq:oslo})
is exactly the same as eq.~(\ref{lagrchiral}), after writing in
component form using eqs.~(\ref{eq:freelagrphisuperspace}),
(\ref{eq:superfieldPhiPhi}),
(\ref{eq:superfieldPhiPhiPhi}) and integrating out the auxiliary fields.
It is instructive to obtain the superfield equations of motion
from the Lagrangian eq.~(\ref{eq:oslo}). The quickest way to do this
is to first use the remarks at the very end of section \ref{superspacelagr}
to rewrite the Lagrangian density as:
\beq
{\cal L}(x) &=&
\int d^2\theta \left [ \frac{1}{4} \Dcon\Dcon \Phi^{*i} \Phi_i +
W(\Phi_i) \right ]
\,+ \,
\int d^2\theta^\dagger \left [W(\Phi_i)\right ]^*.
\eeq
Now varying with respect to $\Phi_i$ immediately gives the superfield equation of
motion:
\beq
0 &=& \frac{1}{4} \Dcon\Dcon \Phi^{*i} + \frac{\delta W}{\delta \Phi_i} ,
\label{eq:superfieldeqmot}
\eeq
and its complex conjugate,
\beq
0&=& \frac{1}{4} DD \Phi_i + \frac{\delta W^*}{\delta \Phi^{*i}} .
\label{eq:superfieldeqmotCON}
\eeq
These are equivalent to the componentlevel equations of motion as can be found from the
Lagrangian in section \ref{subsec:susylagr.chiral}. To verify this,
it is easiest to write eq.~(\ref{eq:superfieldeqmot})
in the coordinate system $(y^\mu, \theta, \theta^\dagger)$,
in which the first term has the simple form
\beq
\frac{1}{4} \Dcon\Dcon \Phi^{*i}
= F^*(y)  i \sqrt{2} \theta\sigma^\mu \partial_\mu \psi^{\dagger i}(y)
\BDminus \theta\theta \partial_\mu \partial^\mu \phi^{*i}(y) .
\eeq
Because this is a chiral (not antichiral) superfield, it is simpler to
write the components as functions of $y^\mu$ as shown,
not $y^{\mu *}$, even though the lefthand
side involves $\Phi^*$.
For an alternate method, consider a Lagrangian density $V$ on
the full superspace, so that the action is
\beq
A &=& \int d^4 x \hspace{0.5pt} \int d^2\theta d^2\theta^\dagger\, V,
\eeq
with $V(S_i, \, D_\alpha S_i, \, \Dcon_{\dot\alpha} S_i)$ assumed to be
a function of
general dynamical superfields $S_i$ and their chiral and antichiral
first derivatives. Then the superfield equations of motion obtained by
variation of the action are
\beq
0 &=& \frac{\partial V}{\partial S_i}
 D_\alpha \left ( \frac{\partial V}{\partial (D_\alpha S_i)}
\right )
 \Dcon_{\dot\alpha} \left (\frac{\partial V}{\partial
(\Dcon_{\dot\alpha} S_i)} \right )
.
\label{eq:superspaceeqmo}
\eeq
In the case of the Lagrangian for chiral superfields eq.~(\ref{eq:oslo}),
Lagrange multipliers
$\Lambda^{*i\dot \alpha}$ and $\Lambda^{\alpha}_i$ can be introduced
to enforce the chiral and antichiral superfield constraints
on $\Phi_i$ and $\Phi^{*i}$ respectively.
The Lagrangian density on superspace is then:
\beq
V &=&
\Lambda^{*i\dot \alpha} \Dcon_{\dot \alpha} \Phi_i
+ \Lambda^{\alpha}_i D_{\alpha} \Phi^{*i}
+ \Phi^{*i} \Phi_i
+ \delta^{(2)}(\theta^\dagger) W(\Phi_i)
+ \delta^{(2)}(\theta) [W(\Phi_i)]^* .
\eeq
Variation with respect to the Lagrange multipliers just gives the
constraints
$\Dcon_{\dot \alpha} \Phi_i = 0$ and
$D_{\alpha} \Phi^{*i} = 0$. Applying eq.~(\ref{eq:superspaceeqmo}) to the
superfields $\Phi_i$ and $\Phi^{*i}$ leads to equations of motion:
\beq
0 &=&
\Phi^{*i} +
\delta^{(2)}(\theta^\dagger) \frac{\delta W}{\delta \Phi_i}
 \Dcon_{\dot\alpha} \Lambda^{*i\dot \alpha},
\\
0 &=&
\Phi_{i} +
\delta^{(2)}(\theta) \frac{\delta W^*}{\delta \Phi^{*i}}
 D_{\alpha} \Lambda^{\alpha}_i
.
\eeq
Now acting on these equations
with $\frac{1}{4} \Dcon\Dcon$ and
$\frac{1}{4} DD$ respectively, and applying
eqs.~(\ref{eq:deltathetatheta}) and (\ref{eq:DDthetatheta}),
one again obtains eqs.~(\ref{eq:superfieldeqmot}) and (\ref{eq:superfieldeqmotCON}).
\subsection{Superspace Lagrangians for Abelian
gauge theory\label{subsec:superspacelagrabelian}}
\setcounter{equation}{0}
\setcounter{footnote}{2}
Now consider the superspace Lagrangian for a gauge theory, treating the
$U(1)$ case first for simplicity. The nonAbelian case will be considered
in the next subsection.
The vector superfield $V(x,\theta,\theta^\dagger)$ of
eq.~(\ref{eq:vectorsuperfieldexpansion})
contains the gauge potential $A^\mu$.
Define corresponding gaugeinvariant Abelian
field strength superfields by
\beq
{\cal W}_\alpha = \frac{1}{4} \Dcon\Dcon D_\alpha V,
\qquad\qquad
{\cal W}^\dagger_{\dot\alpha} = \frac{1}{4} DD \Dcon_{\dot\alpha} V.
\label{eq:defineWalpha}
\eeq
These
are respectively chiral and antichiral by construction
[see eq.~(\ref{eq:chiralfromgeneral})], and are examples of
superfields that carry spinor indices and are anticommuting. They carry
dimension [mass]$^{3/2}$.
To see that ${\cal W}_\alpha$ is gauge invariant, note that under
a supergauge transformation of the form eq.~(\ref{eq:u1superfieldgt}),
\beq
{\cal W}_\alpha \,\rightarrow\,
\frac{1}{4} \Dcon\Dcon D_\alpha \bigl [V
+ i (\Omega^*  \Omega) \bigr ]
\label{eq:Walphagione}
&=&
{\cal W}_\alpha + \frac{i}{4} \Dcon\Dcon D_\alpha \Omega
\\
&=&
{\cal W}_\alpha  \frac{i}{4} \Dcon^{\dot\beta}\bigl \lbrace
\Dcon_{\dot\beta}, D_\alpha \bigr \rbrace \Omega \phantom{xxxxxx}
\\
&=&
{\cal W}_\alpha + \frac{1}{2} \sigma^\mu_{\alpha\dot\beta}
\partial_\mu \Dcon^{\dot\beta}\Omega
\\
&=&
{\cal W}_\alpha
\label{eq:Walphagitwo}
\eeq
The first equality follows from eq.~(\ref{eq:antichiralsuperfieldconstraint})
because $\Omega^*$ is antichiral, the second and fourth equalities from
eq.~(\ref{eq:leftchiralsuperfieldconstraint}) because $\Omega$ is
chiral, and the third from eq.~(\ref{eq:diffopderivs}).
To see how the component fields fit into ${\cal W}_\alpha$, it is convenient
to temporarily specialize to WessZumino gauge as in eq.~(\ref{eq:WZgauge}), and
then convert to the coordinates $(y^\mu, \theta, \theta^\dagger)$
as defined in eq.~(\ref{eq:defineycoord}), with the result
\beq
V(y^\mu, \theta,\theta^\dagger) &=&
\thetasigmamuthetadagger A_\mu(y)
+ \thdthd \theta \lambda(y)
+ \theta \theta \theta^\dagger\hspace{1pt} \lambda^\dagger(y)
+ \frac{1}{2} \theta\theta\thdthd
\left [D(y)
+ i \partial_\mu A^\mu(y) \right ].\phantom{xxx}
\eeq
Now application of eqs.~(\ref{eq:Dinyrep}), (\ref{eq:Ddaggerinyrep})
yields
\beq
{\cal W}_\alpha(y,\theta,\theta^\dagger)
&=&
\lambda_\alpha + \theta_\alpha D
\BDminus \frac{i}{2} (\sigma^\mu\sigmabar^\nu\theta)_\alpha F_{\mu\nu}
+ i \theta\theta (\sigma^\mu\partial_\mu \lambda^\dagger)_\alpha ,
\label{eq:gottawearshades}
\\
{\cal W}^{\dagger\dot\alpha}(y^*,\theta,\theta^\dagger)
&=&
\lambda^{\dagger\dot \alpha} + \theta^{\dagger\dot\alpha} D
\BDplus \frac{i}{2} (\sigmabar^\mu\sigma^\nu \theta^\dagger)^{\dot\alpha}
F_{\mu\nu}
+ i \thdthd (\sigmabar^\mu\partial_\mu \lambda )^{\dot\alpha},
\label{eq:futuresobright}
\eeq
where
all fields on the right side are understood to be functions of
$y^\mu$ and $y^{\mu *}$ respectively, and
\beq
F_{\mu\nu} \,=\, \partial_\mu A_\nu  \partial_\nu A_\mu
\eeq
is the ordinary component field strength.
Although it was convenient to derive eqs.~(\ref{eq:gottawearshades}) and
(\ref{eq:futuresobright}) in WessZumino gauge, they must
be true in general, because ${\cal W}_\alpha$ and ${\cal W}^{\dagger\dot\alpha}$
are supergauge invariant.
Equation (\ref{eq:gottawearshades}) implies
\beq
[{\cal W}^\alpha {\cal W}_\alpha]_F = D^2 +
2 i \lambda \sigma^\mu \partial_\mu \lambda^\dagger
\frac{1}{2} F^{\mu\nu} F_{\mu\nu}
+ \frac{i}{4} \epsilon^{\mu\nu\rho\sigma} F_{\mu\nu} F_{\rho\sigma}
,
\label{eq:Bushwickblues}
\eeq
where now all fields on the right side are functions of $x^\mu$.
Integrating, and eliminating total derivative parts, one obtains
the action
\beq
\int d^4 x\, {\cal L}
&=&
\int d^4 x\, \frac{1}{4} [{\cal W}^\alpha {\cal W}_\alpha]_F + {\rm c.c.}
\,=\,
\int d^4 x\,
\left [
\frac{1}{2} D^2 +
i \lambda^\dagger \sigmabar^\mu \partial_\mu \lambda
 \frac{1}{4} F^{\mu\nu} F_{\mu\nu}
\right ],
\phantom{xx}
\eeq
in agreement with eq.~(\ref{lagrgauge}).
Additionally,
the integral of the $D$term component of $V$ itself is invariant under
both supersymmetry [see eq.~(\ref{eq:sqrttwodeltaD})] and supergauge [see
eq.~(\ref{eq:DgaugetransfromLambda})]
transformations. Therefore, one can include a FayetIliopoulos term
\beq
{\cal L}_{\mbox{FI}} &=& 2 \kappa [V]_D \>=\>\kappa D,
\eeq
again dropping a total derivative. This type of term can play a role in spontaneous
supersymmetry breaking, as we will discuss in section \ref{subsec:origins.Dterm}.
It is also possible to write the Lagrangian density eq.~(\ref{eq:Bushwickblues})
as a $D$term rather
than an $F$term. Since ${\cal W}^\alpha$ is a chiral superfield, with
$\Dcon_{\dot \beta} {\cal W}^\alpha = 0$, one can use
eq.~(\ref{eq:defineWalpha}) to write
\beq
{\cal W}^\alpha {\cal W}_\alpha =
\frac{1}{4} \Dcon\Dcon ({\cal W}^\alpha D_\alpha V).
\eeq
Therefore, using eq.~(\ref{eq:rewriteDasF}),
the Lagrangian for $A^\mu$, $\lambda$, and $D$ can be rewritten as:
\beq
{\cal L}(x) = \int d^2\theta d^2\theta^\dagger \, \left [
\frac{1}{4}
\left (
{\cal W}^\alpha D_\alpha V
+ {\cal W}^\dagger_{\dot\alpha} \Dcon^{\dot\alpha} V \right )  2 \kappa V
\right ].
\eeq
Next consider the coupling of the Abelian gauge field to a set of chiral
superfields $\Phi_i$ carrying $U(1)$ charges $q_i$. Supergauge transformations, as in
eqs.~(\ref{eq:u1superfieldgt})(\ref{eq:DgaugetransfromLambda}), are
parameterized by a nondynamical chiral
superfield $\Omega$,
\beq
\Phi_i &\rightarrow& e^{2ig q_i \Omega} \Phi_i ,
\qquad\qquad
\Phi^{*i} \>\rightarrow\> e^{2ig q_i \Omega^*} \Phi^{*i} ,
\eeq
where $g$ is the gauge coupling.
In the special case that $\Omega$ is
just a real function $\phi(x)$, independent of $\theta$ and $\theta^\dagger$,
this reproduces the usual gauge transformations with $A^\mu \rightarrow
A^\mu \BDminus 2 \partial^\mu \phi$.
The kinetic term
from eq.~(\ref{eq:freelagrphisuperspace})
involves the superfield
$\Phi^{*i} \Phi_i$, which is not supergauge invariant:
\beq
\Phi^{*i} \Phi_i &\rightarrow&
e^{2ig q_i(\Omega  \Omega^*)} \Phi^{*i} \Phi_i .
\label{eq:gtofPhistarPhi}
\eeq
To remedy this, we modify the chiral superfield kinetic term in the Lagrangian to
\beq
\left [ \Phi^{*i} e^{2 g q_i V} \Phi_i \right ]_D .
\eeq
The gauge transformation of the
$e^{2 g q_i V}$ factor, found from
eq.~(\ref{eq:u1superfieldgt}),
exactly cancels that
of eq.~(\ref{eq:gtofPhistarPhi}).
The presence of an exponential of $V$ in the Lagrangian is possible
because $V$ is dimensionless. It might appear to be dangerous, because
normally such a nonpolynomial term would be nonrenormalizable. However,
the gauge dependence of $V$ comes to the rescue: the higher order
terms can be supergauged away. In particular, evaluating $e^{2 g q_i V}$
in the WessZumino gauge, the power series expansion of the exponential
is simple and terminates, because
\beq
V^2 &=& \BDpos\frac{1}{2}
\theta\theta\thdthd A_\mu A^\mu,
\\
V^n &=& 0\qquad(n\geq 3),
\eeq
so that
\beq
e^{2 g q_i V} = 1 + 2 g q_i (\thetasigmamuthetadagger A_\mu +
\thdthd \theta \lambda
+ \theta\theta \theta^\dagger\hspace{1pt} \lambda^\dagger +
\frac{1}{2} \theta\theta\thdthd D )
\BDplus g^2 q_i^2 \theta\theta\thdthd A_\mu A^\mu.
\eeq
Using this, one can work out that, in WessZumino gauge and up to total
derivative terms,
\beq
\left [ \Phi^{*i} e^{2 g q_i V} \Phi_i \right ]_D
&=&
F^{*i} F_i
\BDplus \nablasubmu \phi^{*i} \nabla^\mu \phi_i
+ i \psi^{\dagger i} \sigmabar^\mu \nablasubmu \psi_i
 \sqrt{2} g q_i
( \phi^{*i} \psi_i \lambda + \lambda^\dagger \psi^{\dagger i} \phi_i)
\phantom{xx}
\nonumber \\ &&
+ g q_i \phi^{*i} \phi_i D,
\label{eq:superfieldchiralkinetic}
\eeq
where $\nablasubmu$ is the gaugecovariant spacetime derivative:
\beq
\nablasubmu \phi_i &=& \partial_\mu \phi_i \BDplus i g q_i A_\mu \phi_i,
\qquad\qquad
\nablasubmu \phi^{*i} \>=\> \partial_\mu \phi^{*i} \BDminus i g q_i A_\mu
\phi^{*i},
\\
\nablasubmu \psi_i &=& \partial_\mu \psi_i \BDplus i g q_i A_\mu \psi_i .
\eeq
Equation (\ref{eq:superfieldchiralkinetic}) agrees with the
specialization of eq.~(\ref{gensusylagr}) to the Abelian case.
In summary, the superspace Lagrangian
\beq
{\cal L} &=&
\left [ \Phi^{*i} e^{2 g q_i V} \Phi_i \right ]_D
+ \left (\left [W(\Phi_i)\right ]_F + {\rm c.c.} \right )
+
\frac{1}{4}\left (\left [{\cal W}^\alpha {\cal W}_\alpha \right ]_F
+ {\rm c.c.} \right )
2 \kappa [V]_D
\eeq
reproduces the component form Lagrangian found in
subsection \ref{subsec:susylagr.gaugeinter}
in the special case of matter fields coupled to each other and to a
$U(1)$ gauge symmetry, plus a FayetIliopoulos parameter $\kappa$.
\subsection{Superspace Lagrangians for general
gauge theories\label{subsec:superspacelagrnonabelian}}
\setcounter{equation}{0}
\setcounter{footnote}{2}
Now consider a general gauge symmetry realized on chiral superfields $\Phi_i$ in a representation $R$
with matrix generators $T^{aj}_i$:
\beq
\Phi_i \rightarrow \bigl (e^{2 i g_a \Omega^a T^a} \bigr )_i{}^j
\Phi_j,
\qquad\qquad
\Phi^{*i} \rightarrow \Phi^{*j}
\bigl (e^{2 i g_a \Omega^a T^a} \bigr )_j{}^i.
\eeq
The gauge couplings for the irreducible components of the Lie algebra are $g_a$. As in the Abelian case, the supergauge transformation parameters are chiral superfields $\Omega^a$.
For each Lie algebra generator, there is a
vector superfield $V^a$, which contains the vector gauge boson and gaugino.
The Lagrangian then contains a supergaugeinvariant term
\beq
{\cal L} &=& \Bigl [ \Phi^{*i} (e^{2 g_a T^a V^a})_i{}^j \Phi_j \Bigr ]_D.
\eeq
It is convenient to define matrixvalued vector and gauge parameter
superfields in the representation $R$:
\beq
V_i{}^j &=& 2 g_a T_i^{aj} V^a ,
\qquad\qquad
\Omega_i{}^j \>\,=\,\> 2 g_a T_i^{aj} \Omega^a ,
\label{eq:yourenotthebossofmenow}
\eeq
so that one can write
\beq
\Phi_i \rightarrow \bigl (e^{i \Omega} \bigr )_i{}^j
\Phi_j,
\qquad\qquad
\Phi^{*i} \rightarrow \Phi^{*j}
\bigl (e^{i \Omega^\dagger} \bigr )_j{}^i ,
\eeq
and
\beq
{\cal L} &=& \Bigl [ \Phi^{*i} (e^{V})_i{}^j \Phi_j \Bigr ]_D.
\label{eq:bigbadbeluva}
\eeq
For this to be supergauge invariant,
the nonAbelian gauge transformation rule for the vector superfields must be
\beq
e^V &\rightarrow& e^{i \Omega^\dagger} e^V e^{i\Omega}
.
\label{eq:kentuckyavenue}
\eeq
[Here chiral supermultiplet representation indices $i,j,\ldots$ are suppressed;
$V$ and $\Omega$ with no indices stand for the matrices defined in
eq.~(\ref{eq:yourenotthebossofmenow}).]
Equation (\ref{eq:kentuckyavenue}) can be expanded, keeping terms linear in
$\Omega$, $\Omega^\dagger$, using the BakerCampbellHausdorff formula, to find
\beq
V &\rightarrow & V + i (\Omega^\dagger  \Omega)
 \frac{i}{2} \bigl [V,\, \Omega + \Omega^\dagger ]
+ i\sum_{k=1}^{\infty}
\frac{B_{2k}}{(2k)!} \left [ V, \left [ V,\ldots \left [V,\, \Omega^\dagger  \Omega \right ]\ldots \right]\right]
,
\phantom{xxx}
\label{eq:nonAbelianVtransexpansion}
\eeq
where the $k$th term in the sum involves $k$ matrix commutators of
$V$, and $B_{2k}$ are the Bernoulli numbers defined by
\beq
\frac{x}{e^x  1} = \sum_{n=0}^\infty \frac{B_n}{n!} x^n.
\eeq
Equation~(\ref{eq:nonAbelianVtransexpansion}) is equivalent to
\beq
V^a &\rightarrow& V^a + i (\Omega^{a*}  \Omega^a)
+ g_a f^{abc} V^b (\Omega^{c*} + \Omega^c)
 \frac{i}{3} g_a^2 f^{abc} f^{cde} V^b V^d (\Omega^{e*}  \Omega^e) +
\ldots \phantom{xxx}
\label{eq:nonAbelianVtransexpansiona}
\eeq
where eq.~(\ref{eq:yourenotthebossofmenow}) and $[T^a, T^b] = i f^{abc} T^c$ have been used.
This supergauge transformation includes ordinary gauge transformations as
the special case
$\Omega^{a*} = \Omega^a$.
Because the second term on the right side of eq.~(\ref{eq:nonAbelianVtransexpansiona}) is
independent of $V^a$, one can always do a supergauge transformation to WessZumino gauge
by choosing $\Omega^{a*}  \Omega^a$ appropriately,
just as in the
Abelian case, so that
\beq
\bigl (V^a \bigr )_{\mbox{WZ gauge}}
&=&
\thetasigmamuthetadagger A^a_\mu
+ \thdthd \theta \lambda^a
+ \theta \theta \theta^\dagger\hspace{1pt} \lambda^{\dagger a}
+ \frac{1}{2} \theta\theta\thdthd D^a .
\eeq
After fixing the supergauge to WessZumino gauge, one still has the freedom to do ordinary
gauge transformations.
In the WessZumino gauge, the Lagrangian contribution eq.~(\ref{eq:bigbadbeluva}) is polynomial,
in agreement with what was found in component language in section
\ref{subsec:susylagr.gaugeinter}:
\beq
\left [ \Phi^{*i} \bigl (e^{V} \bigr )_i{}^j \Phi_j \right ]_D
&=&
F^{*i} F_i
\BDplus \nablasubmu \phi^{*i} \nabla^\mu \phi_i
+ i \psi^{\dagger i} \sigmabar^\mu \nablasubmu \psi_i
 \sqrt{2} g_a (\phi^{*} T^a \psi) \lambda^a
 \sqrt{2} g_a \lambda^\dagger (\psi^{\dagger } T^a \phi)
\phantom{xx}
\nonumber \\ &&
+ g_a (\phi^{*} T^a \phi) D^a,
\eeq
where $\nablasubmu$ is the gaugecovariant derivative
defined in eqs.~(\ref{ordtocovphi})(\ref{ordtocovpsi}).
To make kinetic terms and selfinteractions for the vector
supermultiplets in the nonAbelian case,
define a fieldstrength chiral superfield
\beq
{\cal W}_\alpha \>=\> \frac{1}{4} \Dcon\Dcon
\left (e^{V} D_\alpha e^V \right),
\label{eq:definenonabelianfieldstrengthW}
\eeq
generalizing the Abelian case.
Using eq.~(\ref{eq:kentuckyavenue}),
one can show that it transforms under supergauge transformations as
\beq
{\cal W}_\alpha &\rightarrow& e^{i\Omega} {\cal W}_\alpha e^{i\Omega}
.
\eeq
(The proof makes use of the fact that $\Omega$ is chiral and $\Omega^\dagger$ is antichiral, so that $\Dcon_{\dot\alpha} \Omega = 0$
and $D_\alpha \Omega^\dagger = 0$.) This implies that Tr$[W^\alpha W_\alpha]$
is a supergaugeinvariant chiral superfield.
The contents of the parentheses in
eq.~(\ref{eq:definenonabelianfieldstrengthW}) can be expanded as
\beq
e^{V} D_\alpha e^V &=& D_\alpha V  \frac{1}{2} [V, D_\alpha V]
+ \frac{1}{6} \left [V, \left [V, D_\alpha V\right ] \right ] + \ldots ,
\eeq
where again the commutators apply in the matrix sense, and only the first two terms
contribute in WessZumino gauge.
The field strength chiral superfield ${\cal W}_\alpha$ defined in
eq.~(\ref{eq:definenonabelianfieldstrengthW})
is matrixvalued in the representation $R$. One can recover an adjoint representation
field strength superfield ${\cal W}^a_\alpha$ from the matrixvalued
one by writing
\beq
{\cal W}_\alpha = 2 g_a T^a {\cal W}^a_\alpha ,
\eeq
leading to
\beq
{\cal W}^a_\alpha &=&
\frac{1}{4} \Dcon\Dcon \left (
D_\alpha V^a  i g_a f^{abc} V^b D_\alpha V^c + \ldots \right ) .
\eeq
The terms shown explicitly are enough to evaluate this in components in
WessZumino gauge, with the result
\beq
({\cal W}^a_\alpha)_{\mbox{WZ gauge}}
&=&
\lambda^a_\alpha + \theta_\alpha D^a
\BDminus \frac{i}{2} (\sigma^\mu
\sigmabar^\nu \theta)_\alpha F^a_{\mu\nu}
+ i \theta\theta (\sigma^\mu \nablasubmu \lambda^{\dagger a})_\alpha
,
\label{eq:GeneAmmons}
\eeq
where $F^a_{\mu\nu}$ is the nonAbelian field strength of eq.~(\ref{eq:YMfs})
and $\nablasubmu$ is the usual gauge covariant derivative from eq.~(\ref{ordtocovlambda}).
The kinetic terms and selfinteractions for the gauge supermultiplet fields are obtained
from
%the supergaugeinvariant term
\beq
\frac{1}{4 k_a g_a^2} {\rm Tr}[{\cal W}^{\alpha} {\cal W}_\alpha]_F &=&
[{\cal W}^{a\alpha} {\cal W}^a_\alpha]_F,
\label{eq:foxforce}
\eeq
which is invariant under both supersymmetry and supergauge transformations.
Here the normalization of generators is assumed to be
$
{\rm Tr}[T^a T^b] = k_a \delta_{ab},
$
with $k_a$ usually set to $1/2$ by convention for the
defining representations of simple groups.
Equation (\ref{eq:foxforce}) is most easily
evaluated in WessZumino gauge using eq.~(\ref{eq:GeneAmmons}), yielding
\beq
[{\cal W}^{a\alpha} {\cal W}^a_\alpha]_F &=& D^{a}D^a +
2 i \lambda^a \sigma^\mu \nablasubmu \lambda^{\dagger a}
\frac{1}{2} F^{a\mu\nu} F^a_{\mu\nu}
+ \frac{i}{4} \epsilon^{\mu\nu\rho\sigma} F^a_{\mu\nu} F^a_{\rho\sigma} .
\label{eq:CharlieParker}
\eeq
Since eq.~(\ref{eq:CharlieParker}) is supergauge invariant, the same expression is valid
even outside of WessZumino gauge.
Now we can write the general renormalizable Lagrangian for a supersymmetric gauge theory (including superpotential interactions for the chiral supermultiplets when allowed by gauge invariance):
\beq
{\cal L} &=& \left ( \frac{1}{4}  i \frac{g_a^2 \Theta_a}{32 \pi^2}
\right )
\left [
{\cal W}^{a\alpha} {\cal W}^a_\alpha
\right ]_F
+ {\rm c.c.}
+
\left [
\Phi^{*i} (e^{2g_a T^a V^a})_i{}^j \Phi_j \right]_D
+
\left ( [W(\Phi_i)]_F + {\rm c.c.} \right ) .
\phantom{xxxx}
\eeq
This introduces and defines
$\Theta_a$, a CPviolating parameter, whose effect is to include a total derivative term
in the Lagrangian density:
\beq
{\cal L}_{\Theta_a} &=&
\frac{g_a^2 \Theta_a}{64\pi^2} \epsilon^{\mu\nu\rho\sigma} F^a_{\mu\nu} F^a_{\rho\sigma}.
\eeq
In the nonAbelian case, this can have physical
effects due to topologically nontrivial field configurations (instantons).
For a globally nontrivial
gauge configuration with integer winding number $n$, one has
$\int d^4 x\, \epsilon^{\mu\nu\rho\sigma} F^a_{\mu\nu} F^a_{\rho\sigma}
= 64 \pi^2 n/g_a^2$ for a simple gauge group,
so that the contribution to the path integral is
$\exp({i\int d^4 x\,{\cal L}_{\scriptstyle \Theta_a}})
=e^{i n \Theta_a}$.
Note that for nonAbelian gauge groups, a FayetIliopoulos term $2\kappa [V^a]_D$ is not
allowed, because it is not a gauge singlet.
When the superfields are restricted to the WessZumino gauge, the supersymmetry transformations
are not realized linearly in superspace, but the Lagrangian is polynomial. The
nonpolynomial form of the superspace Lagrangian is thus seen to be a supergauge artifact.
Within WessZumino gauge, supersymmetry transformations are still realized, but
nonlinearly, as we found in sections \ref{subsec:susylagr.gauge} and
\ref{subsec:susylagr.gaugeinter}.
The gauge coupling $g_a$ and CPviolating angle $\Theta_a$ are often
combined into a single holomorphic
coupling:
\beq
\tau_a &=& \frac{1}{g_a^2} i \frac{\Theta_a}{8 \pi^2}
%\frac{\Theta_a}{2\pi} + \frac{4\pi i}{g_a^2}.
\eeq
(There are several different normalization conventions for $\tau_a$ in the literature.)
Then, with redefined
vector and field strength superfields that include $g_a$ as part of their normalization,
\beq
\widehat V^{a} &\equiv& g_a V^{a},
\label{eq:holonorm}
\\
\widehat {\cal W}^{a}_{\alpha} &\equiv&
g_a {\cal W}^{a}_{\alpha}
\>\,=\>\,
\frac{1}{4} \Dcon\Dcon \left (
D_\alpha \widehat V^a  i f^{abc} \widehat V^b D_\alpha \widehat V^c + \ldots \right )
,
\label{eq:holonormW}
\eeq
the gauge part of the Lagrangian is written as
\beq
{\cal L} &=&
%\frac{1}{16\pi i}
\frac{1}{4}
\left [ \tau_a \widehat{\cal W}^{a \alpha}\widehat{\cal W}^{a}_{\alpha} \right ]_F
+ {\rm c.c.} +
\left [
\Phi^{*i} (e^{2 T^a \widehat V^a})_i{}^j \Phi_j \right]_D .
\label{eq:tauaholo}
\eeq
An advantage of this normalization convention is that
when written in terms of $\widehat V^a$, the only appearance of the gauge coupling
and $\Theta_a$ is in the $\tau_a$ in eq.~(\ref{eq:tauaholo}).
It is then sometimes useful to treat the complex holomorphic coupling $\tau_a$
as a
chiral superfield with an expectation value for its scalar component.
An expectation value for the
$F$term component of $\tau_a$ will give gaugino masses;
this is sometimes
a useful way to implement the effects of explicit soft supersymmetry breaking.
\subsection{Nonrenormalizable
supersymmetric Lagrangians\label{superspacenonrenorm}}
\setcounter{equation}{0}
\setcounter{footnote}{2}
So far, we have discussed only renormalizable supersymmetric Lagrangians.
However, integrating out the effects
of heavy states will generally lead to
nonrenormalizable interactions in the lowenergy
effective description. Furthermore,
when any realistic supersymmetric theory is extended to include gravity,
the resulting supergravity theory is
nonrenormalizable as a quantum field theory. Fortunately,
the nonrenormalizable interactions can be neglected
for most phenomenological purposes, because they
involve couplings of negative mass dimension, proportional to
powers of $1/\MPlanck$ (or perhaps $1/\Lambda_{\rm UV}$, where
$\Lambda_{\rm UV}$ is some other cutoff scale associated with new
physics). This means that their effects at energy scales $E$ ordinarily
accessible to experiment are typically suppressed by powers of
${E/\MPlanck}$ (or $E/\Lambda_{\rm UV}$). For energies
$E\lsim 1$ TeV, the consequences of nonrenormalizable interactions are
therefore usually far too small to be interesting.
Still, there are several reasons why one may need to include
nonrenormalizable contributions to supersymmetric Lagrangians. First,
some very rare processes (like proton decay) might only be described using
an effective MSSM Lagrangian that includes nonrenormalizable terms.
Second, one may be interested in understanding physics at very high
energy scales where the suppression associated with nonrenormalizable
terms is not enough to stop them from being important. For example, this
could be the case in the study of the very early universe, or in
understanding how additional gauge symmetries get broken. Third, the
nonrenormalizable interactions may play a crucial role in understanding
how supersymmetry breaking is transmitted to the MSSM. Finally, it is
sometimes useful to treat strongly coupled supersymmetric gauge theories
using nonrenormalizable effective Lagrangians, in the same way that
chiral effective Lagrangians are used to study hadron physics in QCD.
Unfortunately, we will not be able to treat these subjects in any sort of
systematic way. Instead, we will merely sketch a few of the key elements
that go into defining a nonrenormalizable supersymmetric Lagrangian.
More detailed treatments and pointers to the literature may be found for example in
refs.~\cite{WessBaggerbook,Westbook,BailinLovebook,Buchbinder:1998qv,
Weinbergbook,Freedman:2012zz,Shifman:2012zz,Nillesreview,GGRS,VNreview,Bertolini:2013via}.
A nonrenormalizable gaugeinvariant theory involving chiral and vector
superfields can be constructed as:
\beq
{\cal L} &=& \left [ K(\Phi_i, \, \tilde \Phi^{*j}) \right ]_D
+ \left (
\left [\frac{1}{4} f_{ab}(\Phi_i) \widehat{\cal W}^{a\alpha} \widehat{\cal W}^b_\alpha
\,+\, W(\Phi_i)
\right ]_F
+ {\rm c.c.}
\right ),
\label{eq:Lnonren}
\eeq
where, in order to preserve supergauge invariance, we define
\beq
\tilde \Phi^{*j} \equiv \bigl ( \Phi^{*} e^{V} \bigr )^j ,
\eeq
with $V = 2 g_a T^a V^a = 2 T^a \widehat V^a$ as above,
and the hatted normalization of the fieldstrength superfields indicated in
(\ref{eq:holonormW}) has been used.
Equation (\ref{eq:Lnonren})
depends on couplings encoded in three functions of the
superfields:
\begin{itemize}
\item[$\bullet$]
The superpotential $W$, which we have already encountered in the
special case of renormalizable supersymmetric Lagrangians. More
generally, it can be an arbitrary
holomorphic function of the chiral superfields treated as complex
variables, and
must be invariant under the gauge symmetries of the theory, and has
dimension [mass]$^3$.
\item[$\bullet$]
The {\em K\"ahler potential} $K$.
Unlike the
superpotential, the K\"ahler potential is a function of both chiral and
antichiral superfields, and includes the vector superfields in such a way
as to be supergauge invariant. It is real, and has dimension
[mass]$^2$. In the special
case of renormalizable theories, we did not have to discuss the K\"ahler
potential explicitly, because at treelevel it is always just
$K = \Phi_i \tilde \Phi^{i*}$.
Any additive part of $K$ that is a chiral (or antichiral)
superfield does not contribute to the
action, since the $D$term of a chiral superfield is a total derivative
on spacetime.
\item[$\bullet$]
The {\it gauge kinetic function} $f_{ab}(\Phi_i)$. Like the
superpotential, it is itself a chiral superfield, and is a holomorphic
function of the chiral superfields
treated as
complex variables. It is dimensionless and symmetric under interchange of
its two indices $a,b$, which run over the adjoint representations of the
simple and Abelian component gauge groups of the model. For the
nonAbelian components of the gauge
group, it is always just proportional to $\delta_{ab}$, but if there are
two or more Abelian components, the gauge invariance of the fieldstrength
superfield [see eqs.~(\ref{eq:Walphagione})(\ref{eq:Walphagitwo})]
allows kinetic mixing so that
$f_{ab}$ is not proportional to $\delta_{ab}$ in general.
In the special case of renormalizable
supersymmetric Lagrangians at tree level, it is
independent of the chiral superfields, and just
equal to $f_{ab} = \delta_{ab}(1/g_a^2  i \Theta_a/8 \pi^2)$,
(for fewer than two Abelian components in the gauge group). More generally, it also
encodes the nonrenormalizable couplings of the gauge supermultiplets
to the chiral supermultiplets.
\end{itemize}
It should be emphasized that eq.~(\ref{eq:Lnonren}) is still not the
most general nonrenormalizable supersymmetric Lagrangian,
even if one restricts to chiral and
gauge vector superfields. One can also include chiral, antichiral, and spacetime
derivatives acting on the superfields, so that for example the K\"ahler potential
can be generalized to include dependence on $D_\alpha \Phi_i$,
$\Dcon_{\dot\alpha} \Phi^{*i}$, $DD\Phi_i$, $\Dcon\Dcon\Phi^{*i}$, etc.
Such terms typically have an extra suppression at low energies
compared to terms without derivatives,
because of the positive mass dimension of the chiral covariant derivatives.
I will not discuss these possibilities below, but will only
make a remark on how supergauge invariance is maintained.
The chiral covariant derivative of a chiral superfield,
$D_\alpha \Phi_i$ is not gauge covariant unless $\Phi_i$ is a gauge singlet;
the ``covariant" in the name refers to supersymmetry transformations, not gauge
transformations. However, one can define
a ``gauge covariant chiral covariant" derivative $\nabla_\alpha$,
whose action on a chiral superfield $\Phi$ is defined by:
\beq
\nabla_\alpha \Phi &\equiv&
e^{V} D_\alpha (e^{V} \Phi) ,
\eeq
where the representation indices $i$ are suppressed.
From eq.~(\ref{eq:kentuckyavenue}), the supergauge transformation
for $e^{V}$ is
\beq
e^{V} \rightarrow e^{i \Omega} e^{V} e^{i \Omega^\dagger},
\eeq
so that
\beq
e^{V} D_\alpha (e^{V} \Phi) &\rightarrow&
e^{i \Omega} e^{V} e^{i \Omega^\dagger} D_\alpha (e^{i \Omega^\dagger} e^{V} \Phi)
\>=\> e^{i \Omega} e^{V} D_\alpha (e^{V} \Phi),
\eeq
where the equality follows from the fact that $\Omega^\dagger$ is antichiral,
and thus ignored by $D_\alpha$. This is the correct covariant transformation law
under supergauge transformations.
So, using $\nabla_\alpha \Phi_i$
as a building block instead of $D_\alpha \Phi_i$, one can
maintain supergauge covariance along with manifest supersymmetry.
Similarly, one can define building blocks:
\beq
\overline\nabla_{\dot\alpha} \Phi^* &\equiv& \Dcon_{\dot\alpha} (\Phi^*e^V ) e^{V}
,
\\
\nabla\nabla \Phi &\equiv& e^{V} DD (e^V \Phi)
\\
\overline\nabla\overline\nabla \Phi^* &\equiv& \Dcon\Dcon (\Phi^*e^{V} ) e^{V}
\eeq
which each have covariant supergauge transformation rules.
Returning to the globally supersymmetric nonrenormalizable theory defined
by eq.~(\ref{eq:Lnonren}), with no extra derivatives,
the part of the Lagrangian coming from the superpotential is
\beq
\left [ W(\Phi_i) \right ]_F = W^i F_i  \frac{1}{2} W^{ij} \psi_i \psi_j ,
\eeq
with
\beq
W^i = \frac{\delta W}{\delta\Phi_i}
\biggl _{\Phi_i \rightarrow \phi_i},
\qquad\qquad
W^{ij} = \frac{\delta^{2} W}{\delta\Phi_i\delta \Phi_j}
\biggl _{\Phi_i \rightarrow \phi_i}
,
\eeq
where the superfields have been replaced by their scalar components after
differentiation.
[Compare eqs.~(\ref{tryint}), (\ref{expresswij}), (\ref{wiwiwi}) and the
surrounding discussion.] After integrating out the auxiliary
fields $F_i$, the part of the scalar potential coming from the
superpotential is
\beq
V = W^i W_j^* (K^{1})_i^j,
\eeq
where $K^{1}$ is the inverse matrix of the K\"ahler metric:
\beq
K^i_j = \frac{\delta^2 K}{\delta \Phi_i \delta \tilde \Phi^{*j}} \biggl _{\Phi_i
\rightarrow \phi_i,\> \tilde \Phi^{*i}
\rightarrow \phi^{*i}}.
\eeq
More generally, the whole component field Lagrangian after integrating
out the auxiliary fields is determined in
terms of the functions $W$, $K$ and $f_{ab}$ and their derivatives with
respect to the chiral superfields, with the remaining chiral superfields
replaced by their scalar components.
The complete form of this is straightforward to evaluate, but somewhat
complicated. In supergravity, there are additional contributions, some of
which are discussed in section \ref{subsec:origins.sugra} below.
\subsection{$R$ symmetries\label{Rsymmetry}}
\setcounter{equation}{0}
\setcounter{footnote}{2}
Some supersymmetric Lagrangians are also invariant under a global
$U(1)_R$ symmetry. The defining feature of a continuous $R$ symmetry is
that the anticommuting coordinates $\theta$ and $\theta^\dagger$
transform under it with charges $+1$ and $1$ respectively, so
\beq
\theta \> \rightarrow \> e^{i \alpha} \theta,
\qquad\qquad
\theta^\dagger \> \rightarrow \> e^{i \alpha} \theta^\dagger
\label{eq:Rtranstheta}
\eeq
where $\alpha$ parameterizes the global $R$ transformation. It follows that
\beq
\hat Q \> \rightarrow \> e^{i \alpha} \hat Q,
\qquad\qquad
\hat Q^\dagger \> \rightarrow \> e^{i \alpha} \hat Q^\dagger,
\eeq
which in turn implies that the supersymmetry generators have
$U(1)_R$ charges $1$ and $+1$, and so
they do not commute
with the $R$ symmetry generator:
\beq
[R, Q] = Q, \qquad\qquad
[R, Q^\dagger] = Q^\dagger
\eeq
Thus the distinct components within a superfield always have different $R$ charges.
If the theory is invariant under an $R$ symmetry, then each superfield $S(x,\theta,\theta^\dagger)$ can be assigned an $R$ charge, denoted $r_S$, defined by its transformation rule
\beq
S(x,\,\theta,\,\theta^\dagger) &\rightarrow&
e^{i r_S \alpha} S(x,\, e^{i \alpha} \theta,\, e^{i\alpha}\theta^\dagger).
\eeq
The $R$ charge of a product of superfields is the sum of the individual $R$ charges.
For a chiral superfield $\Phi$ with $R$ charge $r_\Phi$, the $\phi$, $\psi$, and $F$
components transform with charges $r_\Phi$, $r_\Phi 1$, and $r_\Phi2$, respectively:
\beq
\phi \rightarrow e^{ir_\Phi\alpha} \phi,\qquad\quad
\psi \rightarrow e^{i(r_\Phi  1)\alpha} \psi,\qquad\quad
F \rightarrow e^{i(r_\Phi  2)\alpha} F.
\eeq
The components of $\Phi^*$ carry the opposite charges.
Gauge vector superfields will always have vanishing $U(1)_R$ charge, since they are
real. It follows that the components that are nonzero in WessZumino
gauge transform as:
\beq
A^\mu \rightarrow A^\mu,\qquad\quad
\lambda \rightarrow e^{i\alpha} \lambda,\qquad\quad
D \rightarrow D.\qquad\quad
\label{eq:Rchargesvector}
\eeq
and so have $U(1)_R$ charges $0$, $1$, and $0$ respectively.
Therefore, a Majorana gaugino mass term $\frac{1}{2} M_\lambda \lambda\lambda$,
which will appear when supersymmetry is broken, also always breaks the
continuous $U(1)_R$ symmetry.
The superspace integration measures $d^2\theta$ and $d^2\theta^\dagger$
and the chiral covariant derivatives $D_\alpha$ and
$\Dcon_{\dot\alpha}$ carry $U(1)_R$ charges $2$, $+2$, $1$, and $+1$
respectively. It follows that the gauge fieldstrength superfield
${\cal W}_\alpha$ carries $U(1)_R$ charge $+1$. (The $U(1)_R$ charges of various
objects are collected in Table \ref{table:Rcharges}.)
It is then not hard to check that all
supersymmetric Lagrangian terms found above that involve gauge superfields are
automatically and necessarily $R$symmetric, including the couplings to
chiral superfields. This is also true of the canonical K\"ahler potential
contribution.%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\arraystretch}{1.4}
\begin{table}[tb]
\begin{center}
%
\begin{tabular}{cccccccccccccc}
\hline
& $\theta_\alpha$
& $\theta^\dagger_{\dot\alpha}$
& $d^2\theta$
& $\hat Q_\alpha$
& $D_\alpha$
& ${\cal W}_\alpha$
& $A^\mu$
& $\lambda_\alpha$
& $D$
& $W$
& $\phi$
& $\psi_\alpha$
& $F_\Phi$
\\ \hline
$U(1)_R$ charge
& $+1$
& $1$
& $2$
& $1$
& $1$
& $+1$
& $0$
& $+1$
& $0$
& $+2$
& $r_\Phi$
& $r_\Phi1$
& $r_\Phi2$
\\ \hline
\end{tabular}
%
\caption{$U(1)_R$ charges of various objects.\label{table:Rcharges}}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
However, the superpotential $W(\Phi_i)$ must carry
$U(1)_R$ charge $+2$ in order to conserve the $R$ symmetry,
and this is certainly not
automatic, and often not true. As a simple toy example,
with a single gaugesinglet superfield $\Phi$, the allowed renormalizable
terms in the superpotential are $W(\Phi) = L \Phi + \frac{M}{2} \Phi^2 +
\frac{y}{6} \Phi^3$. If one wants to impose a continuous $U(1)_R$ symmetry,
then one can have at most one of these terms; $L$ is allowed only if
$r_\Phi = 2$, $M$ is allowed only if $r_\Phi = 1$, and $y$ is allowed
only if $r_\Phi = 2/3$. The MSSM superpotential
does turn out to conserve a global $U(1)_R$ symmetry, but it is both anomalous
and broken by Majorana gaugino masses and other supersymmetry breaking effects.
Since continuous $R$ symmetries do not commute
with supersymmetry, and are not conserved in the MSSM after anomalies
and supersymmetry breaking effects are
included, one might wonder why they are considered at all.
Perhaps the most important answer to this involves the role of $U(1)_R$
symmetries in models that break global supersymmetry spontaneously, as
will be discussed in section \ref{subsec:origins.Fterm} below.
It is also possible to extend the particle content of the MSSM in such a way as to
preserve a continuous, nonanomalous $U(1)_R$ symmetry,
but at the cost of introducing Dirac gaugino masses and extra Higgs
fields \cite{Kribs:2007ac}.
Another possibility is that a superpotential could
have a discrete $Z_n$ $R$ symmetry, which can be
obtained by restricting the transformation parameter $\alpha$ in
eqs.~(\ref{eq:Rtranstheta})(\ref{eq:Rchargesvector}) to integer
multiples of $2\pi/n$. The $Z_n$ $R$ charges of all fields are then
integers modulo $n$. However, note that the case $n=2$ is always trivial, in the
sense that any $Z_2$ $R$ symmetry is exactly equivalent to a
corresponding ordinary (non$R$) $Z_2$ symmetry under which all components of
each supermultiplet transform the same way. This is because when $\alpha$
is an integer multiple of $\pi$, then both $\theta$ and $\theta^\dagger$
always just transform by changing sign, which means that fermionic fields just
change sign relative to their bosonic partners. The number of fermionic fields
in any Lagrangian term, in any theory, is always even, so the extra sign
change for fermionic fields has no effect.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Soft supersymmetry breaking interactions}\label{sec:soft}
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A realistic phenomenological model must contain supersymmetry breaking.
From a theoretical perspective, we expect that supersymmetry, if it exists
at all, should be an exact symmetry that is broken spontaneously. In other
words, the underlying model should have a Lagrangian density that is
invariant under supersymmetry, but a vacuum state that is not. In this
way, supersymmetry is hidden at low energies in a manner analogous
to the fate of the electroweak symmetry in the ordinary Standard Model.
Many models of spontaneous symmetry breaking have indeed been proposed and
we will mention the basic ideas of some of them in section
\ref{sec:origins}. These always involve extending the MSSM to include new
particles and interactions at very high mass scales, and there is no
consensus on exactly how this should be done. However, from a practical
point of view, it is extremely useful to simply parameterize our ignorance
of these issues by just introducing extra terms that break supersymmetry
explicitly in the effective MSSM Lagrangian. As was argued in the
Introduction, the supersymmetrybreaking couplings should be soft (of
positive mass dimension) in order to be able to naturally maintain a
hierarchy between the electroweak scale and the Planck (or any other very
large) mass scale. This means in particular that dimensionless
supersymmetrybreaking couplings should be absent.
The possible soft supersymmetrybreaking terms in the Lagrangian of a
general theory are
\beq
\lagr_{\rm soft}\! &=& \!
\left (
\half M_a\, \lambda^a\lambda^a
+ {1\over 6}a^{ijk} \phi_i\phi_j\phi_k
+ \half b^{ij} \phi_i\phi_j
+ t^i \phi_i \right )
+ \conj
 (m^2)_j^i \phi^{j*} \phi_i , \phantom{xxxx}
\label{lagrsoft}
\\
\lagr_{{\rm maybe}\>\,{\rm soft}}\! &=& \!
{1\over 2}c_i^{jk} \phi^{*i}\phi_j\phi_k + \conj
\label{lagrsoftprime}
\eeq
They consist of gaugino masses $M_a$ for each gauge group, scalar
squaredmass terms $(m^2)_i^j$ and $b^{ij}$, and (scalar)$^3$ couplings
$a^{ijk}$ and $c_i^{jk}$, and ``tadpole" couplings $t^i$. The last of
these requires $\phi_i$ to be a gauge singlet, and so $t^i$ does not occur
in the MSSM. One might wonder why we have not included possible soft mass
terms for the chiral supermultiplet fermions, like ${\cal L} = \half
m^{ij} \psi_i \psi_j + {\rm c.c.}$~~Including such terms would be
redundant; they can always be absorbed into a redefinition of the
superpotential and the terms $(m^2)_j^{i}$ and $c_i^{jk}$.
It has been shown rigorously that a softly broken supersymmetric theory
with $\lagr_{\rm soft}$ as given by eq.~(\ref{lagrsoft}) is indeed free of
quadratic divergences in quantum corrections to scalar masses, to all
orders in perturbation theory \cite{softterms}. The situation is slightly
more subtle if one tries to include the nonholomorphic (scalar)$^3$
couplings in $\lagr_{{\rm maybe}\>\,{\rm soft}}$. If any of the chiral
supermultiplets in the theory are singlets under all gauge symmetries,
then nonzero $c_i^{jk}$ terms can lead to quadratic divergences, despite
the fact that they are formally soft. Now, this constraint need not apply
to the MSSM, which does not have any gaugesinglet chiral supermultiplets.
Nevertheless, the possibility of $c_i^{jk}$ terms is nearly always
neglected. The real reason for this is that it is difficult to
construct models of spontaneous supersymmetry breaking in which the
$c_i^{jk}$ are not negligibly small.
In the special case of a theory that has
chiral supermultiplets that are singlets or
in the adjoint representation of a simple factor of the gauge group,
then there are also possible soft supersymmetrybreaking Dirac mass
terms between the corresponding fermions $\psi_a$ and the gauginos
\cite{Polchinski:1982an}\cite{Fox:2002bu}:
\beq
{\cal L} \,=\, M_{\rm Dirac}^a \lambda^a \psi_a + {\rm c.c.}
\label{eq:Diracgauginos}
\eeq
This is not relevant for the MSSM with minimal field
content, which does not have adjoint representation chiral
supermultiplets. Therefore, equation (\ref{lagrsoft}) is usually taken to
be the general form of the soft supersymmetrybreaking Lagrangian. For
some interesting exceptions, see
refs.~\cite{Polchinski:1982an}\cite{Plehn:2008ae}.
The terms in $\lagr_{\rm soft}$ clearly do break supersymmetry,
because they involve
only scalars and gauginos and not their respective superpartners. In fact,
the soft terms in $\lagr_{\rm soft}$ are capable of giving masses to all
of the scalars and gauginos in a theory, even if the gauge bosons and
fermions in chiral supermultiplets are massless (or relatively light). The
gaugino masses $M_a$ are always allowed by gauge symmetry. The $(m^2)_j^i$
terms are allowed for $i,j$ such that $\phi_i$, $\phi^{j*}$ transform in
complex conjugate representations of each other under all gauge
symmetries; in particular this is true of course when $i=j$, so every
scalar is eligible to get a mass in this way if supersymmetry is broken.
The remaining soft terms may or may not be allowed by the symmetries.
The $a^{ijk}$, $b^{ij}$, and $t^i$
terms have the same form as the $y^{ijk}$, $M^{ij}$, and $L^i$ terms in
the superpotential [compare eq.~(\ref{lagrsoft}) to
eq.~(\ref{superpotentialwithlinear})
or eq.~(\ref{superpot})], so they will each be allowed by gauge
invariance if and only if a corresponding superpotential term is allowed.
The Feynman diagram interactions corresponding to the allowed soft terms
in eq.~(\ref{lagrsoft}) are shown in Figure~\ref{fig:soft}.%
\begin{figure}
\begin{center}
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\Text(39.6,13.2)[c]{(a)}
\end{picture}
%
\hspace{1.2cm}
%
\begin{picture}(72,56)(0,4)
\SetScale{1.1}
\SetWidth{0.85}
\DashLine(0,12)(36,12){4}
\DashLine(72,12)(36,12){4}
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\Text(0.5,20.8)[c]{$i$}
\Text(77.5,21.2)[c]{$j$}
\Text(39.6,13.2)[c]{(b)}
\end{picture}
%
\hspace{1.2cm}
%
\begin{picture}(72,56)(0,4)
\SetScale{1.1}
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\DashLine(0,12)(36,12){4}
\DashLine(72,12)(36,12){4}
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\ArrowLine(55.01,12)(55,12)
\Line(33,9)(39,15)
\Line(39,9)(33,15)
\Text(0.5,20.8)[c]{$i$}
\Text(77.5,21.2)[c]{$j$}
\Text(39.6,13.2)[c]{(c)}
\end{picture}
%
\hspace{1.2cm}
%
\begin{picture}(66,56)(0,4)
\SetScale{1.1}
\SetWidth{0.85}
\DashLine(33,52.5)(33,12){4}
\DashLine(0,0)(33,12){4}
\DashLine(66,0)(33,12){4}
\ArrowLine(33,32.2501)(33,32.25)
\ArrowLine(16.5,6)(16.5165,6.006)
\ArrowLine(49.5,6)(49.4835,6.006)
\Text(1,10.7)[c]{$j$}
\Text(70,9.7)[c]{$k$}
\Text(30.5,53)[c]{$i$}
\Text(36.3,13.2)[c]{(d)}
\end{picture}
\end{center}
\caption{Soft supersymmetrybreaking terms:
(a) Gaugino mass $M_a$;
(b) nonholomorphic scalar squared mass $(m^2)_j^i$;
(c) holomorphic scalar squared mass $b^{ij}$;
and
(d) scalar cubic coupling $a^{ijk}$.
\label{fig:soft}}
\end{figure}
For each of
the interactions in Figures~\ref{fig:soft}a,c,d there is another with all
arrows reversed, corresponding to the complex conjugate term in the
Lagrangian. We will apply these general results to the specific case of
the MSSM in the next section.
\section{The Minimal Supersymmetric Standard Model}\label{sec:mssm}
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In sections \ref{sec:susylagr} and \ref{sec:soft}, we have found a general
recipe for constructing Lagrangians for softly broken supersymmetric
theories. We are now ready to apply these general results to the MSSM. The
particle content for the MSSM was described in the Introduction. In this
section we will complete the model by specifying the superpotential and
the soft supersymmetrybreaking terms.
\subsection{The superpotential and supersymmetric
interactions}\label{subsec:mssm.superpotential}
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The superpotential for the MSSM is
\beq
W_{\rm MSSM} =
\sbar u {\bf y_u} Q H_u 
\sbar d {\bf y_d} Q H_d 
\sbar e {\bf y_e} L H_d +
\mu H_u H_d \> .
\label{MSSMsuperpot}
\eeq
The objects $H_u$, $H_d$, $Q$, $L$, $\sbar u$, $\sbar d$, $\sbar e$
appearing here are chiral superfields corresponding to the chiral
supermultiplets in Table \ref{tab:chiral}. (Alternatively, they can be
just thought of as the corresponding scalar fields, as was done in section
\ref{sec:susylagr}, but we prefer not to put the tildes on $Q$, $L$,
$\sbar u$, $\sbar d$, $\sbar e$ in order to reduce clutter.) The
dimensionless Yukawa coupling parameters ${\bf y_u}, {\bf y_d}, {\bf y_e}$
are 3$\times 3$ matrices in family space. All of the gauge [$SU(3)_C$
color and $SU(2)_L$ weak isospin] and family indices in
eq.~(\ref{MSSMsuperpot}) are suppressed. The ``$\mu$ term", as it is
traditionally called, can be written out as $\mu (H_u)_\alpha (H_d)_\beta
\epsilon^{\alpha\beta}$, where $\epsilon^{\alpha\beta}$ is used to tie
together $SU(2)_L$ weak isospin indices $\alpha,\beta=1,2$ in a
gaugeinvariant way. Likewise, the term $\sbar u {\bf y_u} Q H_u$ can be
written out as $\sbar u^{ia}\, {({\bf y_u})_i}^j\, Q_{j\alpha a}\,
(H_u)_\beta \epsilon^{\alpha \beta}$, where $i=1,2,3$ is a family index,
and $a=1,2,3$ is a color index which is lowered (raised) in the $\bf 3$
($\bf \overline 3$) representation of $SU(3)_C$.
The $\mu$ term in eq.~(\ref{MSSMsuperpot}) is the supersymmetric version
of the Higgs boson mass in the Standard Model. It is unique, because terms
$H_u^* H_u$ or $H_d^* H_d$ are forbidden in the superpotential, which must
be holomorphic in the chiral superfields (or equivalently in the scalar
fields) treated as complex variables, as shown in section
\ref{subsec:susylagr.chiral}. We can also see from the form of
eq.~(\ref{MSSMsuperpot}) why both $H_u$ and $H_d$ are needed in order to
give Yukawa couplings, and thus masses, to all of the quarks and leptons.
Since the superpotential must be holomorphic, the $\sbar u Q H_u $ Yukawa
terms cannot be replaced by something like $\sbar u Q H_d^*$. Similarly,
the $\sbar d Q H_d$ and $\sbar e L H_d$ terms cannot be replaced by
something like $\sbar d Q H_u^*$ and $\sbar e L H_u^*$. The analogous
Yukawa couplings would be allowed in a general nonsupersymmetric two
Higgs doublet model, but are forbidden by the structure of supersymmetry.
So we need both $H_u$ and $H_d$, even without invoking the argument based
on anomaly cancellation mentioned in the Introduction.
The Yukawa matrices determine the current masses and CKM mixing angles of
the ordinary quarks and leptons, after the neutral scalar components of
$H_u$ and $H_d$ get VEVs. Since the top quark, bottom quark and tau lepton
are the heaviest fermions in the Standard Model, it is often useful to
make an approximation that only the $(3,3)$ family components of each of
${\bf y_u}$, ${\bf y_d}$ and ${\bf y_e}$ are important:
\beq
{\bf y_u} \approx \pmatrix{0&0&0\cr 0&0&0 \cr 0&0&y_t},\qquad
{\bf y_d} \approx \pmatrix{0&0&0\cr 0&0&0 \cr 0&0&y_b},\qquad
{\bf y_e} \approx \pmatrix{0&0&0\cr 0&0&0 \cr 0&0&y_\tau}.\>\>{}
\label{heavytopapprox}
\eeq
In this limit, only the third family and Higgs fields contribute to the
MSSM superpotential. It is instructive to write the superpotential in
terms of the separate $SU(2)_L$ weak isospin components [$Q_3 = (t\, b)$,
$L_3 = (\nu_\tau\, \tau)$, $H_u = (H_u^+\, H_u^0)$, $H_d = (H_d^0\,
H_d^)$, $\sbar u_3 = \sbar t$, $\sbar d_3 = \sbar b$, $\sbar e_3 = \sbar
\tau$], so:
\beq
W_{\rm MSSM}\! &\approx & \!
y_t (\sbar t t H_u^0  \sbar t b H_u^+) 
y_b (\sbar b t H_d^  \sbar b b H_d^0) 
y_\tau (\sbar \tau \nu_\tau H_d^  \sbar \tau \tau H_d^0)
\> \nonumber \\
&& +
\mu (H_u^+ H_d^  H_u^0 H_d^0).
\label{Wthird}
\eeq
The minus signs inside the parentheses appear because of the antisymmetry
of the $\epsilon^{\alpha\beta}$ symbol used to tie up the $SU(2)_L$
indices. The other minus signs in eq.~(\ref{MSSMsuperpot}) were chosen
(as a convention) so
that the terms $y_t \sbar t t H_u^0$, $y_b \sbar b b H_d^0$, and $y_\tau
\sbar \tau \tau H_d^0$, which will become the top, bottom and tau masses
when $H_u^0$ and $H_d^0$ get VEVs, each have overall positive signs in
eq.~(\ref{Wthird}).
Since the Yukawa interactions $y^{ijk}$ in a general supersymmetric theory
must be completely symmetric under interchange of $i,j,k$, we know that
${\bf y_u}$, ${\bf y_d}$ and ${\bf y_e}$ imply not only Higgsquarkquark
and Higgsleptonlepton couplings as in the Standard Model, but also
squarkHiggsinoquark and sleptonHiggsinolepton interactions. To
illustrate this, Figures~{\ref{fig:topYukawa}}a,b,c show some of the
interactions involving the topquark Yukawa coupling $y_t$.%
\begin{figure}
\begin{center}
\begin{picture}(66,60)(0,0)
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\Text(0,11)[c]{$t_L$}
\Text(65.5,11)[c]{$t_R^\dagger$}
\Text(22.5,51)[c]{$H_u^0$}
\Text(33,13)[c]{(a)}
\end{picture}
%
\hspace{1.9cm}
%
\begin{picture}(66,60)(0,0)
\SetWidth{0.85}
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\Text(0,11)[c]{$\stilde t_L$}
\Text(65.5,11)[c]{$t_R^\dagger$}
\Text(22.7,51)[c]{$\widetilde H_u^0$}
\Text(33,13)[c]{(b)}
\end{picture}
%
\hspace{1.9cm}
%
\begin{picture}(66,60)(0,0)
\SetWidth{0.85}
\ArrowLine(0,0)(33,12)
\DashLine(66,0)(33,12){4}
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\ArrowLine(49.5,6)(49.4835,6.006)
\Text(0,11)[c]{$t_L$}
\Text(65.5,11)[c]{$\stilde t_R^*$}
\Text(22.7,51)[c]{$\widetilde H_u^0$}
\Text(33,13)[c]{(c)}
\end{picture}
\end{center}
\caption{The topquark Yukawa coupling (a) and its ``supersymmetrizations"
(b), (c), all of strength~$y_t$.\label{fig:topYukawa}}
\end{figure}
Figure \ref{fig:topYukawa}a is the Standard Modellike coupling of the top
quark to the neutral complex scalar Higgs boson, which follows from the
first term in eq.~(\ref{Wthird}). For variety, we have used $t_L$ and
$t_R^\dagger$ in place of their synonyms $t$ and $\sbar t$
(see the discussion near the end of section
\ref{sec:notations}). In Figure~\ref{fig:topYukawa}b, we have the coupling
of the lefthanded top squark $\stilde t_L$ to the neutral higgsino field
${\stilde H}_u^0$ and righthanded top quark, while in
Figure~\ref{fig:topYukawa}c the righthanded top antisquark field (known
either as $\stilde {\sbar t}$ or $\stilde t_R^*$ depending on taste)
couples to ${\stilde H}^0_u$ and $t_L$. For each of the three
interactions, there is another with $H_u^0\rightarrow H_u^+$ and $t_L
\rightarrow b_L$ (with tildes where appropriate), corresponding to the
second part of the first term in eq.~(\ref{Wthird}). All of these
interactions are required by supersymmetry to have the same strength
$y_t$. These couplings are dimensionless and can be modified by the
introduction of soft supersymmetry breaking only through finite (and
small) radiative corrections, so this equality of interaction strengths is
also a prediction of softly broken supersymmetry. A useful mnemonic is
that each of Figures~{\ref{fig:topYukawa}}a,b,c can be obtained from any
of the others by changing two of the particles into their superpartners.
