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%% section 2 The Minimal Model of GaugeMediated Supersymmetry Breaking [slacpub7148002 in slacpub7148002: slacpub7148003]
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\section{\usemenu{slacpub7148::context::slacpub7148002}{The Minimal Model of GaugeMediated Supersymmetry Breaking}}\label{section::slacpub7148002}
If supersymmetry is broken at a low scale, the ordinary gauge
interactions can act as messengers of supersymmetry breaking.
The simplest possible messenger sector,
which preserves the successful prediction
of $\sin^2 \theta_W$ at low energy, are fields which possess the
quantum numbers of a single $\bf{5} + \bar{\bf{5}}$ of $SU(5)$.
The triplets, $q$ and $\bar{q}$, and doublets $\ell$ and $\bar{\ell}$,
of $\bf{5} + \bar{\bf{5}}$, couple to a single background field, $S$,
through a superpotential
$W = S( \lambda_1 q \bar{q} + \lambda_2 \ell \bar{\ell})$.
The field $S$ breaks both $U(1)_R$ and supersymmetry through
its scalar and auxiliary components respectively.
Integrating out the messenger sector fields gives rise
radiatively to
both scalar and gaugino masses.
The visible sector
gluino and squarks in this model are heavy enough to be beyond
the reach of the Tevatron.
The masses of the
lefthanded sleptons, $W$inos (partners of the $SU(2)_L$ gauge bosons),
righthanded sleptons, and $B$ino (partner of the $U(1)_Y$ gauge boson),
are in the ratio $2.5~:~2~:~1.1~:~1$.
We will refer to this model as the Minimal GaugeMediated (MGM) model
of supersymmetry breaking.
The dimensionful terms in the Higgs sector required
to break the $U(1)_{PQ}$ and $U(1)_{RPQ}$ symmetries,
$W=\mu H_1 H_2$ and $V=m_{12}^2 H_1 H_2 + h.c.$,
must arise from additional interactions
\cite{4,10,11}, and may be taken as free
parameters in the minimal model.
%$m_{12}^2$ and
Values of $\mu $ larger than roughly 150 GeV
are mildly preferred in order to suppress
charged Higgs contributions to ${\rm Br}(b \to s \gamma)$ \cite{11}.
For the mass ranges considered below, the lightest two neutralinos,
$\na$ and $\nb$,
and lightest chargino $\ca$,
are then mostly gaugino, with small Higgsino mixtures.
In the $\mu \gg m_{\na}$
limit, the spectrum of light states is in the ratios given
above, and the most important parameter which determines the
phenomenology at the Tevatron is just the overall scale.
The production rate for the light states depends on both
the masses and charges.
If the lightest neutralinos are mostly gaugino,
$\na$ is mostly $B$ino.
Pair production of $\na \na$ through offshell $Z^*$ exchange
is then suppressed by a small coupling,
and through $t$ and $u$ channel squark exchange
by the large squark masses.
However,
pair production of $\lR \lR$ through offshell $\gamma^*$ and $Z^*$,
and subsequent cascade decay
$\lR \to \l \na$, leads to the final state $\llgg + \EmissT$
\cite{5,6}.
In addition, pair production of charginos
and neutralinos through an offshell $W^*$
(via coupling to the $W$ino components) leads to
comparable production rates for $\chi^+_1 \chi^_1$ and
$\nb \ca$.
For large $\mu$ the neutralino $\nb$ decays predominantly by
$\nb \to \lR l$.
For any reasonable $\mu$ and
$m_{\ca} > m_{\na} + m_{W}$, the chargino $\ca$
decays predominantly
through its Higgsino components to the Higgsino components of $\na$
by $\ca \to \na W$.
On the other hand, for $m_{\ca} < m_{\na} + m_{W}$,
$\ca$ decays to three body final
states predominantly through offshell $W^*$ and $\lR^*$.
The total cross sections which arise at the Tevatron
in this model with $m_{\chi^0_1} = 100$ GeV and $\mu \gg m_{\chi^0_1}$
are given in Table~1.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\centering
\begin{tabular}{lccc}
\hline \hline
& $\llgg$ & $W\leplep \gag$ & $WW \gag$ \\ \hline
$\lR \lR$ & 6 &  &  \\
$\na \cb$ &  & 11.5 &  \\
$\capos \caneg$ &  &  & 18.8 \\ \hline
Total & 18 & 34.4 & 18.8 \\
\hline \hline
\end{tabular}
\label{tableMGM}
\caption{Production cross sections (fb) for each lepton flavor
within the MGM for $m_{\chi_1^0} = 100$ GeV, $\mu \gg m_{\chi_1^0}$,
and $m_{\tilde l_R} =110$ GeV, as discussed in section 2.
The center of mass energy is 1.8 TeV. Each final
state has $\not \! \! E_{T}$. The total cross sections in each channel
are summed over all lepton flavors.}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this case $\na$ is pure $B$ino and $\nb$ and $\ca$ are
pure $W$ino.
In the $\sin^2 \theta_W \to 0$ limit
$\sigma(\nb \ca) = 2 \sigma( \capos \caneg)$.
For finite $\mu$ constructive or destructive interference with
the Higgsino mixtures in $\nb$ and $\ca$ can significantly
affect the cross sections.
For example, with $\mu = 250$ GeV, $m_{B} = 100$ GeV, and
$\tan \beta \equiv \langle H_2 \rangle / \langle H_1 \rangle = 2$,
$\sigma(\nb \ca) \simeq 25.4$ fb and
$\sigma(\capos \caneg) \simeq 13.7$ fb.
