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\subsubsection{\usemenu{slacpub7237::context::slacpub723700521}{Missing Energy Signatures}}\label{subsubsection::slacpub723700521}
\label{misssig}
The minimal model has a conserved
$R$parity by assumption.
At moderate $\tan \beta$, $\chi_1^0$ is the lightest
standard model superpartner.
If decay to the Goldstino takes place well outside the detector
the classic signature of missing energy results.
However,
the form of the low lying spectrum largely dictates
the modes which can be observed.
The lightest charged states are the right handed sleptons,
$\lR^{\pm}$.
At an $e^+e^$ collider the most relevant mode
is then $e^+ e^ \to \lR^+ \lR^$
with $\lR^{\pm} \to l^{\pm} \na$.
For small $\tan \beta$ all the the sleptons are essentially
degenerate so the rates
%to $\tilde{e}_R$, $\tilde{\mu}_R$, and $\tilde{\tau}_R$
to each lepton flavor should be essentially identical.
For large $\tan \beta$ the $\stau_1$ can become
measurably lighter than $\tilde{e}_R$ and
$\tilde{\mu}_R$ (cf. Fig. \docLink{slacpub7237.tcx}[sfig17n]{7}).
If sleptons receive masses at the messenger scale
only from standard model gauge interactions, the only
source for splitting of $\tau_1$ from $\eR$ and $\tilde{\mu}_R$
is the $\tau$ Yukawa in renormalization group evolution and
mixing.
As discussed in section \docLink{slacpub7237003.tcx}[electroweaksection]{3.2.1} the largest
effect is from $\stau_L  \stau_R$
mixing proportional to $\tan \beta$.
A precision measurement of $m_{\stau_1}$ therefore
provides an indirect probe of
whether
$\tan \beta$ is large or not.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jfig{sfig2x}{fig19.ps}
{Production cross sections (fb) for $p \bar{p}$ initial state to the
final states $\chi_1^{\pm} \chi_2^0$ (upper solid line),
$\chi_1^+ \chi_1^$ (lower solid line),
$\lR^+ \lR^$ (dotdashed line),
$\nL \lL^{\pm}$ (upper dashed line), and
$\lL^+ \lL^$ (lower dashed line).
Lepton flavors are not summed.
The center of mass energy is 2 TeV, ${\rm sgn}(\mu)=+1$,
and $\Lambda=M$.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
At a hadron collider both the mass and gauge quantum numbers determine
the production rate for supersymmetric states.
The production cross sections for electroweak states in
$p \bar{p}$ collisions at $\sqrt{s} = 2$ TeV
(appropriate for the main injector upgrade at the Tevatron)
are shown in Fig. \docLink{slacpub7237.tcx}[sfig2x]{19} as a function of
$m_{\na}$ for MGM boundary conditions with $\Lambda=M$
and ${\rm sgn}(\mu)=+1$.
The largest cross section is for pairs of the mostly $W$ino
$SU(2)_L$ triplet
$(\chi_1^+, \chi_2^0, \chi_1^)$ through offshell $W^{\pm *}$ and
$Z^{0*}$.
Pair production of $\lR^+ \lR^$ is relatively
suppressed even though $m_{\lR} < m_{\chi_1^{\pm}}$ because
scalar production suffers a $\beta^3$ suppression near threshold,
and the right handed sleptons couple only through $U(1)_Y$
interactions via offshell $\gamma^*$ and $Z^{0*}$.
However,
as the overall scale of the superpartner masses is increased
$\lR^+ \lR^$ production becomes relatively more important
as can be seen in Fig. \docLink{slacpub7237.tcx}[sfig2x]{19}.
This is because production of the more massive $\ca \na$ and
$\chi_1^+ \chi_1^$ is reduced by the rapidly falling
parton distribution functions.
Pair production of $\lL^+ \lL^$, $\lL^{\pm} \nL$, and
$\nL \nL$ through offshell $\gamma^*$, $Z^{0*}$, and
$W^{\pm*}$ is suppressed relative to $\lR^+ \lR^$
by the larger left handed slepton
masses.
%Pair production of $\na \na$ through offshell $Z^{0*}$
%is suppressed by the small Higgsino admixtures of $na$,
%and through $t$ and $u$ channel squark exchange by
%the large squark masses \cite{signatures}.
