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%%%% LR macros
\newcommand{\newc}{\newcommand}
\newc\eg{{\it {e.g.}}}	\newc\vs{{\it {vs.}}}	\newc\etal{{\it {et al.}}}

\newcommand\fa{f_{a}}
\newcommand\mchi{m_{\chi}}              \newcommand\nchi{n_{\chi}}

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\newcommand\photino{\widetilde{\gamma}} \newcommand\mphotino{m_{\photino}}
\newcommand\gluino{\widetilde{g}} \newcommand\mgluino{m_{\gluino}}

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\newc\brbsgamma{BR(B\rightarrow X_s\gamma)}

\newc{\tanb}{\tan\beta}
\newc{\azero}{A_0}
\newc{\at}{A_t} \newc{\abot}{A_b} \newc{\atau}{A_\tau}
\newc{\bmu}{B\mu}           \newc{\sgn}{{\rm sgn}}
\newc{\mone}{M_1}           \newc{\mtwo}{M_2}

\newc{\bino}{\widetilde B}              \newc{\wino}{\widetilde W_3}
\newc{\higgsinob}{{\widetilde H}^0_b}   \newc{\higgsinot}{{\widetilde H}^0_t}


\newc{\mw}{m_{\rm W}}
\newc\msusy{M_{\rm SUSY}}
\newc{\mplanck}{M_{\rm P}}

\newc{\mub}{\mu_{\rm b}}	\newc{\muw}{\mu_{\rm W}}
\newc{\mususy}{\mu_{\rm SUSY}}

\newc{\Ci}{C_i}	\newc{\Cip}{C_i^{\prime}}

\newc{\deltadll}{\delta^d_{LL}}	\newc{\deltadlr}{\delta^d_{LR}}
\newc{\deltadrl}{\delta^d_{RL}}	\newc{\deltadrr}{\delta^d_{RR}}

\newc{\abund}{\Omega h^2}
\newc{\abundchi}{\Omega_\chi h^2}
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\newc{\VEV}[1]{\langle #1 \rangle}

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\newcommand\gev{\,\mbox{GeV}}
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\newcommand\kev{\,\mbox{keV}}
\newcommand\ev{\,\mbox{eV}}
%\newcommand\pc{\,\mbox{pc}}
\newcommand\mpc{\,\mbox{Mpc}}

%% macros to produce the symbols "less than or of order of"
%% and "greater than or of order of" %
\newc{\ra}{\rightarrow}
\newc{\beq}{\begin{equation}}
\newc{\eeq}{\end{equation}}
\newc{\bea}{\begin{eqnarray}}
\newc{\eea}{\end{eqnarray}}
%%%% end LR macros

\renewcommand\({\left(}
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%       \newcommand\eqs[2]{Eqs.~(\ref{#1}) and (\ref{#2})}
%       \newcommand\eqss[3]{Eqs.~(\ref{#1}), (\ref{#2}) and (\ref{#3})}
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%       (\ref{#4}) and (\ref{#5})}
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%Fields and their VEVs
%       \def\so{S_1}
%       \def\st{S_3}
%       \def\stb{\overline{S}_3}
%       \def\se{S_8}
%       \def\see{S'_8}
%       \def\sf{S_{15}}
%       \def\sfp{S'_{15}}

%       \def\vo{|S_1|^2}
%       \def\vt{|S_3|^2}
%       \def\vtb{|\overline{S}_3|^2}
%       \def\ve{|S_8|^2}
%       \def\vee{|S'_8|^2}
%       \def\vf{|S_{15}|^2}
%       \def\vfp{|S'_{15}|^2}

%units
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% $\slash D$
% {\slashfraction={.075} $\slash{\cal A}$}
% $\slash B$
% $\slash a$
% {\slashfraction={.09} $\slash p$}
% $\slash q$

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%       \def\mco{\multicolumn}
%       \def\epp{\epsilon^{\prime}}
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%       \def\ra{\rightarrow}
%       \def\ppg{\pi^+\pi^-\gamma}
%       \def\vp{{\bf p}}
%       \def\ko{K^0}
%       \def\kb{\bar{K^0}}
%       \def\al{\alpha}
%       \def\ab{\bar{\alpha}}

