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\begin{document}
\preprint{OUTP-02-42P}
\preprint{CERN-TH/2002-335}
\title{Yukawa structure, flavour and $CP$ violation in Supergravity.}
\author{G. G. Ross}
\affiliation{Dep. of Physics, Theoretical Physics, U. of Oxford, Oxford, OX1 3NP, UK}
\affiliation{Theory Division, CERN, CH-1211, Geneva 23, Switzerland}
\author{O. Vives}
\affiliation{Dep. of Physics, Theoretical Physics, U. of Oxford, Oxford, OX1 3NP, UK}

\begin{abstract}
The hierarchical structure of fermion masses and mixings strongly suggests
an underlying family symmetry. In supergravity any familon field
spontaneously breaking this symmetry necessarily acquires an $F$-term which
contributes to the soft trilinear couplings. We show, as a result, $\mu
\rightarrow e\gamma $ decay can receive large contributions from this source
at the level of current experimental bounds and thus this channel may
provide the first indication of supersymmetry and a clue to the structure of
the soft breaking sector. Using the mercury EDM\ bounds we find strong
bounds on the right handed down quark mixing angles that are
inconsistent with models relating them to neutrino mixing angles and favour
a near-symmetric form for the magnitude of the down quark mass matrix.
\end{abstract}

\maketitle

%\email{graham.ross@cern.ch}

%\email{vives@thphys.ox.ac.uk}

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%  called in {figure} environment.
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Our knowledge of the Supersymmetry breaking sector in the Minimal
Supersymmetric extension of the Standard Model (MSSM) is still very limited.
Only after the discovery of SUSY particles and the measurement of the
Supersymmetric spectrum we will be able to explore it in detail.
Nevertheless, we have already a lot of useful information on this sector
from experiments looking for indirect effects of SUSY particles in
low-energy experiments \cite{Annrev}. In fact, it was readily realized at
the beginning of the SUSY phenomenology era that large contributions to
Flavour Changing Neutral Currents (FCNC) and $CP$ violation phenomena were
expected in Supersymmetric theories with a generic soft breaking sector. The
absence of these effects much below the level most theorists considered
reasonable came to be known as the SUSY flavour and $CP$ problems. These
problems are closely related to the flavour structure of the soft breaking
terms and therefore to the source of flavour itself. The main solutions to
these problems are either universality of the soft breaking terms or
alignment with the Yukawa matrices.

In this paper we ask whether either of these solutions is sufficent in the
case the theory possesses a broken family symmetry which generates the
hierarchical structure of Yukawa couplings through a superpotential of the
form%
\begin{equation}
W=W^{hid}(\eta _{k})+\left( \frac{\displaystyle{\theta }}{\displaystyle{M}}%
\right) ^{\alpha _{ij}}H_{a}Q_{Li}q_{Rj}^{c}+\dots  \label{yukawas}
\end{equation}%
The hierarchy arises through effective operators ordered in terms of
powers of $(\theta /M)$ where $\theta $ is a familon field which
spontaneously breaks the family symmetry and $M$ is the mass of the mediator
transmitting the family symmetry breaking to the quarks and leptons ($M\leq
M_{Planck}$). We will show that neither universality of the soft breaking
terms nor alignment with the Yukawa matrices solutions can be exact and that
the soft breaking terms are necessarily nonuniversal and not aligned with
the Yukawa couplings. This is due to the fact that, in a theory with broken
local Supersymmetry, the familon field(s) necessarily acquires a
non-vanishing $F$-term\cite{abelservant}. 
This $F$ term then induces soft SUSY breaking $A$
terms which are not diagonalised when the fermion masses are diagonalised,
leading to FCNC and $CP$ violating processes.

We first prove the existence of a non-vanishing $F$ term. Consider the
supergravity lagrangian specified in terms of the superpotential $W$ and the
K\"{a}hler potential, $K=\hat{K}(\eta ,\eta ^{\ast })+\sum_{i}K_{i}^{i}(\eta
,\eta ^{\ast })|\varphi _{i}|^{2}$ where $\eta $ are general fields 
belonging to the hidden
sector. The $F$-term contribution to the scalar potential is simply given by, 
\begin{equation}
V=e^{K}\left[ \sum_{i}(K^{-1})_{i}^{j}\tilde{F}^{i}\tilde{F}_{j}-3|W|^{2}%
\right]  \label{scalpot}
\end{equation}%
where $\tilde{F}^{i}=\partial W/\partial \phi _{i}+K_{i}^{i}\phi ^{i\ast }W$
and $\tilde{F}_{i}=\partial W/\partial \phi ^{i\ast }+K_{i}^{i}\phi
_{i}W^{\ast }$ are related to the normalized supergravity $F$-terms by $%
F^{i}=e^{K/2}(K^{-1})_{i}^{i}\tilde{F}^{i}$ and $(K^{-1})_{j}^{i}$ is the
inverse of the K\"{a}hler metric. We study the case of broken supergravity
in which $m_{3/2}=e^{K/2}\langle W\rangle \neq 0$. We will also have a
non-vanishing vev for a certain familon field $\theta $ after minimization
of the scalar potential. This field should have a small vev in units of the
Plank mass to generate the hierarchy in the Yukawa couplings which are
proportional to powers of $\theta $. The field $\theta $ belongs to a
non-trivial representation of the family group and is a singlet under the
Standard Model gauge group. We will also allow for further familon fields, $%
\chi _{i},$ transforming non-trivially under the family symmetry to acquire
vevs. Given this, the form of the $\theta $ and $\chi $
dependence of the K\"{a}hler potential is fixed by the requirement that the
potential should be invariant under the family symmetry and the leading term
is given by $K=\theta \theta ^{\ast }+\sum_{i}\chi _{i}\chi _{i}^{\ast }$
where we have absorbed the constants of proportionality in a redefinition of
the fields. With this form of K\"{a}hler potential we have $F^{\theta
}=\langle \partial W/\partial \theta ~+~m_{3/2}\theta ^{\ast }\rangle $, and
the only way to avoid the conclusion that $F^{\theta }\geq m_{3/2}~\theta $
is if a cancellation occurs between the two terms. 

