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%% Title Page Stuff
  \preprint{UTEXAS-HEP-156}
  \preprint
  \title{$Z \rightarrow b\bar{b}b\bar{b}$ in the light gluino and light sbottom scenario}
  \date{January 10, 2003}
  \author{Rahul Malhotra}
  \email{rahul@math.utexas.edu}
  \author{Duane A. Dicus}
  \affiliation{Center for Particle Physics, University of Texas, Austin, Texas, 78712, USA}
  \pacs{12.60.Jv, 13.87.Ce, 14.65.Fy, 14.80.Ly}
  \begin{abstract}
The contribution of the process $Z \rightarrow
b\bar{b}\tilde{g}\tilde{g}$ to $\Gamma(Z \rightarrow
b\bar{b}b\bar{b})$ is examined in the light gluino $(12 \sim 16$
GeV$)$ and light sbottom $(2 \sim 6$ GeV$)$ scenario. We find that
the ratio $\Gamma(Z \rightarrow b\bar{b}\tilde{g}\tilde{g})/
\Gamma(Z \rightarrow b\bar{b}b\bar{b})$ can be large due to
sensitivity to $b$-quark mass, the sbottom mixing angle and the
gluino mass. We calculate it to be in the range $0.05 - 0.41$
inclusive of the entire parameter space.
  \end{abstract}

  \maketitle


%% INTRODUCTION
  \section{Introduction}
The Standard Model (SM) has been successful in explaining a host
of experimental observations on electroweak and QCD phenomenon at
LEP as well as hadron colliders. But it is generally believed that
the SM is an effective theory valid at the electroweak energy
scale with some new physics lying beyond it. Among the leading
candidates for a theory beyond the SM is the Minimal
Supersymmetric Standard Model (MSSM) \cite{haber} which has been
extensively studied in the past few decades.

The MSSM predicts the existence of SUSY partners of quarks, gluons
and other known particles of the Standard Model. These so-called
``sparticles'' have not been observed which has led to speculation
that they might be too heavy to have observable production rates
at present collider energies. However, it has been suggested in
\cite{lightsb} that a light sbottom ($\tilde{b}_1$) with a mass of
$\mathcal{O}(5$ GeV$)$ is not ruled out by electroweak precision
data if its coupling to the $Z$-boson is tuned to be small in the
MSSM. Berger {\it et al} \cite{berger1} have also recently used
low-energy supersymmetry to explain the long-standing puzzle of
overproduction of $b\bar{b}$ pairs at the Tevatron \cite{excessb}.
They suggest a light gluino ($12 - 16$ GeV) decaying into a
$b$-quark and a light sbottom ($2 - 6$ GeV), and use this model to
fit the Tevatron data at NLO level. Alternatively Cacciari and
Nason \cite{cacciari} suggest that using the latest $b$-quark
fragmentation functions reduces the discrepancy at the Tevatron.
In this article we will work in Berger {\it et al}'s light gluino
scenario.

Their proposal has recently led to a careful re-examination of
$Z$-pole precision data. It has been shown by Cao {\it et al}
\cite{cao} that to account for $R_{b}$ data the other mass
eigenstate of the sbottom ($\tilde{b}_2$) must be lighter than 125
(195) GeV at $2\sigma$ ($3\sigma$) level. Cho \cite{cho} extended
that analysis to the entire range of electroweak precision data
and found that the $\tilde{b}_2$ must be lighter than $180$ GeV at
the $5\sigma$ level. A $\tilde{b}_2$ in such a mass range would
have been produced in association with a $\tilde{b}_1$ at LEPII
energies via the couplings $Z \tilde{b}_1 \bar{\tilde{b}}_2$ and
$Z \tilde{b}_2 \bar{\tilde{b}}_1$. Since such a $\tilde{b}_2$ has
not been observed it would seem that LEP data disfavors Berger
{\it et al}'s scenario. However experimental searches for SUSY
particles are heavily model-dependent and, to the authors'
knowledge, an exhaustive search of LEP data for a $\tilde{b}_2$ in
this particular scenario has not been done. It has also been shown
recently by Baek \cite{baek} that a heavier $\tilde{b}_2$
($\gtrsim 200$ GeV), beyond the energy range of LEPII, is allowed
if large $CP$-violating phases are present in the model.