There are also scalar quartic interactions with strength proportional to
$y_t^2$, as can be seen from Figure~\ref{fig:dim0}c or the last term
in eq.~(\ref{ordpot}). Three of them are shown in Figure~{\ref{fig:stop}}.%
\begin{figure}
\begin{center}
\begin{picture}(56,55)(0,0)
\SetScale{1.1}
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\Text(5,10)[c]{$\tilde t_R^*$}
\Text(61.75,9)[c]{$\tilde t_R$}
\Text(4.25,49)[c]{$\tilde t_L$}
\Text(62,49)[c]{$\tilde t_L^*$}
\Text(27.5,12.1)[c]{(a)}
\end{picture}
%
\hspace{2.25cm}
%
\begin{picture}(56,55)(0,0)
\SetScale{1.1}
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\DashLine(0,0)(25,25){4}
\DashLine(50,0)(25,25){4}
\DashLine(0,50)(25,25){4}
\DashLine(50,50)(25,25){4}
\ArrowLine(12.5,12.5)(12.501,12.501)
\ArrowLine(37.499,12.501)(37.5,12.5)
\ArrowLine(12.499,37.501)(12.5,37.5)
\ArrowLine(37.5,37.5)(37.501,37.501)
\Text(4.5,10)[c]{$\tilde t_L$}
\Text(61,9)[c]{$\tilde t_L^*$}
\Text(8,49)[c]{$H_u^0$}
\Text(64.3,49)[c]{$H_u^{0*}$}
\Text(27.5,12.1)[c]{(b)}
\end{picture}
%
\hspace{2.25cm}
%
\begin{picture}(56,55)(0,0)
\SetScale{1.1}
\SetWidth{0.8}
\DashLine(0,0)(25,25){4}
\DashLine(50,0)(25,25){4}
\DashLine(0,50)(25,25){4}
\DashLine(50,50)(25,25){4}
\ArrowLine(12.5,12.5)(12.501,12.501)
\ArrowLine(37.499,12.501)(37.5,12.5)
\ArrowLine(12.499,37.501)(12.5,37.5)
\ArrowLine(37.5,37.5)(37.501,37.501)
\Text(4.5,10)[c]{$\tilde t_R^*$}
\Text(62,9)[c]{$\tilde t_R$}
\Text(8,49)[c]{$H_u^0$}
\Text(64.3,49)[c]{$H_u^{0*}$}
\Text(27.5,12.1)[c]{(c)}
\end{picture}
\end{center}
\caption{Some of the (scalar)$^4$ interactions with strength
proportional to $y_t^2$.
\label{fig:stop}}
\end{figure}
Using eq.~(\ref{ordpot}) and eq.~(\ref{Wthird}), one can see that
there are
five more, which can be obtained by replacing $\stilde t_L \rightarrow
\stilde b_L$ and/or $H_u^0 \rightarrow H_u^+$ in each vertex. This
illustrates the remarkable economy of supersymmetry; there are many
interactions determined by only a single parameter. In a similar way, the
existence of all the other quark and lepton Yukawa couplings in the
superpotential eq.~(\ref{MSSMsuperpot}) leads not only to
Higgsquarkquark and Higgsleptonlepton Lagrangian terms as in the
ordinary Standard Model, but also to squarkhiggsinoquark and
sleptonhiggsinolepton terms, and scalar quartic couplings [(squark)$^4$,
(slepton)$^4$, (squark)$^2$(slepton)$^2$, (squark)$^2$(Higgs)$^2$, and
(slepton)$^2$(Higgs)$^2$]. If needed, these can all be obtained in terms
of the Yukawa matrices $\bf y_u$, $\bf y_d$, and $\bf y_e$ as outlined
above.
However, the dimensionless interactions determined by the superpotential
are usually not the most important ones of direct interest for
phenomenology. This is because the Yukawa couplings are already known to
be very small, except for those of the third family (top, bottom, tau).
Instead, production and decay processes for superpartners in the MSSM are
typically dominated by the supersymmetric interactions of gaugecoupling
strength, as we will explore in more detail in sections \ref{sec:decays}
and \ref{sec:signals}. The couplings of the Standard Model gauge bosons
(photon, $W^\pm$, $Z^0$ and gluons) to the MSSM particles are determined
completely by the gauge invariance of the kinetic terms in the Lagrangian.
The gauginos also couple to (squark, quark) and (slepton, lepton) and
(Higgs, higgsino) pairs as illustrated in the general case in
Figure~\ref{fig:gauge}g,h and the first two terms in the second line in
eq.~(\ref{gensusylagr}). For instance, each of the squarkquarkgluino
couplings is given by $\sqrt{2} g_3 (\stilde q \, T^{a} q \stilde g +
\conj)$ where $T^a = \lambda^a/2$ ($a=1\ldots 8$) are the matrix
generators for $SU(3)_C$. The Feynman diagram for this interaction is
shown in Figure~\ref{fig:gaugino}a.%
\begin{figure}
\begin{center}
\begin{picture}(66,60)(0,0)
\Photon(0,0)(33,12){1.8}{4}
\SetWidth{0.85}
\ArrowLine(33,12)(0,0)
\ArrowLine(33,12)(66,0)
\DashLine(33,50.5)(33,12){4}
\ArrowLine(33,32.2501)(33,32.25)
\Text(0,11)[c]{$\stilde g$}
\Text(65.5,10)[c]{$q$}
\Text(40,51)[c]{$\stilde q$}
\Text(33,13)[c]{(a)}
\end{picture}
%
\hspace{1.75cm}
%
\begin{picture}(124,60)(0,0)
\Photon(0,0)(33,12){1.8}{4}
\SetWidth{0.85}
\ArrowLine(33,12)(0,0)
\ArrowLine(33,12)(66,0)
\DashLine(33,50.5)(33,12){4}
\ArrowLine(33,32.2501)(33,32.25)
\Text(0,11)[c]{$\stilde W$}
\Text(63.5,11)[l]{$q_L$, $\ell_L$, $\stilde H_u$, $\stilde H_d$}
\Text(38,51)[l]{$\stilde q_L$, $\stilde \ell_L$, $H_u$, $H_d$}
\Text(33,13)[c]{(b)}
\end{picture}
%
\hspace{1.8cm}
%
\begin{picture}(124,60)(0,0)
\Photon(0,0)(33,12){1.8}{4}
\SetWidth{0.85}
\ArrowLine(33,12)(0,0)
\ArrowLine(33,12)(66,0)
\DashLine(33,50.5)(33,12){4}
\ArrowLine(33,32.2501)(33,32.25)
\Text(0,11)[c]{$\stilde B$}
\Text(63.5,11)[l]{$q$, $\ell$, $\stilde H_u$, $\stilde H_d$}
\Text(38,51)[l]{$\stilde q$, $\stilde \ell$, $H_u$, $H_d$}
\Text(33,13)[c]{(c)}
\end{picture}
\end{center}
\caption{Couplings of the gluino, wino, and bino to MSSM (scalar,
fermion) pairs.
\label{fig:gaugino}}
\end{figure}
In Figures~\ref{fig:gaugino}b,c we
show in a similar way the couplings of (squark, quark), (lepton, slepton)
and (Higgs, higgsino) pairs to the winos and bino, with strengths
proportional to the electroweak gauge couplings $g$ and $g^\prime$
respectively. For each of these diagrams, there is another with all arrows
reversed. Note that the winos only couple to the lefthanded squarks and
sleptons, and the (lepton, slepton) and (Higgs, higgsino) pairs of course
do not couple to the gluino. The bino coupling to each (scalar, fermion)
pair is also proportional to the weak hypercharge $Y$ as
given in Table \ref{tab:chiral}.
The interactions shown in Figure~\ref{fig:gaugino} provide, for example,
for decays $\stilde q \rightarrow q\stilde g$ and $\stilde q \rightarrow
\stilde W q^\prime$ and $\stilde q \rightarrow \stilde B q$ when the final
states are kinematically allowed to be onshell. However, a complication
is that the $\stilde W$ and $\stilde B$ states are not mass eigenstates,
because of splitting and mixing due to electroweak symmetry breaking, as
we will see in section \ref{subsec:MSSMspectrum.inos}.
There are also various scalar quartic interactions in the MSSM that are
uniquely determined by gauge invariance and supersymmetry, according to
the last term in eq.~(\ref{fdpot}), as illustrated in
Figure~\ref{fig:gauge}i. Among them are (Higgs)$^4$ terms proportional to
$g^2$ and $g^{\prime 2}$ in the scalar potential. These are the direct
generalization of the last term in the Standard Model Higgs potential,
eq.~(\ref{higgspotential}), to the case of the MSSM. We will have occasion
to identify them explicitly when we discuss the minimization of the MSSM
Higgs potential in section \ref{subsec:MSSMspectrum.Higgs}.
The dimensionful couplings in the supersymmetric part of the MSSM
Lagrangian are all dependent on $\mu$. Using the general result of
eq.~(\ref{lagrchiral}), $\mu$ provides for higgsino fermion
mass terms
\beq
\lagr_{\mbox{higgsino mass}}= \mu (\stilde H_u^+ \stilde H_d^  \stilde
H_u^0 \stilde
H_d^0)+ \conj,
\label{poody}
\eeq
as well as Higgs squaredmass terms in the scalar potential
\beq
\lagr_{\mbox{supersymmetric Higgs mass}} \,=\, \mu^2
\bigl
(
H_u^0^2 + H_u^+^2 + H_d^0^2 + H_d^^2 \bigr ).
\label{movie}
\eeq
Since eq.~(\ref{movie}) is nonnegative with a minimum at
$H_u^0 = H_d^0 = 0$, we cannot understand
electroweak symmetry breaking without including a negative
supersymmetrybreaking squaredmass soft term for the Higgs scalars. An
explicit treatment of the Higgs scalar potential will therefore have to
wait until we have introduced the soft terms for the MSSM. However, we can
already see a puzzle: we expect that $\mu$ should be roughly of order
$10^2$ or $10^3$ GeV, in order to allow a Higgs VEV of order 174 GeV
without too much miraculous cancellation between $\mu^2$ and the
negative soft squaredmass terms that we have not written down yet. But
why should $\mu^2$ be so small compared to, say, $\MPlanck^2$, and in
particular why should it be roughly of the same order as $m^2_{\rm soft}$?
The scalar potential of the MSSM seems to depend on two types of
dimensionful parameters that are conceptually quite distinct, namely the
supersymmetryrespecting mass $\mu$ and the supersymmetrybreaking soft
mass terms. Yet the observed value for the electroweak breaking scale
suggests that without miraculous cancellations, both of these apparently
unrelated mass scales should be within an order of magnitude or so of 100
GeV. This puzzle is called ``the $\mu$ problem". Several different
solutions to the $\mu$ problem have been proposed, involving extensions of
the MSSM of varying intricacy. They all work in roughly the same way; the
$\mu$ term is required or assumed to be absent at treelevel before
symmetry breaking, and then it arises from the VEV(s) of some new
field(s). These VEVs are in turn determined by minimizing a potential that
depends on soft supersymmetrybreaking terms. In this way, the value of
the effective parameter $\mu$ is no longer conceptually distinct from the
mechanism of supersymmetry breaking; if we can explain why $m_{\rm soft}
\ll \MPlanck$, we will also be able to understand why $\mu$ is of the same
order. In sections \ref{subsec:variations.NMSSM} and
\ref{subsec:variations.munonrenorm}
we will study three such
mechanisms: the NexttoMinimal Supersymmetric Standard Model, the KimNilles mechanism
\cite{KimNilles}, and the GiudiceMasiero
mechanism \cite{GiudiceMasiero}.
Another solution based on loop effects was
proposed in ref.~\cite{muproblemGMSB}.
From the point of view of the MSSM, however,
we can just treat $\mu$ as an independent parameter, without
committing to a specific mechanism.
The $\mu$term and the Yukawa couplings in the superpotential
eq.~(\ref{MSSMsuperpot}) combine to yield (scalar)$^3$ couplings [see the
second and third terms on the righthand side of eq.~(\ref{ordpot})] of
the form
\beq
\lagr_{\mbox{supersymmetric (scalar)$^3$}} &= &
\mu^* (
{\stilde{\sbar u}} {\bf y_u} \stilde u H_d^{0*}
+ {\stilde{\sbar d}} {\bf y_d} \stilde d H_u^{0*}
+ {\stilde{\sbar e}} {\bf y_e} \stilde e H_u^{0*}
\cr
&&
+{\stilde{\sbar u}} {\bf y_u} \stilde d H_d^{*}
+{\stilde{\sbar d}} {\bf y_d} \stilde u H_u^{+*}
+{\stilde{\sbar e}} {\bf y_e} \stilde \nu H_u^{+*}
)
+ \conj
\label{striterms}
\eeq
Figure~\ref{fig:stri} shows some of these couplings,
\begin{figure}
\begin{center}
\begin{picture}(66,62)(0,0)
\SetWidth{0.85}
\DashLine(33,52.5)(33,12){4}
\DashLine(0,0)(33,12){4}
\DashLine(66,0)(33,12){4}
\ArrowLine(33,32.25)(33,32.2501)
\ArrowLine(16.5,6)(16.5165,6.006)
\ArrowLine(49.5,6)(49.4835,6.006)
\Text(0,11)[c]{$\stilde t_L$}
\Text(66.5,10)[c]{$\stilde t_R^*$}
\Text(23,56)[c]{$H_d^{0*}$}
\Text(33,12)[c]{(a)}
\end{picture}
%
\hspace{1.7cm}
%
\begin{picture}(66,62)(0,0)
\SetWidth{0.85}
\DashLine(33,52.5)(33,12){4}
\DashLine(0,0)(33,12){4}
\DashLine(66,0)(33,12){4}
\ArrowLine(33,32.25)(33,32.2501)
\ArrowLine(16.5,6)(16.5165,6.006)
\ArrowLine(49.5,6)(49.4835,6.006)
\Text(0,11)[c]{$\stilde b_L$}
\Text(66.5,10)[c]{$\stilde b_R^*$}
\Text(23,56)[c]{$H_u^{0*}$}
\Text(33,12)[c]{(b)}
\end{picture}
%
\hspace{1.7cm}
%
\begin{picture}(66,62)(0,0)
\SetWidth{0.85}
\DashLine(33,52.5)(33,12){4}
\DashLine(0,0)(33,12){4}
\DashLine(66,0)(33,12){4}
\ArrowLine(33,32.25)(33,32.2501)
\ArrowLine(16.5,6)(16.5165,6.006)
\ArrowLine(49.5,6)(49.4835,6.006)
\Text(0,11)[c]{$\stilde \tau_L$}
\Text(66.5,10)[c]{$\stilde \tau_R^*$}
\Text(23,56)[c]{$H_u^{0*}$}
\Text(33,12)[c]{(c)}
\end{picture}
\end{center}
\caption{Some of the supersymmetric (scalar)$^3$ couplings proportional to
$\mu^* y_t$, $\mu^* y_b$, and $\mu^* y_\tau$. When $H_u^0$ and $H_d^0$ get
VEVs, these contribute to
(a) $\stilde t_L,\stilde t_R$ mixing,
(b) $\stilde b_L,\stilde b_R$ mixing,
and (c) $\stilde \tau_L,\stilde \tau_R$ mixing.
\label{fig:stri}}
\end{figure}
proportional to $\mu^* y_t$, $\mu^* y_b$, and $\mu^* y_\tau$ respectively.
These play an important role in determining the mixing of top squarks,
bottom squarks, and tau sleptons, as we will see in section
\ref{subsec:MSSMspectrum.sfermions}.
\subsection{$R$parity (also known as matter parity) and its
consequences}\label{subsec:mssm.rparity}
\setcounter{equation}{0}
The superpotential eq.~(\ref{MSSMsuperpot}) is minimal in the sense that
it is sufficient to produce a phenomenologically viable model. However,
there are other terms that one can write that are gaugeinvariant
and holomorphic in the chiral superfields, but are not included in the MSSM
because they violate either baryon number ($\Baryon$) or total lepton
number ($\Lepton$). The most general gaugeinvariant and renormalizable
superpotential would include not only eq.~(\ref{MSSMsuperpot}), but also
the terms
\beq
W_{\Delta {\rm L} =1} &=&
{1\over 2} \lambda^{ijk} L_iL_j{\sbar e_k}
+ \lambda^{\prime ijk} L_i Q_j {\sbar d_k}
+ \mu^{\prime i} L_i H_u
\label{WLviol} \\
W_{\Delta {\rm B}= 1} &=& {1\over 2} \lambda^{\prime\prime ijk}
{\sbar u_i}{\sbar d_j}{\sbar d_k}
\label{WBviol}
\eeq
where family indices $i=1,2,3$ have been restored. The chiral
supermultiplets carry baryon number assignments $\Baryon=+1/3$ for $Q_i$;
$\Baryon=1/3$ for $\sbar u_i, \sbar d_i$; and $\Baryon=0$ for all others.
The total lepton number assignments are $\Lepton=+1$ for $L_i$,
$\Lepton=1$ for $\sbar e_i$, and $\Lepton=0$ for all others. Therefore,
the terms in eq.~(\ref{WLviol}) violate total lepton number by 1 unit (as
well as the individual lepton flavors) and those in eq.~(\ref{WBviol})
violate baryon number by 1 unit.
The possible existence of such terms might seem rather disturbing, since
corresponding $\Baryon$ and $\Lepton$violating processes have not been
seen experimentally. The most obvious experimental constraint comes from
the nonobservation of proton decay, which would violate both $\Baryon$
and $\Lepton$ by 1 unit. If both $\lambda^\prime$ and
$\lambda^{\prime\prime}$ couplings were present and unsuppressed, then
the lifetime of the proton would be extremely short.
For example, Feynman diagrams like the one in
Figure~\ref{fig:protondecay}\footnote{In this diagram and
others below, the arrows
on propagators are often
omitted for simplicity, and external fermion labels refer to physical
particle states rather than 2component fermion fields.}
would lead to%
\begin{figure}
\begin{minipage}[]{0.41\linewidth}
\caption{Squarks would mediate disastrously rapid proton
decay if $R$parity were violated by both $\Delta {\rm B} = 1$ and
$\Delta {\rm L} = 1$ interactions. This example shows
$p \rightarrow e^+ \pi^0$ mediated by a strange (or bottom) squark.
\label{fig:protondecay}}
\end{minipage}
\begin{minipage}[]{0.585\linewidth}
\begin{picture}(200,60)(56,0)
\SetScale{1.5}
\Line(5,0)(125,0)
\Line(5,12)(42.5,26)
\Line(5,40)(42.5,26)
\Line(125,12)(87.5,26)
\Line(125,40)(87.5,26)
\DashLine(42.5,26)(87.5,26){4}
\Text(4,5)[c]{$u$}
\Text(4,26)[c]{$u$}
\Text(4,53)[c]{$d$}
\Text(98,50)[c]{$\tilde s_R^*$}
\Text(18,30)[c]{$p^+$
$\displaystyle \left\{\vphantom{\displaystyle
{{\Biggl \{ \Biggr \}}\atop{\Biggl \{ \Biggr \}}}}\right.$}
\Text(204,10)[c]{$\biggr \}{\pi^0}$}
\Text(185,5)[c]{$u$}
\Text(185,28)[c]{$u^*$}
\Text(198,58)[c]{$e^+$}
\Text(65,27)[c]{$ \lambda^{\prime\prime*}_{112}$}
\Text(140,27)[c]{$ \lambda'_{112}$}
\end{picture}
\end{minipage}
\end{figure}
$p^+ \rightarrow e^+ \pi^0$ (shown) or
$\mu^+\pi^0$ or $\bar\nu \pi^+$ or $\bar\nu K^+$ etc.~depending on which
components of $\lambda^{\prime}$ and $\lambda^{\prime\prime}$ are
largest.\footnote{The coupling $\lambda^{\prime\prime}$ must be
antisymmetric in its last two flavor indices, since the color indices are
combined antisymmetrically. That is why the squark in
Figure~\ref{fig:protondecay} can be $\stilde{s}$
or $\stilde{b}$, but not $\stilde{d}$, for $u,d$ quarks in the proton.}
Also, diagrams with $t$channel squark exchange can lead to final states
$e^+ K^0$, $\mu^+ K^0$, $\nu\pi^+$, or $\nu K^+$,
with the last two relying on leftright squark mixing.
As a rough estimate based on dimensional analysis, for example,
\beq
\Gamma_{p \rightarrow e^+ \pi^0} \>\sim\> m_{{\rm proton}}^5 \sum_{i=2,3}
\lambda^{\prime 11i}\lambda^{\prime\prime 11i}^2/m_{\stilde d_i}^4,
\eeq
which would be a tiny fraction of a second if the couplings were of order
unity and the squarks have masses of order 1 TeV. In contrast, the decay
time of the proton into lepton+meson final states is known experimentally
to be in excess of $10^{32}$ years. Therefore, at least one of
$\lambda^{\prime ijk}$ or $\lambda^{\prime\prime 11k}$ for each of
$i=1,2$; $j=1,2$; $k=2,3$ must be extremely small. Many other processes
also give strong constraints on the violation of lepton and baryon
numbers \cite{rparityconstraints,RPVreviews}.
One could simply try to take $\Baryon$ and $\Lepton$ conservation as a
postulate in the MSSM. However, this is clearly a step backward from the
situation in the Standard Model, where the conservation of these quantum
numbers is {\it not} assumed, but is rather a pleasantly ``accidental"
consequence of the fact that there are no possible renormalizable
Lagrangian terms that violate $\Baryon$ or $\Lepton$. Furthermore, there
is a quite general obstacle to treating $\Baryon$ and $\Lepton$ as
fundamental symmetries of Nature, since they are known to be necessarily
violated by nonperturbative electroweak effects \cite{tHooft}
(even though those
effects are calculably negligible for experiments at ordinary energies).
Therefore, in the MSSM one adds a new symmetry, which has the effect of
eliminating the possibility of $\Baryon$ and $\Lepton$ violating terms in
the renormalizable superpotential, while allowing the good terms in
eq.~(\ref{MSSMsuperpot}). This new symmetry is called ``$R$parity"
\cite{Rparity} or equivalently ``matter parity" \cite{matterparity}.
Matter parity is a multiplicatively conserved quantum number defined as
\beq
P_M = (1)^{3 (\Baryon\Lepton)}
\label{defmatterparity}
\eeq
for each particle in the theory. It follows that the quark and
lepton supermultiplets all have $P_M=1$, while the Higgs supermultiplets
$H_u$ and $H_d$ have $P_M=+1$. The gauge bosons and gauginos of course do
not carry baryon number or lepton number, so they are assigned matter
parity $P_M=+1$. The symmetry principle to be enforced is that a candidate
term in the Lagrangian (or in the superpotential) is allowed only if the
product of $P_M$ for all of the fields in it is $+1$. It is easy to see
that each of the terms in eqs.~(\ref{WLviol}) and (\ref{WBviol}) is thus
forbidden, while the good and necessary terms in eq.~(\ref{MSSMsuperpot})
are allowed. This discrete symmetry commutes with supersymmetry, as all
members of a given supermultiplet have the same matter parity. The
advantage of matter parity is that it can in principle be an {\it exact}
and fundamental symmetry, which B and L themselves cannot, since they are
known to be violated by nonperturbative electroweak effects. So even with
exact matter parity conservation in the MSSM, one expects that baryon
number and total lepton number violation can occur in tiny amounts, due to
nonrenormalizable terms in the Lagrangian. However, the MSSM does not have
renormalizable interactions that violate B or L, with the standard
assumption of matter parity conservation.
It is often useful to recast matter parity in terms of $R$parity,
defined for each particle as
\beq
P_R = (1)^{3(\Baryon\Lepton) + 2 s}
\label{defRparity}
\eeq
where $s$ is the spin of the particle. Now, matter parity conservation and
$R$parity conservation are precisely equivalent, since the product of
$(1)^{2s}$ for the particles involved in any interaction vertex in a
theory that conserves angular momentum is always equal to $+1$. However,
particles within the same supermultiplet do not have the same $R$parity.
In general, symmetries with the property that fields within the same
supermultiplet have different transformations are called $R$ symmetries;
they do not commute with supersymmetry. Continuous $U(1)$ $R$ symmetries
were described in section \ref{Rsymmetry}, and
are often encountered in the modelbuilding literature; they should not be
confused with $R$parity, which is a discrete $Z_2$ symmetry. In fact, the
matter parity version of $R$parity makes clear that there is really
nothing intrinsically ``$R$" about it; in other words it secretly does
commute with supersymmetry, so its name is somewhat suboptimal.
Nevertheless, the $R$parity assignment is very useful for phenomenology
because all of the Standard Model particles and the Higgs bosons have even
$R$parity ($P_R=+1$), while all of the squarks, sleptons, gauginos, and
higgsinos have odd $R$parity ($P_R=1$).
The $R$parity odd particles are known as ``supersymmetric particles" or
``sparticles" for short, and they are distinguished by a tilde (see Tables
\ref{tab:chiral} and \ref{tab:gauge}).
If $R$parity is exactly conserved, then there can be no mixing
between the sparticles and the $P_R=+1$ particles. Furthermore, every
interaction vertex in the theory contains an even number of $P_R=1$
sparticles. This has three extremely important phenomenological
consequences:
\begin{itemize}
\item[$\bullet$] The lightest sparticle with $P_R=1$, called the
``lightest supersymmetric particle" or LSP, must be absolutely stable. If
the LSP is electrically neutral, it interacts only weakly with ordinary
matter, and so can make an attractive candidate \cite{neutralinodarkmatter}
for the nonbaryonic dark matter that seems to be required by cosmology.
%
\item[$\bullet$] Each sparticle other than the LSP must eventually decay
into a state that contains an odd number of LSPs (usually just one).
%
\item[$\bullet$] In collider experiments, sparticles can only be produced
in even numbers (usually twoatatime).
\end{itemize}
We {\it define} the MSSM to conserve $R$parity or equivalently matter
parity. While this decision seems to be wellmotivated phenomenologically
by proton decay constraints and the hope that the LSP will provide a good
dark matter candidate, it might appear somewhat artificial from a
theoretical point of view. After all, the MSSM would not suffer any
internal inconsistency if we did not impose matter parity conservation.
Furthermore, it is fair to ask why matter parity should be exactly
conserved, given that the discrete symmetries in the Standard Model
(ordinary parity $P$, charge conjugation $C$, time reversal $T$, etc.) are
all known to be inexact symmetries. Fortunately, it {\it is} sensible to
formulate matter parity as a discrete symmetry that is exactly conserved.
In general, exactly conserved, or ``gauged" discrete symmetries \cite{KW}
can exist provided that they satisfy certain anomaly cancellation
conditions \cite{discreteanomaly} (much like continuous gauged
symmetries). One particularly attractive way this could occur is if B$$L
is a continuous gauge symmetry that is spontaneously broken at some
very high energy scale. A continuous $U(1)_{\Baryon\Lepton}$
forbids the renormalizable terms that violate B and L
\cite{Rparityoriginone,Rparityorigintwo},
but this gauge symmetry must be spontaneously broken, since there
is no corresponding massless vector boson.
However, if gauged $U(1)_{\Baryon  \Lepton}$ is only broken
by scalar VEVs (or other order parameters) that carry even integer
values of $3($B$$L$)$, then $P_M$ will automatically survive as an
exactly conserved discrete remnant subgroup \cite{Rparityorigintwo}.
A variety of extensions of the MSSM in which exact $R$parity
conservation is guaranteed in just this way have been proposed
(see for example \cite{Rparityorigintwo,Rparityoriginthree}).
It may also be possible to have gauged discrete symmetries that do not owe
their exact conservation to an underlying continuous gauged symmetry, but
rather to some other structure such as can occur in string theory. It is
also possible that $R$parity is broken, or is replaced by some
alternative discrete symmetry. We will briefly consider these as
variations on the MSSM in section \ref{subsec:variations.RPV}.
\subsection{Soft supersymmetry breaking in the
MSSM}\label{subsec:mssm.soft}
\setcounter{equation}{0}
To complete the description of the MSSM, we need to specify the soft
supersymmetry breaking terms. In section \ref{sec:soft}, we learned how to
write down the most general set of such terms in any supersymmetric
theory. Applying this recipe to the MSSM, we have:
\beq
\lagr_{\rm soft}^{\rm MSSM} &=& \half\left ( M_3 \stilde g\stilde g
+ M_2 \stilde W \stilde W + M_1 \stilde B\stilde B
+\conj \right )
\nonumber
\\
&&
\left ( \stilde {\sbar u} \,{\bf a_u}\, \stilde Q H_u
 \stilde {\sbar d} \,{\bf a_d}\, \stilde Q H_d
 \stilde {\sbar e} \,{\bf a_e}\, \stilde L H_d
+ \conj \right )
\nonumber
\\
&&
\stilde Q^\dagger \, {\bf m^2_{Q}}\, \stilde Q
\stilde L^\dagger \,{\bf m^2_{L}}\,\stilde L
\stilde {\sbar u} \,{\bf m^2_{{\sbar u}}}\, {\stilde {\sbar u}}^\dagger
\stilde {\sbar d} \,{\bf m^2_{{\sbar d}}} \, {\stilde {\sbar d}}^\dagger
\stilde {\sbar e} \,{\bf m^2_{{\sbar e}}}\, {\stilde {\sbar e}}^\dagger
\nonumber \\
&&
 \, m_{H_u}^2 H_u^* H_u  m_{H_d}^2 H_d^* H_d
 \left ( b H_u H_d + \conj \right ) .
\label{MSSMsoft}
\eeq
In eq.~(\ref{MSSMsoft}), $M_3$, $M_2$, and $M_1$ are the gluino, wino, and
bino mass terms. Here, and from now on, we suppress the adjoint
representation gauge indices on the wino and gluino fields, and the gauge
indices on all of the chiral supermultiplet fields. The second line in
eq.~(\ref{MSSMsoft}) contains the (scalar)$^3$ couplings [of the type
$a^{ijk}$ in eq.~(\ref{lagrsoft})]. Each of ${\bf a_u}$, ${\bf a_d}$,
${\bf a_e}$ is a complex $3\times 3$ matrix in family space, with
dimensions of [mass]. They are in onetoone correspondence with the
Yukawa couplings of the superpotential. The third line of
eq.~(\ref{MSSMsoft}) consists of squark and slepton mass terms of the
$(m^2)_i^j$ type in eq.~(\ref{lagrsoft}). Each of ${\bf m^2_{ Q}}$, ${\bf
m^2_{{\sbar u}}}$, ${\bf m^2_{{\sbar d}}}$, ${\bf m^2_{L}}$, ${\bf
m^2_{{\sbar e}}}$ is a $3\times 3$ matrix in family space that can have
complex entries, but they must be hermitian so that the Lagrangian is
real. (To avoid clutter, we do not put tildes on the $\bf Q$ in $\bf
m^2_Q$, etc.) Finally, in the last line of eq.~(\ref{MSSMsoft}) we have
supersymmetrybreaking contributions to the Higgs potential; $m_{H_u}^2$
and $m_{H_d}^2$ are squaredmass terms of the $(m^2)_i^j$ type, while $b$
is the only squaredmass term of the type $b^{ij}$ in eq.~(\ref{lagrsoft})
that can occur in the MSSM.\footnote{The parameter called $b$ here is often
seen elsewhere as $B\mu$ or $m_{12}^2$ or $m_3^2$.} As argued in
the Introduction, we expect
\beq
&&\!\!\!\!\! M_1,\, M_2,\, M_3,\, {\bf a_u},\, {\bf a_d},\, {\bf a_e}\,
\sim\, m_{\rm soft},\\
&&\!\!\!\!\! {\bf m^2_{ Q}},\,
{\bf m^2_{L}},\,
{\bf m^2_{{\sbar u}}},\,
{\bf m^2_{{\sbar d}}},\,
{\bf m^2_{{\sbar e}}},\, m_{H_u}^2,\, m_{H_d}^2,\, b\, \sim \,
m_{\rm soft}^2 ,
\eeq
with a characteristic mass scale $m_{\rm soft}$ that is not much larger
than $10^3$ GeV. The expression eq.~(\ref{MSSMsoft}) is the most general
soft supersymmetrybreaking Lagrangian of the form eq.~(\ref{lagrsoft})
that is compatible with gauge invariance and matter parity conservation in
the MSSM.
Unlike the supersymmetrypreserving part of the Lagrangian, the above
$\lagr_{\rm soft}^{\rm MSSM}$ introduces many new parameters that were not
present in the ordinary Standard Model. A careful count \cite{dimsut}
reveals that there are 105 masses, phases and mixing angles in the MSSM
Lagrangian that cannot be rotated away by redefining the phases and flavor
basis for the quark and lepton supermultiplets, and that have no
counterpart in the ordinary Standard Model. Thus, in principle,
supersymmetry {\em breaking} (as opposed to supersymmetry itself) appears
to introduce a tremendous arbitrariness in the Lagrangian.
\subsection{Hints of an Organizing Principle}\label{subsec:mssm.hints}
\setcounter{equation}{0}
Fortunately, there is already good experimental evidence that some
powerful organizing principle must govern the soft supersymmetry breaking
Lagrangian. This is because most of the new parameters in
eq.~(\ref{MSSMsoft}) imply flavor mixing or CP violating processes of the
types that are severely restricted by experiment
%\cite{FCNCs, morestuff, muegamma, demon, bsgamma, muegammatwo,
%flavorreview, Hewett:1996ct, Grossman:1997pa, demon2, demon3,
%Misiak:1997ei, Bagger:1997gg, Nir:1997tf, Ciuchini:1997bw,
%Ciuchini:1998ix, Masiero:1999ub, Khalil:1999zn, Buras:1999da,
%Borzumati:1999qt, Buras:2000qz, Chankowski:2000ng, Besmer:2001cj,
%Burdman:2001tf, Buras:2002vd, Ciuchini:2002uv}.
\cite{FCNCs}\cite{Ciuchini:2002uv}.
For example, suppose that $\bf m_{\sbar e}^2$ is not diagonal in the basis
$(\stilde e_R, \stilde \mu_R, \stilde \tau_R)$ of sleptons whose
superpartners are the righthanded parts of the Standard Model mass
eigenstates $e,\mu,\tau$. In that case, slepton mixing occurs, so the
individual lepton numbers will not be conserved, even for
processes that only involve the sleptons as virtual particles. A
particularly strong limit on this possibility comes from the experimental
bound on the process
$\mu\rightarrow e \gamma$, which could arise from the oneloop
diagram shown in Figure~\ref{fig:flavormuegamma}a.
The symbol ``$\times$" on the slepton line
represents an insertion coming from
$({\bf m^2_{\sbar e}})_{21}\stilde \mu^*_R\stilde e_R$ in
$\lagr_{\rm soft}^{\rm MSSM}$, and
the sleptonbino vertices are determined by the weak hypercharge gauge
coupling [see Figures~\ref{fig:gauge}g,h and eq.~(\ref{gensusylagr})].%
%
\begin{figure}
%
\begin{center}
\begin{picture}(138,58)(0,5)
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\SetWidth{0.8}
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\Line(62,26)(67.6,31.6)
\Line(62,31.6)(67.6,26)
\Text(68,16.5)[c]{(a)}
\Text(132,48)[c]{${\gamma}$}
\Text(136,9)[c]{$e^$}
\Text(6,9)[c]{$\mu^$}
\Text(68,10)[c]{$\stilde B$}
\Text(46,34)[c]{$\stilde\mu_R$}
\Text(84,34)[c]{$\stilde e_R$}
\end{picture}
%
\hspace{0.75cm}
%
\begin{picture}(138,58)(0,5)
\SetScale{1.05}
\PhotonArc(64.8,0)(28.8,0,180){2}{8}
\SetWidth{0.8}
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\Text(68,16.5)[c]{(b)}
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\Text(136,9)[c]{$e^$}
\Text(6,9)[c]{$\mu^$}
\Text(68.5,42.5)[c]{$\stilde W^$}
\Text(56,8)[c]{$\stilde\nu_\mu$}
\Text(82,8)[c]{$\stilde\nu_e$}
\end{picture}
%
\hspace{0.75cm}
%
\begin{picture}(138,58)(0,5)
\SetScale{1.05}
\Photon(36,0)(93.6,0){1.75}{5}
\SetWidth{0.8}
\DashCArc(64.8,0)(28.8,0,180){5}
\Photon(85.1647,20.3647)(117.748,52.9481){2.4}{5}
\Line(0,0)(36,0)
\Line(93.6,0)(129.6,0)
\Line(93.6,0)(36,0)
\Line(62,26)(67.6,31.6)
\Line(62,31.6)(67.6,26)
\Text(68,16.5)[c]{(c)}
\Text(132,48)[c]{${\gamma}$}
\Text(136,9)[c]{$e^$}
\Text(6,9)[c]{$\mu^$}
\Text(68,10)[c]{$\stilde B$}
\Text(46,34)[c]{$\stilde\mu_L$}
\Text(84,34)[c]{$\stilde e_R$}
\end{picture}
\end{center}
\caption{Some of the diagrams that contribute to the
process $\mu^ \rightarrow e^ \gamma$ in
models with lepton flavorviolating
soft supersymmetry breaking parameters (indicated by $\times$).
Diagrams (a), (b), and (c) contribute to constraints on the
offdiagonal elements of ${\bf m^2_{\sbar e}}$,
${\bf m^2_{L}}$, and ${\bf a_e}$, respectively.
\label{fig:flavormuegamma}}
\end{figure}
The result of calculating this diagram gives \cite{muegamma,muegammatwo},
approximately,
\beq
\Branching (\mu \rightarrow e \gamma)
&=&
\left ( \frac{ m^2_{\tilde \mu^*_R \tilde e_R} }{m^2_{\tilde \ell_R}}
\right )^2
\left (\frac{100\>{\rm GeV}}{m_{\tilde \ell_R}} \right )^4
10^{6} \times
\left \{ \begin{array}{ll}
15 & {\rm for}\>\, m_{\tilde B} \ll m_{\tilde \ell_R},
\\[5pt]
5.6 & {\rm for}\>\, m_{\tilde B} = 0.5 m_{\tilde \ell_R},
\\[5pt]
1.4 & {\rm for}\>\, m_{\tilde B} = m_{\tilde \ell_R},
\\[5pt]
0.13 & {\rm for}\>\, m_{\tilde B} = 2 m_{\tilde \ell_R},
\end{array}
\right.
\phantom{xxx}
\label{eq:muegamma}
\eeq
where it is assumed for simplicity that both $\tilde e_R$ and $\tilde
\mu_R$ are nearly mass eigenstates with almost degenerate squared masses
$m^2_{\tilde \ell_R}$, that $m^2_{\tilde \mu_R^* \tilde e_R} \equiv ({\bf
m^2_{\sbar e}})_{21} = [({\bf m^2_{\sbar e}})_{12}]^*$ can be treated as a
perturbation, and that the bino $\stilde B$ is nearly a mass eigenstate. This
result is to be compared to the present experimental upper limit
$\Branching (\mu \rightarrow e \gamma)_{\rm exp} < 5.7\times 10^{13}$ from
\cite{muegammaexperiment}. So, if the righthanded slepton squaredmass
matrix ${\bf m^2_{\sbar e}}$ were ``random", with all entries of
comparable size, then the prediction for $\Branching (\mu\rightarrow e\gamma)$
would be too large even if the sleptons and bino masses were at 1 TeV.
For lighter superpartners, the constraint on $\tilde \mu_R, \tilde e_R$
squaredmass mixing becomes correspondingly more severe. There are also
contributions to $\mu \rightarrow e \gamma$ that depend on the
offdiagonal elements of the lefthanded slepton squaredmass matrix $\bf
m^2_L$, coming from the
diagram shown in fig.~\ref{fig:flavormuegamma}b
involving the charged wino and the sneutrinos, as well as
diagrams just like fig.~\ref{fig:flavormuegamma}a but with lefthanded
sleptons and either $\stilde B$ or $\stilde W^0$ exchanged.
Therefore, the slepton
squaredmass matrices must not have significant mixings for $\stilde
e_L,\stilde\mu_L$ either.
Furthermore, after the Higgs scalars get VEVs, the $\bf a_e$ matrix could
imply squaredmass terms that mix lefthanded and righthanded sleptons
with different lepton flavors. For example, $\lagr_{\rm soft}^{\rm MSSM}$
contains ${\stilde{\sbar e}} {\bf a_e} {\stilde L} H_d + \conj$ which
implies terms $ \langle H_d^0 \rangle ({\bf a_e})_{12} \stilde e_R^*
\stilde \mu_L \langle H_d^0 \rangle ({\bf a_e})_{21} \stilde \mu_R^*
\stilde e_L + \conj$~~These also contribute to $\mu \rightarrow e \gamma$,
as illustrated in fig.~\ref{fig:flavormuegamma}c.
So the magnitudes of $({\bf a_e})_{12}$ and $({\bf a_e})_{21}$ are also
constrained by experiment to be small, but in a way that is more strongly
dependent on other model parameters \cite{muegammatwo}. Similarly,
$({\bf a_e})_{13}, ({\bf a_e})_{31}$ and $({\bf a_e})_{23}, ({\bf
a_e})_{32}$ are constrained, although more weakly \cite{flavorreview}, by
the experimental limits on $\Branching (\tau \rightarrow e \gamma)$ and
$\Branching (\tau \rightarrow \mu \gamma)$.
There are also important experimental constraints on the squark
squaredmass matrices. The strongest of these come from the neutral kaon
system. The effective Hamiltonian for $K^0\leftrightarrow \overline K^0$
mixing gets contributions from the diagrams in Figure~\ref{fig:flavor},
among others, if $\lagr_{\rm soft}^{\rm MSSM}$ contains terms
that mix down squarks and strange squarks.
The gluinosquarkquark
vertices in Figure~\ref{fig:flavor} are all fixed by supersymmetry to be
of QCD interaction strength. (There are similar diagrams in which the
bino and winos are exchanged, which can be important depending on the
relative sizes of the gaugino masses.) For example, suppose that there is
a nonzero righthanded downsquark squaredmass mixing
$({\bf m^2_{\sbar d}})_{21}$
in the basis corresponding to the quark mass eigenstates.%
\begin{figure}
%
\begin{center}
\begin{picture}(144,58)(0,2)
\SetScale{0.8}
\Photon(40,0)(40,55){1.8}{5}
\Photon(104,0)(104,55){1.8}{5}
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\Line(104,55)(104,0)
\Line(40,0)(40,55)
\Line(69.2,2.8)(74.8,2.8)
\Line(69.2,2.8)(74.8,2.8)
\Line(69.2,52.2)(74.8,57.8)
\Line(69.2,57.8)(74.8,52.2)
\Text(25.5,23.7)[c]{${\stilde g}$}
\Text(89,24.7)[c]{${\stilde g}$}
\Text(46,8.5)[c]{${\tilde d_R}$}
\Text(72.5,8)[c]{${\tilde s_R}$}
\Text(46,52)[c]{${\tilde s_R}^*$}
\Text(72.5,52.5)[c]{${\tilde d_R}^*$}
\Text(0,7)[c]{${d}$}
\Text(115,6)[c]{${s}$}
\Text(0,50)[c]{$\bar{s}$}
\Text(115,52)[c]{$\bar{d}$}
\Text(57.6,16.5)[c]{(a)}
\end{picture}
\begin{picture}(144,58)(0,2)
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\Line(69.2,52.2)(74.8,57.8)
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\Text(25.5,23.7)[c]{${\stilde g}$}
\Text(89,24.7)[c]{${\stilde g}$}
\Text(46,8.5)[c]{${\tilde d_L}$}
\Text(72.5,8)[c]{${\tilde s_L}$}
\Text(46,52)[c]{${\tilde s_R}^*$}
\Text(72.5,52.5)[c]{${\tilde d_R}^*$}
\Text(0,7)[c]{${d}$}
\Text(115,6)[c]{${s}$}
\Text(0,50)[c]{$\bar{s}$}
\Text(115,52)[c]{$\bar{d}$}
\Text(57.6,16.5)[c]{(b)}
\end{picture}
\begin{picture}(144,58)(0,2)
\SetScale{0.8}
\Photon(40,0)(40,55){1.8}{5}
\Photon(104,0)(104,55){1.8}{5}
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\DashLine(40,55)(104,55){5}
\Line(40,0)(0,0)
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\Line(104,55)(144,55)
\Line(104,55)(104,0)
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\Line(69.2,52.2)(74.8,57.8)
\Line(69.2,57.8)(74.8,52.2)
\Text(25.5,23.7)[c]{${\stilde g}$}
\Text(89,24.7)[c]{${\stilde g}$}
\Text(46,8.5)[c]{${\tilde d_L}$}
\Text(72.5,8)[c]{${\tilde s_R}$}
\Text(46,52)[c]{${\tilde s_R}^*$}
\Text(72.5,52.5)[c]{${\tilde d_L}^*$}
\Text(0,7)[c]{${d}$}
\Text(115,6)[c]{${s}$}
\Text(0,50)[c]{$\bar{s}$}
\Text(115,52)[c]{$\bar{d}$}
\Text(57.6,16.5)[c]{(c)}
\end{picture}
\end{center}
\caption{Some of the diagrams that contribute to
$K^0\leftrightarrow \overline K^0$ mixing in
models with strangenessviolating
soft supersymmetry breaking parameters (indicated by $\times$).
These diagrams contribute to constraints on the
offdiagonal elements of (a) ${\bf m^2_{\sbar d}}$, (b)
the combination of ${\bf m^2_{\sbar d}}$ and ${\bf m^2_{Q}}$, and
(c) ${\bf a_d}$.
\label{fig:flavor}}
\end{figure}
Assuming that the supersymmetric correction to $\Delta m_K \equiv m_{K_L}
 m_{K_S}$ following from fig.~\ref{fig:flavor}a and others
does not exceed, in absolute value, the experimental value
$3.5 \times 10^{12}$ MeV, ref.~\cite{Ciuchini:1998ix} obtains:
\beq
\frac{{\rm Re}[(m^2_{\tilde s_R^* \tilde d_R})^2]^{1/2}}{
m^2_{\tilde q}}
&<&
\left ( \frac{m_{\tilde q}}{1000 \>{\rm GeV}} \right )
\times
\left \{ \begin{array}{ll}
0.04 & {\rm for}\>\, m_{\tilde g} = 0.5 m_{\tilde q},
\\[5pt]
0.10 & {\rm for}\>\, m_{\tilde g} = m_{\tilde q},
\\[5pt]
0.22 & {\rm for}\>\, m_{\tilde g} = 2 m_{\tilde q}.
\end{array}
\right.
\eeq
Here nearly degenerate squarks with mass $m_{\tilde q}$ are assumed for
simplicity, with $m^2_{\tilde s_R^* \tilde d_R} = ({\bf m^2_{\sbar
d}})_{21}$ treated as a perturbation. The same limit applies when
$m^2_{\tilde s_R^* \tilde d_R}$ is replaced by $m^2_{\tilde s_L^* \tilde
d_L} = ({\bf m^2_{Q}})_{21}$, in a basis corresponding to the
downtype quark mass eigenstates. An even more striking limit applies to
the combination of both types of flavor mixing when they are comparable in
size, from diagrams including fig.~\ref{fig:flavor}b.
The numerical constraint is \cite{Ciuchini:1998ix}:
\beq
\frac{{\rm Re}[m^2_{\tilde s_R^* \tilde d_R}
m^2_{\tilde s_L^* \tilde d_L}]^{1/2}}{
m^2_{\tilde q}}
&<&
\left ( \frac{m_{\tilde q}}{1000 \>{\rm GeV}} \right )
\times
\left \{
\begin{array}{ll}
0.0016 & {\rm for}\>\, m_{\tilde g} = 0.5 m_{\tilde q},
\\[5pt]
0.0020 & {\rm for}\>\, m_{\tilde g} = m_{\tilde q},
\\[5pt]
0.0026 & {\rm for}\>\, m_{\tilde g} = 2 m_{\tilde q}.
\end{array}
\right.
\label{eq:striking}
\eeq
An offdiagonal contribution from ${\bf a_d}$ would cause flavor mixing
between lefthanded and righthanded squarks, just as discussed above for
sleptons, resulting in a strong constraint from diagrams like
fig.~\ref{fig:flavor}c. More generally, limits on $\Delta m_K$ and
$\epsilon$ and $\epsilon'/\epsilon$ appearing in the neutral kaon
effective Hamiltonian severely restrict the amounts of $\stilde
d_{L,R},\,\stilde s_{L,R}$ squark mixings (separately and in various
combinations), and associated CPviolating complex phases, that one can
tolerate in the soft squared masses.
Weaker, but still interesting, constraints come from the $D^0, \overline
D^0$ system, which limits the amounts of $\stilde u,\stilde c$ mixings
from ${\bf m_{\sbar u}^2}$, ${\bf m_Q^2}$ and ${\bf a_u}$. The $B_d^0,
\overline B_d^0$ and $B_s^0, \overline B_s^0$ systems similarly limit the
amounts of $\stilde d,\stilde b$ and $\stilde s,\stilde b$ squark mixings
from soft supersymmetrybreaking sources. More constraints follow from
rare $\Delta F=1$ meson decays, notably those involving the partonlevel
processes $b\rightarrow s\gamma$ and $b \rightarrow s \ell^+ \ell^$ and
$c \rightarrow u \ell^+ \ell^$ and $s \rightarrow d e^+ e^$ and $s
\rightarrow d \nu \bar \nu$, all of which can be mediated by flavor mixing
in soft supersymmetry breaking. There are also strict constraints on
CPviolating phases in the gaugino masses and (scalar)$^3$ soft couplings
following from limits on the electric dipole moments of the neutron and
electron \cite{demon}. Detailed limits can be found in the literature
%\cite{FCNCs, morestuff, muegamma, demon, bsgamma, muegammatwo,
%flavorreview, Hewett:1996ct, Grossman:1997pa, demon2, demon3,
%Misiak:1997ei, Bagger:1997gg, Nir:1997tf, Ciuchini:1997bw,
%Ciuchini:1998ix, Masiero:1999ub, Khalil:1999zn, Buras:1999da,
%Borzumati:1999qt, Buras:2000qz, Chankowski:2000ng, Besmer:2001cj,
%Burdman:2001tf, Buras:2002vd, Ciuchini:2002uv},
\cite{FCNCs}\cite{Ciuchini:2002uv},
but the essential lesson from experiment is that the soft
supersymmetrybreaking Lagrangian cannot be arbitrary or random.
\vspace{0.6mm}
All of these potentially dangerous flavorchanging and CPviolating
effects in the MSSM can be evaded if one assumes (or can explain!) that
supersymmetry breaking is suitably ``universal". Consider an idealized
limit in which the squark and slepton squaredmass matrices are
flavorblind, each proportional to the $3\times 3$ identity matrix in
family space:
\beq
{\bf m^2_{Q}} = m^2_{Q} {\bf 1},
\qquad\!\!\!\!
{\bf m^2_{\sbar u}} = m^2_{\sbar u} {\bf 1},
\qquad\!\!\!\!
{\bf m^2_{\sbar d}} = m^2_{\sbar d} {\bf 1},
\qquad\!\!\!\!
{\bf m^2_{L}} = m^2_{L} {\bf 1},
\qquad\!\!\!\!
{\bf m^2_{\sbar e}} = m^2_{\sbar e} {\bf 1}
.\>\>\>\>{}
\label{scalarmassunification}
\eeq
Then all squark and slepton mixing angles are rendered trivial, because
squarks and sleptons with the same electroweak quantum numbers will be
degenerate in mass and can be rotated into each other at will.
Supersymmetric contributions to flavorchanging neutral current processes
will therefore be very small in such an idealized limit, up to mixing
induced by $\bf a_u$, $\bf a_d$, $\bf a_e$. Making the further assumption
that the (scalar)$^3$ couplings are each proportional to the corresponding
Yukawa coupling matrix,
\beq
{\bf a_u} = A_{u0} \,{\bf y_u}, \>\>\>\qquad
{\bf a_d} = A_{d0} \,{\bf y_d}, \>\>\>\qquad
{\bf a_e} = A_{e0} \,{\bf y_e},
\label{aunification}
\eeq
will ensure that only the squarks and sleptons of the third family can
have large (scalar)$^3$ couplings. Finally, one can avoid disastrously
large CPviolating effects by assuming that the soft parameters do not
introduce new complex phases. This is automatic for $m_{H_u}^2$ and
$m_{H_d}^2$, and for $m_Q^2$, $m_{\sbar u}^2$, etc.~if
eq.~(\ref{scalarmassunification}) is assumed; if they were not real
numbers, the Lagrangian would not be real. One can also fix $\mu$ in the
superpotential and $b$ in eq.~(\ref{MSSMsoft}) to be real, by appropriate
phase rotations of fermion and scalar components of the $H_u$ and $H_d$
supermultiplets. If one then assumes that
\beq
{\rm Im} (M_1),\, {\rm Im} (M_2),\, {\rm Im} (M_3),\,
{\rm Im} (A_{u0}),\, {\rm Im} (A_{d0}),\, {\rm Im} (A_{e0})
\>\> = \>\>
0,
\qquad{}
\label{commonphase}
\eeq
then the only CPviolating phase in the theory will be the usual CKM phase
found in the ordinary Yukawa couplings. Together, the conditions
eqs.~(\ref{scalarmassunification})(\ref{commonphase}) make up a rather
weak version of what is often called the hypothesis of {\it soft
supersymmetrybreaking universality}. The MSSM with these flavor and
CPpreserving relations imposed has far fewer parameters than the most
general case. Besides the usual Standard Model gauge and Yukawa coupling
parameters, there are 3 independent real gaugino masses, only 5 real
squark and slepton squared mass parameters, 3 real scalar cubic coupling
parameters, and 4 Higgs mass parameters (one of which can be traded for
the known electroweak breaking scale).
There are at least three other
possible types of explanations for the suppression of flavor violation in
the MSSM that could replace the universality hypothesis of
eqs.~(\ref{scalarmassunification})(\ref{commonphase}). They can be
referred to as the ``irrelevancy", ``alignment", and ``$R$symmetry"
hypotheses for the soft masses. The ``irrelevancy" idea is that the
sparticles masses are {\it extremely} heavy, so that their contributions
to flavorchanging and CPviolating diagrams like
Figures~\ref{fig:flavor}a,b are suppressed, as can be seen for example in
eqs.~(\ref{eq:muegamma})(\ref{eq:striking}). In practice, however,
if there is no flavorblind structure, the
degree of suppression needed typically requires $m_{\rm soft}$ much
larger than 1 TeV for at least some of the scalar masses. This seems to
go directly against the motivation for supersymmetry as a cure for the
hierarchy problem as discussed in the Introduction. Nevertheless, it has
been argued that this is a sensible possibility
\cite{Moreminimal,splitsusy}. The fact that the LHC searches conducted so
far have eliminated many models with lighter squarks anyway tends to make these
models seem more attractive. Perhaps a combination of approximate flavor
blindness and heavy superpartner masses is the true explanation
for the suppression of flavorviolating effects.
The ``alignment" idea is that the squark
squaredmass matrices do not have the flavorblindness indicated in
eq.~(\ref{scalarmassunification}), but are arranged in flavor space to be
aligned with the relevant Yukawa matrices in just such a way as to avoid
large flavorchanging effects \cite{cterms,alignmentmodels}. The
alignment models typically require rather special flavor symmetries.
The third possibility is that the theory is (approximately) invariant under a
continuous $U(1)_R$ symmetry \cite{Kribs:2007ac}. This requires that the
MSSM is supplemented, as in \cite{Fox:2002bu}, by additional chiral
supermultiplets in the adjoint representations of $SU(3)_c$, $SU(2)_L$,
and $U(1)_Y$, as well as an additional pair of Higgs chiral
supermultiplets. The gaugino masses in this theory are purely Dirac, of
the type in eq.~(\ref{eq:Diracgauginos}), and the couplings $\bf a_u$,
$\bf a_d$, and $\bf a_e$ are absent. This implies a very efficient
suppression of flavorchanging effects
\cite{Kribs:2007ac,Blechman:2008gu}, even if the squark and slepton mass
eigenstates are light, nondegenerate, and have large mixings in the
basis determined by the Standard Model quark and lepton mass eigenstates.
This can lead to unique and intriguing collider signatures
\cite{Kribs:2007ac,Plehn:2008ae}. However, we will not consider these
possibilities further here.
The softbreaking universality relations
eqs.~(\ref{scalarmassunification})(\ref{commonphase}), or stronger (more
special) versions of them, can be presumed to be the result of some
specific model for the origin of supersymmetry breaking, although there is
no consensus among theorists as to what the specific model
should actually be. In any case, they are indicative of an assumed
underlying simplicity or symmetry of the Lagrangian at some very high
energy scale $Q_0$. If we used this Lagrangian to compute masses and
crosssections and decay rates for experiments at ordinary energies near
the electroweak scale, the results would involve large logarithms of order
ln$(Q_0/m_Z)$ coming from loop diagrams. As is usual in quantum field
theory, the large logarithms can be conveniently resummed using
renormalization group (RG) equations, by treating the couplings and masses
appearing in the Lagrangian as running parameters. Therefore,
eqs.~(\ref{scalarmassunification})(\ref{commonphase}) should be
interpreted as boundary conditions on the running soft parameters at the
scale $Q_0$, which is likely very far removed from direct experimental
probes. We must then RGevolve all of the soft parameters, the
superpotential parameters, and the gauge couplings down to the electroweak
scale or comparable scales where humans perform experiments.
At the electroweak scale, eqs.~(\ref{scalarmassunification}) and
(\ref{aunification}) will no longer hold, even if they were exactly true
at the input scale $Q_0$. However, to a good approximation, key flavor
and CPconserving properties remain. This is because, as we will see in
section \ref{subsec:RGEs} below, RG corrections due to gauge interactions
will respect the form of
eqs.~(\ref{scalarmassunification}) and (\ref{aunification}),
while RG corrections due to Yukawa interactions are quite small except for
couplings involving the top, bottom, and tau flavors. Therefore, the
(scalar)$^3$ couplings and scalar squaredmass mixings should be quite
negligible for the squarks and sleptons of the first two families.
Furthermore, RG evolution does not introduce new CPviolating phases.
Therefore, if universality can be arranged to hold at the input scale,
supersymmetric contributions to flavorchanging and CPviolating
observables can be acceptably small in comparison to present limits
(although quite possibly measurable in future experiments).
One good reason to be optimistic that such a program can succeed is the
celebrated apparent unification of gauge couplings in the MSSM
\cite{gaugeunification}. The 1loop RG equations for the Standard Model
gauge couplings $g_1, g_2, g_3$ are
\beq
\beta_{g_a} \equiv {d\over dt} g_a = {1\over 16\pi^2} b_a g_a^3,
\qquad\quad
(b_1, b_2, b_3) =
\left \{ \begin{array}{ll}
(41/10,\> 19/6,\> 7) & \mbox{Standard Model}\\
(33/5,\> 1,\> 3) & \mbox{MSSM}
\end{array}
\right.
\label{mssmg}
\eeq
where $t= {\rm ln} (Q/Q_0)$, with $Q$ the RG scale. The MSSM
coefficients are larger because of the extra MSSM
particles in loops. The normalization for $g_1$ here is chosen to agree
with the canonical covariant derivative for grand unification of the gauge
group $SU(3)_C \times SU(2)_L\times U(1)_Y$ into $SU(5)$ or $SO(10)$. Thus
in terms of the conventional electroweak gauge couplings $g$ and
$g^\prime$ with $e = g\sin\theta_W = g^\prime \cos\theta_W$, one has
$g_2=g$ and $g_1 = \sqrt{5/3} g^\prime$. The quantities $\alpha_a =
g_a^2/4\pi$ have the nice property that their reciprocals run linearly
with RG scale at oneloop order:
\beq
{d\over dt} \alpha_a^{1} = {b_a\over 2\pi} \qquad\qquad (a=1,2,3)
\qquad
\label{mssmgrecip}
\eeq
Figure \ref{fig:gaugeunification} compares the RG evolution of the
$\alpha_a^{1}$, including twoloop effects, in the Standard Model (dashed
lines) and the MSSM (solid lines).
\begin{figure}
\begin{minipage}[]{0.355\linewidth}
\caption{Twoloop renormalization group evolution of the inverse gauge
couplings
$\alpha_a^{1}(Q)$ in
the Standard Model (dashed lines) and the MSSM (solid lines). In the MSSM
case, the sparticle masses are treated as a common threshold varied
between 750 GeV and 2.5 TeV, and $\alpha_3(m_Z)$ is varied between
$0.117$ and $0.120$.
\label{fig:gaugeunification}}
\end{minipage}
\begin{minipage}[]{0.64\linewidth}
\hspace{.06\linewidth}
{\psfig{figure=unification.eps,width=0.9\linewidth}}
\end{minipage}
\end{figure}%
Unlike the Standard Model, the MSSM
includes just the right particle content to ensure that the gauge
couplings can unify, at a scale $M_U \sim 1.5\times 10^{16}$ GeV.
This unification is of course not perfect; $\alpha_3$ tends to be
slightly smaller than the common value of $\alpha_1(M_U) = \alpha_2(M_U)$
at the point where they meet, which is often taken to be the definition
of $M_U$. However, this small difference can easily be ascribed to
threshold corrections due to whatever new particles exist near $M_U$.
Note that $M_U$ decreases slightly as the superpartner masses are
raised. While
the apparent approximate unification of gauge couplings at $M_U$ might be
just an
accident, it may also be taken as a strong hint in favor of a grand
unified theory (GUT) or superstring models, both of which can naturally
accommodate gauge coupling unification below $\MPlanck$. Furthermore, if
this hint is taken seriously, then we can reasonably expect to be able to
apply a similar RG analysis to the other MSSM couplings and soft masses as
well. The next section discusses the form of the necessary RG equations.
\subsection{Renormalization Group equations for the MSSM\label{subsec:RGEs}}
\setcounter{footnote}{1}
\setcounter{equation}{0}
In order to translate a set of predictions at an input scale into
physically meaningful quantities that describe physics near the
electroweak scale, it is necessary to evolve the gauge couplings,
superpotential parameters, and soft terms using their renormalization
group (RG) equations. This ensures that the loop expansions for
calculations of observables will not suffer from very large logarithms.
As a technical aside, some care is required in choosing
regularization and renormalization procedures in supersymmetry.
