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%% subsection 3.1 Electroweak Symmetry Breaking [slac-pub-7237-0-0-3-1 in slac-pub-7237-0-0-3: ^slac-pub-7237-0-0-3 >slac-pub-7237-0-0-3-2]
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\sectionLink{slac-pub-7237-0-0-3}{slac-pub-7237-0-0-3}{Above: 3. Renormalization Group Analysis}%
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\subsection{\usemenu{slac-pub-7237::context::slac-pub-7237-0-0-3-1}{Electroweak Symmetry Breaking}}\label{subsection::slac-pub-7237-0-0-3-1}
\label{EWSB}
The most significant effect of renormalization group evolution
is the negative contribution to the
up-type Higgs boson mass squared from the large
top quark Yukawa coupling.
As can be seen in Fig. \docLink{slac-pub-7237.tcx}[sfig7n]{4} this leads to a negative
mass squared for $H_u$.
The $\beta$-function for $\mHu^2$ is dominated by the
heavy stop mass
\beq
\frac{dm^2_{H_u}}{dt} \simeq
\frac{1}{16\pi^2} (6h^2_t (m^2_{\tilde{t}_L} +m^2_{\tilde{t}_R}
+ \mHu^2) +\cdots )
\eeq
where $t=\ln(Q)$, with a small
${\cal O}(g_2^2 m_2^2 / h_t^2 m^2_{\tilde{t}})$
correction from gauge interactions.
For the parameters given in Fig. \docLink{slac-pub-7237.tcx}[sfig7n]{4}
the full $\beta$-function for $\mHu^2$
is approximately constant above the stop
thresholds,
differing by less than 1\% between $M$ and $m_{\tilde t}$.
The evolution of $\mHu^2$ is therefore approximately
linear in this region
(the non-linear feature in Fig. \docLink{slac-pub-7237.tcx}[sfig7n]{4} is a square-root
singularity because the quantity plotted is
${\rm sgn}(\mHu) \sqrt{ | \mHu |^2}$).
It is worth noting that
for $\tan \beta$ not too large, and
$M$ not too much larger than $\Lambda$,
$\mHu^2(m_{\tilde t})$ is then well approximated by
\beq
\mHu^2(m_{\tilde t}) \simeq \mHu^2(M) - {3 \over 8 \pi^2} h_t^2
( m^2_{\tilde{t}_L} +m^2_{\tilde{t}_R} ) \ln(M/ m_{\tilde t})
\label{Huapp}
\eq
(although throughout we use numerical integration of the
full renormalization group equations).
For the parameters of Fig. \docLink{slac-pub-7237.tcx}[sfig7n]{4}
the approximation (\docLink{slac-pub-7237-0-0-3.tcx}[Huapp]{15})
differs from the full numerical integration by 2\%
(using the messenger scale values of $h_t$,
$m_{\tilde t_L}$, and $m_{\tilde t_R}$).
The magnitude of the small positive gauge contribution
can be seen below the stop thresholds in Fig. \docLink{slac-pub-7237.tcx}[sfig7n]{4}.
The negative value of $\mHu^2$ leads to electroweak symmetry
breaking in the low energy theory.
The mechanism of radiative symmetry breaking \cite{3}
is similar to that for high scale supersymmetry breaking with universal
boundary conditions.
With high scale supersymmetry breaking, $\mHu^2 < 0$ develops
because of the large logarithm.
Here $\mHu^2 < 0$ results not because the logarithm is large,
but because the stop masses are large \cite{3}.
Notice in Fig. \docLink{slac-pub-7237.tcx}[sfig7n]{4} that $\mHu^2$ turns negative
in less than a decade of running.
The negative stop correction %to $\mHu^2$
effectively amounts to an
${\cal O}((\alpha_3 / 4 \pi)^2
(h_t/4 \pi)^2 \ln(M / m_{\tilde t}))$
three-loop contribution which is larger than the
${\cal O}((\alpha_2 / 4 \pi)^2 )$ two-loop contribution
\cite{3}.
Naturally large stop masses which lead automatically to
radiative electroweak symmetry breaking are one of the
nice features of low scale
gauge-mediated supersymmetry breaking.
Imposing electroweak symmetry breaking gives relations
among the Higgs sector mass parameters.
In the approach taken here we solve for the electroweak
scale values of $\mu$ and $m_{12}^2$
in terms of $\tan \beta$ and $\mZ$
using the minimization
conditions
\bea
|\mu|^2+\frac{\mZ^2}{2} & = &
\frac{(m^2_{H_d}+\Sigma_d)-(m^2_{H_u}+\Sigma_u)\tan^2\beta}
{\tan^2\beta -1}
%\nonumber
\label{mincona} \\
\label{minconb}
\sin 2\beta & = &
\frac{-2m^2_{12}}{(m^2_{H_u}+\Sigma_u)+(m^2_{H_d}+\Sigma_d)+2|\mu|^2}
\eea
where
$\Sigma_{u,d}$ represent finite one-loop corrections
%to $\mHu^2$ and $\mHd^2$
from gauge interactions and top and bottom Yukawas
\cite{22}.
These corrections are
necessary to reduce substantially the scale dependence
of the minimization conditions.
In order to minimize the stop contributions to the finite corrections,
the renormalization scale is taken to be the geometric mean
of the stop masses, $Q^2 = m_{\tilde{t}_1} m_{\tilde{t}_2}$.
The finite corrections to the one-loop effective potential
make a non-negligible contribution
to (\docLink{slac-pub-7237-0-0-3.tcx}[mincona]{16}) and (\docLink{slac-pub-7237-0-0-3.tcx}[minconb]{17}).