%       \def\calm{{\cal M}}
%       \def\calp{{\cal P}}
%       \def\calr{{\cal R}}

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%       \newcommand\sub[1]{_{\rm #1}}
%       \newcommand\su[1]{^{\rm #1}}

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%       \newcommand\supf{^{(f) }}
%       \newcommand\supw{^{(W) }}
%       \newcommand\Tr{{\rm Tr}\,}

%       \newcommand\msinf{M\sub{inf}}

%%%%%%%%%%%%     KO's macros %%%%%%%%%%%%%%%%
\newcommand{\bb}{$B^0$--$\bar{B}^0$}
\newcommand{\kk}{$K^0$--$\bar{K}^0$}
\newcommand{\ek}{$\varepsilon_K$}
\newcommand{\bsbs}{$B_s$--$\bar{B}_s$}
\newcommand{\bdbd}{$B_d$--$\bar{B}_d$}
\newcommand{\dmbs}{$\Delta m_s$}
\newcommand{\dmbd}{$\Delta m_d$}
\newcommand{\dmbsd}{$\Delta m_s/\Delta m_d$}
\newcommand{\bsg}{$b \to s\,\gamma$}
\newcommand{\meg}{$\mu \to e\,\gamma$}
\newcommand{\tmg}{$\tau \to \mu\,\gamma$}
\newcommand{\teg}{$\tau \to e\,\gamma$}
\newcommand{\Bbsg}{$\text{B}(b \to s\,\gamma)$}
\newcommand{\Bmeg}{$\text{B}(\mu \to e\,\gamma)$}
\newcommand{\Btmg}{$\text{B}(\tau \to \mu\,\gamma)$}
\newcommand{\Bteg}{$\text{B}(\tau \to e\,\gamma)$}
\newcommand{\dlt}{$\delta_{13}$}

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\newcommand{\Rn}[1]{{\uppercase\expandafter{\romannumeral#1}}}
% \newcommand{\gsim}
%  {\mathrel{\mbox{\raisebox{-1.0ex}%
%  {$\stackrel{\textstyle >}{\textstyle \sim}$}}}}
% \newcommand{\lsim}%
% {\mathrel{\mbox{\raisebox{-1.0ex}%
% {$\stackrel{\textstyle <}{\textstyle \sim}$}}}}
\newcommand{\ovl}[1]{\overline{#1}}
\newcommand{\wt}[1]{\widetilde{#1}}

\newcommand{\msd}{m_{\wt{d}}}
\newcommand{\msu}{m_{\wt{u}}}
\newcommand{\mn}{m_{\chi^0}}
%\newcommand{\mc}{m_{\chi}}
\newcommand{\mg}{m_{\wt{g}}}

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\newcommand{\psu}{p_{\wt{u}}}
\newcommand{\pn}{p_{\chi^0}}
\newcommand{\pc}{p_{\chi}}
\newcommand{\pg}{p_{\wt{g}}}

%%%%%%%%%%%% end KO's macros %%%%%%%%%%%%%%%%


%%%%% for Bibliography %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newcommand{\Journal}[4]{{#1} {\bf #2} {(#3)} {#4}}
\newcommand{\Journal}[4]{{#1} {\bf #2}, {#4} {(#3)}}
%
\newcommand{\pl}{\sl Phys.~Lett.}
\newcommand{\plb}{\sl Phys.~Lett.~{\bf B}}
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\newcommand{\pr}{\sl Phys.~Rev.}
\newcommand{\pra}{\sl Phys.~Rev.~{\bf A}}
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\newcommand{\mpla}{\sl Mod.~Phys.~Lett.~{\bf A}}
\newcommand{\sjnp}{\sl Sov.~J.~Nucl.~Phys.}
\newcommand{\ibid}{\it ibid.}

