%% ****** Start of file zgg.tex ****** %%
% Article on Decay Width of Z to 2 Gluinos:  Zumin Luo
% Draft as of February 28, 2003
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\def\etal{{\it et.~al.}}
\def\ie{{\it i.e.}}
\def\eg{{\it e.g.}}

\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\bean{\begin{eqnarray*}}
\def\eean{\end{eqnarray*}}
\def\bary{\begin{array}}
\def\eary{\end{array}}
\def\bi{\bibitem}
\def\bit{\begin{itemize}}
\def\eit{\end{itemize}}

\def\lan{\langle}
\def\ran{\rangle}
\def\lra{\leftrightarrow}
\def\la{\leftarrow}
\def\ra{\rightarrow}
\def\dash{\mbox{-}}

\def\re{\rm Re}
\def\im{\rm Im}
\def\eps{\epsilon}
\def\sg{\tilde g}
\def\sb{\tilde b}
\def\vk{\bf k}
\def\vp{\bf p}
\def\ua{\uparrow}
\def\da{\downarrow}

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\Large \centerline {\bf LOWER BOUND ON $\Gamma(Z \to \sg \sg)$}
\centerline{\bf BASED ON UNITARITY \footnote{Enrico Fermi
Institute preprint EFI 03-02}} \normalsize

\vskip 2.0cm

\centerline {Zumin Luo~\footnote{zuminluo@midway.uchicago.edu}}
\vskip 0.5cm

\centerline {\it Enrico Fermi Institute and Department of Physics}
\centerline{\it University of Chicago, 5640 S. Ellis Avenue,
Chicago, IL 60637}

\vskip 4.0cm

\begin{quote}
The $Z$ boson can decay to a pair of light (12--16 GeV) gluinos
through loop-mediated processes.  In the presence of a light
(2--5.5 GeV) bottom squark, the decay width of $Z \to \sg \sg$ is
found to be at least of the order 0.01 MeV based on unitarity of
the $S$-matrix. Implications of this lower bound are discussed.

\end{quote}

\bigskip

\noindent

PACS Categories: 11.30.Pb, 12.60.Jv, 13.38.Dg, 14.80.Ly

\vfill
\newpage


\section{INTRODUCTION \label{sec:int}}

A relatively light (12--16 GeV) gluino $\sg$, along with a lighter
(2--5.5 GeV) bottom squark $\sb$, has been proposed
\cite{Berger:2000mp} to explain the excess of the cross section
for bottom quark production at hadron colliders. The $\sb$ squark
is assumed to be a mixture of $\sb_L$ and $\sb_R$, the
superparterners of $b_L$ and $b_R$. Other supersymmetric (SUSY)
particles, except the other sbottom $\sb'$ and one of the top
squarks, are assumed to be sufficiently heavy. The masses of
$\sb'$ and the light stop are constrained by the electroweak data
to be below 180 GeV and 98 GeV, respectively \cite{Cho:2002mt}. We
follow the convention in \cite{Berger:2000mp} to define
%
\be \left( \begin{array}{c} \sb \\ \sb' \end{array} \right) =
\left(
\begin{array}{c c} \cos \theta_{\sb} & \sin \theta_{\sb} \cr
                        - \sin \theta_{\sb} & \cos \theta_{\sb}
\end{array} \right)
\left( \begin{array}{c} \tilde b_R \\ \tilde b_L \end{array}
\right)~~~.
\ee
%
The introduction of these new particles gives rise to new
interactions in various processes. For example, the total decay
width of the $\Upsilon$ is raised since the decay $\Upsilon \to
\sb\sb^*$ \cite{Berger:2001jb} is now permitted; the decay width
of the $Z$ boson is also changed \cite{Clavelli:1996zm,
Cao:2001rz}. As a result, the extraction of the strong coupling
constant $\alpha_s$ at these two mass scales will be affected. By
contributing to the $\beta$-function, these SUSY particles slow
down the evolution of $\alpha_s$ with energy scale
\cite{Clavelli:zv}. The situation has recently been studied in
detail by Chiang {\it et al.} in \cite{Chiang:2002wi} and no
clear-cut decision can be made in favor of either the Standard
Model evolution or the evolution in the light-gluino/light-sbottom
scenario. The partial decay width $\Gamma(Z \to \sg \sg)$ remains
a key quantity to be determined. A better evaluation of $\Gamma(Z
\to \sg \sg)$, among other things, can improve our understanding
of the effect of this scenario on the electroweak measurables at
the $Z$ pole and hence the determination of $\alpha_s(M_Z)$ in the
scenario.

To validate the proposition of these new particles, direct
searches for light gluinos and light sbottoms at $e^-e^+$
colliders will definitely play a key role. An analysis has been
presented recently by Berge and Klasen \cite{Berge:2002ev} of
gluino pair production at linear $e^-e^+$ colliders. However, they
only considered the mass range $m_{\sg} \ge 200$ GeV. Production
of light gluino pairs was studied, though, by
Ref.~\cite{Nelson:1982cu} and its updated version
\cite{Kileng:1994vc}. However, a light sbottom was not included in
either of these calculations. Production of light gluinos at
$p{\bar p}$ colliders was considered by Terekhov and Clavelli
\cite{Terekhov:1996rp} but without inclusion of the light sbottom
either. Therefore, an analysis of light gluino production in the
presence of a light sbottom will be very useful for gluino
searches.

Previous calculations \cite{Kileng:1994vc, Kane:xp,
Djouadi:1994js} indicate that the branching ratio of $Z \to \sg
\sg$ falls in the range of $10^{-5}$ to $10^{-4}$ for a wide range
of MSSM parameters but without the light sbottom $\sb$. This gives
a partial width of less than ${\cal O}(1)$ MeV. It is argued in
\cite{Chiang:2002wi} that inclusion of $\sb$ should only change
the partial width by a very small amount. It is the purpose of
this paper to verify this argument and give a reasonable lower
bound on $\Gamma(Z \to \sg \sg)$. A full calculation involves
evaluation of triangle Feynman diagrams \cite{Nelson:1982cu,
Kileng:1994vc}. Though ultraviolet divergences cancel within a
complete isodoublet, one has to remove singularities due to
on-shell particles \cite{Djouadi:1994js}. However, since $2 m_b <
M_Z$ and $2 m_{\sb} < M_Z$, the Feynman amplitudes have an
imaginary part which is finite and can be calculated precisely in
an easier way. It is likely that the imaginary parts provide a
fairly good estimate of the full amplitudes as long as
cancellations of loop contributions with high internal momenta are
implemented. The situation is analogous to the $K_S$--$K_L$ mass
difference and the decay $K_L \to \mu^+ \mu^-$ \cite{Sehgal:db}.
In each case the high-momentum components of the loop diagrams are
suppressed (here, through the presence of the charmed quark
\cite{Gaillard:1974hs}), leaving the low-mass on-shell states
($\pi \pi$ or $\gamma \gamma$, respectively) to provide a good
estimate of the matrix element.

The paper is organized as follows: Section~\ref{sec:unit}
establishes the unitarity relation of the ${\cal M}$ matrix
elements; explicit expressions for Dirac and Majorana spinors are
given in Section~\ref{sec:dm}; amplitudes of the cut diagrams
(Fig.~\ref{fig:Zgg}) are calculated in Section~\ref{sec:amps} and
the results are listed in the Appendix; a lower bound on $\Gamma(Z
\to \sg \sg)$ is presented in Section~\ref{sec:lb}; implications
of this lower bounds are discussed in Sections~\ref{sec:gs} and
\ref{sec:ra}; Section~\ref{sec:sum} summarizes.

