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%% subsection 4.3 Additional Soft Terms in the Higgs Sector [slac-pub-7237-0-0-4-3 in slac-pub-7237-0-0-4: slac-pub-7237-0-0-4-4]
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\subsection{\usemenu{slac-pub-7237::context::slac-pub-7237-0-0-4-3}{Additional Soft Terms in the Higgs Sector}}\label{subsection::slac-pub-7237-0-0-4-3}
\label{addhiggs}
The Higgs sector parameters $\mu$ and $m_{12}^2$ require
additional interactions
with the messenger sector beyond the standard model gauge
interactions.
In the minimal model the precise form of these interactions is
not specified, and $\mu$ and $m_{12}^2$ are taken
as free parameters.
%As discussed in section \ref{subvariations},
The additional interactions which couple to the Higgs sector
are likely to contribute to the Higgs soft masses
$\mHu^2$ and $\mHd^2$, and split these from
the left handed sleptons, $m_{\lL}^2$.
Splittings of ${\cal O}(1)$ are not unreasonable since
the additional interactions must generate $\mu$ and $m_{12}^2$
of the same order.
The Higgs splitting may be parameterized by $\Delta_{\pm}^2$,
defined in Eq. (\docLink{slac-pub-7237-0-0-2.tcx}[splithiggs]{11}) of section \docLink{slac-pub-7237-0-0-2.tcx}[subvariations]{2.2}.
It is possible that other scalars also receive additional
contributions to soft masses.
The right handed sleptons
receive a gauge-mediated mass only from
$U(1)_Y$ coupling, and are therefore most
susceptible to a shift in mass from additional
interactions.
Right handed sleptons represent the potentially
most sensitive probe for such interactions \cite{30}.
Note that additional messenger sector interactions do not modify
at lowest order
the relations among gaugino masses.
Since additional interactions {\it must} arise in the Higgs sector,
we focus in this section on
the effect of
additional contributions to the Higgs soft masses
on electroweak
symmetry breaking and the superpartner spectrum.
We also consider the possibility that $m_{12}^2$ is generated
entirely from renormalization group evolution \cite{31}.
Additional contributions to Higgs sector masses can in principle
have large effects on electroweak symmetry breaking.
With the Higgs bosons split from the left handed sleptons
the minimization condition (\docLink{slac-pub-7237-0-0-3.tcx}[mincona]{16}) is modified
to
\beq
\label{newEWSB}
|\mu|^2+\frac{m_Z^2}{2} =
\frac{(m^2_{H_d,0}+\Sigma_d)+
(m^2_{H_u,0}+\Sigma_u)\tan^2\beta}{\tan^2\beta -1}
-\frac{\Delta^2_+}{2}+\frac{\Delta^2_-}{2}\left(
\frac{\tan\beta^2 +1}{\tan\beta^2-1}\right)
\label{minsplit}
\eq
where all quantities are evaluated at the minimization scale,
and
$m^2_{H_u,0}$ and $m^2_{H_d,0}$ are the gauge-mediated contributions
to the soft masses.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jfig{sfig5n}{fig15.eps}
{The relation between the low scale $|\mu|$ parameter and
$\Delta_-\equiv {\rm sgn}(\Delta^2_-(M))(|\Delta^2_-(M)|)^{1/2}$
at the messenger scale
imposed by electroweak symmetry breaking for
$\Delta_+(M)=0$,
$m_{\tilde B}(M)=115,180,250$, GeV $\tan\beta =3$,
and $\Lambda=M$.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The relation between $\mu$ at the minimization scale and
$\Delta_-\equiv {\rm sgn}(\Delta^2_-(M) )\sqrt{|\Delta^2_-(M)|}$
at the messenger scale is shown in
Fig. \docLink{slac-pub-7237.tcx}[sfig5n]{15} for
$\Delta_+(M)=0$,
$m_{\tilde B}(M)=115,180,250$, GeV, $\tan\beta =3$,
and $\Lambda=M$.
For moderate values of $\Delta_-$, the additional Higgs splittings
contribute in quadrature
with the gauge-mediated contributions,
and only give ${\cal O}(\Delta^2_- / \mu^2)$ corrections
to the minimization condition (\docLink{slac-pub-7237-0-0-4.tcx}[minsplit]{33}).