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

\newcommand{\epsfile}[1]{\relax}

\begin{document}
\baselineskip 18pt
\begin{titlepage}
\preprintnumber{
\\
August 2002
}
\vspace*{\titlesep}
\begin{center}
{\Large\bf
De-Constraining Supersymmetry from \boldmath{$b\to s\gamma$}?
}
\\
\vspace*{\titlesep}
{
Ken-ichi Okumura and Leszek Roszkowski
}\\
\vspace*{\authorsep}
{\it 
Department of Physics, Lancaster University,
LANCASTER LA1 4YB, England
}\\
\end{center}
\vspace*{\titlesep}
%
\begin{abstract}

We examine the process $B \to X_s \gamma$ in minimal supersymmetry
(SUSY) with general flavor mixing in the squark sector.  We include
all important next--to--leading (NLO) QCD corrections as well as
dominant NLO SUSY effects from the gluino field.
% In addition to including NLO QCD corrections to the SM/2HDM
% contributions, we introduce NLO QCD resummation above the $m_W$
% scale and SUSY threshold corrections from the gluino field.
We find a generic alignment between
the gluino corrections to the down--type quark masses and to
the dominant Wilson coefficients, which,  for $\mu>0$, leads to a
``focusing effect'' of reducing NLO SUSY corrections
relative to the LO.
%We find much sensitivity to the mixings within the
%left--left and between left--right squark sectors, but little in the
%right--right and right--left sectors.
As a result, stringent constraints from $\brbsgamma$ on the gluino
mass can become significantly reduced relative to the minimal flavor
violation scenario, especially at large $\tanb$, even for small values
of flavor mixing parameters.  The case of $\mu<0$ can also become
allowed. Constraints on general flavor mixings also become
significantly relaxed relative to the LO case.

\end{abstract}
\end{titlepage}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent
{\it 1. Introduction}.\
%%%%% Motivation
%Theoretical&Experimental achievement
The radiative decay $B \to X_s \gamma$ provides a powerful tool for
testing various extensions of the Standard Model (SM), such as
supersymmetry. This is because, in $\bsgamma$ ``new physics''effects
can appear at 1--loop level, which is the same where
the lowest--order SM flavor changing neutral current (FCNC) processes
become allowed. ``New physics'' contributions in $\bsgamma$ can therefore be
of comparable size to the SM one.

Early on it was hoped that, the process could therefore
provide a clear hint for new physics~\cite{bbmr90}. However, as
experimental results~\cite{exp} have
reached an error of $\sim10\%$:
\begin{equation}
BR(B \to X_s\gamma)=(3.41 \pm0.36)\times10^{-4},
\label{bsgexptvalue:ref}
\end{equation}
%
and a similar accuracy has also been recently achieved for a SM prediction
at the full NLO QCD level~\cite{gm01,bcmu02}
\begin{equation}
BR(B \to X_s\gamma)=(3.70 \pm0.30)\times10^{-4},
\label{bsgsmvalue:ref}
\end{equation}
it has become clear that, with the two
ranges partially overlapping, little room has been left for
``new physics'' contributions.  This has in turn been used to
impose severe constraints on flavor--violating (FV) interactions beyond
the SM. These are usually related to some scale $\Lambda$ where ``new
physics'' appears. In the case of low--energy SUSY, $\Lambda$ is set
by the SUSY--breaking scale $\msusy$, which is expected to remain
within ${\cal O}(1\tev)$ on the grounds on naturalness.

A SM contribution to $\bsgamma$ involves an 1--loop exchange of the
top quark and the $W^{-}$. In 2--Higgs doublet models (2HDM), one in
addition has a constructive contribution from the charged Higgs
$H^{-}$ (and the unphysical state $\phi^-$) replacing the
$W$--boson. In the SM, a full NLO analysis has been performed in
several stages and recently completed in~\cite{bcmu02}.
%The remaining uncertainty in Eq.~\ref{bsgsmvalue:ref} comes mostly
%from perturbative and non--perturbative effects related to the charm
%quark~\cite{bcmu02}.
A complete NLO analysis
has also been performed in the 2HDM~\cite{2hdnlo,crs98,cdgg98}.

