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\sectionLink{slac-pub-7237-0-0-2}{slac-pub-7237-0-0-2}{Above: 2. The Minimal Model of Gauge-Mediated Supersymmetry Breaking}%
\subsectionLink{slac-pub-7237-0-0-2}{slac-pub-7237-0-0-2-2}{Next: 2.2. Variations of the Minimal Model}%
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\subsection{\usemenu{slac-pub-7237::context::slac-pub-7237-0-0-2-1}{The Minimal Model}}\label{subsection::slac-pub-7237-0-0-2-1}
\label{minimalsection}
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\sfig{gauginoloop}{fig1.eps}
{One-loop messenger sector supergraph which gives rise to visible sector
gaugino masses.}
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The minimal model of gauge-mediated supersymmetry breaking
(which preserves the successful predictions of perturbative
gauge unification) consists of messenger fields which transform
as a single flavor of ${\bf 5} + \overline{\bf 5}$ of $SU(5)$,
i.e. there are $SU(2)_L$ doublets
$\ell$ and $\bar{\ell}$, and $SU(3)_C$ triplets $q$ and $\bar{q}$.
In order to introduce supersymmetry breaking into the messenger
sector, these fields may be coupled to a gauge
singlet spurion, $S$, through the superpotential
\beq
W = \lambda_2 S \ell \bar{\ell} + \lambda_3 S q \bar{q}
\label{SQQbar}
\eq
The scalar expectation value of $S$ sets the overall scale
for the messenger sector, and the auxiliary component, $F$,
sets the supersymmetry breaking scale.
For $F \neq 0$ the messenger spectrum is not supersymmetric,
$$
m_b = M \sqrt{ 1 \pm {\Lambda \over M} }
$$
\beq
m_f = M
\eq
where $M = \lambda S$ and $\Lambda = F/S$.
The parameter $\Lambda /M$ sets the scale for the fractional
splitting between bosons and fermions.
Avoiding electroweak and color breaking in the messenger
sector requires $M > \Lambda$.
In the models of Ref. \cite{3} the field $S$ is an
elementary singlet which couples through a secondary
messenger sector to the supersymmetry breaking sector.
In more realistic models the messenger fields are
embedded directly in the supersymmetry breaking sector.
This may be accomplished within a model of dynamical supersymmetry
breaking by identifying an unbroken global symmetry with the
standard model gauge group.
In the present context,
the field $S$ should be thought of as a spurion which
represents the dynamics which break supersymmetry.
The physics discussed in this paper does not depend on the
details of the dynamics represented by the spurion.
Because (\docLink{slac-pub-7237-0-0-2.tcx}[SQQbar]{1}) amounts to tree level breaking, the
messenger spectrum satisfies the sum rule
${\cal S}Tr~m^2 = 0$.
With a dynamical supersymmetry breaking sector, this sum rule
need not be satisfied.
The precise value of ${\cal S}Tr~m^2$ in the messenger sector, however,
does not significantly affect the radiatively generated
visible sector soft parameters discussed below.
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\sfig{scalarloop}{fig2.eps}
{Two-loop messenger sector supergraph which gives rise to visible sector
scalar masses. The one-loop subgraph gives rise to visible sector
gaugino wave function renormalization. Other graphs related by
gauge invariance are not shown.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Integrating out the non-supersymmetric messengers gives rise
to effective operators, which lead to supersymmetry breaking
in the visible sector.
Gaugino masses arise at one-loop from the operator
\beq
\int d^2\theta~\ln S\, W^\alpha W_\alpha ~+~ h.c.
\label{SSWDV}
\eq
as shown in Fig. 1.
In superspace this operator amounts to a shift of the
gauge couplings in the presence of the background spurion.
Inserting a single spurion auxiliary component gives a
gaugino mass.