In addition to $Z$-precision data, production of $b\bar{b}$-pairs
at the $Z$-pole via gluon splitting has also been re-examined
recently by Cheung and Keung \cite{cheung}. They calculate the
contribution of $Z \rightarrow q\bar{q}\tilde{g}\tilde{g}$ to the
process $Z \rightarrow q\bar{q}g^{*} \rightarrow b\bar{b}$ (q =
u,d,c,s,b) in the massless $q$ approximation and find that the
former is only around $4 - 15\%$ of the latter. They do not go
further and consider the ratio
 $\Gamma(Z \rightarrow b\bar{b}\tilde{g}\tilde{g})/\Gamma(Z \rightarrow b\bar{b}b\bar{b})$
 as they expect it to be very similar. Like them we note that the process $Z
\rightarrow b\bar{b}\tilde{g}\tilde{g}$ contributes to $\Gamma (Z
\rightarrow b\bar{b}b\bar{b})$ via the final states
$b\bar{b}b\bar{b}\tilde{b}_1\bar{\tilde{b}}_1$,
$b\bar{b}bb\bar{\tilde{b}}_1\bar{\tilde{b}}_1$ and
$b\bar{b}\bar{b}\bar{b}\tilde{b}_1\tilde{b}_1$ which have four
$b$-quarks and two sbottoms arising from the gluino decays
$\tilde{g} \rightarrow b\bar{\tilde{b_{1}}},
\bar{b}\tilde{b_{1}}$. But $Z \rightarrow
b\bar{b}\tilde{g}\tilde{g}$ can arise not only from the gluon
splitting diagrams in Fig 1(b) but also from ``sbottom
 splitting'' diagrams in Fig 1(c). We find that the latter can significantly
 enhance the width for this process. We also calculate
 $\Gamma (Z \rightarrow b\bar{b}b\bar{b})$ to
leading order in the SM over a range of $b$-quark masses. The
final result is a wide theoretical range for the ratio $\Gamma(Z
\rightarrow b\bar{b}\tilde{g}\tilde{g})/ \Gamma(Z \rightarrow
b\bar{b}b\bar{b})$ of $5\%$ to $41\%$ inclusive of the entire
parameter space.
%% CALCULATIONS
  \section{Calculations}
The tree level diagrams for evaluating $\Gamma_{4b} = \left.
\Gamma (Z \rightarrow b\bar{b}b\bar{b}) \right|_{SM}$ and
$\Gamma_{b\tilde{g}} = \Gamma (Z \rightarrow
b\bar{b}\tilde{g}\tilde{g})$ are shown in Fig. 1. Feynman rules
for the MSSM given by Rosiek \cite{rosiek} are used to evaluate
these diagrams. Their formalism allows us to write the lighter
sbottom mass eigenstate as a superposition $\tilde{b}_{1} =
\sin\theta_{b}$ $\tilde{b}_{L}$ $+$ $\cos\theta_{b}$
$\tilde{b}_{R}$ of the left and right-handed states where
$\theta_{b}$ is the sbottom mixing angle. This angle appears in
the coupling:
$$Z\tilde{b}_1\bar{\tilde{b}}_1 \propto (\frac{1}{2}
{\sin}^2\theta_b - \frac{1}{3} {\sin}^2\theta_W)$$ where
$\theta_W$ is the Weinberg angle. However since electroweak data
excludes the process $Z \rightarrow \tilde{b}_1\bar{\tilde{b}}_1$
to a high precision, the mixing angle must be fine-tuned to make
the coupling small i.e., $s_b^2 = \frac{2}{3}{\sin}^2\theta_W$,
$|s_b| \approx 0.38$ \cite{lightsb} where the short-hand notation
$s_b \equiv \sin (\theta_b)$ is used. We vary $|s_b|$ in the
narrow range $0.30 - 0.45$.

At constant scale we find that the diagrams in Fig. 1(c) enhance
the width for $Z \rightarrow b\bar{b}\tilde{g}\tilde{g}$ by $
10-60\%$ in the range $|s_b| = 0.3-0.45$. We therefore choose the
invariant mass of the two gluinos, $m_{\tilde{g}\tilde{g}}$, as
the running scale $Q$ since the diagrams in Fig. 1(b) are still
dominant.

