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\section{\usemenu{slacpub7180::context::slacpub7180001}{Introduction}}\label{section::slacpub7180001}
Most precision measurements of electroweak parameters agree quite
well with the predictions of the standard model \cite{1}.
These experiments rule out many possibilities for physics beyond
the standard model, but they have not touched supersymmetry,
which evades precision constraints because it decouples from
standardmodel physics if the scale of supersymmetry breaking
is more than a few hundred GeV. Definitive tests of supersymmetry
will probably have to wait for direct searches at future colliders.
Once supersymmetry is discovered, a host of new questions arise.
For example, one would like to test the low energy supersymmetric
relations between the particle masses and couplings by making
precision measurements of the supersymmetric parameters. One
would like to measure the supersymmetric masses as accurately
as possible to shed light on the origin of supersymmetry breaking.
Furthermore, one would also like to know whether weakscale
supersymmetry sheds any light on physics at even higher energies.
Indeed, the successful unification of gauge couplings encourages
hope that other supersymmetric parameters might unify as well. It
is important to measure these parameters precisely at low energies
so that they can be extrapolated with confidence to higher energies.
It is in this spirit that we present our calculation of oneloop
corrections to the minimal supersymmetric standard model (MSSM). We
define the MSSM to be the minimally supersymmetrized standard model,
with no right handed neutrinos, and all possible softbreaking terms.
We believe that this minimal model provides an appropriate framework
for analyzing the phenomenology of supersymmetry and supersymmetric
unification.
We approach our calculation in the standard fashion associated
with precision electroweak measurements. We take as inputs
the electromagnetic coupling at zero momentum, $\alpha_{\rm em}
= 1/137.036$, the Fermi constant, $G_\mu = 1.16639\times 10^{5}$
GeV$^{2}$, the $Z$boson pole mass, $M_Z = 91.188$ GeV, the
strong coupling in the \mbox{\footnotesize$\overline{\rm MS}~$}
scheme at the scale $M_Z$, $\alpha_s(M_Z) = 0.118$, as well as
the quark and lepton masses, $m_t = 175$ GeV, $m_b = 4.9$ GeV,
and $m_\tau = 1.777$ GeV \cite{2}.
{}From these inputs, for any treelevel supersymmetric spectrum,
we compute the oneloop $W$boson pole mass, $M_W$, as well as
the oneloop values of the effective weak mixing angle,
$\sin^2\theta^{\rm lept}_{\rm eff}$, and the \mbox{\footnotesize
$\overline{\rm DR}~$} \cite{3} weak mixing angle, $\hat s^2$.
We also compute the oneloop corrections to the quark and lepton
Yukawa couplings, as well as the masses of all the supersymmetric
and Higgs particles.
We work in the \mbox{\footnotesize$\overline{\rm DR}~$} scheme,
and take the treelevel masses to be given in terms of the running
\mbox{\footnotesize$\overline{\rm DR}~$} parameters. For each
(bosonic) particle, we determine the oneloop pole mass,
%
\begin{equation}
M^2 \ =\ \hat M^2(Q)\ \ {\cal R}e\,\Pi(M^2)\ ,
\label{pole mass}
\end{equation}
%
where $\hat M(Q)$ is the treelevel \mbox{\footnotesize$\overline{\rm
DR}~$} mass, evaluated at the \mbox{\footnotesize$\overline{\rm DR}~$}
scale $Q$, and $\Pi(p^2)$ is the oneloop selfenergy. (As usual,
$\Pi(p^2)$ depends on $Q$ and on the masses and couplings of the
particles in the loop. There is a similar expression for the fermion
pole mass.)
In all our computations we include the full selfenergies, which
contain both logarithmic and finite contributions. The logarithmic
corrections can be absorbed by changes in the scale $Q$. Therefore we
checked our logarithmic results against the oneloop supersymmetric
renormalization group equations. Since we write our results using
PassarinoVeltman functions \cite{4}, some of our finite terms are
automatically correct. As a further check, we verified that our
corrections decouple from electroweak observables.
We present our complete calculations in a series of Appendices. These
appendices include the full oneloop corrections to the gauge and
Yukawa couplings, as well as the complete oneloop corrections to the
entire MSSM mass spectrum. While some of these results are not new
(the gaugeboson \cite{5,6}, Higgsboson \cite{6} and
gluino \cite{7,8,9,10} selfenergies and the gaugecoupling
corrections \cite{5,11} already appear in the literature),
we include the full set of corrections in order to provide a complete,
selfcontained and more useful reference.
In Appendix A we write the treelevel masses in terms of the
parameters of the MSSM, and in Appendix B we define the
PassarinoVeltman functions that we use to present our oneloop
results. In Appendix C we compute the oneloop radiative corrections
to the gauge couplings of the MSSM, and in Appendix D we write the
oneloop corrections to the masses. Where appropriate, we evaluate
the corrections to the mass {\em matrices} to account for full
oneloop superparticle mixing. This allows for an accurate
determination of the masses and mixing through the entire parameter
space. Finally, in Appendix E we discuss the radiative corrections to
the Higgs boson masses.
The results in the Appendices hold for the MSSM with the most general
pattern of (flavor diagonal) soft supersymmetry breaking.\footnote{
Our results can be readily extended to include intergenerational
mixing at the cost of additional mixing matrices.}\ The parameter
space is huge because of the large number of operators that softly
break supersymmetry. Therefore in the body of the paper we illustrate
our results in a reduced parameter space, obtained by assuming that
the soft breaking parameters unify at some high scale.