In SUSY, additional 1--loop diagrams come from
%superpartners of the SM fields, and can be comparable in size. The
an exchange of the charged Higgs--stop and the chargino--stop. The latter
contributes constructively (destructively), depending on whether the
Higgs/higgsino mass parameter $\mu$ is negative or positive,
respectively.  Detailed studies of SUSY contributions to $\brbsgamma$
have been performed beyond the
%LO~\cite{cdgg98susy,bmu00,bhgk00,dgg00,cgnw00}. In particular,
LO~\cite{cdgg98susy,bmu00,cgnw00,dgg00}. In particular,
dominant NLO contributions have been calculated~\cite{cgnw00,dgg00}
and shown to be important. They come from two sources: they are
enhanced by factors involving large $\tanb$ - the usual ratio of the
neutral Higgs expectation values, and/or by the logarithm of
$\msusy/\mw$.

The good agreement between the
ranges~(\ref{bsgexptvalue:ref})--(\ref{bsgsmvalue:ref}) has been used
to derive stringent constraints on the mass spectra of the
superpartners in specific popular models, like the Minimal
Supersymmetric SM (MSSM), or its Constrained version (CMSSM).  For
example, in the CMSSM, the case of $\mu<0$ has been shown to be ruled
out, except for very large common gaugino $\mhalf$ and scalar $\mzero$
masses ($\sim$ few~$\tev$), where the SUSY contributions effectively
switch off. For $\mu>0$, stringent lower bounds of a few
hundred~$\gev$ have also been derived on $\mhalf$ and $\mzero$,
especially at large $\tanb$~\cite{cmssmbsgamma}.

Such bounds, even if model dependent, have clear implications for
searches for SUSY in accelerators, for neutralino as dark matter WIMP,
and in other non--accelerator processes.  For example, in dark matter
searches, the constraint from $\bsgamma$ often forbids larger values
of the spin--independent scattering cross section of WIMPs on proton
$\sigma^{SI}_p$, which would otherwise be accessible to today's
experimental sensitivity~\cite{knrr1}.  Generally, it has been concluded that,
suppressing SUSY contributions to $\bsgamma$ can be achieved by taking
larger $\msusy$, and therefore typically larger masses of
superpartners.

The above conclusions have often been reached by assuming, sometimes
implicitly, that the mixings among the mass eigenstates of
supersymmetric partners of the quarks closely resemble the CKM
structure of the quark sector. This scenario, often called Minimal
Flavor Violation (MFV), can be considered as among the simplest ways
of addressing the nagging flavor problem of SUSY.  FV interactions are
in general not forbidden by gauge symmetry and, without assuming any
organizing principle, can lead to exceeding experimental bounds by
even many orders of magnitude~\cite{agis02}.  The mixings of the first
two generations of squarks are strongly constrained by the $K^0-\bar
K^0$ system but bounds on sfermions of the third generation are
much weaker. In the case of $\bsgamma$, rather stringent constraints
have been derived, in the LO approximation, on the magnitude of
possible FV terms beyond MFV in specific scenarios for flavor
structure, but shown to be highly model--dependent~\cite{gluino,ekrww02}.
The question thus arises
about the robustness of various implication for SUSY derived in the
context of MFV or other specific scenarios.

In this Letter, we examine the process $\bsgamma$ in the framework of
the MSSM with {\em general} flavor mixing (GFM) in the down--type
squark sector. In addition to including the full NLO QCD corrections
from the SM+2HDM case~\cite{bcmu02}, we introduce NLO QCD resummation
above $\mw$ and threshold corrections from the heavy gluino field
taking into account all the sources of flavor mixings in the
down--type sector.  In SUSY, the NLO--level analysis has so far been
done in the case of MFV~\cite{cgnw00,dgg00}. On the other hand, most
of analyses of GFM have been conducted with only LO matching
conditions for SUSY contributions~\cite{gluino}, in which case the
effect of NLO--level corrections becomes an issue. We have calculated
all $\tanb$--enhanced NLO contributions in the framework of GFM, as
discussed below.  This level of accuracy allows us to compare SM+SUSY
contributions to $\brbsgamma$ in the case of GFM with analogous
predictions obtained with MFV and in LO.