For $F \ll \lambda S^2$ the gaugino masses %at the messenger scale
are \cite{3}
\beq
m_{\lambda_i}(M) = \frac{\alpha_i(M)}{4\pi}~ \Lambda
\label{gauginomass}
\eq
where $\Lambda = F /S$ and
GUT normalized gauge couplings are assumed
($\alpha_1 = \alpha_2 = \alpha_3$
at the unification scale).\footnote{The standard model normalization
of hypercharge is related to the GUT normalization
by $\alpha^{\prime} = (3/5) \alpha_1$.}
The dominant loop momenta in Fig. 1 are ${\cal O}(M)$, so
%the operator (\ref{SSWDV})
(\docLink{slac-pub-7237-0-0-2.tcx}[gauginomass]{4})
amounts to a boundary condition for
the gaugino masses at the messenger scale.
Visible sector scalar masses arise at two-loops from the
operator
\beq
\int d^4 \theta~ \ln(S^{\dagger} S) ~ \Phi^{\dagger} e^V \Phi
\label{SSPP}
\eq
as shown in Fig. \docLink{slac-pub-7237.tcx}[scalarloop]{2}.
In superspace this operator represents wave function
renormalization from the background spurion.
Inserting two powers of the auxiliary component of the
spurion gives a scalar mass squared.
For $F \ll \lambda S^2$ the scalar masses are \cite{3}
\beq
m^2(M) = 2 \Lambda^2~ \sum_{i=1}^3 ~ k_i
\left( \alpha_i(M) \over 4 \pi \right)^2
\label{scalarmass}
\eq
where the sum is over $SU(3) \times SU(2)_L \times U(1)_Y$, with
$k_1 = (3/5) (Y/2)^2$ where the hypercharge is
normalized as $Q = T_3 + {1 \over 2} Y$,
$k_2 = 3/4$ for $SU(2)_L$ doublets and zero for singlets,
and $k_3 = 4/3$ for $SU(3)_C$ triplets and zero for singlets.
Again, the dominant loop momenta in Fig. \docLink{slac-pub-7237.tcx}[scalarloop]{2}
are ${\cal O}(M)$, so
%the operator (\ref{SSPP})
(\docLink{slac-pub-7237-0-0-2.tcx}[scalarmass]{6})
amounts to a boundary condition for
the scalar masses at the messenger scale.
It is interesting to note that for $F \ll \lambda S^2$ the soft masses
(\docLink{slac-pub-7237-0-0-2.tcx}[gauginomass]{4}) and (\docLink{slac-pub-7237-0-0-2.tcx}[scalarmass]{6}) are independent
of the magnitude of the Yukawa couplings (\docLink{slac-pub-7237-0-0-2.tcx}[SQQbar]{1}).
This is because the one-loop graph of Fig. \docLink{slac-pub-7237.tcx}[gauginoloop]{1}
has an infrared divergence, $k^{-2}$, which is cut off by the
messenger mass $M=\lambda S$, thereby cancelling the $\lambda F$ dependence
in the numerator.
The one-loop subgraph of Fig. (\docLink{slac-pub-7237.tcx}[scalarloop]{2})
has a similar infrared divergence which cancels the
$\lambda$ dependence.
For finite $F/ (\lambda S^2)$ the corrections to (\docLink{slac-pub-7237-0-0-2.tcx}[gauginomass]{4})
and (\docLink{slac-pub-7237-0-0-2.tcx}[scalarmass]{6})
are small unless $M$ is very close to $\Lambda$
\cite{11,12,13}.
Since the gaugino masses arise at one-loop and scalar masses
squared at two-loops, superpartners masses are generally
the same order for particles with similar gauge charges.
If the messenger scale is well below the GUT scale,
then $\alpha_3 \gg \alpha_2 > \alpha_1$, so the squarks and gluino
receive mass predominantly from $SU(3)_C$ interactions,
the left handed sleptons and $W$-ino from $SU(2)_L$ interactions,
and the right handed sleptons and $B$-ino from $U(1)_Y$ interactions.
The gaugino and scalar masses are then related at the messenger
scale by $m_3^2 \simeq {3 \over 8} m_{\tilde{q}}^2$,
$m_2^2 \simeq {2 \over 3} m_{\lL}^2$, and
$m_1^2 = {5 \over 6} m_{\lR}^2$.
This also leads to a hierarchy in mass between electroweak and
strongly interacting states.
