An alternative approach to \bsg in the unconstrained MSSM
Stefano Rigolin
The gluino contributions to the $C'_{7,8}$ Wilson coefficients for \bsg are calculated within the unconstrained MSSM. New stringent bounds on the \dRLbs and \dRRbs mass insertion parameters are obtained in the limit in which the SM and SUSY contributions to $C_{7,8}$ approximately cancel. Such a cancellation can plausibly appear within several classes of SUSY breaking models. %in which the trilinear couplings exhibit a factorized structure %proportional to the Yukawa matrices. Assuming this cancellation takes place, we perform an analysis of the \bsg decay. We show that, in the uMSSM such an alternative is reasonable and it is possible to saturate the \bsg branching ratio and produce a CP asymmetry of up to $20\%$, from only the gluino contribution to $C'_{7,8}$ coefficients. Using photon polarization a LR asymmetry can be defined that in principle allows the $C_{7,8}$ and $C'_{7,8}$ contributions to the \bsg decay to be disentangled.
rigmor02.tex
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\newcommand{\sinb}{\mbox{$\sin 2\beta \ $}}
\newcommand{\dmbd}{\mbox{$\Delta M_{B_d} \ $}}
\newcommand{\dmbs}{\mbox{$\Delta M_{B_s} \ $}}
\newcommand{\bbbar}{\mbox{$B-\bar{B} \ $}}
\newcommand{\bbbard}{\mbox{$B_d-\bar{B}_d \ $}}
\newcommand{\bbbars}{\mbox{$B_s-\bar{B}_s \ $}}
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\newcommand{\eps}{\epsilon}
\newcommand{\tm}{\mbox{$\tilde m$}}
\newcommand{\gtil}{\mbox{$\tilde{g}$}}
\newcommand{\bsg}{\mbox{$b \rightarrow s \gamma \ $}}
\newcommand{\bsgb}{\mbox{$\bar{b} \rightarrow \bar{s} \gamma \ $}}
\newcommand{\delRL}{\mbox{$\Delta^{RL}_{13}$}}
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\begin{document}
\vspace*{4cm}

\title{An alternative approach to \bsg in the unconstrained MSSM}

\author{Stefano Rigolin} 
\address{Theoretical Physics Division, CERN, CH-1211, Geneva 23, 
         Switzerland}

\maketitle\abstracts{
The gluino contributions to the $C'_{7,8}$ Wilson coefficients for 
\bsg are calculated within the unconstrained MSSM. New stringent 
bounds on the \dRLbs and \dRRbs mass insertion parameters are 
obtained in the limit in which the SM and SUSY contributions to 
$C_{7,8}$ approximately cancel. Such a cancellation can plausibly 
appear within several classes of SUSY breaking models. 
%in which the trilinear couplings exhibit a factorized structure 
%proportional to the Yukawa matrices. 
Assuming this cancellation takes place, we perform an analysis of the 
\bsg decay. We show that, in the uMSSM 
such an alternative is reasonable and it is possible to saturate the 
\bsg branching ratio and produce a CP asymmetry of up to $20\%$, from 
only the gluino contribution to $C'_{7,8}$ coefficients. Using photon 
polarization a LR asymmetry can be defined that in principle allows  
the $C_{7,8}$ and $C'_{7,8}$ contributions to the \bsg decay to be 
disentangled.
}

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
The precision measurements of the inclusive radiative decay $B \raw X_s 
\gamma$ provides an important benchmark for the Standard Model (SM) and 
New Physics (NP) models at the weak scale, such as low-energy 
supersymmetric (SUSY) models. In the SM, flavour changing neutral 
currents (FCNC) are forbidden at tree level. The first SM contribution to 
the \bsg transition appears at one loop level from the CKM flavour 
changing structure, showing the characteristic Cabibbo suppression. 
NP contributions to \bsg typically also arise at one loop, and in 
general can be  much larger than the SM contributions if no mechanisms 
for suppressing the new sources of flavour violation exist (see \cite{all} 
for a complete set of references). 

Experimentally, the inclusive $B \raw X_s \gamma$ Branching Ratio (BR) has 
been measured by ALEPH, BELLE and CLEO, resulting in the current 
experimental weighted average $BR(B\rightarrow X_s \gamma)_{exp} = 
(3.23 \pm 0.41) \times 10^{-4}$,
with new results expected shortly from BABAR and BELLE which could further
reduce the experimental errors. Squeezing the theoretical uncertainties
down to the 10\% level has been (and still is) a crucial task. The SM
theoretical prediction has been the subject of intensive theoretical
investigation in the past several years, leading to the completion of 
the NLO QCD calculations.
The original SM NLO calculation \cite{misiak} gives, for $\sqrt{z} = 
m_c/m_b=0.29$, the following result: $BR(B\rightarrow X_s \gamma)_{SM} = 
(3.28 \pm 0.33) \times 10^{-4}$. The main source of theoretical uncertainty 
is due to NNLO QCD ambiguities. In \cite{gambino} it was shown that using 
$\sqrt{z} = 0.22$ (i.e. the running charm mass instead of the pole mass) 
is more justifiable and causes an enhancement of about 10\% of the \bsg 
BR, leading to the current preferred value: 
$BR(B\rightarrow X_s \gamma)_{SM} = (3.73 \pm 0.30) \times 10^{-4}$. 
Although these theoretical uncertainties can be addressed only with a 
complete NNLO calculation, the SM value for the BR is in 
agreement with the experimental measurement within the $1-2\sigma$ level.

The general agreement between the SM theoretical prediction and the 
experimental results has provided useful guidelines for constraining 
the parameter space of models with NP present at the electroweak 
scale, such as the 2HDM and the minimal supersymmetric standard model 
(MSSM). In SUSY models superpartners and charged Higgs loops contribute 
to \bsg$\!$, with contributions that typically rival or exceed the SM 
one in size. For calculational ease only simplified MSSM scenarios 
(like cMSSM or MSSM with minimal flavour violation (MFV)) have usually been 
assumed. Netherveless, as the origin and dynamical mechanism of SUSY 
breaking are unknown, there is no reason {\it a priori} to expect that the 
soft parameters will be flavour-blind (or violate flavour in the same way 
as the SM). Of course, the kaon system has provided strong FCNC constraints 
for the mixing of the first and second generations, which severely limit 
the possibility of flavour violation in that sector. However the 
constraints for third generation mixings are significantly weaker, with 
\bsg usually providing the most stringent constraints.

A discussion of the \bsg process in the general unconstrained MSSM is in 
principle possible, but it is necessary to deal with two unavoidable 
problems: (i) a large number of free, essentially unconstrained 
parameters; (ii) the need to achieve a quite accurate cancellation 
between the sizeable different contributions (SM, Higgs, chargino, 
neutralino and gluino). %to the Wilson Coefficient (WC) $C_7$ associated 
%with the $Q_7 \propto m_b \bar{s}_L \sigma^{\mu\nu} b_R F_{\mu \nu}$ 
%operator in such a way that the experimental measurement, which 
%approximately saturated solely by the SM result, is satisfied. Moreover, 
%in general MSSM models with non-minimal flavour violation, the gluino 
%loop can also contribute significantly to the WC $C'_7$ associated with 
%the chirality-flipped operator, $Q'_7 \propto m_b \bar{s}_R \sigma^{\mu\nu} 
%b_L F_{\mu \nu}$. However, as the SM, Higgs, and chargino contributions 
%to $C'_7$ are typically suppressed by a factor of $O(m_s/m_b)$, it is not 
%possible in general to achieve a cancellation between the different terms 
%in $C'_7$ and thus a stronger fine-tuning has to be imposed. 
In the following we'll provide a particularly interesting and simple 
analysis of \bsg in the uMSSM.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Alternative solution to \bsg branching ratio}
\label{sectiond}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
The low-energy effective Hamiltonian, at the bottom mass scale $\mu_b$, is 
defined as 
%\be 
${\cal H}_{eff} = - (4 G_F/\sqrt{2}) V_{tb} V^*_{ts} 
   \sum_i C_i(\mu_b) Q_i(\mu_b)$.  
%\ee 
The operators relevant to the \bsg process are: \newline
\parbox{0.5\textwidth}{
\bea 
Q_2 & = & \bar{s}_L \gamma_\mu c_L \bar{c}_L \gamma^\mu b_L \ , \nn \\
Q_7 & = & \frac{e}{16 \pi^2} m_b \bar{s}_L \sigma^{\mu \nu} b_R F_{\mu \nu} \ , 
        \nn \\
Q_8 & = & \frac{g_s}{16\pi^2} m_b \bar{s}_L \sigma^{\mu \nu} G^a_{\mu \nu} 
        T_a b_R \ , \nn
\eea }
\parbox{0.5\textwidth}{
\bea 
Q'_2 & = & \bar{s}_R \gamma_\mu c_R \bar{c}_R \gamma^\mu b_R \ , \nn \\ 
Q'_7 & = &\frac{e}{16\pi^2} m_b \bar{s}_R \sigma^{\mu \nu} b_L F_{\mu \nu} \ , 
         \nn \\ 
Q'_8 & = &\frac{e}{16\pi^2} m_b \bar{s}_R \sigma^{\mu \nu} G^a_{\mu \nu} 
         T_a b_L \ . \label{opQp}
\eea }
%
Effects of NP generally appear as modifications of the Wilson Coefficients 
(WC) $C^{(_{'})}_{7,8}$ associated to the operators $Q_{7,8}$ and their 
chirality conjugated ones. In the majority of the previous studies of the 
\bsg process, the main focus was to calculate the SM or NP contributions 
to $C_{7,8}$. 
The contributions coming from $C'_{7,8}$ have usually been neglected on the 
assumption that they are suppressed with respect to $C_{7,8}$ by the ratio 
$m_s/m_b$. While this is always valid in the SM, in the 2HDM 
or within specific MSSM scenarios (with MFV), this mass suppression can be 
absent in the uMSSM where the gluino contributions to $C_{7,8}$ and 
$C'_{7,8}$ are naturally of the same order\cite{borzumati}. 
%in the case of the uMSSM this is not generally the case. 

