%% slac-pub-7180: page file slac-pub-7180-0-0-2-1.tcx.
%% subsection 2.1 Effective weak mixing angle [slac-pub-7180-0-0-2-1 in slac-pub-7180-0-0-2: ^slac-pub-7180-0-0-2 >slac-pub-7180-0-0-2-2]
%%%% latex2techexplorer block:
%% latex2techexplorer page setup:
\newmenu{slac-pub-7180::context::slac-pub-7180-0-0-2-1}{
\docLink{slac-pub-7180.tcx}[::Top]{Top}%
\sectionLink{slac-pub-7180-0-0-2}{slac-pub-7180-0-0-2}{Above: 2. The weak mixing angle and the $W$-boson mass}%
\subsectionLink{slac-pub-7180-0-0-2}{slac-pub-7180-0-0-2-2}{Next: 2.2. $W$-boson mass}%
}
%%%% end of latex2techexplorer block.
\subsection{\usemenu{slac-pub-7180::context::slac-pub-7180-0-0-2-1}{Effective weak mixing angle}}\label{subsection::slac-pub-7180-0-0-2-1}
We start by considering the effective weak mixing angle, $s^2_\ell
\equiv \sin^2\theta^{\rm lept}_{\rm eff}$. The full one-loop
calculation is presented\footnote{We do not include the
supersymmetric nonuniversal $Z$-vertex contribution in the
Appendix. It is a negligible correction in the parameter
space we consider.}
for completeness in Appendix C. The complete result is rather
involved; for now we
simply say that the calculation follows the outline presented above:
We take $\alpha_{\rm em}$, $G_\mu$, $M_Z$, $\alpha_s(M_Z)$, and the
fermion masses as inputs, and compute $s^2_\ell$ as a function of the
supersymmetric masses.
Because we compute the experimental observable $s^2_\ell$ in terms of
other low-energy observables, its one-loop supersymmetric corrections
decouple as the supersymmetric masses become larger than $M_Z$. From
Fig.~\docLink{slac-pub-7180-0-0-2.tcx}[sw]{1} we see that $s^2_\ell$ is especially sensitive to light
sleptons, and that the sum of the supersymmetric corrections is always
negative. We did not plot the Higgs boson and first two generation
squark contributions. They are negligible, less than $1\times10^{-4}$
and $4\times10^{-5}$ in magnitude, respectively. The corrections to
$\mu$-decay and the corrections to the $Z$-$\ell^+$-$\ell^-$ vertex
comprise the nonuniversal corrections to $s_\ell^2$. The former
contributes between $-3$ and $1.5\times10^{-4}$, the latter between
$\pm1.5\times10^{-4}$, and their sum is in the range $-4$ to
$1\times10^{-4}$.
With $m_t=175$ GeV, we find the standard model value of $s^2_\ell$
varies between 0.2311 to 0.2315 for Higgs masses in the range $60 <
m_h < 130$ GeV. This is subject to an error of $2.5 \times 10^{-4}$
from the experimental uncertainty in the electromagnetic coupling
evaluated at $M_Z$, and to corrections of this same order from higher
loop effects \cite{18}. Furthermore, increasing $m_t$ by $10$ GeV
decreases $s^2_\ell$ by $3.3\times 10^{-4}.$
These predictions for $s^2_\ell$ should be compared with the LEP and
SLD average\footnote{The number quoted here assumes lepton
universality.} \cite{1} of 0.23165 $\pm$ 0.00024. Clearly, the
standard model calculation agrees quite well with experiment. The
additional contribution from supersymmetry can lower the value of
$s_\ell^2$ slightly below 0.2300, or about 6$\sigma$ below the
experimental central value. However, we note that higher-order
standard-model corrections, changes in $\alpha_{\rm em}$ and $m_t$,
and other precision observables should all be
systematically taken into account to delineate which regions of
parameter space are ruled out by these measurements. We do not
attempt such a study here.
The corrections to $s^2_\ell$ diminish rapidly as the superpartner
masses become heavy. For example, if we require $m_{\tilde\chi^+},
\ m_{\tilde\ell^+},\ m_h\ > 90$ GeV, we find that $s^2_\ell$ is shifted
by at most $-8\times10^{-4}$ relative to the standard model value.
\begin{figure}[t]
\epsfysize=2.5in \epsffile[-140 220 0 535]{F02.ps}
\begin{center}
\parbox{5.5in}{
\caption[]{\small Finite corrections to $M_W$, in MeV. Figures (a-d)
are as in Fig.~\docLink{slac-pub-7180-0-0-2.tcx}[sw]{1}.
\label{mw}}}
\end{center}
\end{figure}