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\subsubsection{\usemenu{slacpub7237::context::slacpub723700325}{Relations Among the Superpartner Masses}}\label{subsubsection::slacpub723700325}
\label{relations}
The minimal model of gaugemediated supersymmetry breaking
represents a very constrained theory of the soft terms.
In this section we present some quantitative
relations among the superpartner masses.
These can be used to distinguish the MGM from other theories
of the soft terms and within the MGM can be logarithmically
sensitive to the messenger scale.
The gaugino masses at the messenger scale are in proportion
to the gauge couplings squared,
$m_1 : m_2 : m_3 = \alpha_1 : \alpha_2 : \alpha_3$.
Since $\alpha_i m_{\lambda_i}^{1}$ is a renormalization group
invariant at one loop, this relation is preserved to
lowest order at the electroweak scale,
where
$m_{\lambda_i}$ are the $\overline{DR}$ masses.
%the pole masses are related to the $\overline{\rm DR}$ masses by
%$m^{\rm pole}_{\lambda_i} = m_{\lambda_i}(1 + \delta_{\lambda_i})$
%and $ \delta_{\lambda_i}$ are finite corrections which depend
%in detail on the low scale spectrum \cite{finitegluino}.
The MGM therefore yields, in leading log approximation, the
same ratios of gaugino masses as high scale supersymmetry breaking
with universal gaugino boundary conditions.
``Gaugino unification'' is a generic feature of any
weakly coupled gaugemediated messenger sector
which forms a representation of any GUT group and
which has a single spurion.
The gaugino mass ratios are independent of $\Rslash$.
However, as discussed in section \docLink{slacpub7237002.tcx}[subvariations]{2.2},
with multiple sources of supersymmetry breaking and/or messenger
fermion masses the gaugino masses can be sensitive to
messenger Yukawa couplings.
``Gaugino unification'' therefore does {\it not} follow
just from the anzatz
of gaugemediation, even for messenger sectors which can
be embedded in a GUT theory.
An example of such a messenger sector is given in appendix
\docLink{slacpub7237008.tcx}[appnonmin]{B}.
Of course, a messenger sector which forms an incomplete GUT
multiplet (and modifies gauge coupling unification
unification at oneloop under renormalization
group evolution) does not in general yield
``gaugino unification'' \cite{29}.
%The scalar masses at the low scale receive
%classical electroweak $D$term corrections and oneloop renormalization
%group corrections from gaugino masses and the radiatively
%generated $U(1)_Y$ $D$term.
%The $\beta$functions may be integrated at oneloop to
%give the low scale scalar masses
%ratio stuff .....
With gaugemediated supersymmetry breaking
the scalar and gaugino masses are related at the messenger scale.
For a messenger sector well below the GUT scale,
$\alpha_3 \gg \alpha_2 > \alpha_1$, so the most important
scalargaugino correlations
are between squarks and gluino, left handed sleptons and
$W$ino, and right handed sleptons and $B$ino.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jfig{sfig4x}{fig13.ps}
{Ratios of $\overline{\rm DR}$ mass parameters
with MGM boundary conditions as a function
of the messenger scale:
$m_{\tilde{q}_R} / m_{\lR}$ (upper dashed line),
$m_{\lL} / m_{\lR}$ (lower dashed line),
$m_3 / m_{\tilde{q}_R}$ (upper solid line),
$m_2 / m_{\lL}$ (middle solid line), and
$m_1 / m_{\lR}$ (lower solid line).}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The ratios are of course proportional to $\Rslash$ which
determines the overall scale of the gaugino masses at
the messenger scale,
and are modified by renormalization group evolution
to the low scale.
Ratios of the $\overline{\rm DR}$ masses
$m_3 / m_{\tilde{q}_R}$,
$m_2 / m_{\lL}$, and
$m_1 / m_{\lR}$ in the minimal model
are shown in Fig. \docLink{slacpub7237.tcx}[sfig4x]{13} as a function
of the messenger scale.
As discussed in the next section, these ratios can be altered
with nonminimal messenger sectors.
In the minimal model, with $\Rslash=1$, a measurement
of any ratio $m_{\lambda_i}/ m $ gives a logarithmically
sensitive measurement of the messenger scale.
Because of the larger magnitude of the $U(1)_Y$
gauge $\beta$function the ratio
$m_{\lR} / m_1$ is most sensitive to the messenger scale.
%It increases from roughly for $M=\Lambda$ to
%for $M = 10^{5} \Lambda$.
Notice also that $m_3 / m_{\tilde{q}}$ is larger
for a larger messenger scale, while
$m_{\lL} / m_2$ and
$m_{\lR} / m_1$ decrease with the messenger scale.