This is the origin of the shallow plateau in Fig.
\docLink{slac-pub-7237.tcx}[sfig5n]{15} along which $\mu$ does not significantly vary.
The plateau extends over the range $|\Delta_-^2| \lsim |m_{H_u,0}^2|$
at the messenger scale.
For $\tan \beta \gg 1$ the minimization condition
(\docLink{slac-pub-7237-0-0-4.tcx}[minsplit]{33}) becomes
$|\mu|^2 \simeq -\mHu^2 + {1 \over 2}
(\Delta_-^2 - \Delta_+^2 - \mZ^2)$.
For very large $(\Delta_-^2 - \Delta_+^2)$ this reduces to
$\sqrt{2}|\mu| \simeq (\Delta_-^2 - \Delta_+^2)^{1/2}$.
This linear correlation between $\mu$ and $\Delta_-$
for $\Delta_-$ large and $\Delta_+=0$ is apparent
in Fig. \docLink{slac-pub-7237.tcx}[sfig5n]{15}.
The non-linear behavior at small $\Delta_-$
arises from ${\cal O}(\mZ^2 / \mu^2)$ contributions to the
minimization condition (\docLink{slac-pub-7237-0-0-4.tcx}[minsplit]{33}).
The physical correlation between $\mu$ and $\Delta_{\pm}$ is
easily understood in terms of $\mHu^2$ at the messenger scale.
For $\Delta_+=0$ and $\Delta_- > 0$, $\mHu^2$
%at the low scale
is more negative than in the minimal model,
leading to a deeper minimum
in the Higgs potential.
In fact, for the $\mbino(M)=115$ GeV case shown in Fig.
\docLink{slac-pub-7237.tcx}[sfig5n]{15}, $\mHu^2 <0$ already at the messenger scale
for $\Delta_- \gsim 260$ GeV.
Obtaining correct electroweak symmetry breaking for $\Delta_- > 0$
therefore requires
a larger value of $|\mu|$,
as can be seen in Fig. \docLink{slac-pub-7237.tcx}[sfig5n]{15}.
%The pseudo-scalar Higgs mass, $\mA$, also grows for $\Delta_- > 0$.
Conversely,
for $\Delta_+=0$ and $\Delta_- < 0$, $\mHu^2$
%at the low scale
is less
negative than in the minimal model, leading to a more shallow minimum
in the Higgs potential.
Obtaining correct electroweak symmetry breaking in this limit
therefore requires
a smaller value of $|\mu|$,
as can also be seen in Fig. \docLink{slac-pub-7237.tcx}[sfig5n]{15}.
%This also decreases the pseudo-scalar mass.
Eventually,
for $\Delta_-$ very negative,
$\mHu^2$ at the messenger scale is large enough that
the negative
renormalization group evolution from the top Yukawa is
insufficient to drive electroweak symmetry breaking.
In Fig. \docLink{slac-pub-7237.tcx}[sfig5n]{15} this corresponds to $|\mu| < 0$.
With $\Delta_-=0$ and $\Delta_+ > 0$ both $\mHu^2$ and $\mHd^2$
are larger at the messenger scale than in the minimal model,
leading to a more shallow minimum in the Higgs potential.
This results in smaller values of $\mu$, and conversely
larger values of $\mu$ for $\Delta_+ < 0$.
Again, there is only a significant effect for
$|\Delta_+^2| \gsim |\mHu^2|$.
The pseudo-scalar Higgs mass also depends on additional
contributions to the Higgs soft masses,
$\mA^2 = 2 |\mu|^2 + (m^2_{H_u,0} + \Sigma_u) +
(m^2_{H_d,0} + \Sigma_d) + \Delta_+^2$.
For large $\tan \beta$ the minimization condition (\docLink{slac-pub-7237-0-0-4.tcx}[minsplit]{33})
gives
$\mA^2 \simeq - (m^2_{H_u,0} + \Sigma_u) + (m^2_{H_d,0} + \Sigma_d)
+ \Delta_-^2$.
Again, for $|\Delta_-^2| \lsim |m^2_{H_u,0}|$ the pseudo-scalar
mass is only slightly affected, but can be altered significantly
for $\Delta_-$ very large in magnitude.