The unification assumption is useful because it reduces the size
of the parameter space. Moreover, it implies certain mass
relations that can be tested once supersymmetry is discovered. In
addition, for any set of parameters, it allows us to determine the
unification scale thresholds that are necessary to achieve unification.
As we will see, the present set of precision measurements is sufficient
to begin to constrain the physics at the unification scale.
We implement the unification assumption by solving
the twoloop supersymmetric renormalization group equations
subject to twosided boundary conditions. At the weak scale, we
assume a supersymmetric spectrum, and for a given value of the
ratio of vacuum expectation values, $\tan\beta$, we use our oneloop
corrections to extract the \mbox{\footnotesize$\overline{\rm DR}~$}
couplings $g_1,\ g_2, \ g_3,\ \lambda_t,\ \lambda_b$, and $\lambda_\tau$
at the scale $M_Z$. We then use the twoloop \mbox{\footnotesize$
\overline{\rm DR}~$} renormalization group equations \cite{12} to
run these six parameters to the scale $M_{\rm GUT}$, which we define
to be the scale where $g_1$ and $g_2$ meet.
We require that the soft breaking parameters unify at the scale
$M_{\rm GUT}$. Therefore at $M_{\rm GUT}$ we fix a universal scalar
mass, $M_0$, a universal gaugino mass, $M_{1/2}$, and a universal
trilinear scalar coupling, $A_0$. We then run all the soft parameters
back down to the scale $M^2_{\tilde q} = M^2_0 + 4 M^2_{1/2}$, where
we calculate the supersymmetric spectrum using the full oneloop
threshold corrections that we present in this paper. In section 4
we show that this scale is essentially the scale of the squark masses,
and that the other scalar masses and the Higgsino mass are correlated
with it as well.
We require radiative electroweak symmetry breaking \cite{13}, so
the CPodd Higgs mass, $m_A$, and the supersymmetric Higgs mass
parameter, $\mu$, are determined in a full oneloop calculation at
the scale $M_{\tilde q}$. The sign of $\mu$ is left undetermined. We
then iterate the entire procedure to determine a selfconsistent
solution. Typically, convergence to an accuracy of better than
$10^{4}$ is achieved after four iterations.
Once we have a consistent solution, we use the results of the
Appendices to
illustrate the oneloop corrections in the reduced parameter space
associated with unification. We display results for a randomly chosen
sample of 4000 points. Our sample is chosen with a logarithmic
measure in the range $1 < \tan\beta < 60$, $50 < M_{1/2} < 500$ GeV,
$10 < M_0 < 1000$ GeV, and with a linear measure in the range $3
M_{\tilde q} < A_0 < 3 M_{\tilde q}.$ (The upper limits on $M_0$ and
$M_{1/2}$ are chosen so that the squark masses are less than about 1
TeV. While larger squark masses are certainly possible, they
reintroduce the fine tunings that supersymmetry is designed to avoid.)
Each of these points corresponds to a (local) minimum of the oneloop
scalar potential with the correct electroweak symmetry breaking, and
each passes a series of phenomenological constraints: We require the
first and secondgeneration squark masses to be larger than 220 GeV
\cite{14}, the gluino mass to be greater than 170 GeV
\cite{14}, the light Higgs mass\footnote{The light Higgs boson is
similar to that of the standard model in almost all of our parameter
space, so we apply the standard model bound.} to be greater than 60
GeV \cite{2}, the slepton masses to be greater than 45 GeV
\cite{2}, and the chargino masses to be greater than 65 GeV
\cite{15}. We also require all the Yukawa couplings to remain
perturbative ($\lambda<3.5$) up to the unification scale, and since we
assume that $R$parity is unbroken, we enforce the cosmological
requirement that the lightest supersymmetric particle be neutral.
We derive approximations to the radiative corrections that hold with
reasonable accuracy over the unified parameter space. Where
appropriate, we use scatter plots to illustrate the effectiveness of
our approximations. The approximations consist of two parts. First
we identify the most important contributions to the oneloop
corrections. In most cases these are the loops that involve the
strong and/or third generation Yukawa couplings. Then we derive
approximations to the loop expressions that hold over the unified
parameter space.
In the next section, we discuss the radiative corrections to the
effective weak mixing angle, $\sin^2\theta^{\rm lept}_{\rm eff}$, and
the $W$boson pole mass, $M_W$. We illustrate the magnitudes of the
different supersymmetric contributions to these observables. We also
discuss the renormalization of the \mbox{\footnotesize$\overline{\rm
DR}~$} weak mixing angle, $\hat s^2$, and comment on the way that it
affects the gauge thresholds at the unification scale. In section 3
we examine radiative corrections to the third generation quark and
lepton masses. We illustrate the different contributions and present
approximations which hold to a few percent. We also examine Yukawa
unification and demonstrate the size of the unificationscale Yukawa
thresholds. In section 4 we present our results for the radiative
corrections to the supersymmetric and Higgs boson particle masses. We
find large corrections to the masses of the light superparticles. We
compare our results with those of the leading logarithmic
approximation and find significant improvements over much of the
unified parameter space. These corrections are important for
unraveling the underlying supersymmetric structure from the
supersymmetric mass spectrum.
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