The gaugino masses at the messenger scale are in the ratios
$m_1 : m_2 : m_3 = \alpha_1 : \alpha_2 : \alpha_3$,
while the scalar masses squared are in the approximate ratios
$m_{\tilde{q}}^2 : m_{\lL}^2 : m_{\lR}^2 \simeq {4 \over 3} \alpha_3^2 :
{3 \over 4} \alpha_2^2 : {3 \over 5} \alpha_1^2$.
The masses of particles with different gauge charges are tightly
correlated in the minimal model.
These correlations are reflected in the constraints of electroweak
symmetry breaking on the low energy spectrum, as
discussed in section \docLink{slac-pub-7237-0-0-3.tcx}[RGEanalysis]{3}.
The parameter $(\alpha / 4 \pi) \Lambda$ sets the scale for
the soft masses.
This should be of order the weak scale, implying
$\Lambda \sim {\cal O}(100\tev)$.
The messenger scale $M$ is, however, arbitrary in the minimal model,
subject to $M > \Lambda$.
In models in which
the messenger sector is embedded directly in a renormalizable
dynamical supersymmetry breaking sector \cite{14},
the messenger and
effective supersymmetry breaking scales are
the same order, $M \sim \Lambda \sim {\cal O}(100\tev)$,
up to small hierarchies from messenger sector Yukawa couplings.
This is also true of models
with a secondary messenger sector \cite{3}.
The messenger scale can, however, be well separated from the
supersymmetry breaking scale.
This can arise in models with large ratios of dynamical scales.
Alternatively with non-renormalizable supersymmetry breaking,
which vanishes in the flat space limit,
expectation values intermediate between the Planck and
supersymmetry breaking
scale can develop, leading to $M \gg \Lambda$.
A noteworthy feature of the minimal messenger sector
is that it is invariant under charge conjugation and parity,
up to electroweak radiative corrections.
This has the important effect of enforcing the vanishing
of the $U(1)_Y$ Fayet-Iliopoulos $D$-term
at all orders in interactions that involve gauge interactions
and messenger fields only.
This is crucial since a non-zero $U(1)_Y$ D-term at one-loop
would induce soft scalar masses much larger
in magnitude than the
two-loop contributions (\docLink{slac-pub-7237-0-0-2.tcx}[scalarmass]{6}),
and lead to $SU(3)_C$ and $U(1)_Q$ breaking.
This vanishing is unfortunately not an automatic feature of
models in which the messenger fields also transform
under a chiral representation of the gauge group responsible for
breaking supersymmetry.
In the minimal model a $U(1)_Y$ $D$-term is generated only
by gauge couplings to chiral standard model fields at three loops.
The leading log contribution comes from renormalization group
evolution and is discussed in section~\docLink{slac-pub-7237-0-0-3.tcx}[electroweaksection]{3.2.1}.
The dimensionful parameters within the Higgs sector
\beq
W = \mu H_u H_d
\eq
and
\beq
V = m_{12}^2 H_u H_d ~+~ h.c.
\eq
do not follow from the anzatz of gauge-mediated supersymmetry
breaking.
These terms require additional interactions which violate
$U(1)_{PQ}$ and $U(1)_{R-PQ}$ symmetries.
A number of models have been proposed to generate these
terms, including,
additional messenger quarks and
singlets \cite{3}, singlets with an
inverted hierarchy \cite{3}, and singlets with
an approximate global symmetry \cite{15}.
In the minimal model the mechanisms for generating the
parameters $\mu$ and $m_{12}^2$ are not specified, and
they are taken as free parameters at the messenger scale.
As discussed below, upon imposing electroweak symmetry breaking,
these parameters may be eliminated in favor of
$\tan \beta = v_u / v_d$ and $\mZ$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sfig{Aloop}{fig3.eps}
{One-loop visible sector supergraph which contains both
logarithmic and finite contributions to visible sector
$A$-terms. The cross on the visible sector gaugino line
represents the gaugino mass insertion shown in Fig. 1.}
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Soft tri-linear $A$-terms require interactions which
violate both $U(1)_R$ and visible sector chiral
flavor symmetries.
Since the messenger sector does not violate visible sector
flavor symmetries, $A$-terms are not generated at one-loop.
However, two-loop contributions involving a visible sector
gaugino supermultiplet do give rise to
operators of the form
\beq
\int d^4 \theta~ \ln S ~ {D^2 \over \Dbox} Q H_u \bar{u}
~+~h.c.