It is instructive to
consider first the case that the $\theta $ dependent part of the
superpotential contributing to the potential does not involve the vev any
other fields, giving $V(\theta )=\left| F^{\theta }\right| ^{2}-3|W|^{2}$.
Minimisation with respect to $\theta $ leads to the result 
\begin{equation}
F^{\theta }\simeq -m_{3/2}^{2}~\theta ~\left( \frac{\displaystyle{\partial
^{2}W}}{\displaystyle{\partial \theta ^{2}}}\right) ^{-1}  \label{fmin}
\end{equation}%
where we have set $\langle W\rangle =m_{3/2}$ and used the condition 
\begin{equation}
\langle \frac{\partial W}{\partial \theta }\rangle \simeq -m_{3/2}\langle
\theta ^{\ast }\rangle  \label{fsmall}
\end{equation}%
that is needed if $F^{\theta }$ is to be reduced below its natural value $%
\geq m_{3/2}\theta .$ From Eq.~(\ref{fmin}) we see that the condition for an
anomalously small $F^{\theta }$ is 
\begin{equation}
|\frac{\partial ^{2}W}{\partial \theta ^{2}}|>>m_{3/2}  \label{ineq}
\end{equation}%
It is now straightforward to check whether this possibility is realised for
various forms for $W(\theta ).$

We first consider the case that there is only one scale in the problem,
namely the Planck scale and that the fermion mass hierarchy is due to an
expansion in $\theta /M_{Planck.}$ \ One possibility, which has been widely
explored, is that there is an anomalous $U(1)$ family symmetry and only the
field $\theta $ acquires a vev close to the Planck scale, driven by the
requirement the anomalous $D-$term should be small. In this case the
superpotential cannot depend on $\theta $ except in combination with fields
which, for the moment, we require to have vanishing vevs, leading
immediately to the conclusion $F^{\theta }=\theta m_{3/2}$

A second possibility is that the family symmetry is discrete and in this
case one can have a dependence on $\theta $ of the form%
\begin{equation}
W=a+(\theta /M_{Planck})^{p}  \label{monomial}
\end{equation}%
where we have allowed for a term, $a,$ coming from the hidden supersymmetry
breaking sector and $p$ is the order of the discrete
group. In this case $\partial ^{2}W/\partial \theta ^{2}\varpropto
p(p-1)\theta ^{p-2}=(p-1)m_{3/2},$ where we have used Eq.~(\ref{fsmall}) to
determine the magnitude of $\theta .$ Inserted in Eq.~(\ref{fmin}) this
suggests $F^{\theta }$ is reduced by a factor $1/(p-1)$ compared to its
natural value. However, for $p>3,$ the true minimum of the potential
following from Eq.~(\ref{monomial}) is at $\langle\theta \rangle=0.$

If $\theta $ has a renormalisable coupling $\theta XY$\ to other fields in
the theory, there are radiative corrections to the soft SUSY breaking terms
which must be included in the effective potential, namely the soft mass
squared and the soft trilinear term \cite{radiative}. They can lead to 
the global minimum
having nonzero $\theta $ as desired. How do they affect our conclusions? The
soft mass only changes $\partial V/\partial \theta $ at ${\mathcal{O}}%
(m_{3/2}^{2}\theta )$ and so, in the large $p$ limit, does not affect the
derivation of Eq.~(\ref{fmin}). However the the trilinear term contributes
to $\partial V/\partial \theta $ at ${\mathcal{O}}((p-1)m_{3/2}\theta )$
leading to a shift in the position of the minimum at ${\mathcal{O}}(1).$ In
turn this leads to ${\mathcal{O}}(\theta m_{3/2})$ corrections to $F^{\theta
}$ in Eq.~(\ref{fmin}), i.e. $F^{\theta }$ is still of ${\mathcal{O}}(\theta
m_{3/2})$. This discussion can be repeated in the case there is a continuous
symmetry with two charged fields obtaining vevs of a similar size along a $D$%
-flat direction so that $W=a+\chi ^{p}\bar{\chi}^{q}$. In this case $D$%
-flatness requires that $Q\langle \chi \rangle +\bar{Q}\langle \bar{\chi}%
\rangle =0$ with $Q$ and $\bar{Q}$ the charges under the continuous group.
Using this relation we can follow the previous analysis and we arrive 
again to the same result for this case.