Because of this disparate sensitivity,
within the anzatz of minimal gaugemediation,
both $\Rslash$ and $\ln M$ could be extracted from a precision
measurement of all three ratios.
For $\Rslash \leq 1$ the ratio of scalar mass
to associated gaugino mass is always $ \geq 1$
for any messenger scale.
Observation of a first or second generation scalar
lighter than the associated gaugino is therefore a
signal for $\Rslash >1$.
As discussed in section \docLink{slacpub7237004.tcx}[multiple]{4.2},
$\Rslash >1$ is actually possible with larger messenger
sector representations.
In fact, as discussed in appendix \docLink{slacpub7237007.tcx}[appgeneral]{A} in models
with a single spurion in the messenger sector,
$\Rslash$ is senstive to the index of the messenger sector
matter.
Additional matter which transforms under the standard model
gauge group between the electroweak and messenger scales
would of course modify these relations slightly through
renormalization group evolution contributions.
Ratios of
scalar masses at the messenger scale are related by ratios
of gauge couplings squared.
These ratios are reflected in the low energy spectrum.
In particular, since $\alpha_3 \gg \alpha_1$ if the messenger
scale is well below the GUT scale, the ratio
$m_{\tilde{q}} / m_{\lR}$ is sizeable.
Ratios of the $\overline{\rm DR}$ masses
$m_{\tilde{q}_R} / m_{\lR}$ and
$m_{\lL} / m_{\lR}$ are shown in Fig. \docLink{slacpub7237.tcx}[sfig4x]{13} as a function
of the messenger scale.
For $\Lambda = M$, $m_{\tilde{q_R}} / m_{\lR} \simeq 6.3$.
Notice that $m_{\tilde{q}_R} / m_{\lR}$ is smaller for larger
messenger scales,
and is fairly sensitive to $\ln M$.
This is because
$\alpha_3$ decreases rapidly at larger scales,
while $\alpha_1$ increases.
This sensitivity allows an indirect measure of $\ln M$.
%For $M = \Lambda $ $m_{\tilde{q_R}} / m_{\lR} \simeq 6.3$.
%for $M = 10^5 \Lambda$
%$m_{\tilde{q_R}} / m_{\lR} \simeq 4.0$.
The ratio $m_{\lL} / m_{\lR}$ is also fairly sizeable
but not as sensitive to the messenger scale.
For $\Lambda = M$, $m_{\lL} / m_{\lR} \simeq 2.1$.
It is important to note that with
$SU(3)_C \times SU(2)_L \times U(1)_Y$
gaugemediated supersymmetry
breaking, any parity and charge conjugate invariant
messenger sector
which forms a representation of any GUT group and
which has a single spurion
yields, at leading order, the same
scalar mass ratios as in the minimal model.
These mass ratios therefore represent a fairly generic
feature of minimal gaugemediation.
The sizeable hierarchy which arises in gaugemediated supersymmetry
breaking between
scalar masses of particles with different
gauge charges generally does not arise with universal
boundary conditions with a large overall scalar mass.
With gravity mediated supersymmetry breaking and universal
boundary conditions the largest hierarchy results
for the noscale boundary condition $m_0=0$.
In this case the scalar masses are ``gauginodominated,''
being generated in proportion to the gaugino masses under
renormalization group evolution.
The scalar mass
ratios turn out to be just slightly smaller
than the maximum gaugemediated ratios.
With noscale boundary conditions at $M_{GUT}$,
$m_{\tilde{q}_R} / m_{\eR} \simeq 5.6$ and
$m_{\lL} / m_{\lR} \simeq 1.9$.
However, the scalars in this case
are just slightly lighter than the
associated gauginos, in contrast to the MGM with $\Rslash=1$,
in which they are heavier.
It is interesting to note,
however, that for $M \sim 1000$ TeV
and $N=2$ or
$\Rslash \simeq \sqrt{2}$, gauge mediation coincidentally
gives almost identical
mass ratios as high scale supersymmetry breaking with
the noscale boundary condition at the GUT scale.
With gaugemediation,
scalar masses at the messenger scale receive contributions
proportional to gauge couplings squared.
Splitting among squarks with different gauge charges
can therefore be related to
right and left handed slepton masses (cf. Eq. \docLink{slacpub7237002.tcx}[scalarmass]{6}).
This can be quantified in the form of sum rules which involve
various linear combinations of
all the first generation scalar masses squared \cite{11}.
The splitting due to $U(1)_Y$ interactions with the messenger
sector can be quantified by
${\rm Tr}(Ym^2)$, where ${\rm Tr}$ is over
first generation sleptons and squarks.