We find that, at the  level of NLO, even small departures from the MFV
scenario often lead to considerably relaxed constraints on the gluino
mass, especially in the case of large $\tanb$ and $\mu>0$, and that
even the case of $\mu<0$ becomes again readily allowed. Furthermore,
we demonstrate that some constraints on FV terms become considerably
weaker than in the LO case.  We identify the underlying ``focusing
effect'' which generally reduces the gluino NLO contributions to
$\brbsgamma$.


We will first summarize the procedure which we follow. We describe LO
and NLO corrections and matching conditions in the SM and in SUSY with
MFV. Next we identify and compute new leading $\tanb$--enhanced
contributions which arise beyond the MFV framework. We then identify
an alignment between the gluino contribution to $\brbsgamma$ and the
down--type quark mass--matrix which leads to the focusing effect
mentioned above. Finally, we illustrate our results in the framework
of the Constrained MSSM.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.15cm}
%\noindent
{\it 2. Procedure}.\
The contributions to $\bsgamma$ from all states that are much heavier
than $m_b$ are described by the effective Lagrangian~\cite{cmm98}
%\beq
%{\cal L} = \frac{4 G_F}{\sqrt{2}} V_{ts}^*V_{tb}
% \sum^8_{i=1}\left(\Ci(\mu){\it O}_{i}+\Cip(\mu){\it
%O}_{i}^{\prime}\right).
%\eeq
\beq
{\cal L} = \frac{4 G_F}{\sqrt{2}} V_{ts}^*V_{tb}
 \sum^8_{i=1}\left(\Ci(\mu){\it P}_{i}+\Cip(\mu){\it
P}_{i}^{\prime}\right).
\eeq
The Wilson coefficients $C_i(\mu)$ and $C_{i}^{\prime}(\mu)$, which are
associated with operators ${\it P}_{i}$ and their chirality--conjugate
partners ${\it P}_{i}^{\prime}$, play the
role of effective coupling constants. In the SM, it is natural to
choose the scale at which the heavy (SM) states decouple at
$\muw\simeq\mw$.  Their values at $\muw$ are
found by imposing so-called matching conditions
% by comparing Greens' functions of
between the effective and the underlying theory.
% in the limit of external momenta much smaller than the masses of the
% decoupling heavy states.
One then evaluates the Wilson coefficients at $\mub\sim m_b$ from the
running of the Renormalization Group Equations (RGEs)
%RGEs
from $\muw$ down to $\mub$. In order to remove the
dependence on the scale $\mu_b$, at $\muw$ NLO matching conditions
need to be imposed~\cite{bmmp94}. They have been computed in full
in~\cite{NLOmatchingSM,cdgg98,bmu00}, and we have followed~\cite{gm01}
to apply them here.


``New physics'' effects usually appear in the Wilson coefficients
$C_{7,8}$ and $C_{7,8}^{\prime}$, which are associated with the
$\Delta B=1$ magnetic and chromo--magnetic operators $P_{7,8}$ and
$P_{7,8}^{\prime}$, respectively.
In the MFV case, as well as in the SM/2HDM, one usually
neglects $C_{7,8}^{\prime}$ as suppressed by the ratio $m_s/m_b$. This
can no longer be done in the case of GFM, where
the gluino contributions to $C_{7,8}^{\prime}$ are of a
similar strength~\cite{gluino}.
%The limit of the contribution to $\bsgamma$ from SUSY coming only
%from $C_{7,8}^{\prime}$ has been considered in~\cite{ekrww02}.