The most popular regularization method for computations of radiative
corrections within the Standard Model is dimensional regularization
(DREG), in which the number of spacetime dimensions is continued to
$d=42\epsilon$. Unfortunately, DREG introduces a spurious violation of
supersymmetry, because it has a mismatch between the numbers of gauge
boson degrees of freedom and the gaugino degrees of freedom offshell.
This mismatch is only $2\epsilon$, but can be multiplied by factors up to
$1/\epsilon^n$ in an $n$loop calculation. In DREG, supersymmetric
relations between dimensionless coupling constants (``supersymmetric Ward
identities") are therefore not explicitly respected by radiative
corrections involving the finite parts of oneloop graphs and by the
divergent parts of twoloop graphs. Instead, one may use the slightly
different scheme known as regularization by dimensional reduction, or
DRED, which does respect supersymmetry \cite{DRED}. In the DRED method,
all momentum integrals are still performed in $d=42\epsilon$ dimensions,
but the vector index $\mu$ on the gauge boson fields $A^a_\mu$ now runs
over all 4 dimensions to maintain the match with the gaugino degrees of
freedom. Running couplings are then renormalized using DRED with modified
minimal subtraction ($\drbar$) rather than the usual DREG with modified
minimal subtraction ($\msbar$). In particular, the boundary conditions at
the input scale should presumably be applied in a supersymmetrypreserving
scheme like $\drbar$. One loop $\beta$functions are always the same in
these two schemes, but it is important to realize that the $\msbar$ scheme
does violate supersymmetry, so that $\drbar$ is preferred\footnote{Even
the DRED scheme may not provide a supersymmetric regulator, because of
either ambiguities or inconsistencies (depending on the precise method)
appearing at fiveloop order at the latest \cite{DREDdies}. Fortunately,
this does not seem to cause practical difficulties
\cite{JJperspective,Stockinger}. See also ref.~\cite{Woodard} for an
interesting proposal that avoids doing violence to the number of spacetime
dimensions.} from that point of view. (The NSVZ scheme \cite{Shifman} also
respects supersymmetry and has some very useful properties, but with a
less obvious connection to calculations of physical observables. It is
also possible, but not always very practical, to work consistently within
the $\overline{\rm MS}$ scheme, as long as one translates all
$\overline{\rm DR}$ couplings and masses into their $\overline{\rm MS}$
counterparts
%\cite{mstodrone,gluinopolemass,mstodrmore}.)
\cite{mstodrone}\cite{mstodrmore}.)
A general and powerful result known as the {\it supersymmetric
nonrenormalization theorem} \cite{nonrentheo} governs the form of the
renormalization group equations for supersymmetric theories. This theorem
implies that the logarithmically divergent contributions to a particular
process can always be written in terms of wavefunction renormalizations,
without any coupling vertex renormalization.\footnote{Actually, there {\it
is} vertex renormalization working in a supersymmetric gauge theory in
which auxiliary fields have been integrated out, but the sum of divergent
contributions for a process always has the form of wavefunction
renormalization. This is related to the fact that the anomalous dimensions
of the superfields differ, by gaugefixing dependent terms, from the
anomalous dimensions of the fermion and boson component fields
\cite{Jonesreview}.} It can be proved most easily using superfield
techniques. For the parameters appearing in the superpotential
eq.~(\ref{superpotentialwithlinear}), the implication is that
\beq
\beta_{y^{ijk}} \equiv \frac{d}{dt}y^{ijk} \!&=&\!
\gamma^i_n y^{njk} + \gamma^j_n y^{ink} + \gamma^k_n y^{ijn},
\label{eq:genyrge}
\\
\beta_{M^{ij}} \equiv \frac{d}{dt}M^{ij} \!&=&\!
\gamma^i_n M^{nj} + \gamma^j_n M^{in},
\label{eq:genMrge}
\\
\beta_{L^{i}} \equiv \frac{d}{dt}L^{i} \!&=&\!
\gamma^i_n L^{n},
\eeq
where the $\gamma^i_j$ are anomalous dimension matrices associated with
the superfields, which generally have to be calculated in a perturbative
loop expansion. [Recall $t = \ln(Q/Q_0)$, where $Q$ is the renormalization
scale, and $Q_0$ is a reference scale.] The anomalous dimensions and RG
equations for softly broken supersymmetry are now known up to 3loop
order, with some partial 4loop results; they have been given in
%refs.~\cite{rges2gauge,rges2superpot,rges1,twoloopsoft,threeloops,fourloops}.
refs.~\cite{rges2gauge}\cite{fourloops}.
There are also relations, good to all orders in perturbation
theory, that give the RG equations for soft supersymmetry couplings in
terms of those for the supersymmetric couplings \cite{Shifman,allorders}.
Here, for simplicity, only the 1loop approximation will be shown
explicitly.
In general, at 1loop order,
\beq
\gamma^i_j \,=\, \frac{1}{16 \pi^2} \left [
\half y^{imn} y^*_{jmn}  2 g_a^2 C_a(i)
\delta_j^i
\right ],
\label{eq:gengamma}
\eeq
where $C_a(i)$ are the quadratic Casimir group theory invariants for the
superfield $\Phi_i$, defined in terms of the Lie algebra generators $T^a$
by
\beq
(T^aT^a)_i{}^{j}= C_a(i) \delta_i^j
\label{eq:defCasimir}
\eeq
with gauge couplings $g_a$.
Explicitly, for the MSSM supermultiplets:
\beq
&&
C_3(i) =
\Biggl \{ \begin{array}{ll}
4/3 & {\rm for}\>\,\Phi_i = Q, \sbar u, \sbar d,
\\
0 & {\rm for}\>\,\Phi_i = L, \sbar e, H_u, H_d,
\end{array}
\label{defC3}
\\
&&
C_2(i) =
\Biggl \{ \begin{array}{ll}
3/4 & {\rm for}\>\,\Phi_i = Q, L, H_u, H_d,\\
0 & {\rm for}\>\,\Phi_i = \sbar u, \sbar d, \sbar e
,\end{array}
\\
&&
C_1(i) = \>
3 Y_i^2/5 \>\>\>{\rm for~each}\>\,\Phi_i\>\,{\rm
with~weak~hypercharge}\>\, Y_i.
\label{defC1}
\eeq
For the oneloop renormalization of gauge couplings, one has in
general
\beq
\beta_{g_a} =
{d\over dt} g_a
\!&=&\!
{1\over 16\pi^2} g_a^3 \Bigl [\sum_i I_a(i)  3 C_a(G) \Bigr ],
\eeq
where $C_a(G)$ is the quadratic Casimir invariant of the
group [0 for $U(1)$, and $N$ for $SU(N)$], and
$I_a(i)$ is the Dynkin index of the chiral supermultiplet $\phi_i$
[normalized to $1/2$ for each fundamental representation of $SU(N)$ and
to $3 Y_i^2/5$ for $U(1)_Y$]. Equation (\ref{mssmg})
is a special case of this.
The 1loop renormalization group equations for the
general soft supersymmetry breaking Lagrangian parameters appearing in
eq.~(\ref{lagrsoft}) are:
\beq
\beta_{M_a} \equiv
{d\over dt} M_a
\!&=&\!
{1\over 16\pi^2} g_a^2 \Bigl [2 \sum_n I_a(n)  6 C_a(G) \Bigr ] M_a
,
\\
\beta_{a^{ijk}} \equiv \frac{d}{dt} a^{ijk}
\!&=&\!
\frac{1}{16 \pi^2} \left [
\frac{1}{2} a^{ijp} y^*_{pmn} y^{kmn}
+ y^{ijp} y^*_{pmn} a^{mnk}
+ g_a^2 C_a(i) (4 M_a y^{ijk}  2 a^{ijk})
\right ]\phantom{xxxx}
\nonumber \\ &&
+ (i \leftrightarrow k)
+ (j \leftrightarrow k)
,\\
\beta_{b^{ij}} \equiv \frac{d}{dt} b^{ij}
\!&=&\!
\frac{1}{16 \pi^2}
\biggl [
\frac{1}{2} b^{ip} y^*_{pmn} y^{jmn}
+ \frac{1}{2} y^{ijp} y^*_{pmn} b^{mn}
+ M^{ip} y^*_{pmn} a^{mnj}
\nonumber \\ &&
+ g_a^2 C_a(i) (4 M_a M^{ij}  2 b^{ij})
\biggr ]
+ (i \leftrightarrow j)
,
\\
\beta_{t^{i}} \equiv \frac{d}{dt} t^{i}
\!&=&\!
\frac{1}{16 \pi^2} \left [
\frac{1}{2} y^{imn} y^*_{mnp} t^p
+ a^{imn} y^*_{mnp} L^p
+ M^{ip} y^*_{pmn} b^{mn}
\right ],
\\
\beta_{(m^2)_{i}^j} \equiv \frac{d}{dt} (m^2)_{i}^j
\!&=&\!
\frac{1}{16 \pi^2} \biggl [
\frac{1}{2} y_{ipq}^* y^{pqn} (m^2)_n^j
+ \frac{1}{2} y^{jpq} y_{pqn}^* (m^2)_i^n
+ 2 y^*_{ipq} y^{jpr} (m^2)_r^q
\nonumber \\ &&
+ a^*_{ipq} a^{jpq}
 8 g_a^2 C_a(i) M_a^2 \delta_i^j
+ 2 g_a^2 (T^a)_i{}^j
{\rm Tr}(T^a m^2)
\biggr ]
.
\eeq
Applying the above results to the special case of the MSSM, we will use the
approximation that only the thirdfamily Yukawa
couplings are significant, as in eq.~(\ref{heavytopapprox}). Then the
Higgs and thirdfamily superfield anomalous dimensions are diagonal
matrices, and from eq.~(\ref{eq:gengamma}) they are, at 1loop
order:
\beq
\gamma_{H_u} \!&=&\!
\frac{1}{16 \pi^2} \left [
3 y_t^* y_t  \frac{3}{2} g_2^2  \frac{3}{10} g_1^2
\right ],
\label{eq:gammaHu}
\\
\gamma_{H_d} \!&=&\! \frac{1}{16 \pi^2} \left [
3 y_b^* y_b + y_\tau^* y_\tau  \frac{3}{2} g_2^2
 \frac{3}{10} g_1^2
\right ],
\\
\gamma_{Q_3} \!&=&\! \frac{1}{16 \pi^2} \left [
y_t^* y_t + y_b^* y_b  \frac{8}{3} g_3^2
 \frac{3}{2} g_2^2  \frac{1}{30} g_1^2
\right ],
\\
\gamma_{\sbar{u}_3} \!&=&\! \frac{1}{16 \pi^2} \left [
2 y_t^* y_t \frac{8}{3} g_3^2 \frac{8}{15}
g_1^2
\right ],
\\
\gamma_{\sbar{d}_3} \!&=&\! \frac{1}{16 \pi^2} \left [
2 y_b^* y_b \frac{8}{3} g_3^2 \frac{2}{15}
g_1^2
\right ],
\\
\gamma_{L_3} \!&=&\! \frac{1}{16 \pi^2} \left [
y_\tau^* y_\tau  \frac{3}{2} g_2^2  \frac{3}{10}
g_1^2
\right ],
\\
\gamma_{\sbar{e}_3} \!&=&\! \frac{1}{16 \pi^2} \left [
2 y_\tau^* y_\tau \frac{6}{5} g_1^2
\right ].
\label{eq:gammae}
\eeq
[The first and second family anomalous dimensions in the approximation of
eq.~(\ref{heavytopapprox}) follow by setting $y_t$, $y_b$, and $y_\tau$ to
$0$ in the above.] Putting these into eqs.~(\ref{eq:genyrge}),
(\ref{eq:genMrge}) gives the running of the superpotential parameters with
renormalization scale:
\beq
\beta_{y_t} \equiv
{d\over dt} y_t \!&=&\! {y_t \over 16 \pi^2} \Bigl [ 6 y_t^* y_t + y_b^* y_b
 {16\over 3} g_3^2  3 g_2^2  {13\over 15} g_1^2 \Bigr ],
\label{eq:betayt}
\\
\beta_{y_b} \equiv
{d\over dt} y_b \!&=&\! {y_b \over 16 \pi^2}
\Bigl [ 6 y_b^* y_b + y_t^* y_t +
y_\tau^* y_\tau  {16\over 3} g_3^2  3 g_2^2  {7\over 15} g_1^2 \Bigr ],
\\
\beta_{y_\tau} \equiv
{d\over dt} y_\tau \!&=&\! {y_\tau \over 16 \pi^2}
\Bigl [ 4 y_\tau^* y_\tau
+ 3 y_b^* y_b  3 g_2^2  {9\over 5} g_1^2 \Bigr ],
\\
\beta_{\mu} \equiv
{d\over dt} \mu \!&=&\! {\mu \over 16 \pi^2}
\Bigl [ 3 y_t^* y_t + 3 y_b^* y_b
+ y_\tau^* y_\tau  3 g_2^2  {3\over 5} g_1^2 \Bigr ].
\label{eq:betamu}
\eeq
The oneloop RG equations for the gauge couplings $g_1$, $g_2$, and $g_3$
were already listed in eq.~(\ref{mssmg}). The presence of soft
supersymmetry breaking does not affect eqs.~(\ref{mssmg}) and
(\ref{eq:betayt})(\ref{eq:betamu}). As a result of the
supersymmetric nonrenormalization theorem, the $\beta$functions
for each supersymmetric
parameter are proportional to the parameter itself. One consequence of
this is that once we have a theory that can explain why $\mu$ is of order
$10^2$ or $10^3$ GeV at treelevel, we do not have to worry about $\mu$
being made very large by radiative corrections involving the masses of
some very heavy unknown particles; all such RG corrections to $\mu$ will
be directly proportional to $\mu$ itself and to some combinations of
dimensionless couplings.
The oneloop RG equations for the three gaugino mass parameters in the
MSSM are determined by the same quantities $b_a^{\rm MSSM}$ that appear in
the gauge coupling RG eqs.~(\ref{mssmg}):
\beq
\beta_{M_a} \equiv
{d\over dt} M_a \,=\, {1\over 8\pi^2} b_a g_a^2 M_a\qquad\>\>\>
(b_a = 33/5, \>1,\>3)
\label{gauginomassrge}
\eeq
for $a=1,2,3$. It follows that the three ratios $M_a/g_a^2$ are each
constant (RG scale independent) up to small twoloop corrections. Since
the gauge couplings are observed to unify at $Q = M_U = 1.5 \times 10^{16}$
GeV, it is a popular assumption that the gaugino masses also
unify\footnote{In GUT models, it is automatic that the gauge couplings
and gaugino masses are unified at all scales $Q\geq M_U$, because in the
unified theory the gauginos all live in the same representation of the
unified gauge group. In many superstring models, this can also be a good
approximation.} near that scale, with a value called $m_{1/2}$.
If so, then it follows that
\beq
{M_1 \over g_1^2} =
{M_2 \over g_2^2} =
{M_3 \over g_3^2} = {m_{1/2} \over g_U^2}
\label{gauginomassunification}
\eeq
at any RG scale, up to small (and known) twoloop effects and possibly
much larger (and unknown) threshold effects near $M_U$. Here $g_U$ is
the unified gauge coupling at $Q = M_U$. The hypothesis of
eq.~(\ref{gauginomassunification}) is particularly powerful because the
gaugino mass parameters feed strongly into the RG equations for all of the
other soft terms, as we are about to see.
Next we consider the 1loop RG equations for the holomorphic soft parameters
${\bf a_u}$, ${\bf a_d}$, ${\bf a_e}$. In models obeying
eq.~(\ref{aunification}), these matrices start off proportional to the
corresponding Yukawa couplings at the input scale. The RG evolution
respects this property. With the approximation of
eq.~(\ref{heavytopapprox}), one can therefore also write, at any RG scale,
\beq
{\bf a_u} \approx \pmatrix{0&0&0\cr 0&0&0 \cr 0&0&a_t},\qquad\!\!
{\bf a_d} \approx \pmatrix{0&0&0\cr 0&0&0 \cr 0&0&a_b},\qquad\!\!
{\bf a_e} \approx \pmatrix{0&0&0\cr 0&0&0 \cr 0&0&a_\tau},\>\>{}
\label{heavyatopapprox}
\eeq
which defines\footnote{Rescaled soft parameters $A_t = a_t/y_t$,
$A_b=a_b/y_b$, and $A_\tau=a_\tau/y_\tau$ are often used in the
literature. We do not follow this notation, because it cannot be
generalized beyond the approximation of eqs. (\ref{heavytopapprox}),
(\ref{heavyatopapprox}) without introducing horrible complications such as
nonpolynomial RG equations, and because $a_t$, $a_b$ and $a_\tau$ are the
couplings that actually appear in the Lagrangian anyway.} running
parameters $a_t$, $a_b$, and $a_\tau$. In this approximation,
the RG equations for these
parameters and $b$ are
\beq
16\pi^2 {d\over dt} a_t \!&=&\! a_t \Bigl [ 18 y_t^* y_t + y_b^* y_b
 {16\over 3} g_3^2  3 g_2^2  {13\over 15} g_1^2 \Bigr ]
+ 2 a_b y_b^* y_t
\nonumber\\ &&
\!+ y_t \Bigl [ {32\over 3} g_3^2 M_3 + 6 g_2^2 M_2 + {26\over 15} g_1^2 M_1
\Bigr ],
\label{atrge}
\\
16\pi^2{d\over dt} a_b \!&=&\! a_b \Bigl [ 18 y_b^* y_b + y_t^* y_t +
y_\tau^* y_\tau
 {16\over 3} g_3^2  3 g_2^2  {7\over 15} g_1^2 \Bigr ]
+ 2 a_t y_t^* y_b + 2 a_\tau y_\tau^* y_b
\phantom{xxxx}
\nonumber \\&&
\!+ y_b \Bigl [ {32\over 3} g_3^2 M_3 + 6 g_2^2 M_2 + {14 \over 15} g_1^2 M_1
\Bigr ],\qquad{}
\\
16\pi^2{d\over dt} a_\tau \!&=&\! a_\tau \Bigl [ 12 y_\tau^* y_\tau
+ 3 y_b^* y_b  3 g_2^2  {9\over 5} g_1^2 \Bigr ]
+ 6 a_b y_b^* y_\tau
+ y_\tau \Bigl [ 6 g_2^2 M_2 + {18\over 5} g_1^2 M_1 \Bigr ],
\\
16\pi^2{d\over dt} b \!&=&\! b \Bigl [ 3 y_t^* y_t + 3 y_b^* y_b
+ y_\tau^* y_\tau  3 g_2^2  {3\over 5} g_1^2 \Bigr ]
\nonumber \\ &&
\!+ \mu \Bigl [ 6 a_t y_t^* + 6 a_b y_b^* + 2 a_\tau y_\tau^* +
6 g_2^2 M_2 + {6\over 5} g_1^2 M_1 \Bigr ] .
\label{brge}
\eeq
The $\beta$function for each of these soft
parameters is {\it not} proportional to the parameter itself, because
couplings that violate supersymmetry are not protected by the
supersymmetric nonrenormalization theorem. So, even if $a_t$, $a_b$,
$a_\tau$ and $b$ vanish at the input scale, the RG corrections
proportional to gaugino masses appearing in
eqs.~(\ref{atrge})(\ref{brge}) ensure that they will not vanish at
the electroweak scale.
Next let us consider the RG equations for the scalar squared masses in the
MSSM. In the approximation of eqs.~(\ref{heavytopapprox}) and
(\ref{heavyatopapprox}), the squarks and sleptons of the first two
families have only gauge interactions. This means that if the scalar
squared masses satisfy a boundary condition like
eq.~(\ref{scalarmassunification}) at an input RG scale, then when
renormalized to any other RG scale, they will still be almost diagonal,
with the approximate form
\beq
{\bf m_Q^2} \approx \pmatrix{
m_{Q_1}^2 & 0 & 0\cr
0 & m_{Q_1}^2 & 0 \cr
0 & 0 & m_{Q_3}^2 \cr},\qquad\>\>\>
{\bf m_{\sbar u}^2} \approx \pmatrix{
m_{\sbar u_1}^2 & 0 & 0\cr
0 & m_{\sbar u_1}^2 & 0 \cr
0 & 0 & m_{\sbar u_3}^2 \cr},
\eeq
etc. The first and second family squarks and sleptons with given gauge
quantum numbers remain very nearly degenerate, but the thirdfamily
squarks and sleptons feel the effects of the larger Yukawa couplings and
so their squared masses get renormalized differently. The oneloop RG
equations for the first and second family squark and slepton squared
masses are
\beq
16 \pi^2 {d\over dt} m_{\phi_i}^2 \>\,=\>\,
\!\sum_{a=1,2,3} 8 C_a (i) g_a^2 M_a^2
+ \frac{6}{5} Y_i g^{2}_1 S
\label{easyscalarrge}
\eeq
for each scalar $\phi_i$, where the $\sum_a$ is over the three gauge
groups $U(1)_Y$, $SU(2)_L$ and $SU(3)_C$, with Casimir invariants $C_a(i)$
as in eqs.~(\ref{defC3})(\ref{defC1}), and $M_a$ are the corresponding
running gaugino mass parameters. Also,
\beq
S \equiv {\rm Tr}[Y_j m^2_{\phi_j}] =
m_{H_u}^2  m_{H_d}^2 + {\rm Tr}[
{\bf m^2_Q}  {\bf m^2_L}  2 {\bf m^2_{\overline u}}
+ {\bf m^2_{\overline d}} + {\bf m^2_{\overline e}}] .
\label{eq:defS}
\eeq
An important feature of eq.~(\ref{easyscalarrge}) is that the terms on the
righthand sides proportional to gaugino squared masses are negative,
so\footnote{The contributions
proportional to $S$ are
relatively small in most known realistic models.} the scalar squaredmass
parameters grow as they are RGevolved from the input scale down to the
electroweak scale. Even if the scalars have zero or very small masses at
the input scale, they can obtain large positive squared masses at the
electroweak scale, thanks to the effects of the gaugino masses.
\setcounter{footnote}{1}
The RG equations for the squaredmass parameters of the Higgs scalars and
thirdfamily squarks and sleptons get the same gauge contributions as in
eq.~(\ref{easyscalarrge}), but they also have contributions due to the
large Yukawa ($y_{t,b,\tau}$) and soft ($a_{t,b,\tau}$) couplings. At
oneloop order, these only appear in three combinations:
\beq
X_t \!&=&\! 2 y_t^2 (m_{H_u}^2 + m_{Q_3}^2 + m_{\sbar u_3}^2) +2 a_t^2,
\\
X_b \!&=& \! 2 y_b^2 (m_{H_d}^2 + m_{Q_3}^2 + m_{\sbar d_3}^2) +2 a_b^2,
\\
X_\tau\! &=&\! 2 y_\tau^2 (m_{H_d}^2 + m_{L_3}^2 + m_{\sbar e_3}^2)
+ 2 a_\tau^2.
\eeq
In terms of these quantities, the RG equations for the soft Higgs
squaredmass parameters $m_{H_u}^2$ and $m_{H_d}^2$ are
\beq
16 \pi^2 {d\over dt} m_{H_u}^2 \!&=&\!
3 X_t  6 g_2^2 M_2^2  {6\over 5} g_1^2 M_1^2 + \frac{3}{5} g^{2}_1 S
,
\label{mhurge}
\\
16\pi^2{d\over dt} m_{H_d}^2 \!&=&\!
3 X_b + X_\tau  6 g_2^2 M_2^2  {6\over 5} g_1^2 M_1^2  \frac{3}{5}
g^{2}_1 S
.
\label{mhdrge}
\eeq
Note that $X_t$, $X_b$, and $X_\tau$ are generally positive, so their
effect is to decrease the Higgs squared masses as one evolves the RG equations
down from the input scale to the electroweak scale. If $y_t$ is the
largest of the Yukawa couplings, as suggested by the experimental fact
that the top quark is heavy, then $X_t$ will typically be much larger than
$X_b$ and $X_\tau$. This can cause the RGevolved $m_{H_u}^2$ to run
negative near the electroweak scale, helping to destabilize the point $H_u
= H_d = 0$ and so provoking a Higgs VEV (for a linear combination of $H_u$
and $H_d$, as we will see in section \ref{subsec:MSSMspectrum.Higgs}),
which is just what we want.\footnote{One should think of ``$m_{H_u}^2$" as
a parameter unto itself, and not as the square of some mythical real
number $m^{\phantom{2}}_{H_u}$. So there is nothing strange about having
$m_{H_u}^2 < 0$.
%However, strictly speaking $m_{H_u}^2< 0$ is neither
%necessary nor sufficient for electroweak symmetry breaking; see section
%\ref{subsec:MSSMspectrum.Higgs}.
} Thus a large top Yukawa coupling favors
the breakdown of the electroweak symmetry breaking because it induces
negative radiative corrections to the Higgs squared mass.
The thirdfamily squark and slepton squaredmass parameters also get
contributions that depend on $X_t$, $X_b$ and $X_\tau$. Their RG equations
are given by
\beq
16\pi^2{d\over dt} m_{Q_3}^2 \!&=&\!
X_t +X_b{32\over 3} g_3^2 M_3^2 6 g_2^2 M_2^2 {2\over 15} g_1^2 M_1^2
+ \frac{1}{5} g^{2}_1 S ,
\label{mq3rge}
\\
16\pi^2 {d\over dt} m_{\sbar u_3}^2 \!&=&\!
2 X_t  {32\over 3} g_3^2 M_3^2  {32\over 15} g_1^2M_1^2
 \frac{4}{5} g^{2}_1 S ,
\label{mtbarrge}
\\
16\pi^2 {d\over dt} m_{\sbar d_3}^2 \!&=&\!
2 X_b  {32\over 3} g_3^2 M_3^2  {8\over 15} g_1^2M_1^2
+ \frac{2}{5} g^{2}_1 S ,
\label{md3rge}
\\
16\pi^2 {d\over dt} m_{L_3}^2 \!&=&\!
X_\tau  6 g_2^2 M_2^2  {6\over 5} g_1^2 M_1^2  \frac{3}{5} g^{2}_1 S,
\\
16\pi^2 {d\over dt} m_{\sbar e_3}^2 \!&=&\!
2 X_\tau  {24\over 5} g_1^2 M_1^2 + \frac{6}{5} g^{2}_1 S .
\label{mstaubarrge}
\eeq
In eqs.~(\ref{mhurge})(\ref{mstaubarrge}), the terms proportional to
$M_3^2$, $M_2^2$, $M_1^2$, and $S$ are just the same ones as in
eq.~(\ref{easyscalarrge}). Note that the terms proportional to $X_t$ and
$X_b$ appear with smaller numerical coefficients in the $m^2_{Q_3}$,
$m^2_{\sbar u_3}$, $m^2_{\sbar d_3}$ RG equations than they did for the
Higgs scalars, and they do not appear at all in the $m^2_{L_3}$ and
$m^2_{\sbar e_3}$ RG equations. Furthermore, the thirdfamily squark
squared masses get a large positive contribution proportional to $M_3^2$
from the RG evolution, which the Higgs scalars do not get. These facts
make it plausible that the Higgs scalars in the MSSM get VEVs, while the
squarks and sleptons, having large positive squared mass, do not.
An examination of the RG equations (\ref{atrge})(\ref{brge}),
(\ref{easyscalarrge}), and (\ref{mhurge})(\ref{mstaubarrge}) reveals that
if the gaugino mass parameters $M_1$, $M_2$, and $M_3$ are nonzero at the
input scale, then all of the other soft terms will be generated too. This
implies that models in which gaugino masses dominate over all other
effects in the soft supersymmetry breaking Lagrangian at the input scale
can be viable. On the other hand, if the gaugino masses were to vanish at
treelevel, then they would not get any contributions to their masses at
oneloop order; in that case the gauginos would be extremely light and the
model would not be phenomenologically acceptable.
Viable models for the origin of supersymmetry breaking typically make
predictions for the MSSM soft terms that are refinements of
eqs.~(\ref{scalarmassunification})(\ref{commonphase}). These predictions
can then be used as boundary conditions for the RG equations listed above.
In the next section we will study the ideas that go into making such
predictions, before turning to their implications for the MSSM spectrum in
section \ref{sec:MSSMspectrum}.
\section{Origins of supersymmetry breaking}\label{sec:origins}
\subsection{General considerations for spontaneous
supersymmetry breaking}\label{subsec:origins.general}
\setcounter{equation}{0}
\setcounter{figure}{0}
\setcounter{table}{0}
\setcounter{footnote}{1}
In the MSSM, supersymmetry breaking is simply introduced explicitly.
However, we have seen that the soft parameters cannot be arbitrary. In
order to understand how patterns like eqs.~(\ref{scalarmassunification}),
(\ref{aunification}) and (\ref{commonphase}) can emerge, it is necessary
to consider models in which supersymmetry is spontaneously broken. By
definition, this means that the vacuum state $\vac$ is not invariant under
supersymmetry transformations, so $Q_\alpha \vac \not= 0$ and
$Q^\dagger_{\dot{\alpha}}\vac \not=0$. Now, in global supersymmetry, the
Hamiltonian operator $H$ is related to the supersymmetry generators
through the algebra eq.~(\ref{nonschsusyalg1}):
\beq
H=P^0 =
{1\over 4}( Q_1 Q_{{1}}^\dagger + Q_{{1}}^\dagger Q_1
+ Q_2 Q_{{2}}^\dagger + Q_{{2}}^\dagger Q_2 ) .
\eeq
If supersymmetry is unbroken in the vacuum state, it follows that $H\vac =
0$ and the vacuum has zero energy. Conversely, if supersymmetry is
spontaneously broken in the vacuum state, then the vacuum must have
positive energy, since
\beq
\antivac H \vac = {1\over 4} \Bigl (\ Q_1^\dagger \vac \^2 +
\ Q_{{1}} \vac \^2
+ \ Q^\dagger_{2} \vac \^2
+ \ Q_{{2}} \vac \^2
\Bigr ) > 0
\eeq
if the Hilbert space is to have positive norm. If spacetimedependent
effects and fermion condensates can be neglected, then $\antivac H\vac =
\antivac V \vac $, where $V$ is the scalar potential in eq.~(\ref{fdpot}).
Therefore, supersymmetry will be spontaneously broken if the
expectation value of $F_i$ and/or
$D^a$ does not vanish in the vacuum state.
If any state exists in which all $F_i$ and $D^a$ vanish, then it will
have zero energy, implying that supersymmetry is not spontaneously broken
in the true ground state. Conversely, one way to guarantee spontaneous
supersymmetry breaking is to look for models in which the equations
$F_i=0$ and $D^a=0$ cannot all be simultaneously satisfied for {\it any}
values of the fields. Then the true ground state necessarily has broken
supersymmetry, as does the vacuum state we live in (if it is different).
However, another possibility is that the vacuum state in which we live is
not the true ground state (which may preserve supersymmetry), but is
instead a higher energy metastable supersymmetrybreaking state with
lifetime at least of order the present age of the universe
\cite{Ellis:1982vi}\cite{Intriligator:2006dd}. Finite temperature
effects can indeed cause the early universe to prefer the metastable
supersymmetrybreaking local minimum of the potential over the
supersymmetrybreaking global minimum \cite{metastableearlyuniverse}.
Scalar potentials for the
three possibilities are illustrated qualitatively in
Figure \ref{fig:susybreakingpotentials}.
\begin{figure}
\begin{picture}(130,130)(0,0)
\SetScale{0.85}
\LongArrow(75,30)(150,30)
\LongArrow(75,30)(0,30)
\LongArrow(75,30)(75,150)
\LongArrow(75,30)(75,10)
\Text(48,128)[c]{$V(\phi)$}
\Text(130,35)[c]{$\phi$}
\SetWidth{1.1}
\Curve{(0,150)(75,30)(150,150)}
\rText(66,4)[][]{(a)}
\end{picture}
\hspace{1.1cm}
\begin{picture}(130,130)(0,0)
\SetScale{0.85}
\LongArrow(75,30)(150,30)
\LongArrow(75,30)(0,30)
\LongArrow(75,30)(75,150)
\LongArrow(75,30)(75,10)
\Text(48,128)[c]{$V(\phi)$}
\Text(130,35)[c]{$\phi$}
\SetWidth{1.1}
\Curve{(5,150)(75,56)(145,150)}
\rText(66,4)[][]{(b)}
\end{picture}
\hspace{1.1cm}
\begin{picture}(130,130)(0,0)
\SetScale{0.85}
\LongArrow(75,30)(150,30)
\LongArrow(75,30)(0,30)
\LongArrow(75,30)(75,150)
\LongArrow(75,30)(75,10)
\Text(48,128)[c]{$V(\phi)$}
\Text(130,35)[c]{$\phi$}
\SetWidth{1.1}
%\Curve{(5,150)(41,87)(87,60)(100,100)(120,30.5)(150,150)}
\Curve{(5,150)(40,87)(86,60)(100,80)(120,30.5)(150,150)}
\rText(66,4)[][]{(c)}
\end{picture}
\vspace{0.15cm}
\caption{Scalar potentials for (a) unbroken supersymmetry,
(b) spontaneously broken supersymmetry, and (c) metastable supersymmetry
breaking, as functions of an order parameter $\phi$.
\label{fig:susybreakingpotentials}}
\end{figure}
Regardless of whether the vacuum state is stable or metastable,
the spontaneous breaking of a global symmetry always implies a
massless NambuGoldstone mode with the same quantum numbers as the broken
symmetry generator. In the case of global supersymmetry, the broken
generator is the fermionic charge $Q_\alpha$, so the NambuGoldstone
particle ought to be a massless neutral Weyl fermion, called the {\it
goldstino}. To prove it, consider a general supersymmetric model with
both gauge and chiral supermultiplets as in section \ref{sec:susylagr}.
The fermionic degrees of freedom consist of gauginos ($\lambda^a$) and
chiral fermions ($\psi_i$). After some of the scalar fields in the theory
obtain VEVs, the fermion mass matrix has the form:
\beq
{\bf m}_{\rm F} =
\pmatrix{
0 & \sqrt{2} g_b (\langle \phi^{*}\rangle T^b)^i
\cr
\sqrt{2} g_a (\langle \phi^{*}\rangle T^a)^j & \langle W^{ji} \rangle
}
\label{eq:MFwithSSB}
\eeq
in the $(\lambda^a,\,\psi_i)$ basis. [The offdiagonal entries in this
matrix come from the first term in the second line of
eq.~(\ref{gensusylagr}), and the lower right entry can be seen in
eq.~(\ref{noFlagr}).] Now observe that ${\bf m}_{\rm F}$ annihilates the
vector
\beq
{\stilde G} \,=\, \pmatrix{
{\langle D^a \rangle /\sqrt{2}} \cr \langle F_i\rangle }.
\label{explicitgoldstino}
\eeq
The first row of ${\bf m}_{\rm F}$ annihilates $\stilde G$ by virtue of
the requirement eq.~(\ref{wgaugeinvar}) that the superpotential is gauge
invariant, and the second row does so because of the condition $ \langle
{\partial V/ \partial \phi_i} \rangle = 0 , $ which must be satisfied at any
local minimum of the scalar potential. Equation (\ref{explicitgoldstino})
is therefore proportional to the goldstino wavefunction; it is
nontrivial if
and only if at least one of the auxiliary fields has a VEV, breaking
supersymmetry. So we have proved that if global supersymmetry is
spontaneously broken, then there must be a massless goldstino, and that
its components among the various fermions in the theory are just
proportional to the corresponding auxiliary field VEVs.
There is also a useful sum rule that governs the treelevel squared masses
of particles in theories with spontaneously broken supersymmetry. For a
general theory of the type discussed in section \ref{sec:susylagr}, the
squared masses of the real scalar degrees of freedom are the eigenvalues
of the matrix
\beq
{\bf m}^2_{\rm S} =
\pmatrix{
W_{jk}^* W^{ik} + g^2_a (T^a \phi)_j (\phi^* T^a)^i g_a T^{ai}_j D^a
&
W_{ijk}^* W^k + g^2_a (T^a \phi)_i (T^a \phi)_j
\vspace{0.1cm}
\cr
W^{ijk} W_k^* + g^2_a (\phi^* T^a)^i (\phi^* T^a)^j
&
W_{ik}^* W^{jk} + g^2_a (T^a \phi)_i (\phi^* T^a)^j  g_a T^{aj}_i D^a
},\phantom{x}
\label{eq:rescalarmasssq}
\eeq
which can be obtained from writing the quadratic part of the treelevel potential as
\beq
V \,=\, \frac{1}{2} \pmatrix{\phi^{*j} \!\!& \phi_j} {\bf m}^2_{\rm S}
\pmatrix{\phi_{i} \cr \phi^{*i}} .
\eeq
In eq.~(\ref{eq:rescalarmasssq}),
$W^{ijk} = \delta^3 W/\delta \phi_i\delta \phi_j\delta \phi_k$, and
the scalar fields
are understood to be replaced by their VEVs. It follows that the sum of
the real scalar squaredmass eigenvalues is
\beq
{\rm Tr}({\bf m}^2_{\rm S}) \,=\,
2 W_{ik}^* W^{ik} + 2 g^2_a C_a (i) \phi^{*i} \phi_i
2 g_a {\rm Tr}(T^{a}) D^a ,
\eeq
with the Casimir invariants $C_a(i)$ defined by eq.~(\ref{eq:defCasimir}).
Meanwhile, the squared masses of the twocomponent fermions are given by
the eigenvalues of
\beq
{\bf m}_{\rm F}^\dagger {\bf m}_{\rm F}\,=\,
\pmatrix{
2 g_a g_b (\phi^* T^a T^b \phi)
&
\sqrt{2} g_b (T^b \phi)_k W^{ik}
\vspace{0.1cm}
\cr
\sqrt{2} g_a (\phi^* T^a)^k W_{jk}^*
&
\phantom{x} W_{jk}^* W^{ik} + 2 g^2_c (T^c \phi)_j (\phi^* T^c)^i},
\label{eq:fermionmasssquared}
\eeq
so the sum of the twocomponent fermion squared masses is
\beq
{\rm Tr}({\bf m}_{\rm F}^\dagger {\bf m}_{\rm F}) \,=\,
W_{ik}^* W^{ik} + 4 g^2_a C_a (i) \phi^{*i} \phi_i .
\eeq
Finally, the vector squared masses are:
\beq
{\bf m}^2_{\rm V} \,=\, g_a^2 (\phi^* \{T^a, T^b\} \phi),
\eeq
so
\beq
{\rm Tr}({\bf m}^2_{\rm V}) \,=\, 2 g_a^2 C_a(i) \phi^{*i}\phi_i .
\eeq
It follows that the {\em supertrace} of the treelevel squaredmass
eigenvalues, defined in general by a weighted sum over all particles with
spin $j$:
\beq
{\rm STr}(m^2)
\,\equiv\,
\sum_j (1)^{2j} (2j +1) {\rm Tr}(m^2_j),
\label{eq:supertracedef}
\eeq
satisfies the sum rule
\beq
{\rm STr}(m^2)
\,=\,
{\rm Tr}({\bf m^2_{\rm S}})
 2 {\rm Tr}({\bf m}_{\rm F}^\dagger {\bf m}_{\rm F})
+ 3 {\rm Tr}({\bf m^2_{\rm V}})
\,=\, 2g_a {\rm Tr}(T^{a}) D^a\> =\> 0.
\label{eq:supertracesumrule}
\eeq
The last equality assumes that the traces of the $U(1)$ charges over the
chiral superfields are 0. This holds for $U(1)_Y$ in the MSSM, and more
generally for any nonanomalous gauge symmetry. The sum rule
eq.~(\ref{eq:supertracesumrule}) is often a useful check on models of
spontaneous supersymmetry breaking.
\subsection{FayetIliopoulos ($D$term) supersymmetry breaking
\label{subsec:origins.Dterm}}
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Supersymmetry breaking with a nonzero $D$term VEV can occur through the
FayetIliopoulos mechanism \cite{FayetIliopoulos}. If the gauge symmetry
includes a $U(1)$ factor, then, as noted in section \ref{subsec:superspacelagrabelian},
one can introduce a term linear in the
auxiliary field of the corresponding gauge supermultiplet,
\beq
\lagr_{\rm FI} \,=\, \kappa D,
\label{FI}
\eeq
where $\kappa$ is a constant with dimensions of [mass]$^2$. This
term is gaugeinvariant and supersymmetric by itself. [Note that for a
$U(1)$ gauge symmetry, the supersymmetry transformation $\delta D$ in
eq.~(\ref{Dtransf}) is a total derivative.] If we include it in the
Lagrangian, then $D$ may be forced to get a nonzero VEV. To see this,
consider the relevant part of the scalar potential from
eqs.~(\ref{lagrgauge}) and (\ref{gensusylagr}):
\beq
V = \kappa D {1\over 2} D^2  g D \sum_i q_i \phi_i^2 .
\eeq
Here the $q_i$ are the charges of the scalar fields $\phi_i$ under the
$U(1)$ gauge group in question. The presence of the FayetIliopoulos term
modifies the equation of motion eq.~(\ref{solveforD}) to
\beq
D = \kappa  g \sum_i q_i \phi_i^2.
\label{booya}
\eeq
Now suppose that the scalar fields $\phi_i$ that are charged under the
$U(1)$ all have nonzero superpotential masses $m_i$. (Gauge invariance
then requires that they come in pairs with opposite charges.) Then the
potential will have the form
\beq
V = \sum_i m_i^2 \phi_i^2 +
{1\over 2} (\kappa g \sum_i q_i \phi_i^2)^2 .
\eeq
Since this cannot vanish, supersymmetry must be broken; one can check that
the minimum always occurs for nonzero $D$. For the simplest case in which
$m_i^2 > g q_i \kappa$ for each $i$, the minimum is realized for all
$\phi_i=0$ and $D = \kappa$, with the $U(1)$ gauge symmetry unbroken. As
further evidence that supersymmetry has indeed been spontaneously broken,
note that the scalars then have squared masses $m_i^2  g q_i \kappa$,
while their fermion partners have squared masses $m_i^2$. The gaugino
remains massless, as can be understood from the fact that
it is the goldstino, as argued on general
grounds in section \ref{subsec:origins.general}.
For nonAbelian gauge groups, the analog of eq.~(\ref{FI}) would not be
gaugeinvariant and is therefore not allowed, so only $U(1)$ $D$terms can
drive spontaneous symmetry breaking. In the MSSM, one might imagine that
the $D$ term for $U(1)_Y$ has a FayetIliopoulos term as the principal
source of supersymmetry breaking. Unfortunately, this cannot work, because
the squarks and sleptons do not have superpotential mass terms. So, at
least some of them would just get nonzero VEVs in order to make
eq.~(\ref{booya}) vanish. That would break color and/or electromagnetism,
but not supersymmetry. Therefore, a FayetIliopoulos term for $U(1)_Y$
must be subdominant compared to other sources of supersymmetry breaking in
the MSSM, if not absent altogether. One could instead attempt to trigger
supersymmetry breaking with a FayetIliopoulos term for some other $U(1)$
gauge symmetry, which is as yet unknown because it is spontaneously broken
at a very high mass scale or because it does not couple to the Standard
Model particles. However, if this is the dominant source for supersymmetry
breaking, it proves difficult to give appropriate masses to all of the
MSSM particles, especially the gauginos. In any case, we will not discuss
$D$term breaking as the ultimate origin of supersymmetry violation any
further (although it may not be ruled out \cite{dtermbreakingmaywork}).
\subsection{O'Raifeartaigh ($F$term) supersymmetry breaking
\label{subsec:origins.Fterm}}
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Models where spontaneous supersymmetry breaking is ultimately due to a
nonzero $F$term VEV, called O'Rai\f\ear\taigh models
\cite{ORaifeartaigh}, have brighter phenomenological prospects. The idea
is to pick a set of chiral supermultiplets $\Phi_i\supset (\phi_i, \psi_i,
F_i)$ and a superpotential $W$ in such a way that the equations $F_i =
\delta W^*/\delta \phi^{*i} = 0$ have no simultaneous solution within some
compact domain. Then
$V=\sum_i F_i^2$ will have to be positive at its minimum, ensuring that
supersymmetry is broken. The supersymmetry breaking minimum may be
a global minimum of the potential as in Figure \ref{fig:susybreakingpotentials}(b), or only
a local minimum as in Figure \ref{fig:susybreakingpotentials}(c).
The simplest example with a supersymmetry breaking
global minimum has three chiral supermultiplets $\Phi_{1,2,3}$, with superpotential
\beq
W = k \Phi_1 + m \Phi_2 \Phi_3 + {y\over 2} \Phi_1 \Phi_3^2 .
\label{oraif}
\eeq
Note that $W$ contains a linear term, with $k$ having dimensions of
[mass]$^2$. Such a term is allowed if the corresponding chiral
supermultiplet is a gauge singlet. In fact, a linear term is necessary to
achieve $F$term breaking at treelevel in renormalizable
superpotentials,\footnote{Nonpolynomial superpotential terms, which
arise from nonperturbative effects in strongly coupled gauge theories,
avoid this requirement.} since
otherwise setting all $\phi_i=0$ will always give a supersymmetric global
minimum with all $F_i=0$. Without loss of generality, we can choose $k$,
$m$, and $y$ to be real and positive (by phase rotations of the fields).
The scalar potential following from eq.~(\ref{oraif}) is
\beq
&& V_{\rm treelevel} = F_1^2 + F_2^2 + F_3^2, \\
&& F_1 =
k  {y\over 2} \phi_3^{*2} ,\qquad
F_2 = m \phi^*_3 ,\qquad
F_3 = m \phi^*_2  y \phi^*_1 \phi^*_3 .
\eeq
Clearly, $F_1=0$ and $F_2=0$ are not compatible, so supersymmetry must
indeed be broken. If $m^2 > yk$ (which we assume from now on), then the
absolute minimum of the classical potential is at $\phi_2=\phi_3=0$ with $\phi_1$
undetermined, so $F_1 = k$ and $V_{\rm treelevel}=k^2$ at the minimum. The fact that
$\phi_1$ is undetermined at tree level is an example of a ``flat direction" in the
scalar potential; this is a common feature of supersymmetric
models.\footnote{More generally, flat directions, also known as moduli, are noncompact lines
and surfaces in the space of scalar fields along which the scalar
potential vanishes. The classical renormalizable scalar potential of the MSSM would have
many flat directions if supersymmetry were not broken
\cite{flatdirections}.}
The flat direction parameterized by $\phi_1$ is an accidental feature of
the classical scalar potential, and in this case it is removed (``lifted")
by quantum corrections. This can be seen by computing the ColemanWeinberg
oneloop effective potential \cite{ColemanWeinberg}. In a loop expansion,
the effective potential can be written as
\beq
V_{\rm eff} &=& V_{\rm treelevel} + V_{\rm 1loop} + \ldots
\label{eq:Veffexp}
\eeq
where the oneloop contribution is a supertrace over the
scalarfielddependent squaredmass eigenstates labeled $n$, with spin $s_n$:
\beq
V_{\rm 1loop} &=& \sum_n (1)^{2 s_n} (2 s_n + 1) h(m_n^2),
\\
h(z) &\equiv& \frac{1}{64 \pi^2} z^2 \left [\ln(z/Q^2) + a \right].
\label{eq:ColemanWeinberg}
\eeq
Here $Q$ is the renormalization scale and
$a$ is a renormalization schemedependent constant.\footnote{Actually, $a$ can be
different for the different spin contributions, if one chooses a renormalization
scheme that does not respect
supersymmetry. For example, in
the $\msbar$ scheme, $a = 3/2$ for the spin0 and spin$1/2$ contributions,
but $a = 5/6$ for
the spin1 contributions. See ref.~\cite{twoloopEP}
for a discussion, and the extension to twoloop order.}
In the $\drbar$ scheme based on dimensional reduction, $a=3/2$.
Using eqs.~(\ref{eq:rescalarmasssq}) and
(\ref{eq:fermionmasssquared}), the squared mass eigenvalues for the 6 real scalar and
3 twocomponent fermion
states are found to be, as a function of varying $x = \phi_1^2$, with $\phi_2=\phi_3=0$:
\beq
%\mbox{spin 0:}
\mbox{scalars:}
\quad&&
0
,\>\>
0
,\>\>\>
m^2 + \frac{y}{2} \Bigl (y x  k + \sqrt{4 m^2 x + (y x  k)^2} \Bigr ),
\nonumber
\\
&& m^2 + \frac{y}{2} \Bigl (y x + k  \sqrt{4 m^2 x + (y x + k)^2} \Bigr ),
\nonumber
\\
&& m^2 + \frac{y}{2} \Bigl (y x  k  \sqrt{4 m^2 x + (y x  k)^2} \Bigr ),
\nonumber
\\
&&
m^2 + \frac{y}{2} \Bigl (y x + k + \sqrt{4 m^2 x + (y x + k)^2} \Bigr ),
\\
%\mbox{spin}~1/2
\mbox{fermions}:\quad&&
0
,\>\>\>
m^2 + \frac{y}{2} \Bigl (y x + \sqrt{4 m^2 x + y^2 x^2} \Bigr )
,\>\>\>
m^2 + \frac{y}{2} \Bigl (y x  \sqrt{4 m^2 x + y^2 x^2} \Bigr ).\phantom{xxxx}
\eeq
[Note that the sum rule
eq.~(\ref{eq:supertracesumrule}) is indeed satisfied by these squared
masses.]
Now, plugging these into eq.~(\ref{eq:ColemanWeinberg}), one finds that the global minimum of the
oneloop
effective potential is at $x=0$, so $\phi_1 = \phi_2 = \phi_3 = 0$.
The treelevel mass
spectrum of the theory at this point in field space simplifies to
\beq
0,\>\> 0,\>\> m^2,\>\> m^2,\>\> m^2  yk,\>\> m^2 + yk ,
\label{ORscalars}
\eeq
for the scalars, and
\beq
0,\>\> m^2,\>\> m^2
\label{ORfermions}
\eeq
for the fermions.
The nondegeneracy of scalars and fermions is a clear check that
supersymmetry has been spontaneously broken.
The 0 eigenvalues in eqs.~(\ref{ORscalars}) and
(\ref{ORfermions}) correspond to the complex scalar $\phi_1$ and its
fermionic partner $\psi_1$. However, $\phi_1$ and $\psi_1$ have different
reasons for being massless. The masslessness of $\phi_1$ corresponds to
the existence of the classical flat direction, since any value of $\phi_1$ gives the
same energy at treelevel.
The oneloop potential
lifts this flat direction, so that $\phi_1$ gains a
mass once
quantum corrections are included. Expanding $V_{\rm 1loop}$ to first order in
$x$, one finds that
the complex scalar $\phi_1$ receives a
positivedefinite squared mass equal to
\beq
m_{\phi_1}^2 = {y^2 m^2\over 16 \pi^2} \left [
\ln (1  r^2)  1 + \frac{1}{2} \left ( r + 1/r \right )
\ln \left ( \frac{1 + r}{1r} \right )
\right ],
\label{eq:ORmsq}
\eeq
where $r = y k/m^2$. [This reduces to $m_{\phi_1}^2 =
y^4 k^2/48 \pi^2 m^2$ in the limit $yk\ll m^2$.]
In contrast, the Weyl
fermion $\psi_1$ remains exactly massless, to all orders in perturbation theory,
because it is the goldstino, as
predicted in section \ref{subsec:origins.general}.
The O'Rai\f\ear\taigh superpotential eq.~(\ref{oraif}) yields a Lagrangian that
is invariant under a
$U(1)_R$ symmetry (see section \ref{Rsymmetry}) with charge assignments
\beq
r_{\Phi_1} \,=\, r_{\Phi_2} \,=\, 2,\qquad
r_{\Phi_3} \,=\, 0.
\eeq
This illustrates a general result, the NelsonSeiberg theorem \cite{Nelson:1993nf},
which says that
if a theory has a scalar potential with a global minimum that breaks supersymmetry by a nonzero
$F$term, and the superpotential is generic
(contains all terms not forbidden by symmetries),
then the theory must have an exact $U(1)_R$
symmetry. If the $U(1)_R$ symmetry remains unbroken when supersymmetry breaks,
as is the case in the O'Rai\f\ear\taigh model discussed above, then
there is a
problem of explaining how gauginos get masses, because nonzero gaugino mass
terms have $R$charge
$2$. On the other hand, if the $U(1)_R$ symmetry is spontaneously broken,
then there results a pseudoNambuGoldstone boson (the $R$axion) which is problematic
experimentally, although gravitational effects may give it a large enough mass to avoid being
ruled out \cite{Raxion}.
If the supersymmetry breaking vacuum is only metastable, then one does not need an exact $U(1)_R$
symmetry. This can be illustrated by adding to the O'Rai\f\ear\taigh superpotential
eq.~(\ref{oraif}) a term $\Delta W$ that explicitly breaks the continuous $R$ symmetry. For example, consider
\cite{Intriligator:2007py}:
\beq
\Delta W &=& \frac{1}{2} \epsilon m \Phi_2^2,
\eeq
where $\epsilon$ is a small dimensionless parameter, so that the treelevel scalar potential is
\beq
&&
V_{\rm treelevel} \>=\> F_1^2 + F_2^2 + F_3^2,
\\
&& F_1 = k  {y\over 2} \phi_3^{*2} ,\qquad
F_2 = \epsilon m \phi_2^* m \phi^*_3 ,\qquad
F_3 = m \phi^*_2  y \phi^*_1 \phi^*_3 .
\eeq
In accord with the NelsonSeiberg theorem,
there are now (two) supersymmetric minima, with
\beq
\phi_1 \,=\, m/\epsilon y,
\qquad
\phi_2 \,=\, \pm \frac{1}{\epsilon} \sqrt{2k/y},
\qquad
\phi_3 \,=\, \mp \sqrt{2k/y}.
\eeq
However, for small enough $\epsilon$, the local supersymmetrybreaking minimum at
$\phi_1=\phi_2=\phi_3=0$
is also still present and stabilized by
the oneloop effective potential, with potential barriers
between it and the supersymmetric minima, so the situation is qualitatively like
Figure \ref{fig:susybreakingpotentials}(c).
As $\epsilon \rightarrow 0$, the supersymmetric global minima
move off to infinity in field space,
and there is negligible effect on the supersymmetrybreaking local minimum.
One can show \cite{Intriligator:2007py} that the lifetime of the metastable vacuum state
due to quantum tunneling can be made
arbitrarily large. The same effect can be realized by a variety of other
perturbations to the O'Rai\f\ear\taigh model; by eliminating the continuous
$R$ symmetry using small additional contributions to the Lagrangian,
the stable supersymmetry breaking vacuum is converted to a metastable one.
(In some cases, the Lagrangian remains invariant under a discrete $R$ symmetry.)
The O'Rai\f\ear\taigh superpotential determines the mass scale of
supersymmetry breaking $\sqrt{F_1}$ in terms of a dimensionful parameter
$k$ put in by hand. This appears somewhat artificial, since $k$ will have
to be tiny compared to $\MPlanck^2$ in order to give the right order of
magnitude for the MSSM soft terms. It may be more plausible to have a mechanism that
can instead generate such scales naturally. This can be done in models of
dynamical supersymmetry breaking, in which the small
mass scales associated with supersymmetry breaking arise by
dimensional transmutation. In other words, they generally feature a new
asymptotically free nonAbelian gauge symmetry with a gauge coupling $g$
that is perturbative at $\MPlanck$ and gets strong in the infrared at some
smaller scale $\Lambda \sim e^{8\pi^2/b g_0^2} \MPlanck$, where $g_0$
is the running gauge coupling at $\MPlanck$ with negative beta function $
b g^3/16 \pi^2$. Just as in QCD, it is perfectly natural for $\Lambda$
to be many orders of magnitude below the Planck scale. Supersymmetry
breaking may then be best described in terms of the effective dynamics of
the strongly coupled theory.
Supersymmetry is still broken by the VEV of an $F$ field, but it may be
the auxiliary field of a composite chiral supermultiplet built out
of fields that are charged under the new strongly coupled gauge group.
The construction of such models that break supersymmetry through strongcoupling dynamics is
nontrivial if one wants a stable supersymmetrybreaking ground state. In addition to the
argument from the NelsonSeiberg theorem that a $U(1)_R$ symmetry should be present, one can
prove using the Witten index \cite{Wittenindex,AffleckDineSeiberg} that any strongly coupled
gauge theory with only vectorlike, massive matter cannot spontaneously break supersymmetry in
its true ground state. However, things are easier if one only requires a local (metastable)
minimum of the potential. Intriligator, Seiberg, and Shih showed \cite{Intriligator:2006dd} that
supersymmetric YangMills theories with vectorlike matter can have metastable vacuum states with
nonvanishing $F$terms that break supersymmetry, and lifetimes that can be arbitrarily long. The
simplest model that does this is remarkably economical; it is just supersymmetric $SU(N_c)$ gauge
theory, with $N_f$ massive flavors of quark and antiquark supermultiplets, with $N_c + 1 \leq N_f
< 3 N_c/2$. The recognition of the advantages of a metastable vacuum state
opens up many new model building possibilities and ideas
\cite{Intriligator:2006dd,Intriligator:2007py,metastablemodels}.
The topic of known ways of breaking supersymmetry spontaneously through
strongly coupled gauge theories is a big subject that is in danger of becoming vast, and is beyond
the scope of this primer. Fortunately,
there are several excellent reviews, including
\cite{susybreakingrevs} for the more recent developments and \cite{dynamicalsusybreaking}
for older models with stable vacua.
Finding the ultimate cause of supersymmetry breaking is one of the
most important goals for the future. However, for many purposes,
one can simply assume that an $F$term has obtained a VEV, without
worrying about the specific dynamics that caused it. For
understanding collider phenomenology, the most immediate concern is usually
the nature of the couplings of the $F$term VEV to the MSSM fields.
This is the subject we turn to next.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The need for a separate supersymmetrybreaking sector
\label{subsec:origins.sector}}
\setcounter{equation}{0}
\setcounter{footnote}{1}
It is now clear that spontaneous supersymmetry breaking (dynamical or not)
requires us to extend the MSSM. The ultimate supersymmetrybreaking order
parameter cannot belong to any of the MSSM supermultiplets; a $D$term VEV
for $U(1)_Y$ does not lead to an acceptable spectrum, and there is no
candidate gauge singlet whose $F$term could develop a VEV. Therefore one
must ask what effects {\it are} responsible for spontaneous supersymmetry
breaking, and how supersymmetry breakdown is ``communicated" to the MSSM
particles. It is very difficult to achieve the latter in a
phenomenologically viable way working only with renormalizable
interactions at treelevel, even if the model is extended to involve new
supermultiplets including gauge singlets. First, on general grounds it
would be
problematic to give masses to the MSSM gauginos, because the results of
section \ref{sec:susylagr} inform us that renormalizable supersymmetry
never has any (scalar)(gaugino)(gaugino) couplings that could turn into
gaugino mass terms when the scalar gets a VEV. Second, at least some of
the MSSM squarks and sleptons would have to be unacceptably light, and
should have been discovered already. This can be understood from the
existence of sum rules that can be obtained in the same way as
eq.~(\ref{eq:supertracesumrule}) when the restrictions imposed by flavor
symmetries are taken into account. For example, in the limit in which
lepton flavors are conserved, the selectron mass eigenstates $\tilde e_1$
and $\tilde e_2$ could in general be mixtures of $\tilde e_L$ and $\tilde
e_R$. But if they do not mix with other scalars, then part of the sum rule
decouples from the rest, and one obtains:
\beq
m_{\tilde e_1}^2 + m_{\tilde e_2}^2 = 2 m_e^2,
\label{eq:sumrulee}
\eeq
which is of course ruled out by experiment. Similar sum rules follow for
each of the fermions of the Standard Model, at treelevel and in the
limits in which the corresponding flavors are conserved. In principle, the
sum rules can be evaded by introducing flavorviolating mixings, but it is
very difficult to see how to make a viable model in this way. Even
ignoring these problems, there is no obvious reason why the resulting MSSM
soft supersymmetrybreaking terms in this type of model should satisfy
flavorblindness conditions like eqs.~(\ref{scalarmassunification}) or
(\ref{aunification}).
For these reasons, we expect that the MSSM soft terms arise indirectly or
radiatively, rather than from treelevel renormalizable couplings to the
supersymmetrybreaking order parameters. Supersymmetry breaking evidently
occurs in a ``hidden sector" of particles that have no (or only very
small) direct couplings to the ``visible sector" chiral supermultiplets of
the MSSM. However, the two sectors do share some interactions that are
responsible for mediating supersymmetry breaking from the hidden sector to
the visible sector, resulting in the MSSM soft terms.
(See Figure~\ref{fig:structure}.)%
\begin{figure}
\centerline{\psfig{figure=structure.eps,height=.8in}}
\caption{The presumed schematic structure for supersymmetry breaking.
\label{fig:structure}}
\end{figure}
%
In this scenario, the treelevel squared
mass sum rules need not hold, even approximately, for the physical masses
of the visible sector fields, so that a phenomenologically viable
superpartner mass spectrum is, in principle, achievable. As a bonus, if
the mediating interactions are flavorblind, then the soft terms appearing
in the MSSM will automatically obey conditions like
eqs.~(\ref{scalarmassunification}), (\ref{aunification}) and
(\ref{commonphase}).
There have been two main competing proposals for what the mediating
interactions might be. The first (and historically the more popular) is
that they are gravitational. More precisely, they are associated with the
new physics, including gravity, that enters near the Planck scale. In this
``gravitymediated", or {\it Planckscalemediated supersymmetry breaking}
(PMSB) scenario, if supersymmetry is broken in the hidden sector by a VEV
$\langle F\rangle$, then the soft terms in the visible sector should be
roughly
\beq
m_{\rm soft} \>\sim\> {\langle F \rangle / \MPlanck},
\label{mgravusual}
\eeq
by dimensional analysis. This is because we know that $m_{\rm soft}$ must
vanish in the limit $\langle F \rangle \rightarrow 0$ where supersymmetry
is unbroken, and also in the limit $\MPlanck \rightarrow \infty$
(corresponding to $G_{\rm Newton} \rightarrow 0$) in which gravity becomes
irrelevant. For $m_{\rm soft}$ of order a few hundred GeV, one would
therefore expect that the scale associated with the origin of
supersymmetry breaking in the hidden sector should be roughly
${\sqrt{\langle F\rangle}} \sim 10^{10}$ or $10^{11}$ GeV.
A second possibility is that the flavorblind mediating interactions for
supersymmetry breaking are the ordinary electroweak and QCD gauge
interactions. In this {\it gaugemediated supersymmetry breaking} (GMSB)
scenario, the MSSM soft terms come from loop diagrams involving some {\it
messenger} particles. The messengers are new chiral supermultiplets that
couple to a supersymmetrybreaking VEV $\langle F\rangle$, and also have
$SU(3)_C \times SU(2)_L \times U(1)_Y$ interactions, which provide the
necessary connection to the MSSM. Then, using dimensional analysis, one
estimates for the MSSM soft terms
\beq
m_{\rm soft} \sim {\alpha_a\over 4\pi} {\langle F \rangle
\over M_{\rm mess}}
\label{mgravgmsb}
\eeq
where the $\alpha_a/4\pi$ is a loop factor for Feynman diagrams involving
gauge interactions, and $M_{\rm mess}$ is a characteristic scale of the
masses of the messenger fields. So if $M_{\rm mess}$ and $\sqrt{\langle
F\rangle}$ are roughly comparable, then the scale of supersymmetry
breaking can be as low as about ${\sqrt{\langle F\rangle}} \sim 10^4$ GeV
(much lower than in the gravitymediated case!) to give $m_{\rm soft}$ of
the right order of magnitude.