As discussed in section \docLink{slacpub7237003.tcx}[electroweaksection]{3.2.1}
this quantity vanishes with gaugemediated boundary conditions
as the result of anomaly cancelation.
It is therefore interesting to consider the low scale quantity
$$
M_Y^2 =
{1 \over 2} \left( m_{\tilde{u}_L}^2 + m_{\tilde{d}_L}^2 \right)
2 m_{\tilde{u}_R}^2 + m_{\tilde{d}_L}^2
 {1 \over 2} \left( m_{\tilde{e}_L}^2 + m_{\tilde{\nu}_L}^2
\right)
+ m_{\tilde{e}_R}^2
$$
\beq
+ {10 \over 3} \sin^2 \theta_W \cos 2 \beta \mZ^2
\label{Ysumrule}
\eq
where the the sum of the $m^2$ terms is
${1 \over 2}{\rm Tr}(Ym^2)$ over the first generation, and
the ${\cal O}(\mZ^2)$ term is a correction for
classical $U(1)_Y$ $D$terms.
%Renormalization group contributions to (\ref{Ysumrule})
The contribution of the gaugino masses to $M_Y^2$ under
renormalization group evolution cancels at oneloop.
So this quantity is independent of the gaugino spectrum.
%The quantity $M_Y^2$ is not affected by gaugino masses
%under renormalization group evolution,
%and is therefore insensitive to the gaugino spectrum.
In addition, the $\beta$function for ${\rm Tr}(Ym^2)$ is homogeneous
\cite{24}
and independent of the Yukawa couplings at oneloop, even
though the individual masses are affected.
So if %${\rm Tr}(Ym^2) =0$
$M_Y^2=0$ at the messenger scale,
it is not generated above scalar thresholds.
It only receives very small contributions below the squark
thresholds of ${\cal O}((\alpha_1 / 4 \pi) m_{\tilde{q}}^2
\ln( m_{\tilde{q}} / m_{\tilde{l}} )) $.
The relation $M_Y^2 \simeq 0$ tests the assumption that
splittings within the squark and slepton spectrum
are related to $U(1)_Y$ quantum numbers.
The quantity $M_Y^2$ also vanishes in any model in which
soft scalar masses are univeral within GUT multiplets.
This is because ${\rm Tr} Y=0$ over any GUT multiplet.
%A very large class of models therefore give $M_Y^2 \simeq 0$,
%including all GUT models with flavor symmetric
%gravitymediated supersymmetry breaking
%with a hidden sector that does not transform under
%the GUT symmetry.
Within the anzatz of gaugemediation, a violation of
$M_Y^2 \simeq 0$ can result from a number of sources.
First, the messengers might not transform under
$U(1)_Y$.
In this case the $B$ino should also be very light.
Second, a large $U(1)_Y$ $D$term can be generated
radiatively if the messenger sector is not parity and
charge conjugate invariant.
Finally, the squarks and/or sleptons might
transform under additional gauge interactions
which couple with the messenger sector
so that ${\rm Tr}(Ym^2)$ does not vanish over
any generation.
This implies the existence of additional
electroweak scale matter in order to cancel the
${\rm Tr}(Y \{T^a, T^b \})$ anomaly, where $T^a$ is a
generator of the extra gauge interactions.
%It does vanish in models in which the soft scalar masses
%are universal within a GUT multiplet (which includes the
%case of universal boundary conditions).
Unfortunately, sum rules which involve near cancelation among squark
and slepton
masses squared, such as $M_Y^2 =0$, if in fact satisfied, are
often not particularly useful experimentally.
This is because the squark masses are split only at
${\cal O}(m_{\tilde{l}}^2 / m_{\tilde{q}}^2)$
by $SU(2)_L$ and $U(1)_Y$ interactions with the messenger
sector, and at ${\cal O}(\mZ^2 / m_{\tilde{q}}^2)$ from
classical $SU(2)_L$ and $U(1)_Y$ $D$terms.
Testing such sum rules therefore requires,
in general, measurements of
squark masses at the subGeV level, as can be determined
from the masses given in Table 1.
It is more useful to
consider sum rules, such as the ones given below,
which isolate the dominant splitting arising from $SU(2)_L$ interactions,
and are only violated by $U(1)_Y$ interactions.
These violations
are typically smaller than the experimental resolution.
The sum rules may then be tested with somewhat less precise
determinations of squark masses.