\section{UNITARITY RELATION
\label{sec:unit}}

Let us first review the decays $K_{L,S} \to l^- l^+$ considered in
Ref.~\cite{Sehgal:db}. As is the case with $Z \to \sg \sg$, both
decays are forbidden at the tree level. However, they can occur
through a two-photon ($\gamma\gamma$) intermediate state. Other
intermediate states such as $\pi\pi\gamma$ and $3\pi$ are much
less important. As a consequence of the unitarity of the
$S$-matrix ($S^{\dagger}S\equiv (1+ i T)^{\dagger}(1+i T)=1$), the
$T$-matrix element between the initial state $K_{L,S}$ and the
final state $l^- l^+$ satisfies the following relation
%
\bea {\rm Im}\left [\langle l^-l^+|T|K_{L,S}\rangle \right ] & = &
\frac{1}{2}\left [\langle l^-l^+|T^{\dagger}T|K_{L,S}\rangle
\right ] \nonumber \\
& = & \frac{1}{2}\sum_{\epsilon, \epsilon'}
\int\frac{d^3k}{(2\pi)^3}\frac{d^3k'}{(2\pi)^3}\frac{1}{2E}\frac{1}{2E'}\langle
l^- l^+ | T^{\dagger}
|\gamma(k,\epsilon)\gamma'(k',\epsilon')\rangle \nonumber \\
&& \hspace{15 ex} \times \langle
\gamma(k,\epsilon)\gamma'(k',\epsilon')|T|K_{L,S}\rangle ,
\label{eqn:unit1} \eea
%
where Im denotes the imaginary part and
$|\gamma(k,\epsilon)\gamma'(k',\epsilon')\rangle$ is a real
two-photon state with $k$ and $k'$, $\epsilon$ and $\epsilon'$
specifying the 4-momenta and 4-polarizations, respectively. Since
the $T$-matrix elements can be expressed as the invariant ${\cal
M}$ matrix elements multiplied by 4-momentum-conserving
$\delta$-functions, Eq.~(\ref{eqn:unit1}) becomes
%
\bea {\rm Im}\left [{\cal M}(K_{L,S} \to l^-l^+) \right ] & = &
\frac{1}{2}\sum_{\epsilon, \epsilon'}
\int\frac{d^3k}{(2\pi)^3}\frac{d^3k'}{(2\pi)^3}\frac{1}{2E}\frac{1}{2E'}
\langle \gamma(k,\epsilon)\gamma'(k',\epsilon')|{\cal
M}|K_{L,S}\rangle \nonumber \\
&& \hspace{2 ex} \langle \gamma(k,\epsilon)\gamma'(k',\epsilon')|
{\cal M} | l^-l^+ \rangle^* (2\pi)^4 \delta^{(4)}(p-k_1-k_2) ,
\label{eqn:unit2} \eea
%
times an overall $\delta^{(4)}(p-p_1-p_2)$, with $p$, $p_1$ and
$p_2$ being the 4-momenta of $K_{L,S}$, $l^-$ and $l^+$,
respectively. It is expected that the real part of the amplitude
is roughly of the same order as the imaginary part, so that the
actual decay width will exceed the lower bound based on the
imaginary part by only a small factor.

Quite similarly, the imaginary part of the invariant matrix
element ${\cal M}(Z \to {\tilde g}{\tilde g})$ can be written as
%
\be \label{eqn: unit} {\rm Im} \left[{\cal M}(Z \to {\tilde
g}{\tilde g}) \right] = \frac{1}{2} \sum_f \int d\Pi_f {\cal M}(Z
\to f){\cal M}^*({\tilde g}{\tilde g} \to
f)(2\pi)^4\delta^{(4)}(p-\sum_{i=1}^{n_f} p_i) , \ee
%
where the sum runs over all possible intermediate on-shell states
$f$ and $d\Pi_f = \prod_{i=1}^{n_f}\frac{d^3
p_i}{(2\pi)^3}\frac{1}{2E_i}$ with $n_f$ being the numbers of
particles in state $f$ and $p_i$ being the 3-momenta of the
particles. Since ${\tilde b}$ is the lightest supersymmetric
particle in the scenario and all other supersymmetric particles
(except ${\tilde g}$) are expected to be heavier than $M_Z/2$, we
only need to consider the cases where $f$ is $b {\bar b}$ and
${\tilde b} {\tilde b}^*$.

The integral over the phase space $\Pi_f$ can be simplified to the
integral over the solid angle $\Omega$. In the case where $n_f=2$,
we have $\int d\Pi_f (2\pi)^4\delta^{(4)}(p-\sum_{i=1}^{n_f} p_i)
= \frac{1}{32\pi^2}\int d\Omega$, where $d\Omega = \sin\theta
d\theta d\phi$. Here $\theta$ and $\phi$ are polar angles of ${\bf
p}_1$ with respect to the $z$-axis.


\section{DIRAC AND MAJORANA SPINORS
\label{sec:dm}}
%
\begin{figure}
\centerline{\epsfysize = 6 in
\epsffile{cut.eps}
}
\caption[]{\label{fig:Zgg}
  Cut Feynman diagrams for $Z \to \sg \sg$: (a) $Z \to (b{\bar b})^*
  \to \sg \sg$, (b) $Z \to (\sb \sb^*)^* \to \sg \sg$. Similar
  diagrams with $\sg(k_1) \leftrightarrow \sg(k_2)$ are not shown but
  should be included in the calculation with an overall minus sign. }
\end{figure}

The uncrossed cut Feynman diagrams that contribute to the
imaginary part of the full amplitude are shown in
Fig.~\ref{fig:Zgg}. The crossed diagrams $\sg(k_1) \leftrightarrow
\sg(k_2)$ are not shown but should be included in the calculation
with an overall minus sign. In the center-of-mass frame of the $Z$
boson, the 4-momenta of the final gluinos are $k_1 = (E, \vk)$ and
$k_2 = (E, -\vk)$, where $E=M_Z/2$ and ${\vk} = (0, 0, |{\vk}|)$.
Suppose $\vk$ is along the $z$-axis and the polarizations of the
$Z$ are quantized along this axis, i.e., $\epsilon^\mu = (0, 1,
\pm i, 0)/\sqrt{2}$ or $(0, 0, 0, 1)$, corresponding to helicities
$\lambda = \pm 1$ or $0$ respectively. The 4-momenta of the
intermediate bottom quarks are $p_1 = (E, \vp)$ and $p_2 = (E,
-\vp)$, with $\vp = |\vp|(\sin\theta\cos\phi, \sin\theta\sin\phi,
\cos\theta)$.