To avoid this conclusion we must consider the case $\theta $ gets a mass in
the Supersymmetric limit in order to satisfy Eq.(\ref{ineq}). This requires
the existence of additional mass scales in the theory different from $%
M_{Plank}$ or $m_{3/2}$. We assume, as is expected in a string theory, that
all mass scales in the theory must be generated through spontaneous symmetry
breaking and therefore we consider the case there are additional fields in
the theory which acquire different vevs. This leads to the possibility that
more than one term in the superpotential is important \footnote{%
In the previous case one term dominates since $\theta /M_{Planck}$ is the
only dimensionless parameter and is small.}. To demonstrate the
possibilities it is sufficient to consider just one extra field, $\phi ,$
and two terms in the superpotential (setting the Planck scale to unity for
clarity) 
\begin{equation}
W(\theta ,\phi )=\theta ^{p}\phi ^{q}+\theta ^{p^{\prime }}\phi ^{q^{\prime
}}  \label{poly}
\end{equation}%
Given that the field vevs are typically much greater than the electroweak
scale, we take $\phi ,$ like $\theta ,$ to be neutral under the SM group. The
form of the two terms in the superpotential must be determined by a symmetry
which may be the original family symmetry or may be a new one. The symmetry
can be either discrete, with $p$ and $p^{\prime }$ multiples of the order
parameter of the discrete group, or continuous if the fields acquires vevs
along a $D$-flat direction (in which case, to avoid $\langle \theta\rangle
\approx \langle \phi \rangle$ which would cause one of the terms in 
Eq.~(\ref{poly}) to
dominate, there must be additional\ non singlet fields which acquire vevs).
We include in the effective potential for $\phi $ a term $V_{\phi }$ which
forces it to acquires a vacuum expectation value. Replacing $\phi $ by its
vev Eq.~(\ref{poly}) becomes%
\begin{equation}
W(\theta )=M_{1}\theta ^{p}+M_{2}\theta ^{p^{\prime }}  \label{polyth}
\end{equation}%
with $M_{1}=\langle \phi \rangle ^{q}/M_{Planck}^{p+q-3},$ $M_{2}=\langle
\phi \rangle ^{q^{\prime }}/M_{Planck}^{p^{\prime }+q^{\prime }-3}$ \ The
potential following from $W(\theta )$ has a non-trivial minima in the
globally supersymmetric limit given by 
\begin{equation}
\langle \theta _{0}\rangle ^{p^{\prime }-p}=-pM_{1}/p^{\prime }M_{2}
\label{theta0}
\end{equation}%
For a minimum with $\langle \theta \rangle $ and $\langle \phi \rangle $
less than the Planck mass $(q-q^{\prime })$ and $(p^{\prime }-p)$ must be
positive. The minimum occurs through a cancellation between the two terms in 
$\partial W/\partial \theta .$ One can readily show that, due to the
relatively small effect of the last term in Eq.~(\ref{scalpot}), the global
minimum of the full supergravity potential can be close to this minimum. In
this case, due to the cancellation between the two terms, one cannot use
Eq.~(\ref{fsmall}) to relate the magnitudes of $\partial ^{2}W/\partial
\theta ^{2}$ and $m_{3/2}~\theta $, allowing the $\theta $ mass to be much
larger than $m_{3/2}$ and thus suppressing $F^{\theta }.$

Solving for the minimum of the potential following from Eq.~(\ref{polyth})
one finds (in Planck units) 
\begin{eqnarray}
F^{\theta } &=&\left( \frac{3m_{3/2}}{W^{\prime \prime }(\theta _{0})}%
\right) m_{3/2}\theta _{0} \\
&\simeq &\left( \frac{3m_{3/2}}{p^{\prime }(p^{\prime }-p)\left( \frac{p}{%
p^{\prime }}\right) ^{\frac{\left( p^{\prime }-2\right) }{(p^{\prime }-p)}%
}\langle\phi \rangle^{q^{\prime }+\frac{\left( p^{\prime }-2\right) \left(
q-q^{\prime }\right) }{(p^{\prime }-p)}}}\right) m_{3/2}\theta _{0} 
\nonumber
\end{eqnarray}

The suppression depends on $\langle\phi \rangle.$ To bound $\langle\phi
\rangle$ requires minimisation of the full $V(\theta ,\phi )$ potential
following from Eq.~(\ref{poly}) and $V_{\phi }.$ In the global limit%
\begin{eqnarray}
V(\theta ,\phi )=\left| F^{\theta }\right| ^{2}+\left| F^{\phi }\right|
^{2}+V_{\phi }
\end{eqnarray}
If $\theta <\phi $ (requiring $(q-q^{\prime })/(p^{\prime }-p)>1)$ the
individual terms in $F^{\theta }$ and their derivatives are greater than
those in $F^{\phi }.$ Thus minimisation with respect to $\theta $ will be
approximately equivalent to minimisation of $\left| F^{\theta }\right| ^{2}$
keeping $\phi $ fixed as was done following Eq.~(\ref{polyth}). Since $%
F^{\phi }$ breaks supersymmetry and contributes to scalar masses, the
solution to the hierarchy problem requires $\left| F^{\phi }\right| \leq
m_{3/2},$ bounding the allowed vevs. This, together with $\theta <\phi ,$
requires, up to a constant factor 
\begin{equation}
\phi ^{\frac{qp^{\prime }-pq^{\prime }}{p^{\prime }-p}-1}<m_{3/2}
\label{bound}
\end{equation}%
Using Eq.~(\ref{theta0}) we see that this bound allows $F^{\theta }$ to be
smaller than $m_{3/2}\theta _{0}$ provided 2$(q-q^{\prime })/(p^{\prime
}-p)>1,$ consistent with the condition needed for $\theta <\phi .$ To take a
simple example, the case $p=3,q=2,p^\prime =4,q^\prime =0$ gives $\theta
=3\phi ^{2}/4$, with 2$\theta ^{3}\phi <m_{3/2}.$ Thus $\theta
/M_{Planck}\leq 10^{-4}$ and $F^{\theta }\geq \sqrt{3}\left( \theta
/M_{Planck}\right) ^{3/2}m_{3/2}\theta _{0}$ allowing for a strong
suppression of $F^{\theta }.$