In SUSY, the new mass scale $\msusy$ arises, which can be much
larger than $\mw$. The usual procedure in this case is to assume that
$\msusy$ sets the mass scale for all the color--carrying
superpartners, which tend to be heavier, while all the other SUSY
states play a role at $\mw$~\cite{dgg00}.
To include QCD corrections beyond LO,
we extend the treatment of~\cite{cdgg98susy,dgg00} to the case of general
flavor mixing among squarks, and follow~\cite{bmu00} to
compute hard gluon contributions to the vertices involving the
$W$--boson, $H^-$, and $\phi^-$ at $\mw$.  Resummation of
QCD correction between the two scales is performed with an NLO
anomalous dimension~\cite{cmm98}
and NLO matching
condition~\cite{bmu00} within the SM operator basis.
At the NLO--level, SUSY QCD contributions come from ${\cal
O}(\alpha_s)$ corrections involving the gluino field.  For the states
at $\muw$ ($W$, $\phi^-$ and $H^-$), these are absorbed into the
effective vertices by integrating out the gluino
field~\cite{cdgg98susy}. We have
calculated these vertices for GFM in the down--squark sector.

For the fields at $\mususy$, this effective approach cannot be
applied.  Instead of calculating full 2--loop diagrams (a rather
formidable task), we include a finite threshold corrections to the
matrix of the Yukawa couplings of the down--type (up--type) quarks
which are (are not) enhanced in the case of large
$\tanb$~\cite{bottomcorr,cgnw00,dgg00}.
In our case, it is
again crucial to include the effect of flavor mixing.  Details of the
calculation will be presented elsewhere~\cite{or2}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.15cm}
\noindent
{\it 3. General Flavor Mixings}.  In the absence of an underlying
theory of flavor in the squark sector, all the entries of the
$6\times6$ mass--matrix--square of down--type squarks are in general
non--zero (and likewise for the up--type squarks).
It is natural
to break it into four $3\times3$ sub--matrices of the $LL$, $LR$, $RL$
and $RR$ sectors. For the mixing of the 2nd and 3rd generations of
relevance to $\bsgamma$, the departure from the MFV case can be
parameterized by introducing
\beq
\deltadll= \frac{(m_Q^2)_{23}}{\sqrt{(m_Q^2)_{22} (m_Q^2)_{33}}}~~~
\deltadlr= \frac{m_b(A_d)_{32}^*}{\sqrt{(m_D^2)_{33}(m_Q^2)_{22}}},
\eeq
and analogously for $\deltadrl$ and $\deltadrr$. Above,
$\left(m^2_{Q,D}\right)_{2,3}$ denotes the $(2,3)$-element of the soft
mass matrix $m^2_{Q,D}$, while $(A_d)$ stands for the $3\times3$ soft
trilinear term matrix.


In the framework of GFM, these mixings induce new 1--loop
contributions to $\bsgamma$, with one internal line involving the
gluino (and neutralino) and the other from the superpartner of the
bottom turning into the one of the strange quark due to the 2nd--3rd
generation mixing. We insert the $\delta^d$s at $\mususy$ and all
the corrections described above are generalized to the case of
GFM. The $6\times6$--mass matrices are diagonalized numerically,
instead of using a mass-insertion approximation.

\vspace{0.15cm}
\noindent
{\it 4. Focusing Effect}.  We find that, relative to the LO matching
condition at $\muw$, NLO corrections generally reduce SUSY
contributions (mostly from the gluino) to $\brbsgamma$. As a result,
constraints on the mass of the gluino, and related SUSY parameters
(like the chargino and neutralino soft mass parameters), and on FV
couplings become considerably relaxed. The {\em focusing effect}
becomes particularly strong at large $\tanb$. It is illustrated in
Fig.~\ref{brmhalft40ldfmup:fig} in the framework of the CMSSM. We
present the ranges of the $\brbsgamma$ as a function of $\mhalf\simeq
0.35 \mgluino$ in the case of MFV ($\deltadlr=0$) and for
$\deltadlr=\pm 0.02$. We scan over the ranges of $\mzero<1\tev$ and
apply collider bounds on superpartner masses. The blue (darker) bands
are obtained by using our expressions for dominant NLO--level
contributions, while the green (light) bands correspond to the
approximation of LO matching conditions. One can clearly see a strong
suppression of the SUSY contribution at the NLO--level. While some
focusing is already present in the MFV case, the effect becomes
strongly enhanced in the case of GFM.