Therefore, in the following we present an alternative approach to \bsg 
in the uMSSM. We assume a particular scenario in which the {\it total} 
contribution to $C_{7,8}$ is negligible and the main contribution to the 
\bsg BR is given by $C'_{7,8}$. This 
``$C'_7$-dominated'' scenario is realized when the chargino, neutralino, 
%\footnote{Neutralino contribution are always subleading respect to 
%the gluino contribution.} 
and gluino contributions to $C_{7,8}$ sum up in such a way as to cancel 
the W and Higgs contributions almost completely. %\footnote{The main 
%constraint on this scenario is the requirements of the $C_7$ 
%cancellation. The $C_8$ contribution enters in the \bsg branching 
%ratio at $O(\alpha_s)$ and usually cannot account for more 
%than $10\%$ of the measured branching ratio.}. 
In our opinion this situation does not require substantially more fine 
tuning than what is required in the usual MFV scenario, where conversely 
the NP contributions to $C_{7,8}$ essentially cancel between themselves. 
%(or are almost decoupled), so that all the measured \bsg BR 
%is produced by the W diagram. 

In the following we will focus on the gluino contribution to $C_{7,8}$ and 
$C'_{7,8}$. There is only one gluino diagram that contributes to $C_{7}$ and 
$C'_7$, with the external photon line attached to the down-squark line, while 
two diagrams can contribute to the $C_8$ and $C'_8$ coefficients, as the 
gluon external line can be attached to the squark or the gluino lines. The 
one-loop gluino contributions to the $C'_{7,8}$ coefficients 
are given, at first and second order in the MI, respectively by:
\bea
%C^{\tilde{g}}_7 (1) &=& \frac{8 g_s^2 }{3 g^2} {{Q_d}\over{V_{tb} V^*_{ts}}} 
%   {{m_W^2}\over{\tm^2_D}} \left\{ \delta^{LL}_{23} F^{(1)}_2(x^g_D) - 
%   \frac{\tm_{\tilde{g}}}{m_b} \delta^{LR}_{23} F^{(1)}_4(x^g_D) \right\},
%\label{glC7MI} \\
%C^{\tilde{g}}_8 (1) &=& - \frac{g_s^2}{3 g^2} {{Q_d}\over{V_{tb} V^*_{ts}}} 
%   {{m_W^2}\over{\tm^2_D}} \left\{ \delta^{LL}_{23} F^{(1)}_{21}(x^g_D) - 
%   \frac{\tm_{\tilde{g}}}{m_b} \delta^{LR}_{23} F^{(1)}_{43}(x^g_D) \right\},
%\label{glC8MI} \\
C^{' \tilde{g}}_7 (1) &=& \frac{8 g_s^2}{3 g^2}{{Q_d}\over{V_{tb} V^*_{ts}}}
   {{m_W^2}\over{\tm^2_D}} \left\{ \delta^{RR}_{23} F^{(1)}_2(x^g_D) - 
   \frac{\tm_{\tilde{g}}}{m_b} \delta^{RL}_{23} F^{(1)}_4(x^g_D) \right\},
\label{glC7MI} \\
C^{' \tilde{g}}_8 (1) &=& - \frac{g_s^2}{3 g^2} {{Q_d}\over{V_{tb} V^*_{ts}}} 
   {{m_W^2}\over{\tm^2_D}} \left\{ \delta^{RR}_{23} F^{(1)}_{21}(x^g_D) - 
  \frac{\tm_{\tilde{g}}}{m_b} \delta^{RL}_{23} F^{(1)}_{43}(x^g_D) 
  \right\}
\label{glC8pMI}
\eea
and 
\bea
%C^{\tilde{g}}_7 (2) &=& \frac{4 g_s^2 }{3 g^2} {{Q_d}\over{V_{tb} V^*_{ts}}} 
%   {{m_W^2}\over{\tm^2_D}} \frac{m_b (A_b - \mu \tz)}{\tm^2_D} 
%   \left\{ \delta^{LR}_{23} F^{(2)}_2(x^g_D) -
%   \frac{\tm_{\tilde{g}}}{m_b} \delta^{LL}_{23} F^{(2)}_4(x^g_D) \right\},~
%\label{glC7MI2} \\
%C^{\tilde{g}}_8 (2) &=& - \frac{g_s^2}{6 g^2} {{Q_d}\over{V_{tb} V^*_{ts}}} 
%   {{m_W^2}\over{\tm^2_D}} \frac{m_b (A_b - \mu \tz)}{\tm^2_D} 
%   \left\{ \delta^{LR}_{23} F^{(2)}_{21}(x^g_D) - 
%   \frac{\tm_{\tilde{g}}}{m_b} \delta^{LL}_{23} F^{(2)}_{43}(x^g_D) \right\},~
%\label{glC8MI2} \\
C^{' \tilde{g}}_7 (2) &=& \frac{4 g_s^2}{3 g^2}{{Q_d}\over{V_{tb} V^*_{ts}}}
   {{m_W^2}\over{\tm^2_D}} \frac{m_b (A_b - \mu \tz)}{\tm^2_D} 
   \left\{ \delta^{RL}_{23} F^{(2)}_2(x^g_D) - 
   \frac{\tm_{\tilde{g}}}{m_b} \delta^{RR}_{23} F^{(2)}_4(x^g_D) \right\},~
\label{glC7MI2} \\
C^{' \tilde{g}}_8 (2) &=& - \frac{g_s^2}{6 g^2} {{Q_d}\over{V_{tb} V^*_{ts}}} 
   {{m_W^2}\over{\tm^2_D}} \frac{m_b (A_b - \mu \tz)}{\tm^2_D} 
   \left\{ \delta^{RL}_{23} F^{(2)}_{21}(x^g_D) - 
   \frac{\tm_{\tilde{g}}}{m_b} \delta^{RR}_{23} F^{(2)}_{43}(x^g_D) \right\}~. 
\label{glC8pMI2}
\eea
The gluino contributio to $C_{7,8}$ can be obtained exchanging 
L $\lraw$ R in eqs.~(\ref{glC7MI}--\ref{glC8pMI2}).
In deriving eqs.(\ref{glC7MI}--\ref{glC8pMI2}) to the second order in 
the MI parameters, we have kept only the dominant term proportional to \tz 
(the $A_b$ term is retained in the above expression for defining our 
convention for the $\mu$ term; see later) and neglected all of the other 
off-diagonal MIs\footnote{In \cite{all} a complete derivation of the general 
results and conventions used in eqs.~(\ref{glC7MI}-\ref{glC8pMI2}) is 
presented.}. Clearly the dominant terms in eqs.~(\ref{glC7MI}-\ref{glC8pMI2}) 
are those proportional to the 
gluino chirality flip, so that the gluino contribution to $C_7$ ($C'_7$) 
depends, at first order, only on the MI term \dLRbs (\dRLbs$\!$). However, 
for large \tz and $\mu \approx \tm_A$, the second order MI terms in 
eqs.~(\ref{glC7MI2},\ref{glC8pMI2}) can become comparable in size with 
the first order mass insertions. Thus, two different MI parameters are 
relevant in the L/R sectors: (\dLRbs$\!$, \dLLbs$\!$) and (\dRLbs$\!$, 
\dRRbs$\!$), contrary to common wisdom. To which extent the LL and RR MIs 
are relevant depends of course on the values chosen for $\mu$ and $\tan 
\beta$, but in 
a large part of the allowed SUSY parameter space they cannot in general be 
neglected. Moreover, the fact that the gluino WCs depend on two different 
MI parameters will have important consequences in the study of the \bsg 
CP asymmetry.
%\footnote{Specifically, if only the first order term in the MI is taken, 
%the \bsg CP asymmetry vanishes, as discussed in greater detail in section 
%\ref{sectiondc}.} in the scenario presented in the following sections. 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsection{Single and multiple MI-dominance analysis}
\label{sectionda}
%
From eqs.(\ref{glC7MI}--\ref{glC8pMI2}), one can read (in MI language) 
the off-diagonal entries that are relevant to the gluino contribution 
to the $C_{7,8}$ and $C'_{7,8}$ WCs. %\footnote{From now on, for the sake 
%of simplicity, the symbol $\delta^{AB}_{ij}$ will be used 
%instead of $(\delta^{AB}_{ij})^d$ for referring to the down-squark MIs.}. 
Note that limits on \dLRbs $\approx O(10^{-2})$ have previously 
been obtained in \cite{gabbiani}. No stringent bound has been derived 
there for \dLLbs, as this term does not come, at lowest order, with the 
$\tm_{\tilde{g}}/m_b$ enhancement (see eqs.(\ref{glC7MI})). No limits 
were showed on \dRLbs and \dRRbs because the MI formula are symmetric 
in the $L \leftrightarrow R$ exchange and in the scenarios generally adopted 
in the literature the ``opposite chirality'' MIs are suppressed by a factor 
$m_s/m_b$ and so negligible. An analysis of the \dRLbs dependence has 
been performed in \cite{borzumati}, in which the $W$ contribution to 
$C_{7,8}$ was not set to zero (sometimes also Higgs and MFV chargino 
contributions to  $C_{7,8}$ were included). Consequently their bounds on 
the down-squark off-diagonal MIs contributing to $C'_{7,8}$ are 
more stringent than the bounds we derive in our scenario, for which 
the total contribution to $C_{7,8}$ is assumed to be negligible. 
It is clearly only in the scenario we study that an {\em absolute} 
constraint on these MIs can be derived. Moreover no analysis on \dLLbs 
and \dRRbs was performed in \cite{borzumati} as these contributions are 
not relevant in the small \tz region, as can be seen from 
eqs.(\ref{glC7MI2},\ref{glC8pMI2}). 
%
%___________________________________________________________________________
%\FIGURE[t]{
\begin{figure}[t]
\begin{tabular}{cc}
\hspace{-0.5cm}
\epsfig{file=dRL23_20.ps,width=6.25cm,angle=-90} &
\hspace{-0.5cm}
\epsfig{file=dRR23_20.ps,width=6.25cm,angle=-90} 
\end{tabular}
\caption{The dependence of \bsg branching ratio on $\delta^{RL}_{23}$ 
and $\delta^{RR}_{23}$ for different values of $\tm_{\tilde{g}}/\tm_{D}$, 
for $\tz = 20$ and $\mu = 350$ GeV. All of the other off-diagonal entries, 
except the one displayed on the axes, are assumed to vanish. 
$C_{7,8}(\mu_W) = 0$ is assumed. The horizontal lines represent the $1 
\sigma$ experimentally allowed region.}
\label{fig1} 
\end{figure}
%        }
%___________________________________________________________________________
%
In fig.~\ref{fig1} we show the dependence of the \bsg BR on the MI terms 
\dRLbs and \dRRbs for different values of $x^g_D=\tm^2_{\tilde{g}}/\tm^2_{D}$ 
and for $\tz = 20$ and $\mu = 350$ GeV. All the other off-diagonal entries 
in the down-squark mass matrix are assumed to vanish. ``Individual'' limits 
\dRLbs $< 10^{-2}$ and \dRRbs $< 1.5 \times 10^{-1}$ can be obtained 
respectively from the left and right side plot of fig.~\ref{fig1}. 
%Horizontal full lines represent $1 \sigma$ deviations from the 
%experimental results. 
Of course, the required cancellation of the total $C_{7,8}$ contribution 
may in general need non-vanishing 
off-diagonal entries of the up and down squark mass matrices. However, the 
specific values of these entries do not significantly affect the absolute 
limits on the \dRLbs and \dRRbs MIs shown in fig.~\ref{fig1}. As expected 
from eqs.(\ref{glC7MI2},\ref{glC8pMI2}), the bounds obtained 
for \dRRbs are strongly dependent on the product $\mu \tan \beta$. 
%In fig.~\ref{fig2} we show the \tz dependence of this limit, for fixed 
%$\tm_{\tilde{g}}/\tm_{\tilde{q}}=350/500$ and $\mu = 350$ GeV. 
More stringent bounds on \dRRbs can be obtained for larger \tz$\!$. For 
\tz$>35$ the bounds on \dRRbs can become as stringent as the \dRLbs bounds. 
%Similar considerations and bounds obviously hold also for the \dLLbs MI. 
%As we are only interested here in the gluino contributions to $C'_{7,8}$, 
%we do not discuss this sector in detail. Clearly this term must be 
%taken into consideration if a similar analysis was performed for the 
%$C_{7,8}$ coefficient in the large \tz region.
%
%___________________________________________________________________________
%\FIGURE[t]{
%\begin{figure}[t]
%\centerline{
%\epsfig{file=dRR23_tz.ps,width=6.25cm, angle=-90} }
%\caption{Dependence of \bsg branching ratio on $\delta^{RR}_{23}$ 
%for three different values of \tz$\!$, with the other parameters fixed to 
%$\tm_{\tilde{g}}/\tm_{\tilde{q}} = 350/500$ and $\mu = 350$ GeV. All of 
%the other off-diagonal entries, except the one displayed on the axes, 
%are assumed to vanish. The horizontal lines represent the $1 \sigma$ 
%experimentally allowed region.}
%\label{fig2} 
%\end{figure}
%       }
%___________________________________________________________________________
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\subsection{Multiple MI dominance analysis}
%\label{sectiondb}