\subsection{The goldstino and the gravitino}\label{subsec:origins.gravitino}
\setcounter{equation}{0}
\setcounter{footnote}{1}
As shown in section \ref{subsec:origins.general}, the spontaneous breaking
of global supersymmetry implies the existence of a massless Weyl fermion,
the goldstino. The goldstino is the fermionic component of the
supermultiplet whose auxiliary field obtains a VEV.
We can derive an important property of the goldstino by considering the
form of the conserved supercurrent eq.~(\ref{supercurrent}). Suppose for
simplicity\footnote{More generally, if supersymmetry is spontaneously
broken by VEVs for several auxiliary fields $F_i$ and $D^a$, then one
should make the replacement $\langle F \rangle \rightarrow ( \sum_i
\langle F_i \rangle^2 + {1\over 2} \sum_a \langle D^a \rangle^2 )^{1/2}$
everywhere in the following.} that the only nonvanishing auxiliary field
VEV is $\langle F \rangle$ with goldstino superpartner $\stilde G$. Then
the supercurrent conservation equation tells us that
\beq
0 = \partial_\mu J^\mu_\alpha =
i \langle F \rangle (\sigma^\mu \partial_\mu \stilde G^\dagger)_\alpha +
\partial_\mu j^\mu_\alpha + \ldots
\label{beezlebub}
\eeq
where $j^\mu_\alpha$ is the part of the supercurrent that involves all of
the other supermultiplets, and the ellipses represent other contributions
of the goldstino supermultiplet to $\partial_\mu J^\mu_\alpha$, which we
can ignore. [The first term in eq.~(\ref{beezlebub}) comes from the second
term in eq.~(\ref{supercurrent}), using the equation of motion $F_i =
W^{*}_i$ for the goldstino's auxiliary field.] This equation of motion
for the goldstino field allows us to write an effective Lagrangian
\beq
\lagr_{\rm goldstino}
= i \stilde G^\dagger \sigmabar^\mu \partial_\mu \stilde G
 {1\over \langle F \rangle}(\stilde G \partial_\mu j^\mu
+ \conj) ,
\label{goldstinointeraction}
\eeq
which describes the interactions of the goldstino with all of the other
fermionboson pairs \cite{Fayetsupercurrent}. In particular, since
$j^\mu_\alpha =
(\sigma^\nu\sigmabar^\mu \psi_i)_\alpha \partial_\nu\phi^{*i}
\BDplus \sigma^\nu \sigmabar^\rho \sigma^\mu \lambda^{\dagger a}
F_{\nu\rho}^a/2\sqrt{2} + \ldots$, there are goldstinoscalarchiral
fermion and goldstinogauginogauge boson vertices as shown in
Figure~\ref{fig:goldstino}. Since this derivation depends only on
%
\begin{figure}
\begin{center}
\begin{picture}(66,60)(0,0)
\SetWidth{0.85}
\ArrowLine(0,0)(33,12)
\ArrowLine(66,0)(33,12)
\DashLine(33,52.5)(33,12){4}
\ArrowLine(33,32.2501)(33,32.25)
\Text(0,10)[c]{$\psi$}
\Text(66,10)[c]{$\stilde G$}
\Text(26,52)[c]{$\phi$}
\Text(33,12)[c]{(a)}
\end{picture}
%
\hspace{2.5cm}
%
\begin{picture}(66,60)(0,0)
\Photon(0,0)(33,12){1.75}{4}
\SetWidth{0.85}
\Photon(33,52.5)(33,12){2}{4.5}
\ArrowLine(0,0)(33,12)
\ArrowLine(66,0)(33,12)
\ArrowLine(33,32.2501)(33,32.25)
\Text(0,11)[c]{$\lambda$}
\Text(66,10)[c]{$\stilde G$}
\Text(25,52)[c]{$A$}
\Text(33,12)[c]{(b)}
\end{picture}
\end{center}
\caption{Goldstino/gravitino $\tilde G$ interactions with superpartner
pairs $(\phi,\psi)$ and $(\lambda,A)$.
\label{fig:goldstino}}
\end{figure}
%
supercurrent conservation, eq.~(\ref{goldstinointeraction}) holds
independently of the details of how supersymmetry breaking is communicated
from $\langle F \rangle$ to the MSSM sector fields $(\phi_i,\psi_i)$ and
$(\lambda^a, A^a)$. It may appear strange at first that the interaction
couplings in eq.~(\ref{goldstinointeraction}) get larger in the limit
$\langle F \rangle$ goes to zero. However, the interaction term $\stilde G
\partial_\mu j^\mu$ contains two derivatives, which turn out to always
give a kinematic factor proportional to the squaredmass difference of the
superpartners when they are onshell, i.e.~$m_{\phi}^2  m_{\psi}^2$
and $m^2_{\lambda}  m_{A}^2$ for Figures~\ref{fig:goldstino}a and
\ref{fig:goldstino}b respectively. These can be nonzero only by virtue of
supersymmetry breaking, so they must also vanish as $\langle F\rangle
\rightarrow 0$, and the interaction is welldefined in that limit.
Nevertheless, for fixed values of $m_{\phi}^2  m_{\psi}^2$ and
$m^2_{\lambda}  m_{A}^2$, the interaction term in
eq.~(\ref{goldstinointeraction}) can be phenomenologically important if
$\langle F \rangle $ is not too large
%\cite{Fayetsupercurrent,eeGMSBsignal,DDRT,AKKMM2}.
\cite{Fayetsupercurrent}\cite{AKKMM2}.
The above remarks apply to the breaking of global supersymmetry. However,
taking into account gravity, supersymmetry must be promoted to a local
symmetry. This means that the spinor parameter $\epsilon^\alpha$, which
first appeared in section \ref{subsec:susylagr.freeWZ}, is no longer a
constant, but can vary from point to point in spacetime. The resulting
locally supersymmetric theory is called {\it supergravity}
\cite{supergravity,superconformalsupergravity}.
It necessarily unifies the spacetime
symmetries of ordinary general relativity with local supersymmetry
transformations. In supergravity, the spin2 graviton has a spin3/2
fermion superpartner called the gravitino, which we will denote $\stilde
\Psi_\mu^\alpha$. The gravitino has odd $R$parity ($P_R=1$), as can be
seen from the definition eq.~(\ref{defRparity}). It carries both a vector
index ($\mu$) and a spinor index ($\alpha$), and transforms
inhomogeneously under local supersymmetry transformations:
\beq
\delta \stilde\Psi_\mu^\alpha \>=\>
\partial_\mu\epsilon^\alpha +\ldots
\label{gravitinoisgauge}
\eeq
Thus the gravitino should be thought of as the ``gauge" field of local
supersymmetry transformations [compare eq.~(\ref{Agaugetr})]. As long as
supersymmetry is unbroken, the graviton and the gravitino are both
massless, each with two spin helicity states. Once supersymmetry is
spontaneously broken, the gravitino acquires a mass by absorbing
(``eating") the goldstino, which becomes its longitudinal (helicity $\pm
1/2$) components. This is called the {\it superHiggs} mechanism, and it
is analogous to the ordinary Higgs mechanism for gauge theories, by which the
$W^\pm$ and $Z^0$ gauge bosons in the Standard Model gain mass by
absorbing the NambuGoldstone bosons associated with the spontaneously
broken electroweak gauge invariance. The massive spin3/2 gravitino now
has four helicity states, of which two were originally assigned to the
wouldbe goldstino. The gravitino mass is traditionally called $m_{3/2}$,
and in the case of $F$term breaking it can be estimated as
\cite{gravitinomassref}
\beq
m_{3/2} \>\sim\> {\langle F \rangle / \MPlanck},
\label{gravitinomass}
\eeq
This follows simply from dimensional analysis, since $m_{3/2}$ must vanish
in the limits that supersymmetry is restored ($\langle F \rangle
\rightarrow 0$) and that gravity is turned off ($\MPlanck \rightarrow
\infty$).
Equation (\ref{gravitinomass}) implies very different expectations for the
mass of the gravitino in gravitymediated and in gaugemediated models,
because they usually make very different predictions for $\langle F
\rangle$.
In the Planckscalemediated supersymmetry breaking case, the gravitino
mass is comparable to the masses of the MSSM sparticles [compare
eqs.~(\ref{mgravusual}) and (\ref{gravitinomass})]. Therefore $m_{3/2}$ is
expected to be at least of order 100 GeV or so. Its interactions will be
of gravitational strength, so the gravitino will not play any role in
collider physics, but it can be important in cosmology
\cite{cosmogravitino}. If it is the LSP, then it is stable and its
primordial density could easily exceed the critical density, causing the
universe to become matterdominated too early. Even if it is not the LSP,
the gravitino can cause problems unless its density is diluted by
inflation at late times, or it decays sufficiently rapidly.
In contrast, gaugemediated supersymmetry breaking models predict that the
gravitino is much lighter than the MSSM sparticles as long as $M_{\rm
mess} \ll \MPlanck$. This can be seen by comparing eqs.~(\ref{mgravgmsb})
and (\ref{gravitinomass}). The gravitino is almost certainly the LSP in
this case, and all of the MSSM sparticles will eventually decay into final
states that include it. Naively, one might expect that these decays are
extremely slow. However, this is not necessarily true, because the
gravitino inherits the nongravitational interactions of the goldstino it
has absorbed. This means that the gravitino, or more precisely its
longitudinal (goldstino) components, can play an important role in
collider physics experiments. The mass of the gravitino can generally be
ignored for kinematic purposes, as can its transverse (helicity $\pm 3/2$)
components, which really do have only gravitational interactions.
Therefore in collider phenomenology discussions one may interchangeably
use the same symbol $\stilde G$ for the goldstino and for the gravitino of
which it is the longitudinal (helicity $\pm 1/2$) part. By using the
effective Lagrangian eq.~(\ref{goldstinointeraction}), one can compute
that the decay rate of any sparticle $\stilde X$ into its Standard Model
partner $X$ plus a goldstino/gravitino $\stilde G$ is
\beq
\Gamma(\stilde X \rightarrow X\stilde G) \,=\,
{m^5_{\stilde X} \over 16 \pi \langle F \rangle^2}
\left ( 1  {m_X^2/ m_{\stilde X}^2} \right )^4 .
\qquad\>\>{}
\label{generalgravdecay}
\eeq
This corresponds to either Figure~\ref{fig:goldstino}a or
\ref{fig:goldstino}b, with $(\stilde X,X) = (\phi,\psi)$ or $(\lambda,A)$
respectively. One factor $(1  m_X^2/m_{\stilde X}^2 )^2$ came from the
derivatives in the interaction term in eq.~(\ref{goldstinointeraction})
evaluated for onshell final states, and another such factor comes from
the kinematic phase space integral with $m_{3/2} \ll m_{\stilde X}, m_X$.
If the supermultiplet containing the goldstino and $\langle F \rangle$ has
canonically normalized kinetic terms, and the treelevel
vacuum energy is required to
vanish, then the estimate eq.~(\ref{gravitinomass}) is
sharpened to
\beq
m_{3/2} \>=\> {\langle F \rangle / \sqrt{3} \MPlanck} .
\label{gravitinomassbogus}
\eeq
In that case, one can rewrite eq.~(\ref{generalgravdecay}) as
\beq
\Gamma(\stilde X \rightarrow X\stilde G) \,=\,
{m_{\stilde X}^5 \over 48 \pi \MPlanck^2 m_{3/2}^2}
\left ( 1  {m_X^2/ m_{\stilde X}^2} \right )^4 ,
\qquad\>\>{}
\label{specificgravdecay}
\eeq
and this is how the formula is sometimes presented, although it is less
general since it assumes eq.~(\ref{gravitinomassbogus}). The
decay width is larger for smaller $\langle F \rangle$, or equivalently for
smaller $m_{3/2}$, if the other masses are fixed. If $\stilde X$ is a
mixture of superpartners of different Standard Model particles $X$, then
each partial width in eq.~(\ref{generalgravdecay}) should be multiplied by a
suppression factor equal to the square of the cosine of the appropriate
mixing angle. If $m_{\stilde X}$ is of order 100 GeV or more, and $\sqrt{
\langle F\rangle } \lsim$ few $\times 10^6$ GeV [corresponding to
$m_{3/2}$ less than roughly 1 keV according to
eq.~(\ref{gravitinomassbogus})], then the decay $\stilde X \rightarrow X
\stilde G$ can occur quickly enough to be observed in a modern collider
detector. This implies some interesting phenomenological signatures,
which we will discuss further in sections \ref{subsec:decays.gravitino}
and \ref{sec:signals}.
We now turn to a more systematic analysis of the way in which the MSSM
soft terms arise.
\subsection{Planckscalemediated supersymmetry breaking
models}\label{subsec:origins.sugra}
\setcounter{equation}{0}
\setcounter{footnote}{1}
Consider models in which the spontaneous supersymmetry breaking sector
connects with our
MSSM sector mostly through gravitationalstrength interactions, including the
effects of supergravity \cite{MSUGRA,rewsbtwo}.
Let $X$ be the chiral superfield whose $F$ term
auxiliary field breaks supersymmetry, and consider first a globally supersymmetric
effective Lagrangian, with the Planck scale suppressed effects
that communicate between the two sectors included as
nonrenormalizable
terms of
the types discussed in section \ref{superspacenonrenorm}. The superpotential, the K\"ahler
potential, and the gauge kinetic function, expanded for large $\MPlanck$, are:
\beq
W &=& W_{\rm MSSM}  \frac{1}{\MPlanck} \left (
\frac{1}{6} y^{Xijk} X \Phi_i \Phi_j \Phi_k +
\frac{1}{2} \mu^{Xij} X \Phi_i \Phi_j \right ) + \ldots
,
\label{eq:WnonrenormX}
\\
K &=& \Phi^{*i} \Phi_i
%+ \frac{1}{\MPlanck} \bigl (
%n_i^j X \Phi^{*i} \Phi_j + \overline n_i^j X^* \Phi^{*i} \Phi_j \bigr )
+ \frac{1}{\MPlanck} \bigl (
n_i^j X + \overline n_i^j X^* \bigr ) \Phi^{*i} \Phi_j
 \frac{1}{\MPlanck^2} k_i^j X X^* \Phi^{*i}
\Phi_j +
\ldots ,
\label{eq:KnonrenormX}
\\
f_{ab} &=& \frac{\delta_{ab}}{g_a^2}
\Bigl (1  \frac{2}{\MPlanck} f_a X + \ldots
\Bigr ) .
\label{eq:fnonrenormX}
\eeq
Here $\Phi_i$ represent the chiral superfields of the MSSM or an
extension of it,
and $y^{Xijk}$, $k_i^j$, $n_i^j$, $\overline n_i^j$ and $f_a$ are dimensionless
couplings while $\mu^{Xij}$ has the dimension of mass.
The leading term in the K\"ahler potential is chosen to give canonically normalized
kinetic terms.
The matrix $k_i^j$ must be Hermitian, and $\overline n_i^j = (n^i_j)^*$,
in order for the Lagrangian to be real.
To find the resulting soft supersymmetry breaking terms in the lowenergy
effective theory, one can apply the superspace formalism of section
\ref{sec:superfields}, treating $X$ as a ``spurion" by making the
replacements:
\beq
X \rightarrow \theta\theta F ,\qquad\qquad
X^* \rightarrow \theta^\dagger\theta^\dagger F^*,
\eeq
where $F$ denotes $\langle F_X
\rangle$. The resulting supersymmetrybreaking Lagrangian, after integrating out the auxiliary fields in $\Phi_i$, is:
\beq
\lagr_{\rm soft} &=&
{F\over 2\MPlanck} f_a \lambda^a \lambda^a
 {F \over 6\MPlanck} y^{Xijk} \phi_i \phi_j \phi_k
 {F\over 2 \MPlanck} \mu^{Xij}\phi_i \phi_j
 {F \over \MPlanck} n_i^j \phi_j W^i_{\rm MSSM}
+ \conj
\phantom{xxx}
\nonumber
\\
&&
 \frac{F^2}{\MPlanck^2} ( k^i_j + n^i_p \overline n^p_j) \phi^{*j} \phi_i ,
\label{hiddengrav}
\eeq
where $\phi_i$ and $\lambda^a$ are the scalar and gaugino
fields in the MSSM sector.
Now if one
assumes that $\sqrt{F} \sim 10^{10}$ or $10^{11}$ GeV,
then eq.~(\ref{hiddengrav})
has the same form as eq.~(\ref{lagrsoft}), with MSSMsector soft terms of
order $m_{\rm soft} \sim F/\MPlanck$, perhaps of order a few hundred
GeV.
In particular, if we write the visible sector superpotential as
\beq
W_{\rm MSSM} &=& \frac{1}{6} y^{ijk} \Phi_i \Phi_j \Phi_k
+ \frac{1}{2} \mu^{ij} \Phi_i \Phi_j,
\eeq
then the soft terms in that sector, in the
notation of eq.~(\ref{lagrsoft}), are:
\beq
M_a &=& {F\over \MPlanck} f_a,
\\
a^{ijk} &=& {F\over \MPlanck}( y^{Xijk}
+ n^i_p y^{pjk}+ n^j_p y^{pik}+ n^k_p y^{pij} )
,
\\
b^{ij} &=& {F\over \MPlanck} (\mu^{Xij} + n^i_p \mu^{pj}+ n^j_p \mu^{pi}),
\\
(m^2)^i_j &=& \frac{F^2}{\MPlanck^2} (k^i_j + n^i_p \overline n^p_j)
.
\label{eq:hiddengravcoup}
\eeq
Note that couplings of the form $\lagr_{\rm maybe~soft}$ in
eq.~(\ref{lagrsoftprime}) do not arise from eq.~(\ref{hiddengrav}).
Although they
actually are expected to occur, the largest possible sources for them
are nonrenormalizable K\"ahler potential terms, which lead to:
\beq
{\cal L} &=& {F^2 \over \MPlanck^3} x^{jk}_i
\phi^{*i} \phi_j \phi_k
+ {\rm c.c.},
\eeq
where $x^{jk}_i$ is dimensionless.
This explains why, at least within this model framework, the couplings $c_i^{jk}$
in eq.~(\ref{lagrsoftprime}) are of order
$F^2/\MPlanck^3
\sim m^2_{\rm soft}/\MPlanck$, and therefore
negligible.
In principle, the parameters $f_a$, $k^i_j$, $n_i^j$, $y^{Xijk}$ and $\mu^{Xij}$
ought to be
determined by the fundamental underlying theory. The familiar flavor
blindness of
gravity expressed in Einstein's equivalence principle does not, by
itself, tell us anything about their form.
% oops, haven't gotten there yet.
%, and in particular need not
%imply eqs.~(\ref{gauginounificationsugra})(\ref{aunificationsugra}).
Therefore, the requirement of approximate flavor blindness
in ${\cal L}_{\rm soft}$ is a new
assumption in this framework, and is not guaranteed without further
structure. Nevertheless, it has
historically been popular to make a dramatic simplification by assuming a
``minimal" form for the normalization of kinetic terms and gauge
interactions in the nonrenormalizable Lagrangian.
Specifically, it is often assumed that there is a common $f_a=f$ for the
three gauginos, that $k_i^j = k \delta_i^j$ and $n_i^j = n \delta_i^j$
are the same for all scalars, with $k$ and $n$ real,
and that the other couplings are proportional to the corresponding
superpotential parameters, so that $y^{Xijk} = \alpha y^{ijk}$ and
$\mu^{Xij} = \beta \mu^{ij}$ with universal real dimensionless constants
$\alpha$ and $\beta$. Then the soft terms in $\lagr_{\rm soft}^{\rm MSSM}$
are all determined by just four parameters:
\beq
m_{1/2} = f{\langle \FX \rangle\over \MPlanck},\qquad\!\!\!
m^2_{0} = (k + n^2) {\langle \FX \rangle^2\over \MPlanck^2},\qquad\!\!\!
A_0 = (\alpha + 3n){\langle \FX \rangle\over \MPlanck},\qquad\!\!\!
B_0 = (\beta + 2 n) {\langle \FX \rangle\over \MPlanck}.\>\>\phantom{xx}
\label{sillyassumptions}
\eeq
In terms of these, the parameters appearing in
eq.~(\ref{MSSMsoft}) are:
\beq
&&\!\!\!\! M_3 = M_2 = M_1 = m_{1/2},
\label{gauginounificationsugra}
\\
&&\!\!\!\! {\bf m^2_{Q}} =
{\bf m^2_{{\sbar u}}} =
{\bf m^2_{{\sbar d}}} =
{\bf m^2_{ L}} =
{\bf m^2_{{\sbar e}}} =
m_0^2\, {\bf 1},
\>\>\>\>\>\>\>\> \> m_{H_u}^2 = m^2_{H_d} = m_0^2, \>\>\>\qquad\qquad{}
\label{scalarunificationsugra}
\\
&&\!\!\!\! {\bf a_u} = A_0 {\bf y_u},\qquad
{\bf a_d} = A_0 {\bf y_d},\qquad
{\bf a_e} = A_0 {\bf y_e},
\label{aunificationsugra}
\\
&&\!\!\!\! b = B_0 \mu ,
\label{bsilly}
\eeq
at a renormalization scale $Q \approx \MPlanck$. It is a matter of some
controversy whether the assumptions going into this parameterization are
wellmotivated on purely theoretical grounds, but from
a phenomenological perspective they are clearly very nice. This framework
successfully evades the most dangerous types of flavor changing and CP
violation as discussed in section \ref{subsec:mssm.hints}. In particular,
eqs.~(\ref{scalarunificationsugra}) and (\ref{aunificationsugra}) are just
stronger versions of eqs.~(\ref{scalarmassunification}) and
(\ref{aunification}), respectively. If $m_{1/2}$, $A_0$ and $B_0$ all have
the same complex phase, then eq.~(\ref{commonphase}) will also be
satisfied.
Equations (\ref{gauginounificationsugra})(\ref{bsilly}) also have the
virtue of being extraordinarily predictive, at least in principle.
[Of course,
eq.~(\ref{bsilly}) is
contentfree unless one can relate $B_0$ to the other parameters in some
nontrivial way.] As discussed in sections \ref{subsec:mssm.hints} and
\ref{subsec:RGEs}, they should be applied as RG boundary conditions at the
scale $\MPlanck$. The RG evolution of the soft parameters down to the
electroweak scale will then allow us to predict the entire MSSM spectrum
in terms of just five parameters $m_{1/2}$, $m_0^2$, $A_0$, $B_0$, and
$\mu$ (plus the alreadymeasured gauge and Yukawa couplings of the MSSM).
A popular approximation is to start this RG running from the unification
scale $M_U\approx 1.5\times 10^{16}$ GeV instead of $\MPlanck$. The reason
for this is more practical than principled; the apparent unification of
gauge couplings gives us a strong hint that we know something about how
the RG equations behave up to $M_U$, but unfortunately gives us little
guidance about what to expect at scales between $M_U$ and $\MPlanck$. The
errors made in neglecting these effects are proportional to a loop
suppression factor times ln$(\MPlanck/M_U)$. These corrections hopefully
can be partly absorbed into a redefinition of $m_0^2$, $m_{1/2}$, $A_0$
and $B_0$ at $M_U$, but in many cases will lead to other important effects
\cite{PP} that are difficult to anticipate.
The framework described in the previous two paragraphs has been
the subject of the bulk of phenomenological and experimental studies of supersymmetry,
and has become a benchmark scenario for experimental collider
search limits. It
is sometimes referred to as the {\it minimal supergravity} (MSUGRA)
or {\it Constrained Minimal Supersymmetric Standard Model} (CMSSM)
scenario for the soft terms.
Particular models of gravitymediated supersymmetry breaking can be even
more predictive, relating some of the parameters $m_{1/2}$, $m_0^2$, $A_0$
and $B_0$ to each other and to the mass of the gravitino $m_{3/2}$. For
example, three popular kinds of models for the soft terms are:
\vspace{.08in}
$\bullet$
Dilatondominated: \cite{dilatondominated}~~~$m^2_0 =
m^2_{3/2}$,~~~~$m_{1/2} = A_0 = {\sqrt 3} m_{3/2}$.
\vspace{.02in}
$\bullet$
Polonyi: \cite{polonyi}
{}~~~$m^2_0 = m^2_{3/2}$,
{}~~~~$A_0 = (3 {\sqrt 3}) m_{3/2}$,
{}~~~~$m_{1/2} = {\cal O}(m_{3/2})$.
\vspace{.08in}
$\bullet$ ``Noscale": \cite{noscale}~~~$m_{1/2} \gg
m_0, A_0, m_{3/2}$.
\vspace{.1in}
\noindent Dilaton domination arises in a particular limit of
superstring theory. While it appears to be highly predictive, it can
easily be generalized in other limits \cite{stringsoft}. The Polonyi model
has the advantage of being the simplest possible model for supersymmetry
breaking in the hidden sector, but it is rather {\it ad hoc} and does not
seem to have a special place in grander schemes like superstrings. The
``noscale" limit may appear in a lowenergy limit of superstrings in which
the gravitino mass scale is undetermined at treelevel (hence the name).
It implies that the gaugino masses dominate over other sources of
supersymmetry breaking near $\MPlanck$. As we saw in section
\ref{subsec:RGEs}, RG evolution feeds the gaugino masses into the squark, slepton,
and Higgs squaredmass parameters with sufficient magnitude to give
acceptable phenomenology at the electroweak scale. More recent versions of
the noscale scenario, however, also can give significant $A_0$ and
$m_0^2$ at the input scale. In many cases $B_0$ can also be predicted in terms
of the other parameters, but this is quite sensitive to model assumptions.
For phenomenological studies, $m_{1/2}$, $m_0^2$, $A_0$ and $B_0$ are
usually just taken to be convenient but imperfect (and perhaps
downright misleading) parameterizations
of our ignorance of the supersymmetry breaking mechanism.
In a more perfect world, experimental searches might be conducted and reported
using something like the larger 15dimensional flavorblind parameter
space of eqs.~(\ref{scalarmassunification})(\ref{commonphase}),
but such a higher dimensional parameter space is difficult
to simulate comprehensively,
for practical reasons.
Let us now review in a little more detail how the soft supersymmetry
breaking terms can arise in supergravity models. The part of the scalar
potential that does not depend on the gauge kinetic function can be found
as follows. First, one may define the real, dimensionless {\em K\"ahler
function} in terms of the K\"ahler potential and superpotential with the
chiral superfields replaced by their scalar components:
\beq
G \>=\>
{K/\MPlanck^2} + {\rm ln}({W/\MPlanck^3}) +{\rm ln}{(W^*/\MPlanck^3)}.
\label{eq:defKahlerfun}
\eeq
Many references use units with $\MPlanck=1$,
which simplifies the expressions but can slightly obscure the
correspondence with the global supersymmetry limit of large $\MPlanck$.
From $G$, one can construct its derivatives with respect to the scalar
fields and their complex conjugates: $G^i = {\delta G/\delta \phi_i}$;
$G_i = {\delta G/\delta \phi^{* i}}$; and $G_i^j = {\delta^2
G/\delta\phi^{* i}\delta\phi_j}$. As in section
\ref{subsec:susylagr.chiral}, raised (lowered) indices $i$
correspond to derivatives with respect to $\phi_i$ ($\phi^{*i}$).
Note that $G_i^j =
K_i^j/\MPlanck^2$, which is often called the K\"ahler metric, does not
depend on the superpotential. The inverse of this matrix is denoted
$(G^{1})_i^j$, or equivalently $\MPlanck^2 (K^{1})_i^j$, so that
$(G^{1})^k_i G_k^j = (G^{1})^j_k G_i^k = \delta_i^j$. In terms of these
objects, the generalization of the $F$term contribution to the
scalar potential in ordinary renormalizable global supersymmetry turns out
\cite{supergravity,superconformalsupergravity} to be:
\beq
V_F \>=\> \MPlanck^4 \, e^G \Bigl [ G^i (G^{1})_i^j G_j 3 \Bigr ]
\label{vsugra}
\eeq
in supergravity. It can be rewritten as
\beq
V_F \>=\> K_i^j F_j F^{*i}  3 e^{K/\MPlanck^2} WW^*/\MPlanck^2 ,
\label{compactvsugra}
\eeq
where
\beq
F_i \>=\> \MPlanck^2\, e^{G/2} \, (G^{1})_i^j G_j
\>=\> e^{K/2 \MPlanck^2}\, (K^{1})_i^j
\Bigl ( W^*_j + W^* K_j/\MPlanck^2 \Bigr ),
\label{fisugra}
\eeq
with $K^i = \delta K/\delta \phi_i$ and $K_j = \delta K/\delta \phi^{*j}$.
The $F_i$ are order parameters for supersymmetry breaking in supergravity
(generalizing the auxiliary fields in the renormalizable global
supersymmetry case). In other words, local supersymmetry will be broken if
one or more of the $F_i$ obtain a VEV. The gravitino then absorbs the
wouldbe goldstino and obtains a squared mass
\beq
m^2_{3/2} \,=\, \langle K_j^i F_i F^{*j}\rangle/3\MPlanck^2.
\label{sugragravitinomass}
\eeq
Taking a minimal K\"ahler potential $ K = \phi^{*i} \phi_i $,
one has $K_i^j=(K^{1})_i^j = \delta_i^j$, so that expanding
eqs.~(\ref{compactvsugra}) and (\ref{fisugra}) to lowest order in
$1/\MPlanck$ just reproduces the results $F_i = W^*_i$ and $V = F_i
F^{*i} = W^i W_i^*$, which were found in section
\ref{subsec:susylagr.chiral} for renormalizable global supersymmetric
theories [see eqs.~(\ref{replaceF})(\ref{ordpot})].
Equation~(\ref{sugragravitinomass}) also reproduces the expression for the
gravitino mass that was quoted in eq.~(\ref{gravitinomass}).
The scalar potential eq.~(\ref{vsugra}) does not include the $D$term
contributions from gauge interactions, which are given by
\beq
V_D \>=\> {1\over 2}{\rm Re}[f_{ab}\, {\widehat D}^a {\widehat
D}^b],
\label{eq:defVD}
\eeq
with $\widehat D^a = f_{ab}^{1} \widetilde D^b$, where
\beq
{\widetilde D}^a \equiv
G^i (T^a)_i{}^j \phi_j = \phi^{*j} (T^a)_j{}^i G_i =
K^i (T^a)_i{}^j \phi_j = \phi^{*j} (T^a)_j{}^i K_i,
\eeq
are real order parameters of supersymmetry breaking, with the last three
equalities following from the gauge invariance of $W$ and $K$.
Note that in the treelevel global supersymmetry case $f_{ab} =
\delta_{ab}/g_a^2$ and $K^i = \phi^{*i}$, eq.~(\ref{eq:defVD}) reproduces
the result of section \ref{subsec:susylagr.gaugeinter} for the
renormalizable global supersymmetry $D$term scalar potential, with
$\widehat{D}^a = g_a D^a$ (no sum on $a$).
The full scalar
potential is
\beq
V = V_F + V_D,
\eeq
and it depends on $W$ and $K$ only through the combination $G$ in
eq.~(\ref{eq:defKahlerfun}). There are many other contributions to the
supergravity Lagrangian involving fermions and vectors, which
can be found in
ref.~\cite{supergravity,superconformalsupergravity}, and
also turn
out to depend
only on $f_{ab}$ and
$G$. This allows one
to consistently
redefine $W$ and $K$ so that there are no purely holomorphic or purely
antiholomorphic terms appearing in the latter.
Unlike in the case of global supersymmetry, the scalar potential in
supergravity is {\it not} necessarily nonnegative, because of the $3$
term in eq.~(\ref{vsugra}). Therefore, in principle, one can have
supersymmetry breaking with a positive, negative, or zero vacuum energy.
Results in experimental cosmology \cite{cosmokramer} imply
a positive vacuum energy associated with the acceleration of the
scale factor of the observable universe,
\beq
\rho_{\rm vac}^{\rm observed} = \frac{\Lambda}{8\pi G_{\rm Newton}} \approx
(2.3 \times 10^{12}\>{\rm GeV})^4,
\eeq
but this is also certainly tiny compared to the scales associated with
supersymmetry breaking. Therefore, it is tempting to simply assume that
the vacuum energy is 0 within the approximations pertinent for working out
the supergravity effects on particle physics at collider energies. However, it
is notoriously unclear {\em why} the terms in the scalar potential in a
supersymmetrybreaking vacuum should conspire to give $\langle V \rangle
\approx 0$ at the minimum. A naive estimate, without miraculous
cancellations, would give instead $ \langle V \rangle$ of order $\langle F
\rangle^2$, so at least roughly ($10^{10}$ GeV)$^4$ for Planckscale
mediated supersymmetry breaking, or ($10^4$ GeV)$^4$ for gaugemediated
supersymmetry breaking. Furthermore, while $\rho_{\rm vac} = \langle V
\rangle$ classically, the former is a very largedistance scale measured
quantity, while the latter is associated with effective field theories at
length scales comparable to and shorter than those familiar to high energy
physics. So, in the absence of a compelling explanation for the tiny value
of $\rho_{\rm vac}$, it is not at all clear that $\langle V \rangle
\approx 0$ is really the right condition to impose \cite{cosmock}.
Nevertheless, with $\langle V \rangle = 0$ imposed as a
constraint,
eqs.~(\ref{compactvsugra})(\ref{sugragravitinomass}) tell us that $
\langle K_j^i F_i F^{*j} \rangle = 3 \MPlanck^4 e^{\langle G \rangle} = 3
e^{\langle K \rangle/\MPlanck^2} \langle W \rangle^2/\MPlanck^2$, and an
equivalent formula for the gravitino mass is therefore $m_{3/2} =
e^{\langle G\rangle/2} \MPlanck$.
An interesting special case arises if we assume a minimal K\"ahler
potential and divide the fields $\phi_i$ into a visible sector including
the MSSM fields $\varphi_i$, and a hidden sector containing a field $X$
that breaks supersymmetry for us (and other fields that we need not treat
explicitly). In other words, suppose that the superpotential and the
K\"ahler potential
%(expressed as functions of the scalar fields)
have the forms
\beq
W &=& W_{\rm vis}(\varphi_i) + W_{\rm hid}(X),
\label{minw}\\
K &=& \varphi^{*i} \varphi_i + X^* X .
\label{mink}
\eeq
Now let us further assume that the dynamics of the hidden sector fields
provides nonzero VEVs
\beq
\langle X \rangle = x \MPlanck,\qquad
\langle W_{\rm hid}\rangle = w \MPlanck^2,\qquad
\langle \delta W_{\rm hid}/\delta X \rangle = w^\prime \MPlanck ,
\eeq
which define a dimensionless quantity $x$, and $w$, $w^\prime$ with
dimensions of [mass]. Requiring\footnote{We do this only
to follow popular example; as just
noted we cannot endorse this imposition.} $\langle V \rangle = 0$ yields $w^\prime
+ x^* w^2 = 3 w^2$, and
\beq
m_{3/2} \>=\> {\langle F_X \rangle /\sqrt{3} \MPlanck} \>=\> e^{x^2/2}w.
\eeq
Now we suppose that it is valid to expand the scalar potential in powers
of the dimensionless quantities $w/\MPlanck$, $w^\prime/\MPlanck$,
$\varphi_i/\MPlanck$, etc., keeping only terms that depend on the visible
sector fields $\varphi_i$. In
leading order the result is:
\beq
V &=& (W^*_{\rm vis})_i (W_{\rm vis})^i + m_{3/2}^2
\varphi^{*i}\varphi_{i}
\nonumber \\ && \!\!
+ e^{x^2/2} \left [w^* \varphi_i (W_{\rm vis})^i\, +\,
(x^* w^{\prime *} + x^2 w^*  3 w^*) W_{\rm vis} + \conj \right
].\qquad{}
\label{yapot}
\eeq
A tricky point here is that we have rescaled the visible sector
superpotential $W_{\rm vis} \rightarrow e^{x^2/2} W_{\rm vis}$
everywhere, in order that the first term in eq.~(\ref{yapot}) is the
usual, properly normalized, $F$term contribution in global supersymmetry.
The next term is a universal soft scalar squared mass of the form
eq.~(\ref{scalarunificationsugra}) with
\beq
m_0^2 \>=\> {\langle F_X \rangle^2/ 3 \MPlanck^2}
\>=\> m_{3/2}^2 .
\eeq
The second line of eq.~(\ref{yapot}) just gives soft (scalar)$^3$ and
(scalar)$^2$ holomorphic couplings of the form
eqs.~(\ref{aunificationsugra}) and (\ref{bsilly}), with
\beq
A_0 \,=\,  x^* {\langle F_X \rangle / \MPlanck},
\qquad\>\>
B_0 \,=\, \Bigl (
{1\over x + w^{\prime *}/w^*} x^*\Bigr ){\langle F_X \rangle / \MPlanck}
\qquad{}
\label{a0b0x}
\eeq
since $\varphi_i (W_{\rm vis})^i$ is equal to $3 W_{\rm vis}$ for the
cubic part of $W_{\rm vis}$, and to $2 W_{\rm vis}$ for the quadratic
part. [If the complex phases of $x$, $w$, $w^\prime$ can be rotated away,
then eq.~(\ref{a0b0x}) implies $B_0 = A_0  m_{3/2}$, but there are many
effects that can ruin this prediction.] The Polonyi model mentioned in
section \ref{subsec:origins.sugra} is just the special case of this
exercise in which $W_{\rm hid}$ is assumed to be linear in $X$.
However, there is no reason why $W$ and $K$ must have the
simple form eq.~(\ref{minw}) and eq.~(\ref{mink}). In general, the
superpotential and K\"ahler potential will have terms coupling $X$ to the MSSM
fields as in eqs.~(\ref{eq:WnonrenormX}) and (\ref{eq:KnonrenormX}).
If one now plugs such terms into eq.~(\ref{vsugra}), one obtains a
general form like eq.~(\ref{hiddengrav}) for the soft terms. It is only
when special assumptions are made [like eqs.~(\ref{minw}), (\ref{mink})]
that one gets the phenomenologically desirable results in
eqs.~(\ref{sillyassumptions})(\ref{bsilly}). Thus
supergravity by itself does not guarantee universality or
even flavorblindness of the soft
terms.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Gaugemediated supersymmetry breaking
models}\label{subsec:origins.gmsb}
\setcounter{equation}{0}
\setcounter{footnote}{1}
In gaugemediated supersymmetry breaking (GMSB) models
\cite{oldgmsb,newgmsb}, the ordinary gauge interactions, rather than
gravity, are responsible for the appearance of soft supersymmetry breaking
in the MSSM. The basic idea is to introduce some new chiral
supermultiplets, called messengers, that couple to the ultimate source of
supersymmetry breaking, and also couple indirectly to the (s)quarks and
(s)leptons and higgs(inos) of the MSSM through the ordinary $SU(3)_C\times
SU(2)_L\times U(1)_Y$ gauge boson and gaugino interactions. There is
still gravitational communication between the MSSM and the source of
supersymmetry breaking, of course, but that effect is now relatively
unimportant compared to the gauge interaction effects.
In contrast to Planckscale mediation, GMSB can be understood entirely in
terms of loop effects in a renormalizable framework. In the simplest
such model, the messenger fields are a set of lefthanded chiral
supermultiplets $q$, $\overline q$, $\ell$, $\overline \ell$
transforming under $SU(3)_C\times SU(2)_L\times U(1)_Y$ as
\beq
q\sim({\bf 3},{\bf 1}, {1\over 3}),\qquad\!\!\!
\overline q\sim({\bf \overline 3},{\bf 1}, {1\over 3}),\qquad\!\!\!
\ell \sim({\bf 1},{\bf 2}, {1\over 2}),\qquad\!\!\!
\overline \ell\sim({\bf 1},{\bf 2}, {1\over 2}).\qquad\!\!\!{}
\label{minimalmess}
\eeq
These supermultiplets contain messenger quarks $\psi_q, \psi_{\overline
q}$ and scalar quarks $q, \overline q$ and messenger leptons $\psi_\ell,
\psi_{\overline \ell}$ and scalar leptons $\ell, \overline \ell$. All of
these particles must get very large masses so as not to have been
discovered already. Assume they do so by coupling to a gaugesinglet
chiral supermultiplet $S$ through a superpotential:
\beq
W_{\rm mess} \, = \, y_2 S \ell \overline \ell + y_3 S q \overline q .
\eeq
The scalar component of $S$ and its auxiliary ($F$term) component are
each supposed to acquire VEVs, denoted $\langle S \rangle $ and $\langle
F_S \rangle $ respectively. This can be accomplished either by putting $S$
into an O'Rai\f\ear\taightype model \cite{oldgmsb}, or by a dynamical
mechanism \cite{newgmsb}. Exactly how this happens is an interesting and
important question, with many possible answers but no clear
favorite at present. Here, we will
simply parameterize our ignorance of the precise mechanism of
supersymmetry breaking by asserting that $S$ participates in another part
of the superpotential, call it $W_{\rm breaking}$, which provides for the
necessary spontaneous breaking of supersymmetry.
Let us now consider the mass spectrum of the messenger fermions and
bosons. The fermionic messenger fields pair up to get mass terms:
\beq
\lagr &=&
 y_2 \langle S \rangle \psi_\ell \psi_{\overline \ell}
 y_3 \langle S \rangle \psi_q \psi_{\overline q} + \conj
\label{messfermass}
\eeq
as in eq.~(\ref{lagrchiral}). Meanwhile, their scalar messenger partners
$\ell,\overline\ell$ and $q,\overline q$ have a scalar potential given by
(neglecting $D$term contributions, which do not affect the following
discussion):
\beq
V &=&
\left  {\delta \Wmess \over \delta \ell} \right ^2 +
\left  {\delta \Wmess \over \delta \overline\ell} \right ^2 +
\left  {\delta \Wmess \over \delta q} \right ^2 +
\left  {\delta \Wmess \over \delta \overline q} \right ^2 +
\left  {\delta \over \delta S} (\Wmess + W_{\rm breaking}) \right ^2
\>\phantom{xxx}
\eeq
as in eq.~(\ref{ordpot}). Now, suppose that, at the minimum of the
potential,
\beq
\langle S \rangle &\not=& 0,\\
\langle \delta W_{\rm breaking}/\delta S \rangle
&=& \langle F_S^* \rangle \>\not=\> 0,\\
\langle \delta \Wmess /\delta S \rangle &=& 0.
\eeq
Replacing $S$ and $F_S$ by their VEVs, one finds quadratic mass terms in
the potential for the messenger scalar leptons:
\beq
V &=&
y_2 \langle S \rangle^2 \bigl ( \ell^2 +
\overline \ell ^2 \bigr ) +
y_3 \langle S \rangle^2 \bigl ( q^2 +
\overline q^2 \bigr )
\nonumber \\
&&\left (y_2 \langle F_S \rangle \ell\overline \ell
+ y_3 \langle F_S \rangle q\overline q + \conj \right )
\nonumber
\\&& +\> {\rm quartic}\> {\rm terms}.
\label{nosteenkinlabel}
\eeq
The first line in eq.~(\ref{nosteenkinlabel}) represents supersymmetric
mass terms that go along with eq.~(\ref{messfermass}), while the second
line consists of soft supersymmetrybreaking masses. The complex scalar
messengers $\ell,\overline\ell$ thus obtain a squaredmass matrix equal
to:
\beq
\pmatrix{ y_2 \langle S \rangle ^2
& y^*_2 \langle F^*_S \rangle \cr
y_2 \langle F_S \rangle & \phantom{x}y_2 \langle S \rangle ^2 }
\eeq
with squared mass eigenvalues $y_2 \langle S\rangle ^2 \pm y_2 \langle
F_S \rangle $. In just the same way, the scalars $q,\overline q$ get
squared masses $y_3 \langle S\rangle ^2 \pm y_3 \langle F_S \rangle $.
So far, we have found that the effect of supersymmetry breaking is to
split each messenger supermultiplet pair apart:
\beq
\ell,\overline\ell : \qquad & m_{\rm fermions}^2 = y_2 \langle S\rangle
^2\, ,
\qquad & m_{\rm scalars}^2 = y_2 \langle S\rangle ^2
\pm y_2 \langle F_S \rangle  \, , \\
q,\overline q : \qquad & m_{\rm fermions}^2 = y_3 \langle S\rangle
^2\, ,
\qquad & m_{\rm scalars}^2 = y_3 \langle S\rangle ^2
\pm y_3 \langle F_S \rangle  \> .
\eeq
The supersymmetry violation apparent in this messenger spectrum for
$\langle F_S \rangle \not= 0$ is communicated to the MSSM sparticles
through radiative corrections. The MSSM gauginos obtain masses
from the 1loop Feynman diagram shown in Figure~\ref{fig:1loop}.%
\begin{figure}
\begin{minipage}[]{0.55\linewidth}
\caption{Contributions to the MSSM gaugino masses in gaugemediated
supersymmetry breaking models come from oneloop graphs involving
virtual messenger particles.\label{fig:1loop}}
\end{minipage}
\hspace{0.01\linewidth}
\begin{minipage}[]{0.4\linewidth}
\begin{picture}(170,74)(100,44)
\SetScale{0.8}
\SetWidth{0.475}
\Photon(75,0)(32,0){2.2}{4}
\Photon(75,0)(32,0){2.2}{4}
\SetWidth{0.8}
\Line(3,35)(3,29)
\Line(3,35)(3,29)
\Line(3,35)(3,29)
\Line(3,35)(3,29)
\ArrowLine(75,0)(32,0)
\ArrowLine(75,0)(32,0)
\ArrowArc(0,0)(32,270,360)
\ArrowArcn(0,0)(32,270,180)
\DashArrowArc(0,0)(32,0,90){4}
\DashArrowArcn(0,0)(32,180,90){4}
\Text(70,13)[c]{$\stilde B, \stilde W, \tilde g$}
\Text(0,38)[c]{$\langle F_S \rangle$}
\Text(0,38)[c]{$\langle S \rangle$}
\end{picture}
\end{minipage}
\end{figure}
%
The scalar and fermion lines in the loop are messenger fields. Recall that
the interaction vertices in Figure~\ref{fig:1loop} are of gauge coupling
strength even though they do not involve gauge bosons; compare
Figure~\ref{fig:gauge}g. In this way, gaugemediation provides that
$q,\overline q$ messenger loops give masses to the gluino and the bino,
and $\ell,\overline \ell$ messenger loops give masses to the wino and bino
fields. Computing the 1loop diagrams, one finds \cite{newgmsb} that the
resulting MSSM gaugino masses are given by
\beq
M_a \,=\, {\alpha_a\over 4\pi} \Lambda , \qquad\>\>\>(a=1,2,3) ,
\label{gauginogmsb}
\eeq
in the normalization for $\alpha_a$ discussed in section
\ref{subsec:mssm.hints}, where we have introduced a mass parameter
\beq
\Lambda \,\equiv\, \langle F_S\rangle/\langle S \rangle \> .
\label{defLambda}
\eeq
(Note that if $\langle F_S\rangle$ were 0, then $\Lambda=0$ and the
messenger scalars would be degenerate with their fermionic superpartners
and there would be no contribution to the MSSM gaugino masses.) In
contrast, the corresponding MSSM gauge bosons cannot get a corresponding
mass shift, since they are protected by gauge invariance. So
supersymmetry breaking has been successfully communicated to the MSSM
(``visible sector"). To a good approximation, eq.~(\ref{gauginogmsb})
holds for the running gaugino masses at an RG scale $Q_0$ corresponding to
the average characteristic mass of the heavy messenger particles, roughly
of order $M_{\rm mess} \sim y_I \langle S \rangle$ for $I = 2,3$. The
running mass parameters can then be RGevolved down to the electroweak
scale to predict the physical masses to be measured by future experiments.
The scalars of the MSSM do not get any radiative corrections to their
masses at oneloop order. The leading contribution to their masses comes
from the twoloop graphs shown in Figure~\ref{fig:2loops}, with the
messenger fermions (heavy solid lines) and messenger scalars (heavy dashed
lines) and ordinary gauge bosons and gauginos running around the loops.%
\begin{figure}
\vspace{0.2cm}
\centerline{\psfig{figure=GMSBscalars.eps,height=1.55in}}
\caption{MSSM scalar squared masses in gaugemediated
supersymmetry breaking models arise in leading order from these twoloop
Feynman graphs. The heavy dashed lines are messenger scalars, the solid
lines are messenger fermions, the wavy lines are ordinary Standard Model
gauge bosons, and the solid lines with wavy lines superimposed are the
MSSM gauginos. \label{fig:2loops}}
\end{figure}
%
By computing these graphs,
one finds that each MSSM scalar $\phi_i$ gets a squared mass given by:
\beq
m^2_{\phi_i} \,=\,
2 {\Lambda^2}
\left [ \left ({\alpha_3\over 4\pi}\right )^2 C_3(i) +
\left ({\alpha_2\over
4 \pi}\right )^2 C_2(i) +
\left ({\alpha_1\over 4 \pi}\right )^2 C_1(i) \right ] ,
\label{scalargmsb}
\eeq
with the quadratic Casimir invariants $C_a(i)$ as in
eqs.~(\ref{eq:defCasimir})(\ref{defC1}). The squared masses in
eq.~(\ref{scalargmsb}) are positive (fortunately!).
The terms $\bf a_u$, $\bf a_d$, $\bf a_e$ arise first at twoloop order,
and are suppressed by an extra factor of $\alpha_a/4 \pi$ compared to the
gaugino masses. So, to a very good approximation one has, at the messenger
scale,
\beq
{\bf a_u} = {\bf a_d} = {\bf a_e} = 0,
\label{aaagmsb}
\eeq
a significantly stronger condition than eq.~(\ref{aunification}). Again,
eqs.~(\ref{scalargmsb}) and (\ref{aaagmsb}) should be applied at an RG
scale equal to the average mass of the messenger fields running in the
loops. However, evolving the RG equations down to the electroweak scale
generates nonzero $\bf a_u$, $\bf a_d$, and $\bf a_e$ proportional to the
corresponding Yukawa matrices and the nonzero gaugino masses, as
indicated in section \ref{subsec:RGEs}. These will only be large for the
thirdfamily squarks and sleptons, in the approximation of
eq.~(\ref{heavytopapprox}). The parameter $b$ may also be taken to vanish
near the messenger scale, but this is quite modeldependent, and in any
case $b$ will be nonzero when it is RGevolved to the electroweak scale.
In practice, $b$ can be fixed in terms of the other parameters
by the requirement of correct electroweak
symmetry breaking, as discussed below in section
\ref{subsec:MSSMspectrum.Higgs}.
Because the gaugino masses arise at {\it one}loop order and the scalar
squaredmass contributions appear at {\it two}loop order, both
eq.~(\ref{gauginogmsb}) and (\ref{scalargmsb}) correspond to the estimate
eq.~(\ref{mgravgmsb}) for $m_{\rm soft}$, with $M_{\rm mess} \sim y_I
\langle S \rangle$. Equations (\ref{gauginogmsb}) and (\ref{scalargmsb})
hold in the limit of small $\langle F_S \rangle /y_I\langle S \rangle^2$,
corresponding to mass splittings within each messenger supermultiplet that
are small compared to the overall messenger mass scale. The subleading
corrections in an expansion in $\langle F_S \rangle /y_I\langle S
\rangle^2$ turn out \cite{gmsbcorrA}\cite{gmsbcorrC}
to be quite small unless there
are very large messenger mass splittings.
The model we have described so far is often called the minimal model of
gaugemediated supersymmetry breaking. Let us now generalize it to a more
complicated messenger sector. Suppose that $q, \overline q$ and $\ell,
\overline \ell $ are replaced by a collection of messengers
$\Phi_I,\overline \Phi_I$ with a superpotential
\beq
W_{\rm mess} \,=\, \sum_I y_I S \Phi_I \overline \Phi_I .
\eeq
The bar is used to indicate that the lefthanded chiral superfields
$\overline \Phi_I$ transform as the complex conjugate representations of
the lefthanded chiral superfields $\Phi_I$. Together they are said to
form a ``vectorlike" (selfconjugate) representation of the Standard Model gauge
group. As before, the fermionic components of each pair $\Phi_I$ and
$\overline\Phi_I$ pair up to get squared masses $y_I \langle S
\rangle^2$ and their scalar partners mix to get squared masses $y_I
\langle S \rangle^2 \pm y_I \langle F_S \rangle  $. The MSSM gaugino
mass parameters induced are now
\beq
M_a \,=\, {\alpha_a\over 4\pi} \Lambda \sum_I n_a(I) \qquad\>\>\>(a=1,2,3)
\label{gauginogmsbgen}
\eeq
where $n_a(I)$ is the Dynkin index for each $\Phi_I+\overline \Phi_I$, in
a normalization where $n_3 = 1$ for a ${\bf 3} + {\bf \overline 3}$ of
$SU(3)_C$ and $n_2 = 1$ for a pair of doublets of $SU(2)_L$. For $U(1)_Y$,
one has $n_1 = 6Y^2/5$ for each messenger pair with weak hypercharges $\pm
Y$. In computing $n_1$ one must remember to add up the contributions for
each component of an $SU(3)_C$ or $SU(2)_L$ multiplet. So, for example,
$(n_1, n_2, n_3) = (2/5, 0, 1)$ for $q+\overline q$ and $(n_1, n_2, n_3) =
(3/5, 1, 0)$ for $\ell+\overline \ell$. Thus the total is $\sum_I (n_1,
n_2, n_3) = (1, 1, 1)$ for the minimal model, so that
eq.~(\ref{gauginogmsbgen}) is in agreement with eq.~(\ref{gauginogmsb}).
On general grouptheoretic grounds, $n_2$ and $n_3$ must be integers, and
$n_1$ is always an integer multiple of $1/5$ if fractional electric
charges are confined.
The MSSM scalar masses in this generalized gauge mediation framework are
now:
\beq
m^2_{\phi_i} \,=\,
2 \Lambda^2
\left [ \left ({\alpha_3\over 4\pi}\right )^2 C_3(i) \sum_I n_3(I) +
\left ({\alpha_2\over 4 \pi}\right )^2 C_2(i) \sum_I n_2(I)+
\left ({\alpha_1\over 4 \pi}\right )^2 C_1(i) \sum_I n_1(I)
\right ] .\phantom{xx}
\label{scalargmsbgen}
\eeq
In writing eqs.~(\ref{gauginogmsbgen}) and (\ref{scalargmsbgen}) as simple
sums, we have implicitly assumed that the messengers are all approximately
equal in mass, with
\beq
M_{\rm mess} \,\approx\, y_I \langle S \rangle .
\eeq
Equation (\ref{scalargmsbgen}) is still not a bad approximation if the
$y_I$ are not very different from each other, because the dependence of
the MSSM mass spectrum on the $y_I$ is only logarithmic (due to RG
running) for fixed $\Lambda$. However, if large hierarchies in the
messenger masses are present, then the additive contributions to the
gaugino masses and scalar squared masses from each individual messenger
multiplet $I$ should really instead be incorporated at the mass scale of
that messenger multiplet. Then RG evolution is used to run these various
contributions down to the electroweak or TeV scale; the individual
messenger contributions to scalar and gaugino masses as indicated above
can be thought of as threshold corrections to this RG running.
Messengers with masses far below the GUT scale will affect the running of
gauge couplings and might therefore be expected to ruin the apparent
unification shown in Figure~\ref{fig:gaugeunification}. However, if the
messengers come in complete multiplets of the $SU(5)$ global
symmetry\footnote{This $SU(5)$ may or may not be promoted to a local gauge
symmetry at the GUT scale. For our present purposes, it is used only as
a classification scheme, since the global $SU(5)$ symmetry is only
approximate in the effective theory at the (much lower) messenger mass
scale where gauge mediation takes place.} that contains the Standard Model
gauge group, and are not very different in mass, then approximate
unification of gauge couplings will still occur when they are extrapolated
up to the same scale $M_U$ (but with a larger unified value for the gauge
couplings at that scale). For this reason, a popular class of models is
obtained by taking the messengers to consist of $\nmess$ copies of the
${\bf 5}+{\bf \overline 5}$ of $SU(5)$, resulting in
\beq
\sum_I n_1(I) = \sum_I n_2(I) =\sum_I n_3(I) = \nmess\> .
\eeq
Equations~(\ref{gauginogmsbgen}) and
(\ref{scalargmsbgen}) then reduce to
\beq
&&M_a \,=\, {\alpha_a \over 4 \pi} \Lambda \nmess ,
\label{gmsbgauginonmess}\\
&&m^2_{\phi_i} \,=\, 2 \Lambda^2 \nmess
\sum_{a=1}^3 C_a(i) \left ({\alpha_a\over 4\pi}\right )^2 ,
\label{gmsbnmess}
\eeq
since now there are $\nmess$ copies of the minimal messenger sector
particles running around the loops. For example, the minimal model in
eq.~(\ref{minimalmess}) corresponds to $\nmess = 1$. A single copy of
${\bf 10} + {\bf \overline{ 10}}$ of $SU(5)$ has Dynkin indices $\sum_I
n_a(I) = 3$, and so can be substituted for 3 copies of ${\bf 5}+{\bf
\overline 5}$. (Other combinations of messenger multiplets can also
preserve the apparent unification of gauge couplings.) Note that the
gaugino masses scale like $\nmess$, while the scalar masses scale like
$\sqrt{\nmess}$. This means that sleptons and squarks will tend to be
lighter relative to the gauginos for larger values of $\nmess$ in
nonminimal models. However, if $\nmess$ is too large, then the running
gauge couplings will diverge before they can unify at $M_U$. For messenger
masses of order $10^6$ GeV or less, for example, one needs $\nmess\leq 4$.
There are many other possible generalizations of the basic gaugemediation
scenario as described above; see for example
refs.~\cite{gmsbcorrB}\cite{GGMb}. The common feature that makes all such models
attractive is that the masses of the squarks and sleptons depend only on
their gauge quantum numbers, leading automatically to the degeneracy of
squark and slepton masses needed for suppression of flavorchanging
effects. But the most distinctive phenomenological prediction of
gaugemediated models may be the fact that the gravitino is the LSP. This
can have crucial consequences for both cosmology and collider physics, as
we will discuss further in sections \ref{subsec:decays.gravitino} and
\ref{sec:signals}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Extradimensional and anomalymediated
supersymmetry breaking}\label{subsec:origins.amsb}
\setcounter{equation}{0}
\setcounter{footnote}{1}
It is also possible to take the partitioning of the MSSM and supersymmetry
breaking sectors shown in fig.~\ref{fig:structure} seriously as geography.
This can be accomplished by assuming that there are extra spatial
dimensions of
the KaluzaKlein or warped type \cite{warped}, so that a physical distance
separates the visible and hidden\footnote{The name ``sequestered" is often
used instead of ``hidden" in this context.} sectors. This general idea
opens up numerous possibilities, which are hard to classify in a detailed
way. For example, string theory suggests six such extra dimensions, with
a staggeringly huge number of possible solutions.
Many of the popular models used to explore this
extradimensional mediated supersymmetry breaking (the acronym XMSB is
tempting) use just one single hidden extra dimension with the MSSM chiral
supermultiplets confined to one 4dimensional spacetime brane and the
supersymmetrybreaking sector confined to a parallel brane a distance
$R_5$ away, separated by a 5dimensional bulk, as in
fig.~\ref{fig:branes}.
\begin{figure}
\begin{minipage}[]{0.51\linewidth}
\caption{The separation of the supersymmetrybreaking sector from the MSSM
sector could take place along a hidden spatial dimension, as in the simple
example shown here. The branes are 4dimensional parallel spacetime
hypersurfaces in
a 5dimensional spacetime.\label{fig:branes}}
\end{minipage}
\hspace{0.025\linewidth}
\begin{minipage}[]{0.435\linewidth}
\begin{picture}(135,115)(50,0)
\SetScale{1.25}
\rText(43,66)[][]{``the bulk"}
\rText(49,116)[][]{$R_5 $}
\Line(5,20)(5,75)
\Line(15,40)(15,95)
\Line(5,20)(15,40)
\Line(5,75)(15,95)
\Line(65,20)(65,75)
\Line(75,40)(75,95)
\Line(65,20)(75,40)
\Line(65,75)(75,95)
\LongArrow(40,99)(74,99)
\LongArrow(40,99)(16.6,99)
\rText(1,13)[][]{MSSM brane}
\rText(1,1)[][]{(we live here)}
\rText(89,13)[][]{Hidden brane}
\rText(84,1)[][]{$\langle F \rangle \not= 0$}
\end{picture}
\end{minipage}
\end{figure}%
Using this as an illustration, the dangerous flavorviolating terms
proportional to $y^{Xijk}$ and
$k^i_j$ in eq.~(\ref{hiddengrav}) are suppressed by
factors like $e^{R_5 M_5}$, where $R_5$ is the size of the 5th dimension
and $M_5$ is the 5dimensional fundamental (Planck) scale, and it is
assumed that the MSSM chiral supermultiplets are confined to their brane.
Therefore, it should be enough to require that $R_5 M_5 \gg 1$, in other
words that the size of the 5th dimension (or, more generally, the volume
of the compactified space) is relatively large in units of the fundamental
length scale. Thus the suppression of flavorviolating effects does not
require any finetuning or extreme hierarchies, because it is exponential.
One possibility is that the gauge supermultiplets of the MSSM propagate in
the bulk, and so mediate supersymmetry breaking
%\cite{MirabelliPeskin,gauginomediationoriginal,gauginomediationlater,
%deconstructedgauginomediation}.
\cite{MirabelliPeskin}\cite{deconstructedgauginomediation}.
This mediation is direct for gauginos, with
\beq
M_a \sim \frac{\langle F \rangle}{M_5(R_5 M_5)} ,
\eeq
but is loopsuppressed for the soft terms involving
scalars. This implies that in the simplest
version of the idea, often called ``gaugino mediation", soft supersymmetry
breaking is dominated by the gaugino masses. The phenomenology is
therefore quite similar to that of the ``noscale" boundary conditions
mentioned in section \ref{subsec:origins.sugra} in the context of PMSB
models. Scalar squared masses and the scalar cubic couplings come from
renormalization group running down to the electroweak scale. It is useful
to keep in mind that gaugino mass dominance is really the essential
feature that defeats flavor violation, so it may well
turn out to be more robust
than any particular model that provides it.
It is also possible that the gauge supermultiplet fields are also confined
to the MSSM brane, so that the transmission of supersymmetry breaking is
due entirely to supergravity effects. This leads to
anomalymediated supersymmetry breaking (AMSB) \cite{AMSB}, sonamed
because the resulting MSSM soft terms can be understood in terms of the
anomalous violation of a local superconformal invariance, an extension of
scale invariance. In one formulation of supergravity
\cite{superconformalsupergravity}, Newton's constant (or equivalently, the
Planck mass scale) is set by the VEV of a scalar field $\phi$ that is part
of a nondynamical chiral supermultiplet (called the ``conformal
compensator"). As a gauge fixing, this field obtains a VEV of $\langle
\phi \rangle = 1$, spontaneously breaking the local superconformal
invariance. Now, in the presence of spontaneous supersymmetry breaking
$\langle F \rangle \not= 0$, for example on the hidden brane, the
auxiliary field component also obtains a nonzero VEV, with
\beq
\langle F_\phi \rangle
\,\sim\,
\frac{\langle F \rangle}{\MPlanck}
\,\sim\,
m_{3/2} .
\label{eq:AMSBgravitino}
\eeq
The nondynamical conformal compensator field $\phi$ is taken to
be dimensionless, so that $F_\phi$ has dimensions of [mass].
