%% slacpub7127: page file slacpub7127002.tcx.
%% section 2 Light Higgsinos and FCNC [slacpub7127002 in slacpub7127002: slacpub7127003]
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\section{\usemenu{slacpub7127::context::slacpub7127002}{Light Higgsinos and FCNC}}\label{section::slacpub7127002}
The importance of light higgsinos and squarks to FCNC processes is
simply that the presence of the quark masses at the
quarksquarkhiggsino vertices removes the super GIM cancellation
present in the usual supersymmetric box graphs. Thus higgsino mediated
box graphs give nonzero contributions to neutral meson mixing
independent of details of the squark mass matrices.
Given the factor of $m_t/\sin\beta$
that appears in the $d_L  \tilde{u}_R  \tilde{h}$ vertex, we are
gauranteed to get large supersymmetric contributions to $K\bar K$,
$B_d\bar B_d$ and $B_s\bar B_s$ mixing for light righthanded
uptype squarks, higgsinos, and small $\tan\beta$.{\footnote{A similar
argument could be made for charged Higgs boson mediated boxes. These,
however, are generally smaller because of the heavy top quark in the
loops. We have checked that over the parameter range we study here,
the charged Higgs bosons can be made heavy enough to satisfy
constraints from other processes like $b \rightarrow s \gamma$, and
not affect our analysis. }}
Consider the basic quarksquarkhiggsino vertex:
%
\beq
{\cal L}_I = \frac{g}{\sqrt 2 M_W \sin\beta}
\bar d_L[V^{\dagger}_{KM}\hat M_U \tilde V_U]
\tilde u_R \tilde h
\label{vertex}
\eeq
%
where $V_{KM}$ is the CKM matrix, $\hat M_U$ is the diagonal matrix of
uptype quark masses, and $\tilde V_U = V^{\dagger}_{UR}\tilde V_{UR}$
is a product of the unitary matrices that diagonalize the righthanded
uptype quark and squark mass matrices respectively. Starting with
Eq.~(\docLink{slacpub7127002.tcx}[vertex]{1}) we can derive the following very simple formula for
the supersymmetric contribution to the off diagonal terms in the
mass matrix of the neutral
meson consisting of the quarks $(\bar a b)$ with $a, ~b=1,2,3$:
%
\beq
(M_{ab})_{12} = K_{\bar a b}[V^{\dagger}_{KM}\hat M_U \tilde V_U
\frac{{\tilde M}^{1}}{2\sqrt 3\sin^2\beta}
{\tilde V}^{\dagger}_U\hat M_U V_{KM}]_{ab}^2
\label{master}
\eeq
%
where
%
\beq
K_{\bar a b} = \frac{G_F^2}{12\pi^2}(B_{\bar a b}
f_{\bar a b}^2m_{\bar a b}\eta)
\label{defK}
\eeq
%
with $B_{\bar a b},~f_{\bar a b},~m_{\bar a b}$ being
the bag factor, decay constant and mass of the meson,
and $\eta$ a QCD correction factor which we always set equal to the
corresponding QCD correction for the Standard Model box diagram with
top quarks in the loop.
$\tilde M$ is the diagonal matrix of righthanded uptype squark
masses. We have ignored any difference between the charged higgsino
mass and the squark masses in deriving Eq.~(\docLink{slacpub7127002.tcx}[master]{2}).
This approximation does not significantly
affect the accuracy of our results for the range of masses we
consider (this was noted in \cite{8}), while allowing
us to derive the simple expression of Eq.~(\docLink{slacpub7127002.tcx}[master]{2}).
Let us now assume the following form
for the mixing matrix $\tilde V$:
%
\beq
\tilde V = \left(\begin{array}{ccc}
1& 0& 0\\
0& 1/\sqrt 2& 1/\sqrt 2 \\
0& 1/\sqrt 2& 1/\sqrt 2 \end{array} \right)
\eeq
%
{\it i.e.} the righthanded scalar charm and top are maximally mixed.
This form of the mixing matrix could be
motivated in some of the models for fermion masses based on Abelian
horizontal symmetries ~\cite{7,11}.