\begin{center} \begin{picture} (450,170) (0,0)
\Text(28,157)[l]{(a)} \Text(0,105)[l]{$Z^{0}$}
\Photon(18,105)(48,105){3}{4} \ArrowLine(48,105)(78,135)
\Text(83,140)[l]{$b$} \ArrowLine(78,75)(48,105)
\Text(83,70)[l]{$\bar{b}$} \Gluon(65,122)(95,122){3}{4}
\ArrowLine(95,122)(125,152) \Text(130,157)[l]{$b$}
\ArrowLine(125,92)(95,122) \Text(130,87)[l]{$\bar{b}$}
\Text(188,157)[l]{(b)} \Text(160,105)[l]{$Z^{0}$}
\Photon(178,105)(208,105){3}{4} \ArrowLine(208,105)(238,135)
\Text(243,140)[l]{$b$} \ArrowLine(238,75)(208,105)
\Text(243,70)[l]{$\bar{b}$} \Gluon(225,122)(255,122){3}{4}
\Gluon(255,122)(285,152){3}{5} \Line(255,122)(285,152)
\Text(290,157)[l]{$\tilde{g}$} \Gluon(285,92)(255,122){3}{5}
\Line(285,92)(255,122) \Text(290,87)[l]{$\tilde{g}$}
\Text(348,157)[l]{(c)} \Text(320,105)[l]{$Z^{0}$}
\Photon(338,105)(368,105){3}{4} \ArrowLine(368,105)(380,117)
\Gluon(380,117)(422,117){3}{5} \Line(380,117)(422,117)
\Text(427,117)[l]{$\tilde{g}$} \DashArrowLine(380,117)(400,137){2}
\Text(378,132)[l]{$\tilde{b}_{1}$} \Gluon(400,137)(442,137){3}{5}
\Line(400,137)(442,137) \Text(445,137)[l]{$\tilde{g}$}
\ArrowLine(400,137)(415,152) \Text(420,157)[l]{$b$}
\ArrowLine(398,75)(368,105) \Text(403,70)[l]{$\bar{b}$}
\Text(0,20)[l]{FIG. 1: Feynman diagrams contributing to (a) $Z
\rightarrow b\bar{b}b\bar{b}$ and (b),(c) $Z \rightarrow
b\bar{b}\tilde{g} \tilde{g}$. Diagrams} \Text(0,5)[l]{with
gluon/gluino emission off the $\bar{b}$-leg and the crossing of
identical particles are not shown.}
\end{picture} \end{center}

In calculating $\Gamma_{4b}$ the $b\bar{b}$-pair produced by gluon
splitting cannot be isolated due to interference terms between
crossed diagrams in Fig. 1(a), making the off-shellness of the
virtual gluon indeterminate. This is in contrast to the gluon
splitting processes $Z \rightarrow q\bar{q}g^{*} \rightarrow
b\bar{b}$ and $Z \rightarrow b\bar{b}g^{*} \rightarrow q\bar{q}$,
$q \neq b$, where the secondary production of $b$-quarks does not
interfere with primary production at leading order \cite{kniehl}.
Therefore to calculate $\Gamma_{4b}$ we first find the ratio
$\Gamma_{4b}$/$\Gamma (Z \rightarrow q\bar{q}g^{*} \rightarrow
b\bar{b})$ at constant $Q$-scale, summing the denominator over $q
= u,d,s,c,b$ in the massless $q$ approximation. Then $\Gamma_{4b}$
is evaluated over a running $Q$-scale as follows:
\begin{equation}
\Gamma_{4b} = \left. \frac{\Gamma_{4b}}{\Gamma (Z \rightarrow q\bar{q}g^{*} \rightarrow b\bar{b})} \right|_{Q = const.}
\times \left. \Gamma (Z \rightarrow q\bar{q}g^{*} \rightarrow b\bar{b}) \right|_{Q = m_{b\bar{b}}}
\end{equation}
The strong coupling value $\alpha_{S}(M_Z) = 0.118$ is used. In
the absence of a higher-order calculation, there is no choice but
to vary the $b$-quark mass between the lower limit of the
$\overline{MS}$ value $\bar{m}_b(\bar{m}_b) = 4$ GeV and the
$B$-meson mass $m_B = 5.25$ GeV \cite{PDG}.

The results are given in terms of the ratios:
$$R_{4b} \equiv \left. \frac{\Gamma_{4b}}{\Gamma(Z \rightarrow hadrons)} \right|_{SM},\quad R_{b\tilde{g}}
\equiv \frac{\Gamma_{b\tilde{g}}}{\Gamma(Z \rightarrow hadrons)}$$
The total hadronic width of the $Z$ is taken to be $\Gamma(Z \rightarrow hadrons) = 1.744$ GeV \cite{PDG}.
%% RESULTS
\section{Results}
Using Eqn (1) we numerically calculate $R_{4b} = (6.06 - 3.04)
\times 10^{-4}$ for $m_b = 4 - 5.25$ GeV. This value has been
measured by DELPHI \cite{delphi} and OPAL \cite{opal} to be
$(6.0\pm 1.9\pm 1.4) \times 10^{-4}$ and $(3.6\pm 1.7\pm 2.7)
\times 10^{-4}$ respectively and the uncorrelated average is given
by the PDG to be $R^{exp}_{4b} = (5.2\pm 1.9) \times 10^{-4}$
\cite{PDG}. Our entire calculated range is within $1.2\sigma$ of
the experimental average. The central value of $R^{exp}_{4b}$ is
obtained for $m_b \sim 4.3$ GeV which agrees well with $m_b =
4.25$ GeV used by Cheung and Keung in \cite{cheung} to fit the
full gluon-splitting process.