A general analysis of the gluino contribution to $C'_{7,8}$ depends 
simultaneously on both the \dRLbs and \dRRbs MIs. For a complete 
specification of our scenario the only other free parameters that need be 
fixed are the ratio between the gluino mass and the common down-squark mass, 
$\tm_{\tilde{g}}/\tm_D$, the product $\mu$ \tz$\!$, and the relative phase 
between \dRLbs and \dRRbs. The influence of all the other down-sector squark 
matrix off-diagonal entries and MSSM parameters in the $C'_{7,8}$ sector 
can safely be neglected. %Thus, we can have a complete description in terms 
%of only five free parameters of the \bsg phenomenology in our MSSM 
%``$C'_7$-dominated'' scenario. 
In fig.~\ref{fig5} (left) we show the 
$1 \sigma$ experimentally allowed region in the (\dRLbs$\!$, \dRRbs$\!$) 
parameter space for a specific choice of $\tm_{\tilde{g}}/\tm_D=350/500$, 
$\mu=350$ GeV, and for three different values of \tz=$3, 20$ and $35$. 
For \dRLbs or \dRRbs vanishing, one obtains the regions depicted in 
fig.~\ref{fig1}. 
Larger regions in the (\dRLbs$\!$, \dRRbs$\!$) parameter space are 
obtained when both the MIs take non-vanishing values. It is clear that 
no absolute limit can be derived for the two MIs simultaneously. 
The values (\dRRbs$\!$, \dRRbs$\!$) $\approx (1,0.1)$ are, for example, 
possible %\footnote{One should check if, for such large MI values, charge 
%and colour breaking minima appear. Anyway as these are usually rather 
%model-dependent assumptions, we do not introduce here the constraints 
%discussed, for example, in \cite{ccb}.} 
for \tz$=35$. In fact, as can be seen in fig.~\ref{fig5} (left), there is 
always a ``flat direction'' where large values of \dRLbs and \dRRbs can be 
tuned in such a way that the gluino contribution to $C'_{7,8}$ is consistent 
with the experimental bound. This flat direction clearly depends on the 
chosen values for $\tm_{\tilde{g}}/\tm_{\tilde{q}}$ and $\mu$ \tz$\!$. 
The presence of this particular direction is explained by the fact that 
we are allowing complex off-diagonal entries. Hence the relative phase 
between \dRLbs and \dRRbs can be fixed in such a way that the needed 
amount of cancellation can be obtained between the first and second order 
MI contribution. In the notation used in eqs.(\ref{glC7MI}--\ref{glC8pMI2}) 
the line of maximal cancellation is obtained for 
$\varphi=\arg[$\dRLbs \dRRbs$] =\pm \pi$.  
%
%___________________________________________________________________________
%\FIGURE[t]{
\begin{figure}[t]
\vspace{0.1cm}
\begin{tabular}{cc}
\hspace{-0.5cm}
\epsfig{file=dRLRR23col_tz.ps, height=8.15cm, angle=-90} & 
\hspace{-0.5cm}
\epsfig{file=brasycol.ps, height=8.15cm, angle=-90} 
\end{tabular}
\caption{$1\sigma$-allowed region in the (\dRRbs$\!$, \dRLbs$\!$) 
parameter space (left) and Asymmetry vs Branching Ratio (right) for three 
different values of $\tan \beta$, with the other parameters fixed to 
$\tm_{\tilde{g}}/\tm_{\tilde{q}}=350/500$ and $\mu=350$ GeV. All the other 
off-diagonal entries, except the one displayed on the axes, are assumed 
to vanish.}
\label{fig5} 
\end{figure}
%       }
%___________________________________________________________________________

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\subsection{CP asymmetry and LR asymmetry}
\label{sectiondc}
%
In addition to the \bsg BR, the experimental collaborations 
will provide in the coming years more precise measurements of the \bsg 
CP asymmetry.
%\be
%{\cal A}_{CP}(\bsg) = \frac{BR(\bsg)-BR(\bsgb)}{BR(\bsg)+BR(\bsgb)}~.
%\label{asyCP}
%\ee 
The present experimental value gives, at $90\%$ CL, the range $-0.27 < 
{\cal A}_{CP}(\bsg) < 0.10$ which is still too imprecise to provide useful 
tests for NP, although the measurement is expected to be upgraded soon.