We will denote the common mass of the lightest squark and charged
higgsino by $\tilde m$, and assume no special degeneracy between the
physical squark masses
In this case, we
can use Eq.~(\docLink{slacpub7127002.tcx}[master]{2}) to obtain the
following expressions for the meson mixing to first order in $m_c/m_t$:
%
\beq
M_{12}^{B_d} ={\displaystyle{
\frac{G_F^2}{12\pi^2}B_{B_d}f_{B_d}^2m_{B_d}\eta_{B_d}m_t^2
[a V_{td}^{\ast 2}V_{tb}^2}}
+{\displaystyle{b V_{td}^{\ast 2}V_{tb}^2 +
c V_{cd}^{\ast}V_{td}^{\ast}V_{tb}^2]}}
\label{massbd}
\eeq
%
\beq
M_{12}^{B_s} ={\displaystyle{
\frac{G_F^2}{12\pi^2}B_{B_s}f_{B_s}^2m_{B_s}\eta_{B_s}m_t^2
[a V_{ts}^{\ast 2}V_{tb}^2}}
+{\displaystyle{b V_{ts}^{\ast 2}V_{tb}^2 +
c V_{cs}^{\ast}V_{ts}^{\ast}V_{tb}^2]}}
\label{massbs}
\eeq
%
\beqa
M_{12}^{K} =&{\displaystyle{
\frac{G_F^2}{12\pi^2}B_{K}f_{K}^2m_K\eta_{K}m_t^2
[a V_{td}^{\ast 2}V_{ts}^2}}
+{\displaystyle{b V_{td}^{\ast 2}V_{ts}^2 +
c V_{cd}^{\ast}V_{td}^{\ast}V_{ts}^2
+ c V_{td}^{\ast 2}V_{cs}V_{ts}}} \nonumber \\
&+{\displaystyle{\frac{m_c^2\eta_{cc}}{m_t^2\eta_{K}}
f_2(y_c) V_{cd}^{\ast 2}V_{cs}^2 +
\frac{m_c^2\eta_{ct}}{m_t^2\eta_{K}}
f_3(y_c,y_t)
V_{cd}^{\ast}V_{cs}V_{td}^{\ast}V_{ts}}}]
\label{massk}
\eeqa
%
where
%
\beq
a=f_2(y_t),~
b=\frac{1}{48\sin^4\beta}\frac{m_t^2}{{\tilde m}^2},~
c=\frac{1}{24\sin^4\beta}\frac{m_cm_t}{{\tilde m}^2},
\label{defabc}
\eeq
%
$y_i=m_i^2/M_W^2$,
and the functions $f_2(x),~f_3(x,y)$ are defined in ~\cite{12,
13}:
%
\beqa
f_2(x)&=&{\displaystyle{\frac{1}{4}+\frac{9}{4(1x)}
\frac{3}{2(1x)^2}\frac{3x^2\ln x}{2(1x)^3}}}\nonumber \\
f_3(x,y)&=&{\displaystyle{\ln(\frac{y}{x})\frac{3y}{4(1y)}
(1+\frac{y\ln y}{1y})}}
\eeqa
%
The terms proportional to $b$ and $c$ in
Eqs.~(\docLink{slacpub7127002.tcx}[massbd]{5}\docLink{slacpub7127002.tcx}[massk]{7}) are the
supersymmetric contributions, and have important consequences for the
determination of the CKM matrix elements as we show in the next section.
The dominant supersymmetric
contribution proportional to $b$ is present also in the usual analyses
based on the constrained MSSM, and is always in phase with the
Standard Model contribution proportional to $m_t^2$. Although we
started with only the CKM phase, in this model,
both the $B_d\bar B_d$ and the $K\bar K$
mass matrices have a second out of phase contribution given by the
term proportional to $c$. This
contribution is a result of the mixing between the righthanded scalar
top and charm, and should be observable at the $B$ factory $CP$
violating experiments. An estimate of the importance of this term
compared to the ``in phase'' supersymmetric contribution is given by
%
\beq
\frac{cV_{cd}}{bV_{td}}
\simeq \frac{2m_c}{A\lambda^2 m_t}
\simeq 50 \frac{m_c}{m_t}.
\eeq
%
where $A$ and $\lambda$ parametrize elements of the CKM matrix as
shown below, and we have used $(A\lambda^2)^{1}=V_{cb}^{1}\simeq
25$.
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