Thus we see that it is possible in the general case to suppress $F^{\theta }$
below $\langle \theta \rangle ~m_{3/2}.$ However this is not the case in
many models. If the expansion parameter determining the fermion hierarchy is 
$\theta /M_{Planck},$ $q$ must be large ($q\geq 8$ for $\theta
/M_{Planck}\simeq 0.1).$ If the family symmetry is responsible for making $q$
large in Eq.~(\ref{poly}) there must be no fields acquiring vevs with family
charge allowing lower order terms. Effectively this means that the family
structure will be determined by the field $\theta $ carrying a single sign
of family charge. Such models have been used to generate fermion mass
structure but they cannot reproduce the phenomenologically successful Gatto,
Sartori Tonin (GST) relation which requires familon fields of both charges 
\cite{tasi}. Moreover the multiplet structure needed to get $q\geq 8$ 
looks very contrived.

It is possible to arrange for contributions from fields of both sign of
family charge, for example if $\chi $ and $\overline{\chi }$ acquire vevs
along the $D$-flat direction, which can generate the GST relation. In this
case the analysis above applies in a modified form. To get a reduction in $%
F^{\chi }$ again requires further field(s), $\phi $. One can reproduce the
argument following from Eq.~(\ref{poly}) interpreting $\theta ^{p}=\sqrt{%
\chi\overline{\chi }}$. This time there must be
additional symmetries requiring $q\geq 8$ for $\chi /M_{Planck}\simeq 0.1$ .
To avoid this conclusion requires the introduction of yet another scale $%
M<M_{Planck}$ to order the family hierarchy which complicates the theory
further, perhaps making it less believable. 

Given this, we consider it very likely that the familon field(s) will have
an $F$-term greater than or equal to its natural value and so we turn now to a
discussion of the phenomenological implications that follow if $F^{\theta }$%
acquires a vev which is given by $\beta \langle \theta \rangle m_{3/2}$ with 
$\beta ={\cal{O}}(1)$. In this case terms involving $\theta $ also 
contribute to the
soft SUSY breaking terms and these terms violate flavour conservation and $CP$
via the couplings in Eq.~(\ref{yukawas}). Using this and Eq.~(\ref{scalpot})
we can determine the soft breaking terms in the observable sector after SUSY
breaking. This leads to the trilinear terms \cite{soni}, 
\begin{equation}
A_{ij}\hat{Y}^{ij}=F^{\eta}\hat{K}_{\eta}Y^{ij}+\alpha _{ij}\ 
\frac{\displaystyle{e^{K/2}}}{\displaystyle{M}}\left( \frac{\displaystyle{%
\theta }}{\displaystyle{M}}\right) ^{\alpha _{ij}-1}\beta m_{3/2}\theta 
\label{tri}
\end{equation}%
with $\hat{K}_{\eta}=\partial \hat{K}/\partial \eta$ and $%
Y^{ij}=e^{K/2}(\theta /M)^{\alpha _{ij}}$. The presence of the $\alpha _{ij}$
in the right hand side is due to the dependence of the effective Yukawa
couplings on $\theta $. Note the fact that the small parameter $\theta $
appears in $F^{\theta }$ does not affect the trilinear couplings because is
reabsorbed in the Yukawa coupling itself, 
\[
m_{3/2}\ \langle \theta \rangle \ \frac{\displaystyle{\partial Y^{ij}}}{%
\displaystyle{\partial \theta }}=\alpha _{i,j}\ m_{3/2}\ Y^{ij}.
\]%
From Eq.~(\ref{tri}) we see that, in any model which explains the hierarchy in
the Yukawa textures through nonrenormalizable operators, the trilinear
couplings are necessarily\ of the nonuniversal. In a similar way, we also
expect non-renormalizable contributions to the K\"{a}hler potential of the
kind $(\theta \theta ^{\ast }/M^{2})^{\alpha (i,j)}$. However these
contributions appear only at order $2\alpha (i,j)$ in $\theta /M$ with
respect to the dominant term $\mathcal{O}(1)$. So, in the following we
concentrate in the nonuniversal trilinear couplings.