In the classical limit, there is still no supersymmetry breaking in the
MSSM sector, due to the exponential suppression provided by the extra
dimensions.\footnote{AMSB can also be realized without invoking extra
dimensions. The suppression of flavorviolating MSSM soft terms can
instead be achieved using a stronglycoupled conformal field theory near
an infraredstable fixed point \cite{AMSBinfourd}.} However, there is an
anomalous violation of superconformal (scale) invariance manifested in the
running of the couplings. This causes supersymmetry breaking to show up in
the MSSM by virtue of the nonzero beta functions and anomalous dimensions
of the MSSM brane couplings and
fields. The resulting soft terms are \cite{AMSB} (using
$\mAMSB$ to denote its VEV from now on):
\beq
M_a &=& \mAMSB \beta_{g_a}/g_a ,
\label{eq:AMSBgauginos}
\\
(m^2)_j^i &=& \frac{1}{2} \mAMSB^2 \frac{d}{dt} \gamma_j^i
\>=\> \frac{1}{2} \mAMSB^2 \left [
\beta_{g_a} \frac{\partial}{\partial g_a}
+ \beta_{y^{kmn}} \frac{\partial}{\partial y^{kmn}}
+ \beta_{y^*_{kmn}} \frac{\partial}{\partial y^*_{kmn}} \right] \gamma_j^i,
\label{eq:AMSBscalars}
\\
a^{ijk} &=& \mAMSB \beta_{y^{ijk}},
\label{eq:AMSBscalar}
\eeq
where the anomalous dimensions $\gamma^i_j$ are normalized as in
eqs.~(\ref{eq:gengamma}) and (\ref{eq:gammaHu})(\ref{eq:gammae}). As in
the GMSB scenario of the previous subsection, gaugino masses arise at
oneloop order, but scalar squared masses arise at twoloop order. Also,
these results are approximately flavorblind for the first two families,
because the nontrivial flavor structure derives only from the MSSM Yukawa
couplings.
There are several unique features of the AMSB scenario. First, there is no
need to specify at which renormalization scale
eqs.~(\ref{eq:AMSBgauginos})(\ref{eq:AMSBscalar}) should be applied as
boundary conditions. This is because they hold at every renormalization
scale, exactly, to all orders in perturbation theory. In other words,
eqs.~(\ref{eq:AMSBgauginos})(\ref{eq:AMSBscalar}) are not just boundary
conditions for the renormalization group equations of the soft parameters,
but solutions as well. (These AMSB renormalization group trajectories can
also be found from this renormalization group invariance property alone
\cite{AMSBtrajectories}, without reference to the supergravity
derivation.) In fact, even if there are heavy supermultiplets in the
theory that have to be decoupled, the boundary conditions hold both above
and below the arbitrary decoupling scale. This remarkable insensitivity to
ultraviolet physics in AMSB ensures the absence of flavor
violation in the lowenergy MSSM soft terms. Another interesting
prediction is that the gravitino mass $m_{3/2}$ in these models is
actually much larger than the scale $m_{\rm soft}$ of the MSSM soft terms,
since the latter are loopsuppressed compared to
eq.~(\ref{eq:AMSBgravitino}).
There is only one unknown parameter, $\mAMSB$, among the MSSM soft terms
in AMSB. Unfortunately, this exemplary falsifiability is
marred by the fact that it is already falsified. The dominant
contributions to the firstfamily squark and slepton squared masses are:
\beq
m^2_{\tilde q} &= & \frac{\mAMSB^2}{(16 \pi^2)^2}
\left (8 g_3^4 + \ldots \right ),
\\
m^2_{\tilde e_L} &= & \frac{\mAMSB^2}{(16 \pi^2)^2}
\left (\frac{3}{2} g_2^4 + \frac{99}{50} g_1^4 \right )
\\
m^2_{\tilde e_R} &= &
\frac{\mAMSB^2}{(16 \pi^2)^2} \frac{198}{25} g_1^4
\eeq
The squarks have large positive squared masses, but the sleptons have
negative squared masses, so the AMSB model in its simplest form is not
viable. These signs come directly from those of the beta functions of the
strong and electroweak gauge interactions, as can be seen from the
right side of eq.~(\ref{eq:AMSBscalars}).
The characteristic ultraviolet insensitivity to physics at high mass
scales also makes it somewhat nontrivial to modify the theory to escape
this tachyonic slepton problem by deviating from the AMSB trajectory. There
can be large deviations from AMSB provided by supergravity
\cite{isAMSBrobust}, but then in general the flavorblindness is also
forfeit. One way to modify AMSB is to introduce additional supermultiplets
that contain supersymmetrybreaking mass splittings that are large
compared to their average mass \cite{AMSBlightstates}. Another way is to
combine AMSB with gaugino mediation \cite{AMSBhybrid}. Some other
proposals can be found in \cite{otherAMSBattempts}. Finally, there is a
perhaps less motivated approach in which a common
parameter $m_0^2$ is added to all of the scalar squared masses at some
scale, and chosen large enough to allow the sleptons to have positive
squared masses above bounds from the CERN LEP $e^+e^$ collider.
This allows the phenomenology to be
studied in a framework conveniently parameterized by just:
\beq
\mAMSB,\, m^2_0,\, \tan\beta,\, {\rm arg}(\mu),
\eeq
with $\mu$ and $b$ determined by requiring correct electroweak symmetry
breaking as described in the next section. (Some sources use $m_{3/2}$ or
$M_{\rm aux}$ to denote $ \mAMSB$.) The MSSM gaugino masses at the leading
nontrivial order are unaffected by the {\it ad hoc} addition of $m_0^2$:
\beq
M_1 &=& \frac{\mAMSB}{16 \pi^2} \frac{33}{5} g_1^2
\\
M_2 &=& \frac{\mAMSB}{16 \pi^2} g_2^2
\\
M_3 &=& \frac{\mAMSB}{16 \pi^2} 3 g_3^2
\eeq
This implies that $M_2 \ll M_1 \ll M_3$, so the lightest neutralino
is actually mostly wino, with a lightest chargino that is only of order
200 MeV heavier, depending on the values of $\mu$ and $\tan\beta$.
The decay $\stilde C_1^\pm \rightarrow \stilde N_1 \pi^\pm$
produces a very soft pion, implying
unique and difficult signatures in colliders
\cite{Chen:1996ap}\cite{AMSBphenothree}.
Another large general class of models breaks supersymmetry using the
geometric or topological
properties of the extra dimensions. In the ScherkSchwarz
mechanism \cite{ScherkSchwarz}, the symmetry is broken by
assuming different boundary
conditions for the fermion and boson fields on the compactified space. In
supersymmetric models where the size of the extra dimension is
parameterized by a modulus (a massless or nearly massless excitation)
called a radion, the $F$term component of the radion chiral
supermultiplet can obtain a VEV, which becomes a source for supersymmetry
breaking in the MSSM. These two ideas turn out to be often related.
Some of the variety of models proposed along these lines can be found in
\cite{SSandradion}. These mechanisms
can also be combined with gauginomediation and AMSB.
It seems likely that the possibilities are not yet fully explored.
\section{The mass spectrum of the MSSM}\label{sec:MSSMspectrum}
\renewcommand{\theequation}{\arabic{section}.\arabic{subsection}.\arabic{equation}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Electroweak symmetry breaking and the Higgs
bosons}\label{subsec:MSSMspectrum.Higgs}
\setcounter{equation}{0}
\setcounter{footnote}{1}
In the MSSM, the description of electroweak symmetry breaking is slightly
complicated by the fact that there are two complex Higgs doublets $H_u =
(H_u^+,\> H_u^0)$ and $H_d = (H_d^0,\> H_d^)$ rather than just one in the
ordinary Standard Model. The classical scalar potential for the Higgs
scalar fields in the MSSM is given by
\beq
V\! &=&\!
(\mu^2 + m^2_{H_u}) (H_u^0^2 + H_u^+^2)
+ (\mu^2 + m^2_{H_d}) (H_d^0^2 + H_d^^2)
\nonumber \\ &&
+\, [b\, (H_u^+ H_d^  H_u^0 H_d^0) + \conj]
\nonumber \\ &&
+ {1\over 8} (g^2 + g^{\prime 2})
(H_u^0^2 + H_u^+^2  H_d^0^2  H_d^^2 )^2
+ \half g^2 H_u^+ H_d^{0*} + H_u^0 H_d^{*}^2 .
\phantom{xxx}
\label{bighiggsv}
\eeq
The terms proportional to $\mu ^2$ come from $F$terms [see
eq.~(\ref{movie})]. The terms proportional to $g^2$ and $g^{\prime 2}$
are the $D$term contributions, obtained from the general
formula eq.~(\ref{fdpot}) after some rearranging. Finally, the terms
proportional to $m_{H_u}^2$, $m_{H_d}^2$, and $b$ are just a
rewriting of the last three terms of eq.~(\ref{MSSMsoft}). The full scalar
potential of the theory also includes many terms involving the squark and
slepton fields that we can ignore here, since they do not get VEVs because
they have large positive squared masses.
We now have to demand that the minimum of this potential should break
electroweak symmetry down to electromagnetism $SU(2)_L\times U(1)_Y
\rightarrow U(1)_{\rm EM}$, in accord with observation. We can use the
freedom to make gauge transformations to simplify this analysis. First,
the freedom to make $SU(2)_L$ gauge transformations allows us to rotate
away a possible VEV for one of the weak isospin components of one of the
scalar fields, so without loss of generality we can take $H_u^+=0$ at the
minimum of the potential. Then one can check that a minimum of the potential
satisfying $\partial V/\partial H_u^+=0$ must also have $H_d^ = 0$. This
is good, because it means that at the minimum of the potential
electromagnetism is necessarily unbroken, since the charged components of
the Higgs scalars cannot get VEVs. After setting $H_u^+=H_d^=0$, we are
left to consider the scalar potential
\beq
V \!&=&\!
(\mu^2 + m^2_{H_u}) H_u^0^2 + (\mu^2 + m^2_{H_d}) H_d^0^2
 (b\, H_u^0 H_d^0 + \conj)
\nonumber \\ &&
+ {1\over 8} (g^2 + g^{\prime 2}) ( H_u^0^2  H_d^0^2 )^2 .
\label{littlehiggsv}
\eeq
The only term in this potential that depends on the phases of the fields
is the $b$term. Therefore, a redefinition of the phase of $H_u$ or $H_d$
can absorb any phase in $b$, so we can take $b$ to be real and positive.
Then it is clear that a minimum of the potential $V$ requires that $H_u^0
H_d^0$ is also real and positive, so $\langle H_u^0\rangle$ and $\langle
H_d^0\rangle$ must have opposite phases. We can therefore use a $U(1)_Y$
gauge transformation to make them both be real and positive without loss
of generality, since $H_u$ and $H_d$ have opposite weak hypercharges ($\pm
1/2$). It follows that CP cannot be spontaneously broken by the Higgs
scalar potential, since the VEVs and $b$ can be simultaneously chosen
real, as a convention. This implies that the Higgs scalar mass eigenstates
can be assigned welldefined eigenvalues of CP, at least at treelevel.
(CPviolating phases in other couplings can induce loopsuppressed CP
violation in the Higgs sector, but do not change the fact that $b$,
$\langle H_u^0 \rangle$,
and $\langle H_d \rangle$ can always be chosen real and positive.)
In order for the MSSM scalar potential to be viable, we must first make
sure that the potential is bounded from below for arbitrarily large values
of the scalar fields, so that $V$ will really have a minimum. (Recall from
the discussion in sections \ref{subsec:susylagr.chiral} and
\ref{subsec:susylagr.gaugeinter} that scalar potentials in purely
supersymmetric theories are automatically nonnegative and so clearly
bounded from below. But, now that we have introduced supersymmetry
breaking, we must be careful.) The scalar quartic interactions in $V$ will
stabilize the potential for almost all arbitrarily large values of $H_u^0$
and $H_d^0$. However, for the special directions in field space $H_u^0 =
H_d^0$, the quartic contributions to $V$ [the second line in
eq.~(\ref{littlehiggsv})] are identically zero. Such directions in field
space are called $D$flat directions, because along them the part of the
scalar potential coming from $D$terms vanishes. In order for the
potential to be bounded from below, we need the quadratic part of the
scalar potential to be positive along the $D$flat directions. This
requirement amounts to
\beq
2 b < 2 \mu ^2 + m^2_{H_u} + m^2_{H_d}.
\label{eq:boundedfrombelow}
\eeq
Note that the $b$term always favors electroweak symmetry breaking.
Requiring that one linear combination of $H_u^0$ and $H_d^0$ has a
negative squared mass near $H_u^0=H_d^0=0$ gives
\beq
b^2 > (\mu^2 + m^2_{H_u} )(\mu^2 + m^2_{H_d}).
\label{eq:destabilizeorigin}
\eeq
If this inequality is not satisfied, then $H_u^0 = H_d^0 = 0$ will be a
stable minimum of the potential (or there will be no stable minimum at
all), and electroweak symmetry breaking will not occur.
Interestingly, if $m_{H_u}^2 = m_{H_d}^2$ then the constraints
eqs.~(\ref{eq:boundedfrombelow}) and (\ref{eq:destabilizeorigin}) cannot
both be satisfied. In models derived from the MSUGRA or
GMSB boundary conditions, $m_{H_u}^2 = m_{H_d}^2$ is supposed to
hold at tree level at the input scale, but the $X_t$ contribution to the
RG equation for $m_{H_u}^2$ [eq.~(\ref{mhurge})] naturally pushes it to
negative or small values $m_{H_u}^2 < m_{H_d}^2$ at the electroweak scale.
Unless this effect is significant, the parameter space in which the
electroweak symmetry is broken would be quite small. So, in these models
electroweak symmetry breaking is actually driven by quantum corrections;
this mechanism is therefore known as {\it radiative electroweak symmetry
breaking}. Note that although a negative value for $\mu^2 + m_{H_u}^2$
will help eq.~(\ref{eq:destabilizeorigin}) to be satisfied, it is not
strictly necessary. Furthermore, even if $m_{H_u}^2<0$, there may be no
electroweak symmetry breaking if $\mu$ is too large or if $b$ is too
small. Still, the large negative contributions to $m_{H_u}^2$ from the RG
equation are an important factor in ensuring that electroweak symmetry
breaking can occur in models with simple boundary conditions for the soft
terms. The realization that this works most naturally with a large
topquark Yukawa coupling provides additional motivation for these models
\cite{rewsbone,rewsbtwo}.
Having established the conditions necessary for $H_u^0$ and $H_d^0$ to get
nonzero VEVs, we can now require that they are compatible with the
observed phenomenology of electroweak symmetry breaking, $SU(2)_L \times
U(1)_Y \rightarrow U(1)_{\rm EM}$. Let us write
\beq
v_u = \langle H_u^0\rangle,
\qquad\qquad
v_d = \langle H_d^0\rangle.
\label{defvuvd}
\eeq
These VEVs are related to the known mass of the $Z^0$ boson and the
electroweak gauge couplings:
\beq
v_u^2 + v_d^2 = v^2 = 2 m_Z^2/(g^2 + g^{\prime 2}) \approx (174\>{\rm
GeV})^2.
\label{vuvdcon}
\eeq
The ratio of the VEVs is traditionally written as
\beq
\tan\beta \equiv v_u/v_d.
\label{deftanbeta}
\eeq
The value of $\tan\beta$ is not fixed by present experiments, but it
depends on the Lagrangian parameters of the MSSM in a calculable way.
Since $v_u = v \sin\beta$ and $v_d = v \cos\beta$ were taken to be real
and positive by convention,
we have $0 < \beta < \pi/2$, a requirement that will be
sharpened below. Now one can write down the conditions $\partial
V/\partial H_u^0= \partial V/\partial H_d^0 = 0$ under which the potential
eq.~(\ref{littlehiggsv}) will have a minimum satisfying
eqs.~(\ref{vuvdcon}) and (\ref{deftanbeta}):
\beq
&&m_{H_u}^2 + \mu ^2 b \cot\beta  (m_Z^2/2) \cos (2\beta)
\>=\> 0 ,
\label{mubsub2}
\\
&&m_{H_d}^2 + \mu ^2 b \tan\beta + (m_Z^2/2) \cos (2\beta) \>=\> 0.
\label{mubsub1}
\eeq
It is easy to check that these equations indeed satisfy the necessary
conditions eqs.~(\ref{eq:boundedfrombelow}) and
(\ref{eq:destabilizeorigin}). They allow us to eliminate two of the
Lagrangian parameters $b$ and $\mu$ in favor of $\tan\beta$, but do not
determine the phase of $\mu$. Taking $\mu^2$, $b$, $m_{H_u}^2$ and
$m_{H_d}^2$ as input parameters, and $m_Z^2$ and $\tan\beta$ as output
parameters obtained by solving these two equations, one obtains:
\beq
\sin (2\beta) &=& \frac{2 b}{m^2_{H_u} + m^2_{H_d} + 2\mu^2},
\label{eq:solvesintwobeta}
\\
m_Z^2 &=& \frac{m^2_{H_d}  m^2_{H_u}}{\sqrt{1  \sin^2(2\beta)}}
 m^2_{H_u}  m^2_{H_d} 2\mu^2
.
\label{eq:solvemzsq}
\eeq
Note that $\sin (2\beta)$ is always positive. If $m^2_{H_u} < m^2_{H_d}$,
as is usually assumed, then $\cos(2\beta)$ is negative; otherwise it is
positive.
As an aside, eqs.~(\ref{eq:solvesintwobeta}) and
(\ref{eq:solvemzsq}) highlight the ``$\mu$ problem" already mentioned in
section \ref{subsec:mssm.superpotential}. Without miraculous
cancellations, all of the input parameters ought to be within an order of
magnitude or two of $m^2_Z$. However, in the MSSM, $\mu$ is a
supersymmetryrespecting parameter appearing in the superpotential, while
$b$, $m_{H_u}^2$, $m_{H_d}^2$ are supersymmetrybreaking parameters. This
has lead to a widespread belief that the MSSM must be extended at very
high energies to include a mechanism that relates the effective value of
$\mu$ to the supersymmetrybreaking mechanism in some way; see sections
\ref{subsec:variations.NMSSM} and
\ref{subsec:variations.munonrenorm} and
ref.~\cite{muproblemGMSB} for examples.
Even if the value of $\mu$ is set by soft supersymmetry breaking, the
cancellation needed by eq.~(\ref{eq:solvemzsq}) is often remarkable when
evaluated in specific model frameworks, after constraints from direct
searches for the superpartners are taken into account.
For example, expanding for large $\tan\beta$, eq.~(\ref{eq:solvemzsq})
becomes
\beq
m_Z^2 = 2 (m^2_{H_u} + \mu^2)
+ \frac{2}{\tan^2\beta} (m^2_{H_d}  m^2_{H_u})
+ {\cal O}(1/\tan^4\beta).
\eeq
Typical viable solutions for the MSSM have $m^2_{H_u}$ and $\mu^2$ each
much larger than $m_Z^2$, so that significant cancellation is needed. In
particular, large top squark squared masses, needed to avoid having the
Higgs boson mass turn out too small [see eq.~(\ref{hradcorr}) below]
compared to the observed value of 125 GeV, will feed into $m^2_{H_u}$.
The cancellation needed in the minimal model may therefore be at the
several per cent level, or worse. It is impossible to objectively characterize
whether this should be considered worrisome, but it certainly causes
subjective worry as the LHC bounds on superpartners increase.
Equations~(\ref{mubsub2})(\ref{eq:solvemzsq})
are based on the treelevel potential, and involve
running renormalized Lagrangian parameters, which depend on the choice of
renormalization scale. In practice, one must include radiative corrections
at oneloop order, at least, in order to get numerically stable results.
To do this, one can compute the loop corrections $\Delta V$ to the
effective potential $V_{\rm eff}(v_u, v_d) = V + \Delta V$ as a function
of the VEVs. The impact of this is that the equations governing the VEVs
of the full effective potential are obtained by simply replacing
\beq
m^2_{H_u} \rightarrow m^2_{H_u} + \frac{1}{2 v_u}
\frac{\partial (\Delta V)}{\partial v_u},\qquad\qquad
m^2_{H_d} \rightarrow m^2_{H_d} + \frac{1}{2 v_d}
\frac{\partial (\Delta V)}{\partial v_d}
\label{eq:Vradcor}
\eeq
in eqs.~(\ref{mubsub2})(\ref{eq:solvemzsq}), treating $v_u$ and $v_d$
as real variables in the differentiation.
The result for $\Delta V$ has now been obtained through twoloop
order in the MSSM \cite{twoloopEP,twoloopEPMSSM}. The most important
corrections come from
the oneloop diagrams involving the top squarks and top quark, and
experience shows that the validity of the treelevel approximation and
the convergence of perturbation theory are therefore improved by choosing
a renormalization scale roughly of order the average of the top squark masses.
The Higgs scalar fields in the MSSM consist of two complex
$SU(2)_L$doublet, or eight real, scalar degrees of freedom. When the
electroweak symmetry is broken, three of them are the wouldbe
NambuGoldstone bosons $G^0$, $G^\pm$, which become the longitudinal modes
of the $Z^0$ and $W^\pm$ massive vector bosons. The remaining five Higgs
scalar mass eigenstates consist of two CPeven neutral scalars $h^0$ and
$H^0$, one CPodd neutral scalar $A^0$, and a charge $+1$ scalar $H^+$ and
its conjugate charge $1$ scalar $H^$. (Here we define $G^ = G^{+*}$ and
$H^ = H^{+*}$. Also, by convention, $h^0$ is lighter than $H^0$.) The
gaugeeigenstate fields can be expressed in terms of the mass eigenstate
fields as:
%
\renewcommand{\arraystretch}{1.4}
\beq
\pmatrix{H_u^0 \cr H_d^0} &=&
\pmatrix{v_u \cr v_d}
+ {1\over \sqrt{2}} R_\alpha \pmatrix{h^0 \cr H^0}
+ {i\over \sqrt{2}} R_{\beta_0} \pmatrix{G^0 \cr A^0}
\eeq
\beq
\pmatrix{H_u^+ \cr H_d^{*}} &=& R_{\beta_\pm} \pmatrix{G^+ \cr H^+}
\eeq
where the orthogonal rotation matrices
\beq
&&R_{\alpha} = \pmatrix{\cos\alpha & \sin\alpha \cr
\sin\alpha & \cos\alpha},
\\
&&R_{\beta_0} = \pmatrix{\sin\beta_0 & \cos\beta_0 \cr
\cos\beta_0 & \sin\beta_0},
\qquad\quad
R_{\beta_\pm} = \pmatrix{\sin\beta_\pm & \cos\beta_\pm \cr
\cos\beta_\pm & \sin\beta_\pm}
,
\phantom{xxxxx}
\eeq
are chosen so that the quadratic part of the potential has diagonal
squaredmasses:
\beq
V &=&
\half m_{h^0}^2 (h^{0})^2 + \half m_{H^0}^2 (H^{0})^2
+ \half m_{G^0}^2 (G^{0})^2 + \half m_{A^0}^2 (A^{0})^2
\nonumber \\ &&
+ m_{G^\pm}^2 G^+^2 + m_{H^\pm}^2 H^+^2 + \ldots ,
\phantom{xxx}
\eeq
Then, provided that $v_u,v_d$ minimize the treelevel
potential,\footnote{It is often more
useful to expand around VEVs $v_u, v_d$
that do not minimize the treelevel potential, for example to minimize the
loopcorrected effective potential instead. In that case, $\beta$,
$\beta_0$, and $\beta_\pm$ are all slightly different.} one finds that
$\beta_0 = \beta_\pm = \beta$, and $m^2_{G^0} = m^2_{G^\pm}=0$, and
\beq
m_{A^0}^2 \!\!&=&\!\! 2 b/\sin (2\beta)
\,=\, 2\mu^2 + m^2_{H_u} + m^2_{H_d}
\\
m^2_{h^0, H^0} \!\!\!&=\!\!& \half
\Bigl (
m^2_{A^0} + m_Z^2 \mp
\sqrt{(m_{A^0}^2  m_Z^2)^2 + 4 m_Z^2 m_{A^0}^2 \sin^2 (2\beta)}
\Bigr ),\>\>\>\>\>{}
\label{eq:m2hH}
\\
m^2_{H^\pm} \!\!&=&\!\! m^2_{A^0} + m_W^2 .
\label{eq:m2Hpm}
\eeq
The mixing angle $\alpha$ is determined, at treelevel, by
\beq
{\sin 2\alpha\over \sin 2 \beta} \,=\,
\left ( {m_{H^0}^2 + m_{h^0}^2 \over
m_{H^0}^2  m_{h^0}^2} \right ) ,
\qquad\quad
{\tan 2\alpha\over \tan 2 \beta} \,=\,
\left ( {m_{A^0}^2 + m_{Z}^2 \over
m_{A^0}^2  m_{Z}^2} \right ) ,
\eeq
and is traditionally chosen to be negative; it follows that $\pi/2
<\alpha < 0$ (provided $m_{A^0} > m_Z$). The Feynman rules for couplings
of the mass eigenstate Higgs scalars to the Standard Model quarks and
leptons and the electroweak vector bosons, as well as to the various
sparticles, have been worked out in detail in ref.~\cite{GunionHaber,HHG,Haber:1997dt}.
The masses of $A^0$, $H^0$ and $H^\pm$ can be arbitrarily
large, in principle, since they all grow with $b/\sin (2\beta)$. In contrast, the mass of
$h^0$ is bounded above. From eq.~(\ref{eq:m2hH}), one finds
at treelevel \cite{treelevelhiggsbound}:
\beq
m_{h^0} \><\> m_Z \cos (2\beta) 
\label{eq:higgsineq}
\eeq
This corresponds to a shallow direction in the %scalar
potential, along the
direction $(H_u^0v_u, H_d^0v_d) \propto (\cos\alpha,\sin\alpha)$. The
existence of this shallow direction can be traced to the supersymmetric fact that the
quartic Higgs couplings are small, being given by the squares of the electroweak gauge
couplings, via the $D$term. A contour map of the potential, for a typical
case with $\tan\beta \approx \cot\alpha \approx 10$, is shown in figure
\ref{fig:contourmap}.%
\begin{figure}
\centerline{\psfig{figure=higgscon.eps,width=0.9\linewidth}}
\vspace{0.32cm}
\caption{A contour map of the Higgs potential, for a typical case with
$\tan\beta \approx \cot\alpha \approx 10$. The minimum of the potential
is marked by $+$, and the contours are equally spaced equipotentials.
Oscillations along the shallow direction, with $H^0_u/H_d^0 \approx 10$,
correspond to the mass eigenstate $h^0$, while the orthogonal steeper
direction corresponds to the mass eigenstate $H^0$.\label{fig:contourmap}}
\end{figure}
%
If the treelevel inequality (\ref{eq:higgsineq}) were robust, the
lightest Higgs boson of the MSSM would have been discovered in the previous century
at the CERN LEP2 $e^+e^$ collider, and its mass obviously could not approach the
observed value of 125 GeV.
However, the treelevel formula for the squared mass of $h^0$ is subject
to quantum corrections that are relatively drastic. The largest such
contributions typically come from top and stop loops, as
shown\footnote{In general, oneloop 1particlereducible
tadpole diagrams should also be included.
However, they exactly cancel against the treelevel
tadpoles, and so both can be omitted,
if the VEVs $v_u$ and $v_d$ are taken at the minimum of the
loopcorrected effective potential (see previous footnote).}
in fig.~\ref{fig:MSSMhcorrections}.%
\begin{figure}[!t]
\begin{center}
\begin{picture}(338,38)(38,1)
\Text(38,0)[c]{$\Delta(m_{h^0}^2)\> =\, $}
\SetWidth{0.7}
\DashLine(0,0)(25,0){5}
\DashLine(65,0)(90,0){5}
\SetWidth{1.3}
\CArc(45,0)(20,0,360)
\Text(3,8)[c]{$h^0$}
\Text(45,28)[c]{$t$}
\Text(105,0)[c]{$+$}
\SetWidth{0.7}
\DashLine(120,0)(145,0){4.5}
\DashLine(185,0)(210,0){4.5}
\SetWidth{1.3}
\DashCArc(165,0)(20,180,360){4}
\DashCArc(165,0)(20,0,180){4}
\Text(124,8)[c]{$h^0$}
\Text(165,28)[c]{$\tilde t$}
\SetWidth{0.7}
\Text(225,0)[c]{$+$}
\DashLine(240,5)(275,5){4.5}
\DashLine(310,5)(275,5){4.5}
\SetWidth{1.3}
\DashCArc(275,11)(16,90,270){4}
\Text(243,3)[c]{$h^0$}
\Text(275,19)[c]{$\tilde t$}
\end{picture}
\end{center}
\caption{Contributions to the MSSM lightest Higgs squared mass from topquark and
topsquark oneloop diagrams. Incomplete cancellation, due to soft
supersymmetry breaking, leads to a large positive correction to
$m_{h^0}^2$ in the limit of heavy top squarks.\label{fig:MSSMhcorrections}}
\end{figure}
In the limit of topsquark
masses $m_{\stilde t_1}$, $m_{\stilde t_2}$ much greater than the top
quark mass $m_t$, the largest radiative correction
to $m_{h^0}^2$ in eq.~(\ref{eq:m2hH}) is:
\beq
\Delta (m^2_{h^0}) &=&
{3\over 4 \pi^2} \cos^2\!\alpha\>\, y_t^2 m_t^2
\left [
{\rm ln}(m_{\tilde t_1} m_{\tilde t_2} / m_t^2 )
+ \Delta_{\rm threshold} \right ], \phantom{xx}
\label{hradcorr}
\eeq
where
\beq
\Delta_{\rm threshold} &=&
c_{\tilde t}^2 s_{\tilde t}^2 [(m_{\tilde t_2}^2  m_{\tilde t_1}^2)/m_t^2]
\, {\rm ln}(m_{\tilde t_2}^2/m_{\tilde t_1}^2)
\nonumber
\\
&&
+ c_{\tilde t}^4 s_{\tilde t}^4 \left [
(m_{\tilde t_2}^2  m_{\tilde t_1}^2)^2  \half
(m_{\tilde t_2}^4  m_{\tilde t_1}^4)
\, {\rm ln}(m_{\tilde t_2}^2/m_{\tilde t_1}^2)
\right ]/m_t^4,
\label{eq:detDeltathreshold}
\eeq
with $c_{\tilde t}$ and $s_{\tilde t}$ equal to the cosine and sine of a topsquark mixing angle $\theta_{\tilde t}$, defined below following eq.~(\ref{pixies}).
One way to understand eq.~(\ref{hradcorr}) is by thinking in terms of the
low energy effective Standard Model theory obtained by integrating out the top
squarks at a renormalization scale equal to the geometric mean of their masses.
Then
%eq.~(\ref{eq:detDeltathreshold})
$\Delta_{\rm threshold}$
comes from the finite threshold correction to the supersymmetric
Higgs quartic coupling, via the first three
diagrams shown in fig.~\ref{fig:MSSMVcorrections}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\begin{center}
%\begin{picture}(435,30)(0,6)
\begin{picture}(435,30)(140,6)
%\DashLine(0,15)(30,15){5}
%\DashLine(0,15)(30,15){5}
%\DashLine(90,15)(60,15){5}
%\DashLine(90,15)(60,15){5}
%\SetWidth{1.3}
%\Line(30,15)(30,15)
%\Line(60,15)(60,15)
%\Line(30,15)(60,15)
%\Line(30,15)(60,15)
%\Text(45,23)[c]{$t$}
\SetWidth{0.7}
\DashLine(350,15)(380,15){5}
\DashLine(350,15)(380,15){5}
\DashLine(440,15)(410,15){5}
\DashLine(440,15)(410,15){5}
\SetWidth{1.3}
\DashLine(380,15)(380,15){4}
\DashLine(410,15)(410,15){4}
\DashLine(380,15)(410,15){4}
\DashLine(380,15)(410,15){4}
\Text(395,23)[c]{$\tilde t$}
\SetWidth{0.7}
\DashLine(235,15)(260,0){4}
\DashLine(235,15)(260,0){4}
\DashLine(315,15)(285,15){5}
\DashLine(315,15)(285,15){5}
\SetWidth{1.3}
\DashLine(285,15)(285,15){4}
\DashLine(260,0)(285,15){4}
\DashLine(260,0)(285,15){4}
\Text(270,15.5)[c]{$\tilde t$}
\SetWidth{0.7}
\DashLine(122,15)(147,0){4}
\DashLine(122,15)(147,0){4}
\DashLine(199,15)(179,0){4}
\DashLine(199,15)(179,0){4}
\SetWidth{1.3}
\DashCArc(163,0)(16,0,180){4}
\DashCArc(163,0)(16,180,360){4}
\Text(165,24)[c]{$\tilde t$}
%
\SetWidth{0.7}
\DashLine(500,15)(530,15){5}
\DashLine(500,15)(530,15){5}
\DashLine(590,15)(560,15){5}
\DashLine(590,15)(560,15){5}
\SetWidth{1.3}
\Line(530,15)(530,15)
\Line(560,15)(560,15)
\Line(530,15)(560,15)
\Line(530,15)(560,15)
\Text(545,23)[c]{$t$}
%
%\DashLine(0,15)(30,15){5}
%\DashLine(0,15)(30,15){5}
%\DashLine(90,15)(60,15){5}
%\DashLine(90,15)(60,15){5}
%\SetWidth{1.3}
%\Line(30,15)(30,15)
%\Line(60,15)(60,15)
%\Line(30,15)(60,15)
%\Line(30,15)(60,15)
%\Text(45,23)[c]{$t$}
%
\end{picture}
\end{center}
\caption{Contributions to the lowenergy
Standard Model effective Higgs quartic interaction.
Integrating out the
%top quark and
top squarks
yields threshold contributions to the quartic Higgs coupling
in the lowenergy effective theory
from the first three oneloop diagrams. The last diagram, involving
the top quark, provides
renormalization group running of the lowenergy effective
Higgs quartic coupling proportional to $y_t^4$.
\label{fig:MSSMVcorrections}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The term with
${\rm ln}(m_{\tilde t_1} m_{\tilde t_2} / m_t^2)$
in eq.~(\ref{hradcorr}) then from comes the renormalization group running of
the Higgs quartic coupling (due to the last diagram in fig.~\ref{fig:MSSMVcorrections})
down to the topquark mass scale, which turns out to be a good renormalization scale at
which to evaluate $m_{h^0}^2$ within the Standard Model effective theory.
For small or moderate topsquark mixing, the logarithmic running term is largest, but
$\Delta_{\rm threshold}$ can also be quite important. These corrections to the
Higgs effective quartic coupling increase the steepness of the Higgs potential, thus
raising $m_{h^0}$ compared to the naive treelevel prediction.
The term proportional to $c_{\tilde t}^2 s_{\tilde t}^2$
in eq.~(\ref{eq:detDeltathreshold}) is positive definite,
while the term proportional to $c_{\tilde t}^4 s_{\tilde t}^4$ is negative definite.
For fixed topsquark masses, the maximum possible $h^0$ mass therefore
occurs for rather large topsquark mixing,
$c_{\tilde t}^2 s_{\tilde t}^2 = m_t^2/[m_{\tilde t_2}^2 + m_{\tilde t_1}^2
 2 (m_{\tilde t_2}^2  m_{\tilde t_1}^2)/{\rm ln}(m_{\tilde t_2}^2/m_{\tilde t_1}^2)
]$ or 1/4, whichever is less. This is often referred to as the ``maximal mixing"
scenario for the MSSM Higgs sector. (What is being maximized
is not the mixing, but $m_{h^0}$ with respect to the topsquark mixing.)
It follows that the quantity in
square brackets in eq.~(\ref{hradcorr}) is always less than
$\ln(m_{\tilde t_2}^2/m_t^2) + 3$.
Equation (\ref{hradcorr}) already shows that $m_{h^0}$
can easily exceed the $Z$ boson mass, and the
observed value of $m_{h^0} = 125$ GeV can in principle be accommodated. However,
the above is a highly simplified account; to get reasonably accurate predictions for the Higgs scalar masses and mixings
for a given set of model parameters, one must also include
the remaining oneloop corrections and even the dominant twoloop and threeloop
effects
%\cite{hcorrections,HHH,HHWHiggs,EZHiggs,ComplexHiggs,reconciliation,
%ENHiggs,BDSZHiggs,
%Dedes:2003km
%Allanach:2004rh,
%SPMHiggs,FeynHiggs,
%Borowka:2014wla,
%Hollik:2014bua,
%Degrassi:2014pfa,
%Martin:2007pg,
%Harlander:2008ju,
%Kant:2010tf,
%Draper:2013oza,
%Bagnaschi:2014rsa,
%Vega:2015fna,
%DraperRzehak}.
\cite{hcorrections}\cite{Vega:2015fna}.
The theoretical
uncertainties associated with the prediction of $m_{h^0}$, given all of the
soft supersymmetry breaking parameters, are still quite large, especially
when the topsquarks are heavy, and are of order several GeV.
For a recent review, see \cite{DraperRzehak}.
Including such corrections, it had been estimated long
before the discovery of the 125 GeV Higgs boson that
\beq
m_{h^0} \lsim 135\>{\rm GeV}
\label{mssmhiggsbound}
\eeq
in the MSSM. This prediction assumed that all of the sparticles that can contribute
to $m_{h^0}^2$ in loops have masses that do not exceed 1 TeV,
and the bound increases logarithmically with the topsquark masses.
However, in many specific model
frameworks with small or moderate topsquark mixing,
the bound eq.~(\ref{mssmhiggsbound}) is very far from saturated, and
it turns out to be a severe challenge to accommodate values even as large as the observed
$m_{h^0} = 125$ GeV, unless the top squarks are extremely heavy, or else highly mixed.
Unfortunately, it is difficult to make this statement very
precise, due both to the high dimensionality of the
supersymmetric parameter space and the theoretical errors in the $m_{h^0}$ prediction.
In the MSSM, the masses and CKM mixing angles of the quarks and leptons
are determined not only by the Yukawa couplings of the superpotential but
also the parameter $\tan\beta$. This is because the top, charm and up
quark mass matrix is proportional to $v_u = v \sin\beta$ and the bottom,
strange, and down quarks and the charge leptons get masses proportional to
$v_d = v \cos\beta$. At treelevel,
\beq
m_t \,=\, y_t v \sin\beta
,
\qquad\quad
m_b \,=\, y_b v \cos\beta
,
\qquad\quad
m_\tau = y_\tau v \cos\beta .\phantom{xxx}
\label{eq:ytbtaumtbtau}
\eeq
These relations hold for the running masses rather than the physical pole
masses, which are significantly larger for $t,b$ \cite{polecat}. Including
those corrections, one can relate the Yukawa couplings to $\tan\beta$ and
the known fermion masses and CKM mixing angles. It is now clear why we
have not neglected $y_b$ and $y_\tau$, even though $m_b,m_\tau\ll m_t$. To
a first approximation, $y_b/y_t = (m_b/m_t)\tan\beta$ and $y_\tau/y_t =
(m_\tau/m_t)\tan\beta$, so that $y_b$ and $y_\tau$ cannot be neglected if
$\tan\beta$ is much larger than 1. In fact, there are good theoretical
motivations for considering models with large $\tan\beta$. For example,
models based on the GUT gauge group $SO(10)$
can unify the running top, bottom and tau Yukawa couplings at the
unification scale; this requires $\tan\beta$ to be very roughly of order
$m_t/m_b$ \cite{so10,copw}.
Note that if one tries to make $\sin\beta$ too small, then $y_t$ will be
nonperturbatively large. Requiring that $y_t$ does not blow up above the
electroweak scale, one finds that $\tan\beta \gsim 1.2$ or so, depending
on the mass of the top quark, the QCD coupling, and other details. In
principle, there is also a constraint on $\cos\beta$ if one requires
that $y_b$ and $y_\tau$ do not become nonperturbatively large. This
gives a rough upper bound of $\tan\beta \lsim$ 65. However, this is
complicated somewhat by the fact that the bottom quark mass gets
significant oneloop nonQCD corrections in the large $\tan\beta$ limit
\cite{copw}. One can obtain a stronger upper bound on $\tan\beta$ in some
models where $m_{H_u}^2 = m_{H_d}^2$ at the input scale, by requiring that
$y_b$ does not significantly exceed $y_t$. [Otherwise, $X_b$ would be
larger than $X_t$ in eqs.~(\ref{mhurge}) and (\ref{mhdrge}), so one would
expect $m_{H_d}^2 < m_{H_u}^2$ at the electroweak scale, and the minimum
of the potential would have $\langle H_d^0 \rangle > \langle H_u^0
\rangle$. This would be a contradiction with the supposition that
$\tan\beta$ is large.] The parameter $\tan\beta$ also
directly impacts the masses and mixings of the MSSM sparticles,
as we will see below.
It is interesting to write the dependences on the angles $\beta$ and $\alpha$ of
the treelevel couplings of the neutral MSSM Higgs bosons. The
bosonic couplings are proportional to:
\beq
h^0 W^+W^,\quad h^0 ZZ,\quad Z H^0 A^0,\quad W^\pm H^0 H^\mp &\propto& \sin(\beta\alpha)
,
\\
H^0 W^+W^,\quad H^0 ZZ,\quad
Z h^0 A^0,\quad W^\pm h^0 H^\mp\> &\propto& \cos(\beta\alpha),
\eeq
and the couplings to fermions are proportional to
\beq
h^0b\bar b,\quad h^0\tau^+\tau^\> &\propto&\frac{\sin\alpha}{\cos\beta}\>=\>
\sin(\beta\alpha)  \tan\beta \cos(\beta\alpha),\phantom{xxxx}
\\
h^0t\bar t\> &\propto& \frac{\cos\alpha}{\sin\beta}\>=\>
\sin(\beta\alpha) + \cot\beta \cos(\beta\alpha),
\\
H^0b\bar b,\quad H^0\tau^+\tau^\> &\propto&\frac{\cos\alpha}{\cos\beta}\>=\>
\cos(\beta\alpha) + \tan\beta \sin(\beta\alpha),
\\
H^0t\bar t\> &\propto& \frac{\sin\alpha}{\sin\beta}\>=\>
\cos(\beta\alpha)  \cot\beta \sin(\beta\alpha),
\\
A^0b\bar b,\quad A^0\tau^+\tau^\> &\propto& \tan\beta,
\\
A^0t\bar t\> &\propto& \cot\beta.
\eeq
An important case, often referred to as the ``decoupling limit", occurs
when $m_{A^0} \gg m_Z$. Then the treelevel prediction for
$m_{h^0}$ saturates its upper bound
mentioned above, with $m^2_{h^0} \approx m_Z^2 \cos^2 (2\beta) + $ loop
corrections. The particles $A^0$, $H^0$, and $H^\pm$ will be much heavier
and nearly degenerate, forming an isospin doublet that decouples from
sufficiently lowenergy processes. The angle $\alpha$ is very nearly
$\beta\pi/2$, with
\beq
\cos(\beta\alpha) &=&
%\frac{m_Z^2}{m_{A^0}^2}
\sin(2\beta) \cos(2\beta)\, m_Z^2/m_{A^0}^2+
{\cal O}(m_Z^4/m_{A^0}^4),
\\
\sin(\beta\alpha) &=& 1  {\cal O}(m_Z^4/m_{A^0}^4),
\eeq
%\beq
%\tan\beta\, \tan\alpha &=& 1 + 2 \cos(2\beta) m_Z^2/m_{A^0}^2 + {\cal O}(m_Z^4/m_{A^0}^2) ,
%\eeq
so that
$h^0$ has nearly the same couplings to quarks and leptons and
electroweak gauge bosons as would the Higgs boson of the ordinary
Standard Model without supersymmetry. Radiative corrections modify these treelevel
predictions, but modelbuilding experiences
have shown that it is not uncommon for $h^0$ to behave in a way nearly
indistinguishable from a Standard Modellike Higgs boson, even if
$m_{A^0}$ is not too huge. The measurements of the 125 GeV Higgs boson observed
at the LHC are indeed consistent, so far, with the Standard Model predictions,
and it is sensible to identify this particle with $h^0$.
However, it should be kept in mind that the
couplings of $h^0$ might still turn out to deviate in measurable ways from those
of a Standard Model Higgs boson. After including the effects of radiative corrections,
the most significant effect for moderately
large $m_{A^0}$ is a possible enhancement of the $h^0 b \overline b$
coupling compared to the value it would have in the Standard Model.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Neutralinos and charginos}\label{subsec:MSSMspectrum.inos}
\setcounter{equation}{0}
\setcounter{footnote}{1}
The higgsinos and electroweak gauginos mix with each other because of the
effects of electroweak symmetry breaking. The neutral higgsinos ($\stilde
H_u^0$ and $\stilde H_d^0$) and the neutral gauginos ($\stilde B$,
$\stilde W^0$) combine to form four mass eigenstates called {\it
neutralinos}. The charged higgsinos ($\stilde H_u^+$ and $\stilde H_d^$)
and winos ($\stilde W^+$ and $\stilde W^$) mix to form two mass
eigenstates with charge $\pm 1$ called {\it charginos}. We will
denote\footnote{Other common notations use $\stilde \chi_i^0$ or $\stilde
Z_i$ for neutralinos, and $\stilde \chi^\pm_i$ or $\stilde W^\pm_i$ for
charginos.} the neutralino and chargino mass eigenstates by $\stilde N_i$
($i=1,2,3,4$) and $\stilde C^\pm_i$ ($i=1,2$). By convention, these are
labeled in ascending order, so that $m_{\stilde N_1} < m_{\stilde N_2}
=\> \pmatrix{M_2 & \sqrt{2} \sbeta\, m_W\cr
\sqrt{2} \cbeta\, m_W & \mu \cr }.
\label{charginomassmatrix}
\eeq
The mass eigenstates are related to the gauge eigenstates by two unitary
2$\times$2 matrices $\bf U$ and $\bf V$ according to
\beq
\pmatrix{\stilde C^+_1\cr
\stilde C^+_2} = {\bf V}
\pmatrix{\stilde W^+\cr
\stilde H_u^+},\quad\qquad\>\>\>\>\>\>
\pmatrix{\stilde C^_1\cr
\stilde C^_2} = {\bf U}
\pmatrix{\stilde W^\cr
\stilde H_d^}.\phantom{xxx}
\eeq
Note that the mixing matrix for the positively charged lefthanded
fermions is different from that for the negatively charged lefthanded
fermions. They are chosen so that
\beq
{\bf U}^* {\bf X} {\bf V}^{1} =
\pmatrix{m_{\stilde C_1} & 0\cr
0 & m_{\stilde C_2}},
\eeq
with positive real entries $m_{\stilde C_i}$. Because these are only
2$\times$2 matrices, it is not hard to solve for the masses analytically:
\beq
m^2_{{\stilde C}_{1}}, m^2_{{\stilde C}_{2}}
& = & {1\over 2}
\Bigl [ M_2^2 + \mu^2 + 2m_W^2
\nonumber
\\
&&\mp
\sqrt{(M_2^2 + \mu ^2 + 2 m_W^2 )^2  4  \mu M_2  m_W^2 \sin 2
\beta ^2 }
\Bigr ] .
\eeq
These are the (doubly degenerate) eigenvalues of the $4\times 4$ matrix
${\bf M}_{\stilde C}^\dagger {\bf M}_{\stilde C}$, or equivalently the
eigenvalues of ${\bf X}^\dagger {\bf X}$, since
\beq
{\bf V} {\bf X}^\dagger {\bf X} {\bf V}^{1} =
{\bf U}^* {\bf X} {\bf X}^\dagger {\bf U}^{T} =
\pmatrix{m^2_{{\stilde C}_1} & 0 \cr 0 & m^2_{{\stilde C}_2}}.
\eeq
(But, they are {\it not} the squares of the eigenvalues of $\bf X$.) In
the limit of eq.~(\ref{gauginolike}) with real $M_2$ and $\mu$,
the chargino mass eigenstates consist of a winolike $\stilde
C_1^\pm$ and and a higgsinolike $\stilde C_2^\pm$, with masses
\beq
m_{{\stilde C}_1} &=& M_2 
{ m_W^2 (M_2 + \mu \sin 2 \beta ) \over \mu^2  M_2^2 } +\ldots
\\
m_{{\stilde C}_2}
&=& \mu  + {I m_W^2 (\mu + M_2 \sin 2 \beta) \over \mu^2  M^2_2 }
+\ldots .
\eeq
Here again the labeling assumes $M_2<\mu$, and $I$ is the sign of $\mu$.
Amusingly, $\stilde C_1$ is degenerate with the neutralino $\stilde
N_2$ in the approximation shown, but that is not an exact result. Their
higgsinolike colleagues $\stilde N_3$, $\stilde N_4$ and $\stilde C_2$
have masses of order $\mu$. The case of $M_1 \approx 0.5 M_2 \ll \mu$
is not uncommonly found in viable models following from the boundary
conditions in section \ref{sec:origins}, and it has been elevated to the
status of a benchmark framework in many phenomenological studies. However
it cannot be overemphasized that such expectations are not mandatory.
The Feynman rules involving neutralinos and charginos may be inferred in
terms of $\bf N$, $\bf U$ and $\bf V$ from the MSSM Lagrangian as
discussed above; they are collected in
refs.~\cite{HaberKanereview}, \cite{GunionHaber}.
Feynman rules
based on twocomponent spinor notation have also been given in
\cite{DHM}.
In practice, the masses and mixing angles for the
neutralinos and charginos are best computed numerically. Note that the
discussion above yields the treelevel masses. Loop corrections to these
masses can be significant, and have been found systematically at oneloop
order in ref.~\cite{PBMZ}, with partial twoloop results in
\cite{GNCpoletwo,NCpole}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The gluino\label{subsec:MSSMspectrum.gluino}}
\setcounter{equation}{0}
\setcounter{footnote}{1}
The gluino is a color octet fermion, so it cannot mix with any other
particle in the MSSM, even if $R$parity is violated. In this regard, it
is unique among all of the MSSM sparticles. In models with MSUGRA
or GMSB boundary conditions,
the gluino mass
parameter $M_3$ is related to the bino and wino mass parameters $M_1$ and
$M_2$ by eq.~(\ref{gauginomassunification}), so
\beq
M_3 \,=\, {\alpha_s\over \alpha} \sin^2\theta_W\,
M_2 \,=\, {3\over 5} {\alpha_s \over \alpha} \cos^2\theta_W\, M_1
\label{eq:TiaEla}
\eeq
at any RG scale, up to small twoloop corrections. This implies a rough
prediction
\beq
M_3 : M_2 : M_1 \approx 6:2:1
\eeq
near the TeV scale. It is therefore
reasonable to suspect that the gluino
is considerably heavier than the lighter neutralinos and charginos
(even in many models where the gaugino mass unification condition is not
imposed).
For more precise estimates, one must take into account the fact that $M_3$
is really a running mass parameter with an implicit dependence on the RG
scale $Q$. Because the gluino is a strongly interacting particle, $M_3$
runs rather quickly with $Q$ [see eq.~(\ref{gauginomassrge})]. A more
useful quantity physically is the RG scaleindependent mass $m_{\tilde g}$
at which the renormalized gluino propagator has a pole. Including oneloop
corrections to the gluino propagator due to gluon exchange and
quarksquark loops, one finds that the pole mass is given in terms of the
running mass in the $\drbar$ scheme by \cite{gluinopolemass}
\beq
m_{\tilde g} = M_3(Q) \Bigl ( 1 + {\alpha_s\over 4 \pi}
\bigl [ 15 + 6\> {\rm ln}(Q/ M_3) + \sum A_{\tilde q}\bigr ] \Bigr )
\label{gluinopole}
\eeq
where
\beq
A_{\tilde q} \>=\> \int_0^1 \, dx \, x \, {\rm ln}
\bigl [
x m_{\tilde q}^2/M_3^2 + (1x) m_{q}^2/M_3^2  x(1x)  i \epsilon
\bigr ].
\eeq
The sum in eq.~(\ref{gluinopole}) is over all 12 squarkquark
supermultiplets, and we have neglected small effects due to squark mixing.
[As a check, requiring $m_{\tilde g}$ to be independent of $Q$ in
eq.~(\ref{gluinopole}) reproduces the oneloop RG equation for $M_3(Q)$ in
eq.~(\ref{gauginomassrge}).] The correction terms proportional to
$\alpha_s$ in eq.~(\ref{gluinopole}) can be quite significant,
because the gluino is strongly interacting, with a large group
theory factor [the 15 in eq.~(\ref{gluinopole})] due to its color octet
nature, and because it couples to all of the squarkquark pairs. The leading
twoloop corrections to the gluino pole mass have also been found
\cite{gluinopoletwo,GNCpoletwo,Martin:2006ub}, and are implemented in the latest
version of the {\tt SOFTSUSY} program \cite{SOFTSUSY}.
They typically increase the prediction by another 1 or 2\%.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The squarks and sleptons\label{subsec:MSSMspectrum.sfermions}}
\setcounter{equation}{0}
\setcounter{footnote}{1}
In principle, any scalars with the same electric charge, $R$parity, and
color quantum numbers can mix with each other. This means that with
completely arbitrary soft terms, the mass eigenstates of the squarks and
sleptons of the MSSM should be obtained by diagonalizing three $6\times 6$
squaredmass matrices for uptype squarks ($\stilde u_L$, $\stilde c_L$,
$\stilde t_L$, $\stilde u_R$, $\stilde c_R$, $\stilde t_R$), downtype
squarks ($\stilde d_L$, $\stilde s_L$, $\stilde b_L$, $\stilde d_R$,
$\stilde s_R$, $\stilde b_R$), and charged sleptons ($\stilde e_L$,
$\stilde \mu_L$, $\stilde \tau_L$, $\stilde e_R$, $\stilde \mu_R$,
$\stilde \tau_R$), and one $3\times 3$ matrix for sneutrinos ($\stilde
\nu_e$, $\stilde \nu_\mu$, $\stilde \nu_\tau$). Fortunately, the general
hypothesis of flavorblind soft parameters
eqs.~(\ref{scalarmassunification}) and (\ref{aunification}) predicts that
most of these mixing angles are very small. The thirdfamily squarks and
sleptons can have very different masses compared to their first and
secondfamily counterparts, because of the effects of large Yukawa ($y_t$,
$y_b$, $y_\tau$) and soft ($a_t$, $a_b$, $a_\tau$) couplings in the RG
equations (\ref{mq3rge})(\ref{mstaubarrge}). Furthermore, they can have
substantial mixing in pairs ($\stilde t_L$, $\stilde t_R$), ($\stilde
b_L$, $\stilde b_R$) and ($\stilde \tau_L$, $\stilde \tau_R$). In
contrast, the first and secondfamily squarks and sleptons have
negligible Yukawa couplings, so they end up in 7 very nearly degenerate,
unmixed pairs $(\stilde e_R, \stilde \mu_R)$, $(\stilde \nu_e, \stilde
\nu_\mu)$, $(\stilde e_L, \stilde \mu_L)$, $(\stilde u_R, \stilde c_R)$,
$(\stilde d_R, \stilde s_R)$, $(\stilde u_L, \stilde c_L)$, $(\stilde d_L,
\stilde s_L)$. As we have already discussed in section
\ref{subsec:mssm.hints}, this avoids the problem of disastrously large
virtual sparticle contributions to flavorchanging processes.
Let us first consider the spectrum of first and secondfamily squarks and
sleptons. In many models, including both MSUGRA
[eq.~(\ref{scalarunificationsugra})] and GMSB
[eq.~(\ref{scalargmsb})] boundary conditions, their running squared masses
can be conveniently parameterized, to a good approximation, as:
\beq
m_{Q_1}^2 = m_{Q_2}^2 \!\!\!&=&\!\!\! m_0^2 + K_3 + K_2 + {1\over 36}K_1,
\label{mq1form} \\
m_{\sbar u_1}^2 = m_{\sbar u_2}^2 \!\!\! &=&\!\!\! m_0^2 + K_3
\qquad\>\>\>
+ {4\over 9} K_1,
\\
m_{\sbar d_1}^2 = m_{\sbar d_2}^2 \!\!\!&=&\!\!\! m_0^2 + K_3
\qquad\>\>\>
+ {1\over 9} K_1,
\\
m_{L_1}^2 = m_{L_2}^2 \!\!\!&=&\!\!\! m_0^2 \qquad\>\>\> + K_2 + {1\over
4} K_1,
\\
m_{\sbar e_1}^2 = m_{\sbar e_2}^2 \!\!\!&=&\!\!\! m_0^2
\qquad\qquad\>\>\>\>\>\> \, +\, K_1.
\label{me1form}
\eeq
A key point is that the same $K_3$, $K_2$ and $K_1$ appear everywhere in
eqs.~(\ref{mq1form})(\ref{me1form}), since all of the chiral
supermultiplets couple to the same gauginos with the same gauge couplings.
The different coefficients in front of $K_1$ just correspond to the
various values of weak hypercharge squared for each scalar.
In MSUGRA models, $m_0^2$ is the same common scalar squared
mass appearing in eq.~(\ref{scalarunificationsugra}). It can be very
small, as in the ``noscale" limit, but it could also be the dominant
source of the scalar masses. The contributions $K_3$, $K_2$ and $K_1$ are
due to the RG running\footnote{The quantity $S$ defined in
eq.~(\ref{eq:defS}) vanishes at the input scale
for both MSUGRA and
GMSB boundary conditions, and remains small under RG evolution.}
proportional to the gaugino masses. Explicitly, they are found at one loop
order by solving eq.~(\ref{easyscalarrge}):
\beq
K_a(Q) = \left\lbrace \matrix{{3/5}\cr {3/4} \cr {4/3}}
\right \rbrace \times
{1\over 2 \pi^2} \int^{{\rm ln} Q_{0}}_{{\rm ln}Q}dt\>\,
g^2_a(t) \,M_a(t)^2\qquad\>\>\> (a=1,2,3).
\label{kintegral}
\eeq
Here $Q_{0}$ is the input RG scale at which the MSUGRA
boundary condition eq.~(\ref{scalarunificationsugra}) is applied, and $Q$
should be taken to be evaluated near the squark and slepton mass under
consideration, presumably less than about 1 TeV. The running parameters
$g_a(Q)$ and $M_a(Q)$ obey eqs.~(\ref{mssmg}) and
(\ref{gauginomassunification}). If the input scale is approximated by the
apparent scale of gauge coupling unification $Q_0 = M_U \approx 1.5 \times
10^{16}$ GeV, one finds that numerically
\beq
K_1 \approx 0.15 m_{1/2}^2,\qquad
K_2 \approx 0.5 m_{1/2}^2,\qquad
K_3 \approx (4.5\>{\rm to}\> 6.5) m_{1/2}^2.
\label{k123insugra}
\eeq
for $Q$ near the electroweak scale. Here $m_{1/2}$ is the common gaugino
mass parameter at the unification scale. Note that $K_3 \gg K_2 \gg K_1$;
this is a direct consequence of the relative sizes of the gauge couplings
$g_3$, $g_2$, and $g_1$. The large uncertainty in $K_3$ is due in part to
the experimental uncertainty in the QCD coupling constant, and in part to
the uncertainty in where to choose $Q$, since $K_3$ runs rather quickly
below 1 TeV. If the gauge couplings and gaugino masses are unified between
$M_U$ and $\MPlanck$, as would occur in a GUT model, then the effect of
RG
running for $M_U < Q < \MPlanck$ can be absorbed into a redefinition of
$m_0^2$. Otherwise, it adds a further uncertainty roughly proportional to
ln$(\MPlanck/M_U)$, compared to the larger contributions in
eq.~(\ref{kintegral}), which go roughly like ln$(M_U/1$~TeV).
In gaugemediated models, the same parameterization
eqs.~(\ref{mq1form})(\ref{me1form}) holds, but $m_0^2$ is always 0. At
the input scale $Q_0$, each MSSM scalar gets contributions to its squared
mass that depend only on its gauge interactions, as in
eq.~(\ref{scalargmsb}). It is not hard to see that in general these
contribute in exactly the same pattern as $K_1$, $K_2$, and $K_3$ in
eq.~(\ref{mq1form})(\ref{me1form}). The subsequent evolution of the
scalar squared masses down to the electroweak scale again just yields more
contributions to the $K_1$, $K_2$, and $K_3$ parameters. It is somewhat
more difficult to give meaningful numerical estimates for these parameters
in GMSB models than in the MSUGRA models without
knowing the messenger mass scale(s) and the multiplicities of the
messenger fields. However, in the gaugemediated case one quite generally
expects that the numerical values of the ratios $K_3/K_2$, $K_3/K_1$ and
$K_2/K_1$ should be even larger than in eq.~(\ref{k123insugra}). There are
two reasons for this. First, the running squark squared masses start off
larger than slepton squared masses already at the input scale in
gaugemediated models, rather than having a common value $m_0^2$.
Furthermore, in the gaugemediated case, the input scale $Q_0$ is
typically much lower than $\MPlanck$ or $M_U$, so that the RG evolution
gives
relatively more weight to RG scales closer to the electroweak scale, where
the hierarchies $g_3>g_2>g_1$ and $M_3>M_2>M_1$ are already in effect.
In general, one therefore expects that the squarks should be considerably
heavier than the sleptons, with the effect being more pronounced in
gaugemediated supersymmetry breaking models than in MSUGRA
models. For any specific choice of model, this effect can be easily
quantified with a numerical RG computation.
The hierarchy $m_{\rm squark} > m_{\rm
slepton}$ tends to hold even in models that do not fit neatly into any of
the categories outlined in section \ref{sec:origins}, because the RG
contributions to squark masses from the gluino are always present and
usually quite large, since QCD has a larger gauge coupling than the
electroweak interactions.