Focusing comes from two sources. Firstly, RGE evolution between
$\mususy$ and $\muw$ reduces SUSY contributions. For example, for
$\msusy\sim1\tev$, one finds $C_7(\muw)\simeq 0.8 C_7(\mususy) +0.06
C_8(\mususy)$ and $C_8(\muw)\simeq 0.8 C_8(\mususy)$.  The NLO QCD
matching condition at $\mususy$ causes some additional reduction.
Secondly, we find a remarkable alignment between gluino correction to
$m_b$ (and $m_s$)~\cite{hrs} and the corresponding penguin operator at
large $\tanb$.  This reduces the gluino contribution to $C_{7,8}$ at
$\mu>0$ and, as a result, SUSY contribution to $\brbsgamma$ in the
case of GFM. The essential point is that, flavor mixing in the soft
SUSY--breaking terms induces flavor off-diagonal elements in the quark
propagator.  To renormalize it so that it keeps flavor diagonal form,
one needs to choose a counter term $\Delta m_{ij}$ ($i,j=d,s,b$) for
the quark mass matrix.  This counter term induces ${\cal O}(\alpha_s)$
correction to $C_{7,8}$ at $\mususy$.
The correction to $C_7$ exhibits a strong correlation with
the initial gluino contribution to $C_7$ in GFM.  For $C_8$, a
similar but milder correlation exists.
The effect is enhanced by
large $\tanb$. For example, $\deltadll$ (or $\deltadlr$) induces a
mass insertion $-\Delta m_{bs}^*$ in the $b_R-s_L$ quark line. This
generates a diagram with an exchange of the $\widetilde c_L$ and the
chargino $\chi^-$, and another one with the $\widetilde b_R -
\widetilde s_L$ (with mass insertion $-\Delta m_{bs}^*$) and the
gluino in the loop. Both are proportional to $\tanb$ and reduce the LO
effect in $C_{7,8}$ at $\mu>0$.  A more technical explanation will be
provided in~\cite{or2}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
\begin{figure}[t!]
\vspace*{-0.4in}
\hspace*{-.70in}
\begin{center}
\begin{minipage}{3in} %{3.5in}
\centerline{\psfig{figure=or1-brmhalf-t40-dlr-mup.eps, angle=0,height=3in}}
%\epsfig{file=or1-brmhalf-t40-dlr-mup.eps,height=3in}
\end{minipage}
\caption{\label{brmhalft40ldfmup:fig} {\small We plot $\brbsgamma$
\vs\ $\mhalf$ in the Constrained MSSM for $\deltadlr=\pm0.02$ and $0$
(MFV). For each case we show the bands obtained by varying $\mzero$ in
the approximation of using only LO matching conditions (green) and of
including NLO corrections (blue). The SM prediction and $2\sigma$~CL
experimental limits are also marked.
}
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

If we take $\deltadrr$ (or $\deltadrl$) instead, analogous diagrams
are induced for the $b_L$ to $s_R$ transition, which
reduces the gluino contribution at $\mu>0$ in a similar manner.
In the case of $\mu<0$, the corrections described above add
constructively and cause a miss--alignment instead.
This competes with the reduction caused by the RGE evolution.