In a similar fashion, $R_{b\tilde{g}}$ is very sensitive to the
gluino mass $m_{\tilde{g}}$ showing a decrease by nearly a factor
of 3.5 as $m_{\tilde{g}}$ varies from $12$ to $16$ GeV [Fig.
2(a)]. The effect of varying $s_b$ is also considerable
($30-40\%$) within the range $|s_b| = 0.30 - 0.45$ as shown in
Fig. 2(b). $R_{b\tilde{g}}$ is lower for negative values of $s_b$
but increases with $|s_b|$ due to constructive interference with
the gluon-splitting diagrams. On the other hand variations in
$b$-quark mass and the sbottom mass ($2 - 6$ GeV) have very little
effect ($\approx 5\%$) [Fig. 3]. Including all parameters, we find
$R_{b\tilde{g}} = (0.25 - 1.33) \times 10^{-4}$.

The net process $R_{4b} + R_{b\tilde{g}}$ equals $(3.3 - 7.3)
\times 10^{-4}$ for the entire parameter space which is still
within $1.2\sigma$ of the experimental value. However the ratio $x
= R_{b\tilde{g}}/R_{4b}$ can be quite large, varying from $5 -
41\%$ [Fig. 4]. As a percent of the total $R_{4b} +
R_{b\tilde{g}}$ the gluino process could be $4 - 30\%$.

The $R^{exp}_{4b}$ data might therefore contain a large admixture
of sbottoms produced by gluino-decay if Berger {\it et al}'s model
is correct. Signatures of a light sbottom have been discussed by
Berger in \cite{berger2}. Although a $\tilde{b}_1$ having $b$-like
semi-leptonic decays is disfavored by CLEO \cite{cleo} the
possibility remains that it could have $R$-parity violating decays
into soft jets: $\bar{\tilde{b}}_1 \rightarrow u + s; c + d; c +
s$. It can also be long-lived and form a charged mesino
($\tilde{B}^{\pm}$) with a $u$-type quark. This can exit the muon
chambers leaving behind fake ``heavy muon"-like tracks including
some activity in the hadron calorimeters. Finally if the sbottom
leaves no signature in the detectors at all, it might still be
observed as missing energy.

%% CONCLUSIONS
\section{Conclusions}
We have shown that Berger {\it et al}'s light gluino model cannot
be constrained through an excess in the measured rate $Z
\rightarrow b\bar{b}b\bar{b}$ at this point due to large
uncertainties in the experimental value as well as tree-level
calculations of $R_{4b}$. However at the same time we show that a
significant fraction of experimental data for $R_{4b}$, ranging
from $4 - 30\%$ can contain sbottom signatures if this model is
correct. We therefore suggest that the data be carefully
re-examined to constrain this scenario and/or further limit the
possible decay channels for a light $\tilde{b}_1$.

%% ACKNOWLEDGEMENTS
\section{Acknowledgements}
This work was supported in part by the United States Department of
Energy under Contract No. DE-FG03-93ER40757.

\begin{thebibliography}{1}
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname
urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

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\bibitem{cheung}
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\end{thebibliography}

\pagebreak

\begin{figure}[t]
\includegraphics{2a.eps}
\linebreak
\linebreak
\includegraphics{2b.eps}
\end{figure}
\begin{flushleft}
FIG. 2: Dependence of $R_{b\tilde{g}}$ on (a) gluino mass and (b) sbottom mixing angle, $s_b = \sin (\theta_b)$. All curves are made for $m_b = 4.5$ GeV and $m_{\tilde{b}_1} = 3.5$ GeV.
\end{flushleft}

\pagebreak

\begin{figure}[t]
\includegraphics{3.eps}
\end{figure}
\begin{flushleft}
FIG. 3: Dependence of $R_{b\tilde{g}}$ on $m_b$, shown for $s_b = 0.38$ and $m_{\tilde{g}} = 14$ GeV.
\end{flushleft}
\begin{figure}[h]
\includegraphics{4.eps}
\end{figure}
\begin{flushleft}
FIG. 4: $x = R_{b\tilde{g}}/R_{4b}$ dependence on gluino mass.
Upper (lower) bound curves are obtained for $m_b = 5.25$ ($4$)
GeV, $m_{\tilde{b}_1} = 6$ ($2$) GeV and $s_b = +0.45$ ($-0.30$).
\end{flushleft}
\end{document}