The only flavour-violating and CP-violating source in the SM (and MFV 
scenarios) is given by the CKM matrix, which results in a very small 
prediction for the CP asymmetry. In the SM an asymmetry of approximatively 
$0.5 \%$. If other sources of CP violation are present, a much larger CP 
asymmetry could be produced. In our $C'_7$-dominated scenario, one can derive 
the following approximate relation for the CP asymmetry, in terms of the 
\dRLbs and \dRRbs MIs:
%\be
%{\cal A}_{CP}(\bsg) \approx  \frac{2 k \, |\dRLbs \dRRbs \!\!| \sin \varphi}
%       {|\dRLbs \! \!|^2 + k^2 |\dRRbs \! \!|^2 + 2 k |\dRLbs \dRRbs \! \!| 
%       \cos \varphi}~, \qquad k = \frac{m_b \mu \tz}{\tm^2_D}
%\label{asymMI}
%\ee
\be
{\cal A}_{CP}(\bsg \!) = -\frac{4}{9}\alpha_s(\mu_b) \frac{ 
   {\rm Im}\left[ C'_7 C'^*_8 \right]}{|C'_7|^2} \approx k(x^g_D)
   \left(\frac{m_b \mu \ \tz}{\tm^2_D} \right) 
   |\dRLbs \dRRbs \!\!| \sin \varphi~,
\label{asymMI}
\ee
where $\varphi$ is the relative phase between \dRLbs and \dRRbs as 
previously defined. %The constant of proportionality $k(x^g_D)$ depends 
%only on the ratio $\tm_{\tilde{g}}/\tm_D$ through the integrals $F_i$ and 
%can be easily obtained from eqs.~(\ref{glC7MI}) and (\ref{glC7MI2}). 
One can immediately note that if only one MI is 
considered, the CP asymmetry is automatically zero. 
%This follows from the fact that the interference terms $C'_2 C'^*_7$ 
%and $C'_2 C'^*_8$ (proportional to only one MI) are vanishing as $C'_2=0$. 
%Moreover the ``standard'' $C_2 C_7^*$ and $C_2 C_8^*$ intereference terms 
%give negligible contributions in our scenario as $C_7 \approx C_8 \approx 
%0$ are imposed. 
In addition a non-vanishing phase in the off-diagonal down-squark mass 
matrix is necessary. No sensitive bounds on this phase can be extracted 
from EDMs in a general flavour-violating scenario. 

%___________________________________________________________________________
%\FIGURE[t]{
%\begin{figure}[t]
%\vspace{0.1cm}
%\centerline{
%\epsfig{file=brasycol.ps, width=11cm, angle=-90} } 
%\caption{Asymmetry as a function of the  branching ratio for three 
%different values of $\tan \beta$, with $\tm_{\tilde{g}}/\tm_{\tilde{q}} = 
%350/500$, and $\mu=350$ GeV. All the off-diagonal entries except \dRLbs 
%and \dRRbs are assumed to vanish. The vertical lines represent the 
%$1\sigma$ experimentally allowed region.}
%\label{fig6} 
%\end{figure}
%       }
%___________________________________________________________________________

In fig.~\ref{fig5} (right), we show the results obtained for the BR 
and CP asymmetry in which \dRLbs$\!$, \dRRbs and the relative phase 
$\varphi$ are varied arbitrarily for fixed value of $\tm_{\tilde{g}}/
\tm_{\tilde{q}}=350/500$ and \tz$=35$. The full vertical lines 
represent the $1 \sigma$ region experimentally allowed by the 
\bsg BR measurements. It is possible, using $C'_{7,8}$ 
alone, to saturate the \bsg measured BR and at the same 
time have a CP asymmetry even larger than $\pm 10\%$, the sign of the 
asymmetry being determined by the sign of $\sin \varphi$. As shown in 
fig.~\ref{fig5} (right), in the relevant BR range the CP asymmetry 
range is constant. No strong dependence from $\tan \beta$, 
in the large \tz region, is present. The points with large asymmetry 
($>5\%$) lie in the ``flat direction" observed in fig.~\ref{fig5} and 
they have almost $\varphi \approx \pm \pi$ (obviously for $\varphi=\pm 
\pi$ the CP asymmetry vanishes). The explanation of this fact is the 
following. The numerator is proportional to $\sin \varphi$ and so 
goes to 0 as $\varphi$ approaches $\pm \pi$. However, at the same time  
it is enhanced for large MI values. This happens when the flat 
direction condition is (almost) satisfied. Here, in fact, a cancellation 
between the two (large) MI terms takes place, providing the enhancement 
of the CP asymmetry as the denominator remains practically constant, fixed 
by the allowed experimental measurement on the BR. Note also 
that for parameter values outside the flat direction condition a CP 
asymmetry of a few per cent can still be observed, about ten times 
bigger than the SM prediction. %The same order of magnitude can be 
%observed in MFV, when large \tz effects are taken into account 
%\cite{demir}. 
In our scenario even smaller values of the CP asymmetry can be obtained, 
e.g. if one of the two off-diagonal entries is negligible, or the two MIs 
are ``aligned''. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\subsection{Distinguishing the ``$C'_7$-dominated'' scenario from the 
%``$C_7$-dominated" one}
%\label{sectiondd}
%
A possible method for disentangling the relative contributions to the 
\bsg BR from the $Q_7$ and $Q'_7$ operators utilizes an   
analysis of the photon polarization. %A detailed analysis of how it
%is possible to extract information from the photon polarization in 
%radiative B decays is given in \cite{bsgpol}. 
For simplicity, let us define the following ``theoretical'' LR asymmetry at LO:
\be
{\cal A}_{LR}(\bsg \! \!) = 
   \frac{BR(b \raw s \gamma_L) - BR(b \raw s \gamma_R)}
        {BR(b \raw s \gamma_L) + BR(b \raw s \gamma_R)} = 
   \frac{|C_7(\mu_b)|^2-|C'_7(\mu_b)|^2}{|C_7(\mu_b)|^2+|C'_7(\mu_b)|^2}~,
\label{asyLR}
\ee
which could in principle disinguish between $C_7$ and $C'_7$ dominated 
scenarios. Here L,R is the polarization of the external photon. This 
quantity is related to the quark chiralities of the $Q_7, Q'_7$ operators. 
Such a measurement is not yet available, as only the average quantity 
$BR(b \raw s \gamma_L) + BR(b \raw s \gamma_R)$ is reported experimentally. 
In the SM case, and in general in all the MFV and mSUGRA scenarios, only 
the $C_7$ coefficient gives a non-negligible contribution to the \bsg 
BR, in such a way that  
%Only the right-handed bottom quark (in the centre of mass reference frame) 
%can decay, producing a photon with left polarization and 
${\cal A}_{LR}(\bsg \!\!) = 1$. 
%Small deviations from unity are possible because of subleading $m_s/m_b$ 
%terms and hadronization effects. 
In our scenario, where the total contribution to $C_7$ is negligible, 
%only left-handed bottom quarks can decay, emitting a photon with 
%right polarization, 
%which in turn predict ${\cal A}_{LR}(\bsg\!\!) = -1$. 
one obtains ${\cal A}_{LR}(\bsg\!\!) = -1$. In any other uMSSM scenario, 
any LR asymmetry between $1$ and $-1$ is allowed. Consequently, a 
measurement of $A_{LR}(\bsg \!\!)$ different from one will be a clear 
indication of physics beyond the SM with a non-minimal flavour structure. 
%It will be very interesting to know 
%if (and how precisely) CLEO, BABAR, and BELLE can measure the LR asymmetry 
%of eq.~\ref{asyLR}). 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\section{Conclusions}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
We have discussed an alternative explanation of the \bsg BR in the 
unconstrained MSSM. We analyzed in particular the gluino contribution to 
the WC $C'_7$ associated with the chirality operator $Q'_7$. We show 
that this coefficient arises mainly from two off-diagonal entries: 
\dRLbs and \dRRbs$\!$. For scenarios where the $C_{7,8}$ contributions 
to \bsg are small (i.e. for regions in the MSSM parameter space where 
Ws, Higgs, chargino and gluino contributions to $C_{7,8}$ tend to cancel 
each other), $C'_{7,8}$ provides the dominant effect. We derived absolute 
bounds separately on each of these coefficients. We then described the 
allowed region of (\dRLbs$\!$, \dRRbs$\!$) parameter space, as a function 
of $\tan \beta$. We observed that (for a fixed ratio 
$\tm_{\tilde{g}}/\tm_{\tilde{q}}$ and for each chosen value of $\mu \tan 
\beta$), there exists a ``flat direction'' where large (even $O(1)$) 
off-diagonal entries are allowed. For the majority of parameter space 
the CP asymmetry is less than $5\%$, even if asymmetries 
as large as $20\%$ can be obtained along these ``flat directions''. Finally, 
we suggested that the measure of the LR asymmetry could  
help to disentangle the $C_7$ from the $C'_7$ contribution to the \bsg BR. 
Any ${\cal A}_{LR}(\bsg) \neq 1$ would be an irrefutable proof, not only 
of physics beyond the SM, but also it would indicate the existence of a
non-minimal flavour violation structure of the down-squark mass matrix.
It would be very interesting if such a quantity could be measured. One 
implication of our analysis is that previous results on MSSM parameters, 
including constraints on the ``sign of $\mu$''\footnote{The relative 
sign between the parameters $\mu$ and $A_t$. See \cite{everett} 
for a discussion about the $g-2$ ``sign of $\mu$''.} which are more 
model-dependent than has generally been assumed.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Acknowledgements}
%
%We thanks A. Donini, D. Demir, F. Feruglio, B. Gavela and A. Masiero 
%for reading the manuscript and for the useful comments provided. L.E., 
%G.K. and S.R. thanks the ``Aspen Center for Physics'' for the warm 
%hospitality and the very nice atmosphere offered during the final 
%stage of this work. 
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
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% 
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                             End of moriond.tex                               %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%To get a sense of the typical magnitudes of the SUSY contribution 
%to \bsg$\!$, it is illustrative to consider the (unphysical) limit of 
%unbroken SUSY but broken electroweak gauge symmetry, which corresponds 
%to the supersymmetric Higgsino mass parameter $\mu$ set to zero and the 
%ratio of Higgs vacuum expectation values \tz$\equiv v_u/v_d$ set to 1. 
%In this limit SM and SUSY contributions are identical in size and 
%cancel each other \cite{ferrara}, due to the usual sign difference 
%between boson and fermion loops. Of course, this limit is unphysical: 
%not only must SUSY be (softly) broken, but $\mu=0$ and \tz$=1$ have 
%been ruled out by direct and indirect searches at LEP. %\cite{LEP}. 
%
%In the realistic case of softly broken SUSY, the contributions to
%\bsg depend strongly on the parameters of the SSB Lagrangian, as well 
%as the values of $\mu$ and $\tan \beta$. 