Next we must check whether this breaking of universality does not contradict
any of the very stringent bounds from low energy phenomenology. Although we
do not have a complete theory of flavour that provides the full field
dependence of the low energy effective Yukawa couplings \cite{flavourth} a
fit to the fermion masses and mixing angles points to a definite texture for
the Yukawa matrices \cite{liliana}, 
\begin{equation}
\frac{M}{m_{3}}=\left( 
\begin{array}{ccc}
0 & b\epsilon ^{3} & c\epsilon ^{3} \\ 
b^{\prime }\epsilon ^{3} & d\epsilon ^{2} & a\epsilon ^{2} \\ 
f\epsilon ^{m} & g\epsilon ^{n} & 1%
\end{array}%
\right) ,  \label{texture}
\end{equation}%
with $\epsilon _{d}=\sqrt{m_{s}/m_{b}}=0.15$ and $\epsilon = _{u}=\sqrt{%
m_{c}/m_{t}}=0.05$. at the unification scale, and $a,b,b^{\prime },d,g,f$
coefficients ${\mathcal{O}}(1)$ and complex in principle. The coefficient $c$
is very sensitive to the magnitude of the elements below the diagonal and
can be of ${\mathcal{O}}(1)$ or smaller. Our analysis will not depend on its
value. In this texture the two undetermined elements, $(3,1)$ and $(3,2)$,
determine the unmeasured right-handed quark mixings. From the magnitude of
the eigenvalues however we can constrain $m\geq 1$. In the context of a
broken flavour symmetry the hierarchy in the Yukawa matrices is generated by
different powers in the vevs of the familon fields, $\epsilon _{a}=\langle
\theta _{a}\rangle /M$. For the case of a single familon we can immediately
calculate the nonuniversality in the trilinear terms, $(Y^{A})_{ij}\equiv
Y_{ij}A_{ij}$, 
\begin{equation}
(Y^{A})_{ij}=A_{0}Y_{ij}+\beta m_{3/2}\ Y_{33}\left( 
\begin{array}{ccc}
0 & 3b\epsilon ^{3} & 3c\epsilon ^{3} \\ 
3b^{\prime }\epsilon ^{3} & 2d\epsilon ^{2} & 2a\epsilon ^{2} \\ 
fm\epsilon ^{m} & gn\epsilon ^{n} & 0%
\end{array}%
\right)  \label{trilinear}
\end{equation}%
with $A_{0}=F^{\eta}\hat{K}_{\eta}$. For the case of several
familons the $F$-terms of different fields are expected to differ and 
although the coefficients will change from Eq.~(\ref{trilinear}) 
the proportionality of $Y^A$ and $Y$ will also be lost. Our
results will not depend sensitively on variations of ${\mathcal{O}}(1)$ in
these coefficients so, to avoid a proliferation of constants, we will
analyse the particular case of Eq.~(\ref{trilinear}). The Yukawa texture in
Eq.~(\ref{texture}) is diagonalized by superfield rotations in the so-called
SCKM basis, $\tilde{Y}=V_{L}^{\dagger }\cdot Y\cdot V_{R}$. However, in this
basis large off-diagonal terms necessarily remain in the trilinear
couplings, $\tilde{Y}^{A}=V_{L}^{\dagger }\cdot Y^{A}\cdot V_{R}$. In first
place we analyze the possible FCNC contributions from these terms. The
phenomenologically relevant flavour off-diagonal entries in the basis of
diagonal Yukawa matrices are, 
\begin{eqnarray}
(\tilde{Y}^{A})_{32} &\simeq &Y_{33}\beta \,m_{3/2}\,g\,\left( n-1\right) \
\epsilon ^{n}+\dots ~~~~~~~~ \\
(\tilde{Y}^{A})_{21} &\simeq &Y_{33}\,\beta m_{3/2}\,2\epsilon
^{3}(b^{\prime }+a\,(\frac{\displaystyle{b^{\prime }}}{\displaystyle{d}}\,g\
n\,\epsilon ^{n}-f\,m\,\epsilon ^{m-1}))  \nonumber
\end{eqnarray}%
The form of these matrices applies at the messenger scale which, being due
to supergravity, is $M_{Planck}$. Then we must use the MSSM Renormalization
Group Equations (RGE) \cite{RGE} to obtain the corresponding matrices at the
electroweak scale. The main effect in this RGE evolution is a large flavour
universal gaugino contribution to the diagonal elements in the sfermion mass