\label{SSQHu}
\eq
as shown in Fig. \docLink{slac-pub-7237.tcx}[Aloop]{3},
and similarly for down-type quarks and leptons.
This operator is related by an integration by parts in superspace
to a correction to
the superpotential in the presence of the background spurion.
Inserting a single auxiliary component of the spurion
(equivalent to a visible sector gaugino mass in the one-loop subgraph)
gives a tri-linear $A$-term.
The momenta in the effective one-loop graph shown in Fig. \docLink{slac-pub-7237.tcx}[Aloop]{3}
are equally weighted in logarithmic intervals between
the gaugino mass and messenger scale.
Over these scales the gaugino mass is ``hard.''
The $A$-terms therefore effectively
vanish at the messenger scale, and
are generated
from renormalization group evolution below
the messenger scale, and finite contributions at
the electroweak scale.
At the low scale the $A$-terms have magnitude
$A \sim (\alpha / 4 \pi) m_{\lambda}\ln(M/m_{\lambda})$.
%where $h$ is a Yukawa coupling.
Note that $A$ is small compared with the scale
of the other soft terms,
%especially for electroweak states
%which only receive contributions proportional to the
%$W$-ino mass.
unless the logarithm is large.
%Gauge-mediated supersymmetry breaking generally gives
%rise to small $A$-terms.
As discussed in section \docLink{slac-pub-7237-0-0-3.tcx}[sspectrum]{3.2} the $A$-terms
do not have a qualitative effect on the superpartner
spectrum unless the messenger scale is large.
The general MSSM has a large number of $CP$-violating phases
beyond those of the standard model.
Since here the soft masses are flavor symmetric (up to very small
GIM suppressed corrections discussed in section (\docLink{slac-pub-7237-0-0-1.tcx}[UVinsensitive]{1.1})),
only flavor symmetric phases are relevant.
A background charge analysis \cite{16} fixes the
basis independent combination of flavor symmetric phases to be
${\rm Arg}(m_{\lambda} \mu (m_{12}^2)^*)$ and
${\rm Arg}(A^* m_{\lambda})$.
Since the $A$-terms vanish at the messenger scale only the first
of these can arise in the soft terms.
In the models of \cite{3} the auxiliary component of a
single field is the source for
all soft terms, giving a correlation among the phases such
that ${\rm Arg}(m_{\lambda} \mu (m_{12}^2)^*) = 0$ mod $\pi$.
In the minimal model, however, the mechanism for generating
$\mu$ and $m_{12}^2$ is not specified, and the phase is
arbitrary.
Below the messenger scale the particle content of the
minimal model is just that of the MSSM, along with the
gravitino discussed in section \docLink{slac-pub-7237-0-0-5.tcx}[collidersection]{5.2}
and appendix \docLink{slac-pub-7237-0-0-9.tcx}[appgoldstino]{C}.
At the messenger scale the boundary conditions for the
visible sector
soft terms are given by (\docLink{slac-pub-7237-0-0-2.tcx}[gauginomass]{4}) and
(\docLink{slac-pub-7237-0-0-2.tcx}[scalarmass]{6}), $\mu$, $m_{12}^2$, and $A=0$.
It is important to note that from the low energy
point of view the minimal model is just a set of
boundary conditions specified at the messenger scale.
These boundary conditions may be traded for the electroweak
scale parameters
\beq
( ~ \tan \beta~,~ \Lambda=F/S~,~{\rm Arg}~\mu~,~\ln M~)
\label{minpar}
\eq
The most important of these is $\Lambda$ which sets
the overall scale for the superpartner spectrum.
Since all the soft masses are related in the minimal model,
$\Lambda$ may be traded for any of these, such
as $m_{\tilde{B}}(M)$.
It may also be traded for a physical mass, such as
$m_{\na}$ or $m_{\lL}$.
In addition, as discussed in section \docLink{slac-pub-7237-0-0-3.tcx}[EWSB]{3.1}
$\tan \beta$ ``determines'' $m_{12}$ and $\mu$
in the low energy theory, and can have important effects
on the superpartner spectrum.
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