Notice that $\mA$ is independent of $\Delta_+$ in this limit.
This is because in the contribution
to $\mA^2$ the change in $|\mu|^2$ induced by $\Delta_+^2$
is cancelled by a
compensating change in $\mHd^2$.
This approximate independence of $\mA$ on $\Delta_+$ persists
for moderate values of $\tan \beta$.
For example,
for the parameters of Fig. \docLink{slac-pub-7237.tcx}[sfig5n]{15} with
$\mbino(M) = 115$ GeV and $\Delta_-=0$, $\mA$ only varies
between 485 GeV and 525 GeV for
$-500$ GeV $< \Delta_+ < $ $500$ GeV, while $\mu$ varies from 510 GeV to
230 GeV over the same range.
The additional contributions to the Higgs soft masses
can, if large enough, change the form of the superpartner
spectrum.
The charginos and neutralinos are affected mainly through
the value of $\mu$ implied by electroweak symmetry breaking.
For very large $|\mu|$, the approximately degenerate
singlet $\chi_3^0$ and triplet $(\chi_2^{+}, \chi_4^0, \chi_2^-)$
discussed in section \docLink{slac-pub-7237-0-0-3.tcx}[electroweaksection]{3.2.1} are mostly
Higgsino, and have mass $\mu$.
%These mass of these states is therefore sensitive to
%additional interactions with the messenger sector
%(or additional matter content at the electroweak scale).
For $\mu \lsim m_2$ the charginos and neutralinos are
a general mixture of gaugino and Higgsino.
A value of $\mu$ in this range,
as evidenced by a sizeable Higgsino component of
$\chi_1^0$, $\chi_2^0$, or $\chi_1^{\pm}$, or
a light $\chi_3^0$ or $\chi_2^{\pm}$,
would be strong evidence for
deviations from the minimal model in the Higgs sector.
The heavy Higgs masses are determined by $\mA$.
Since $\mA^2$ is roughly independent of $\Delta_+^2$, while
$|\mu|$ is sensitive to $( \Delta_-^2 - \Delta_+^2)$,
the relative shift between
the Higgsinos and heavy Higgses
is sensitive to the individual
splittings of $\mHu^2$ and $\mHd^2$ from the left handed sleptons,
$m_{\lL}^2$.
Within the MGM,
given an independent measure of $\tan \beta$ (such as from
left handed slepton - sneutrino splitting,
$m_{\lL}^2 - m_{\nu_L}^2 = -\mW^2 \cos 2 \beta$)
the mass of the Higgsinos and heavy Higgses
therefore provides an indirect probe
for additional contributions to the Higgs soft masses.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jfig{sfig6n}{fig16.eps}
{The ratio $m_{\tilde l_R}/m_{\chi^0_1}$ as a
function of $\Delta_-$ at the messenger scale
for $m_{\tilde B}(M)=115,180,250\gev$, $\tan\beta =3$,
${\rm sgn}(\mu)=+1$, and $\Lambda=M$.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Non-minimal contributions to Higgs soft masses can
also affect the other scalar masses through renormalization
group evolution.
The largest effect comes from the radiative contribution to
the $U(1)_Y$ $D$-term, which is generated in
proportion to $S = {1 \over 2}{\rm Tr}(Ym^2)$.
In the minimal model the Higgs contribution to $S$
vanishes at the messenger scale because the Higgses
are degenerate and have opposite hypercharge.
%For $\Delta_- \neq 0$ they are no longer degenerate and
%contribute to $S$.
For $\Delta_- > 0$ they are no longer degenerate and
give a negative contribution to $S$.
This increases the magnitude of the contribution in the minimal
model from running below the squark thresholds.
To illustrate the effect of the Higgs contribution to $S$ on
the scalar masses,
$m_{\tilde l_R}/m_{\chi^0_1}$ is shown
in Fig. \docLink{slac-pub-7237.tcx}[sfig6n]{16} as a
function of $\Delta_-$ at the messenger scale
for $m_{\tilde B}(M)=115,180,250\gev$, $\tan\beta =3$,
${\rm sgn}(\mu)=+1$, and $\Lambda=M$.
For $\Delta_-$ very large and
positive the radiatively generated
$U(1)_Y$ $D$-term contribution to right handed slepton masses
increases the ratio $m_{\tilde l_R}/m_{\chi^0_1}$.