matrices (see for instance Tables I and IV in \cite{wien}). In this minimal
supergravity scheme we take gaugino masses as $m_{1/2}=\sqrt{3}m_{3/2}$ and
sfermion masses $m_{0}^{2}=m_{3/2}^{2}$. So, the average squark and slepton
masses, 
\begin{eqnarray}
&m_{\tilde{q}}^{2}\simeq 6\cdot m_{1/2}^{2}+m_{0}^{2}\simeq 19\,m_{3/2}^{2}&
\nonumber \\
&m_{\tilde{l}}^{2}\simeq 1.5\cdot m_{1/2}^{2}+m_{0}^{2}\simeq 5.5\
m_{3/2}^{2}&  \label{average}
\end{eqnarray}%
The RG evolution of the trilinear terms is also similarly dominated by
gluino contributions and the third generation Yukawa couplings. However, the
offdiagonal elements in the down and lepton trilinear matrices are basically
unchanged for $\tan \beta \leq 30$ \cite{wien}. From here we can obtain the
full trilinear couplings and compare with the experimental observables at
low energies. The so-called Mass Insertions (MI) formalism is very useful in
this framework. The left--right MI are defined in the SCKM basis as $(\delta
_{LR})_{ij}=(m_{LR}^{2})_{ij}/m_{\tilde{f}}^{2}$, with $m_{\tilde{f}}^{2}$
the average sfermion mass. 
\begin{table}[tbp]
\begin{center}
\begin{tabular}{||c|c|c|c|c||}
\hline\hline
$x$ & ${\sqrt{\left|\mbox{Im} \left(\delta^{d}_{LR} \right)_{12}^{2} \right|}
}$ & $\sqrt{\left|\mbox{Re} \left(\delta^{d}_{13} \right)_{LR}^{2}\right|} $
& ${\left|\left(\delta^{l}_{LR} \right)_{12} \right|}$ & ${%
\left|\left(\delta^{l}_{LR} \right)_{23} \right|}$ \\ \hline
$0.3 $ & $1.1\times 10^{-5} $ & $1.3\times 10^{-2} $ & $6.9\times 10^{-7} $
& $8.7\times 10^{-3} $ \\ 
$1.0 $ & $2.0\times 10^{-5} $ & $1.6\times 10^{-2} $ & $8.4\times 10^{-7} $
& $1.0\times 10^{-2} $ \\ 
$4.0 $ & $6.3\times 10^{-5} $ & $3.0\times 10^{-2} $ & $1.9\times 10^{-6} $
& $2.3\times 10^{-2} $ \\ \hline\hline
\end{tabular}%
\end{center}
\caption{MI bounds from $\protect\varepsilon ^{\prime }/\protect\varepsilon $%
, $b\rightarrow s\protect\gamma $, $\protect\mu \rightarrow e\protect\gamma $
and $\protect\tau \rightarrow \protect\mu \protect\gamma $ for an average
squark mass of $500\mbox{ GeV}$ and different values of $x=m_{\tilde{g}%
}^{2}/m_{\tilde{q}}^{2}$ or an average slepton mass of $100\mbox{ GeV}$ and
different values of $x=m_{\tilde{\protect\gamma}}^{2}/m_{\tilde{l}}^{2}$ in
the hadronic processes and leptonic decays repectively. These bounds scale
as $(m_{\tilde{f}}(\mbox{GeV})/500(100))^{2}$ for different average sfermion
masses}
\label{tab:MI1}
\end{table}
We can estimate the value of the $(2,1)$ LR MI as, 
\begin{eqnarray}
&&\left( \delta _{LR}^{d}\right) _{21}\simeq \frac{\displaystyle{%
m_{b}\,\epsilon _{d}^{3}}}{\displaystyle{19\,m_{3/2}}}(b^{\prime }\,+a\,%
\frac{\displaystyle{b^{\prime }}}{\displaystyle{d}}\,g\,n\,\epsilon
_{d}^{n}-a\,f\,m\,\epsilon _{d}^{m-1})\simeq  \nonumber \\
&&(b^{\prime }\,+a\,\frac{\displaystyle{b^{\prime }}}{\displaystyle{d}}%
\,g\,n\,\epsilon _{d}^{n}-a\,f\,m\,\epsilon _{d}^{m-1})\,7.5\times 10^{-6}
\label{eps'}
\end{eqnarray}%
using $m_{\tilde{q}}\simeq 500$ GeV corresponding to $m_{3/2}\simeq 120$ GeV
and $\epsilon _{d}\simeq 0.15$ \cite{liliana}. We can compare our estimate
for the mass insertion with the phenomenological bounds in Table \ref%
{tab:MI1} with $x=m_{\tilde{g}}^{2}/m_{\tilde{q}}^{2}\simeq 1$. Even
allowing a phase ${\mathcal{O}}(1)$, necessary to contribute to $\varepsilon
^{\prime }/\varepsilon $, we can see the bound requires only $m\geq 1$ which
is already required to fit the fermion masses. Note however that in the
presence of a phase, $\varepsilon ^{\prime }/\varepsilon $ naturally
receives a sizeable contribution from the $b^{\prime }$ term \cite{murayama}.