\begin{itemize}
\item 
The SUSY partners are very heavy and their contribution decouples, so 
that only the Higgs sector contributes to \bsg$\!$. In this scenario, as 
well in general 2HDMs, NLO calculations have been performed 
\cite{degrassi1,ciafaloni,borzumati2h}. As SM and Higgs sectors give 
coherent contributions, a lower bound on the charged Higgs mass can
usually be derived \cite{deboer} in this class of models. In the large 
\tz region the two-loop SUSY correction to the Higgs vertex can produce 
quite sizeable modifications and should be carefully taken into 
account \cite{degrassi3}.  
\item 
The SUSY partners as well as the extra Higgs bosons have masses of order 
the electroweak scale, but the only source of flavour violation is in the 
CKM matrix.  This scenario, known as minimal flavour violation (MFV), is 
motivated for example within minimal supergravity (mSUGRA) models. MFV 
scenarios have been studied at LO \cite{bertolini}--\cite{masiero},  
in certain limits at NLO \cite{degrassi2}, and including large \tz 
enhanced two-loop SUSY contributions \cite{degrassi3,demir}. 
In this scenario, the \bsg decay receives a contribution 
from the chargino sector as well as from the charged Higgs sector.
To avoid overproducing \bsg $\!$, the charged Higgs and chargino loops
must cancel to a good degree.  This cancellation can be achieved for a
particular ``sign of $\mu$'' in the mSUGRA parameter space\footnote{
Specifically the relative sign between the parameters $\mu$ and $A_t$ 
(see refs.\cite{degrassi3,degrassi2}) and so generally different from 
the ``sign of $\mu$'' relevant in 
the case of the muon $g-2$ MSSM contribution \cite{everett}.}, which 
flips the sign of the chargino contribution relative to the SM and 
charged Higgs loops, always interfering constructively. Although 
this cancellation can occur and puts important constraints on the 
mSUGRA parameter space, it is important to note that it is not due to 
any known symmetry but rather should be interpreted, in a certain 
sense, as a fine-tuning.
\item 
There are new sources of flavour violation in the soft breaking terms.  In
this case, additional SUSY loops involving down-type squarks and gluinos 
or neutralinos (hereafter neglected respect to the gluino loops due to 
the weaker coupling) contribute to \bsg$\!$. It is well known that the 
gluino contribution can dominate the amplitude for such non-minimal SUSY 
models, both due to the $\alpha_s/\alpha$ enhancement with respect to the 
other SM and SUSY contributions and due to the $m_{\tilde{g}}/m_b$ 
enhancement from the chirality flip along the gluino line.  Thus in this 
scenario, which is generally noted as the unconstrained MSSM (uMSSM),
usually only the gluino contribution is discussed. It has been shown
\cite{gabbiani,pokorski} that the 23-LR off-diagonal entry of the
down-squark mass matrix is severely constrained by \bsg measurements 
to be of $O(10^{-2})$. Less stringent bounds can be obtained for 
the other 23 off-diagonal entries. No known symmetry assures that these 
constraints can be automatically satisified; again this fact could be 
interpreted at the electroweak scale as a fine-tuning. 
\end{itemize}
%In MFV, the SUSY contributions are loops involving up-type quarks and 
%charged Higgs bosons, and up-type squarks and charginos. These diagrams 
%contribute to the Wilson coefficient $C_7$ of the effective operator 
%$Q_7$ along with the SM. The relevant down-squark mass insertion parameters
%($\delta^{LR}_{23}$, $\delta^{LL}_{23}$)  for the gluino contribution to
%$C_7$ have been bounded by \cite{gabbiani}. 

The structure of the paper is as follows. In section \ref{sectionb}, we
briefly summarize the theoretical framework for the calculation of the 
\bsg branching ratio at LO and NLO. In section \ref{sectionc}, we derive 
useful mass insertion (MI) formulas for the gluino contributions to the 
Wilson coefficients $C_{7,8}$ and $C'_{7,8}$. We demonstrate explicitly 
that in the large \tz region, a good understanding of these expressions 
is obtained only by retaining terms in the MI expansion through the 
second order. For $\mu$ of the same order as the common squark mass 
parameter and large $\tan \beta$, new (previously overlooked) off-diagonal 
terms become relevant in the \bsg process. %After this general derivation
We then devote our attention in section \ref{sectiond} to the analysis of 
the gluino contribution to $C'_{7,8}$ in the general uMSSM. In particular, 
we ask the question of whether the contribution to $C'_7$ alone can 
saturate the \bsg branching ratio, assuming that the SM and SUSY 
contributions to $C_7$ cancel each other to an extent that the effects 
of $C_7$ are subleading. While this scenario may initially appear to be 
unnatural, we will argue that sufficient cancellations in $C_7$ do not 
involve significantly more fine-tuning than the usual cancellation required 
in MFV scenarios. With this analysis, we thus provide an alternative 
interpretation of \bsg$\!$, which is at least as viable as any supersymmetric 
one. This analysis also provides more general mass insertion bounds on 
\dRLbs than those obtained recently \cite{besmer}, where the SM (and 
sometimes Higgs and chargino) contributions to $C_7$ are always retained.  
As we are generally interested in moderate to  large values of \tz$\!$, 
we are able to put rather stringent bounds on the mass insertion parameter 
\dRRbs$\!$. In section \ref{sectiondc}, we study the branching 
ratio and CP asymmetry as functions of the SUSY parameter space within 
this scenario, assuming complex off-diagonal MIs. Throughout the paper, 
to avoid EDM constraints we set the relevant reparametrization-invariant 
combinations of the flavour-independent phases to zero. Finally in section 
\ref{sectiondd} we show that if the photon polarization will be measured, 
it is possible to distinguish such a scenario from the usual $C_7$-dominated 
one through the definition of a LR asymmetry.