Before evaluating the amplitudes of the diagrams, we first specify
the convention we will use and give explicit expressions for the
Dirac and Majorana spinors. We adopt the convention of spinors of Peskin
and Schroeder \cite{peskin}, in which the metric tensor
$g_{\mu\nu}={\rm Diagonal}(1,-1,-1,-1)$ and
%
\be \gamma^0=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0
\end{array} \right), \hspace{4ex} \gamma^5=\left(\begin{array}{cc} -1 & 0 \\ 0 &
1 \end{array} \right) \hspace{2ex} {\rm and} \hspace{2ex}
\gamma^i=\left(\begin{array}{cc} 0 & \sigma^i \\ -\sigma^i & 0
\end{array} \right), i=1, 2, 3, \ee
%
where $\sigma^i$ are the Pauli matrices. The Dirac spinors in this
representation can be written as
%
\be \label{eqn:spinors} u^s(p)=\left( \begin{array}{c}
\sqrt{p\cdot\sigma}\xi^s_p \\
\sqrt{p\cdot{\bar \sigma}}\xi^s_p \end{array}\right),
\hspace{10ex}
v^s(p) = \left( \begin{array}{c} \sqrt{p\cdot\sigma}\eta^s_p \\
-\sqrt{p\cdot{\bar \sigma}}\eta^s_p \end{array}\right), \ee
%
where $\sigma^\mu=(1,\sigma^i)$, ${\bar
\sigma}^\mu=(1,-\sigma^i)$, $s$ is the spin index and $p$
specifies the corresponding 4-momentum. The two-component spinors
$\xi^s_p$ and $\eta^s_p$ are related via a ``spin-flipping''
matrix: $\eta^s_p=-i\sigma^2(\xi^s_p)^*$. We will use the arrows
$\ua$ and $\da$ to denote spin up and spin down along $\vp$,
respectively. After taking the square roots of the $2\times2$ matrices in
Eq.~(\ref{eqn:spinors}), we obtain
%
\bea u^{\ua}(p_1) = \left( \begin{array}{c}
\sqrt{E-|\vp|}\xi^{\ua} \\ \sqrt{E+|\vp|}\xi^{\ua}
\end{array} \right) & \hspace{5ex} & u^{\da}(p_1) = \left(
\begin{array}{c} \sqrt{E+|\vp|}\xi^{\da} \\ \sqrt{E-|\vp|}\xi^{\da}
\end{array} \right) \nonumber \\
v^{\ua}(p_2) = \left( \begin{array}{c} \sqrt{E-|\vp|}\eta^{\ua}
\\ -\sqrt{E+|\vp|}\eta^{\ua}
\end{array} \right) & \hspace{5ex} & v^{\da}(p_2) = \left(
\begin{array}{c} \sqrt{E+|\vp|}\eta^{\da}
\\ -\sqrt{E-|\vp|}\eta^{\da}
\end{array} \right) ,
\eea
%
where
%
\bea \xi^\uparrow \equiv \xi^{\ua}_{p_1}= \left( \begin{array}{c}
\cos \frac{\theta}{2}
\\ e^{i\phi}\sin \frac{\theta}{2} \end{array} \right)
& \hspace{1cm} {\rm and} \hspace{1cm} & \xi^\downarrow \equiv
\xi^{\da}_{p_1} = \left(
\begin{array}{c} -e^{-i\phi}\sin \frac{\theta}{2} \\ \cos
\frac{\theta}{2} \end{array} \right) \nonumber \\
\eta^\uparrow \equiv \eta^{\ua}_{p_2}= \left(\begin{array}{c} -\sin
\frac{\theta}{2} \\
e^{i\phi}\cos \frac{\theta}{2} \end{array} \right) & \hspace{1cm}
{\rm and} \hspace{1cm} & \eta^\downarrow \equiv \eta^{\da}_{p_2}=
\left(\begin{array}{c} e^{-i\phi}\cos \frac{\theta}{2} \\ \sin
\frac{\theta}{2}
\end{array} \right)  \eea
%
It can be easily verified that $u(p, s)=C{\bar v}^T(p, s)$ and
$v(p, s)=C{\bar u}^T(p, s)$, where $T$ means ``transpose'' and
$C=i\gamma^0\gamma^2$ is the charge conjugate matrix. The Majorana
spinors $u_{\sg}$ and $v_{\sg}$ also satisfy these relations
\cite{Haber:1984rc}. Thus we can immediately write
%
\bea u_{\sg}^{\ua}(k_1) = \left( \begin{array}{c}
\sqrt{E-|\vk|}\zeta_+
\\ \sqrt{E+|\vk|}\zeta_+
\end{array} \right) & \hspace{5ex} & u_{\sg}^{\da}(k_1) =
\left( \begin{array}{c} \sqrt{E+|\vk|}\zeta_- \\
\sqrt{E-|\vk|}\zeta_-
\end{array} \right) \nonumber \\
u_{\sg}^{\ua}(k_2) = \left( \begin{array}{c} \sqrt{E+|\vk|}\zeta_+
\\ \sqrt{E-|\vk|}\zeta_+
\end{array} \right) & \hspace{5ex} & u_{\sg}^{\da}(k_2) =
\left( \begin{array}{c} -\sqrt{E-|\vk|}\zeta_- \\
-\sqrt{E+|\vk|}\zeta_-
\end{array} \right) \nonumber \\
v_{\sg}^{\ua}(k_1) = \left( \begin{array}{c} \sqrt{E+|\vk|}\zeta_-
\\ -\sqrt{E-|\vk|}\zeta_-
\end{array} \right) & \hspace{5ex} & v_{\sg}^{\da}(k_1) = \left(
\begin{array}{c}
-\sqrt{E-|\vk|}\zeta_+
\\ \sqrt{E+|\vk|}\zeta_+
\end{array} \right) \nonumber \\
v_{\sg}^{\ua}(k_2) = \left( \begin{array}{c} \sqrt{E-|\vk|}\zeta_-
\\ -\sqrt{E+|\vk|}\zeta_-
\end{array} \right) & \hspace{5ex} & v_{\sg}^{\da}(k_2) = \left(
\begin{array}{c}
\sqrt{E+|\vk|}\zeta_+
\\ -\sqrt{E-|\vk|}\zeta_+
\end{array} \right) ,
\eea
%
with $\zeta_+=\left(\begin{array}{c} 1 \\ 0
\end{array} \right)$ and $\zeta_-=\left(\begin{array}{c} 0 \\ 1
\end{array} \right)$. Here the arrows $\ua$ and $\da$ denote spin up
and spin down along $\vk$ (i.e., the $z$-axis), respectively. The
Feynman rules for the Majorana fields are given in a
representation independent way in \cite{Haber:1984rc}.

\section{AMPLITUDES OF THE CUT DIAGRAMS
\label{sec:amps}}

The ${\cal M}$ matrix element for $Z \to b {\bar b}$ is
%
\be \label{eqn:zbb} {\cal M}(Z \to b {\bar b}) = -\frac{g_W}{2
\cos\theta_W} {\bar u}(p_1) {\not \! \epsilon}(p)(g_L
P_L+g_RP_R)v(p_2) \delta^{ij}, \ee
%
where ${\not \! \epsilon} \equiv \epsilon\cdot\gamma$,
$g_L=g_V+g_A=\frac{2}{3}\sin^2\theta_W-1$,
$g_R=g_V-g_A=\frac{2}{3}\sin^2\theta_W$,
$P_L=\frac{1-\gamma^5}{2}$, $P_R=\frac{1+\gamma^5}{2}$,
$\delta^{ij}$ is a Kronecker delta in the quark color indices and
$p = p_1 + p_2$ is the 4-momentum of the $Z$. Using the notations
in the previous section, we have
%
\bea {\cal M}(Z \to b^\ua {\bar b}^\ua) & = & (0,
i\sin\phi+\cos\phi\cos\theta,
-i\cos\phi+\sin\phi\cos\theta, -\sin\theta)\cdot \epsilon(p) \nonumber \\
& & \hspace{15 ex}  \times \frac{g_W}{2 \cos\theta_W}
\left[(E-|{\vp}|)g_L + (E+|{\vp}|)g_R \right] \delta^{ij} \nonumber \\
{\cal M}(Z \to b^\da {\bar b}^\da) & = & (0,
-i\sin\phi+\cos\phi\cos\theta,
i\cos\phi+\sin\phi\cos\theta, -\sin\theta)\cdot \epsilon(p) \nonumber \\
& & \hspace{15 ex}  \times \frac{g_W}{2 \cos\theta_W}
\left[(E+|{\vp}|)g_L + (E-|{\vp}|)g_R \right] \delta^{ij} \nonumber \\
{\cal M}(Z \to b^\ua {\bar b}^\da) & = & \frac{g_W m_b}{2
\cos\theta_W}e^{-i\phi}\left[ g_L
(-1,\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)
\right. \nonumber \\
& & \hspace{15 ex} \left.
+g_R(1,\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\right]\cdot
\epsilon(p) \delta^{ij} \nonumber \\
{\cal M}(Z \to b^\da {\bar b}^\ua) & = & -\frac{g_W m_b}{2
\cos\theta_W}e^{i\phi}\left[ g_L
(1,\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)
\right. \nonumber \\
& & \hspace{15 ex} \left.
+g_R(-1,\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\right]\cdot
\epsilon(p)\delta^{ij} \nonumber  \eea
%
For $\sin^2\theta_W=0.2311$ and without top quark corrections, the
partial decay width for $Z$ to decay into massless $b{\bar b}$ is
then $\frac{G_F M_Z^3}{4\sqrt{2}\pi}(g_L^2+g_R^2) = 368$ MeV.
%3\left(\frac{g_WM_Z}{\sqrt{2}\cos\theta_W}\right)^2(g_L^2+g_R^2).
Now we consider $b {\bar b} \to \sg \sg$ via exchange of a $\sb$
or $\sb'$, the matrix element for which is denoted ${\cal M}(b
{\bar b} \to \sg \sg)$ or ${\cal M}'(b {\bar b} \to \sg \sg)$,
respectively. We have ${\cal M}(b {\bar b} \to \sg \sg)={\cal
M}^{(1)}(b {\bar b} \to \sg \sg)+{\cal M}^{(2)}(b {\bar b} \to \sg
\sg)$, with
%
\bea {\cal M}^{(1)}(b {\bar b} \to \sg \sg) & = & -2
g_s^2\frac{(t^b t^a)_{ji}}{(p_1-k_1)^2-m_{\sb}^2}{\bar
u}_{\sg}(k_1) \left(P_L\sin\theta_{\sb}-P_R\cos\theta_{\sb}
\right) u(p_1) \nonumber \\
& &  \hspace{20 ex}  {\bar v}(p_2) \left(P_R\sin\theta_{\sb} -
P_L\cos\theta_{\sb} \right) v_{\sg}(k_2) , \label{eqn:uncr}\\
{\cal M}^{(2)}(b {\bar b} \to \sg \sg) & = & -2 g_s^2 \frac{(t^a
t^b)_{ji}}{(p_1-k_2)^2-m_{\sb}^2} v_{\sg}^T(k_2)C^{-1}
\left(P_L\sin\theta_{\sb}-P_R\cos\theta_{\sb}
\right) u(p_1) \nonumber \\
& &  \hspace{20 ex}  {\bar v}(p_2) \left(P_R\sin\theta_{\sb} -
P_L\cos\theta_{\sb} \right) C {\bar u}^T_{\sg}(k_1) ,
\label{eqn:cr}\eea
%
where the superscript (1) denotes the uncrossed diagram and (2)
the crossed diagram;  $a, b$ and $i, j$ are the color indices of
the gluinos and the quarks, respectively; $t^a$ are the
fundamental representation matrices of $SU(3)$. Since
$v_{\sg}^T(k_2)C^{-1}=-{\bar u}_{\sg}(k_2)$ and $C {\bar
u}_{\sg}^T(k_1)=v_{\sg}(k_1)$, Eq.~(\ref{eqn:cr}) can
alternatively be obtained from Eq.~(\ref{eqn:uncr}) by
interchanging $k_1$ and $k_2$ and adding an overall minus sign.
The helicities of the final gluinos are determined by $\lambda$,
the initial helicity of the $Z$. For $\lambda = 1$, both gluinos
have spin up in the $z$-direction, while for $\lambda = -1$, both
have spin down in the $z$-direction. For $\lambda = 0$, one of
them has spin up and the other has spin down in the $z$-direction.
One expects $|{\rm Im}{\cal M}(Z^{\da} \to \sg^{\da}
\sg^{\da})|=|{\rm Im}{\cal M}(Z^{\ua} \to \sg^{\ua} \sg^{\ua})|$,
because the two processes are related by mirror symmetry. One also
expects ${\rm Im}{\cal M}(Z^{(0)} \to \sg^{\ua} \sg^{\da})=0$,
because these final gluinos have the same helicities and should
therefore be excluded by the Pauli principle. Therefore only the
$\cal M$ matrix elements for $\lambda=1$ are listed in the
Appendix. The matrix element ${\cal M}'(b {\bar b} \to \sg \sg)$
can be obtained from ${\cal M}(b {\bar b} \to \sg \sg)$ by
replacing $m_{\sb}$, $\sin\theta_{\sb}$ and $\cos\theta_{\sb}$
with $m_{\sb'}$, $\cos\theta_{\sb}$ and $-\sin\theta_{\sb}$,
respectively.