Similarly, the MI corresponding to the $b \to s \gamma$ decay are, 
\begin{eqnarray}
\left(\delta^d_{LR}\right)_{3 2} \simeq \frac{\displaystyle{%
m_{3/2}\,m_b\,g\,n\, \epsilon_d^n}}{\displaystyle{19\,m_{3/2}^2}} \simeq 2.2
\times 10^{-3}\,g\,n\,\epsilon_d^n
\end{eqnarray}
again with $m_{3/2}\simeq 120$ GeV. This estimate is already of the same
order of the phenomenological bound for any $n$ and we do not get any new
constraint on $n$.

The situation is more interesting in the leptonic sector. Here, the photino
contribution is indeed the dominant one for LR mass insertions. In Table \ref%
{tab:MI1} , we show the rescaled bounds from Ref. \cite{gabbiani} for the
present limits on the branching ratio. In this case, it seems reasonable to
expect some kind of lepton-quark Yukawa unification. To generate the correct
muon and electron mass we follow Georgi and Jarlskog's suggestion and put a
relative factor of 3 in the $(2,2)$ entry. We also put a factor of $3$ in
the $(2,3)$ and $(3,2)$ entries as is required by non-Abelian models
which seek to explain the near equality in the down quark mass matrix of the 
$(2,2)$ and $(2,3)$ matrix elements. As we shall see, even with this factor,
the contribution of these elements is sub-dominant. With this we obtain,

\begin{eqnarray}
&\left( \delta _{LR}^{e}\right) _{12}\simeq \frac{\displaystyle{m_{\tau
}\,\epsilon _{d}^{3}}}{\displaystyle{5.5\,m_{3/2}}}(b^{\prime }+\,9a\,\frac{%
\displaystyle{b^{\prime }}}{\displaystyle{d}}\,g\,n\,\epsilon
_{d}^{n}-3\,a\,f\,m\,\epsilon _{d}^{m-1})\simeq &  \nonumber \\
&(b^{\prime }\,+9\,a\,\frac{\displaystyle{b^{\prime }}}{\displaystyle{d}}%
\,g\,n\,\epsilon _{d}^{n}-3\,a\,f\,m\,\epsilon _{d}^{m-1})\,8.7\times
10^{-6}&  \label{mueg}
\end{eqnarray}%
where we take $m_{3/2}\simeq 120$ GeV corresponding to $m_{\tilde{l}}=280$
GeV. This result should be compared with the experimental bound from the
non-observation of $\mu \rightarrow e\gamma $ given by $\left( \delta
_{LR}^{e}\right) _{12}\leq 7\times 10^{-7}\ (280/100)^{2}=5.5\times 10^{-6}.$
Note that the contributions of the last two terms in Eq.~(\ref{mueg}) does not
lead to new constraints on the value of $m$ and $n$. However there is an
unavoidable contribution from the $b^{\prime }$ entry, $\left( \delta
_{LR}^{e}\right) _{12}\simeq b^{\prime }\,8.7\times 10^{-6},$ exceeding 
the experimental bound. To avoid this requires a
larger value of the slepton mass. For an average slepton mass of 320 GeV our
estimate would be just below the MI bound ($m_{3/2}=136$ GeV, $m_{\tilde{q}%
}=600$ GeV). To summarise, assuming a quark-lepton unification at $M_{GUT},$
nonuniversality in the trilinear terms predicts a large $\mu \rightarrow
e\gamma $ branching ratio even beyond the values expected from other sources
such as the SUSY seesaw \cite{SUSYseesaw}. This illustrates the point that $%
\mu \rightarrow e\gamma $ is a particularly sensitive probe of SUSY and the
soft breaking sector.