Regardless of the type of model,
there is also a ``hyperfine" splitting in the squark and slepton mass
spectrum, produced by electroweak symmetry breaking. Each squark and
slepton $\phi$ will get a contribution $\Delta_\phi$ to its squared mass,
coming from the $SU(2)_L$ and $U(1)_Y$ $D$term quartic interactions [see
the last term in eq.~(\ref{fdpot})] of the form (squark)$^2$(Higgs)$^2$
and (slepton)$^2$(Higgs)$^2$, when the neutral Higgs scalars $H_u^0$ and
$H_d^0$ get VEVs. They are modelindependent for a given value of
$\tan\beta$:
\beq
\Delta_\phi \>=\> \frac{1}{2}
(T_{3\phi} g^2  Y_\phi g^{\prime 2}) (v_d^2  v_u^2) \>=\>
(T_{3\phi}  Q_\phi\sin^2\theta_W)
\cos (2\beta)\, m_Z^2 ,
\label{defDeltaphi}
\eeq
where $T_{3\phi}$, $Y_\phi$, and $Q_\phi$ are the third component of weak
isospin, the weak hypercharge, and the electric charge of the lefthanded
chiral supermultiplet to which $\phi$ belongs. For example,
$\Delta_{\tilde u_L} = ({1\over 2}  {2\over 3} \sin^2\theta_W)\cos
(2\beta)\, m_Z^2$ and $\Delta_{\tilde d_L} = ({1\over 2} + {1\over 3}
\sin^2\theta_W)\cos (2\beta)\, m_Z^2$ and $\Delta_{\tilde u_R} = ({2\over
3} \sin^2\theta_W)\cos (2\beta) \, m_Z^2$. These $D$term contributions
are typically smaller than the $m_0^2$ and $K_1$, $K_2$, $K_3$
contributions, but should not be neglected. They split apart the
components of the $SU(2)_L$doublet sleptons and squarks. Including them,
the firstfamily squark and slepton masses are now given by:
\beq
m_{\tilde d_L}^2 \!\!&=&\!\! m_0^2 + K_3 + K_2 + {1\over 36} K_1 +
\Delta_{\tilde d_L},
\label{msdlform}
\\
m_{\tilde u_L}^2 \!\!&=&\!\! m_0^2 + K_3 + K_2 + {1\over 36} K_1 +
\Delta_{\tilde u_L},
\\
m_{\tilde u_R}^2\!\! &=&\!\! m_0^2 + K_3 \qquad\>\>\> + {4\over 9} K_1 +
\Delta_{\tilde u_R},
\\
m_{\tilde d_R}^2 \!\!&=&\!\! m_0^2 + K_3 \qquad\>\>\> + {1\over 9} K_1 +
\Delta_{\tilde d_R},
\label{msdrform}
\\
m_{\tilde e_L}^2 \!\!&=&\!\! m_0^2 \qquad\>\>\> + K_2 + {1\over 4} K_1 +
\Delta_{\tilde e_L},
\label{mselform}
\\
m_{\tilde \nu}^2 \!&=& \!\! m_0^2 \qquad\>\>\> + K_2 + {1\over 4} K_1 +
\Delta_{\tilde \nu},
\\
m_{\tilde e_R}^2 \!\!&=&\!\! m_0^2 \qquad\qquad\>\>\>\>\>\> \, +\,
K_1
\, + \Delta_{\tilde e_R},
\label{mserform}
\eeq
with identical formulas for the secondfamily squarks and sleptons. The
mass splittings for the lefthanded squarks and sleptons are governed by
modelindependent sum rules
\beq
m_{\tilde e_L}^2 m_{\tilde \nu_e}^2 \,=\,
m_{\tilde d_L}^2 m_{\tilde u_L}^2 \,=\,
g^2 (v_u^2  v_d^2)/2 \,=\, \cos (2\beta)\, m_W^2 .
\eeq
In the allowed range $\tan\beta>1$, it follows that $m_{\tilde e_L} >
m_{\tilde \nu_e}$ and $m_{\tilde d_L} > m_{\tilde u_L}$, with the
magnitude of the splittings constrained by electroweak symmetry breaking.
Let us next consider the masses of the top squarks, for which there are
several nonnegligible contributions. First, there are squaredmass terms
for $\stilde t^*_L \stilde t_L$ and $\stilde t_R^* \stilde t_R$ that are
just equal to $m^2_{Q_3} + \Delta_{\tilde u_L}$ and $m^2_{\sbar u_3} +
\Delta_{\tilde u_R}$, respectively, just as for the first and
secondfamily squarks. Second, there are contributions equal to $m_t^2$
for each of $\stilde t^*_L \stilde t_L$ and $\stilde t_R^* \stilde t_R$.
These come from $F$terms in the scalar potential of the form $y_t^2
H_u^{0*} H_u^0 \stilde t_L^* \stilde t_L$ and $y_t^2 H_u^{0*} H_u^0
\stilde t_R^* \stilde t_R$ (see Figures~\ref{fig:stop}b and
\ref{fig:stop}c), with the Higgs fields replaced by their VEVs. (Of
course, similar contributions are present for all of the squarks and
sleptons, but they are too small to worry about except in the case of the
top squarks.) Third, there are contributions to the scalar potential from
$F$terms of the form $\mu^* y_t \stilde{\sbar t} \stilde t H_d^{0*}
+\conj$; see eqs.~(\ref{striterms}) and Figure~\ref{fig:stri}a. These
become $\mu^* v y_t \cos\beta\, \stilde t^*_R \stilde t_L + \conj$ when
$H_d^0$ is replaced by its VEV. Finally, there are contributions to the
scalar potential from the soft (scalar)$^3$ couplings $a_t \stilde{\sbar
t} \stilde Q_3 H_u^0 + \conj$ [see the first term of the second line of
eq.~(\ref{MSSMsoft}), and eq.~(\ref{heavyatopapprox})], which become $ a_t
v \sin\beta\, \stilde t_L \stilde t_R^* + \conj$ when $H_u^0$ is replaced
by its VEV. Putting these all together, we have a squaredmass matrix for
the top squarks, which in the gaugeeigenstate basis ($\stilde t_L$,
$\stilde t_R$) is given by
\beq
\lagr_{\mbox{stop masses}} = \pmatrix{\stilde t_L^* & \stilde t_R^*}
{\bf m_{\stilde t}^2} \pmatrix{\stilde t_L \cr \stilde t_R}
\eeq
where
\beq
{\bf m_{\stilde t}^2} =
\pmatrix{
m^2_{Q_3} + m_t^2 + \Delta_{\tilde u_L} &
v(a_t^* \sin\beta  \mu y_t\cos\beta )\cr
v (a_t \sin\beta  \mu^* y_t\cos\beta ) &
m^2_{\sbar u_3} + m_t^2 + \Delta_{\tilde u_R}
} .
\label{mstopmatrix}
\eeq
This hermitian matrix can be diagonalized by a unitary matrix to give mass
eigenstates:
\beq
\pmatrix{\stilde t_1\cr\stilde t_2} =
\pmatrix{ c_{\tilde t} & s_{\tilde t}^* \cr
s_{\tilde t} & c_{\tilde t}^*}
\pmatrix{\stilde t_L \cr \stilde t_R} .
\label{pixies}
\eeq
Here $m^2_{\tilde t_1}< m^2_{\tilde t_2}$ are the eigenvalues of
eq.~(\ref{mstopmatrix}), and $c_{\tilde t}^2 + s_{\tilde t}^2 = 1$. If
the offdiagonal elements of eq.~(\ref{mstopmatrix}) are real, then
$c_{\tilde t}$ and $s_{\tilde t}$ are the cosine and sine of a stop mixing
angle $\theta_{\tilde t}$, which can be chosen in the range $0\leq
\theta_{\tilde t} < \pi$. Because of the large RG effects proportional to
$X_t$ in eq.~(\ref{mq3rge}) and eq.~(\ref{mtbarrge}), in MSUGRA and GMSB and similar models one finds that $m_{\sbar u_3}^2 < m_{Q_3}^2$ at the electroweak
scale, and both of these
quantities are usually significantly smaller than the squark squared
masses for the first two families. The diagonal terms $m_t^2$ in
eq.~(\ref{mstopmatrix}) can mitigate this effect slightly, but only slightly,
and the offdiagonal entries will typically induce a significant mixing, which
always reduces the lighter topsquark squaredmass eigenvalue. Therefore,
models often predict that $\stilde t_1$ is the lightest squark of all, and
that it is predominantly $\tilde t_R$.
A very similar analysis can be performed for the bottom squarks and
charged tau sleptons, which in their respective gaugeeigenstate bases
($\stilde b_L$, $\stilde b_R$) and ($\stilde \tau_L$, $\stilde \tau_R$)
have squaredmass matrices:
\beq
{\bf m_{\stilde b}^2} =
\pmatrix{
m^2_{Q_3} + \Delta_{\tilde d_L} &
v (a_b^* \cos\beta  \mu y_b\sin\beta )\cr
v (a_b \cos\beta  \mu^* y_b\sin\beta ) &
m^2_{\sbar d_3} + \Delta_{\tilde d_R} },
\label{msbottommatrix}
\eeq
\beq
{\bf m_{\stilde \tau}^2} =
\pmatrix{
m^2_{L_3} + \Delta_{\tilde e_L} &
v (a_\tau^* \cos\beta \mu y_\tau\sin\beta )\cr
v (a_\tau \cos\beta  \mu^* y_\tau\sin\beta ) &
m^2_{\sbar e_3} + \Delta_{\tilde e_R}}.
\label{mstaumatrix}
\eeq
These can be diagonalized to give mass eigenstates $\stilde b_1, \stilde
b_2$ and $\stilde \tau_1, \stilde \tau_2$ in exact analogy with
eq.~(\ref{pixies}).
The magnitude and importance of mixing in the sbottom and stau sectors
depends on how big $\tan\beta$ is. If $\tan\beta$ is not too large (in
practice, this usually means less than about $10$ or so, depending on the
situation under study), the sbottoms and staus do not get a very large
effect from the mixing terms and the RG effects due to $X_b$ and $X_\tau$,
because $y_b,y_\tau \ll y_t$ from eq.~(\ref{eq:ytbtaumtbtau}). In that
case the mass eigenstates are very nearly the same as the gauge
eigenstates $\stilde b_L$, $\stilde b_R$, $\stilde \tau_L$ and $\stilde
\tau_R$. The latter three, and $\stilde \nu_\tau$, will be nearly
degenerate with their first and secondfamily counterparts with the same
$SU(3)_C \times SU(2)_L \times U(1)_Y$ quantum numbers. However, even in
the case of small $\tan\beta$, $\stilde b_L$ will feel the effects of the
large top Yukawa coupling because it is part of the doublet containing
$\stilde t_L$. In particular, from eq.~(\ref{mq3rge}) we see that $X_t$
acts to decrease $m_{Q_3}^2$ as it is RGevolved down from the input scale
to the electroweak scale. Therefore the mass of ${\stilde b_L}$ can be
significantly less than the masses of $\stilde d_L$ and $\stilde s_L$.
For larger values of $\tan\beta$, the mixing in
eqs.~(\ref{msbottommatrix}) and (\ref{mstaumatrix}) can be quite
significant, because $y_b$, $y_\tau$ and $a_b$, $a_\tau$ are
nonnegligible. Just as in the case of the top squarks, the lighter
sbottom and stau mass eigenstates (denoted $\stilde b_1$ and $\stilde
\tau_1$) can be significantly lighter than their first and secondfamily
counterparts. Furthermore, ${\stilde \nu_\tau}$ can be significantly
lighter than the nearly degenerate ${\stilde \nu_e}$, $\stilde \nu_\mu$.
The requirement that the thirdfamily squarks and sleptons should all have
positive squared masses implies limits on the magnitudes of
$a_t^*\sin\beta \mu y_t \cos\beta$ and $a_b^*\cos\beta  \mu y_b
\sin\beta$ and and $a_\tau^* \cos\beta  \mu y_\tau \sin\beta$. If they
are too large, then the smaller eigenvalue of eq.~(\ref{mstopmatrix}),
(\ref{msbottommatrix}) or (\ref{mstaumatrix}) will be driven negative,
implying that a squark or charged slepton gets a VEV, breaking $SU(3)_C$
or electromagnetism. Since this is clearly unacceptable, one can put
bounds on the (scalar)$^3$ couplings, or equivalently on the parameter
$A_0$ in MSUGRA models. Even if all of the squaredmass
eigenvalues are positive, the presence of large (scalar)$^3$ couplings can
yield global minima of the scalar potential, with nonzero squark and/or
charged slepton VEVs, which are disconnected from the vacuum that
conserves $SU(3)_C$ and electromagnetism \cite{badvacua}. However, it is
not always immediately clear whether the mere existence of such
disconnected global minima should really disqualify a set of model
parameters,
because the tunneling rate from our ``good" vacuum to the ``bad" vacua
can easily be longer than the age of the universe \cite{kusenko}.
Radiative corrections to the squark and slepton masses are potentially important, and are given at oneloop order in ref.~\cite{PBMZ}.
For squarks, the leading twoloop corrections have been found in
refs.~\cite{Martin:2005eg,Martin:2006ub}, and are implemented in the latest
version of the {\tt SOFTSUSY} code \cite{SOFTSUSY}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Summary: the MSSM sparticle
spectrum}\label{subsec:MSSMspectrum.summary}
\setcounter{equation}{0}
\setcounter{footnote}{1}
In the MSSM, there are 32 distinct masses corresponding to undiscovered
particles, not including the gravitino. Above, we have explained
how the masses and mixing angles for these particles can be computed,
given an underlying model for the soft terms at some input scale.
The mass eigenstates of the MSSM are listed in Table
\ref{tab:undiscovered}, assuming
only that the mixing of first and secondfamily squarks and sleptons is
negligible.%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\arraystretch}{1.4}
\begin{table}[tb]
\begin{center}
\begin{tabular}{ccccc}
\hline
Names & Spin & $P_R$ & Gauge Eigenstates & Mass Eigenstates \\
\hline\hline
Higgs bosons & 0 & $+1$ &
$H_u^0\>\> H_d^0\>\> H_u^+ \>\> H_d^$
&
$h^0\>\> H^0\>\> A^0 \>\> H^\pm$
\\ \hline
& & &${\stilde u}_L\>\> {\stilde u}_R\>\> \stilde d_L\>\> \stilde d_R$&(same)
\\
squarks& 0&$1$& ${\stilde s}_L\>\> {\stilde s}_R\>\> \stilde c_L\>\>
\stilde c_R$& (same) \\
& & &
$\stilde t_L \>\>\stilde t_R \>\>\stilde b_L\>\> \stilde b_R$
&
${\stilde t}_1\>\> {\stilde t}_2\>\> \stilde b_1\>\> \stilde b_2$
\\ \hline
& & &${\stilde e}_L\>\> {\stilde e}_R \>\>\stilde \nu_e$&(same)
\\
sleptons& 0&$1$&${\stilde \mu}_L\>\>{\stilde \mu}_R\>\>\stilde\nu_\mu$&(same)
\\
& & &
$\stilde \tau_L\>\> \stilde \tau_R \>\>\stilde \nu_\tau$
&
${\stilde \tau}_1 \>\>{\stilde \tau}_2 \>\>\stilde \nu_\tau$
\\
\hline
neutralinos & $1/2$&$1$ &
$\stilde B^0 \>\>\>\stilde W^0\>\>\> \stilde H_u^0\>\>\> \stilde H_d^0$
&
$\stilde N_1\>\> \stilde N_2 \>\>\stilde N_3\>\> \stilde N_4$
\\
\hline
charginos & $1/2$&$1$ &
$\stilde W^\pm\>\>\> \stilde H_u^+ \>\>\>\stilde H_d^$
&
$\stilde C_1^\pm\>\>\>\stilde C_2^\pm $
\\
\hline
gluino & $1/2$&$1$ &$\stilde g$ &(same) \\
\hline
${\rm goldstino}\atop{\rm (gravitino)}$ & ${1/2}\atop{(3/2)}$&$1$&$\stilde
G$ &(same) \\
\hline
\end{tabular}
\caption{The undiscovered particles in the Minimal Supersymmetric Standard
Model (with sfermion mixing for the first two families assumed to be
negligible).
\label{tab:undiscovered}}
\vspace{0.4cm}
\end{center}
\end{table}%
A complete set of Feynman rules for the
interactions of these particles with each other and with the Standard
Model quarks, leptons, and gauge bosons can be found in
refs.~\cite{HaberKanereview,GunionHaber}.
Feynman rules
based on twocomponent spinor notation have also been given in
\cite{DHM}.
Specific models for the soft
terms can predict the masses and the mixing angles angles for the
MSSM in terms of far fewer parameters. For example,
in the MSUGRA models, the only free
parameters not already measured by
experiment are $m_0^2$, $m_{1/2}$, $A_0$, $\mu$, and $b$. In
GMSB models, the free parameters include
the scale $\Lambda$, the messenger mass scale $M_{\rm mess}$,
the integer number $\nmess$ of copies of the minimal messengers,
the goldstino decay constant $\langle F \rangle $, and the Higgs mass
parameters $\mu$ and $b$.
After RG evolving the soft terms down to the
electroweak scale, one can demand that the scalar potential gives correct
electroweak symmetry breaking. This allows us to trade $\mu$ and $b$
for one parameter $\tan\beta$, as in
eqs.~(\ref{mubsub1})(\ref{mubsub2}). So, to a reasonable approximation,
the entire mass spectrum in MSUGRA models is determined by
only five unknown parameters: $m_0^2$, $m_{1/2}$, $A_0$, $\tan\beta$, and
Arg($\mu$), while in the simplest gaugemediated supersymmetry breaking
models one can pick parameters $\Lambda$, $M_{\rm mess}$, $\nmess$,
$\langle F \rangle $, $\tan\beta$, and Arg($\mu$). Both frameworks are
highly predictive. Of course, it is quite likely that the essential
physics of supersymmetry breaking is not captured by either of these two
scenarios in their minimal forms.
Figure \ref{fig:running} shows the RG running of scalar and gaugino
masses in a sample model based on the MSUGRA boundary
conditions imposed at $Q_0 = 1.5\times 10^{16}$ GeV.
\begin{figure}
\vspace{0.2cm}
\centerline{\psfig{figure=MSSMrun.eps,height=3.65in}}
\vspace{0.24cm}
\caption{RG evolution of scalar and gaugino mass parameters
in the MSSM with MSUGRA boundary conditions imposed at $Q_0 = 1.5\times
10^{16}$ GeV. The parameter
$\mu^2 + m^2_{H_u}$ runs negative, provoking electroweak
symmetry breaking.
\label{fig:running}}
\end{figure}
[The parameter values used for this illustration were $m_0 = 300$ GeV,
$m_{1/2} = A_0 = 1000$ GeV, $\tan\beta = 15$, and
sign($\mu$)$=+$, but these values were chosen more for their
artistic value in Figure \ref{fig:running}, and not as an attempt at realism.
The goal here is to understand the qualitative trends,
rather than guess the correct numerical values.]
The running gaugino masses are solid lines labeled by
$M_1$, $M_2$, and $M_3$. The dotdashed lines labeled $H_u$ and $H_d$ are
the running values of the quantities $(\mu^2 + m_{H_u}^2)^{1/2}$ and
$({\mu^2 + m_{H_d}^2})^{1/2}$, which appear in the Higgs potential. The
other lines are the running squark and slepton masses, with dashed lines
for the square roots of the third family parameters $m^2_{\sbar d_3}$,
$m^2_{Q_3}$, $m^2_{\sbar u_3}$, $m^2_{L_3}$, and $m^2_{\sbar e_3}$ (from
top to bottom), and solid lines for the first and second family sfermions.
Note that $\mu^2 + m_{H_u}^2$ runs negative because of the effects of the
large top Yukawa coupling as discussed above, providing for electroweak
symmetry breaking. At the electroweak scale, the values of the Lagrangian
soft parameters can be used to extract the physical masses,
crosssections, and decay widths of the particles, and other observables
such as dark matter abundances and rare process rates. There are a variety
of publicly available programs that do these tasks, including radiative
corrections; see for example
%\cite{ISAJET,SOFTSUSY,SuSpect,SPheno,SDECAY,HDECAY,NMHDECAY,
%CPsuperH,DarkSUSY,micrOMEGAs,FeynHiggs}.
\cite{ISAJET}\cite{micrOMEGAs},\cite{FeynHiggs}.
Figure \ref{fig:sample} shows deliberately qualitative sketches of sample
MSSM mass spectrum obtained from four different types of models
assumptions.
\begin{figure}[p]
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\caption{Four sample mass spectra for the undiscovered
particles in the MSSM, for
(a) MSUGRA with $m^2_0 \ll m_{1/2}^2$,
(b) MSUGRA with $m^2_0 \gg m_{1/2}^2$,
(c) GMSB with $N_5=1$, and
(d) GMSB with $N_5=3$.
Mass scales are not equal for the four cases, and are deliberately omitted.
These spectra are presented for entertainment purposes only! No warranty, expressed or implied, guarantees that they look anything like the real world. \label{fig:sample}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The first, in Figure \ref{fig:sample}(a), is the output from an
MSUGRA model with
relatively low $m_0^2$ compared to $m^2_{1/2}$ (similar to fig.~\ref{fig:running}).
This model features a neardecoupling limit for the Higgs sector, and
a binolike $\stilde N_1$ LSP, nearly degenerate winolike
$\stilde N_2, \stilde C_1$, and higgsinolike $\stilde N_3, \stilde N_4,
\stilde C_2$. The gluino is the heaviest superpartner. The squarks are
all much heavier than the sleptons, and the lightest sfermion is a stau.
(The secondfamily squarks and sleptons are nearly degenerate with those of the first family, and so are not shown separately.)
Variations in the model
parameters have important and predictable effects. For example,
taking larger values of $\tan\beta$ with other model parameters
held fixed will usually tend to lower $\stilde b_1$ and $\stilde \tau_1$
masses compared to those of the other sparticles.
Taking
larger $m_0^2$ will tend to
squeeze together the spectrum of squarks and sleptons and move
them all higher compared to the
neutralinos, charginos and
gluino. This is illustrated in Figure \ref{fig:sample}(b), which instead has
$m_0^2 \gg m_{1/2}^2$.
%[The MSUGRA parameters used to make this graph were
%$m_{1/2} = A_0 = 320$ GeV, $m_0 = 3200$ GeV, $\tan\beta=10$, $\mu>0$.]
In this model, the heaviest chargino and neutralino are winolike.
The third sample sketch, in fig.~\ref{fig:sample}(c), is obtained from a
typical minimal GMSB model, with $N_5 = 1$
%[and boundary conditions as in
%eq.~(\ref{gmsbgauginonmess}) with $\Lambda = 150$ TeV,
%$\tan\beta = 15$, and sign($\mu$)$=+$ at a scale $Q_0 = M_{\rm mess} =
%300$ TeV for the illustration].
Here we see that the hierarchy between
strongly interacting sparticles and weakly interacting ones is quite
large. Changing the messenger scale or $\Lambda$ does not reduce the
relative splitting between squark and slepton masses, because there is no
analog of the universal $m_0^2$ contribution here. Increasing the number
of messenger fields tends to decrease the squark and slepton masses
relative to the gaugino masses, but still keeps the hierarchy between
squark and slepton masses intact. In the model shown, the LSP is the
nearly massless gravitino and the NLSP is a
binolike neutralino, but for larger number of messenger fields it could
be either a stau, or else coNLSPs $\tilde \tau_1$, $\tilde e_L$, $\tilde
\mu_L$, depending on the choice of $\tan\beta$.
The fourth sample sketch, in fig.~\ref{fig:sample}(d),
is of a typical GMSB model with a nonminimal messenger sector, $N_5=3$
%[and boundary conditions as in
%eq.~(\ref{gmsbgauginonmess}) with $\Lambda = 60$ TeV,
%$\tan\beta = 15$, and sign($\mu$)$=+$ at a scale $Q_0 = M_{\rm mess} =
%120$ TeV for the illustration].
Again the LSP is the nearly massless gravitino, but this
time the NLSP is the lightest stau. The heaviest superpartner is the gluino, and the
heaviest chargino and neutralino
are winolike.
It would be a mistake to rely too heavily on specific scenarios for the
MSSM mass and mixing spectrum, and the above illustrations are only
a tiny fraction of the available possibilities. However, it is also
useful to keep in mind some general trends that often recur in various
different models. Indeed, there has emerged a sort of folklore
concerning likely features of the MSSM spectrum, partly based on
theoretical bias and partly on the constraints inherent in many known viable
softlybroken supersymmetric theories. We remark on these features mainly
because they represent the prevailing prejudices among many supersymmetry
theorists, which is certainly a useful thing to know even if one wisely
decides to remain skeptical. For example, it is perhaps not unlikely that:
%
\begin{itemize}
%
\item[$\bullet$] The LSP is the lightest neutralino $\stilde N_1$, unless
the gravitino is lighter or $R$parity is not conserved. If $M_1 <
M_2,\mu$, then $\stilde N_1$ is likely to be binolike, with a mass
roughly
0.5 times the masses of $\stilde N_2$ and $\stilde C_1$ in many
wellmotivated models. If, instead,
$\mu < M_1,M_2$, then the LSP $\stilde N_1$
has a large higgsino content and
$\stilde N_2$ and $\stilde C_1$ are not much heavier.
And, if $M_2 \ll M_1, \mu$, then the LSP will be a winolike
neutralino, with a chargino only very slightly heavier.
%
\item[$\bullet$] The gluino will be much heavier than the lighter
neutralinos and charginos. This is certainly true in the case of the
``standard" gaugino mass relation eq.~(\ref{gauginomassunification}); more
generally, the running gluino mass parameter grows relatively quickly as
it is RGevolved into the infrared because the QCD coupling is larger than
the electroweak gauge couplings. So even if there are big corrections to
the gaugino mass boundary conditions eqs.~(\ref{gauginounificationsugra})
or (\ref{gauginogmsb}), the gluino mass parameter $M_3$ is likely to come
out larger than $M_1$ and $M_2$.
%
\item[$\bullet$] The squarks of the first and second families are nearly
degenerate and much heavier than the sleptons. This is because each squark
mass gets the same large positivedefinite radiative corrections from
loops involving the gluino. The lefthanded squarks $\stilde u_L$,
$\stilde d_L$, $\stilde s_L$ and $\stilde c_L$ are likely to be heavier
than their righthanded counterparts $\stilde u_R$, $\stilde d_R$,
$\stilde s_R$ and $\stilde c_R$, because of the effect parameterized
by $K_2$ in eqs.~(\ref{msdlform})(\ref{mserform}).
%
\item[$\bullet$] The squarks of the first two families cannot be lighter
than about 0.8 times the mass of the gluino in MSUGRA
models, and about 0.6 times the mass of the gluino in the simplest
gaugemediated models as discussed in section \ref{subsec:origins.gmsb} if
the number of messenger squark pairs is $\nmess \leq 4$.
In the MSUGRA case this is because the gluino mass feeds
into the squark masses through RG evolution; in the gaugemediated case it
is because the gluino and squark masses are tied together by
eqs.~(\ref{gauginogmsbgen}) and (\ref{scalargmsbgen}).
%
\item[$\bullet$] The lighter stop $\stilde t_1$ and the lighter sbottom
$\stilde b_1$ are probably the lightest squarks. This is because stop and
sbottom mixing effects and the effects of $X_t$ and $X_b$ in
eqs.~(\ref{mq3rge})(\ref{md3rge}) both tend to decrease the lighter stop
and sbottom masses.
%
\item[$\bullet$] The lightest charged slepton is probably a stau $\stilde
\tau_1$. The mass difference $m_{\tilde e_R}m_{\tilde \tau_1}$ is
likely to be significant if $\tan\beta$ is large, because of the effects
of a large tau Yukawa coupling. For smaller $\tan\beta$, $\stilde \tau_1$
is predominantly $\stilde \tau_R$ and it is not so much lighter than
$\stilde e_R$, $\stilde \mu_R$.
%
\item[$\bullet$] The lefthanded charged sleptons $\stilde e_L$ and
$\stilde \mu_L$ are likely to be heavier than their righthanded
counterparts $\stilde e_R$ and $\stilde \mu_R$. This is because of the
effect of $K_2$ in eq.~(\ref{mselform}). (Note also that $\Delta_{\tilde
e_L}  \Delta_{\tilde e_R}$ is positive but very small because of the
numerical accident $\sin^2\theta_W \approx 1/4$.)
%
\end{itemize}
It should be kept in mind that each of these prejudices
might be defied by the real world.
The most important point is that by measuring the masses and mixing angles
of the MSSM particles we will be able to gain a great deal of information
that differentiate between competing proposals for the
origin and mediation of supersymmetry breaking.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Implications of the 125 GeV Higgs boson}\label{subsec:MSSMspectrum.h125}
%\setcounter{equation}{0}
%\setcounter{footnote}{1}
\section{Sparticle decays}\label{sec:decays}
\setcounter{equation}{0}
\setcounter{figure}{0}
\setcounter{table}{0}
\setcounter{footnote}{1}
This section contains a brief qualitative overview of the decay patterns
of sparticles in the MSSM, assuming that $R$parity is conserved. We will
consider in turn the possible decays of neutralinos, charginos, sleptons,
squarks, and the gluino. If, as is most often assumed, the lightest
neutralino $\NI$ is the LSP, then all decay chains will end up with it in
the final state. Section \ref{subsec:decays.gravitino} discusses the
alternative possibility that the gravitino/goldstino $\G$ is the LSP.
For the sake of simplicity of notation, we will often not distinguish
between particle and antiparticle names and labels in this section, with
context and consistency (dictated by charge and color conservation)
resolving any ambiguities.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Decays of neutralinos and
charginos}\label{subsec:decays.inos}
\setcounter{equation}{0}
\setcounter{footnote}{1}
Let us first consider the possible twobody decays. Each neutralino and
chargino contains at least a small admixture of the electroweak gauginos
$\stilde B$, $\stilde W^0$ or $\stilde W^\pm$, as we saw in section
\ref{subsec:MSSMspectrum.inos}. So, $\stilde N_i$ and $\stilde C_i$ inherit
couplings of weak interaction strength to (scalar, fermion) pairs, as
shown in Figure~\ref{fig:gaugino}b,c. If sleptons or squarks are
sufficiently light, a neutralino or chargino can therefore decay into
lepton+slepton or quark+squark. To the extent that sleptons are probably
lighter than squarks, the lepton+slepton final states are favored. A
neutralino or chargino may also decay into any lighter neutralino or
chargino plus a Higgs scalar or an electroweak gauge boson, because they
inherit the gauginohiggsinoHiggs (see Figure~\ref{fig:gaugino}b,c) and
$SU(2)_L$ gauginogauginovector boson (see Figure~\ref{fig:gauge}c)
couplings of their components. So, the possible twobody decay modes for
neutralinos and charginos in the MSSM are:
\beq
\stilde N_i \rightarrow
Z\stilde N_j,\>\>\, W\stilde C_j,\>\>\, h^0\stilde N_j,\>\>\, \ell \stilde
\ell,\>\>\,
\nu \stilde \nu,\>\>\,
[A^0 \stilde N_j,\>\>\, H^0 \stilde N_j,\>\>\, H^\pm
\stilde C_j^\mp,\>\>\,
q\stilde q],
\qquad\>\>\>{}
\label{nino2body}
\\
\stilde C_i \rightarrow
W\stilde N_j,\>\>\, Z\stilde C_1,\>\>\, h^0\stilde C_1,\>\>\, \ell \stilde
\nu,\>\>\,
\nu \stilde \ell,\>\>\,
[A^0 \stilde C_1,\>\>\, H^0 \stilde C_1,\>\>\, H^\pm \stilde N_j,\>\>\,
q\stilde q^\prime],
\qquad\>\>\>{}
\label{cino2body}
\eeq
using a generic notation $\nu$, $\ell$, $q$ for neutrinos, charged
leptons, and quarks. The final states in brackets are the more
kinematically implausible ones. (Since $m_{h^0} = 125$ GeV, it
is the most likely of the Higgs scalars to appear in these decays.) For
the heavier neutralinos and chargino ($\stilde N_3$, $\stilde N_4$ and
$\stilde C_2$), one or more of the twobody decays in
eqs.~(\ref{nino2body}) and (\ref{cino2body}) is likely to be kinematically
allowed. Also, if the decays of neutralinos and charginos with a
significant higgsino content into thirdfamily quarksquark pairs are
open, they can be greatly enhanced by the topquark Yukawa coupling,
following from the interactions shown in fig.~\ref{fig:topYukawa}b,c.
It may be that all of these twobody modes are kinematically forbidden for
a given chargino or neutralino, especially for $\stilde C_1$ and $\stilde
N_2$ decays. In that case, they have threebody decays
\beq
\stilde N_i \rightarrow f f \stilde N_j,\>\>\>\,
\stilde N_i \rightarrow f f^\prime \stilde C_j,\>\>\>\,
\stilde C_i \rightarrow f f^\prime \stilde N_j,\>\>\>\,{\rm and}\>\>\>\,
\stilde C_2 \rightarrow f f \stilde C_1,\qquad\>\>\>\>\>{}
\label{cino3body}
\eeq
through the same (but now offshell) gauge bosons, Higgs scalars,
sleptons, and squarks that appeared in the twobody decays
eqs.~(\ref{nino2body}) and (\ref{cino2body}). Here $f$ is generic notation
for a lepton or quark, with $f$ and $f^\prime$ distinct members of the same
$SU(2)_L$ multiplet (and of course one of the $f$ or $f'$ in each of these
decays must actually be an antifermion). The chargino and neutralino
decay widths into the various final states can be found in
%refs.~\cite{inodecays,epprod,neutralinoloopdecays}.
refs.~\cite{inodecays}\cite{neutralinoloopdecays}.
The Feynman diagrams for the neutralino and chargino decays with $\stilde
N_1$ in the final state that seem most likely to be important are shown in
figure~\ref{fig:NCdecays}.
\begin{figure}
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\rText(82,6)[][]{$\scriptstyle f$}
\end{picture}
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~~~~
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\end{center}
%
\vspace{0.03cm}
\begin{center}
\scalebox{1.42}{
\begin{picture}(90,35)(10,0)
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\rText(84,6)[][]{$\scriptstyle\tilde N_1$}
\end{picture}
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~~~~
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~~~~
\scalebox{1.42}{
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\rText(42,6)[][]{$\scriptstyle H^\pm$}
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\rText(84,6)[][]{$\scriptstyle t,\,\nu_\tau,\,\ldots$}
\end{picture}
}
\end{center}
\vspace{0.25cm}
\caption{Feynman diagrams for neutralino and chargino decays with $\tilde
N_1$ in the final state. The intermediate scalar or vector boson in each
case can be either onshell (so that actually there is a sequence of
twobody decays) or offshell, depending on the sparticle mass spectrum.
\label{fig:NCdecays}}
\end{figure}
In many situations, the decays
\beq
\stilde C_1^\pm \rightarrow \ell^\pm \nu \stilde N_1,\qquad\quad
\stilde N_2 \rightarrow \ell^+\ell^ \stilde N_1
\label{eq:CNleptonic}
\eeq
can be particularly important for phenomenology, because the leptons in
the final state might result in clean signals. These decays are more
likely if the intermediate sleptons are relatively light, even if they
cannot be onshell. Unfortunately, the enhanced mixing of staus, common in
models, may well result in larger branching fractions for both $\stilde
N_2$ and $\tilde C_1$ into final states with taus, rather than electrons
or muons. This is one reason why good tau identification may be very helpful
in attempts to discover and study supersymmetry.
In other situations, decays without isolated leptons in the final state
are more useful, so that one will not need to contend with
background events with missing energy coming from leptonic $W$ boson
decays in Standard Model processes. Then the decays of interest
are the ones with quark partons in the final state, leading to
\beq
\stilde C_1 \rightarrow jj \stilde N_1,\qquad\quad
\stilde N_2 \rightarrow jj \stilde N_1,
\label{eq:CNjetdecays}
\eeq
where $j$ means a jet. If the second of these decays goes through an
onshell $h^0$, then these will usually be $b$jets that reconstruct an invariant mass consistent with 125 GeV.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Slepton decays}\label{subsec:decays.sleptons}
\setcounter{equation}{0}
\setcounter{footnote}{1}
Sleptons can have twobody decays into a lepton and a chargino or
neutralino, because of their gaugino admixture, as may be seen directly
from the couplings in Figures~\ref{fig:gaugino}b,c. Therefore, the
twobody decays
\beq
\stilde \ell \rightarrow \ell \stilde N_i,\>\>\>\>\>\,
\stilde \ell \rightarrow \nu \stilde C_i,\>\>\>\>\>\,
\stilde \nu \rightarrow \nu \stilde N_i,\>\>\>\>\>\,
\stilde \nu \rightarrow \ell \stilde C_i
\eeq
can be of weak interaction strength. In particular, the direct decays
\beq
\stilde \ell \rightarrow \ell \stilde N_1
\>\>\>\>\>{\rm and}\>\>\>\>\>
\stilde \nu \rightarrow \nu \stilde N_1
\label{sleptonrightdecay}
\eeq
are (almost\footnote{An exception occurs if the mass difference $m_{\tilde
\tau_1}  m_{\tilde N_1}$ is less than $m_{\tau}$.}) always kinematically
allowed if $\stilde N_1$ is the LSP. However, if the sleptons are
sufficiently heavy, then the twobody decays
\beq
\stilde \ell \rightarrow \nu \stilde C_{1}
,\>\>\>\>\>\,
\stilde \ell \rightarrow \ell \stilde N_{2}
,\>\>\>\>\>\,
\stilde \nu \rightarrow \nu \stilde N_{2}
,\>\>\>\>{\rm and}\>\>\>\>
\stilde \nu \rightarrow \ell \stilde C_{1}
\label{sleptonleftdecay}
\eeq
can be important. The righthanded sleptons do not have a coupling to the
$SU(2)_L$ gauginos, so they typically prefer the direct decay $\stilde
\ell_R \rightarrow \ell\NI$, if $\NI$ is binolike. In contrast, the
lefthanded sleptons may prefer to decay as in
eq.~(\ref{sleptonleftdecay}) rather than the direct decays to the LSP as
in eq.~(\ref{sleptonrightdecay}), if the former is kinematically open and
if $\stilde C_1$ and $\stilde N_2$ are mostly wino. This is because the
sleptonleptonwino interactions in Figure~\ref{fig:gaugino}b are
proportional to the $SU(2)_L$ gauge coupling $g$, whereas the
sleptonleptonbino interactions in Figure~\ref{fig:gaugino}c are
proportional to the much smaller $U(1)_Y$ coupling $g^\prime$. Formulas
for these decay widths can be found in ref.~\cite{epprod}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Squark decays}\label{subsec:decays.squarks}
\setcounter{equation}{0}
\setcounter{footnote}{1}
If the decay $ \stilde q \rightarrow q\stilde g $ is kinematically
allowed, it will usually dominate, because the quarksquarkgluino vertex
in Figure~{\ref{fig:gaugino}}a has QCD strength. Otherwise, the squarks
can decay into a quark plus neutralino or chargino: $ \stilde q
\rightarrow q \stilde N_i$ or $ q^\prime \stilde C_i $. The direct decay
to the LSP $\stilde q \rightarrow q \stilde N_1$ is always kinematically
favored, and for righthanded squarks it can dominate if
$\stilde N_1$ is mostly bino. However, the lefthanded squarks may strongly prefer
to decay into heavier charginos or neutralinos instead, for example
$\stilde q \rightarrow q \stilde N_2$ or $q^\prime \stilde C_1$, because
the relevant squarkquarkwino couplings are much bigger than the
squarkquarkbino couplings. Squark decays to higgsinolike charginos and
neutralinos are less important, except in the cases of stops and sbottoms,
which have sizable Yukawa couplings. The gluino, chargino or neutralino
resulting from the squark decay will in turn decay, and so on, until a
final state containing $\stilde N_1$ is reached. This results in
numerous and complicated decay chain possibilities called cascade decays
\cite{cascades}.
It is possible that the decays $\stilde t_1 \rightarrow t\stilde g$ and
$\stilde t_1 \rightarrow t \stilde N_1$ are both kinematically forbidden.
If so, then the lighter top squark may decay only into charginos, by
$\stilde t_1 \rightarrow b \stilde C_1$, or by a threebody decay
$\stilde t_1 \rightarrow b W \stilde N_1$. If even this decay is
kinematically closed, then it has only the flavorsuppressed decay to a
charm quark, $ \stilde t_1\rightarrow c \stilde N_1$, and the fourbody
decay $ \stilde t_1\rightarrow bff' \stilde N_1 $. These decays can be
very slow \cite{stoptocharmdecay}, so that the lightest stop can be
quasistable on the time scale relevant for collider physics, and can
hadronize into bound states.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Gluino decays}\label{subsec:decays.gluino}
\setcounter{equation}{0}
\setcounter{footnote}{1}
The decay of the gluino can only proceed through a squark, either onshell
or virtual. If twobody decays $ \stilde g \rightarrow q\stilde q $ are
open, they will dominate, again because the relevant gluinoquarksquark
coupling in Figure~\ref{fig:gaugino}a has QCD strength. Since the top and
bottom squarks can easily be much lighter than all of the other squarks,
it is quite possible that $ \stilde g \rightarrow t \stilde t_1$ and/or
$\stilde g \rightarrow b \stilde b_1$ are the only available twobody
decay mode(s) for the gluino, in which case they will dominate over all
others. If instead all of the squarks are heavier than the gluino, the
gluino will decay only through offshell squarks, so $ \stilde g
\rightarrow q q \stilde N_i$ and $ q q^\prime \stilde C_i $. The squarks,
neutralinos and charginos in these final states will then decay as
discussed above, so there can be many competing gluino decay chains. Some
of the possibilities are shown in fig.~\ref{fig:gluinocascades}.
\begin{figure}
\begin{flushleft}
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%just a spaceholder below
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\end{picture}
}
%
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\rText(83,6)[][]{$\scriptstyle\tilde N_1$}
\rText(40,12)[][]{$\scriptstyle {\rm (a)}$}
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%
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\end{flushleft}
%
\vspace{0.25cm}
\begin{flushleft}
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\begin{picture}(172,38)(0,0)
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\end{flushleft}
\vspace{0.05cm}
\caption{Some of the many possible examples of gluino cascade decays
ending with a neutralino LSP in the final state. The squarks appearing in
these diagrams may be either onshell or offshell, depending on the mass
spectrum of the theory.\label{fig:gluinocascades}}
\end{figure}
The cascade decays can have finalstate branching fractions that are
individually small and quite sensitive to the model parameters.
The simplest gluino decays, including the ones shown in
fig.~\ref{fig:gluinocascades}, can have 0, 1, or 2 charged leptons (in
addition to two or more hadronic jets) in the final state. An important
feature is that when there is exactly one charged lepton, it can have
either charge with exactly equal probability. This follows from the fact
that the gluino is a Majorana fermion, and does not ``know" about electric
charge; for each diagram with a given lepton charge, there is always an
equal one with every particle replaced by its antiparticle.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Decays to the gravitino/goldstino}\label{subsec:decays.gravitino}
\setcounter{equation}{0}
\setcounter{footnote}{1}
Most phenomenological studies of supersymmetry assume explicitly or
implicitly that the lightest neutralino is the LSP. This is typically the
case in gravitymediated models for the soft terms. However, in
gaugemediated models (and in ``noscale" models), the LSP is instead the
gravitino. As we saw in section \ref{subsec:origins.gravitino}, a very
light gravitino may be relevant for collider phenomenology, because it
contains as its longitudinal component the goldstino, which has a
nongravitational coupling to all sparticleparticle pairs $(\stilde X,
X$). The decay rate found in eq.~(\ref{generalgravdecay}) for $\stilde
X\rightarrow X\G$ is usually not fast enough to compete with the other
decays of sparticles $\stilde X$ as mentioned above, {\it except} in the
case that $\stilde X$ is the nexttolightest supersymmetric particle
(NLSP). Since the NLSP has no competing decays, it should always decay
into its superpartner and the LSP gravitino.
In principle, any of the MSSM superpartners could be the NLSP in models
with a light goldstino, but most models with gauge mediation of
supersymmetry breaking have either a neutralino or a charged lepton
playing this role. The argument for this can be seen immediately from
eqs.~(\ref{gauginogmsbgen}) and (\ref{scalargmsbgen}); since $\alpha_1 <
\alpha_2,\alpha_3$, those superpartners with only $U(1)_Y$ interactions
will tend to get the smallest masses. The gaugeeigenstate sparticles with
this property are the bino and the righthanded sleptons $\stilde e_R$,
$\stilde \mu_R$, $\stilde \tau_R$, so the appropriate corresponding mass
eigenstates should be plausible candidates for the NLSP.
First suppose that $\stilde N_1$ is the NLSP in light goldstino models.
Since $\stilde N_1$ contains an admixture of the photino (the linear
combination of bino and neutral wino whose superpartner is the photon),
from eq.~(\ref{generalgravdecay}) it decays into photon +
goldstino/gravitino with a partial width
\beq
\Gamma (\NI \rightarrow \gamma \G ) \,=\,
2\times 10^{3} \> \kappa_{1\gamma}\left ({m_{\NI}\over 100\>\rm{
GeV}}\right )^5
\left ( {\sqrt{\langle F \rangle}\over 100\>{\rm TeV}} \right )^{4} \>
{\rm eV}.\qquad{}
\label{neutralinodecaywidth}
\eeq
Here $\kappa_{1\gamma} \equiv {\bf N}_{11}\cos\theta_W + {\bf N}_{12}\sin
\theta_W ^2$ is the ``photino content" of $\stilde N_1$, in terms of the
neutralino mixing matrix ${\bf N}_{ij}$ defined by eq.~(\ref{diagmN}). We
have normalized $m_{\NI}$ and $\sqrt{\langle F \rangle}$ to (very roughly)
minimum expected values in gaugemediated models. This width is much
smaller than for a typical flavorunsuppressed weak interaction decay, but
it is still large enough to allow $\stilde N_1$ to decay before it has
left a collider detector, if $\sqrt{\langle F\rangle}$ is less than a few
thousand TeV in gaugemediated models, or equivalently if $m_{3/2}$ is
less than a keV or so when eq.~(\ref{gravitinomass}) holds. In fact, from
eq.~(\ref{neutralinodecaywidth}), the mean decay length of an $\NI$ with
energy $E$ in the lab frame is
\beq
d = 9.9 \times 10^{3}\> {1\over \kappa_{1\gamma}}\,
({E^2/ m_{\NI}^2}  1)^{1/2}
\left ({m_{\NI}\over 100\>\rm{ GeV}}\right )^{5}
\left({\sqrt{\langle F \rangle}\over 100\>{\rm TeV}} \right )^{4}
\>{\rm cm},
\label{neutralinodecaylength}
\eeq
which could be anything from submicron to multikilometer, depending on
the scale of supersymmetry breaking $\sqrt{\langle F \rangle}$. (In other
models that have a gravitino LSP, including certain ``noscale" models
\cite{noscalephotons}, the same formulas apply with ${\langle F \rangle}
\rightarrow \sqrt{3} m_{3/2} \MPlanck$.)
Of course, $\stilde N_1$ is not a pure photino, but contains also
admixtures of the superpartner of the $Z$ boson and the neutral Higgs
scalars. So, one can also have \cite{DDRT} $\NI\rightarrow Z\G$, $h^0\G$,
$A^0\G$, or $H^0\G$, with decay widths given in ref.~\cite{AKKMM2}. Of
these decays, the last two are unlikely to be kinematically allowed, and
only the $\NI \rightarrow \gamma\G$ mode is guaranteed to be kinematically
allowed for a gravitino LSP. Furthermore, even if they are open, the
decays $\stilde N_1 \rightarrow Z\G$ and $\stilde N_1 \rightarrow h^0 \G$
are subject to strong kinematic suppressions proportional to
$(1m_Z^2/m_{\stilde N_1}^2)^4$ and $(1  m_{h^0}^2/m_{\stilde N_1}^2)^4$,
respectively, in view of eq.~(\ref{generalgravdecay}). Still, these decays
may play an important role in phenomenology if ${\langle F\rangle }$ is
not too large, $\stilde N_1$ has a sizable zino or higgsino content, and
$m_{\stilde N_1}$ is significantly greater than $m_Z$ or $m_{h^0}$.
A charged slepton makes another likely candidate for the NLSP. Actually,
more than one slepton can act effectively as
the NLSP, even though one of them is slightly lighter, if they are
sufficiently close in mass so that each has no kinematically allowed
decays except to the goldstino. In GMSB models, the squared masses
obtained by $\widetilde e_R$, $\widetilde \mu_R$ and $\widetilde \tau_R$
are equal because of the flavorblindness of the gauge couplings. However,
this is not the whole story, because one must take into account mixing
with $\widetilde e_L$, $\widetilde \mu_L$, and $\widetilde \tau_L$ and
renormalization group running. These effects are very small for
$\widetilde e_R$ and $\widetilde \mu_R$ because of the tiny electron and
muon Yukawa couplings, so we can quite generally treat them as degenerate,
unmixed mass eigenstates. In contrast, $\widetilde \tau_R$ usually has a
quite significant mixing with $\widetilde \tau_L$, proportional to the tau
Yukawa coupling. This means that the lighter stau mass eigenstate
$\widetilde \tau_1$ is pushed lower in mass than $\widetilde e_R$ or
$\widetilde \mu_R$, by an amount that depends most strongly on
$\tan\beta$. If $\tan\beta$ is not too large then the stau mixing effect
leaves the slepton mass eigenstates $\widetilde e_R$, $\widetilde \mu_R$,
and $\widetilde \tau_1$ degenerate to within less than $m_\tau \approx 1.8
$ GeV, so they act effectively as coNLSPs. In particular, this means
that even though the stau is slightly lighter, the threebody slepton
decays $\widetilde e_R \rightarrow e\tau^\pm\widetilde \tau_1^\mp$ and
$\widetilde \mu_R \rightarrow \mu\tau^\pm\widetilde \tau_1^\mp$ are not
kinematically allowed; the only allowed decays for the three lightest
sleptons are $\widetilde e_R\rightarrow e \G$ and $\widetilde \mu_R
\rightarrow \mu\G$ and $\widetilde \tau_1 \rightarrow \tau \G$. This
situation is called the ``slepton coNLSP" scenario.
For larger values of $\tan\beta$, the lighter stau eigenstate $\stilde
\tau_1$ is more than $1.8$ GeV lighter than $\widetilde e_R$ and
$\widetilde \mu_R$ and $\NI$. This means that the decays $\NI \rightarrow
\tau\stilde \tau_1$ and $\widetilde e_R \rightarrow e \tau \stilde \tau_1$
and $\widetilde \mu_R \rightarrow \mu \tau \stilde\tau_1$ are open. Then
$\widetilde \tau_1$ is the sole NLSP, with all other MSSM supersymmetric
particles having kinematically allowed decays into it. This is called the
``stau NLSP" scenario.
In any case, a slepton NLSP can decay like $\stilde \ell \rightarrow \ell
\G$ according to eq.~(\ref{generalgravdecay}), with a width and decay
length just given by eqs.~(\ref{neutralinodecaywidth}) and
(\ref{neutralinodecaylength}) with the replacements $\kappa_{1\gamma}
\rightarrow 1$ and $m_{\stilde N_1} \rightarrow m_{\stilde \ell}$. So, as
for the neutralino NLSP case, the decay $\stilde \ell \rightarrow \ell\G$
can be either fast or very slow, depending on the scale of supersymmetry
breaking.
If $\sqrt{\langle F \rangle}$ is larger than roughly $10^3$ TeV (or the
gravitino is heavier than a keV or so), then the NLSP is so longlived
that it will usually escape a typical collider detector. If $\NI$ is the
NLSP, then, it might as well be the LSP from the point of view of collider
physics. However, the decay of $\NI$ into the gravitino is still important
for cosmology, since an unstable $\NI$ is clearly not a good dark matter
candidate while the gravitino LSP conceivably could be. On the other hand,
if the NLSP is a longlived charged slepton, then one can see its tracks
(or possibly decay kinks) inside a collider detector \cite{DDRT}. The
presence of a massive charged NLSP can be established by measuring an
anomalously long timeofflight or high ionization rate for a track in the
detector.
\section{Experimental signals for supersymmetry}\label{sec:signals}
\setcounter{equation}{0}
\setcounter{figure}{0}
\setcounter{table}{0}
\setcounter{footnote}{1}
So far, the experimental study of supersymmetry has unfortunately been
confined to setting limits. As we have already remarked in section
\ref{subsec:mssm.hints}, there can be indirect signals for supersymmetry
from processes that are rare or forbidden in the Standard Model but have
contributions from sparticle loops. These include $\mu\rightarrow
e\gamma$, $b\rightarrow s\gamma$, neutral meson mixing, electric dipole
moments for the neutron and the electron, etc. There are also virtual
sparticle effects on Standard Model predictions like $R_b$ (the fraction
of hadronic $Z$ decays with $b\overline b$ pairs) \cite{Rb} and the
anomalous magnetic moment of the muon \cite{muonmoment}, which
exclude some models that would otherwise be viable.
Extensions of the MSSM (including, but not limited, to GUTs)
can quite easily predict proton decay and neutronantineutron
oscillations at potentially observable rates, even if $R$parity is exactly
conserved. However, it would be impossible to ascribe a positive result
for any of these processes to supersymmetry in an unambiguous way. There
is no substitute for the direct detection of sparticles and verification
of their quantum numbers and interactions. In this section we will give an
incomplete and qualitative review of some of the possible signals for
direct detection of supersymmetry. LHC data and analyses
are presently advancing this subject at a very high rate, so that any detailed and specific discussion would be obsolete on a time scale of weeks or months. The most recent
experimental results from the LHC are available at the
ATLAS and CMS physics results web pages.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Signals at hadron colliders}\label{subsec:signals.TEVLHC}
\setcounter{equation}{0}
\setcounter{footnote}{1}
At this writing, the CERN Large Hadron Collider (LHC)
has already excluded significant chunks of supersymmetric parameter space,
based on protonproton collisions amounting to about 5 fb$^{1}$ at $\sqrt{s} = 7$ TeV,
20 fb$^{1}$ at $\sqrt{s} = 8$ TeV, and 4 fb$^{1}$ at $\sqrt{s} = 13$ TeV.
In many MSUGRA and similar models, gluinos and
squarks with masses well above 1 TeV are already excluded by LHC data,
superseding the results
from the CDF and D$\emptyset$ detectors at the
Fermilab Tevatron $p\overline p$
collider with $\sqrt{s} = 1.96$ TeV.
Future planned increases in LHC integrated luminosity suggest that
if supersymmetry is the solution to the hierarchy problem
discussed in the Introduction, then the LHC
has a good chance of finding direct evidence for it within the next few years.
At hadron colliders, sparticles can be produced in pairs from parton
collisions of electroweak strength:
\beq
q \overline q^{\phantom '}\! &\rightarrow & \stilde
C_i^+ \stilde C_j^,
\>\>
\stilde N_i \stilde N_j,
\qquad\quad
u \overline d \>\rightarrow\> \stilde C_i^+ \stilde N_j,
\qquad\quad
d \overline u \>\rightarrow\> \stilde C_i^ \stilde N_j,
\label{eq:qqbarinos}
\\
q \overline q^{\phantom '}\! &\rightarrow & \stilde \ell^+_i \stilde \ell^_j,
\>\>\>
\stilde \nu_\ell \stilde \nu^*_\ell
\qquad\qquad\>\>
u \overline d \>\rightarrow\> \stilde \ell^+_L \stilde \nu_\ell
\qquad\qquad\>
d \overline u \>\rightarrow\> \stilde \ell^_L \stilde \nu^*_\ell,
\label{eq:qqbarsleptons}
\eeq
as shown in fig.~\ref{fig:qqbarsusy}, and reactions of QCD strength:
\beq
gg &\rightarrow & \stilde g \stilde g,
\>\>\,
\stilde q_i \stilde q_j^*,
\label{eq:gluegluegluinos}
\\
gq &\rightarrow & \stilde g \stilde q_i,
\label{eq:gluequarkgluinosquark}
\\
q \overline q &\rightarrow& \stilde g \stilde g,
\>\>\,
\stilde q_i \stilde q_j^*,
\label{eq:qqbargluinosorsquarks}
\\
q q &\rightarrow& \stilde q_i \stilde q_j,
\label{eq:qqsquarks}
\eeq
as shown in figs.~\ref{fig:ggsusy} and \ref{fig:qqsusy}.
\begin{figure}[p]
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\caption{Feynman diagrams for electroweak
production of sparticles at hadron colliders
from quarkantiquark annihilation. The charginos and neutralinos
in the $t$channel diagrams only couple because of their gaugino
content, for massless initialstate quarks, and so are drawn as
wavy lines superimposed on solid.
\label{fig:qqbarsusy}}
\end{figure}
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\caption{Feynman diagrams for gluino and squark production
at hadron colliders from gluongluon and gluonquark
fusion.\label{fig:ggsusy}}
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%
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%
\hspace{1.5cm}
%
\begin{picture}(120,52.5)(0,13)
\SetWidth{0.85}
\Line(0,50)(55,50)
\Line(0,0)(55,0)
\DashLine(110,0)(55,50){5}
\DashLine(110,50)(55,0){5}
\Line(55,0)(55,50)
\rText(6,43.2)[][]{$q$}
\rText(6,6)[][]{$q$}
\rText(50,27)[r][]{$\stilde g$}
\rText(112,49)[][]{$\stilde q$}
\rText(112,5)[][]{$\stilde q$}
\SetWidth{0.4}
\Photon(55,0)(55,50){2.1}{5.5}
\end{picture}
\end{center}
\caption{Feynman diagrams for gluino and squark production at hadron
colliders from strong quarkantiquark annihilation and quarkquark scattering.
\label{fig:qqsusy}}
\end{figure}%
The reactions in (\ref{eq:qqbarinos}) and (\ref{eq:qqbarsleptons}) get
contributions from electroweak vector bosons in the $s$channel, and those
in (\ref{eq:qqbarinos}) also have $t$channel squarkexchange
contributions that are of lesser importance in most models. The processes
in (\ref{eq:gluegluegluinos})(\ref{eq:qqsquarks}) get contributions from
the $t$channel exchange of an appropriate squark or gluino, and
(\ref{eq:gluegluegluinos}) and (\ref{eq:qqbargluinosorsquarks}) also have
gluon $s$channel contributions. In a crude first approximation, for the
hard parton collisions needed to make heavy particles, one may think of
the Tevatron as a quarkantiquark collider, and the LHC as a gluongluon
and gluonquark collider. However, the signals are always an inclusive
combination of the results of parton collisions of all types, and generally
cannot be neatly separated.
At the Tevatron collider, the chargino and neutralino production processes
(mediated primarily by valence quark annihilation into virtual weak
bosons) tended to have the larger crosssections, unless the squarks or
gluino were rather light (less than 300 GeV or so, which is now clearly
ruled out by the LHC). In a typical model
where $\stilde C_1$ and $\stilde N_2$ are mostly $SU(2)_L$ gauginos and
$\stilde N_1$ is mostly bino, the largest production crosssections in
(\ref{eq:qqbarinos}) belong to the $\stilde C_1^+\stilde C_1^$ and
$\stilde C_1\stilde N_2$ channels, because they have significant couplings
to $\gamma,Z$ and $W$ bosons, respectively, and because of kinematics. At
the LHC, the situation is typically reversed, with production of gluinos
and squarks by gluongluon and gluonquark fusion usually dominating. At both
colliders, one can also have associated production of a chargino or
neutralino together with a squark or gluino, but most models
predict that the
crosssections (of mixed electroweak and QCD strength) are much lower than
for the ones in (\ref{eq:qqbarinos})(\ref{eq:qqsquarks}). Slepton pair
production as in (\ref{eq:qqbarsleptons}) was quite small at the
Tevatron, but might be observable eventually at the LHC \cite{sleptonLHC}.
Crosssections for sparticle production at hadron colliders can be found
in refs.~\cite{gluinosquarkproduction}, and have been incorporated in
computer programs including
%\cite{ISAJET,PYTHIA,COMPHEP,GRACE,Herwig,CATPISS,SMADGRAPH}.
\cite{ISAJET},\cite{PYTHIA}\cite{Alwall:2011uj}.
The decays of the produced sparticles result in final states with two
neutralino LSPs, which escape the detector. The LSPs carry away at
least $2 m_{\NI}$ of missing energy, but at hadron colliders only the
component of the missing energy that is manifest as momenta transverse to
the colliding beams, usually denoted $\Et$ or $E_T^{\rm miss}$
(although $\vec{\slashchar{p}}_T$ or $\vec{p}_T^{\hspace{1.5pt}\rm miss}$
might be more logical names) is observable.
So, in general
the observable signals for supersymmetry at hadron colliders are $n$
leptons + $m$ jets + $\Et$, where either $n$ or $m$ might be 0. There are
important Standard Model backgrounds to these signals, especially
from processes involving production of $W$ and $Z$ bosons that decay to
neutrinos, which provide the $\Et$. Therefore it is important to identify
specific signal region cuts for which the backgrounds can be reduced. Of course,
the optimal choice of cuts
depends on which sparticles are being produced and how they decay, facts that are not known in advance. Depending on the specific object of the search,
backgrounds can be further reduced by
requiring at least some number $n$ of
energetic jets, and imposing a cut on a variable $H_T$,
typically defined to be the sum of the largest
few (or all) of the $p_T$'s of the jets and leptons in each event. (Unfortunately, there is no standard definition of $H_T$.) Different signal regions can be defined by
how many jets are required in the event, the minimum $p_T$ cuts on those jets, how many
jets are included in the definition of $H_T$, and other fine details.
Alternatively, one can cut on $m_{\rm eff} \equiv H_T + \Et$ rather than $H_T$.
Another cut that is often used in searches is to require a minimum value for the ratio of
$\Et$ to either $H_T$ or $m_{\rm eff}$; the backgrounds tend to
have smaller values of this ratio than a supersymmetric signal would.
LHC searches have also made use of more sophisticated kinematic observables,
such as $M_{T2}$ \cite{MT2}, $\alpha_T$ \cite{alphaT}, and razor variables
\cite{razor}.
The classic $\Et$ signal for supersymmetry at hadron colliders is events
with jets and $\Et$ but no energetic isolated leptons. The latter
requirement reduces backgrounds from Standard Model processes with
leptonic $W$ decays, and is obviously most effective if the relevant
sparticle decays have sizable branching fractions into channels with no
leptons in the final state. The most important potential backgrounds are:
\begin{itemize}
\item detector mismeasurements of jet energies,
\item $W$+jets, with the $W$ decaying to $\ell\nu$, when the charged lepton is missed or absorbed into a jet,
\item $Z$+jets, with $Z \rightarrow \nu \bar \nu$,
\item $t\overline t$ production, with $W\rightarrow \ell\nu$, when the charged lepton is
missed.
\end{itemize}
One must choose the $\Et$ cut high enough to reduce these backgrounds, and also to assist in efficient triggering. Requiring at least one very high$p_T$ jet can also satisfy a trigger requirement. In addition, the first (QCD)
background can be reduced by requiring that the transverse direction of the $\Et$ is not
too close to the transverse direction of a jet. The jets$+\Et$ signature is a favorite possibility for the
first evidence for supersymmetry to be found at the LHC.
It can get important contributions from
every type of sparticle pair production, except slepton pair production.
Another important possibility for the LHC is the single lepton plus jets plus $\Et$ signal
\cite{LHCdiscovery}. It has a potentially large Standard Model
background from production of
$W\rightarrow\ell\nu$, either together with jets or from top decays.
However, this background can
be reduced by putting a cut on the transverse mass variable
$m_T = \sqrt{2 p_T^\ell \Et [1  \cos(\Delta \phi)]}$,
where $\Delta \phi$ is the difference in azimuthal angle between the missing
transverse momentum and the lepton. For $W$ decays, this is essentially always less than
100 GeV even after detector resolution effects,
so a cut requiring $m_T > 100$ GeV nearly eliminates those background
contributions at the LHC.
The single lepton plus jets signal can have an extremely large rate from various sparticle
production modes, and may give a good discovery or confirmation signal at the LHC.
The samecharge dilepton signal \cite{likesigndilepton} has the advantage
of relatively small
backgrounds. It can occur if the gluino decays with a
significant branching fraction to hadrons
plus a chargino, which can subsequently decay into a final state with a charged lepton, a
neutrino, and $\stilde N_1$. Since the gluino doesn't know anything about
electric charge, the
charged lepton produced from each gluino decay can have either sign with equal
probability, as discussed in section
\ref{subsec:decays.gluino}. This means that gluino pair
production or gluinosquark production will often lead
to events with two leptons with the same
charge (and uncorrelated flavors) plus jets and $\Et$.
This signal can also arise from squark
pair production, for example if the squarks decay like
$\stilde q \rightarrow q\stilde g$. The physics backgrounds at hadron
colliders are very small,
because the largest Standard Model sources for isolated lepton pairs, notably DrellYan,
$W^+W^$, and $t\overline t$ production, can only yield oppositecharge dileptons.
Despite the backgrounds just mentioned,
oppositecharge dilepton signals, for example from
slepton pair production, or sleptonrich decays
of heavier superpartners, with subsequent decays
$\stilde \ell \rightarrow \ell \NI$, may also eventually
give an observable signal at the LHC.