Since we are interested in analyzing a supersymmetric world where 
the one-loop SUSY effects are of the same order as the SM loops, we 
assume relatively light superpartner masses. Specifically we choose the 
gluino mass $\tm_{\tilde{g}}=350$ GeV and the common diagonal down-squark 
mass $\tm_D=500$ GeV, with the lightest down-squark mass in the 250--500 
GeV range. All of the other sfermion masses, as well the chargino and 
neutralino masses, do not enter directly in our analysis and (some of 
them) can be taken to be reasonably light, as suggested by 
\cite{altarelli}. Motivated by the lower limit on the Higgs boson mass 
\cite{janot} (which suggests $|\cos 2 \beta| \approx 1$) and by the muon 
$g-2$ excess, we focus to some extent on moderate to large values of $\tan 
\beta$, though our formulas and much of the analysis hold in general. 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

However, it has been recently shown \cite{kv} that in many classes of 
SUSY breaking models a particular structure of the soft trilinear 
couplings $\tilde{A}$ of the soft-breaking Lagrangian 
can be derived, which can alleviate these constraints.
Writing these couplings as $\tilde{A}_{ij}=A_{ij} Y_{ij}$ (in which $Y$ 
denotes the fermion Yukawa matrices), the matrices $A$ for the up and 
down sector are given respectively by:
\be
A^{(u)}_{ij} = A^L_{ii} + A^{R,u}_{jj} \qquad , \qquad  
A^{(d)}_{ij} = A^L_{ii} + A^{R,d}_{jj}~.  
\label{tril}
\ee
As shown in \cite{kv}, this factorization holds quite generally in string 
models, for example in Calabi--Yau 
models in the large $T$ limit or in Type I models \cite{imr}, as well as 
in gauge-mediated \cite{giudice1} and anomaly-mediated models 
\cite{randall}--\cite{rattazzi}. If eq.~\ref{tril}) holds, 
specific relations can be derived for the off-diagonal LR entries in the 
squark mass matrix. In particular, the leading contribution to the 
%neglecting 
%all the quark masses with respect to the third generation ones, 
entries of interest for the \bsg process are given in the SCKM basis as:
\bea
\tilde{A}^{(u)}_{23} & \propto & m_t \left[ 
  (A^L_{22}-A^L_{11}) (V^{(u)}_L)_{22} (V^{(u)}_L)^*_{32} +  
  (A^L_{33}-A^L_{11}) (V^{(u)}_L)_{23} (V^{(u)}_L)^*_{33} \right]~, 
  \label{At23} \\
\tilde{A}^{(u)}_{32} & \propto & m_t \left[ 
  (A^{R,u}_{22}-A^{R,u}_{11}) (V^{(u)}_R)_{32} (V^{(u)}_R)^*_{22} +  
  (A^{R,u}_{33}-A^{R,u}_{11}) (V^{(u)}_R)_{33} (V^{(u)}_R)^*_{23} \right]~, 
  \label{At32} \\  
\tilde{A}^{(d)}_{23} & \propto & m_b \left[ 
  (A^L_{22}-A^L_{11}) (V^{(d)}_L)_{22} (V^{(d)}_L)^*_{32} +  
  (A^L_{33}-A^L_{11}) (V^{(d)}_L)_{23} (V^{(d)}_L)^*_{33} \right]~, 
  \label{Ab23}\\
\tilde{A}^{(d)}_{32} & \propto & m_b \left[ 
  (A^{R,d}_{22}-A^{R,d}_{11}) (V^{(d)}_R)_{32} (V^{(d)}_R)^*_{22} +  
  (A^{R,d}_{33}-A^{R,d}_{11}) (V^{(d)}_R)_{33} (V^{(d)}_R)^*_{23} \right]~, 
  \label{Ab32} 
\eea
with $V^{(u,d)}_{L,R}$ the rotation matrices for the up and down quark 
sector from the interaction to the mass eigenstate\footnote{In this 
notation the CKM matrix is $V_{CKM} = V^{(u)}_L (V^{(d)}_L)^\dagger$.}. 
From eqs.(\ref{At23}-\ref{Ab32}), one can realize first that the 
down-sector LR off-diagonal entries are naturally suppressed by a factor 
of $O(m_b/m_t)$ respect to the up-squark sector ones because of the  
particular factorization of the soft trilinear couplings 
given in eq.~\ref{tril}). Second, in these classes of models both the $23$ 
and $32$ entries are of the same order and proportional to the largest 
mass (up or down). 
%\footnote{It is generally reported in literature that 
%the off-diagonal LR entries in the squark mass matrix are proportional to 
%the right quark mass. This is true only in particular models (WHICH??).}. 
%Consequently, in these classes of models, ${\cal O}(10^{-2})$ down-sector 
%entries can be considered a ``prediction'' of the underlying 
%fundamental theory, when ${\cal O}(1)$ up-squark sector off-diagonal 
%entries are considered. 
Consequently, in these classes of models, $O(10^{-2})$ off-diagonal 
entries in the down-squark sector along with $O(1)$ off-diagonal entries 
in the up-squark can be considered in some sense as a 
prediction of the underlying fundamental theory\footnote{It is important
to note, however, that the off-diagonal entries of $\tilde{A}$ in the SCKM
basis contain terms proportional to the products of entries of the
left-handed and right-handed quark rotation matrices, which are largely
unconstrained (except for the CKM constraint for the left-handed up and
down quark rotation matrices which enter (for example) $\tilde{A}_{23}$).
The quark rotation matrices are highly model-dependent. While the
diagonal entries can in general safely be taken $O(1)$, it is
typically assumed that the off-diagonal quark rotation matrices are 
suppressed by powers of the Cabibbo angle in a way that mirrors the CKM
matrix (see e.g. \cite{kv}). Note though that this assumption is not
required, particularly for the right-handed quark rotation matrices
which enter $\tilde{A}_{32}$, which are of particular relevance to 
this paper.}. %note these should be suppressed by quark rotations. 
This fact implies comparable chargino and gluino contributions 
to \bsg$\!$, making the possibility of cancellations between the W and the 
different SUSY contributions to the $Q_7$ operator less unnatural. 
The constraints on the gluino contribution to $Q'_7$ are simultaneously  
alleviated. This flavour structure holds in essentially all attempts to build 
string-motivated models of the soft-breaking Lagrangian.
% and we know of no theoretical arguments that imply stronger assumptions. 
%Thus, we proceed to study \bsg within this framework.

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{\bsg branching ratio at NLO}
\label{sectionb}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
For the purpose of presentation, we summarize the theoretical 
framework for evaluating the \bsg branching ratio at NLO. A complete and 
detailed discussion can be found for example in 
\cite{misiak}--\cite{kagan}. 
The starting point in the calculation of the B meson decay rates is the 
low-energy effective Hamiltonian, at the bottom mass scale $\mu_b$:
\be 
{\cal H}_{eff} = - \frac{4 G_F}{\sqrt{2}} V_{tb} V^*_{ts} 
   \sum_i C_i(\mu_b) Q_i(\mu_b) \ .  
\ee 
The operators relevant to the \bsg process\footnote{This of course depends 
on the basis chosen; we have chosen the one easiest for our 
discussion.} are: 
\bea 
Q_2 & = & \bar{s}_L \gamma_\mu c_L \bar{c}_L \gamma^\mu b_L \ , \nn \\
Q_7 & = & \frac{e}{16 \pi^2} m_b \bar{s}_L \sigma^{\mu \nu} b_R F_{\mu \nu} \ , 
        \nn \\
Q_8 & = & \frac{g_s}{16\pi^2} m_b \bar{s}_L \sigma^{\mu \nu} G^a_{\mu \nu} 
        T_a b_R \ . \label{opQ}
\eea 
and their $L \lraw R$ chirality counterpart:
\bea 
Q'_2 & = & \bar{s}_R \gamma_\mu c_R \bar{c}_R \gamma^\mu b_R \ , \nn \\ 
Q'_7 & = &\frac{e}{16\pi^2} m_b \bar{s}_R \sigma^{\mu \nu} b_L F_{\mu \nu} \ , 
         \nn \\ 
Q'_8 & = &\frac{e}{16\pi^2} m_b \bar{s}_R \sigma^{\mu \nu} G^a_{\mu \nu} 
         T_a b_L \ . \label{opQp}
\eea 

