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%% subsection 4.1 Gluino mass [slac-pub-7180-0-0-4-1 in slac-pub-7180-0-0-4: ^slac-pub-7180-0-0-4 >slac-pub-7180-0-0-4-2]
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\subsection{\usemenu{slac-pub-7180::context::slac-pub-7180-0-0-4-1}{Gluino mass}}\label{subsection::slac-pub-7180-0-0-4-1}
\begin{figure}[t]
\epsfysize=2.5in \epsffile[-140 210 0 525]{F10.ps}
\begin{center}
\parbox{5.5in}{
\caption[]{\small Corrections to the gluino mass versus $m_{\tilde
g}/M_{\tilde q}$. Figure (a) shows the complete one-loop corrections;
(b) shows the leading logarithmic corrections; (c) shows the finite
corrections; and (d) shows the difference between the full one-loop
result and the approximation given in the text.
\label{gl}}}
\end{center}
\end{figure}
The gluino mass corrections are perhaps the simplest of all the mass
renormalizations. They have previously been studied in
Refs.~\cite{7,8,9,10}; for completeness we list the
corrections in Appendix D. The gluino mass corrections arise from
gluon/gluino and quark/squark loops. The corrections can be rather
large, so we include them in a way which automatically incorporates
the one-loop renormalization group resummation,
%
\begin{equation}
m_{\tilde g}\ =\ M_3(Q) \left[\, 1 - \left({\Delta M_3\over M_3}
\right)^{g\tilde g} - \left({\Delta M_3\over M_3}\right)^{q\tilde q}\,
\right]^{-1}\ .
\end{equation}
%
The gluon/gluino loop gives
%
\begin{eqnarray}
\left({\Delta M_3 \over M_3}\right)^{g\tilde g}\ &=& \ {3
g^2_3\over8\pi^2}\ \left[\,2B_0(M_3,M_3,0) -
B_1(M_3,M_3,0)\,\right] \nonumber \\ &=&\ {3 g^2_3 \over 16\pi^2}
\ \left[\,3\ln\left({Q^2\over M_3^2}\right) + 5\,\right]\ .
\end{eqnarray}
%
The quark/squark loop can be simplified by assuming that all quarks
have zero mass, and that all squarks have a common mass, which we take
to be $M_{Q_1}$, the soft mass of the first generation of left-handed
squarks. We find
%
\begin{equation}
\left({\Delta M_3\over M_3}\right)^{q\tilde q}\ = \ -\ {3
g^2_3\over4\pi^2}\ B_{1}(M_3,0,M_{Q_1})\ .
\end{equation}
%
Here
%
\begin{equation}
B_1(p,0,m)\ =\ -{1\over2}\ln\left({M^2\over Q^2 }\right) + 1 - {1\over
2x} \left[\, 1+{(x-1)^2\over x}\ln |x-1|\, \right] +
{1\over2}\theta(x-1)\ln x\ ,
\label{b1_p0m2}
\end{equation}
%
where $M = \max(p^2,m^2)$ and $x=p^2/m^2$. As usual, the full mass
renormalization contains logarithmic and finite contributions.
The gluino mass corrections are shown in Fig.~\docLink{slac-pub-7180-0-0-4.tcx}[gl]{10}. In the figure
we define the tree-level gluino mass to be $M_3(M_3)$, and we evaluate
the one-loop mass at the scale $Q = M_{\tilde q}$. Because we resum
the correction, varying the scale from $M_{\tilde q}/2$ to $2M_{\tilde
q}$ changes the one-loop mass by at most $\pm1\%$.
{}From the figure we see that the leading logarithmic correction can
be as large as 20\%, while the finite correction ranges from 3 to
10\%. The finite contribution is largest in the region where the
logarithm is largest, so the leading logarithm approximation is
nowhere good. On the other hand, the approximation we provide
typically holds to a few percent. It is off by as much as 6\% in the
region where the full correction is 30\%. In this region we expect
the two-loop correction to be of order 6\%.