The trilepton signal \cite{trilepton} is another possible discovery mode,
featuring three leptons plus $\Et$, and possibly hadronic jets. At the
Tevatron, this would most likely have come about from electroweak $\stilde
C_1\stilde N_2$ production followed by the decays indicated in
eq.~(\ref{eq:CNleptonic}), in which case high$p_T$ hadronic activity
should be absent in the event. A typical Feynman diagram for such an event
is shown in fig.~\ref{fig:trilepton}.
It could also come from $\stilde g\stilde g$, $\stilde q\stilde g$, or
$\stilde q \stilde q$ production, with one of the gluinos or squarks
decaying through a $\stilde C_1$ and the other through a $\stilde N_2$
in a variety of different ways.
%
\begin{figure}
\begin{minipage}[]{0.45\linewidth}
\caption{A complete Feynman diagram for a clean (no high$p_T$ hadronic
jets) trilepton event at a hadron collider, from production of an onshell
neutralino and a chargino, with subsequent leptonic decays, leading in
this case to $\mu^+\mu^e^+ + \Et$.\label{fig:trilepton}}
\end{minipage}
\begin{minipage}[]{0.545\linewidth}
\begin{picture}(115,143)(96,68)
\SetScale{1.4}
\SetWidth{0.67}
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\DashLine(15,27)(45,27){3.5}
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\Photon(30,0)(0,0){1.5}{5}
\Line(45,27)(30,0)
\Line(45,27)(30,0)
\Text(18,11)[]{$ W^+$}
\Text(70,40)[]{$ u$}
\Text(71,37)[]{$ \bar d$}
\Text(4,27)[]{$\tilde C_1^+$}
\Text(5,27)[]{$\tilde N_2$}
\Text(42,31)[]{$ \tilde \nu_e$}
\Text(42,29)[]{$ \tilde \mu$}
\Text(115,42)[]{$ \tilde N_1$}
\Text(115,40)[]{$ \tilde N_1$}
\Text(94,72.5)[]{$\mu^$}
\Text(90.7,74.5)[]{$\nu$}
\Text(53,72.5)[]{$\mu^+$}
\Text(52,76.5)[]{$e^+$}
\end{picture}
\end{minipage}
\end{figure}
This is the more likely origin at the LHC, at least in most benchmarks
based on MSUGRA or similar models.
In that case, there will be
very high$p_T$ jets from the decays, in addition to the three leptons and
$\Et$. These signatures rely on the $\stilde N_2$ having a significant
branching fraction for the threebody decay to leptons in
eq.~(\ref{eq:CNleptonic}). The competing twobody decay modes
$\stilde N_2 \rightarrow h^0 \stilde N_1$ and $\stilde N_2 \rightarrow Z
\stilde N_1$ are sometimes called ``spoiler" modes, since if they are
kinematically allowed they can dominate, spoiling the trilepton signal.
This is because if the $\stilde N_2$ decay is
through an onshell $h^0$ or $Z^0$, then the final state will likely include jets
(especially bottomquark jets in the case of $h^0$) rather than isolated leptons.
Although the trilepton signal is lost, supersymmetric events with
$h^0 \rightarrow b \bar b$ following from $\stilde N_2 \rightarrow h^0 \stilde N_1$
could eventually be useful at the LHC, especially since we now know that $M_{h^0} = 125$
GeV.
One should also be aware of interesting signals that can appear for particular ranges of
parameters. Final state leptons appearing in the signals listed above might be
predominantly tau, and so a significant fraction could be realized as hadronic $\tau$ jets.
This is because most models based on lepton universality at the input scale
predict that $\stilde \tau_1$ is lighter than the selectrons
and smuons. Similarly, supersymmetric events may have a preference for bottom jets,
sometimes through decays involving top quarks because $\stilde t_1$ is relatively light,
and sometimes because $\stilde b_1$ is expected to be lighter than the squarks of the first
two families, and sometimes for both reasons. In such cases, there will be at least four
potentially $b$taggable jets in each event. Other things being equal, the larger
$\tan\beta$ is, the stronger the preference for hadronic $\tau$ and $b$ jets will be in
supersymmetric events.
After evidence for the existence of supersymmetry is acquired, the LHC
data can be used to extract sparticle masses by analyzing the kinematics
of the decays. With a neutralino LSP always escaping the detector, there
are no true invariant mass peaks possible. However, various combinations
of masses can be measured using kinematic edges and other reconstruction
techniques. For a particularly favorable possibility, suppose the decay of the secondlightest neutralino
occurs in two stages through a real slepton, $\stilde N_2 \rightarrow \ell
\stilde \ell \rightarrow \ell^+\ell^\stilde N_1$. Then the resulting
dilepton invariant mass distribution is as shown in
fig.~\ref{fig:LHCendpoint}.%
\begin{figure}
\begin{minipage}[]{0.6\linewidth}
\caption{The theoretical shape of the dilepton invariant mass distribution
from events with $\stilde N_2 \rightarrow \ell \stilde \ell \rightarrow
\ell^+\ell^\stilde N_1$. No cuts or detector effects are included.
The endpoint is at $M_{\ell\ell}^{\rm max} = m_{\tilde N_2}
(1  m^2_{\tilde \ell}/m^2_{\tilde N_2})^{1/2}
(1  m^2_{\tilde N_1}/m^2_{\tilde \ell})^{1/2}.$
\label{fig:LHCendpoint}}
\end{minipage}
\begin{minipage}[]{0.339\linewidth}
\begin{picture}(165,95)(30,0)
\SetScale{1.35}
\LongArrow(0,0)(110,0)
\LongArrow(0,0)(0,50)
\Text(9,77)[]{Events/GeV}
\Text(148,10)[]{$M_{\ell\ell}$}
\Text(90,10)[]{$M_{\ell\ell}^{\rm max}$}
\SetWidth{1.3}
\Line(0,0)(65,40)
\Line(65,40)(65,0)
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\end{picture}
\end{minipage}
\end{figure}
It features a sharp edge, allowing a precision measurement of the
corresponding combination of $\stilde N_2$, $\stilde \ell$, and
$\stilde N_1$ masses \cite{LHCN2edge,LHCdileptonedge,ATLASTDR},
cuts will distort the shape, especially on the low end. There are
significant backgrounds to this analysis, for example coming from
$t\overline t$ production. However, the signal from $\stilde N_2$
has sameflavor leptons, while the background has contributions from
different flavors. Therefore the edge can be enhanced by plotting the
combination $[e^+e^] + [\mu^+\mu^]  [e^+\mu^]  [\mu^+e^]$,
subtracting the background.
Heavier sparticle mass combinations can also be reconstructed at the LHC
%\cite{ATLASTDR,Paigereviews}, \cite{HPLHC}\cite{Kawagoe}
\cite{ATLASTDR}\cite{Kawagoe}
using other kinematic
distributions. For example, consider the gluino decay chain $\stilde g
\rightarrow q \stilde q^* \rightarrow q \bar q \stilde N_2$ with $\stilde
N_2 \rightarrow \ell \stilde \ell^* \rightarrow \ell^+ \ell^ \stilde N_1$
as above. By selecting events close to the dilepton mass edge as
determined in the previous paragraph, one can reconstruct a peak in the
invariant mass of the $jj\ell^+\ell^$ system, which correlates well with
the gluino mass. As another example, the decay $\stilde q_L \rightarrow q
\stilde N_2$ with $\stilde N_2 \rightarrow h^0 \stilde N_1$ can be
analyzed by selecting events near the peak from $h^0 \rightarrow b
\overline b$. There will then be a broad $jb\bar b$ invariant mass
distribution, with a maximum value that can be related to $m_{\tilde
N_2}$, $m_{\tilde N_1}$ and $m_{\tilde q_L}$, since $m_{h^0} = 125$ GeV is known.
There are many other similar opportunities, depending on the specific sparticle
spectrum. These techniques may determine the sparticle mass
differences much more accurately than the individual masses, so that the mass of
the unobserved LSP will be constrained but not precisely
measured.\footnote{A possible exception occurs if the lighter
top squark has no
kinematically allowed flavorpreserving 2body decays, which requires
$m_{\tilde t_1} < m_{\tilde N_1} + m_t$ and
$m_{\tilde t_1} < m_{\tilde C_1} + m_b$. Then the $\tilde t_1$ will
live long enough to form
hadronic bound states. Scalar stoponium
might then be observable at the LHC via its rare $\gamma\gamma$ decay,
allowing a uniquely precise measurement of the mass through
a narrow peak (limited by detector resolution) in the diphoton
invariant mass spectrum \cite{Drees:1993yr,stoponium2}.}
Following the 2012 discovery of the 125 GeV Higgs boson, presumably $h^0$, the remaining
Higgs scalar bosons of the MSSM are also targets of searches at the
the LHC. The heavier neutral Higgs scalars can be searched for in decays
\beq
&&
A^0/H^0 \>\rightarrow\> \tau^+\tau^,\> \mu^+\mu^,\> b\overline
b,\>t\overline t,
\\
&&
H^0 \>\rightarrow\> h^0 h^0,
\\
&&
A^0 \>\rightarrow\> Z h^0 \>\rightarrow\> \ell^+ \ell^ b \overline b,
\eeq
with prospects that vary considerably depending on the parameters of the
model. The charged Higgs boson may also appear at the LHC in
topquark decays, if $m_{H^+} < m_t$.
If instead $m_{H^+} > m_t$, then one can look for
\beq
bg \rightarrow t H^
\qquad\mbox{or}\qquad
gg \rightarrow t \overline b H^,
\eeq
followed by the decay $H^ \rightarrow \tau^ \bar \nu_\tau$ or
$H^ \rightarrow \bar t b$ in each case, or the charge conjugates of these processes.
More details on Higgs search projections and experimental results are
available at the ATLAS and CMS physics results web pages.
The remainder of this subsection briefly considers the possibility that
the LSP is the goldstino/gravitino, in which case the sparticle discovery
signals discussed above can be significantly improved. If the NLSP is a
neutralino with a prompt decay, then $\NI\rightarrow \gamma\G$ will yield
events with two energetic, isolated photons plus $\Et$ from the escaping
gravitinos, rather than just $\Et$. So at a hadron collider the signal is
$\gamma\gamma+X+\Et$ where $X$ is any collection of leptons plus jets. The
Standard Model backgrounds relevant for such events are quite small. If
the $\NI$ decay length is long enough, then it may be measurable because
the photons will not point back to the event vertex. This would be
particularly useful, as it would give an indication of the
supersymmetrybreaking scale $\sqrt{\langle F \rangle}$; see
eq.~(\ref{generalgravdecay}) and the discussion in section
\ref{subsec:decays.gravitino}. If the $\NI$ decay is outside of the
detector, then one just has the usual leptons + jets + $\Et$ signals as
discussed above in the neutralino LSP scenario.
In the case that the NLSP is a charged slepton, then the decay $\stilde
\ell \rightarrow \ell\G$ can provide two extra leptons in each event,
compared to the signals with a neutralino LSP. If the $\stilde \tau_1$ is
sufficiently lighter than the other charged sleptons $\stilde e_R$,
$\stilde \mu_R$ and so is effectively the sole NLSP, then events will
always have a pair of taus. If the slepton NLSP is longlived, one can
look for events with a pair of very heavy charged particle tracks or a
long timeofflight in the detector. Since slepton pair production usually
has a much smaller crosssection than the other processes in
(\ref{eq:qqbarinos})(\ref{eq:qqsquarks}), this will typically be
accompanied by leptons and/or jets from the same event vertex, which may
be of crucial help in identifying candidate events. It is also quite
possible that the decay length of $\stilde \ell \rightarrow \ell\G$ is
measurable within the detector, seen as a macroscopic kink in the charged
particle track. This would again be a way to measure the scale
of supersymmetry breaking through eq.~(\ref{generalgravdecay}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Signals at $e^+e^$ colliders}\label{subsec:signals.LEPNLC}
\setcounter{equation}{0}
\setcounter{footnote}{1}
At $e^+e^$ colliders, all sparticles (except the gluino) can be produced
in treelevel reactions:
%
\begin{figure}[p]
\begin{center}
\begin{picture}(140,45)(0,15)
\SetScale{0.9}\SetWidth{0.85}
\Line(0,0)(40,30)
\Line(0,60)(40,30)
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\Line(90,30)(130,60)
\Line(90,30)(130,0)
\Text(7,5)[]{$e^+$}
\Text(7,58)[]{$e^$}
\Text(61,39)[]{$\gamma, Z$}
\Text(129,56)[]{$\stilde C_i^$}
\Text(129,4)[]{$\stilde C_j^+$}
\SetWidth{0.4}
\end{picture}
%
\hspace{1.75cm}
%
\begin{picture}(140,45)(0,15)
\SetScale{0.9}\SetWidth{0.85}
\Line(0,0)(60,0)
\Line(0,60)(60,60)
\Line(60,0)(120,0)
\Line(60,60)(120,60)
\DashLine(60,0)(60,60){5}
\Text(2,6)[c]{$e^+$}
\Text(2,60)[c]{$e^$}
\Text(46,28)[c]{$\stilde \nu_e$}
\Text(122,56)[c]{$\stilde C_i^$}
\Text(122,5)[c]{$\stilde C_j^+$}
\SetWidth{0.45}
\Photon(60,0)(120,0){2.1}{6}
\Photon(60,60)(120,60){2.1}{6}
\end{picture}
\end{center}
\caption{Diagrams for chargino pair production at $e^+e^$ colliders.
\label{fig:eecharginoprod}}
\end{figure}
%
\begin{figure}[p]
\begin{center}
\begin{picture}(130,48)(0,15)
\SetScale{0.9}\SetWidth{0.85}
\Line(0,0)(40,30)
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\Text(5.5,5)[]{$e^+$}
\Text(5.5,58)[]{$e^$}
\Text(59,38)[]{$Z$}
\Text(125,57)[]{$\stilde N_i$}
\Text(125,3)[]{$\stilde N_j$}
\end{picture}
%
\hspace{0.93cm}
%
\begin{picture}(120,48)(0,15)
\SetScale{0.9}\SetWidth{0.85}
\Line(3,0)(60,0)
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\Text(3,5)[c]{$e^+$}
\Text(3,59)[c]{$e^$}
\Text(36,28)[c]{$\stilde e_L$, $\stilde e_R$}
\Text(115,58)[c]{$\stilde N_i$}
\Text(115,4)[c]{$\stilde N_j$}
\SetWidth{0.45}
\Photon(60,0)(117,0){2.1}{6}
\Photon(60,60)(117,60){2.1}{6}
\end{picture}
%
\hspace{0.9cm}
%
\begin{picture}(115,48)(0,15)
\SetScale{0.9}\SetWidth{0.85}
\Line(0,0)(60,0)
\Line(0,60)(60,60)
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\DashLine(60,0)(60,60){5}
\Text(5,5)[c]{$e^+$}
\Text(5,59)[c]{$e^$}
\Text(36,28)[c]{$\stilde e_L$, $\stilde e_R$}
\Text(117,56)[c]{$\stilde N_i$}
\Text(117,3)[c]{$\stilde N_j$}
\SetWidth{0.45}
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\Photon(60,0)(85,25){2.1}{3}
\Photon(120,60)(95,35){2.1}{3}
\end{picture}
\end{center}
\caption{Diagrams for neutralino pair production at $e^+e^$ colliders.
\label{fig:eeneutralinoprod}}
\end{figure}
%
\begin{figure}[p]
\begin{center}
\begin{picture}(140,48)(0,15)
\SetScale{0.9}\SetWidth{0.85}
\Line(0,0)(40,30)
\Line(0,60)(40,30)
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\DashLine(90,30)(130,60){5}
\DashLine(90,30)(130,0){5}
\Text(4,5)[]{$e^+$}
\Text(4,59)[]{$e^$}
\Text(61,38)[]{$\gamma, Z$}
\Text(128,57)[]{$\stilde \ell^$}
\Text(128,5)[]{$\stilde \ell^+$}
\end{picture}
%
\hspace{1.75cm}
%
\begin{picture}(140,48)(0,15)
\SetScale{0.9}\SetWidth{0.85}
\Line(0,0)(60,0)
\Line(0,60)(60,60)
\DashLine(60,0)(120,0){5}
\DashLine(60,60)(120,60){5}
\Line(60,0)(60,60)
\Text(4,8)[c]{$e^+$}
\Text(4,59)[c]{$e^$}
\Text(45,28)[c]{$\stilde N_i$}
\Text(118,59)[c]{$\stilde e^$}
\Text(118,5)[c]{$\stilde e^+$}
\SetWidth{0.45}
\Photon(60,0)(60,60){2.1}{6}
\end{picture}
\end{center}
\caption{Diagrams for charged slepton pair production at $e^+e^$
colliders.
\label{fig:eesleptonprod}}
\end{figure}
%
\begin{figure}[p]
\begin{center}
\begin{picture}(140,48)(0,15)
\SetScale{0.9}\SetWidth{0.85}
\Line(0,0)(40,30)
\Line(0,60)(40,30)
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\DashLine(90,30)(130,60){5}
\DashLine(90,30)(130,0){5}
\Text(5,5)[]{$e^+$}
\Text(5,59)[]{$e^$}
\Text(61,38)[]{$Z$}
\Text(126.2,57)[]{$\tilde \nu_\ell$}
\Text(126.6,2)[]{$\tilde \nu_\ell^*$}
\end{picture}
%
\hspace{1.75cm}
%
\begin{picture}(140,48)(0,15)
\SetScale{0.9}\SetWidth{0.85}
\Line(0,0)(60,0)
\Line(0,60)(60,60)
\DashLine(60,0)(120,0){5}
\DashLine(60,60)(120,60){5}
\Line(60,0)(60,60)
\Text(4,8)[c]{$e^+$}
\Text(4,59)[c]{$e^$}
\Text(45,27)[c]{$\tilde C_i$}
\Text(118.5,57)[c]{$\tilde \nu_e$}
\Text(118.5,4)[c]{$\tilde \nu_e^*$}
\SetWidth{0.45}
\Photon(60,0)(60,60){2.1}{6}
\end{picture}
\end{center}
\caption{Diagrams for sneutrino pair production at $e^+e^$ colliders.
\label{fig:eesneutrinoprod}}
\end{figure}
%
\begin{figure}[p]
\begin{center}
\begin{picture}(140,44)(0,15)
\SetScale{0.9}\SetWidth{0.85}
\Line(0,0)(40,30)
\Line(0,60)(40,30)
\Photon(40,30)(90,30){2.4}{5}
\DashLine(90,30)(130,60){5}
\DashLine(90,30)(130,0){5}
\Text(6,5)[]{$e^+$}
\Text(6,59)[]{$e^$}
\Text(59,38)[]{$\gamma, Z$}
\Text(124.6,57)[]{$\stilde q$}
\Text(126.8,3)[]{$\stilde q^*$}
\end{picture}
%
\end{center}
\caption{Diagram for squark production at $e^+e^$ colliders.
\label{fig:eesquarkprod}}
\end{figure}
%
\begin{figure}[p]
\begin{center}
\begin{picture}(140,61)(0,2)
\SetScale{0.9}\SetWidth{0.85}
\Line(0,0)(40,30)
\Line(0,60)(40,30)
\Photon(40,30)(90,30){2.4}{5}
\DashLine(90,30)(130,60){5}
\Photon(90,30)(130,0){2.4}{4.5}
\Text(7,4)[]{$e^+$}
\Text(7,58)[]{$e^$}
\Text(61,38)[]{$Z$}
\Text(126,58)[]{$h^0$}
\Text(125,3)[]{$Z$}
\Text(63.5,14)[c]{$\propto \,\sin^2(\beta  \alpha)$}
\end{picture}
%
\hspace{1.75cm}
%
\begin{picture}(140,61)(0,2)
\SetScale{0.9}\SetWidth{0.85}
\Line(0,0)(40,30)
\Line(0,60)(40,30)
\Photon(40,30)(90,30){2.4}{5}
\DashLine(90,30)(130,60){5}
\DashLine(90,30)(130,0){5}
\Text(7,4)[]{$e^+$}
\Text(7,58)[]{$e^$}
\Text(61,38)[]{$Z$}
\Text(126,58)[]{$h^0$}
\Text(126,3)[]{$A^0$}
\Text(63.5,14)[c]{$\propto\, \cos^2(\beta  \alpha)$}
\end{picture}
\end{center}
\caption{Diagrams for neutral Higgs scalar boson
production at $e^+e^$ colliders.\label{fig:eehiggs}}
\end{figure}
\beq
e^+e^ \rightarrow
\stilde C_i^+ \stilde C_j^,\>\>\, \stilde N_i \stilde N_j,
\>\>\, \stilde \ell^+ \stilde \ell^,\>\>\, \stilde \nu \stilde \nu^*,
\>\>\, \stilde q \stilde q^* ,
\label{eesignals}
\eeq
as shown in figs.~\ref{fig:eecharginoprod}\ref{fig:eesquarkprod}. The
important interactions for sparticle production are the
gauginofermionscalar couplings shown in Figures~\ref{fig:gaugino}b,c and
the ordinary vector boson interactions. The crosssections are therefore
determined just by the electroweak gauge couplings and the sparticle
mixings. They were calculated in ref.~\cite{epprod}, and are available
in computer programs
%\cite{ISAJET,PYTHIA,COMPHEP,GRACE,Herwig,SUSYGEN}.
\cite{ISAJET}, \cite{PYTHIA}\cite{Herwig}, \cite{SUSYGEN}.
All of the processes in eq.~(\ref{eesignals}) get contributions from the
$s$channel exchange of the $Z$ boson and, for charged sparticle pairs,
the photon. In the cases of $\stilde C_i^+ \stilde C_j^$, $\stilde N_i
\stilde N_j$, $\stilde e_R^+ \stilde e_R^$, $\stilde e_L^+ \stilde
e_L^$, $\stilde e_L^\pm \stilde e_R^\mp$, and $\stilde \nu_e \stilde
\nu_e^*$ production, there are also $t$channel diagrams exchanging a
virtual sneutrino, selectron, neutralino, neutralino, neutralino, and
chargino, respectively. The $t$channel contributions are significant if
the exchanged sparticle is not too heavy. For example, the production of
winolike $\stilde C_1^+ \stilde C_1^$ pairs typically suffers a
destructive interference between the $s$channel graphs with $\gamma,Z$
exchange and the $t$channel graphs with $\stilde \nu_e$ exchange, if the
sneutrino is not too heavy. In the case of sleptons, the pair production
of smuons and staus proceeds only through $s$channel diagrams, while
selectron production also has a contribution from the $t$channel
exchanges of the neutralinos, as shown in Figure~\ref{fig:eesleptonprod}.
For this reason, the selectron production crosssection
may be significantly larger than that of
smuons or staus at $e^+e^$ colliders.
The pairproduced sparticles decay as discussed in section
\ref{sec:decays}. If the LSP is the lightest neutralino, it will always
escape the detector because it has no strong or electromagnetic
interactions. Every event will have two LSPs leaving the detector, so
there should be at least $2m_{\NI}$ of missing energy ($\Etot$). For
example, in the case of $\stilde C_1^+ \stilde C_1^$ production, the
possible signals include a pair of acollinear leptons and $\Etot$, or one
lepton and a pair of jets plus $\Etot$, or multiple jets plus $\Etot$. The
relative importance of these signals depends on the branching fraction of
the chargino into the competing final states, $\stilde C_1 \rightarrow
\ell\nu \NI$ and $qq^\prime\NI$. In the case of slepton pair production,
the signal should be two energetic, acollinear, sameflavor leptons plus
$\Etot$. There is a potentially large Standard Model background for the
acollinear leptons plus $\Etot$ and the lepton plus jets plus $\Etot$
signals, coming from $W^+W^$ production with one or both of the $W$
bosons decaying leptonically. However, these and other Standard Model
backgrounds can be kept under control with angular cuts, and beam
polarization if available. It is not difficult to construct the other
possible signatures for sparticle pairs, which can become quite
complicated for the heavier charginos, neutralinos and squarks.
The MSSM neutral Higgs bosons can also be produced at $e^+e^$ colliders,
with the principal processes of interest at low energies
\beq
e^+e^ \rightarrow h^0 Z,
\qquad\qquad
e^+e^\rightarrow h^0A^0,
\eeq
shown in fig.~\ref{fig:eehiggs}. At treelevel, the first of these has a
crosssection given by the corresponding Standard Model crosssection
multiplied by a factor of $\sin^2(\beta  \alpha)$, which approaches 1 in
the decoupling limit of $m_{A^0} \gg m_Z$ discussed in section
\ref{subsec:MSSMspectrum.Higgs}. The other process is complementary, since
(up to kinematic factors) its crosssection is the same but multiplied
by $\cos^2(\beta  \alpha)$, which is significant if $m_{A^0}$ is not
large. If $\sqrt{s}$ is high enough [note the mass relation
eq.~(\ref{eq:m2Hpm})], one can also have
\beq
e^+e^\rightarrow H^+ H^,
\eeq
with a crosssection that is fixed, at treelevel, in terms of
$m_{H^\pm}$, and also
\beq
e^+e^ \rightarrow H^0Z,
\qquad\qquad
e^+e^\rightarrow H^0A^0,
\eeq
with crosssections proportional to $\cos^2(\beta  \alpha)$ and
$\sin^2(\beta  \alpha)$ respectively. Also, at sufficiently high
$\sqrt{s}$, the process
\beq
e^+ e^ \rightarrow \nu_e \bar \nu_e h^0
\eeq
following from $W^+W^$ fusion provides the best way to study the Higgs
boson decays, which can differ \cite{GunionHaber,HHG,Haber:1997dt}
from those in the Standard Model.
The CERN LEP $e^+e^$ collider conducted searches until November 2000,
with various center of mass energies up to 209 GeV, yielding no firm
evidence for superpartner production. The resulting limits
\cite{LEPSUSYWG} on the charged sparticle masses are of order roughly half
of the beam energy, minus taxes paid for detection and identification
efficiencies, backgrounds, and the suppression of crosssections near
threshold. The bounds become weaker if the mass difference between the
sparticle in question and the LSP (or another sparticle that the produced
one decays into) is less than a few GeV, because then the available
visible energy can be too small for efficient detection and identification.
Despite the strong limits coming from the LHC, some of the limits from LEP are still relevant, especially when the mass differences between supersymmetric particle
are small.
For example, LEP established limits $m_{\tilde e_R} > 99$ GeV and
$m_{\tilde \mu_R} > 95$ GeV at 95\% CL, provided that $m_{\tilde \ell_R} 
m_{\tilde N_1} > 10$ GeV, and that the branching fraction for $\ell_R
\rightarrow \ell \stilde N_1$ is 100\% in each case. The limit for staus
is weaker, and depends somewhat more strongly on the neutralino LSP mass.
The LEP chargino mass bound is approximately $m_{\tilde C_1} > 103$ GeV
for mass differences $m_{\tilde C_1}  m_{\tilde N_1} > 3$ GeV, assuming
that the chargino decays predominantly through a virtual $W$, or with
similar branching fractions. However, this bound reduces to about
$m_{\tilde C_1} > 92$ GeV for $100$ MeV $< m_{\tilde C_1}  m_{\tilde N_1}
< 3$ GeV. For small positive mass differences 0 $< m_{\tilde C_1} 
m_{\tilde N_1} <$ 100 MeV, the limit is again about $m_{\tilde C_1} > 103$
GeV, because the chargino is longlived enough to have a displaced decay
vertex or leave a track as it moves through the detector. These limits
assume that the sneutrino is heavier than about 200 GeV, so that it does
not significantly reduce the production crosssection by interference of
the $s$ and $t$channel diagrams in fig.~\ref{fig:eecharginoprod}. If the
sneutrino is lighter, then the bound reduces, especially if $m_{\tilde C_1} 
m_{\tilde \nu}$ is positive but small, so that the decay $\stilde C_1
\rightarrow \tilde \nu \ell$ dominates but releases very little visible
energy. More details on these and many other legacy limits from the LEP runs can
be found at \cite{LEPSUSYWG} and \cite{RPP}.
If supersymmetry is the solution to the hierarchy problem, then the LHC
may be able to establish strong evidence for it, and measure
some of the sparticle mass differences, as discussed in the previous
subsection. However, many important questions will remain.
Competing theories can also produce missing energy signatures. The overall
mass scale of sparticles may not be known as well as one might like.
Sparticle production will be inclusive and overlapping and might be
difficult to disentangle. A future $e^+e^$ collider
with sufficiently large $\sqrt{s}$
should be able to resolve these issues, and establish more firmly that
supersymmetry is indeed responsible, to the exclusion of other
candidate theories. In
particular, the couplings, spins, gauge quantum numbers, and absolute
masses of the sparticles will all be measurable.
At an $e^+e^$ collider, the processes in eq.~(\ref{eesignals}) can all be
probed close to their kinematic limits, given sufficient integrated
luminosity. (In the case of sneutrino pair production, this assumes that
some of the decays are visible, rather than just
$\stilde\nu\rightarrow\nu\NI$.) Establishing the properties of the
particles can be done by making use of polarized beams and the relatively
clean $e^+e^$ collider environment. For example, consider the production
and decay of sleptons in $e^+e^ \rightarrow \stilde\ell^+\stilde\ell^$
with $\stilde\ell \rightarrow \ell\NI$. The resulting leptons will have
(up to significant but calculable effects of initialstate radiation,
beamstrahlung, cuts, and detector efficiencies and resolutions) a flat
energy distribution as shown in fig.~\ref{fig:ILCendpoints}.%
\begin{figure}
\begin{minipage}[]{0.6\linewidth}
\caption{The theoretical shape of the lepton energy distribution from
events with $e^+e^ \rightarrow \stilde \ell^+\stilde \ell^ \rightarrow
\ell^+\ell^\stilde N_1\stilde N_1$ at an $e^+e^$ collider. No cuts
or initial state radiation or beamstrahlung or detector effects are
included. The endpoints are $E_{{\rm max,min}}
= \frac{\sqrt{s}}{4} (1  m^2_{\tilde
N_1}/m^2_{\tilde \ell})[1 \pm (1  4 m^2_{\tilde \ell}/s)^{1/2}]$,
allowing precision reconstruction of both $\stilde \ell$ and $\stilde N_1$
masses.\label{fig:ILCendpoints}}
\end{minipage}
\begin{minipage}[]{0.339\linewidth}
\begin{picture}(165,80)(30,0)
\SetScale{1.35}
\LongArrow(0,0)(100,0)
\LongArrow(0,0)(0,50)
\Text(9,77)[]{Events/GeV}
\Text(138,10)[]{$E_{\ell}$}
\Text(93,10)[]{$E_{\rm max}$}
\Text(21,10)[]{$E_{\rm min}$}
\SetWidth{1.3}
\Line(0,0)(12,0)
\Line(12,0)(12,40)
\Line(12,40)(65,40)
\Line(65,40)(65,0)
\Line(65,0)(100,0)
\end{picture}
\end{minipage}
\end{figure}
By measuring the endpoints of this distribution, one can precisely and
uniquely determine both $m_{\stilde\ell_R}$ and $m_{\stilde N_1}$. There
is a large $W^+W^ \rightarrow \ell^+ \ell^{\prime } \nu_\ell \bar
\nu_{\ell'}$ background, but this can be brought under control using
angular cuts, since the positively (negatively) charged leptons from the
background tend to go preferentially along the same direction as the
positron (electron) beam. Also, since the background has uncorrelated
lepton flavors, it can be subtracted. Changing the polarization of the
electron beam will even further reduce the background, and will also allow
controlled variation of the production of righthanded and lefthanded
sleptons, to get at the electroweak quantum numbers.
More generally, inclusive sparticle production at a given fixed $e^+e^$
collision energy will result in a superposition of various kinematic
edges in lepton and jet energies, and distinctive distributions in
dilepton and dijet energies and invariant masses. By varying the beam
polarization and changing the beam energy, these observables give
information about the couplings and masses of the sparticles. For example,
in the ideal limit of a righthanded polarized electron beam, the reaction
\beq
e^_R e^+ \rightarrow \stilde C_1^+ \stilde C_1^
\eeq
is suppressed if $\stilde C_1$ is pure wino, because in the first diagram
of fig.~\ref{fig:eecharginoprod} the righthanded electron only couples to
the $U(1)_Y$ gauge boson linear combination of $\gamma,Z$ while the wino
only couples to the orthogonal $SU(2)_L$ gauge boson linear combination,
and in the second diagram the electronsneutrinochargino coupling
involves purely lefthanded electrons. Therefore, the polarized beam
crosssection can be used to determine the charged wino mixing with the
charged higgsino. Even more precise information about the sparticle masses
can be obtained by varying the beam energy in small discrete steps very
close to thresholds, an option unavailable at hadron colliders. The rise
of the production crosssection above threshold provides information about
the spin and ``handedness", because the production crosssections for
$\tilde \ell_R^+ \tilde \ell_R^$ and $\tilde \ell_L^+ \tilde \ell_L^$
are $p$wave and therefore rise like $\beta^3$ above threshold, where
$\beta$ is the velocity of one of the produced sparticles. In contrast,
the rates for $\tilde e_L^\pm \tilde e_R^\mp$ and for chargino and
neutralino pair production are $s$wave, and therefore should rise like
$\beta$ just above threshold. By measuring the angular distributions of
the final state leptons and jets with respect to the beam axis, the spins
of the sparticles can be inferred. These will provide crucial tests that
the new physics that has been discovered is indeed supersymmetry.
A sample of the many detailed studies along these lines can be found in
%refs.~\cite{ilcmassdetrefs,JLC,NLCsusy,ELC,ALC}.
refs.~\cite{ilcmassdetrefs}\cite{ALC}.
In general, a future
$e^+ e^$ collider will provide an excellent way of testing softlybroken
supersymmetry and measuring the model parameters, if it has enough energy.
Furthermore, the processes $e^+e^ \rightarrow$ $h^0Z$, $h^0A^0$, $H^0Z$,
$H^0A^0$, $H^+H^$, and $h^0 \nu_e \bar \nu_e$ should be able to
test the Higgs sector of supersymmetry at an $e^+e^$ collider.
The situation may be qualitatively better if the
gravitino is the LSP as in gaugemediated models, because of the decays
mentioned in section \ref{subsec:decays.gravitino}. If the lightest
neutralino is the NLSP and the decay $\NI\rightarrow\gamma\G$ occurs
within the detector, then even the process $e^+e^\rightarrow \NI\NI$
leads to a dramatic signal of two energetic photons plus missing energy
%\cite{eeGMSBsignal,DDRT,AKKMM2}.
\cite{eeGMSBsignal}\cite{AKKMM2}.
There are significant backgrounds to the
$\gamma\gamma\Etot$ signal, but they are easily removed by cuts. Each of
the other sparticle pairproduction modes eq.~(\ref{eesignals}) will lead
to the same signals as in the neutralino LSP case, but now with two
additional energetic photons, which should make the experimentalists'
tasks quite easy. If the decay length for $\NI\rightarrow\gamma\G$ is much
larger than the size of a detector, then the signals revert back to those
found in the neutralino LSP scenario. In an intermediate regime for the
$\NI\rightarrow\gamma\G$ decay length, one may see events with one or both
photons displaced from the event vertex by a macroscopic distance.
If the NLSP is a charged slepton $\stilde \ell$, then $e^+e^\rightarrow
\stilde \ell^+\stilde \ell^$ followed by prompt decays $\stilde
\ell\rightarrow \ell \G$ will yield two energetic sameflavor leptons in
every event, and with a different energy distribution than the acollinear
leptons that would follow from either $\stilde C_1^+\stilde C_1^$ or
$\stilde \ell^+\stilde \ell^$ production in the neutralino LSP scenario.
Pair production of nonNLSP sparticles will yield unmistakable signals,
which are the same as those found in the neutralino NLSP case but with two
additional energetic leptons (not necessarily of the same flavor). An even
more striking possibility is that the NLSP is a slepton that decays very
slowly \cite{DDRT}. If the slepton NLSP is so longlived that it decays
outside the detector, then slepton pair production will lead to events
featuring a pair of charged particle tracks with high ionization rates
that betray their very large mass. If the sleptons decay within the
detector, then one can look for largeangle
kinks in the charged particle tracks, or a
macroscopic impact parameter. The pair production of any of the other
heavy charged sparticles will also yield heavy charged particle tracks or
decay kinks, plus leptons and/or jets, but no $\Etot$ unless the decay
chains happen to include neutrinos. It may also be possible to identify
the presence of a heavy charged NLSP by measuring its anomalously long
timeofflight through the detector.
In both the neutralino and slepton NLSP scenarios, a measurement of the
decay length to $\stilde G$ would provide a great opportunity to measure
the supersymmetrybreaking scale $\sqrt{\langle F \rangle}$, as discussed
in section \ref{subsec:decays.gravitino}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dark matter and its detection}\label{subsec:signals.darkmatter}
\setcounter{equation}{0}
\setcounter{footnote}{1}
Evidence from experimental cosmology has now solidified to the point that,
with some plausible assumptions, the cold dark matter density is known to
be \cite{cosmokramer,RPP}
\beq
\Omega_{\rm DM} h^2 \approx 0.120,
\label{eq:OmegaDM}
\eeq
with statistical errors of about 2\%, and systematic errors that are less
clear. Here $\Omega_{\rm DM}$ is the average energy density in
nonbaryonic dark matter divided by the total critical density that would
lead to a spatially flat homogeneous universe, and $h$ is the Hubble
constant in units of 100 km sec$^{1}$ Mpc$^{1}$, observed to be $h^2
\approx 0.46$ with an error of order 3\%. This translates into a cold dark
matter density
\beq
\rho_{\rm DM} \approx 1.2 \times 10^{6} \> {\rm GeV}/{\rm cm}^3 ,
\label{eq:rhodm}
\eeq
averaged over very large distance scales.
One of the nice features of supersymmetry with exact $R$parity
conservation is that a stable electrically neutral LSP might be this cold
dark matter. There are three obvious candidates: the lightest sneutrino,
the gravitino, and the lightest neutralino. The possibility of a sneutrino
LSP making up the dark matter with a cosmologically interesting density
has been largely ruled out by direct searches \cite{sneutrinonotLSP} (see
however \cite{sneutrinoLSP}). If the gravitino is the LSP, as in many
gaugemediated supersymmetry breaking models, then gravitinos from
reheating after inflation \cite{gravitinoDMfromreheating} or from other
sparticle decays \cite{gravitinoDMfromdecays} might be the dark matter,
but they would be impossible to detect directly even if they have the
right cosmological density today. They interact too weakly. The most
attractive prospects for direct detection of supersymmetric dark matter,
therefore, are based on the idea that the lightest neutralino $\NI$ is the
LSP \cite{neutralinodarkmatter,darkmatterreviews}.
In the early universe, sparticles existed in thermal equilibrium with the
ordinary Standard Model particles. As the universe cooled and expanded,
the heavier sparticles could no longer be produced, and they
eventually annihilated or
decayed into neutralino LSPs. Some of the LSPs
pairannihilated into final states not containing sparticles. If there are
other sparticles that are only slightly heavier, then they existed in
thermal equilibrium in comparable numbers to the LSP, and their
coannihilations are also important in determining the resulting dark
matter density \cite{GriestSeckel,Gondolo:1990dk}. Eventually, as the
density decreased, the annihilation rate became small compared to the
cosmological expansion, and the $\NI$ experienced ``freeze out", with a
density today determined by this small rate and the subsequent dilution
due to the expansion of the universe.
In order to get the observed dark matter density today, the thermalaveraged
effective annihilation crosssection times the relative speed $v$ of the
LSPs should be about \cite{darkmatterreviews}
\beq
\langle \sigma v \rangle \>\sim\> 1\>{\rm pb} \>\sim\>
\alpha^2/(150\>{\rm GeV})^2,
\eeq
so a neutralino LSP naturally has, very roughly, the correct (electroweak)
interaction strength and mass. More detailed and precise estimates can be
obtained with publicly available computer programs
\cite{DarkSUSY,micrOMEGAs}, so that the predictions of specific candidate
models of supersymmetry breaking can be compared to
eq.~(\ref{eq:OmegaDM}). Some of the diagrams that are typically important
for neutralino LSP pair annihilation are shown in
fig.~\ref{fig:darkmatterannihilation}. Depending on the mass of $\NI$,
various other processes including $\NI\NI\rightarrow$$ZZ$, $Zh^0$,
$h^0h^0$ or even $W^\pm H^\mp$, $Z A^0$, $h^0 A^0$, $h^0 H^0$, $H^0 A^0$,
$H^0H^0$, $A^0 A^0$, or $H^+H^$ may also have been important. Some of the
diagrams that can lead to coannihilation of the LSPs with slightly
heavier sparticles are shown in figs.~\ref{fig:coannihilation} and
\ref{fig:sfermioncoannihilation}.%
\begin{figure}
\begin{picture}(120,72)(0,12)
\SetWidth{0.85}
\Line(0,0)(110,0)
\Line(0,50)(110,50)
\DashLine(55,0)(55,50){5}
\rText(0,9)[][]{$\stilde N_1$}
\rText(0,60)[][]{$\stilde N_1$}
\rText(55.5,27)[][]{$\tilde f$}
\rText(105,59)[][]{$ f$}
\rText(105,8)[][]{$\bar f$}
\Text(55,16)[c]{(a)}
\end{picture}
%
\hspace{0.75cm}
%
\begin{picture}(180,72)(10,12)
\SetWidth{0.85}
\Line(0,0)(30,25)
\Line(0,50)(30,25)
\DashLine(30,25)(93,25){5}
\Line(123,0)(93,25)
\Line(123,50)(93,25)
\rText(10,10)[][]{$\stilde N_1$}
\rText(10,59)[][]{$\stilde N_1$}
\rText(56.8,32.8)[][]{$A^0$ ($h^0,H^0$)}
\rText(146,53)[][]{$b,t,\tau^,\ldots$}
\rText(146,4)[][]{$\bar b,\bar t,\tau^+,\ldots$}
\Text(66.5,16)[c]{(b)}
\end{picture}
%
\hspace{1.0cm}
%
\begin{picture}(120,72)(0,12)
\SetWidth{0.85}
\Line(0,0)(55,0)
\Line(0,50)(55,50)
\Line(55,0)(55,50)
\Photon(55,0)(110,0){2}{5}
\Photon(55,50)(110,50){2}{5}
\rText(0,9)[][]{$\stilde N_1$}
\rText(0,60)[][]{$\stilde N_1$}
\rText(57.5,27)[][]{$\tilde C_i$}
\rText(105.5,60.2)[][]{$W^+$}
\rText(105.5,9.2)[][]{$W^$}
\Text(55,16)[c]{(c)}
\end{picture}
%
\caption{Contributions to the annihilation crosssection for neutralino
dark matter LSPs from (a) $t$channel slepton and squark exchange,
(b) nearresonant annihilation through a Higgs boson
($s$wave for $A^0$, and $p$wave for $h^0$, $H^0$),
and (c) $t$channel chargino exchange.
\label{fig:darkmatterannihilation}}
\end{figure}
\begin{figure}[!t]
\begin{center}
\begin{picture}(120,50)(0,17)
\SetWidth{0.85}
\Line(0,50)(33,25)
\Line(77,25)(110,50)
\Photon(33,25)(77,25){2.1}{4.5}
\Line(0,0)(33,25)
\Line(77,25)(110,0)
\rText(9,10.2)[][]{$\stilde N_2$}
\rText(8,58.5)[][]{$\stilde N_1$}
\rText(51.9,34)[][]{$Z$}
\rText(109.5,56)[][]{$f$}
\rText(112,8)[][]{$\overline{f}$}
\end{picture}
%
\hspace{1.5cm}
%
\begin{picture}(120,50)(0,17)
\SetWidth{0.85}
\Line(0,50)(33,25)
\Line(77,25)(110,50)
\Photon(33,25)(77,25){2.1}{4.5}
\Line(0,0)(33,25)
\Line(77,25)(110,0)
\rText(8.3,10)[][]{$\stilde C_1$}
\rText(8.3,58.9)[][]{$\stilde N_1$}
\rText(51.9,34)[][]{$W$}
\rText(109.5,56)[][]{$f$}
\rText(112,8)[][]{$\overline{f'}$}
\end{picture}
%
\hspace{1.5cm}
%
\begin{picture}(120,50)(0,17)
\SetWidth{0.85}
\Line(0,50)(33,25)
\Photon(77,25)(110,50){2.1}{4.5}
\Photon(33,25)(77,25){2.1}{4.5}
\Line(0,0)(33,25)
\Photon(77,25)(110,0){2.1}{4.5}
\rText(8.3,10)[][]{$\stilde C_1$}
\rText(8.3,58.9)[][]{$\stilde N_1$}
\rText(51.9,34)[][]{$W$}
\rText(109.5,56)[][]{$W$}
\rText(112,8)[][]{$\gamma, Z$}
\end{picture}
\end{center}
\caption{Some contributions to the coannihilation of dark matter
$\stilde N_1$ LSPs with slightly heavier $\stilde N_2$ and $\stilde C_1$.
All three diagrams are particularly important if the LSP is higgsinolike,
and the last two diagrams are important if the LSP is winolike.
\label{fig:coannihilation}}
\end{figure}
\begin{figure}[!t]
\begin{center}
\begin{picture}(120,50)(0,17)
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\rText(8,9)[][]{$\stilde f$}
\rText(6,58.5)[][]{$\stilde N_1$}
\rText(51.5,34)[][]{$f$}
\rText(109.5,56)[][]{$f$}
\rText(112,8)[][]{$\gamma, Z$}
\end{picture}
%
\hspace{1.5cm}
%
\begin{picture}(120,50)(0,16)
\SetWidth{0.85}
\Line(0,50)(110,50)
\DashLine(0,0)(55,0){5}
\DashLine(55,0)(55,50){5}
\Photon(55,0)(110,0){2}{5}
\rText(2,9)[][]{$\stilde f$}
\rText(0,60)[][]{$\stilde N_1$}
\rText(55.7,27)[][]{$\stilde f$}
\rText(105,59)[][]{$ f$}
\rText(107,9)[][]{$\gamma, Z$}
\end{picture}
%
\hspace{1.5cm}
%
\begin{picture}(120,50)(0,17)
\SetWidth{0.85}
\DashLine(0,50)(55,50){5}
\Line(55,50)(110,50)
\DashLine(0,0)(55,0){5}
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\Line(55,0)(110,0)
\rText(0,9)[][]{$\stilde f$}
\rText(0,60)[][]{$\stilde f$}
\rText(57.5,27)[][]{$\stilde N_i$}
\rText(105,59)[][]{$f$}
\rText(105,8)[][]{$f$}
\end{picture}
\end{center}
\caption{Some contributions to the coannihilation of dark matter
$\stilde N_1$ LSPs with slightly heavier sfermions, which in popular
models are most plausibly staus (or perhaps top
squarks).\label{fig:sfermioncoannihilation}}
\end{figure}
If $\stilde N_1$ is mostly higgsino or mostly wino, then the the
annihilation diagram fig.~\ref{fig:darkmatterannihilation}c and the
coannihilation mechanisms provided by fig.~\ref{fig:coannihilation} are
typically much too efficient \cite{Mizuta:1992qp,EdsjoGondolo,DNRY} to
allow the full required cold dark matter density, unless the LSP is very
heavy, of order 1 TeV or more. This is often considered to be somewhat at
odds with the idea that supersymmetry is the solution to the hierarchy
problem; on the other hand, it is consistent with
the lower bounds set on sparticle masses by the LHC.
However, for lighter higgsinolike or winolike LSPs, nonthermal
mechanisms can be invoked to provide the right dark matter abundance
\cite{AMSBphenotwo,winoDM}.
A recurring feature of many models of supersymmetry breaking is that the
lightest neutralino is mostly bino. It turns out that in much of the
parameter space not already ruled out by LEP with a binolike $\stilde
N_1$, the predicted relic density is too high, either because the LSP
couplings are too small, or the sparticles are too heavy, or both, leading
to an annihilation crosssection that is too low. To avoid this, there
must be significant contributions to $\langle \sigma v \rangle$. The
possibilities can be classified qualitatively in terms of the diagrams
that contribute most strongly to the annihilation.
First, if at least one sfermion is not too heavy, the diagram of
fig.~\ref{fig:darkmatterannihilation}a is effective in reducing the dark
matter density. In models with a binolike $\stilde N_1$, the most
important such contribution usually comes from $\stilde e_R$, $\stilde
\mu_R$, and $\stilde \tau_1$ slepton exchange. The region of parameter
space where this works out right is often referred to by the jargon ``bulk
region", because it corresponded to the main allowed region with dark
matter density less than the critical density, before $\Omega_{\rm DM}
h^2$ was accurately known and before the highest energy LEP searches had
happened. However, the diagram of fig.~\ref{fig:darkmatterannihilation}a
is subject to a $p$wave suppression, and so sleptons that are light
enough to reduce the relic density sufficiently are, in many models, also
light enough to be excluded by LEP or LHC searches,
or have difficulties with other indirect constraints.
In the MSUGRA framework described in
section \ref{subsec:origins.sugra}, the viable bulk region
takes $m_0$ and $m_{1/2}$ less than about 100 GeV and 250 GeV
respectively, depending on other parameters. Within MSUGRA, this part of
parameter space has now been thoroughly excluded by the LHC.
%
If the final state of neutralino pair annihilation
is instead $t \overline t$, then there is no $p$wave
suppression. This typically requires a top squark that is less than
about 150 GeV heavier than the LSP, which in turn has $m_{\tilde N_1}$
between about
$m_t$ and $m_t + 100$ GeV. This situation
does not occur in the MSUGRA framework,
but can be natural if the ratio of gluino and wino mass parameters,
$M_3/M_2$, is
smaller than the unification prediction of eq.~(\ref{eq:TiaEla}) by a
factor of a few \cite{Compressed}.
A second way of annihilating excess binolike LSPs to the correct density
is obtained if $2 m_{\tilde N_1} \approx m_{A^0}$, or $m_{h^0}$, or
$m_{H^0}$, as shown in fig.~\ref{fig:darkmatterannihilation}b, so that the
crosssection is near a resonance pole. An $A^0$ resonance annihilation
will be $s$wave, and so more efficient than a $p$wave $h^0$ or $H^0$
resonance. Therefore, the most commonly found realization involves
annihilation through $A^0$. Because the $A^0 b \overline b$ coupling is
proportional to $m_b \tan\beta$, this usually entails large values of
$\tan\beta$ \cite{DNDM}. (Annihilation through $h^0$ is also possible
\cite{hfunnel}, if the LSP mass is close to $m_{h^0}/2 = 62.5$ GeV.)
The region of parameter space where this happens is often
called the ``$A$funnel" or ``Higgs funnel" or ``Higgs resonance region".
A third effective annihilation mechanism is obtained if $\stilde N_1$
mixes to obtains a significant higgsino or wino admixture. Then both
fig.~\ref{fig:darkmatterannihilation}c and the coannihilation diagrams of
fig.~\ref{fig:coannihilation} can be important
\cite{EdsjoGondolo}. In the ``focus point" region of parameter space,
where $\mu$ is not too large, an LSP with a significant higgsino content
can yield the correct relic abundance even for very heavy squarks and
sleptons \cite{focuspointDM}. This is motivated by focusing properties of
the renormalization group equations, which allow $\mu^2 \ll m_0^2$ in
MSUGRA models \cite{hyperbolic,focuspoint}. In fact, within MSUGRA, squarks are required to be
very heavy, typically several TeV. This possibility is attractive, given the
LHC results that exclude most models with squarks lighter than 1 TeV.
It is also possible to arrange for just enough wino content in the LSP to do the
job \cite{winocontentDM}, by choosing $M_1/M_2$ appropriately.
A fourth possibility, the ``sfermion coannihilation region" of parameter
space, is obtained if there is a sfermion that happens to be less than a
few GeV heavier than the LSP \cite{GriestSeckel}. In many model
frameworks, this is most naturally the lightest stau
\cite{staucoannihilation}, but it could also be the lightest top squark
\cite{stopcoannihilation}. A significant density of this sfermion will
then coexist with the LSP around the freezeout time, and so annihilations
involving the sfermion with itself or with the LSP, including those of the
type shown in fig.~\ref{fig:sfermioncoannihilation}, will further dilute
the number of sparticles and so the eventual dark matter density.
It is important to keep in mind that a set of MSSM Lagrangian
parameters that ``fails" to predict the correct relic dark matter abundance
by the standard thermal mechanisms is {\em not} ruled out as a model for
collider physics. This is because simple extensions can completely change
the dark matter relic abundance prediction without changing the predictions for
colliders much or at all. For example, if the model predicts a neutralino
dark matter abundance that is too small, one need only assume another
sector (even a completely disconnected one) with a stable neutral
particle, or that the dark matter is supplied by some nonthermal
mechanism such as outofequilibrium decays of heavy particles. If the
predicted neutralino dark matter abundance appears to be too large, one
can assume that $R$parity is slightly broken, so that the offending LSP
decays before nucleosynthesis; this would require some other unspecified
dark matter candidate. Or, the dark matter LSP might be some particle that
the lightest neutralino decays into. One possibility is a gravitino LSP
\cite{gravitinoDMfromdecays}. Another example is obtained by extending
the model to solve the strong CP problem with an invisible axion, which
can allow the LSP to be a very weaklyinteracting axino \cite{axinoDM}
(the fermionic supersymmetric partner of the axion). In such cases, the
dark matter density after the lightest neutralino decays would be reduced
compared to its naively predicted value by a factor of $m_{\rm
LSP}/m_{\tilde N_1}$, provided that other sources for the LSP relic
density are absent. A correct density for neutralino LSPs can also be
obtained by assuming that they are produced nonthermally in reheating of
the universe after neutralino freezeout but before nucleosynthesis
\cite{gelgon}. Finally, in the absence of a compelling explanation for the
apparent cosmological constant, it seems possible that the standard model
of cosmology will still need to be modified in ways not yet imagined.
If neutralino LSPs really make up the cold dark matter, then their local
mass density in our neighborhood ought to be of order 0.3 GeV/cm$^3$ [much
larger than the density averaged over the largest scales,
eq.~(\ref{eq:rhodm})] in order to explain the dynamics of our own galaxy.
LSP neutralinos could be detectable directly through their weak
interactions with ordinary matter, or indirectly by their ongoing
annihilations. However, the dark matter halo is subject to significant uncertainties
in density, velocity, and clumpiness, so even if the Lagrangian
parameters were known exactly, the signal rates would be quite indefinite,
possibly even by orders of magnitude.
The direct detection of $\NI$ depends on their elastic scattering off of
heavy nuclei in a detector. At the parton level, $\NI$ can interact with a
quark by virtual exchange of squarks in the $s$channel, or Higgs scalars
or a $Z$ boson in the $t$channel. It can also scatter off of gluons
through oneloop diagrams. The scattering mediated by neutral Higgs
scalars is suppressed by tiny Yukawa couplings, but is coherent for the
quarks and so can actually be the dominant contribution for nuclei with
larger atomic weights, if the squarks are heavy. The energy transferred to
the nucleus in these elastic collisions is typically of order tens of keV
per event. There are important backgrounds from natural radioactivity and
cosmic rays, which can be reduced by shielding and pulseshape analysis. A
wide variety of current or future experiments are sensitive to some, but
not all, of the parameter space of the MSSM that predicts a dark matter
abundance in the range of eq.~(\ref{eq:OmegaDM}).
Another, more indirect, way to detect neutralino LSPs is through ongoing
annihilations. This can occur in regions of space where the density is
greatly enhanced. If the LSPs lose energy by repeated elastic scattering
with ordinary matter, they can eventually become concentrated inside
massive astronomical bodies like the Sun or the Earth. In that case, the
annihilation of neutralino pairs into final states leading to neutrinos is
the most important process, since no other particles can escape from the
center of the object where the annihilation is going on. In particular,
muon neutrinos and antineutrinos from $\NI\NI \rightarrow W^+ W^$ or
$ZZ$, (or possibly $\NI\NI \rightarrow \tau^+ \tau^$ or
$\nu\overline\nu$, although these are $p$wave suppressed) will travel
large distances, and can be detected in neutrino telescopes. The neutrinos
undergo a chargedcurrent weak interaction in the earth, water, or ice
under or within the detector, leading to energetic upwardgoing muons
pointing back to the center of the Sun or Earth.
Another possibility is that neutralino LSP annihilation in the galactic
center (or the halo) could result in highenergy photons from cascade
decays of the heavy Standard Model particles that are produced. These
photons could be detected in air Cerenkov telescopes or in spacebased
detectors. There are also interesting possible signatures from neutralino
LSP annihilation in the galactic halo producing detectable quantities of
highenergy positrons or antiprotons.
More information on these possibilities, and the various experiments that
can exploit them, can be found from refs.~\cite{darkmatterreviews} and
papers referred to in them.
\section{Beyond minimal supersymmetry}\label{sec:variations}
\setcounter{equation}{0}
\setcounter{figure}{0}
\setcounter{table}{0}
\setcounter{footnote}{1}
In this section I will briefly outline a few of my favorite variations
on the basic picture of the MSSM discussed above. First, the possibility
of $R$parity violation is considered in section
\ref{subsec:variations.RPV}. Another obvious way to extend the MSSM is
to introduce new chiral supermultiplets, corresponding to scalars and
fermions that are all sufficiently heavy to have avoided discovery so
far. This requires that the new chiral supermultiplets must form a real
representation of the Standard Model gauge group; they can then have a
significant positive effect on the Higgs boson mass through loop
corrections, as described in section \ref{subsec:variations.vectorlike}.
However, the simplest possibility for adding particles is to put them in
just one gaugesinglet chiral supermultiplet; this can raise the Higgs
boson mass at tree level, as discussed in section
\ref{subsec:variations.NMSSM}. The resulting model is also attractive
because it can solve the $\mu$ problem that was described in sections
\ref{subsec:mssm.superpotential} and \ref{subsec:MSSMspectrum.Higgs}.
Two other solutions to the $\mu$ problem, based on including
nonrenormalizable superpotential terms or K\"ahler potential terms, are
discussed in section \ref{subsec:variations.munonrenorm}. The MSSM could
also be extended by introducing new gauge interactions that are
spontaneously broken at high energies. The possibilities here include
GUT models like $SU(5)$ or $SO(10)$ or $E_6$, which unify the Standard
Model gauge interactions, with important implications for rare processes
like proton decay and $\mu \rightarrow e \gamma$. Superstring models
also usually enlarge the Standard Model gauge group at high energies.
One or more Abelian subgroups could survive to the TeV scale, leading to
a $Z'$ massive vector boson. There is a vast literature on these
possibilities, but I will concentrate instead on the implications of
just adding a single $U(1)$ factor that is assumed to be spontaneously
broken at energies beyond the reach of any foreseeable collider. As
described in section \ref{subsec:variations.Dterms}, the broken gauge
symmetry can still leave an imprint on the soft supersymmetrybreaking
Lagrangian at low energies.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$R$parity violation}\label{subsec:variations.RPV}
\setcounter{equation}{0}
In the preceding, it has been assumed that $R$parity (or equivalently matter parity) is
an exact symmetry of the MSSM. This assumption precludes renormalizable
proton decay and predicts that the LSP should be stable, but despite these
virtues $R$parity is not inevitable. Because of the threat of proton
decay, one expects that if $R$parity is violated, then in the
renormalizable Lagrangian either Bviolating or Lviolating couplings are
allowed, but not both, as explained in section \ref{subsec:mssm.rparity}.
There are also upper bounds on the individual $R$parity violating
couplings \cite{RPVreviews}.
One proposal is that matter parity can be replaced by an alternative
discrete symmetry that still manages to forbid proton decay at the level
of the renormalizable Lagrangian. The $Z_2$ and $Z_3$ possibilities have
been cataloged in ref.~\cite{baryontriality}, where it was found that
provided no new particles are added to the MSSM, that the discrete
symmetry is familyindependent, and that it can be defined at the level of
the superpotential, there is only one other candidate besides matter
parity. That other possibility is a $Z_3$ discrete symmetry
\cite{baryontriality}, which was originally called ``baryon parity" but is
more appropriately referred to as ``baryon triality". The baryon triality
of any particle with baryon number B and weak hypercharge $Y$ is defined
to be
\beq
Z_3^{\rm B} = {\rm exp}\left ({2\pi i} [{\rm B}2Y]/3 \right ).
\eeq
This is always a cube root of unity, since B$2Y$ is an integer for every
MSSM particle. The symmetry principle to be enforced is that the product
of the baryon trialities of the particles in any term in the Lagrangian
(or superpotential) must be 1. This symmetry conserves baryon number at
the renormalizable level while allowing lepton number violation; in other
words, it allows the superpotential terms in eq.~(\ref{WLviol}) but
forbids those in eq.~(\ref{WBviol}). In fact, baryon triality conservation
has the remarkable property that it absolutely forbids proton
decay \cite{noprotondecay}. The reason for this is simply that baryon
triality requires that B can only be violated in multiples of 3 units
(even in nonrenormalizable interactions), while any kind of proton decay
would have to violate B by 1 unit. So it is eminently falsifiable.
Similarly, baryon triality conservation predicts that experimental
searches for neutronantineutron oscillations will be negative, since they
would violate baryon number by 2 units. However, baryon triality
conservation does allow the LSP to decay. If one adds some new chiral
supermultiplets to the MSSM (corresponding to particles that are
presumably very heavy), one can concoct a variety of new candidate
discrete symmetries besides matter parity and baryon triality. Some of
these will allow B violation in the superpotential, while forbidding the
lepton number violating superpotential terms in eq.~(\ref{WLviol}).
Another idea is that matter parity is an exact symmetry of the underlying
superpotential, but it is spontaneously broken by the VEV of a scalar with
$P_R=1$. One possibility is that an MSSM sneutrino gets a VEV
\cite{sneutvevRPV}, since sneutrinos are scalars carrying L=1. However,
there are strong bounds \cite{nonsneutvevRPV} on $SU(2)_L$doublet
sneutrino VEVs $\langle \stilde \nu \rangle \ll m_Z$ coming from the
requirement that the corresponding neutrinos do not have large masses. It
is somewhat difficult to understand why such a small VEV should occur,
since the scalar potential that produces it must include soft sneutrino
squaredmass terms of order $m^2_{\rm soft}$. One can get around this by
instead introducing a new gaugesinglet chiral supermultiplet with L=$1$.
The scalar component can get a large VEV, which can induce Lviolating
terms (and in general Bviolating terms also) in the lowenergy effective
superpotential of the MSSM \cite{nonsneutvevRPV}.
In any case, if $R$parity is violated, then the collider searches for
supersymmetry can be completely altered. The new couplings imply
singlesparticle production mechanisms at colliders, besides the usual
sparticle pair production processes. First, one can have $s$channel
single sfermion production. At electronpositron colliders, the $\lambda$
couplings in eq.~(\ref{WLviol}) give rise to $e^+e^\rightarrow \stilde
\nu$. At the LHC, single sneutrino or charged slepton
production, $q \bar q \rightarrow \stilde \nu$ or $\stilde \ell$ are
mediated by $\lambda'$ couplings, and single squark production $qq
\rightarrow \stilde {\bar q}$ is mediated by $\lambda''$ couplings in
eq.~(\ref{WBviol}).