For example, with the parameters of Fig. \docLink{slac-pub-7237.tcx}[sfig7n]{4},
minimization of the tree level conditions with the renormalization
scheme given above gives $\mu = 360$ GeV,
while inclusion of the finite corrections results in $\mu = 395$ GeV.
The minimization conditions (\docLink{slac-pub-7237-0-0-3.tcx}[mincona]{16}) and (\docLink{slac-pub-7237-0-0-3.tcx}[minconb]{17})
depend on
the value of the top quark mass, $m_t$, mainly through
the running of $\mHu^2$, and also through the finite corrections.
For the parameters of Fig. \docLink{slac-pub-7237.tcx}[sfig7n]{4},
a top mass range in the range $m_t = 175 \pm 15$ GeV gives a
$\mu$ parameter in the range $\mu = 395 \pm 50$ GeV.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jfig{sfig16n}{fig5.eps}
{The relation between $m_2$ and $|\mu|$
imposed by electroweak symmetry breaking with MGM boundary conditions
for $\tan\beta =2,3,5,10,30$, and $\Lambda=M$.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The correlation between $\mu$ and $m_{12}^2$ can be obtained from
(\docLink{slac-pub-7237-0-0-3.tcx}[mincona]{16}) and (\docLink{slac-pub-7237-0-0-3.tcx}[minconb]{17})
in terms of $\tan \beta$ and $\mZ$.
The relation between the $W$-ino mass, $m_2$,
and $\mu$ (evaluated at the renormalization scale)
imposed by electroweak symmetry breaking
in the minimal model is shown
in Fig. \docLink{slac-pub-7237.tcx}[sfig16n]{5} for
$\tan \beta = 2,3,5,10$, and $30$, with $\Lambda = M$ and
${\rm sgn}(\mu)=+1$.
The actual correlation is of course between the Higgs sector
mass parameters.
The parameter $m_2$ is plotted just as a representative mass
of states transforming under $SU(2)_L$, and of the overall scale
of the soft masses
(the gaugino masses directly affect electroweak
symmetry breaking only through very small higher order corrections
to renormalized Higgs sector parameters).
The $\mu$ parameter typically lies in the range
$ {3 \over 2} m_2 \lsim |\mu| \lsim 3 m_2$ or
$ m_{\lL} \lsim |\mu| \lsim 2 m_{\lL}$,
depending on the precise values of
$\tan \beta$ and $\ln M$.
The correlation between $\mu$ and the
overall scale of the soft masses arises because
the stop masses set the scale for $\mHu^2$ at the electroweak
scale, and therefore the depth of the Higgs potential.
For $\tan \beta \gg 1$ %the condition on $\mu$ is approximately
the conditions (\docLink{slac-pub-7237-0-0-3.tcx}[mincona]{16}) reduce to
$|\mu|^2 \simeq -\mHu^2 - {1 \over 2} \mZ^2$.
In this limit, for $|\mu|^2 \gg \mZ^2$,
$|\mu| \simeq (- \mHu^2)^{1/2}$, with the small difference determining
the electroweak scale.
At moderate $\tan \beta$ the corrections to this approximation
increase $\mu$ for a fixed overall scale.
At fixed $\tan \beta$, and $M$ not too far
above $\Lambda$, $\mHu^2$ at the renormalization scale
is approximately a linear function of
the overall scale $\Lambda$, as can be seen from Eq. (\docLink{slac-pub-7237-0-0-3.tcx}[Huapp]{15}).
The very slight non-linearity in Fig. \docLink{slac-pub-7237.tcx}[sfig16n]{5} arises
from $\ln M$ dependence, and ${\cal O}(\mZ^2 / \mu^2)$
effects in the minimization conditions.
The limit $|\mu|^2, |\mHu|^2 \gg \mZ^2$ of course represents
a tuning among the Higgs potential parameters
in order to obtain proper electroweak symmetry breaking.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\jfig{sfig11n}{fig6.eps}
{The ratio $m_{12}/\mu$ as a function of
$\tan\beta$ imposed by electroweak symmetry breaking for
$m_{\tilde B}(M)=115, 180,250$ GeV and $\Lambda=M$.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The ratio $m_{12} / \mu$ at the renormalization scale is
plotted in Fig. \docLink{slac-pub-7237.tcx}[sfig11n]{6}
for $m_{\tilde B}(M)=115, 180,250$ GeV and $\Lambda=M$.
Again, the tight correlation, approximately independent of the
overall scale, arises because all soft terms are related to
a single scale, $\Lambda$.
The small splitting between the three cases shown
in Fig. \docLink{slac-pub-7237.tcx}[sfig11n]{6} arises from $\ln M$ dependence, and
${\cal O}(\mZ^2 / \mu^2)$
effects in the minimization conditions.
Ignoring corrections from the bottom Yukawa,
$m_{12} / \mu \rightarrow 0$
for $\tan \beta \gg 1$.
The saturation of $m_{12} / \mu$ at large $\tan \beta$ is
due to bottom Yukawa effects in the renormalization
group evolution and finite corrections to $\mHd^2$.
Any theory for the origin of $\mu$ and $m_{12}^2$ with
minimal gauge mediation, and
only the MSSM degrees of freedom at low energy, would
have to reproduce (or at least be compatible with) the
relation given in Fig. \docLink{slac-pub-7237.tcx}[sfig11n]{6}.
Note that all the low scale Higgs sector mass parameters are quite
similar in magnitude %, differing by not more than a factor of two
over essentially all the parameter space of the MGM.