The renormalization group and classical $U(1)_Y$ $D$term
contributions which slightly increase $m_{\lR}$,
and the renormalization group contribution which decreases
$m_{\na}$, have an impact on the
relative importance $\lR^+ \lR^$ production.
These effects, along with the radiatively generated $U(1)_Y$ $D$term,
``improve'' the kinematics of the leptons arising
from $\lR^{\pm} \to l^{\pm} \na$
since $m_{\lR}  m_{\na}$ is increased \cite{31}.
However, the overall rate is simultaneously reduced
%relative to
%$\chi_1^+ \chi_1^$ and $\chi_1^{\pm} \nb$
to a fairly insignificant level \cite{30}.
For example, with ${\rm sgn}(\mu)=+1$
an overall scale which would give an average of one
$\tilde{l}_R^+ \tilde{l}_R^$ event
in 100 pb$^{1}$ of integrated luminosity, would result
in over 80 %events with $\chi_1^{\pm} \chi_2^0$ final states.
chargino events.
As discussed in section \docLink{slacpub7237003.tcx}[electroweaksection]{3.2.1}, the shift
in the triplet $(\chi_1^+, \chi_2^0, \chi_1^)$ mass
from mixing with the Higgsinos is anticorrelated with
${\rm sgn}(\mu)$.
For ${\rm sgn}(\mu)=1$ the splitting between the right handed
sleptons and triplet is larger, thereby reducing slightly
chargino production.
For example, with ${\rm sgn}(\mu)=1$, a
single $\tilde{l}_R^+ \tilde{l}_R^$ event
in 100 pb$^{1}$ of integrated luminosity, would result
in 30 chargino events.
The relative rate of the $\lR^+ \lR^$ initial state
is increased in the minimal model for $\Rslash > 1$.
However, as discussed in Ref. \cite{30},
obtaining a rate comparable to $\chi_1^{\pm} \chi_2^0$
results in ``poor'' kinematics, in that the leptons
arising from $\lR^{\pm} \to l^{\pm} \na$ are fairly
soft since
$m_{\lR}  m_{\na}$ is reduced.
Note that for $\Rslash < 1$ chargino production becomes even more
important than $\lR^+ \lR^$ production.
%The $U(1)_Y$ $D$term effects which increase
%the fractional splitting between $m_{\lR}$ and $m_{\ca}$ become
%less important
%as the overall scale is increased.
In the minimal model pair production of
$\ca \nb$ and $\chi_1^+ \chi_1^$ are the most important modes at a
hadron collider.
The cascade decays of $\ca$ and $\nb$
are largely fixed by the form of the
superpartner spectrum and couplings.
If open, $\ca$ decays predominantly through its
Higgsino components to the Higgsino components of $\na$ by
$\ca \to \na W^{\pm}$.
Likewise, $\nb$ can also decay by $\nb \to \na Z^0$.
However, if open $\nb \to h^0 \na$ is suppressed by
only a single Higgsino component in
either $\nb$ or $\na$, and represents the dominant
decay mode for $m_{\nb} \gsim \mh + m_{\na}$.
The decay $\nb \to \lR^{\pm} l^{\mp}$ is suppressed by
the very small $B$ino component of $\nb$, and is only important
if the other twobody modes given above are closed.
If the two body decay modes for $\ca$ are closed, it decays
through threebody final states predominantly
through offshell $W^{\pm*}$.
Over much of the parameter space the minimal model therefore
gives rise to the signatures
$p \bar{p} \to W^{\pm} Z^0 + \EmissT$,
$W^{\pm} h^0 + \EmissT$, and
$W^+ W^ + \EmissT$.
If decay to the Goldstino takes place well outside the
detector, the minimal model yields
the ``standard'' chargino signatures at a hadron collider
\cite{43}.
If the intrinsic supersymmetry breaking scale is below a few
1000 TeV, the lightest standard model superpartner can
decay to its partner plus the Goldstino within the
detector \cite{41,42}.
For the case of $\na$ as the lightest standard model superpartner,
this degrades somewhat the missing energy, but leads to
additional visible energy.