Second, one can have $t$channel exchange of sfermions, providing for
gaugino production in association with a standard model fermion. At
electronpositron colliders, one has $e^+ e^ \rightarrow \stilde C_i
\ell$ mediated by $\stilde \nu_e$ in the $t$channel, and $e^+ e^
\rightarrow \stilde N_i \nu$ mediated by selectrons in the $t$channel, if
the appropriate $\lambda$ couplings are present. At the
LHC, one can look for the partonic processes $q\overline q \rightarrow
(\stilde N_i\>{\rm or}\>\stilde C_i\>{\rm or}\>\tilde g)+(\ell\>{\rm
or}\>\nu)$, mediated by $t$channel squark exchange if $\lambda'$
couplings are present. If instead $\lambda''$ couplings are present, then
$qq \rightarrow (\stilde N_i\>{\rm or}\>\stilde C_i\>{\rm or}\>\tilde g)+
q$, again with squarks exchanged in the $t$channel, provides a possible
production mechanism.
Next consider sparticle decays. In many cases, the $R$parity violating
couplings are already constrained by experiment, or expected from more
particular theoretical models, to be smaller than electroweak gauge
couplings \cite{RPVreviews}. If so, then the heavier sparticles will
usually decay to final states containing the LSP, as in section
\ref{sec:decays}. However, now the LSP can also decay; if it is a
neutralino, as most often assumed, then it will decay into three Standard
Model fermions. The collider signals to be found depend on the type of
$R$parity violation.
Lepton number violating terms of the type $\lambda$ as in
eq.~(\ref{WLviol}) will lead to final states from $\stilde N_1$ decay with
two oppositely charged, and possibly different flavor, leptons and a
neutrino, as in Figure~\ref{fig:rparityviolation}a,b.
\begin{figure}
\begin{center}
\begin{picture}(135,61)(0,18)
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\Text(7,10)[c]{$\stilde N_1$}
\Text(70,9)[c]{$\stilde \ell$}
\Text(73,41)[c]{$\ell$}
\Text(118.5,41)[c]{$\ell'$}
\Text(135,7)[c]{$\nu''$}
\Text(90.5,6.5)[c]{$\lambda$}
\Text(67.5,18.3)[c]{(a)}
\end{picture}
%
\hspace{1.15cm}
%
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\Text(7,10)[c]{$\stilde N_1$}
\Text(70,9)[c]{$\stilde \nu''$}
\Text(75,42)[c]{$\nu''$}
\Text(118,41)[c]{$\ell$}
\Text(135,7)[c]{$\ell'$}
\Text(90.5,6.5)[c]{$\lambda$}
\Text(67.5,18.3)[c]{(b)}
\end{picture}
%
\hspace{1.15cm}
%
\begin{picture}(135,61)(0,18)
\SetWidth{0.85}
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\Line(90,0)(135,0)
\Text(7,10)[c]{$\stilde N_1$}
\Text(70,9)[c]{$\stilde \ell$}
\Text(73,41)[c]{$\ell$}
\Text(118,41)[c]{$q$}
\Text(133,8)[c]{$q'$}
\Text(90.5,6.5)[c]{$\lambda'$}
\Text(67.5,18.3)[c]{(c)}
\end{picture}
\vspace{1cm}
%
\begin{picture}(135,51)(0,8)
\SetWidth{0.85}
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\Line(45,0)(67.5,40.5)
\DashLine(45,0)(90,0){4.5}
\Line(90,0)(112.5,40.5)
\Line(90,0)(135,0)
\Text(7,10)[c]{$\stilde N_1$}
\Text(70,8.2)[c]{$\stilde \nu$}
\Text(73,41)[c]{$\nu$}
\Text(118,41)[c]{$q$}
\Text(135,9)[c]{$q'$}
\Text(90.5,6.5)[c]{$\lambda'$}
\Text(67.5,18.3)[c]{(d)}
\end{picture}
%
\hspace{1.15cm}
%
\begin{picture}(135,51)(0,8)
\SetWidth{0.85}
\Line(0,0)(45,0)
\Line(45,0)(67.5,40.5)
\DashLine(45,0)(90,0){4.5}
\Line(90,0)(112.5,40.5)
\Line(90,0)(135,0)
\Text(7,10)[c]{$\stilde N_1$}
\Text(70,9)[c]{$\stilde q$}
\Text(73,41)[c]{$q$}
\Text(117,41)[l]{$\ell$ or $\nu$}
\Text(135,9)[c]{$q'$}
\Text(90.5,6.5)[c]{$\lambda'$}
\Text(67.5,18.3)[c]{(e)}
\end{picture}
\hspace{1.15cm}
%
\begin{picture}(135,51)(0,8)
\SetWidth{0.85}
\Line(0,0)(45,0)
\Line(45,0)(67.5,40.5)
\DashLine(45,0)(90,0){4.5}
\Line(90,0)(112.5,40.5)
\Line(90,0)(135,0)
\Text(7,10)[c]{$\stilde N_1$}
\Text(70,9)[c]{$\stilde q$}
\Text(73,41)[c]{$q$}
\Text(119.5,41)[c]{$q'$}
\Text(133,9)[c]{$q''$}
\Text(90.5,6.5)[c]{$\lambda''$}
\Text(67.5,18.3)[c]{(f)}
\end{picture}
\end{center}
\caption{Decays of the $\NI$ LSP in models with $R$parity violation,
with lepton number not conserved (a)(e) [see eq.~(\ref{WLviol})], and
baryon number not conserved (f) [see eq.~(\ref{WBviol})].
\label{fig:rparityviolation}}
\end{figure}
Couplings of the $\lambda'$ type will cause $\stilde N_1$ to decay to a
pair of jets and either a charged lepton or a neutrino, as shown in
Figure~\ref{fig:rparityviolation}c,d,e. Signals with Lviolating LSP
decays will therefore always include charged leptons or large missing
energy, or both.
On the other hand, if terms of the form $\lambda^{\prime\prime}$ in
eq.~(\ref{WBviol}) are present instead, then there are Bviolating decays
$\stilde N_1 \rightarrow q q^\prime q^{\prime\prime}$ from diagrams like
the one shown in Figure~\ref{fig:rparityviolation}f. In that case,
supersymmetric events will always have lots of hadronic activity, and will
only have physics missing energy signatures when the other parts of the
decay chains happen to include neutrinos.
There are other possibilities, too. The decaying LSP need not be $\NI$.
Sparticles that are not the LSP can, in principle, decay directly to
Standard Models quarks and leptons, if the $R$parity violating couplings
are large enough. The $t$channel exchange of sfermions can produce a pair
of Standard Model fermions, leading to indirect sparticle signatures.
Or, if the $R$parity violating couplings are sufficiently small, then the
LSP will usually decay outside of collider detectors, and the model will
be difficult or impossible to distinguish from the $R$parity conserving
case. Surveys of experimental constraints and future
prospects can be found in \cite{RPVreviews}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Extra vectorlike chiral
supermultiplets}\label{subsec:variations.vectorlike}
\setcounter{equation}{0}
An interesting way to extend the MSSM is by adding extra particles
in chiral supermultiplets.
It has now become clear that together
the new fields must form a vectorlike (selfconjugate) representation
of the Standard Model gauge group. Otherwise, the only way the new
fermions could have masses large enough to have avoided discovery would be through
extremely large Yukawa couplings to the Higgs VEVs.
These couplings would in turn lead to very large
corrections to the 125 GeV Higgs boson production crosssection at the LHC
through loop effects, as well as corrections to electroweak precision observables,
both in contradiction with the observations. In contrast, the addition of
chiral supermultiplets with vectorlike quantum numbers to the MSSM does not lead to such problems, and can help to raise the lightest Higgs boson mass up to 125 GeV
in models where it would otherwise be too light
%\cite{Moroi:1991mg,Babu:2004xg,Babu:2008ge,Martin:2009bg,Graham:2009gy}.
\cite{Moroi:1991mg}\cite{Graham:2009gy}.
If the new vectorlike chiral supermultiplets live in the fundamental representation of
$SU(2)_L$ or $SU(3)_c$, or are charged under $U(1)_Y$, then they must come in pairs with opposite gauge quantum numbers.
If we call such a pair
$\Phi_i$ and $\overline \Phi_i$, then there is an allowed superpotential
mass term of the form
\beq
W = M_i \Phi_i \overline \Phi_i ,
\label{eq:Wvectorlikemass}
\eeq
which does not involve any interactions with the Higgs boson. Note that
such electroweak singlet mass terms
can arise from whatever mechanism also gives rise to the $\mu$ term of the MSSM. Three
such possible mechanisms are described below in sections \ref{subsec:variations.NMSSM} and
\ref{subsec:variations.munonrenorm}. Whatever that mechanism is, it is
reasonable to suppose that it operates the same way to produce the masses $M_i$
with the same order of magnitude as $\mu$, i.e.~at the TeV scale.
Because the new vectorlike particle have mostly electroweak singlet
masses, they do not impact Higgs boson production and decay, and
decouple from precision electroweak observables involving the $Z$ and
$W$ selfenergies and the Standard Model fermions. In order for the
lightest of the new particles to not cause problems as stable relics
from thermal production in the early universe, one may suppose that
either $\Phi_i$ or $\overline \Phi_i$ has the same gauge quantum numbers
as one of the MSSM quark and lepton chiral superfields, allowing small
mixing Yukawa couplings to the Higgs boson. This small mixing allows the
new vectorlike fermions to decay to Standard Model fermions.
If they are indeed at the TeV scale, the new particles can be
pairproduced at the LHC, either through gluon fusion or through
$s$channel $W$ or $Z$ boson diagrams. Thus one can look for heavy
cousins of the top quark, bottom quark, and/or tau lepton; call them
$t'$, $b'$, and $\tau'$. These fermions will have decays that depend on
the choice of mixing terms between them and the Standard Model fermions.
The easiest way to minimize possible flavor problems in lowenergy
experiments is to assume that the mixing is primarily with the third
family. Then the relevant decays will be:
\beq
&&t' \rightarrow Zt,\>\>\>h^0 t,\>\>\> W^+b,
\\
&&b' \rightarrow Zb,\>\>\>h^0 b,\>\>\> W^t,
\\
&&\tau' \rightarrow Z\tau,\>\>\>h^0 \tau,\>\>\> W^\nu,
\eeq
with branching ratios that depend on the type of mixing Yukawa coupling.
The possibilities and the resulting branching ratio predictions
are discussed in detail in \cite{Martin:2009bg}.
If the Yukawa couplings that mix the
new fermions to the Standard Model fermions are larger than about $10^{6}$, these
2body decays will occur promptly within collider detectors. The scalar partners
of these fermionic states are likely to be much heavier, because they have soft
supersymmetrybreaking contributions to their masses. In addition, for a given
mass, production
crosssections for scalars tend to be lower than for fermions, so it is
most likely that the new vectorlike fermions will be discovered first.
In order to raise the Higgs boson mass, one can also introduce a Yukawa
coupling between the new chiral supermultiplets
and the MSSM Higgs fields. As an example, suppose
there are extra vectorlike chiral supermultiplets in the following
representations of $SU(3)_c \times SU(2)_L \times U(1)_Y$:
\beq
{\cal Q} &=& ({\bf 3}, {\bf 2}, +1/6),\qquad\quad
\overline{\cal Q} \>=\> ({\bf \overline{3}}, {\bf 2}, 1/6),
\\
{\cal U} &=& ({\bf 3}, {\bf 1}, +2/3),\qquad\quad
\overline{\cal U} \>=\> ({\bf \overline{3}}, {\bf 1}, 2/3).
\eeq
Then the allowed superpotential terms include:
\beq
W &=& M_{\cal Q} {\cal Q}\overline{\cal Q} +
M_{\cal U} {\cal U}\overline{\cal U}
+ k H_u {\cal Q}\overline{\cal U}
\eeq
where $M_{\cal Q}$ and $M_{\cal U}$ are electroweak singlet masses
as in eq.~(\ref{eq:Wvectorlikemass}), and $k$ is a Yukawa
coupling, which can be large and yet provide only a subdominant contribution to the
masses of the vectorlike states. There is an infraredstable
quasifixed point at $k\approx 1.05$, giving a natural expectation for its magnitude
\cite{Martin:2009bg}.
This coupling mediates a positive 1loop contribution to
lightest Higgs scalar boson mass, provided that
the masses of the new scalars are larger than the masses of the new fermions.
(This is similar to the 1loop contribution from the top/stop sector.)
An approximate formula for this contribution,
with several simplifying assumptions, is \cite{Babu:2008ge}:
\beq
\Delta (m_{h^0}^2) = \frac{3}{4 \pi^2} k^4 v^2 \sin^4\beta \left [
\ln(x)  \frac{1}{6} (51/x)(11/x) \right ]
\label{eq:deltamhvectorlike}
\eeq
Here $x = M_S^2/M_F^2$, and it is assumed
that the scalars in ${\cal Q}, \overline{\cal Q},
{\cal U}$, and $\overline{\cal U}$ are approximately degenerate with
each other with average mass $M_S$, and likewise for the new fermions with
average mass $M_F \approx M_{\cal Q} \approx M_{\cal U} $, that
$k v_u$ is a small perturbation on these masses,
and that the mixing in the new scalar sector is small.
It is also assumed that the Higgs bosons are in
the decoupling limit described at the end of section
\ref{subsec:MSSMspectrum.Higgs}. For $x>1$,
eq.~(\ref{eq:deltamhvectorlike}) is positive definite
and monotonically increasing with $x$. For example, with $x=4$, the correction
to the Higgs boson mass can be about 10 GeV. (Results for the Higgs mass
correction with these assumptions relaxed can be found in \cite{Martin:2009bg}; mixing in the scalar sector increases the Higgs mass correction.)
Note that even in the limit of very large $M_F$, the contribution to $m_{h^0}^2$
does not decouple, provided only that the hierarchy $x>1$ is maintained. Despite
this nondecoupling contribution to $m_{h^0}^2$, the contributions to
precision electroweak observables do decouple quadratically (like $m_W^2/M_F^2$), and so are quite benign \cite{Martin:2009bg}.
The positive contribution to the Higgs mass from extra vectorlike quarks
is a plausible way to rescue supersymmetric theories that
would otherwise have difficulty in accommodating the 125 GeV Higgs boson. For example,
GMSB models typically predict much lower $m_{h^0}$, unless all of the superpartners are
well out of reach of the LHC, because they imply small topsquark mixing.
However, including extra vectorlike quarks with a large Yukawa coupling
allows the MSSM superpartners to be as light as their direct experimental
limits in GMSB models, while still allowing
$m_{h^0} = 125$ GeV
%\cite{Endo:2011mc,Evans:2011uq,Martin:2012dg}.
\cite{Endo:2011mc}\cite{Martin:2012dg}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The nexttominimal supersymmetric standard
model}\label{subsec:variations.NMSSM}
\setcounter{equation}{0}
The simplest possible extension of the particle content of the MSSM is
obtained by adding a new gaugesinglet chiral supermultiplet that is
even under matter parity. The resulting model
%\cite{NMSSM,NMSSMpheno,NMSSMdarkmatter,NMSSMdomainwalls,nMSSM}
\cite{NMSSM}\cite{nMSSM}
is often
called the nexttominimal supersymmetric standard model or NMSSM or
(M+1)SSM. The most general renormalizable superpotential for this field
content is
\beq
W_{\rm NMSSM} \>=\> W_{\rm MSSM}
+ \lambda S H_u H_d + \frac{1}{3} \kappa S^3 + \frac{1}{2} \mu_S S^2 ,
\label{NMSSMwww}
\eeq
where $S$ stands for both the new chiral supermultiplet and its scalar
component. There could also be a term linear in $S$ in $W_{\rm NMSSM}$,
but in global supersymmetry it can always be removed by redefining $S$ by
a constant shift. The soft supersymmetrybreaking Lagrangian is
\beq
{\cal L}_{\rm soft}^{\rm NMSSM} \>=\> {\cal L}_{\rm soft}^{\rm MSSM}
\bigl (
a_{\lambda} S H_u H_d  {1\over 3} a_\kappa S^3 + {1\over 2} b_S S^2
+ t S
+ \conj \bigr ) m_S^2 S^2
.
\label{eq:NMSSMsoft}
\eeq
The tadpole coupling $t$ could be subject to dangerous quadratic
divergences in supergravity \cite{NMSSMtadpole} unless it is highly
suppressed or forbidden by some additional symmetry at very high energies.
One of the virtues of the NMSSM is that it can provide a solution to the
$\mu$ problem mentioned in sections \ref{subsec:mssm.superpotential} and
\ref{subsec:MSSMspectrum.Higgs}. To understand this, suppose we
set\footnote{The even more economical case with only $t \sim m_{\rm
soft}^3$ and $\lambda$ and $a_\lambda$ nonzero is also viable and
interesting \cite{nMSSM}.} $\mu_S = \mu = 0$ so that there are no mass
terms or dimensionful parameters in the superpotential at all, and also
set the corresponding terms $b_S = b = 0$ and $t=0$ in the
supersymmetrybreaking Lagrangian. If
$\lambda$, $\kappa$, $a_\lambda$, and $a_{\kappa}$ are chosen
auspiciously, then phenomenologically acceptable VEVs will be induced for
$S$, $H_u^0$, and $H_d^0$. By doing phase rotations on these fields, all
three of $s \equiv \langle S \rangle$ and $v_u = v \sin\beta = \langle
H_u^0 \rangle$ and $v_d = v \cos\beta = \langle H_d^0 \rangle$ can be made
real and positive. In this convention, $a_\lambda + \lambda \kappa^* s$
and $a_{\kappa} + 3 \lambda^* \kappa v_u v_d/s$ will also be real and
positive.
However, in general, this theory could have unacceptably large CP
violation. This can be avoided by assuming that $\lambda$, $\kappa$,
$a_\lambda$ and $a_\kappa$ are all real in the same convention that makes
$s$, $v_u$, and $v_d$ real and positive; this is natural if the mediation
mechanism for supersymmetry breaking does not introduce new CP violating
phases, and is assumed in the following. To have a stable minimum with
respect to variations in the scalar
field phases, it is required that $a_\lambda +
\lambda \kappa s > 0$ and $a_\kappa (a_\lambda + \lambda \kappa s) + 3
\lambda \kappa a_\lambda v_u v_d/s > 0$. (An obvious sufficient, but not
necessary, way to achieve these two conditions is to assume that
$\lambda \kappa > 0$ and $a_\kappa>0$ and $a_\lambda > 0$.)
An effective $\mu$term for $H_u H_d$ will arise from
eq.~(\ref{NMSSMwww}), with \beq \mu_{\rm eff} \>=\> \lambda s. \eeq It is
determined by the dimensionless couplings and the soft terms of order
$m_{\rm soft}$, instead of being a free parameter conceptually independent
of supersymmetry breaking. With the conventions chosen here, the
sign of $\mu_{\rm eff}$ (or more generally its phase) is the same as that
of $\lambda$. Instead of eqs.~(\ref{mubsub2}), (\ref{mubsub1}), the
minimization conditions for the Higgs potential are now:
\beq
m^2_{H_u} + \lambda^2 (s^2 + v^2 \cos^2\beta)
 (a_\lambda + \lambda \kappa s) s \cot\beta
 (m_Z^2/2) \cos(2\beta) &=& 0,
\\
m^2_{H_d} + \lambda^2 (s^2 + v^2 \sin^2\beta)
(a_\lambda + \lambda \kappa s) s \tan\beta
+ (m_Z^2/2) \cos(2\beta) &=& 0,
\\
m^2_{S} + \lambda^2 v^2 + 2 \kappa^2 s^2  a_\kappa s
(\kappa\lambda + a_\lambda/2 s) v^2 \sin(2\beta) &=& 0.
\eeq
The effects of radiative corrections $\Delta V(v_u,v_d,s)$ to the
effective potential are included by replacing $m_S^2 \rightarrow m_S^2 +
[\partial (\Delta V)/\partial s]/2s$, in addition to
eq.~(\ref{eq:Vradcor}).
The absence of dimensionful terms in $W_{\rm NMSSM}$, and the
corresponding terms in $V_{\rm soft}^{\rm NMSSM}$, can be enforced by
introducing a new symmetry. The simplest way is to notice that the new
superpotential and Lagrangian will be invariant under a $Z_3$ discrete
symmetry, under which every field in a chiral supermultiplet transforms as
$\Phi \rightarrow e^{2 \pi i/3} \Phi$, and all gauge and gaugino fields
are inert. Imposing this symmetry indeed eliminates $\mu$, $\mu_S$, $b$,
$b_S$, and $t$. However, if this symmetry were exact, then because it must
be spontaneously broken by the VEVs of $S$, $H_u$ and $H_d$, domain walls
are expected to be produced in the electroweak symmetry breaking phase
transition in the early universe \cite{NMSSMdomainwalls}. These would
dominate the cosmological energy density, and would cause unobserved
anisotropies in the microwave background radiation. Several ways of
avoiding this problem have been proposed, including late inflation after
the domain walls are formed, embedding the discrete symmetry into a
continuous gauged symmetry at very high energies, or allowing either
higherdimensional terms in the Lagrangian or a very small $\mu$ term to
explicitly break the discrete symmetry.
The NMSSM contains, besides the particles of the MSSM, a real $P_R=+1$
scalar, a real $P_R=+1$ pseudoscalar, and a $P_R=1$ Weyl fermion
``singlino". These fields have no gauge couplings of their own, so they
can only interact with Standard Model particles by mixing with the neutral
MSSM fields with the same spin and charge. The real scalar mixes with the
MSSM particles $h^0$ and $H^0$, and the pseudoscalar mixes with $A^0$.
One of the effects of replacing the $\mu$ term by the dynamical field $S$
is that the lightest Higgs boson squared mass is raised, by an amount bounded
at treelevel by:
\beq
\Delta(m_{h^0}^2) &\leq& \lambda^2 v^2 \sin^2 (2 \beta).
\eeq
This extra contribution comes from the $F_S^2$ contribution to
the scalar potential. Its effect is limited, because there is an upper bound
$\lambda \lsim 0.8$ if one
requires that $\lambda$ not have a Landau pole in its RG running below the
GUT mass scale.
Also, the neutral Higgs scalars have reduced couplings to the electroweak
gauge bosons, compared to those in the Standard Model, because of the
mixing with the singlets. Because the 125 GeV Higgs boson discovered by the LHC
appears to have properties like those of a Standard Model Higgs boson, it seems unlikely
to have a large admixture of the single field $S$. This means that there could be
a yetundiscovered neutral Higgs scalar that is mostly electroweak singlet and
even lighter than 125 GeV.
The odd $R$parity singlino $\stilde S$ mixes with the four MSSM
neutralinos, so there are really five neutralinos now. The singlino could
be the LSP, depending on the parameters of the model, and so could be the
dark matter \cite{NMSSMdarkmatter}. The neutralino mass matrix in the
$\psi^0 = (\stilde B, \stilde W^0, \stilde H_d^0, \stilde H_u^0, \stilde S)$
gaugeeigenstate basis is:
\beq
{\bf M}_{\stilde N} \,=\, \pmatrix{
M_1 & 0 & g' v_d/\sqrt{2} & g' v_u/\sqrt{2} & 0\cr
0 & M_2 & g v_d/\sqrt{2} & g v_u/\sqrt{2} & 0\cr
g' v_d/\sqrt{2} & g v_d/\sqrt{2} & 0 & \lambda s & \lambda v_u\cr
g' v_u/\sqrt{2} & g v_u/\sqrt{2}& \lambda s & 0 & \lambda v_d\cr
0 & 0 & \lambda v_u & \lambda v_d & 2 \kappa s}.
\label{NMSSMNinomassmatrix}
\eeq
[Compare eq.~(\ref{preneutralinomassmatrix}).] For small $v/s$ and
$\lambda v/\kappa s$, mixing effects of the singlet Higgs scalar and the singlino are
small, and they nearly decouple. In that case, the phenomenology of the
NMSSM is almost indistinguishable from that of the MSSM. For larger
$\lambda$, the mixing is important and the experimental signals for
sparticles and the Higgs scalars can be altered in important ways
%\cite{NMSSMpheno,NMSSMdarkmatter,NMSSMdomainwalls,nMSSM,NMHDECAY}.
\cite{NMSSMpheno}\cite{nMSSM}, \cite{NMHDECAY}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The $\mu$term from nonrenormalizable
Lagrangian terms}\label{subsec:variations.munonrenorm}
\setcounter{equation}{0}
The previous subsection described how the NMSSM can provide a solution to the
$\mu$ problem. Another possible solution involves generating $\mu$ from
nonrenormalizable Lagrangian terms. If the nonrenormalizable terms are in the
superpotential,
this is called the KimNilles mechanism\cite{KimNilles},
and if they are in the K\"ahler potential
it is called the GiudiceMasiero mechanism\cite{GiudiceMasiero}.
It is useful to note that when the $\mu$ term is set to zero,
the MSSM superpotential
has a global $U(1)$ PecceiQuinn symmetry, with charges listed in Table
\ref{table:PecceiQuinn}.
This symmetry cannot be an exact symmetry of the Lagrangian, since it has
an $SU(3)_c$ anomaly. However, if all other sources of PecceiQuinn breaking
are small, then there must result a pseudoNambuGoldstone boson, the axion.
If the scale of the breaking is too low, then the axion would be ruled
out by astrophysical observations, so one must introduce an additional
explicit breaking of the PecceiQuinn symmetry. This is what happens in the
NMSSM of the previous section.
On the other hand, if the scale of PecceiQuinn breaking is
such that the axion decay constant
is in the range
\beq
10^{9}\>\mbox{GeV} \> \lsim \,f\, \lsim\> 10^{12}\>\mbox{GeV},
\label{eq:axionfrange}
\eeq
then the resulting axion is of the invisible DFSZ type \cite{DFSZ}
that is consistent with present astrophysical constraints. This is an
enticing possibility, since it links the solution to the strong CP problem to
supersymmetry breaking.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\renewcommand{\arraystretch}{1.4}
\begin{table}[tb]
\begin{center}
%
\begin{tabular}{cccccccc}
\hline
& $H_u$ & $H_d$ & $Q$ & $L$ & $\overline u$ & $\overline d$ &
$\overline e$ \\
\hline PecceiQuinn charge & $+1$ & $+1$ & $1$ & $1$ & $0$ & $0$ &
$0$ \\ \hline
\end{tabular}
%
\caption{PecceiQuinn charges of MSSM chiral superfields.
These charges are not unique,
as one can add to them any multiple of the weak hypercharge or
B$$L.\label{table:PecceiQuinn}}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To illustrate the KimNilles mechanism, consider
the nonrenormalizable superpotential
\beq
W = \frac{\lambda_\mu}{2\MPlanck} S^2 H_u H_d,
\eeq
where $S$ is an $SU(3)_c \times SU(2)_L \times U(1)_Y$ singlet chiral
superfield, and
$\lambda_\mu$
is a dimensionless coupling normalized by the reduced Planck mass $\MPlanck$. From Table
\ref{table:PecceiQuinn}, $S$ has PecceiQuinn charge $1$.
If $S$ obtains a VEV that is parametrically of order
\beq
\langle S \rangle \sim
\sqrt{m_{\rm soft} M_P},
\label{eq:KimNillesSVEV}
\eeq
then the spontaneous breaking of the
PecceiQuinn symmetry gives rise to an invisible
axion of the DFSZ type \cite{DFSZ}, with a decay constant
$f \sim \langle S \rangle$ that will
automatically be in the range eq.~(\ref{eq:axionfrange}).
The lowenergy effective theory will then contain the usual $\mu$ term, with
\beq
\mu = \frac{\lambda_\mu}{2 M_P} \langle S^2 \rangle \sim m_{\rm soft},
\eeq
simultaneously solving
the $\mu$ problem and the strong
CP problem. It is natural to also
have a dimensionless, holomorphic soft supersymmetrybreaking term in the
Lagrangian of the form:
\beq
{\cal L}_{\rm soft} = \frac{a_b}{M_P} S^2 H_u H_d + {\rm c.c.},
\eeq
where $a_b$ is of order $m_{\rm soft}$.
The $b$ term in the MSSM will then arise as \beq
b = \frac{a_b}{M_P} \langle S^2
\rangle,
\eeq
and will be of order $m_{\rm soft}^2$, as required for
electroweak symmetry breaking.
To ensure the required spontaneous breaking with a stable vacuum,
one can introduce an additional
nonrenormalizable superpotential term, in several different possible ways
%\cite{MurayamaSuzukiYanagida,ChunKimNilles,ChoiChunKim,Martinaxinos}.
\cite{MurayamaSuzukiYanagida}\cite{Martinaxinos}.
For example, one could take \cite{Martinaxinos}:
\beq
W = \frac{\lambda_S}{4 M_{P}} S^2 S^{\prime 2},
\eeq
where $S'$ is a chiral superfield with PecceiQuinn charge $+1$.
This implies a scalar potential that stabilizes $S$ and $S'$ at
large field strength:
\beq
V_S \,=\, F_S^2 + F_{S'}^2 \,=\, \frac{\lambda_S^2}{4 M_P^2}
S S'^2 (S^2 + S'^2) .
\eeq
There is also a soft supersymmetrybreaking Lagrangian:
\beq
{\cal L}_{\rm soft} = V_{\rm soft} = m^2_S S^2 + m^2_{S'}
S'^2  \left ( \frac{a_S}{4 M_P} S^2 S^{\prime 2} + {\rm c.c.} \right ),
\eeq
where $m^2_S$ and $m^2_{S'}$ are of order $m_{\rm soft}^2$
and $a_S$ is of order $m_{\rm soft}$. The total scalar potential
$V_S + V_{\rm soft}$
will have
an appropriate VEV of order eq.~(\ref{eq:KimNillesSVEV}) provided that
$m^2_S$, $m^2_{S'}$ are
negative or if $a_S$ is sufficiently large. For example, with $m^2_S =
m^2_{S'}$ for simplicity, there will be a nontrivial minimum of
the potential if $a_S^2  12 m_S^2 \lambda_S^2 > 0$,
and it will be a global minimum of the potential if $a_S^2  16 m_S^2
\lambda_S^2 > 0$.
One pseudoscalar degree of freedom, a mixture of $S$ and $S'$,
is the axion, with a very small mass. The rest of the chiral
supermultiplet from which the axion came will have masses of order
$m_{\rm soft}$, but couplings to the MSSM that are highly suppressed.
However, if one of the fermionic members of this chiral supermultiplet
(a singlino that can be properly called an ``axino'' $\tilde a$, and
which has tiny mixing with the MSSM neutralinos $\tilde N_i$) is lighter
than all of the MSSM odd $R$parity particles,
then it could be the LSP dark matter. If its relic density
arises predominantly from decays of the wouldbe LSP $\tilde N_1$, then today
$\Omega_{\rm DM} h^2$ today can be obtained from that one would have obtained for $\tilde
N_1$ if it were stable, but just
suppressed by a factor of $m_{\tilde a}/m_{\tilde N_1}$. It is
also possible that the decay of $\tilde N_1$ to $\tilde a$ could occur
within a collider detector, rarely and with a macroscopic decay length but
just often enough to provide a signal in a sufficiently large sample of
superpartner pair production events \cite{Martinaxinos}.
There are several variations on the theme given above.
The nonrenormalizable superpotential could instead
have the schematic form
$S^3 S' + S S' H_u H_d$
as in the original explicit model of this type
\cite{MurayamaSuzukiYanagida}, or
$S^3 S' + S^2 H_u H_d$
as in \cite{ChoiChunKim},
or $S S^{\prime 3} + S^2 H_u H_d$ as in \cite{Martinaxinos},
each entailing a different assignment
of PecceiQuinn charges for the gauge singlet fields,
but with qualitatively similar behavior.
One can also introduce more than two new fields that break the
PecceiQuinn symmetry at the intermediate scale.
The GiudiceMasiero mechanism instead relies on
a nonrenormalizable contribution to the
K\"ahler potential in addition to the usual
canonical terms for the MSSM Higgs fields:
\beq
K = H_u H_u^* + H_d H_d^* + \Bigl (\frac{\lambda_\mu}{M_P} H_u H_d X^* +
{\rm c.c.} \Bigr ) + \ldots. \label{eq:KahlerpotentialforGiudiceMasiero}
\eeq
Here $\lambda_\mu$ is a dimensionless coupling parameter and
$X$ has PecceiQuinn charge $+2$, and is a chiral superfield responsible for
spontaneous breaking of supersymmetry through its auxiliary $F$ field.
Giudice and Masiero showed
\cite{GiudiceMasiero} that in supergravity, the presence of such
couplings in the K\"ahler potential
will always give rise to a nonzero $\mu$ with a natural
orderofmagnitude of $m_{\rm soft}$. The $b$ term arises
similarly with orderofmagnitude $m_{\rm soft}^2$. The actual values of
$\mu$ and $b$ depend on contributions to the full superpotential and K\"ahler potential
involving the hiddensector fields including $X$; see
\cite{GiudiceMasiero} for details. These terms do
not have any other direct effect on phenomenology, so without faith in a
complete underlying theory it will be difficult to correlate them with future
experimental results.
One way of understanding the origin of the $\mu$ term in the GiudiceMasiero
class of models is to consider the lowenergy effective theory below $M_P$
involving a nonrenormalizable K\"ahler potential term of the form in
eq.~(\ref{eq:KahlerpotentialforGiudiceMasiero}). Even if not present in
the fundamental theory, this term could arise from radiative corrections
\cite{HallLykkenWeinberg}. If
the auxiliary field for $X$ obtains a VEV, then one obtains \beq
\mu = \frac{\lambda_\mu}{M_P} \langle F_X^* \rangle .
\eeq
This will be of the correct order of magnitude if parametrically
$\langle F_X^* \rangle \sim m_{\rm soft} M_P$, which is indeed the typical
size
assigned to the $F$terms of the hidden sector in Planckscale mediated
models of supersymmetry breaking. The
$b$ term in the soft supersymmetry breaking sector at low energies could
arise in this effective field theory picture from K\"ahler potential terms of
the form $K = \frac{\lambda_b}{M_P^2} Y^* Z H_u H_d$, where $\langle F_Y^*
\rangle \sim \langle F_Z \rangle \sim m_{\rm soft} M_P$.
However, this is not necessary, because with $\mu
\not= 0$, the lowenergy nonzero value of $b$ will arise from
threshold effects and renormalization group running.
One
could also identify both of the fields $Y,Z$ with $X$, at the cost of explicitly violating
the PecceiQuinn symmetry.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Extra $D$term contributions to scalar
masses}\label{subsec:variations.Dterms}
\setcounter{equation}{0}
Another way to generalize the MSSM is to include additional gauge
interactions. The simplest possible gauge extension introduces just one
new Abelian gauge symmetry; call it $U(1)_X$. If it is broken at a very
high mass scale, then the corresponding vector gauge boson and gaugino
fermion will both be heavy and will decouple from physics at the TeV scale
and below. However, as long as the MSSM fields carry $U(1)_X$ charges, the
breaking of $U(1)_X$ at an arbitrarily high energy scale can still leave a
telltale imprint on the soft terms of the MSSM \cite{Dterms}.
To see how this works, let us consider the scalar potential for a model in
which $U(1)_X$ is broken. Suppose that the MSSM scalar fields, denoted
generically by $\phi_i$, carry $U(1)_X$ charges $x_i$. We also introduce a
pair of chiral supermultiplets $\Splus$ and $\Sminus$ with $U(1)_X$
charges normalized to $+1$ and $1$ respectively. These fields are
singlets under the Standard Model gauge group $SU(3)_C \times SU(2)_L
\times U(1)_Y$, so that when they get VEVs, they will just accomplish the
breaking of $U(1)_X$. An obvious guess for the superpotential containing
$\Splus$ and $\Sminus$ is $W = M \Splus \Sminus$, where $M$ is a
supersymmetric mass. However, unless $M$ vanishes or is very small, it
will yield positivesemidefinite quadratic terms in the scalar potential
of the form $V = M^2 (\Splus^2 + \Sminus^2)$, which will force the
minimum to be at $\Splus = \Sminus = 0$. Since we want $\Splus$ and
$\Sminus$ to obtain VEVs, this is unacceptable. Therefore we assume that
$M$ is 0 (or very small) and that the leading contribution to the
superpotential comes instead from a nonrenormalizable term, say:
\beq
W \,=\, {\lambda\over 2 \MPlanck} \Splus^2 \Sminus^2.
\label{wfordterms}
\eeq
The equations
of motion for the auxiliary fields are then $F^*_{\Splus} = \partial
W/\partial \Splus = (\lambda/\MPlanck)\Splus \Sminus^{2}$ and
$F^*_{\Sminus} = \partial W/\partial \Sminus = (\lambda/\MPlanck)\Sminus
\Splus^{2}$, and the corresponding contribution to the scalar potential is
\beq
V_F \>=\> F_{\Splus}^2 + F_{\Sminus}^2 \>=\>
{\lambda^2\over \MPlanck^2}
\Bigl ( \Splus^4 \Sminus^2 + \Splus^2 \Sminus^4 \Bigr ) .
\eeq
In addition, there are supersymmetrybreaking terms that must be taken
into account:
\beq
V_{\rm soft} \,=\, m_+^2 \Splus^2 + m_^2 \Sminus^2 
\left ({a\over 2\MPlanck} \Splus^2 \Sminus^2 + \conj\right ).
\eeq
The terms with $m_+^2$ and $m_^2$ are soft squared masses for
$\Splus$ and $\Sminus$. They could come from a minimal supergravity
framework at the Planck scale, but in general they will be renormalized
differently, due to different interactions for $\Splus$ and $\Sminus$,
which we have not bothered to write down in eq.~(\ref{wfordterms}) because
they involve fields that will not get VEVs. The last term is a ``soft"
term analogous to the $a$ terms in
eq.~(\ref{lagrsoft}), with $a$ of order $m_{\rm soft}$. The coupling
$a/2\MPlanck$ is actually dimensionless, but should be treated as soft
because of its origin and its tiny magnitude. Such terms arise from the
supergravity Lagrangian in an exactly analogous way to the usual soft
terms. Usually one can just ignore them, but this one plays a crucial role
in the gauge symmetry breaking mechanism. The scalar potential for terms
containing $\Splus$ and $\Sminus$ is:
\beq
V =
{1\over 2} g_X^2 \Bigl ( \Splus^2  \Sminus^2 + \sum_i x_i \phi_i^2
\Bigr )^2 + V_F + V_{\rm soft}.
\label{xpotential}
\eeq
The first term involves the square of the $U(1)_X$ $D$term [see
eqs.~(\ref{solveforD}) and (\ref{fdpot})], and $g_X$ is the $U(1)_X$ gauge
coupling. The scalar potential eq.~(\ref{xpotential}) has a nearly
$D$flat direction, because the $D$term part vanishes for $\phi_i=0$ and
any $\Splus = \Sminus$. Without loss of generality, we can take $a$
and $\lambda$ to both be real and positive for purposes of minimizing the
scalar potential. As long as $a^2  6 \lambda^2 (m_+^2 + m_^2) >
0$, there is a minimum of the potential very near the flat direction:
\beq
\langle \Splus \rangle^2 \,\approx\, \langle \Sminus \rangle^2
\, \approx \,
\Bigl [a + \sqrt{ a^2  6 \lambda^2 (m_+^2 + m_^2) } \Bigr ]
\MPlanck/6 \lambda^2
\eeq
(with $\langle \phi_i\rangle = 0$), so $\langle \Splus \rangle \approx
\langle \Sminus \rangle \sim {\cal O}(\sqrt{m_{\rm soft} \MPlanck})$. This
is also a global minimum of the potential if $a^2  8 \lambda^2 (m_+^2
+ m_^2) > 0$. Note that $m_+^2 + m_^2 < 0$ is a sufficient,
but not necessary, condition. The $V_F$ contribution is what stabilizes
the scalar potential at very large field strengths. The VEVs of $\Splus$
and $\Sminus$ will typically be of order $10^{10}$ GeV or so. Therefore
the $U(1)_X$ gauge boson and gaugino, with masses of order $g_X \langle
S_\pm\rangle$, will play no role in collider physics.
However, there is also necessarily a small deviation from $\langle
\Splus\rangle = \langle \Sminus \rangle$, as long as $m_+^2 \not=
m_^2$. At the minimum of the potential with $\partial V/\partial
\Splus = \partial V/\partial \Sminus = 0$, the leading order difference in
the VEVs is given by
\beq
\langle \Splus \rangle^2  \langle \Sminus \rangle^2
\,=\, \langle D_X \rangle/g_X
\,\approx\, (m_^2  m_+^2)/2 g_X^2,
\eeq
assuming that $\langle \Splus \rangle$ and $\langle \Sminus \rangle$ are
much larger than their difference. After integrating out $\Splus$ and
$\Sminus$ by replacing them using their equations of motion expanded
around the minimum of the potential, one finds that the MSSM scalars
$\phi_i$ each receive a squaredmass correction
\beq
\Delta m_i^2 \,=\, x_i g_X \langle D_X \rangle\, ,
\label{dxtermcorrections}
\eeq
in addition to the usual soft terms from other sources. The $D$term
corrections eq.~(\ref{dxtermcorrections}) can be roughly of the order of
$m_{\rm soft}^2$ at most, since they are all proportional to $m_^2
m_+^2$. The result eq.~(\ref{dxtermcorrections}) does not
actually depend on the choice of the nonrenormalizable superpotential, as
long as it produces the required symmetry breaking with large VEVs; this
is a general feature. The most important feature of
eq.~(\ref{dxtermcorrections}) is that each MSSM scalar squared mass
obtains a correction just proportional to its charge $x_i$ under the
spontaneously broken gauge group, with a universal factor $g_X \langle D_X
\rangle$. In a sense, the soft supersymmetrybreaking terms $m_+^2$ and
$m_^2$ have been recycled into a nonzero $D$term for $U(1)_X$,
which then leaves its ``fingerprint" on the spectrum of MSSM scalar
masses. From the point of view of TeV scale physics, the quantity $g_X
\langle D_X \rangle$ can simply be taken to parameterize our ignorance of
how $U(1)_X$ got broken. Typically, the charges $x_i$ are rational numbers
and do not all have the same sign, so that a particular candidate $U(1)_X$
can leave a quite distinctive pattern of mass splittings on the squark and
slepton spectrum. As long as the charges are familyindependent, the
squarks and sleptons with the same electroweak quantum numbers remain
degenerate, maintaining the natural suppression of flavormixing effects.
The additional gauge symmetry $U(1)_X$ in the above discussion can stand
alone, or may perhaps be embedded in a larger nonAbelian gauge group. If
the gauge group for the underlying theory at the Planck scale contains
more than one new $U(1)$ factor, then each can make a contribution like
eq.~(\ref{dxtermcorrections}). Additional $U(1)$ gauge group
factors are quite
common in superstring models, and in the GUT groups $SO(10)$ and $E_6$,
suggesting optimism about the existence of
corresponding $D$term corrections. Once one merely assumes the existence
of additional $U(1)$ gauge groups at very high energies, it is unnatural
to assume that such $D$term contributions to the MSSM scalar masses
should vanish, unless there is an exact symmetry that enforces $m_+^2 =
m_^2$. The only question is whether or not the magnitude of the
$D$term contributions is significant compared to the usual minimal
supergravity and RG contributions. Therefore, efforts to
understand the sparticle spectrum of the MSSM may need to take into
account the possibility of $D$terms from additional gauge groups.
\section{Concluding remarks}\label{sec:outlook}
\setcounter{equation}{0}
\setcounter{figure}{0}
\setcounter{table}{0}
\setcounter{footnote}{1}
In this primer, I have tried to convey some of the more essential
features of supersymmetry as a theory of physics beyond the Standard Model.
One of the nicest qualities of supersymmetry is that so much is known
about its implications already, despite the present lack of direct experimental evidence.
The interactions of the Standard Model particles and their superpartners are
fixed by supersymmetry, up to mass mixing effects due to supersymmetry
breaking. Even the terms and stakes of many of the important outstanding
questions, especially the paramount issue ``How is supersymmetry
broken?", are already rather clear. That this can be so is a testament
to the unreasonably predictive quality of the symmetry itself.
At this writing, LHC searches have been performed
based on 5 fb$^{1}$ at $\sqrt{s} = 7$ TeV,
20 fb$^{1}$ at $\sqrt{s} = 8$ TeV, and 4 fb$^{1}$ at $\sqrt{s} = 13$ TeV.
These searches have not found any evidence for superpartners, and
have put strong lower bounds on the masses of squarks and the gluino in
large classes of models. Even for the weakly interacting
superpartners, the mass limits have begun to exceed those from LEP, in some
cases greatly so. The earliest search strategies used by ATLAS and CMS were
tuned to simple and optimistic templates, including the the MSUGRA
scenario with new parameters $m^2_0$, $m_{1/2}$, $A_0$, $\tan\beta$ and
Arg$(\mu )$, and the GMSB scenario with new parameters $\Lambda$,
$M_{\rm mess}$, $\nmess$, $\langle F \rangle$, $\tan\beta$, and ${\rm
Arg}(\mu )$. However, the only indispensable idea of supersymmetry is
simply that of a symmetry between fermions and bosons. Nature may or may
not be kind enough to realize this beautiful idea within one of the
specific frameworks that have already been explored well by theorists.
More recent searches reported by the LHC experimental collaborations
probe the more general supersymmetric parameter space, including
$R$parity violating models, and models in which small mass differences
or decay modes with softened visible energies
make the detection of supersymmetry more difficult.
While the present lack of direct evidence for sparticles is
disappointing, it is at least consistent with the observation of
$m_{h^0} = 125$ GeV. As noted above, this value of the lightest Higgs
boson mass points to top squarks that are quite heavy, at least within
the MSSM with small or moderate stop mixing. In many model frameworks,
the topsquark masses are correlated, through radiative corrections,
with the masses of the other squarks and the gluino. Therefore, based only on
the information that $m_{h^0} = 125$ GeV, one could have surmised
that supersymmetry probably would not be discovered early at the LHC, and that
perhaps even with $\sqrt{s} =13$ or 14 TeV the discovery of sparticles is
not favored, contrary to earlier expectations. A more
optimistic inference one could draw is that the MSSM is likely to be augmented
with additional particles or interactions that raise the $h^0$ mass, as
discussed for example in sections \ref{subsec:variations.vectorlike}
and \ref{subsec:variations.NMSSM}.
It is also worth nothing that most of the {\em other} theories that had been
put forward as solutions to the hierarchy problem are in no better
shape than supersymmetry is, given the discovery of the
125 GeV Higgs boson as well as the lack of other evidence for
exotic physics at the LHC in the runs at 7 and 8 TeV.
In fact, many of the competitors to supersymmetry
in this regard have now been eliminated. Therefore, based on a belief that
the hierarchy problem needs a solution at the TeV scale, and the alternatives are less than compelling,
I personally maintain a guarded optimism that supersymmetry
will be discovered at the LHC in the higher energy runs that have just begun.
If supersymmetry is experimentally verified, the discovery will not be
an end, but rather a beginning in high energy physics. It seems likely
to present us with questions and challenges that we can only guess at
presently. The measurement of sparticle masses, production
crosssections, and decay modes will rule out some models for
supersymmetry breaking and lend credence to others. We will be able to
test the principle of $R$parity conservation, the idea that
supersymmetry has something to do with the dark matter, and possibly
make connections to other aspects of cosmology including baryogenesis
and inflation. Other fundamental questions, like the origin of the $\mu$
parameter and the rather peculiar hierarchical structure of the Yukawa
couplings may be brought into sharper focus with the discovery of
superpartners. Understanding the precise connection of supersymmetry to
the electroweak scale will surely open the window to even deeper levels
of fundamental physics.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\addcontentsline{toc}{section}{Acknowledgments}
\section*{Acknowledgments} This is an extended and revised version
of a chapter in the volume {\it Perspectives on Supersymmetry}
(World Scientific, 1998) at the kind invitation of Gordy Kane. I am
thankful to him and to James Wells for many helpful comments and
suggestions on this primer. I am also indebted to my other collaborators
on supersymmetry and related matters:
Ben Allanach,
Sandro Ambrosanio,
Nima ArkaniHamed,
Diego Casta\~no,
Ray Culbertson,
Yanou Cui,
Michael Dine,
Manuel Drees,
Herbi Dreiner,
Tony Gherghetta,
Howie Haber,
Ian Jack,
Tim Jones,
Chris Kolda,
Graham Kribs,
Nilanjana Kumar,
Tom LeCompte,
Stefano Moretti,
David Morrissey,
Steve Mrenna,
Jianming Qian,
Dave Robertson,
Roberto Ruiz de Austri,
Scott Thomas,
Kazuhiro Tobe,
Mike Vaughn,
Graham Wilson,
Youichi Yamada,
James Younkin,
2890 members of ATLAS,
and especially Pierre Ramond, for many illuminating and inspiring
conversations.
%Any mistakes in this work are purely their fault for not teaching me
%better.
Corrections to
previous versions have been provided by
Daniel Arnold,
Howie Baer,
Jorge de Blas,
Meike de With,
Herbi Dreiner,
Paddy Fox,
Hajime Fukuda,
Peter Graf,
Gudrun Hiller,
Graham Kribs,
Bob McElrath,
Matt Reece,
Ver\'onica Sanz,
Frank Daniel Steffen,
Shufang Su,
John Terning,
Keith Thomas,
Scott Thomas,
Sean Tulin,
and
Robert Ziegler.
I will be grateful to receive further corrections at
{\tt spmartin@niu.edu}, and a list of them is maintained at
{\tt http://www.niu.edu/spmartin/primer}.
I thank the Aspen Center for Physics, Fermilab, the Kavli Institute
for Theoretical Physics in Santa Barbara, and SLAC for their hospitality,
and the students of
the 2013 and 2005 ICTP Summer Schools on Particle Physics,
the 2011 TASI Summer School,
the 2010 PreSUSY Summer School in Bonn,
the 2008 CERN/Fermilab Hadron Collider Physics Summer School,
and PHYS 686 at NIU in Spring 2004,
for asking interesting questions. This work was supported in part by the U.S.
Department of Energy, and by National Science Foundation
grants PHY9970691, PHY0140129, PHY0456635, PHY0757325, PHY1068369,
and PHY1417028.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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For reviews, see
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%%CITATION = HEPPH 9903544;%%
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%%CITATION = HEPPH 0112235;%%
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%%CITATION = HEPPH 0212397;%%
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% %%CITATION = HEPEX 9905013;%%
% %``Search for the leptonfamilynumber nonconserving decay mu+ $\to$ e+
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% %%CITATION = HEPEX 0111030;%%
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``Split supersymmetry", as in
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%%CITATION = HEPPH 0406088;%%
abandons the motivation of supersymmetry as
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J.R.~Ellis, C.H.~Llewellyn Smith and G.G.~Ross,
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%
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%
%\bibitem{Banks:2005df}
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%%CITATION = JHEPA,0604,021;%%
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%
%\bibitem{Abel:2006cr}
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%
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%
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P.~Fayet and J.~Iliopoulos,
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P.~Fayet,
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\bibitem{dtermbreakingmaywork} However, see for example
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A nonzero FayetIliopoulos term for an anomalous $U(1)$ symmetry
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M.~Dine, N.~Seiberg and E.~Witten,
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J.~Atick, L.~Dixon and A.~Sen,
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Nucl.\ Phys.\ B {\bf 292}, 109 (1987).
%%CITATION = NUPHA,B292,109;%%
%This may even help to explain the
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%L.E.~Ib\'a\~nez,
% Phys.\ Lett.\ B {\bf 303}, 55 (1993)
% [hepph/9205234];
% %%CITATION = HEPPH 9205234;%%
%L.E.~Ib\'a\~nez and G.G.~Ross,
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% %%CITATION = HEPPH 9403338;%%
%P.~Bin\'etruy, S.~Lavignac, P.~Ramond,
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% Nucl.\ Phys.\ B {\bf 477}, 353 (1996)
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% %%CITATION = HEPPH 9601243;%%
%P.~Bin\'etruy, N.~Irges, S.~Lavignac and P.~Ramond,
% %``Anomalous U(1) and lowenergy physics: The power of Dflatness and
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% [hepph/9612442];
% %%CITATION = HEPPH 9612442;%%
%N.~Irges, S.~Lavignac, P.~Ramond,
% %``Predictions from an anomalous U(1) model of Yukawa hierarchies,''
% Phys.\ Rev.\ D {\bf 58}, 035003 (1998)
% [hepph/9802334].
% %%CITATION = HEPPH 9802334;%%
\bibitem{ORaifeartaigh}
L.~O'Raifeartaigh,
%``Spontaneous Symmetry Breaking For Chiral Scalar Superfields,''
Nucl.\ Phys.\ B {\bf 96}, 331 (1975).
%%CITATION = NUPHA,B96,331;%%
\bibitem{flatdirections}
M.A.~Luty and W.I.~Taylor,
%``Varieties of vacua in classical supersymmetric gauge theories,''
Phys.\ Rev.\ D {\bf 53}, 3399 (1996)
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%%CITATION = HEPTH 9506098;%%
M.~Dine, L.~Randall and S.D.~Thomas,
%``Baryogenesis from flat directions of the supersymmetric standard model,''
Nucl.\ Phys.\ B {\bf 458}, 291 (1996)
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%%CITATION = HEPPH 9507453;%%
T.~Gherghetta, C.F.~Kolda and S.P.~Martin,
%``Flat directions in the scalar potential of the supersymmetric standard
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Nucl.\ Phys.\ B {\bf 468}, 37 (1996)
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%%CITATION = HEPPH 9510370;%%
\bibitem{ColemanWeinberg}
S.~Coleman and E.~Weinberg,
%``Radiative Corrections As The Origin Of Spontaneous Symmetry Breaking,''
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%%CITATION = PHRVA,D7,1888;%%
\bibitem{twoloopEP}
S.P.~Martin,
%``Twoloop effective potential for a general renormalizable theory and
%softly broken supersymmetry,''
Phys.\ Rev.\ D {\bf 65}, 116003 (2002)
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%%CITATION = HEPPH 0111209;%%
\bibitem{Nelson:1993nf}
A.E.~Nelson and N.~Seiberg,
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Nucl.\ Phys.\ B {\bf 416}, 46 (1994)
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%%CITATION = NUPHA,B416,46;%%
\bibitem{Raxion}
J.~Bagger, E.~Poppitz and L.~Randall,
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%%CITATION = NUPHA,B426,3;%%
\bibitem{Intriligator:2007py}
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%%CITATION = JHEPA,0707,017;%%
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E.~Witten,
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%%CITATION = NUPHA,B202,253;%%
\bibitem{AffleckDineSeiberg}
I.~Affleck, M.~Dine and N.~Seiberg,
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%``Dynamical Supersymmetry Breaking In FourDimensions And Its
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\bibitem{metastablemodels}
For example, see:
%\bibitem{Franco:2006es}
S.~Franco and A.M.~Uranga,
%``Dynamical SUSY breaking at metastable minima from Dbranes at obstructed
%geometries,''
JHEP {\bf 0606}, 031 (2006)
[hepth/0604136].
%%CITATION = JHEPA,0606,031;%%
%
%\bibitem{Ooguri:2006pj}
H.~Ooguri and Y.~Ookouchi,
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[hepth/0606061].
%%CITATION = NUPHA,B755,239;%%
%
%\bibitem{Banks:2006ma}
T.~Banks,
%``Remodeling the pentagon after the events of 2/23/06,''
[hepph/0606313].
%%CITATION = HEPPH/0606313;%%
%
%\bibitem{Franco:2006ht}
S.~Franco, I.~GarciaEtxebarria and A.M.~Uranga,
%``Nonsupersymmetric metastable vacua from brane configurations,''
JHEP {\bf 0701}, 085 (2007)
[hepth/0607218].
%%CITATION = JHEPA,0701,085;%%
%
%\bibitem{Kitano:2006wz}
R.~Kitano,
%``Gravitational gauge mediation,''
Phys.\ Lett.\ B {\bf 641}, 203 (2006)
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%%CITATION = PHLTA,B641,203;%%
%
%\bibitem{Amariti:2006vk}
A.~Amariti, L.~Girardello and A.~Mariotti,
%``Nonsupersymmetric metastable vacua in SU(N) SQCD with adjoint matter,''
JHEP {\bf 0612}, 058 (2006)
[hepth/0608063].
%%CITATION = JHEPA,0612,058;%%
%
%\bibitem{Dine:2006gm}
M.~Dine, J.L.~Feng and E.~Silverstein,
%``Retrofitting O'Raifeartaigh models with dynamical scales,''
Phys.\ Rev.\ D {\bf 74}, 095012 (2006)
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%%CITATION = PHRVA,D74,095012;%%
%
%\bibitem{Dudas:2006gr}
E.~Dudas, C.~Papineau and S.~Pokorski,
%``Moduli stabilization and uplifting with dynamically generated Fterms,''
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%%CITATION = JHEPA,0702,028;%%
%
%\bibitem{Abe:2006xp}
H.~Abe, T.~Higaki, T.~Kobayashi and Y.~Omura,
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%%CITATION = PHRVA,D75,025019;%%
%
%\bibitem{Dine:2006xt}
M.~Dine and J.~Mason,
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%%CITATION = PHRVA,D77,016005;%%
%
%\bibitem{Kitano:2006xg}
R.~Kitano, H.~Ooguri and Y.~Ookouchi,
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%%CITATION = PHRVA,D75,045022;%%
%
%\bibitem{Murayama:2006yf}
H.~Murayama and Y.~Nomura,
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%
%\bibitem{Csaki:2006wi}
C.~Csaki, Y.~Shirman and J.~Terning,
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%
%\bibitem{Aharony:2006my}
O.~Aharony and N.~Seiberg,
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%%CITATION = JHEPA,0702,054;%%
%
%\bibitem{Shih:2007av}
D.~Shih,
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%%CITATION = JHEPA,0802,091;%%
%
%\bibitem{Giveon:2008ne}
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M.~Dine and J.D.~Mason,
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A.~Nelson,
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%%CITATION = HEPPH 9707442;%%
G.F.~Giudice and R.~Rattazzi,
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Phys.\ Rept.\ {\bf 322}, 419 (1999)
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E.~Poppitz and S.P.~Trivedi,
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P.~Fayet,
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%``Scattering CrossSections Of The Photino And The Goldstino (Gravitino) On
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[hepph/9602239],
%%CITATION = HEPPH 9602239;%%
%``Search for supersymmetry with a light gravitino at the Fermilab Tevatron
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%``Superfield Densities And Action Principle In Curved Superspace,''
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E.~Cremmer et al.,
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J.~Bagger,
%``Coupling The Gauge Invariant Supersymmetric Nonlinear Sigma Model To
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E.~Cremmer et al.,
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T.~Moroi, H.~Murayama, M.~Yamaguchi,
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A.H.~Chamseddine, R.~Arnowitt and P.~Nath,
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Phys.\ Rev.\ Lett.\ {\bf 49}, 970 (1982);
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R.~Barbieri, S.~Ferrara and C.~A.~Savoy,
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L.E.~Ib\'a\~nez,
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L.J.~Hall, J.D.~Lykken and S.~Weinberg,
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N.~Ohta,
%``Grand Unified Theories Based On Local Supersymmetry,''
Prog.\ Theor.\ Phys.\ {\bf 70}, 542 (1983).
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%``Grand Unification In Simple Supergravity,''
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L.~AlvarezGaum\'e, J.~Polchinski, and M.~Wise,
%``Minimal LowEnergy Supergravity,''
Nucl.\ Phys.\ B {\bf 221}, 495 (1983).
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P.~Moxhay and K.~Yamamoto,
%``Effects Of Grand Unification Interactions On Weak Symmetry Breaking In
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Nucl.\ Phys.\ B {\bf 256}, 130 (1985);
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K.~Grassie,
%``Consequences Of A GUT Sector In Minimal N=1 Supergravity Models With
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B.~Gato,
%``Can The SU(5) Running Be Neglected In The Minimal N=1 SUGRA Model?,''
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N.~Polonsky and A.~Pomarol,
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R.~Barbieri, J.~Louis and M.~Moretti,
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%\href{http://www.slac.stanford.edu/spires/find/hep/www?irn=561940}{SPIRES entry}
%See \cite{BailinLovebook} for a pedagogical account.
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\bibitem{stringsoft} For a review, see
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% \perspectives,
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\bibitem{cosmokramer}
P.~A.~R.~Ade {\it et al.} [Planck Collaboration],
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Phys.\ Rev.\ Lett.\ {\bf 73}, 1758 (1994)
[hepph/9404311];
%%CITATION = HEPPH 9404311;%%
K.~Choi, J.E.~Kim and G.T.~Park,
%``Phenomenology of soft terms in the presence of nonvanishing hidden sector
%potential energy,''
Nucl.\ Phys.\ B {\bf 442}, 3 (1995)
[hepph/9412397].
%%CITATION = HEPPH 9412397;%%
See also
N.C.~Tsamis and R.P.~Woodard,
%``Relaxing the cosmological constant,''
Phys.\ Lett.\ B {\bf 301}, 351 (1993);
%%CITATION = PHLTA,B301,351;%%
%``Quantum Gravity Slows Inflation,''
Nucl.\ Phys.\ B {\bf 474}, 235 (1996)
[hepph/9602315];
%%CITATION = HEPPH 9602315;%%
%``The quantum gravitational backreaction on inflation,''
Annals Phys.\ {\bf 253}, 1 (1997)
[hepph/9602316];
%%CITATION = HEPPH 9602316;%%
%``Nonperturbative models for the quantum gravitational backreaction on
%inflation,''
Annals Phys.\ {\bf 267}, 145 (1998)
[hepph/9712331].
%%CITATION = HEPPH 9712331;%%
and references therein, for discussion of nonperturbative quantum
gravitational effects on the effective cosmological constant. This work
suggests that requiring the treelevel vacuum energy to vanish may not be
correct or meaningful. Moreover, perturbative supergravity or superstring
predictions for the vacuum energy may not be relevant to the question of
whether the observed cosmological constant is sufficiently small.
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M.~Dine and W.~Fischler,
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C.R.~Nappi and B.A.~Ovrut,
%``Supersymmetric Extension Of The SU(3) X SU(2) X U(1) Model,''
Phys.\ Lett.\ B {\bf 113}, 175 (1982);
%%CITATION = PHLTA,B113,175;%%
L.~AlvarezGaum\'e, M. Claudson and M.~B.~Wise,
%``LowEnergy Supersymmetry,''
Nucl.\ Phys.\ B {\bf 207}, 96 (1982).
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M.~Dine, A.~E.~Nelson,
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[hepph/9303230];
%%CITATION = HEPPH 9303230;%%
M.~Dine, A.E.~Nelson, Y.~Shirman,
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[hepph/9408384];
%%CITATION = HEPPH 9408384;%%
M.~Dine, A.E.~Nelson, Y.~Nir, Y.~Shirman,
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[hepph/9607225];
%%CITATION = HEPPH 9607225;%%
\bibitem{gmsbcorrB}
S.P.~Martin
%``Generalized messengers of supersymmetry breaking and the sparticle mass
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[hepph/9608224];
%%CITATION = HEPPH 9608224;%%
\bibitem{gmsbcorrC}
E.~Poppitz and S.P.~Trivedi,
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P.~Meade, N.~Seiberg and D.~Shih,
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%%CITATION = PHRVA,D79,035002;%%
M.~Buican, P.~Meade, N.~Seiberg and D.~Shih,
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T.T.~Dumitrescu, Z.~Komargodski, N.~Seiberg and D.~Shih,
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V.A.~Rubakov and M.E.~Shaposhnikov,
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%``Extra SpaceTime Dimensions: Towards A Solution To The Cosmological
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L.~Randall and R.~Sundrum,
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%% Next four entries are orphans.
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\end{thebibliography}
\end{document}