Now we consider the diagram in Fig.~\ref{fig:Zgg} (b) and a
similar diagram with $\sg(k_1) \leftrightarrow \sg(k_2)$, where
the intermediate state is a pair of scalar quarks ($\sb$ and
$\sb^*$). The tree-level $Z\sb\sb^*$ coupling is proportional to
$g_L\sin^2\theta_{\sb}+ g_R\cos^2\theta_{\sb}$, so a mixing angle
of $\theta_{\sb} = \arcsin{\sqrt{2\sin^2\theta_W/3}} \simeq
23^\circ$ or $157^\circ$ will make it vanish. A weak $Z\sb\sb^*$
coupling is assumed \cite{Berger:2000mp} to satisfy the tight
constraints imposed by precision measurements at the $Z$ peak.
Consequently the contribution of the $\sb\sb^*$ intermediate state
to ${\rm Im}{\cal M}(Z \to \sg \sg)$ will also be small. However,
to see how the two types of diagrams shown in Fig.~\ref{fig:Zgg}
interfere with each other, here we take $\theta_{\sb}$ to be a
free parameter. For the first part of the cut diagram
(Fig.~\ref{fig:Zgg} (b)), we have
%
\be {\cal M}(Z \to \sb \sb^*) = -\frac{g_W}{2
\cos\theta_W}\left[g_L\sin^2\theta_{\sb}+ g_R
\cos^2\theta_{\sb}\right] ({\tilde p}_1-{\tilde
p}_2)^\mu\epsilon_\mu(p) \delta^{ij} ,\ee
%
where $i$ and $j$ are the squark color indices. For the other part
of the diagram,
%
\bea {\cal M}^{(1)}(\sb \sb^* \to \sg \sg) & = & 2g_s^2\frac{ (t^b
t^a)_{ji}}{({\tilde p}_1-k_1)^2-m_b^2}v_{\sg}^T(k_2) C^{-1}
\left[P_L\sin\theta_{\sb} - P_R\cos\theta_{\sb}
\right]  \nonumber \\
&& ({\not \! {\tilde p}}_1-{\not \! k}_1+m_b) \left[P_R\sin\theta_{\sb} -
P_L\cos\theta_{\sb} \right] C{\bar u}^T_{\sg}(k_1) \\
{\cal M}^{(2)}(\sb \sb^* \to \sg \sg) & = & 2g_s^2\frac{(t^a
t^b)_{ji}}{({\tilde p}_1-k_2)^2-m_b^2} {\bar
u}_{\sg}(k_1)\left[P_L\sin\theta_{\sb} - P_R\cos\theta_{\sb} \right]
\nonumber \\
&& ({\not \! {\tilde p}}_1-{\not \! k}_2+m_b)
\left[P_R\sin\theta_{\sb} - P_L\cos\theta_{\sb} \right]
v_{\sg}(k_2) ,\eea
%
where (1) denotes the uncrossed diagram and (2) the crossed
diagram. The relevant matrix elements for $\lambda=1$ are
presented in the Appendix.
%