The neutralino $\na$ decays by
$\na \to \gamma + G$ and if kinematically accessible
$\na \to (Z^0, h^0, H^0, A^0) + G$.
In the minimal model $\mA, \mH > m_{\na}$ so the only two
body final states potentially open are
$\na \to (\gamma, Z^0, h^0) + G$.
However, as discussed in section
\docLink{slacpub7237003.tcx}[electroweaksection]{3.2.1}, with MGM boundary conditions,
electroweak symmetry breaking implies %$ 3 m_1 < \mu 6 m_1$, so
that $\na$ is mostly $B$ino,
and therefore decays predominantly to the
gauge boson final states.
The decay $\na \to h^0 + G$
takes place only through the small Higgsino components.
In appendix \docLink{slacpub7237009.tcx}[appgoldstino]{C}
the decay rate to the $h^0$ final state is
shown to be suppressed by
${\cal O}(\mZ^2 m_{\na}^2 / \mu^4)$
compared with the
gauge boson final states, and is therefore
insignificant in the minimal model.
Observation of the decay $\na \to h^0 + G$ would
imply nonnegligible Higgsino components in $\na$,
and be a clear signal for deviations from the minimal
model in the Higgs sector.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jfig{sfig1x}{fig20.ps}
{The branching ratios for $\na \to \gamma + G$ (solid line)
and $\na \to Z^0 + G$ (dashed line) as a function of
$m_{\na}$ for $\Lambda=M$.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For example, as discussed in section \docLink{slacpub7237004.tcx}[addhiggs]{4.3},
$\Delta_$ large and negative leads to a mostly
Higgsino $\na$, which decays predominantly by
$\na \to h^0 + G$.
The branching ratios in the minimal model
for $\na \to \gamma +G$ and
$\na \to Z^0 + G$ are shown in Fig.
\docLink{slacpub7237.tcx}[sfig1x]{20} as a function of $m_{\na}$ for $\Lambda=M$.
%For $m_{\na} \gg \mZ$ the branching ratios approach
%${\rm Br}(\na \to \gamma + G) \simeq \cos^2 \theta_W$ and
%${\rm Br}(\na \to Z^0 + G) \simeq \sin^2 \theta_W$.
In the minimal model, with $\na$ decaying within the detector,
the signatures are the same as those given above, but
with an additional pair of $\gamma \gamma$,
$\gamma Z^0$, or $Z^0 Z^0$.
At an $e^+ e^$ collider $e^+ e^ \to \na \na \to
\gamma \gamma + \Emiss$ becomes the discovery mode
\cite{41,42,44}.
At a hadron collider the reduction in $\EmissT$
from the secondary decay is
more than compensated by the additional
very distinctive visible energy.
The presence of hard photons
significantly reduces the background compared
with standard supersymmetric signals
\cite{41,42,30,45,46}.
In addition, decay of $\na \to \gamma + G$
over a macroscopic distance leads to displaced
photon tracks, and of
$\na \to Z^0 + G$ to displaced charged particle tracks.
Measurement of the displaced vertex distribution
gives a measure of the supersymmetry breaking scale.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jfig{sfig3x}{fig21.ps}
{The ratio $\sigma( p \bar{p} \to \stau_1^+ \stau_1^) /
\sigma( p \bar{p} \to \eR^+ \eR^)$ as a function of
$\tan \beta$ for $\mbino(M)=115$ GeV and $\Lambda=M$.
The center of mass energy is 2 TeV.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the minimal model, for large $\tan \beta$ the
$\stau_1$ can become significantly lighter than
$\eR$ and $\tilde{\mu}_R$.
This enhances the $\stau_1^+ \stau_1^$ production cross
section at a hadron collider.
The ratio $\sigma( p \bar{p} \to \stau_1^+ \stau_1^) /
\sigma( p \bar{p} \to \eR^+ \eR^)$ for $\sqrt{s}=2$ TeV
is shown in Fig. \docLink{slacpub7237.tcx}[sfig3x]{21} as a function of
$\tan \beta$ for $\mbino(M)=115$ GeV and $\Lambda=M$.
Measurement of this ratio gives a measure of the
$\tilde{\tau}_1$ mass.
Within the minimal model this allows an indirect probe
of $\tan \beta$.
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