The Wilson coefficients $C^{ (_{'} ) }_{2,7,8}$ are initially evaluated
at the electroweak or soft SUSY breaking scale, which we generically 
denote as $\mu_0$, and then evolved down to the bottom mass scale $\mu_b$. 
The standard\footnote{In a recent paper \cite{borzumati} it has been 
pointed out that the gluino contribution (and the same argument holds 
also for the chargino and neutralino contributions) is the sum of two 
different pieces, one proportional to the bottom mass and one proportional 
to the gluino mass, which have a different RG evolution. We have found 
that at LO, this is equivalent to the usual SM evolution once the running 
bottom mass $m_b(\mu_0)$ is used instead of the pole mass in the 
$C_i(\mu_0)$ Wilson coefficients.} RG equations for the $C_{2,7,8}$ 
operators from the electroweak scale ($\mu_W < m_t$) to the low-energy 
scale $\mu_b$ are given by:
\bea
C_2 (\mu_b) &=& \frac{1}{2}\left(\eta^{-\frac{12}{23}} + 
   \eta^{\frac{6}{23}} \right) C_2(\mu_W) \ , \label{C2mub} \\
C_7 (\mu_b) &=& \eta^{\frac{16}{23}} C_7(\mu_W) + \frac{8}{3}
   \left( \eta^{\frac{14}{23}} -\eta^{\frac{16}{23}} \right) C_8(\mu_W) + 
   \sum_{i=1}^8 h_i \eta^{a_i} \ , \label{C7mub} \\
C_8 (\mu_b) &=& \eta^{\frac{14}{23}} C_8(\mu_W) + 
   \sum_{i=1}^8 \bar{h}_i \eta^{a_i} \ , \label{C8mub} 
\eea
where $\eta=\alpha_s(\mu_W)/\alpha_s(\mu_b)$ and $h_i,\bar{h}_i$ and 
$a_i$ are constants (see \cite{misiak} for details). The $C'_{2,7,8}$ 
coefficients obey the same running as their chirality-conjugate 
counterparts. If the NP scale is much higher than $m_t$, the running from 
$\mu_{SUSY}$ to $\mu_W$ with six quarks should also be 
taken into account (see the first paper of \cite{degrassi3}).  
The coefficient $C_2$ is dominated by a SM tree-level diagram and is 
normalized such that $C_2(\mu_W) = 1$. Its chirality conjugate, $C'_2$, 
has no SM contribution at tree level and can thus be safely set to 
zero. The NP contributions to $C_2$ and $C'_2$ appear at one-loop 
order and are negligible. The Wilson coefficients $C_7$ and $C'_7$ are 
the only coefficients that contribute directly to the \bsg branching 
ratio at the lowest QCD order ($\alpha_s^0$). These coefficients receive 
contributions from both the SM and NP at one-loop order. The coefficients 
$C_8$ and $C'_8$ receive one-loop SM and NP contributions through the 
same types of diagrams as $C_7$ and $C'_7$, but with the external photon 
line substituted by a gluon line. When the QCD running from the matching 
scale $\mu_0$ to $\mu_b$ is performed, these different coefficients mix, 
as shown in eqs.(\ref{C2mub}-\ref{C8mub}), so that the ``effective'' 
low-energy coefficients $C_{2,7,8}(\mu_b)$ receive contributions from 
different operators.
 
The \bsg branching ratio is usually defined by normalizing it to the 
semileptonic $b \raw c \ e^- \ \bar{\nu}_e$ branching ratio, giving:
\be
BR(B \raw X_s \gamma)|_{E_\gamma > (1-\delta) E^{max}_\gamma} = 
BR(B \raw X_c e \bar{\nu}) \frac{6 \alpha}{\pi f(z)} 
\left| \frac{V_{tb} V_{ts}^*}{V_{cb}} \right|^2 K(\delta,z)~.
\ee
Here $f(z)$ is a phase-space function and should be calculated for 
on-shell masses, namely $\sqrt{z} = m_c/m_b = 0.29$; $\delta$ is the 
experimental photon detection threshold, which for comparison between 
experimental data and theoretical prediction is usually set to $0.9$ 
\cite{kagan}. % $BR(B \raw X_c e \bar{\nu})= 10.5\%$ is taken. 
The dependence of $K_{NLO}$ from the Wilson coefficients $C_i$ and 
$C'_i$ at NLO is given by \cite{kagan}:
\bea
K_{NLO}(\delta,z) = & \sum_{i \leq j=2,7,8} & k^{(0)}_{ij}(\delta,z) \ 
   \left\{ {\rm Re}[C^{(0)}_i(\mu_b) C^{(0)*}_j(\mu_b)] \ +  
   \left( C_{i,j} \raw C'_{i,j} \right) \right\} \ + \nn \\ 
& & k^{(1)}_{77}(\delta,z) \left\{ {\rm Re}[C^{(1)}_7(\mu_b) 
   C^{(0)*}_7(\mu_b)] \ + \left( C_7 \raw C'_7 \right) \right\}~.
\label{knlo}
\eea
In the previous expression $C^{(0)}_i$ and $C^{(1)}_i$ refer respectively 
to the LO and NLO contributions to the Wilson coefficients $C_i$ defined 
as:
\be
C_i (\mu_b) = C_i^{(0)}(\mu_b) + \frac{\alpha_s(\mu_b)}{4\pi} 
   C^{(1)}_i (\mu_b) + O(\alpha, \alpha_s^2)~. 
\ee
As in the following we are deriving only one-loop formulas for the 
Wilson coefficients $C^{(_{'})}_{7,8}$, $C_i \equiv C_i^{(0)}$. 
We will briefly discuss the effects of including $C^{(1)}_7$ in 
section \ref{sectionda}. The coefficients $k_{ij}(\delta,z)$ used in our 
analysis are calculated for $\delta=0.9$ and $\sqrt{z} = 0.22$,using the 
formulas derived in \cite{misiak,kagan}. 
%\footnote{Note that the coefficients $k_{ij}$ reported in \cite{kagan} 
%are obtained for $\sqrt{z} = 0.29$.}
The LO branching ratio expression can be easily derived from eq.~\ref{knlo}), 
setting $k^{(0)}_{77}=1$ and all the other $k^{(0,1)}_{ij}=0$, giving:
\be
K_{LO} = |C_7(\mu_b)|^2 + |C'_7(\mu_b)|^2~,
\ee  
independently of the choice of $\delta$ and $z$.

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

\bea
C^{\tilde{g}}_7(\mu_W) &=&\frac{4g_s^2 }{3g^2}{{Q_d}\over{V_{tb} V^*_{ts}}} 
   \sum_{A} {{m_W^2}\over{\tm^2_A}} \left\{ L_{b} L_{s}^* F_2(x^g_A) + 
   \frac{\tm_{\tilde{g}}}{m_b} R_{b} L_{s}^* F_4(x^g_A) \right\}~, 
\label{glC7} \\
C^{\tilde{g}}_8(\mu_W) &=& -\frac{g_s^2}{6g^2} {{Q_d}\over{V_{tb} V^*_{ts}}} 
   \sum_{A} {{m_W^2}\over{\tm^2_A}} \left\{ L_{b} L_{s}^* F_{21}(x^g_A) + 
   \frac{\tm_{\tilde{g}}}{m_b} R_{b} L_{s}^* F_{43}(x^g_A) \right\}~, 
\label{glC8} \\
C^{' \tilde{g}}_7(\mu_W) &=&\frac{4g_s^2}{3g^2}{{Q_d}\over{V_{tb} V^*_{ts}}}
   \sum_{A} {{m_W^2}\over{\tm^2_A}}  
   \left\{ R_{b} R_{s}^* F_2(x^g_A) + 
   \frac{\tm_{\tilde{g}}}{m_b} L_{b} R_{s}^* F_4(x^g_A) \right\}~,
\label{glC7p} \\
C^{' \tilde{g}}_8(\mu_W) &=&-\frac{g_s^2}{6g^2}{{Q_d}\over{V_{tb} V^*_{ts}}} 
   \sum_{A} {{m_W^2}\over{\tm^2_A}} \left\{ R_{b} R_{s}^* F_{21}(x^g_A) + 
   \frac{\tm_{\tilde{g}}}{m_b} L_{b} R_{s}^* F_{43}(x^g_A) \right\}~, 
\label{glC8p}
\eea
in which $x^g_A = \tm_{\tilde{g}}^2 / \tm_{\tilde{A}}^2$, with 
$\tm_{\tilde{g}}$ the gluino mass and $\tm_{\tilde{A}}$ the mass of 
the A-th down squark eigenstate; $L_d$ and $R_d$ are the Left and 
Right gluino couplings to a generic down quark $d$ given by:
\be
L^{\tilde g}_{d} =  - \sqrt{2} \ U_{A,d} \qquad , \qquad  
R^{\tilde g}_{d} =   \sqrt{2} \ U_{A,d+3} \ , \label{glcoupl}
\ee
in which $U$ is the $6 \times 6$ down-squark rotation matrix. The loop 
integrals $F_{12}$ and $F_{43}$ are defined as:
\be
F_{21} = F_2(x) + 9 F_1(x) \qquad , \qquad  F_{43} = F_4(x) + 9 F_3(x)\ ,
\ee 
using the conventions for the integrals $F_i(x)$ as in \cite{bertolini} 
for an easier connection with the standard convention in the literature. 