\section{LOWER BOUND ON $\Gamma(Z \to \sg \sg)$
\label{sec:lb}}

Now we are ready to put things together and obtain a lower bound
on $\Gamma(Z \to \sg \sg)$. First we consider an extreme case with
$m_b=m_{\sb}=m_{\sg}=0$ and $m_{\sb'}=\infty$. In this limit, the
product ${\cal M}(Z \to f){\cal M}(f \to \sg \sg)$ has an angular
dependence of either $(1+\cos\theta)$ or $(1-\cos\theta)$.
However, the $\cos\theta$ term does not contribute to the
imaginary part of the full amplitude, because integrating it over
the solid angle $\Omega$ gives zero. Note that ${\rm tr}(t^a
t^b)={\rm tr}(t^b t^a)=\delta^{ab}/2$. The only nonvanishing
amplitudes are then
%
\begin{eqnarray*}
{\cal M}(Z^\ua \to b{\bar b}){\cal M}(b{\bar b} \to \sg^\ua
\sg^\ua) & = & - {\cal M}(Z^\da \to b{\bar b}){\cal M}(b{\bar b}
\to \sg^\da \sg^\da) \\
& = & \frac{\delta^{ab}}{2} \frac{M_Z g_W
g_s^2}{\sqrt{2}\cos\theta_W} g_-
\\
{\cal M}(Z^\ua \to \sb\sb^*){\cal M}(\sb\sb^* \to \sg^\ua \sg^\ua)
& = & - {\cal M}(Z^\da \to \sb\sb^*){\cal M}(\sb\sb^* \to \sg^\da
\sg^\da) \\
& = & \frac{\delta^{ab}}{2} \frac{M_Z g_W
g_s^2}{\sqrt{2}\cos\theta_W} g_+ \cos2\theta_{\sb}~~~,
\end{eqnarray*}
%
where we have summed over the four helicity states of $b {\bar
b}$, $g_\pm = g_L\sin^2\theta_{\sb} \pm g_R\cos^2\theta_{\sb}$.
From the above two equations we can see that the two types of
diagrams in Fig.~\ref{fig:Zgg} interfere destructively if $\sb$ is
more left-handed ($45^\circ<\theta_{\sb}<135^\circ$) or dominantly
right-handed ($\theta_{\sb}<23^\circ$ or
$\theta_{\sb}>157^\circ$); the contribution of diagram (b) remains
negligible in the neighborhood of the decoupling angle ($23^\circ$
or $157^{\circ}$). The imaginary parts of the amplitudes are
%
\bea {\rm Im}{\cal M}(Z^{\ua} \to \sg^{\ua} \sg^{\ua}) & = & -
{\rm Im}{\cal M}(Z^{\da} \to \sg^{\da} \sg^{\da})
\label{eqn:mirror}\\
& = & \frac{\delta^{ab}}{16} \frac{M_Z g_W
g_s^2}{\sqrt{2}\cos\theta_W} (g_L-g_R) \sin^2\theta_{\sb}
\cos^2\theta_{\sb}~~~. \nonumber \eea
%
This relation (\ref{eqn:mirror}) also holds when all the particles
have a finite mass. The final result in the limit
$m_b=m_{\sb}=m_{\sg}=0$ and $m_{\sb'}=\infty$ can be expressed as
a ratio
%
\bea \frac{\Gamma(Z \to \sg \sg)}{\Gamma(Z \to b{\bar b})} & \ge &
\frac{1}{2}\frac{\left|{\rm Im}{\cal M}(Z^{\ua} \to \sg^{\ua}
\sg^{\ua})\right|^2+\left|{\rm Im}{\cal M}(Z^{\da} \to \sg^{\da}
\sg^{\da})\right|^2}{\left|{\rm Im}{\cal M}(Z^{\ua} \to b^{\ua}
{\bar b}^{\ua})\right|^2+\left|{\rm Im}{\cal M}(Z^{\da} \to
b^{\da} {\bar b}^{\da})\right|^2} \nonumber \\
& = & \frac{\alpha_s^2}{6}
\frac{(g_L-g_R)^2\sin^4\theta_{\sb}\cos^4\theta_{\sb}}{g_L^2+g_R^2}~~~.
\eea
%
The factor of $1/2$ comes in because the final gluinos are
identical. Taking $\Gamma(Z \to b{\bar b})=375.9$ MeV
\cite{PDGEL}, we plot the lower bound on the decay width $\Gamma(Z
\to \sg \sg)$ as a function of the sbottom mixing angle
$\theta_{\sb}$ in Fig.~\ref{fig:dw}. When all the masses are
finite, we can no longer ignore the $\cos\theta$ terms, because
the denominators of the propagators are no longer of the form
$\sim(1 \pm \cos\theta)$, which previously cancelled with the same
factors in the numerators of the amplitudes and gave only linear
terms in $\cos\theta$. However, it is still not hard to perform
the integration over the angles. Define
%
\begin{eqnarray*}
I_\pm(x,y,z) & = & \frac{1}{2}\int_0^\pi
\frac{(1\pm\cos\theta)^2}{x^2 + y^2 + z^2 + 2 x y\cos\theta}
\sin\theta d\theta \\
I_0(x,y,z) & = & \int_0^\pi \frac{\sin^2\theta}{x^2 + y^2 + z^2 +
2 x y\cos\theta} \sin\theta d\theta~~~,
\end{eqnarray*}
%
and let $c_\pm = I_\pm(v_b, v_{\sg}, r_{\sb})$, $c'_\pm =
I_\pm(v_b, v_{\sg}, r_{\sb'})$, $c_0 =I_0(v_b, v_{\sg}, r_{\sb})$,
$c'_0 =I_0(v_b, v_{\sg}, r_{\sb'})$, ${\tilde c}_0=I_0(v_{\sb},
v_{\sg}, r_b)$, where $r_b = 2m_b/M_Z$, $v_b = \sqrt{1-r_b^2}$ is
the ``velocity'' of an on-shell $b$ quark; $r_{\sb}$, $r_{\sb'}$,
$r_{\sg}$ and $v_{\sg}$ are defined in a similar way. The {\it
exact} final result can then be written as
%
\be \Gamma(Z \to \sg \sg) \ge \frac{G_F M_Z^3 \alpha_s^2}
{96\sqrt{2}\pi} [c_+ (v_b+v_{\sg})g_- + {\tilde c}_0 v_{\sb}^2
v_{\sg} g_+ \cos 2\theta_{\sb} + O(m)]^2 v_{\sg}~~~, \ee
%
where $O(m)$ is the sum of the terms that approach zero when
$m_b$, $m_{\sb}$ and $m_{\sg}$ go to zero. We have $O(m) = c_1 g_V
\cos 2\theta_{\sb} + c_2 g_A \sin2\theta_{\sb} - c_3 g_A$, with
%
\begin{eqnarray*} c_1 & = & (c_- - c'_-) (v_b - v_{\sg}) + (c_0 - c'_0) r_b^2
v_{\sg} \\
c_2 & = & (c_- - c'_-)r_b r_{\sg} v_b \\
c_3 & = & (c_- + c'_-) (v_b - v_{\sg}) + (c_+ + c'_+) r_b^2
v_{\sg}~~~.
\end{eqnarray*}
%
The lower bound is plotted in Fig.~\ref{fig:dw} as a function of
$\theta_{\sb}$ for a specific set of values for the masses:
$m_b=4.1$ GeV, $m_{\sb}=4.5$ GeV, $m_{\sb'}=170$ GeV, $m_{\sg}=15$
GeV. The shape of the curve is changed significantly by the
masses, especially in the large $\theta_{\sb}$ range. This is
because a right-handed $b$ quark couples only to a left-handed
$\sg$ (and vice versa) in the massless limit while they are
chirally mixed if massive. The contribution from a finite
$m_{\sb'}$ ($\le 180$ GeV) is less than 5\% for most
$\theta_{\sb}$ (except those around which the lower bound
vanishes). The lower bound is about 0.019 MeV near
$\theta_{\sb}=23^\circ$ and 0.007 MeV near
$\theta_{\sb}=157^\circ$; the largest lower bound is 0.064 MeV
attained around $\theta_{\sb}=43^\circ$. We expect the full width
to be of this order of magnitude. This can be seen from comparison
with the decay $Z \to q{\bar q}g^* \to q{\bar q}\sg\sg$
\cite{Cheung:2002rk}. Both decay widths are $\sim \alpha
\alpha_s^2$ and the phase spaces are similar if $q$ is a light
quark. However, $Z \to q{\bar q}g^* \to q{\bar q}\sg\sg$ is mainly
a tree process, while $Z \to \sg \sg$ can only occur through loop
processes. The decay width $\Gamma(Z \to q{\bar q}g^* \to q{\bar
q}\sg\sg)$ is calculated in a model-independent way to be (0.20 --
0.74) MeV \cite{Cheung:2002rk} for $m_{\sg}=$12--16 GeV. (A recent
analysis shows that $\Gamma(Z \to b {\bar b}\sg \sg)$ can be
enhanced by 1\% -- 26 \% due to additional ``sbottom splitting''
diagrams \cite{Malhotra:2003da}. This will raise $\Gamma(Z \to q
{\bar q} \sg \sg)$ by 0.01 -- 0.23 MeV). $\Gamma(Z \to \sg \sg)$
is expected to be less. A very conservative estimate of the upper
bound is taken to be $1$ MeV \cite{Chiang:2002wi}.

\begin{figure}
\centerline{\epsfysize = 6 in
\epsffile{dw.eps}
}
\caption[]{\label{fig:dw}
   Lower bound on $\Gamma(Z \to \sg \sg)$ as a function of the
sbottom mixing angle $\theta_{\sb}$. Solid curve: $m_b=4.1$ GeV,
$m_{\sb}=4.5$ GeV, $m_{\sg}=15$ GeV, $m_{\sb'}=170$ GeV; Dotted
curve: $m_b= m_{\sb}= m_{\sg}=0$ and $m_{\sb'}=\infty$. }
\end{figure}

\section{IMPLICATIONS FOR GLUINO SEARCHES IN $Z$ DECAYS
\label{sec:gs}}

Aside from $Z \to \sg \sg$ and $Z \to q{\bar q}g^* \to q{\bar
q}\sg\sg$ in which gluinos are produced in pairs, there exist two
other gluino-producing $Z$ decays, $Z \to b {\bar b}^* \to b \sb^*
\sg$ and $Z \to {\bar b}b^* \to {\bar b}\sb\sg$. These two
processes are $\sim$ $\alpha \alpha_s$ at the tree level and have
a combined decay width of 2.5 -- 8.0 MeV \cite{Cheung:2002na}
depending on the sign of $\sin 2\theta_{\sb}$. The new SUSY
particles do not always contribute positively to the $Z$ width,
however. Cao {\it et al.} \cite{Cao:2001rz} and S.w. Baek
\cite{Baek:2002xf} showed that the decay width $\Gamma(Z \to b
{\bar b})$ can be reduced by as much as $7.8$ MeV. By fine-tuning
the parameters in the light gluino and light sbottom scenario, all
the electroweak measurables ($\Gamma_Z$, $\Gamma_{\rm had}(Z)$,
$R_b$, $R_c$) at the $Z$ pole can be still within the $1 \sigma$
bounds of the experimental values. Thus, existence of the new
particles can only be verified through direct searches for gluinos
or sbottoms. The light sbottom is assumed to be long-lived at the
collider scale or to decay promptly to light hadrons in this
scenario. In either case, it forms a hadronic jet within the
detector due to its color charge. $\sg$ decays exclusively to
$b\sb^*$ or ${\bar b} \sb$ and becomes two hadronic jets. The
smallness of the lower bound on $\Gamma(Z \to \sg \sg)$ implies
the insignificance of $Z \to \sg \sg$ in gluino searches. Searches
for signals of $Z \to q{\bar q}\sg\sg$ and $Z \to b \sb^* \sg +
{\bar b}\sb\sg$ will be expected to play a pivotal role.