The branching ratios can also be modified for finite $\mu$.
For the above parameters, $m_{\nb}  m_{\na} > m_Z$ and so
$\nb$ decays predominantly through its Higgsino components to
the Higgsino components of $\na$ by $\nb \to \na Z$.
The final states $W \llgg + \EmissT$
are then replaced by $WZ \gag + \EmissT$.
The total rates of course depend on the overall
scale, but the relative rates in the various channels
are a slow function of the overall scale.
The final states $\llgg + \EmissT$, $W \leplep \gag+\EmissT$,
and $WW\gag + \EmissT$, therefore represent an important
test of the MGM in the large $\mu$ limit.
The relative rates in the $W \llgg + \EmissT$ and
$WZ \gag + \EmissT$ are sensitive to the magnitude
of $\mu$, as discussed above.
In addition, if the usual gauge interactions are the dominant
messengers of supersymmetry breaking, it follows that
the righthanded sleptons
are essentially degenerate.
Final states for each lepton flavor should have equal rate.
%Violations of lepton universality in two photon events
%would likely indicate a much richer familydependent
%messenger sector.
Because of the relatively large mass of the lefthanded sleptons,
pair production of $\lL \lL$ through offshell $\gamma^*$ and $Z^*$,
and $\nL \lL$ through offshell $W^*$,
are suppressed in the MGM.
For example, with the parameters given in Table~1,
$\sigma(\nL \lL) / \sigma(\lR \lR) \simeq 0.04$ and
$\sigma(\lL \lL) / \sigma(\lR \lR) \simeq 0.025$.
An important feature of the MGM is the kinematics of the
partons in the final states.
Since the mass splitting between $\lR$ and the
$B$ino is so small, the decay $\lR \to l \na$ results in
fairly soft leptons.
In contrast, for the decay $\na \to \gamma G$, the photon
receives half the $\na$ mass in the rest frame, resulting
in a larger average photon energy.
In addition, since $\na$ is generally boosted in the lab frame, the
photon $E_T$ spectrum is much flatter than that of the leptons.
The $E_T$ and $\EmissT$ for the $\llgg + \EmissT$ final state
with the parameters of Table 1 are shown in Fig. 1 \cite{12}.
This illustrates how the kinematics can be used to infer mass
splittings within a decay chain.
%It is important to reemphasize the distinct kinematic features
%of the MGM. When the lightest neutralino decays to the massless
%photon and Goldstino, half of its mass goes into the total photon
%energy resulting in a large $E_T$ of the photons. In general,
%the neutralino is boosted with respect to the lab frame which
%will further increase the maximum photon energy and
%flatten out its $E_T$ distribution. These direct decays will show
%up in the detector as very hard photons. Also, since the $\tilde l_R$
%is so close in mass to the $\chi^0_1$, decays such as
%$\tilde l_R\to l\chi^0_1$ will generate soft leptons.
%These kinematic features are both important experimental flags
%for MGM models: {\it hard photons, and soft leptons.} In Fig.~\ref{kineReR}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jfig{kineReR}%{/afs/slac/u/th/jwells/fortran/isajet/plots/fig1.ps}
{dtfig1.ps}
{The
$E_T$ and $\EmissT$ spectra for the $\llgg + \EmissT$ channel
in the MGM model with the parameters given in Table 1.
%$m_{\lR} = 110$ GeV, $m_{\na} = 100\gev$,
%as discussed in section 2.
%The center of mass energy is 1.8 TeV.
The two solid lines are
the $E_T$ distributions of the hard and soft electron.
Similarly, the dashed lines are the $E_T$ distributions of the
hard and soft photon. The dotted line is the $\EmissT$ distribution.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%we have constructed~\cite{isajet} a kinematic profile of the decay
%products for $\tilde l_R\tilde l_R$ production for a typical set of
%parameters. One can quickly see that the photons are indeed much
%more energetic, on average, than the electrons, and the photon
%transverse energy is broader.
%Finally, the MGM is the most predictive and simplest model of
%gauge mediated supersymmetry breaking. The slepton crosssections,
%decays, and kinematics are largely determined by just one scale parameter,
%and so it is straightforward to raise the mass scale and find discovery limits
%on the slepton mass which are possible at the Tevatron.
%We estimate that with $10\xfb^{1}$ at a high luminosity Tevatron upgrade
%running at $2\tev$ center of mass energy, $\tilde l_R$ pair
%production can be probed
%up to $m_{\tilde l_R}\lsim 210\gev$. This mass reach is significantly
%higher than the capability at LEP190 ($m_{\tilde l_R}\lsim 90\gev$).
%Note that in the gravity mediated supersymmetry breaking scenario,
%which does not have two hard photons present in slepton pair production,
%the prospects of detecting sleptons is much more daunting~\cite{sleptev}.
%To summarize this section, we have found in the Minimal Gauge Mediated
%supersymmetry breaking model (MGM) that (1) the masses of the
%scalars and gauginos are fixed by one scale, (2) the lefthanded slepton
%production crosssection is always much less than the righthanded slepton
%production, (3) the small mass difference
%between $\tilde l_R$ and $\chi^0_1$ leads to soft electrons in the
%final state, and (4) the massive neutralino decaying into massless
%photon and Goldstino leads to hard photons with a broad energy distribution.
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