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


\vspace{0.15cm}
\noindent
{\it 5. Relaxation of Constraints on SUSY and FV Terms}.  The
focusing effect leads to striking implications for SUSY contributions
to $\brbsgamma$ in the case of GFM. We will illustrate this again the
the framework of the CMSSM.  In the four windows of
Fig.~\ref{dellrvsmhalf:fig} we show the contours of the $\brbsgamma$
as a function of $\mhalf$ and $\deltadlr$ (upper left and right) and
$\deltadll$ (lower left and right), for $\tanb=40$, $\mzero=500\gev$,
$A_0=0$ and $\mu<0$ ($\mu>0$). The light green and dark yellow bands
agree with the experimental range~(\ref{bsgexptvalue:ref}) at the
$1\sigma$ ($2\sigma$)~CL. Larger departures are denoted as
``excluded''. In the case of $\deltadlr$, in MFV one finds
$\mgluino\gsim600\gev$ ($\mhalf\gsim200\gev$) at the $1\sigma$~CL for
$\mu>0$.  In contrast, even relatively small non--zero values of
$|\deltadlr|\simeq0.01$ remove the constraint from $\bsgamma$
altogether.  The effect is even more striking for negative $\mu$ where
relatively light gluino becomes allowed again.  
We have numerically checked that, relative to
LO, constraints on FV terms at NLO become also considerably relaxed. For
example, for the case of Fig.~\ref{dellrvsmhalf:fig}, at
$\mhalf=500\gev$ and $\mu>0$, we find
$\deltadlr(NLO)/\deltadlr(LO)\simeq7.5$~\cite{or2}.
Similar de--constraining can also be obtained by introducing
non--zero $\deltadll$s instead, both for positive and negative $\mu$.

It is interesting to note the existence of two branches of allowed
solutions in each window of Fig.~\ref{dellrvsmhalf:fig}.  By
simultaneously allowing for more than one $\delta^d$ to become
non--zero, one can relax the constraint from $\bsgamma$ even further,
even at LO~\cite{gluino}. In addition, in general FV contributions can
also come from the up--squark sector~\cite{up-mixing}. We have checked
that its effect is numerically less important.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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

\begin{figure}%[t!]
\vspace*{-0.3in}	%{-0.25in}
%	\hspace*{-.70in}
\begin{center}
\begin{minipage}{9.2cm} %{8.6cm}
\centerline{\hspace*{-.2cm}\psfig{figure=or1-t40-dlr-mun.eps, angle=0,width=4.6cm} %4.3cm}
	    \hspace*{-.3cm}\psfig{figure=or1-t40-dlr-mup.eps, angle=0,width=4.6cm}
	    %4.3cm}
}
\end{minipage}
\end{center}
\vspace*{-0.2in}
\begin{center}
\begin{minipage}{9.2cm} %{8.6cm}
\centerline{\hspace*{-.2cm}\psfig{figure=or1-t40-dll-mun.eps, angle=0,width=4.6cm} %4.3cm}
	    \hspace*{-.3cm}\psfig{figure=or1-t40-dll-mup.eps, angle=0,width=4.6cm}
	    %4.3cm}
}
\end{minipage}
\caption{\label{dellrvsmhalf:fig} {\small We plot contours of $\brbsgamma$
\vs\ $\mhalf$ and $\deltadlr$ (upper windows) and $\deltadll$ (lower
windows) for $\mu<0$ (left side) and $\mu>0$ (right side)  in the
Constrained MSSM. Regions (in grey)
beyond the $1\sigma$ (green) and $2\sigma$~CL (dark yellow) agreement with
experiment (Eq.~(\ref{bsgexptvalue:ref})) are marked ``excluded''.
}}
\end{center}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.15cm}
\noindent
{\it 6. Conclusions and Outlook}.
Contributions from ``new physics''
to rare processes like $\bsgamma$, do not necessarily have to be
suppressed by the largeness of the effective energy scale beyond the
SM. In the case of SUSY, we have pointed out the existence of the
alignment mechanism which, at the NLO level, leads to reducing SUSY
contributions to $\brbsgamma$, with most sensitivity to
flavor mixing terms $\deltadll$ and $\deltadlr$.  The end result is that of
significantly relaxing the constraints from $\brbsgamma$ on the mass
of the gluino and also other SUSY parameters, depending on a model. It
appears that the alignment mechanism is actually generic to processes
involving (chromo--) magnetic operators. We are in the process of
exploring its role in other processes, like $\mu \to e \gamma$ or $B
\to X_s ll$, for which stringent constraints on ``new physics''
contributions either exist or will soon become available.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\acknowledgments
\section*{Acknowledgments}
We would like to thank P.~Gambino, M.~Misiak and L.~Silvestrini for
helpful comments.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
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\end{document}




 



