%
\section{IMPLICATIONS FOR RUNNING OF $\alpha_s$
\label{sec:ra}}

Both the light gluino ($\sg$) and the light sbottom ($\sb$) can
change the $\beta$-function that governs the energy-scale
dependence (``running") of the strong coupling constant
$\alpha_s$. At two-loop level, $\alpha_s(M_Z)$ can be raised by
$0.014 \pm 0.001$ \cite{Chiang:2002wi} with respect to its
standard model value if extrapolated from the mass scale $m_b$. A
natural question arises: are values of $\alpha_s(M_Z)$ determined
from measurements at different energy scales still in accordance
in the presence of $\sg$ and $\sb$? To answer this question, the
effects of the new SUSY particles on measurements at different
scales must be analyzed. For example, the hadronic width of the
$Z$ is changed in two ways: 1) the interference of the standard
model diagrams and the diagrams with the SUSY particles in loops
will reduce the partial width of $Z \to b {\bar b}$; 2) the
existence of the new decay channels $Z \to \sb \sb$, $Z \to \sg
\sg$, $Z \to q{\bar q}\sg\sg$ and $Z \to b \sb^* \sg /{\bar
b}\sb\sg$ will raise the hadronic width.  The squark mixing angle
$\theta_{\sb}$ is constrained by the first channel to be near
$23^{\circ}$ or $157^{\circ}$. The lower bound on the decay width
of $Z \to \sg \sg$ is only of the order 0.01 MeV at either of
these two angles (Fig.~\ref{fig:dw}). Thus both channels combined
will change the predicted hadronic width of the $Z$ by a
negligible amount if this lower bound provides a good estimate of
the actual width for $Z \to \sg \sg$. Actually, even if the actual
width is about 1 MeV, its effect is still small compared to that
of the decrease in $\Gamma(Z \to b {\bar b})$ and the increase in
$\Gamma_{\rm had}(Z)$ due to $Z \to q{\bar q}\sg\sg$ and $Z \to b
\sb^* \sg /{\bar b}\sb\sg$. A better determination of $\Gamma(Z
\to b {\bar b})$, $\Gamma(Z \to q{\bar q}\sg\sg)$ and $\Gamma(Z
\to b \sb^* \sg /{\bar b}\sb\sg)$, or a more precise measurement
of $R_b$ (which will constrain the value of $\Gamma(Z \to b {\bar
b})$ more tightly), is needed for a clear-cut decision in favor of
either the Standard Model or the light gluino/light sbottom
scenario.


\section{SUMMARY
\label{sec:sum}}

Instead of calculating the full decay width $\Gamma(Z \to \sg
\sg)$ which depends on many other unknown parameters (e.g.,
$m_{\tilde t}$, $m_{{\tilde t}'}$, $\theta_{\tilde t}$, etc), we
have obtained its lower bound as the function of a single
parameter $\theta_{\sb}$. The lower bound is of the order 0.01 MeV
around $\theta_{\sb}=23^\circ$ or $157^\circ$, the decoupling
angles for $Z \sb \sb^*$. This lower bound is valid as long as all
other SUSY particles are heavy. We expect the full width to be not
far from this lower bound. Compared with other decay processes
like $\Gamma(Z \to q{\bar q}\sg\sg)$ and $\Gamma(Z \to b \sb^* \sg
/{\bar b}\sb\sg)$, $Z \to \sg \sg$ will only play a moderate role
in searches for gluinos and analysis of effects of the SUSY
scenario on $\alpha_s(M_Z)$.

\section*{ACKNOWLEDGMENTS}

I would like to thank Jonathan L. Rosner and Cheng-Wei Chiang for
very useful discussions and suggestions. This work was supported
in part by the U.\ S.\ Department of Energy through Grant Nos.\
DE-FG02-90ER-40560.

\section*{APPENDIX: RELEVANT MATRIX ELEMENTS}

We define
%
\begin{eqnarray*}
A^{(1)} & = &-2 g_s^2\frac{(t^b
t^a)_{ji}}{(p_1-k_1)^2-m_{\sb}^2} \\
A^{(2)} & = &-2 g_s^2\frac{(t^a
t^b)_{ji}}{(p_1-k_2)^2-m_{\sb}^2} \\
{\tilde A}^{(1)} & = &-2 g_s^2\frac{(t^b
t^a)_{ji}}{({\tilde p}_1-k_1)^2-m_b^2} \\
{\tilde A}^{(2)} & = &-2 g_s^2\frac{(t^a
t^b)_{ji}}{({\tilde p}_1-k_2)^2-m_b^2} \\
B_{\pm\pm} & = &\sqrt{(E \pm |{\vk}|)(E \pm |{\vp}|)} \\
{\tilde B}_{\pm\pm} & =& \sqrt{(E \pm |{\vk}|)(E \pm |{\vk}|)} \\
S_{\sb} & = & \sin\theta_{\sb} \\
C_{\sb} & = & \cos\theta_{\sb} \\
\hline\hline
\end{eqnarray*}
%
%
${\cal M}$ matrix elements for $Z^{\ua} \to b {\bar b} \to
\sg^{\ua} \sg^{\ua}$:
%
\begin{eqnarray*}
{\cal M}(Z^{\ua} \to b^{\ua}{\bar b}^{\ua}) & = &
-\frac{g_W}{\sqrt{2}\cos\theta_W}e^{i\phi}\left[
(E-|{\vp}|)g_L+(E+|{\vp}|)g_R\right]\frac{1+\cos\theta}{2}\delta^{ij} \\
{\cal M}^{(1)}(b^{\ua}{\bar b}^{\ua} \to \sg^{\ua} \sg^{\ua}) & =
& -A^{(1)}
e^{-i\phi}(B_{+-}S_{\sb}-B_{-+}C_{\sb})(B_{+-}S_{\sb}-B_{-+}C_{\sb})
\frac{1+\cos\theta}{2}\\
{\cal M}^{(2)}(b^{\ua}{\bar b}^{\ua} \to \sg^{\ua} \sg^{\ua}) & =
& A^{(2)}
e^{-i\phi}(B_{--}S_{\sb}-B_{++}C_{\sb})(B_{--}S_{\sb}-B_{++}C_{\sb})
\frac{1+\cos\theta}{2}\\
{\cal M}(Z^{\ua} \to b^{\da}{\bar b}^{\da}) & = &
\frac{g_W}{\sqrt{2}\cos\theta_W}e^{i\phi}\left[
(E+|{\vp}|)g_L+(E-|{\vp}|)g_R\right]\frac{1-\cos\theta}{2}\delta^{ij} \\
{\cal M}^{(1)}(b^{\da}{\bar b}^{\da} \to \sg^{\ua} \sg^{\ua}) & =
& A^{(1)}
e^{-i\phi}(B_{++}S_{\sb}-B_{--}C_{\sb})(B_{++}S_{\sb}-B_{--}C_{\sb})
\frac{1-\cos\theta}{2}\\
{\cal M}^{(2)}(b^{\da}{\bar b}^{\da} \to \sg^{\ua} \sg^{\ua}) & =
& -A^{(2)}
e^{-i\phi}(B_{-+}S_{\sb}-B_{+-}C_{\sb})(B_{-+}S_{\sb}-B_{+-}C_{\sb})
\frac{1-\cos\theta}{2}\\
{\cal M}(Z^{\ua} \to b^{\ua}{\bar b}^{\da}) & = &
-\frac{g_Wm_b}{\sqrt{2}\cos\theta_W}(g_L+g_R)\frac{\sin\theta}{2}\delta^{ij} \\
{\cal M}^{(1)}(b^{\ua}{\bar b}^{\da} \to \sg^{\ua} \sg^{\ua}) & =
& -A^{(1)}
(B_{+-}S_{\sb}-B_{-+}C_{\sb})(B_{++}S_{\sb}-B_{--}C_{\sb})\frac{\sin\theta}{2}\\
{\cal M}^{(2)}(b^{\ua}{\bar b}^{\da} \to \sg^{\ua} \sg^{\ua}) & =
& A^{(2)}
(B_{--}S_{\sb}-B_{++}C_{\sb})(B_{-+}S_{\sb}-B_{+-}C_{\sb})\frac{\sin\theta}{2}\\
{\cal M}(Z^{\ua} \to b^{\da}{\bar b}^{\ua}) & = &
\frac{g_Wm_b}{\sqrt{2}\cos\theta_W}e^{2i\phi}(g_L+g_R)
\frac{\sin\theta}{2}\delta^{ij} \\
{\cal M}^{(1)}(b^{\da}{\bar b}^{\ua} \to \sg^{\ua} \sg^{\ua}) & =
& A^{(1)}e^{-2i\phi}
(B_{++}S_{\sb}-B_{--}C_{\sb})(B_{+-}S_{\sb}-B_{-+}C_{\sb})\frac{\sin\theta}{2}\\
{\cal M}^{(2)}(b^{\da}{\bar b}^{\ua} \to \sg^{\ua} \sg^{\ua}) & =
& -A^{(2)}e^{-2i\phi}
(B_{-+}S_{\sb}-B_{+-}C_{\sb})(B_{--}S_{\sb}-B_{++}C_{\sb})\frac{\sin\theta}{2}
\\
\hline\hline
\end{eqnarray*}
%
${\cal M}$ matrix elements for $Z^{\ua} \to \sb \sb^* \to
\sg^{\ua} \sg^{\ua}$:
%
\begin{eqnarray*}
{\cal M}(Z^{\ua} \to \sb \sb^*) & = &
\frac{g_W}{\sqrt{2}\cos\theta_W} e^{i\phi}|{\tilde
{\vp}}|\left[g_LS^2_{\sb}+ g_R
C^2_{\sb}\right]\sin\theta \\
{\cal M}^{(1)}(\sb \sb^* \to \sg^{\ua} \sg^{\ua}) & = & {\tilde
A}^{(1)} e^{-i\phi} |{\tilde {\vp}}|({\tilde
B}_{--}S^2_{\sb}+{\tilde
B}_{++}C^2_{\sb})\sin\theta \\
{\cal M}^{(2)}(\sb \sb^* \to \sg^{\ua} \sg^{\ua}) & = & -{\tilde
A}^{(2)} e^{-i\phi} |{\tilde {\vp}}|({\tilde
B}_{++}S^2_{\sb}+{\tilde
B}_{--}C^2_{\sb})\sin\theta \\
\hline\hline
\end{eqnarray*}
%