It is illustrative to write the gluino contribution to the $C_{7,8}$ and 
$C'_{7,8}$ Wilson coefficients using the MI approximation. First, note 
that the set of integrals used in \cite{bertolini} is not the most 
appropriate for dealing with the MI formulas. However, for the sake of 
simplicity we will retain these conventions and further define the 
integrals $F_i$ and their ``derivatives'' through the following 
self-consistent relations:
\bea
\hskip 0.5cm
F_i(\frac{x}{y}) \equiv \frac{1}{y} f_i(x,y) \ , \quad
F^{(1)}_i(\frac{x}{y}) \equiv \frac{1}{y^2} 
   \frac{\partial}{\partial y} f_i(x,y) \ , \ ... \ , \quad
F^{(n)}_i(\frac{x}{y}) \equiv \frac{1}{n!} \frac{1}{y^{n+1}}
\frac{\partial^n}{\partial y^n} f_i(x,y)\ . & & \nn  
\eea 


In the previous formulas we have $x^g_D=\tm^2_{\tilde{g}}/\tm^2_D$, with 
$\tm_D$ the average down-squark mass related to the down-squark mass 
eigenstates via the relation $\tm^2_A = \tm^2_D +\delta m^2_A$. The 
definitions of the MI parameters are:
\bea
\delta^{LL}_{ij} = \frac{1}{\tm^2_D} \sum_{A=1}^6 U^\dagger_{i,A} 
   \delta m^2_A U_{A,j} \quad &,& \quad
\delta^{RR}_{ij} = \frac{1}{\tm^2_D} \sum_{A=1}^6 U^\dagger_{i+3,A} 
   \delta m^2_A U_{A,j+3} \ , \quad \nn \\ 
\delta^{LR}_{ij} = \frac{1}{\tm^2_D} \sum_{A=1}^6 U^\dagger_{i,A} 
   \delta m^2_A U_{A,j+3} \quad &,& \quad 
\delta^{RL}_{ij} = \frac{1}{\tm^2_D} \sum_{A=1}^6 U^\dagger_{i+3,A} 
   \delta m^2_A U_{A,j} \ .
\label{MIdef}
\eea

% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

As previously discussed, many classes of 
SUSY breaking models \cite{kv} lead to off-diagonal LR entries of the 
down-squark sector that are naturally suppressed compared with those 
of the up-squark sector:
\be
(\delta^{LR}_{ij})^d \approx \frac{\max(m_i,m_j)}{m_t} (\delta^{LR}_{ij})^u
\ee
in which $m_{i,j}$ are down-quark masses. In particular, the 
$(\delta^{LR}_{23})^d$ 
%``23-LR'' 
entries, which are 
relevant to the \bsg process, receive a $O(m_b/m_t)$ suppression as can 
be derived from eqs.~(\ref{At23}--\ref{Ab32}). For $(\delta^{LR}_{23})^u 
\approx O(1)$, a natural value $(\dRLbs)^d \approx O(m_b/m_t) \approx 
10^{-2}$ is obtained. With this mechanism at work, off-diagonal chargino 
and gluino contributions to flavour changing processes are naturally of the 
same order. The $\alpha_s/\alpha_w$ enhancement of the gluino contribution 
with respect to the chargino one is compensated by the $m_b/m_t$ suppression 
of the LR off-diagonal entries.
Clearly a complete analysis of the regions of uMSSM parameter space where 
the $C_{7,8}$ cancellation takes place is an important task, 
necessary for studying the details of this scenario. However, a detailed 
analysis is beyond the scope of this paper and will be discussed elsewhere 
\cite{ekrww2}. It is worth mentioning that in preliminary scans we checked 
that it is not difficult to %reaasonable to 
find a candidate set of parameters where $C_{7,8}$ numerically yield 
small contributions to the \bsg branching ratio. Of course, this 
set is obviously not expected to be unique, and further checking 
that any such parameter sets are consistent with all the other existing 
measurements of FCNC and CP-violating observables will impose 
further strong constraints.

Finally, we stress that in the following analysis we do not 
make any specific assumptions as to the size of the off-diagonal 
entries of the down-squark mass matrix. In particular, we are not using 
any of the relations described in eqs.~(\ref{tril}--\ref{Ab32}). The 
previous arguments have been intended as a theoretical framework for the 
following model-independent analysis. A general discussion of the 
CP-violating sector, using the factorization ansatz of eq.~\ref{tril}), 
will be the subject of a forthcoming paper \cite{ekrww2}.

% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%

In Figs.~\ref{fig1} and \ref{fig2}, we set $C_7 = C_8 = 0$, so that the 
only contribution to the $\bsg$ branching ratio is due to  the gluino 
contribution to $C'_7$ and $C'_8$. Thus one should think that for 
vanishing \dRLbs and/or \dRRbs the branching ratio in our scenario should 
vanish. The reason for the finite, non-zero contribution is the fact 
that we are using a NLO formula for the \bsg branching ratio \cite{kagan}. 
At NLO, imposing the condition $C_{7,8}(\mu_W)=0$ still leaves constant 
terms that arise from the mixing of the SM operators 
(specifically, in our chosen basis, $C_2$) that do not contribute to the 
branching ratio at LO. In fig.~\ref{fig3} (left side), we compare the 
results obtained using the LO and NLO expressions for the \bsg branching 
ratio imposing the condition $C_{7,8}(\mu_W)=0$. As can be seen 
explicitly, the difference in using the LO or NLO is sizeable. In 
fig.~\ref{fig3} (right side), 
we compare the results obtained using the LO and NLO expression for the 
\bsg branching ratio imposing the condition 
$C_{7,8}(\mu_b)=0$. 
As one be seen now, the LO contribution to the \bsg branching ratio 
vanishes for vanishing MIs. This does not happen for the LO contribution 
of the left plot, as a finite contribution to the branching ratio appears 
from the running $\mu_W \raw \mu_b$ when the condition $C_{7,8}(\mu_W)=0$ 
is taken. In all the plots, except fig.~\ref{fig3} (right side), we 
use $C_{7,8}(\mu_W)=0$, as this is the natural scale where cancellations 
could be explained in terms 
of the underlying fundamental theory, while the choice $C_{7,8}(\mu_b)=0$ 
seems highly accidental. Finally, it should be noted that the strongest 
restriction comes from imposing the condition $C_{7}=0$. The same 
requirement on $C_8$ could easily be relaxed, and our results 
would remain almost unchanged. The $C_8$ contribution to the \bsg 
branching ratio represents in fact only $10\%$ of the total effect.  
%
%___________________________________________________________________________
%\FIGURE[t]{
\begin{figure}[t]
\vspace{0.1cm}
\begin{tabular}{cc}
\hspace{-0.5cm}
\epsfig{file=muWLO.ps, height=8.15cm, angle=-90} & 
\hspace{-0.5cm}
\epsfig{file=mubLO.ps, height=8.15cm, angle=-90} 
\end{tabular}
\caption{Dependence of \bsg branching ratio on $\delta^{RL}_{23}$ 
for $\tm_{\tilde{g}}/\tm_{\tilde{q}}=350/500$, \tz$=20$ and $\mu=350$ 
GeV. All the other off-diagonal entries, except the one displayed on the 
axes, are assumed to vanish. In the plots we show the result 
obtained using LO (dashed line) and NLO (full line) formula for the 
\bsg branching ratio, setting respectively $C_{7,8}(\mu_W)=0$ (left plot) 
and $C_{7,8}(\mu_b)=0$ (right plot). The horizontal lines represent the $1 
\sigma$ experimentally allowed region.}
\label{fig3} 
\end{figure}
%      }
%___________________________________________________________________________
%
It is important to notice at this point that a consistent analysis of 
\bsg at NLO would require the calculation of the two-loop (QCD and SQCD) 
contribution to the $C'_7$ coefficient. In the general uMSSM the 
calculation of the $O(\alpha_s^2)$ contribution to $C'_7$ (and obviously 
$C_7$) is extremely complicated. In \cite{bobeth}, the contribution to 
$C_7$ from the two-loop diagrams with one gluino and one gluon internal 
line has been calculated. This represents the dominant MSSM two-loop 
contribution only in the limit of very large gluino mass of $O(1{\rm TeV})$ 
and a small \tz ($\approx 1$). Thus it cannot be applied to our analysis, in 
which SUSY masses (and the gluino mass in particular) below $500$ GeV 
and large \tz are assumed. In fact, if the gluino mass is light  
the two-loop diagrams with two gluino internal lines should also be taken 
into account. Moreover, if \tz is large, diagrams with internal 
Higgsino lines can no longer be neglected, as Yukawa couplings can 
become of $O(1)$. Using the results of \cite{bobeth}, one obtains 
an effect of a few per cent in the \bsg branching ratio. It should be 
remembered, however, that in our analysis this provides only a very crude 
estimation. It seems reasonable to expect a possible $10\%$ modification 
of the \bsg branching ratio results from the inclusion of the complete NLO 
calculation of the $C'_7$ coefficient. Moreover, while the two-loop 
diagrams with gluino/gluon internal lines have the same MI structure and 
as such are proportional to the one-loop gluino contribution to $C'_7$, 
this is not the case for the diagrams with gluino/Higgsino internal 
lines, for which the CKM flavour changing structure also enters.   


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