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%% subsection 4.2 Multiple Messenger Generations [slac-pub-7237-0-0-4-2 in slac-pub-7237-0-0-4: slac-pub-7237-0-0-4-3]
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\subsection{\usemenu{slac-pub-7237::context::slac-pub-7237-0-0-4-2}{Multiple Messenger Generations}}\label{subsection::slac-pub-7237-0-0-4-2}
\label{multiple}
The minimal model contains a single messenger generation of
${\bf 5} + \overline{\bf 5}$ of $SU(5)$.
%As discussed in section \ref{subvariations}
This can be extended to any vector
representation of the standard model gauge group.
Such generalizations may be parameterized by the equivalent number of
${\bf 5} + \overline{\bf 5}$ messenger generations,
$N = C_3$, where $C_3$ is %the third Casimir
defined in Appendix \docLink{slac-pub-7237-0-0-7.tcx}[appgeneral]{A}.
For a ${\bf 10} + \overline{\bf 10}$ of $SU(5)$ $N=3$.
{}From the general expressions given in appendix \docLink{slac-pub-7237-0-0-7.tcx}[appgeneral]{A}
for gaugino and scalar masses, it is apparent that
gaugino masses grow like $N$ while scalar masses grow
like $\sqrt{N}$ \cite{42}.
This corresponds roughly to the gaugino mass parameter
$\Rslash = \sqrt{N}$.
Messenger sectors with larger matter representations
therefore result
in gauginos which are heavier relative to the scalars than
in the minimal model.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jfig{sfig4n}{fig14.eps}
{Renormalization group evolution of the
${\overline{\rm DR}}$ mass parameters with boundary conditions
of the two generation messenger sector.
The messenger scale is $M = 54$ TeV, $\Lambda=M$,
$\mbino(M) = 163$ GeV, and $\tan \beta = 3$.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The renormalization group evolution of the $\overline{\rm DR}$
parameters for $N=2$ with a messenger scale of
$M=54$ TeV, $\Lambda = M$,
$\mbino(M) = 163$ GeV, and $\tan \beta =3$ is
shown in Fig. \docLink{slac-pub-7237.tcx}[sfig4n]{14}.
The renormalization group contribution to the scalar masses
proportional to the gaugino masses is slightly larger than
for $N=1$.
Notice that at the low scale the renormalized
right handed slepton masses are slightly smaller than the
$B$-ino mass.
The physical slepton masses, however, receive a positive
contribution from the classical $U(1)_Y$ $D$-term, while
the physical $\na$ mass receives a negative contribution from
mixing with the Higgsinos
for ${\rm sgn}(\mu)=+1$.
With $N=2$ and the messenger scale not too far above $\Lambda$,
the $\lR$ and $\na$ are therefore very close in mass.
For the parameters of Fig. \docLink{slac-pub-7237.tcx}[sfig4n]{14}
$m_{\na} = 138$ GeV and $m_{\lR} = 140$ GeV,
so that $\na$ remains the lightest standard model superpartner.
The $D$-term and Higgsino mixing contributions become smaller
for a larger overall scale.
For $M=60$ TeV, $\Lambda=M$, and $\tan \beta =3$, the
$\na$ and $\lR$ masses cross
at $m_{\na} = m_{\lR} \simeq 153$ GeV.
Since the $B$-ino mass decreases while the right handed slepton
masses increase under renormalization,
$\meR > m_{\na}$ for a messenger scale well above $\Lambda$.
The near degeneracy of $\lR$ and $\na$ is just a coincidence
of the numerical boundary conditions for $N=2$
and ${\rm sgn}(\mu)=+1$.
For messenger sectors with $N \geq 3$ a right handed slepton
is naturally the lightest standard model superpartner.
The heavier gauginos which result for $N \geq 2$ only
slightly modify electroweak symmetry breaking through
a larger positive renormalization group
contribution to the Higgs soft masses,
and finite corrections at the low scale.
The negative contribution to the $\stau_1$ mass
relative to $\eR$ and $\tilde{\mu}_R$ from
mixing and Yukawa contributions to renormalization
are therefore also only slightly modified.
For a given physical scalar mass at the low scale, the
ratio $m_{\stau_1} / m_{\eR}$ is very similar to the $N=1$
case.
For $N \geq 3$, and the regions of the $N=2$ parameter space
in which $m_{\lR} < m_{\na}$, the $\stau_1$
is the lightest standard model superpartner.
As discussed in section \docLink{slac-pub-7237-0-0-5.tcx}[collidersection]{5.2}, collider signatures for
these cases are much different than for the MGM with $N=1$
with $\na$ as the lightest standard model superpartner.