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%%CITATION = ;%%

%\cite{Sehgal:db}
\bibitem{Sehgal:db}
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Anti-Lepton And K(L) $\to$ Lepton Anti-Lepton,''
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1582 (1971)].
%%CITATION = PHRVA,183,1511;%%

%\cite{Gaillard:1974hs}
\bibitem{Gaillard:1974hs}
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%``Rare Decay Modes Of The K - Mesons In Gauge Theories,''
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%%CITATION = PHRVA,D10,897;%%

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%\cite{Haber:1984rc}
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\end{thebibliography}
\end{document}

${\cal M}$ matrix elements for $Z^{\da} \to b {\bar b} \to
\sg^{\da} \sg^{\da}$:
%
\begin{eqnarray*}
{\cal M}(Z^{\da} \to b^{\ua}{\bar b}^{\ua}) & = &
\frac{g_W}{\sqrt{2}\cos\theta_W}e^{-i\phi}\left[
(E-|{\vp}|)g_L+(E+|{\vp}|)g_R\right]\frac{1-\cos\theta}{2}\delta^{ij} \\
{\cal M}^{(1)}(b^{\ua}{\bar b}^{\ua} \to \sg^{\da} \sg^{\da}) & =
& A^{(1)}e^{i\phi}
(B_{--}S_{\sb}-B_{++}C_{\sb})(B_{--}S_{\sb}-B_{++}C_{\sb})
\frac{1-\cos\theta}{2}\\
{\cal M}^{(2)}(b^{\ua}{\bar b}^{\ua} \to \sg^{\da} \sg^{\da}) & =
& -A^{(2)}e^{i\phi}
(B_{+-}S_{\sb}-B_{-+}C_{\sb})(B_{+-}S_{\sb}-B_{-+}C_{\sb})
\frac{1-\cos\theta}{2}\\
{\cal M}(Z^{\da} \to b^{\da}{\bar b}^{\da}) & = &
-\frac{g_W}{\sqrt{2}\cos\theta_W}e^{-i\phi}\left[
(E+|{\vp}|)g_L+(E-|{\vp}|)g_R\right]\frac{1+\cos\theta}{2}\delta^{ij} \\
{\cal M}^{(1)}(b^{\da}{\bar b}^{\da} \to \sg^{\da} \sg^{\da}) & =
& -A^{(1)}e^{i\phi}
(B_{-+}S_{\sb}-B_{+-}C_{\sb})(B_{-+}S_{\sb}-B_{+-}C_{\sb})
\frac{1+\cos\theta}{2}\\
{\cal M}^{(2)}(b^{\da}{\bar b}^{\da} \to \sg^{\da} \sg^{\da}) & =
& A^{(2)}e^{i\phi}
(B_{++}S_{\sb}-B_{--}C_{\sb})(B_{++}S_{\sb}-B_{--}C_{\sb})
\frac{1+\cos\theta}{2}\\
{\cal M}(Z^{\da} \to b^{\ua}{\bar b}^{\da}) & = &
-\frac{g_Wm_b}{\sqrt{2}\cos\theta_W}e^{-2i\phi}(g_L+g_R)
\frac{\sin\theta}{2}\delta^{ij}
\\
{\cal M}^{(1)}(b^{\ua}{\bar b}^{\da} \to \sg^{\da} \sg^{\da}) & =
& -A^{(1)}e^{2i\phi}
(B_{--}S_{\sb}-B_{++}C_{\sb})(B_{-+}S_{\sb}-B_{+-}C_{\sb})
\frac{\sin\theta}{2}\\
{\cal M}^{(2)}(b^{\ua}{\bar b}^{\da} \to \sg^{\da} \sg^{\da}) & =
& A^{(2)}e^{2i\phi}
(B_{+-}S_{\sb}-B_{-+}C_{\sb})(B_{++}S_{\sb}-B_{--}C_{\sb})\frac{\sin\theta}{2}\\
{\cal M}(Z^{\da} \to b^{\da}{\bar b}^{\ua}) & = &
\frac{g_Wm_b}{\sqrt{2}\cos\theta_W}(g_L+g_R)\frac{\sin\theta}{2}\delta^{ij}
\\
{\cal M}^{(1)}(b^{\da}{\bar b}^{\ua} \to \sg^{\da} \sg^{\da}) & =
& A^{(1)}
(B_{-+}S_{\sb}-B_{+-}C_{\sb})(B_{--}S_{\sb}-B_{++}C_{\sb})\frac{\sin\theta}{2}\\
{\cal M}^{(2)}(b^{\da}{\bar b}^{\ua} \to \sg^{\da} \sg^{\da}) & =
& -A^{(2)}
(B_{++}S_{\sb}-B_{--}C_{\sb})(B_{+-}S_{\sb}-B_{-+}C_{\sb})\frac{\sin\theta}{2}
\\
\hline\hline
\end{eqnarray*}
%
${\cal M}$ matrix elements for $Z^{(0)} \to b {\bar b} \to
\sg^{\ua} \sg^{\da} / \sg^{\da} \sg^{\ua} $ (the superscript (0)
means $Z$ has helicity $\lambda=0$):
%
\begin{eqnarray*}
{\cal M}(Z^{(0)} \to b^{\ua}{\bar b}^{\ua}) & = &
\frac{g_W}{2\cos\theta_W}\left[
(E-|{\vp}|)g_L+(E+|{\vp}|)g_R\right]\sin\theta\delta^{ij} \\
{\cal M}^{(1)}(b^{\ua}{\bar b}^{\ua} \to \sg^{\ua} \sg^{\da}) & =
& A^{(1)}
(B_{+-}S_{\sb}-B_{-+}C_{\sb})(B_{--}S_{\sb}-B_{++}C_{\sb})\frac{\sin\theta}{2}\\
{\cal M}^{(2)}(b^{\ua}{\bar b}^{\ua} \to \sg^{\ua} \sg^{\da}) & =
& A^{(2)}
(B_{--}S_{\sb}-B_{++}C_{\sb})(B_{+-}S_{\sb}-B_{-+}C_{\sb})\frac{\sin\theta}{2}\\
{\cal M}^{(1)}(b^{\ua}{\bar b}^{\ua} \to \sg^{\da} \sg^{\ua}) & =
& -A^{(1)}
(B_{--}S_{\sb}-B_{++}C_{\sb})(B_{+-}S_{\sb}-B_{-+}C_{\sb})\frac{\sin\theta}{2}\\
{\cal M}^{(2)}(b^{\ua}{\bar b}^{\ua} \to \sg^{\da} \sg^{\ua}) & =
& -A^{(2)}
(B_{+-}S_{\sb}-B_{-+}C_{\sb})(B_{--}S_{\sb}-B_{++}C_{\sb})\frac{\sin\theta}{2}\\
{\cal M}(Z^{(0)} \to b^{\da}{\bar b}^{\da}) & = &
\frac{g_W}{2\cos\theta_W}\left[
(E+|{\vp}|)g_L+(E-|{\vp}|)g_R\right]\sin\theta\delta^{ij} \\
{\cal M}^{(1)}(b^{\da}{\bar b}^{\da} \to \sg^{\ua} \sg^{\da}) & =
& A^{(1)}
(B_{++}S_{\sb}-B_{--}C_{\sb})(B_{-+}S_{\sb}-B_{+-}C_{\sb})\frac{\sin\theta}{2}\\