For $\Delta_-$ very negative,
the rapid increase in $m_{\tilde l_R}/m_{\chi^0_1}$
occurs because $|\mu|$ is so small that $\chi_1^0$ becomes
mostly Higgsino with mass $\mu$.
All these modifications of the
form of the superpartner spectrum are significant
only if the Higgs bosons receive
additional contributions to the soft masses which are
roughly larger in magnitude than the gauge-mediated contribution.
%Such contributions are not completely unreasonable since
The $\mu$ parameter is renormalized multiplicatively while
$m_{12}^2$ receives renormalization group contributions proportional
to $\mu m_{\lambda}$, where $m_{\lambda}$ is the $B$-ino
or $W$-ino mass.
As suggested in Ref. \cite{31}, it is therefore interesting
to investigate the possibility that $m_{12}^2$ is generated
only radiatively below the messenger scale, with
the boundary condition $m_{12}^2(M)=0$.
Most models of the Higgs sector interactions actually suffer
from $m_{12}^2 \gg \mu^2$ \cite{3,15}, but
$m_{12}^2(M)=0$ represents a potentially interesting, and
highly constrained subspace of the MGM.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\jfig{sfig8n}{fig17.eps}
{The relation between $m_{12}(M)$ and $\tan\beta$ imposed by electroweak
symmetry breaking for %$M=\Lambda\propto m_{\tilde B}$
$m_{\tilde B}(M) =115,180,250$ GeV, and $\Lambda=M$.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In order to illustrate what constraints this boundary condition
implies, the relation between $m_{12}(M)$ and
$\tan \beta$ imposed by electroweak symmetry breaking is
shown in Fig. \docLink{slac-pub-7237.tcx}[sfig8n]{17} for
$m_{\tilde B}(M) =115,180,250$ GeV and $\Lambda=M$.
The non-linear feature at $m_{12}(M)\simeq 0$ is a square root
singularity since the $\beta$-function for $m_{12}^2$
is an implicit function of $\tan \beta$ only through the slow
dependence of $\mu$ on $\tan \beta$.
The value of $\tan \beta$ for which $m_{12}(M)=0$ is almost
entirely independent of the overall scale of the superpartners.
This is because to lowest order the minimization condition
(\docLink{slac-pub-7237-0-0-3.tcx}[minconb]{17}) fixes $m_{12}^2$ at the low scale
to be a homogeneous function of the overall superpartner scale
(up to $\ln(m_{\tilde{t}_1} m_{\tilde{t}_2} / m_t^2)$ finite
corrections)
$m_{12}^2 \simeq f(\alpha_i,\tan \beta) (\alpha / 4 \pi)^2 \Lambda^2$.
If $m_{12}$ vanishes at any scale, then the function $f$ vanishes
at that scale, thereby determining $\tan \beta$.
For $m_{12}(M)=0$ and $\Lambda=M$ we find $\tan \beta \simeq 46$.
With the boundary condition $m_{12}(M)=0$, the resulting
large value of $\tan \beta$ is natural.
This is because
$m_{12}^2(Q)$ at the minimization scale, $Q$, is small.
With the parameters given above $m_{12}(Q) \simeq -80$ GeV.
For $m_{12}(Q) \rightarrow 0$, $H_d$ does not participate
in electroweak symmetry breaking, and $\tan \beta \to \infty$.
As discussed in section \docLink{slac-pub-7237-0-0-3.tcx}[electroweaksection]{3.2.1},
at large $\tan \beta$,
$m_{\stau_1}$ receives a large negative contribution
from the $\tau$ Yukawa due to renormalization
group evolution and mixing.
For the values of $\tan \beta$ given above
we find $m_{\stau_1} \lsim m_{\na}$.
It is important to note that for such large values of
$\tan \beta$, physical quantities, such as
$m_{\stau_1} / m_{\na}$, depend sensitively on
the precise value of the
$b$ Yukawa through renormalization group and finite
contributions to the Higgs potential.
%For $m_t^{\rm pole} = 175 \pm 15$ GeV we find that
%$\tan \beta = 46 \pm ??$ for $m_{12}(M)=0$ and
%$\Lambda=M$.