%A more useful quantity experimentally is simply
%${1 \over 2}{\rm Tr}(Ym^2)$
%and ${\rm Tr}(T_3 m^2)$, where the trace is over
%the first generation squarks
%\beq
%M_{\tidle{Q}Y}^2 \equiv
%{1 \over 2} \left( m_{\tilde{u}_L}^2 + m_{\tilde{d}_L}^2 \right)
% 2 m_{\tilde{u}_R}^2 + m_{\tilde{d}_L}^2
%\eq
The near degeneracy among squarks may be quantified by
the splitting between right handed squarks
\beq
\Delta_{\tilde{q}_R}^2 = m_{\tilde{u}_R}^2  m_{\tilde{d}_R}^2 .
\label{sumruleright}
\eq
Ignoring $U(1)_Y$ interactions,
this quantity is a renormalization group invariant.
%This quantity is independent of the gluino and $W$ino masses under
%renormalization group evolution
%and is only violated by $U(1)_Y$ interactions.
It receives nonzero contributions at
${\cal O}(m_{\eR}^2/ m_{\tilde{q}}^2 )$
from $U(1)_Y$ interactions with the messenger sector and
renormalization group contributions from the $B$ino mass,
and
${\cal O}(\mZ^2/ m_{\tilde{q}}^2 )$
from classical $U(1)_Y$ $D$terms at the low scale.
Numerically
$\Delta_{\tilde{q}_R}^2 / ( m_{\tilde{u}_R}^2+m_{\tilde{d}_R}^2)
\simeq 0$
to better than 0.3\% with MGM boundary conditions.
The near degeneracy between right handed squarks
is a necessary condition if squarks receive
mass mainly from $SU(3)_C$ interactions.
The quantity $\Delta_{\tilde{q}_R}^2$
also vanishes to the same order with universal boundary
conditions, but need not even approximately
vanish in theories in which
the soft masses are only universal within GUT multiplets.
An experimentally more interesting measure
which quantifies the splitting between left and
right handed squarks is
\beq
M_{LR}^2 =
m_{\tilde{u}_L}^2 + m_{\tilde{d}_L}^2 
\left( m_{\tilde{u}_R}^2 + m_{\tilde{d}_R}^2 \right)
 \left( m_{\lL}^2 + m_{\tilde{\nu}_L}^2
\right)
\eq
%This quantity is also independent of the gluino and $W$ino
%masses under renormalization group evolution
%and receives nonzero contributions only from $U(1)_Y$
%interactions.
This quantity is also a renormalization group invariant ignoring
$U(1)_Y$ interactions.
It formally vanishes at the same order as (\docLink{slacpub7237003.tcx}[sumruleright]{26}).
Numerically
$M_{LR}^2 / ( m_{\tilde{u}_R}^2 + m_{\tilde{d}_R}^2)
\simeq 0$
to better than 1\% with
MGM boundary conditions.
Without the left handed slepton contribution,
$M_{LR}^2 / ( m_{\tilde{u}_R}^2 + m_{\tilde{d}_R}^2)
\simeq 0$
can be violated by up to 10\%.
%This relation is more accessible than
This relation tests the assumption that the splitting between
the left and right handed squarks is due mainly to
$SU(2)_L$ interactions within the messenger sector.
The splitting is therefore correlated with the left handed slepton
masses, which receive masses predominantly from the same source.
If the squarks and sleptons receive mass predominantly from
gauge interactions with the messenger sector,
the masses depend only on gauge quantum numbers, and
are independent of generation up to very small
${\cal O}(m_f^2 / M^2)$ corrections at the messenger scale, where
$m_f$ is the partner fermion mass.
However, third generation masses are modified by Yukawa
contributions under renormalization group evolution and
mixing.
Mixing effects can be eliminated by considering the quantity
${\rm Tr}(m_{LR}^2)$ where $m_{LR}^2$ is the leftright
scalar mass squared
matrix.
In addition, it is possible to choose linear combinations
of masses which are independent of Yukawa couplings
under renormalization group evolution at
oneloop,
$m_{\tilde{u}_L}^2 + m_{\tilde{u}_R}^2  3 m_{\tilde{d}_L}^2$,
and similarly for sleptons
\cite{23}.
The quantities
\beq
M^2_{\tilde{t}  \tilde{q}} =
m_{\tilde{t}_1}^2 + m_{\tilde{t}_2}^2  2 m_t^2
 3 m_{\tilde{b}_2}^2
 \left( m_{\tilde{u}_L}^2 + m_{\tilde{u}_R}^2
 3 m_{\tilde{d}_L}^2 \right)
\label{thirdsquark}
\eq
\beq
M^2_{\tilde{\tau}  \tilde{e}} =
m_{\tilde{\tau}_1}^2 + m_{\tilde{\tau}_2}^2 % m_t^2
 3 m_{\tilde{\nu}_{\tau}}^2
 \left( m_{\tilde{e}_L}^2 + m_{\tilde{e}_R}^2
 3 m_{\tilde{\nu}_e}^2 \right)
\label{thirdslepton}
\eq
only receive contributions at twoloops under renormalization,
and in the case
of $M^2_{\tilde{t}  \tilde{q}}$ from $\tilde{b}$ mixing effects
which are negligible unless $\tan \beta$ is very large.