{\cal M}^{(2)}(b^{\da}{\bar b}^{\da} \to \sg^{\ua} \sg^{\da}) & =
& A^{(2)}
(B_{-+}S_{\sb}-B_{+-}C_{\sb})(B_{++}S_{\sb}-B_{--}C_{\sb})\frac{\sin\theta}{2}\\
{\cal M}^{(1)}(b^{\da}{\bar b}^{\da} \to \sg^{\da} \sg^{\ua}) & =
& A^{(1)}
(B_{-+}S_{\sb}-B_{+-}C_{\sb})(B_{--}S_{\sb}-B_{++}C_{\sb})\frac{\sin\theta}{2}\\
{\cal M}^{(2)}(b^{\da}{\bar b}^{\da} \to \sg^{\da} \sg^{\ua}) & =
& A^{(2)}
(B_{++}S_{\sb}-B_{--}C_{\sb})(B_{+-}S_{\sb}-B_{-+}C_{\sb})\frac{\sin\theta}{2}\\
{\cal M}(Z^{(0)} \to b^{\ua}{\bar b}^{\da}) & = &
-\frac{g_Wm_b}{2\cos\theta_W}e^{-i\phi}(g_L+g_R)\cos\theta\delta^{ij}
\\
{\cal M}^{(1)}(b^{\ua}{\bar b}^{\da} \to \sg^{\ua} \sg^{\da}) & =
& -A^{(1)}e^{i\phi}
(B_{+-}S_{\sb}-B_{-+}C_{\sb})(B_{-+}S_{\sb}-B_{+-}C_{\sb})
\frac{1+\cos\theta}{2}\\
{\cal M}^{(2)}(b^{\ua}{\bar b}^{\da} \to \sg^{\ua} \sg^{\da}) & =
& -A^{(2)}e^{i\phi}
(B_{--}S_{\sb}-B_{++}C_{\sb})(B_{++}S_{\sb}-B_{--}C_{\sb})
\frac{1+\cos\theta}{2}\\
{\cal M}^{(1)}(b^{\ua}{\bar b}^{\da} \to \sg^{\da} \sg^{\ua}) & =
& -A^{(1)}e^{i\phi}
(B_{--}S_{\sb}-B_{++}C_{\sb})(B_{++}S_{\sb}-B_{--}C_{\sb})
\frac{1-\cos\theta}{2}\\
{\cal M}^{(2)}(b^{\ua}{\bar b}^{\da} \to \sg^{\da} \sg^{\ua}) & =
& -A^{(2)}e^{i\phi}
(B_{+-}S_{\sb}-B_{-+}C_{\sb})(B_{-+}S_{\sb}-B_{+-}C_{\sb})
\frac{1-\cos\theta}{2}\\
{\cal M}(Z^{(0)} \to b^{\da}{\bar b}^{\ua}) & = &
\frac{g_Wm_b}{2\cos\theta_W}e^{i\phi}(g_L+g_R)\cos\theta\delta^{ij}
\\
{\cal M}^{(1)}(b^{\da}{\bar b}^{\ua} \to \sg^{\ua} \sg^{\da}) & =
& -A^{(1)}e^{-i\phi}
(B_{++}S_{\sb}-B_{--}C_{\sb})(B_{--}S_{\sb}-B_{++}C_{\sb})
\frac{1-\cos\theta}{2}\\
{\cal M}^{(2)}(b^{\da}{\bar b}^{\ua} \to \sg^{\ua} \sg^{\da}) & =
& -A^{(2)}e^{-i\phi}
(B_{-+}S_{\sb}-B_{+-}C_{\sb})(B_{+-}S_{\sb}-B_{-+}C_{\sb})
\frac{1-\cos\theta}{2}\\
{\cal M}^{(1)}(b^{\da}{\bar b}^{\ua} \to \sg^{\da} \sg^{\ua}) & =
& -A^{(1)}e^{-i\phi}
(B_{-+}S_{\sb}-B_{+-}C_{\sb})(B_{+-}S_{\sb}-B_{-+}C_{\sb})
\frac{1+\cos\theta}{2}\\
{\cal M}^{(2)}(b^{\da}{\bar b}^{\ua} \to \sg^{\da} \sg^{\ua}) & =
& -A^{(2)}e^{-i\phi}
(B_{++}S_{\sb}-B_{--}C_{\sb})(B_{--}S_{\sb}-B_{++}C_{\sb})
\frac{1+\cos\theta}{2}
\\
\hline\hline
\end{eqnarray*}
%
%
${\cal M}$ matrix elements for $Z^{\da} \to \sb \sb^* \to
\sg^{\da} \sg^{\da}$:
%
\begin{eqnarray*}
{\cal M}(Z^{\da} \to \sb \sb^*) & = &
\frac{g_W}{\sqrt{2}\cos\theta_W} e^{-i\phi}|{\tilde
{\vp}}|\left[g_LS^2_{\sb}+ g_R
C^2_{\sb}\right]\sin\theta \\
{\cal M}^{(1)}(\sb \sb^* \to \sg^{\da} \sg^{\da}) & = & {\tilde
A}^{(1)} e^{i\phi} |{\tilde {\vp}}|({\tilde
B}_{++}S^2_{\sb}+{\tilde
B}_{--}C^2_{\sb})\sin\theta \\
{\cal M}^{(2)}(\sb \sb^* \to \sg^{\da} \sg^{\da}) & = & -{\tilde
A}^{(2)} e^{i\phi} |{\tilde {\vp}}|({\tilde
B}_{--}S^2_{\sb}+{\tilde
B}_{++}C^2_{\sb})\sin\theta \\
\hline\hline
\end{eqnarray*}
%
${\cal M}$ matrix elements for $Z^{(0)} \to \sb \sb^* \to
\sg^{\ua} \sg^{\da} / \sg^{\da} \sg^{\ua}$:
%
\begin{eqnarray*}
{\cal M}(Z^{(0)} \to \sb \sb^*) & = &
\frac{g_W}{\cos\theta_W}|{\tilde {\vp}}|\left[g_LS^2_{\sb}+ g_R
C^2_{\sb}\right]\cos\theta \\
{\cal M}^{(1)}(\sb \sb^* \to \sg^{\ua} \sg^{\da}) & = & {\tilde
A}^{(1)} \left[(|{\tilde {\vp}}|\cos\theta-|{\vk}|)({\tilde
B}_{+-}S^2_{\sb}+{\tilde B}_{-+}C^2_{\sb})\right. \\
& & \hspace{15 ex}\left. +m_b({\tilde B}_{++}-{\tilde
B}_{--})S_{\sb}C_{\sb} \right]\\
{\cal M}^{(2)}(\sb \sb^* \to \sg^{\ua} \sg^{\da}) & = & {\tilde
A}^{(2)} \left[(|{\tilde {\vp}}|\cos\theta+|{\vk}|)({\tilde
B}_{-+}S^2_{\sb}+{\tilde B}_{+-}C^2_{\sb})\right. \\
& & \hspace{15 ex}\left. +m_b({\tilde B}_{--}-{\tilde
B}_{++})S_{\sb}C_{\sb} \right]\\
{\cal M}^{(1)}(\sb \sb^* \to \sg^{\da} \sg^{\ua}) & = & -{\tilde
A}^{(1)} \left[(|{\tilde {\vp}}|\cos\theta-|{\vk}|)({\tilde
B}_{-+}S^2_{\sb}+{\tilde B}_{+-}C^2_{\sb})\right. \\
& & \hspace{15 ex}\left. +m_b({\tilde B}_{++}-{\tilde
B}_{--})S_{\sb}C_{\sb} \right]\\
{\cal M}^{(2)}(\sb \sb^* \to \sg^{\da} \sg^{\ua}) & = & -{\tilde
A}^{(2)} \left[(|{\tilde {\vp}}|\cos\theta+|{\vk}|)({\tilde
B}_{+-}S^2_{\sb}+{\tilde B}_{-+}C^2_{\sb})\right. \\
& & \hspace{15 ex}\left. +m_b({\tilde B}_{--}-{\tilde
B}_{++})S_{\sb}C_{\sb} \right]
\end{eqnarray*}