Another interesting constraint is provided by Electric Dipole Moment (EDM)
bounds. Even in the most conservative case, where all soft SUSY breaking
parameters and $\mu $ are real, we know that the Yukawa matrices contain
phases ${\mathcal{O}}(1)$. If the trilinear terms are nonuniversal, these
phases are not completely removed from the diagonal elements of $Y^{A}$ in
the SCKM basis and hence can give rise to large EDMs \cite{abel}. However
the phase in the trilinear terms will be exactly zero at leading order in $%
\theta $ for any diagonal element \footnote{This leads to a 
weaker bound than that found in \cite{abel}.}. To see this
we must take into account of the fact that the eigenvalues, $D(\theta ),$
and mixing matrices, $V_{L,R}(\theta )$ of the Yukawa matrix depend on $%
\theta $, $Y(\theta )=V_{L}^{\dagger }(\theta )\,D(\theta )\,V_{R}(\theta )$%
. The contribution to the trilinear terms is proportional to $\theta \
\partial Y/\partial \theta .$ Evaluating this in the basis in which the
Yukawas are diagonal, one finds 
\begin{eqnarray}
V_{L}\theta \frac{\displaystyle{\partial Y}}{\displaystyle{\partial \theta }}%
V_{R}^{\dagger }=V_{L}\frac{\displaystyle{\theta \partial V_{L}^{\dagger }}}{%
\displaystyle{\partial \theta }}D+\frac{\displaystyle{\theta \partial D}}{%
\displaystyle{\partial \theta }}+D\frac{\displaystyle{\theta \partial V_{R}}%
}{\displaystyle{\partial \theta }}V_{R}^{\dagger }
\end{eqnarray}
In this expression the dominant contribution in $\theta $ to a diagonal
element is controlled by the second term. This follows because $\theta \
\partial V/\partial \theta $ always adds at least a power of $\theta $ to
the diagonal element and therefore the first and third terms in the above
equation can only contribute to subdominant terms in the $\theta $ expansion
for the diagonal elements. The dominant second term is proportional to the
leading $\theta $ term in $Y_{ii}$ with a coefficient of proportionality
equal to its power in $\theta $ i.e. the phase is unchanged and is real in
the basis that the Yukawa couplings are real. This implies that any
observable phase in the diagonal elements will only appear at higher orders,
requiring $n\geq 1$ or $m\geq 2$ or through higher order contributions to
entries of the Yukawa matrix. Using this result, and assuming real $\mu $
and soft breaking terms, the EDMs have the form 
\begin{eqnarray}
\mbox{Im}\left( \delta _{LR}^{q,l}\right) _{11}\simeq \frac{\displaystyle{%
m_{1}}}{\displaystyle{R_{q,l}\,m_{3/2}}}\left( \epsilon ^{n}\,n\,+\epsilon
^{m-1}\,(m-1)\,\right)
\end{eqnarray}
where we use $m_{1}=m_{3}\,\epsilon ^{4}bb^{\prime }/d$ and we have taken
all unknown coefficients of ${\mathcal{O}}(1)$ to be unity. The coefficients 
$R_{q}=19$ and $R_{l}=5.5$ take care of the RGE effects in the eigenvalues
as before. So with $\epsilon _{d}=0.15$, $\epsilon _{u}=0.05$ and $%
m_{d}\simeq 10$ MeV, $m_{u}\simeq 5$ MeV and $m_{e}=0.5$ MeV we get, 
\begin{eqnarray}
\mbox{Im}\left( \delta _{LR}^{d}\right) _{11} \simeq \left( \epsilon
_{d}^{n}\,n\,+\epsilon _{d}^{m-1}\,(m-1)\,\right) \,3.9\times 10^{-6} 
\nonumber \\
\mbox{Im}\left( \delta _{LR}^{u}\right) _{11} \simeq \left( \epsilon
_{u}^{n}\,n\,+\epsilon _{u}^{m-1}\,(m-1)\,\right) \,1.9\times 10^{-6} 
\nonumber \\
\mbox{Im}\left( \delta _{LR}^{e}\right) _{11} \simeq \left( \epsilon
_{d}^{n}\,n\,+\epsilon _{d}^{m-1}\,(m-1)\,\right) \,6.7\times 10^{-7}
\end{eqnarray}%
\begin{table}[tbp]
\begin{center}
\begin{tabular}{|c||c|c|c|c|}
\hline
$x$ & $\vert \mathrm{Im}(\delta_{11}^{d})_{LR}\vert$ & $\vert \mathrm{Im}%
(\delta_{11}^{u})_{LR} \vert$ & $\vert \mathrm{Im}(\delta_{22}^{d})_{LR}
\vert$ & $\vert \mathrm{Im}(\delta_{11}^{l})_{LR} \vert$ \\ \hline\hline
0.3 & $4.3\times 10^{-8}$ & $4.3\times 10^{-8}$ & $3.6\times 10^{-6}$ & $%
4.2\times 10^{-7}$ \\ 
1 & $8.0\times 10^{-8}$ & $8.0\times 10^{-8}$ & $6.7\times 10^{-6}$ & $%
5.1\times 10^{-7}$ \\ 
3 & $1.8\times 10^{-7}$ & $1.8\times 10^{-7}$ & $1.6\times 10^{-5}$ & $%
8.3\times 10^{-7}$ \\ \hline
\end{tabular}%
\end{center}
\caption{MI bounds from the mercury EDM for an average squark mass of $600 
\mbox{ GeV}$ and for the electron EDM with an average slepton mass of $320 
\mbox{ GeV}$ and different values of $x=m_{\tilde{g}}^{2}/m_{\tilde{q}}^{2}$%
. The bounds scale as $(m_{\tilde{q}(\tilde{l})}(\mbox{GeV})/600(320))$.}
\label{tab:MI2}
\end{table}
We compare these estimates with the phenomenological bounds in Table \ref%
{tab:MI2} \cite{hg,gabbiani}. The bounds from the neutron and electron EDM do
not provide any new information on the structure of the Yukawa textures.
However, the mercury EDM bounds are much more restrictive and taking $%
x\simeq 1$ we find that, in the down sector, the case $n=1$, $m=2$ is not
allowed by EDM experiments and we require $n\geq 2$, $m\geq 3$. As we said
above, the same bound applies to the subdominant corrections to $Y_{22}$ and 
$Y_{12},Y_{21}$ where the first correction to the dominant terms can only be 
$\epsilon ^{4}$ or $\epsilon ^{5}$ respectively to satisfy EDM bounds.

In summary, we have shown here that in a broken supergravity theory any
field that acquires a vev also has an $F$-term of order $\langle \theta
\rangle m_{3/2}$. These $F$-term contribute unsuppressed to the trilinear
couplings and have observable effects in low energy phenomenology. The most
significant contribution, assuming a GUT type relation between the quark and
lepton masses, is to $\mu \rightarrow e\gamma .$ To keep this at the level
of current experimental bounds requires a slepton mass greater than $320$ GeV.
At this level $\mu \rightarrow e\gamma $ should be seen by the proposed
experiments in the near future. There are also strong bounds coming
from the mercury EDM bounds which impose significant limits on the down
quark matrix elements below the diagonal responsible for right handed
mixing. These bounds disfavour large right handed mixing and thus disfavour $%
SU(5)$ based models in which the large (left handed) neutrino mixing angles
are related to large down quark right handed mixing angles. Of course these
strong bounds apply only to supergravity models with gravity as the
supersymmetry breaking messenger. Models with light messenger states, such
as gauge mediation models, have a much lower value for $m_{3/2}$ and this
determines the size of these flavour and $CP$ violating effects.

We are grateful for useful conversations with R. Rattazzi. We acknowledge
support from the RTN European project HPRN-CT-2000-0148. O.V. acknowledges
partial support from the Spanish MCYT FPA2002-00612 and DGEUI of the Gen.
Valenciana grant GV01-94

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\end{document}