The relations $M^2_{\tilde{t}  \tilde{q}} \simeq 0$
and $M^2_{\tilde{\tau}  \tilde{e}} \simeq 0$
test the assumption that scalars with different
gauge quantum numbers have a flavor independent mass
at the messenger scale.
They vanish in any theory of the
soft terms with flavor independent masses at the messenger scale,
but need not vanish in theories in which alignment of the
squark mass matrices with the quark masses
is responsible for the lack of supersymmetric
contributions to flavor changing neutral currents.
Within the anzatz of gaugemediation, violations of these relations
would imply additional flavor dependent interactions with the
messenger sector.
%Significant
%splittings of the $\stau$ from $\tilde{e}$ and $\tilde{mu}$
%occur if $\tan \beta$ is very large, as discussed in section
%\ref{electroweaksection}.
If the quantities (\docLink{slacpub7237003.tcx}[thirdsquark]{28}) and (\docLink{slacpub7237003.tcx}[thirdslepton]{29})
are satisfied, implying the masses are generation independent
at the messenger scale, it is possible to extract the
Yukawa contribution to the renormalization group evolution.
The quantities
\beq
M_{h_t}^2 = m_{\tilde{t}_1}^2 + m_{\tilde{t}_2}^2  2 m_t^2
 \left( m_{\tilde{u}_L}^2 + m_{\tilde{u}_R}^2 \right)
\eq
\beq
M_{h_{\tau}}^2 = m_{\tilde{\tau}_1}^2 + m_{\tilde{\tau}_2}^2
%  2 m_{\tau}^2
 \left( m_{\tilde{e}_L}^2 + m_{\tilde{e}_R}^2 \right)
\eq
are independent of third generation mixing effects.
Under renormalization group evolution
$M_{h_t}^2$ receives an ${\cal O}((h_t / 4 \pi)^2 m_{\tilde{t}}^2
\ln(M/m_{\tilde{t}}) )$
negative contribution from the top Yukawa.
For moderate values of $\tan \beta$ this amounts to a
14\% deviation from
$M_{h_t}^2 / (m_{\tilde{t}_1}^2 + m_{\tilde{t}_1}^2)=0$
for $M = \Lambda$ and
grows to 29\% for $M= 10^5 \Lambda$.
Given an independent measure of $\tan \beta$ to fix the value
of $h_t$, this quantity gives an indirect probe of $\ln M$.
Unfortunately it requires a fairly precise measurement of the
squark and stop masses, but is complimentary to the
$\ln M$ dependence of the mass ratios of scalars with different
gauge charges discussed above.
The quantity $M_{h_{\tau}}^2$ is only significant if $\tan \beta$
is very large.
If this is the case, the splitting between $\tilde{\nu}_{\tau}$
and $\tilde{\nu}_e$, $\Delta^2_{\tilde{\nu}_{\tau}  \tilde{\nu}_e} =
m^2_{\tilde{\nu}_{\tau}}  m^2_{\tilde{\nu}_{e}}$,
gives an independent check of the renormalization contribution
through the relation
\beq
M_{h_{\tau}}^2 = 3 \Delta^2_{\tilde{\nu}_{\tau}  \tilde{\nu}_e}
\eq
%Finally, it is also possible to consider the quantities
%\beq
%M_{\tilde{Q}_L}^2 \equiv m_{\tilde{u}_L}^2  m_{\tilde{d}_L}^2
%  \mW^2 \cos 2 \beta
%\eq
%\beq
%M_{\tilde{l}_L}^2 \equiv m_{\tilde{\nu}_L}^2  m_{\tilde{e}_L}^2
%  \mW^2 \cos 2 \beta
%\eq
%where the ${\cal O}(\mW^2)$ term is a correction for
%the classical $SU(2)_L$ $D$term contribution to the masses.
%These quantity are also not affected by gaugino masses under
%renormalization group evolution,
%and must vanish in any supersymmetric theory in which
%$\tilde{u}_L  \tilde{d}_L$
%and $\eL  \nL$
%form $SU(2)_L$ doublets \cite{ssconstraints}.
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