Weakened Constraints from \boldmath{$b\to s\gamma$} on Supersymmetry Flavor Mixing Due\\ to Next--To--Leading--Order Corrections
Ken-ichi Okumura
We examine the process $B \to X_s \gamma$ in minimal supersymmetry (SUSY) with general squark flavor mixings. We include all relevant next--to--leading order (NLO) QCD corrections and dominant NLO SUSY effects from the gluino. We find that gluino--squark corrections to down--type quark masses induce large NLO corrections to the dominant Wilson coefficients whose size is often similar to those at LO, especially at large $\tanb$. For $\mu>0$, destructive interference, and suppression by the renormalization group running lead to a ``focusing effect'' of reducing the size of gluino corrections to the branching ratio, and also of reducing the LO sensitivity to flavor mixings among squarks. Constraints from $\brbsgamma$ on the SUSY--breaking scale can become significantly weakened relative to the minimal flavor violation case, even, at large $\tanb$, for small flavor mixings. The case of $\mu<0$ can also becomes allowed.
or1-hep-resubmit.tex
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\begin{document}
\draft
%\twocolumn[\hsize\textwidth\columnwidth\hsize\csname @twocolumnfalse\endcsname
 \preprint{%\\ {}~hep-ph/9905212
}
%]


\title{Weakened Constraints from \boldmath{$b\to s\gamma$} on
  Supersymmetry Flavor Mixing  Due\\ to Next--To--Leading--Order Corrections}
% \title{Weakened Constraints from \boldmath{$b\to s\gamma$} on
%  Supersymmetry Flavor Mixing\\  Due to NLO Corrections}
%De-Constraining Supersymmetry from \boldmath{$b\to s\gamma$}?}

\author{Ken-ichi Okumura}
%\footnote{Previously at Lancaster University.} 
\altaffiliation
{Previously at Lancaster University.}
\affiliation{Department of Physics, KAIST, Daejeon, 305-701, Korea}
\author{Leszek Roszkowski}
%\address
\altaffiliation
{Previously at Lancaster University.}
\affiliation{Department of Physics and Astronomy, University of Sheffield, 
Sheffield, UK}

%\email[]{Your e-mail address}
%\homepage[]{Your web page}
%\thanks{}
%\altaffiliation{}

%\date{\today}

\begin{abstract}

We examine the process $B \to X_s \gamma$ in minimal supersymmetry
(SUSY) with general squark flavor mixings.  We include
all relevant next--to--leading order (NLO) QCD corrections and
dominant NLO SUSY effects from the gluino.
We find that gluino--squark corrections to down--type quark masses 
induce large NLO corrections to the dominant Wilson
coefficients whose size is often similar to those at LO,
 especially at large $\tanb$.
For $\mu>0$, destructive interference, and suppression by the
renormalization group running 
lead to a ``focusing effect'' of reducing 
the size of 
gluino corrections to the branching ratio, and also of
reducing the LO sensitivity to flavor mixings among squarks.  
 Constraints from $\brbsgamma$ on the SUSY--breaking
scale 
can become significantly weakened relative to the minimal flavor
violation case, even, at large $\tanb$, 
for small flavor mixings. 
The case of $\mu<0$ can also becomes allowed.
\end{abstract}

\pacs{PACS: 12.60.5.Jv, 13.20.He, 12.15.Ff}


\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\noindent
{\it 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 a one--loop level, which is the same as the lowest--order SM
flavor changing neutral current (FCNC), and can therefore be of
comparable size~\cite{bbmr90}.  However, as both experimental
results~\cite{exp} and a SM prediction at the full 
next--to--leading--order (NLO) QCD
level~\cite{gm01,bcmu02} have reached an error of $\sim10\%$,
\begin{eqnarray}
\label{bsgexptvalue:ref}
{\cal B}(B \to X_s\gamma)_{\rm exp} &=&(3.34 \pm0.38)\times10^{-4},\\
{\cal B}(B \to X_s\gamma)_{\rm SM} &=&(3.70 \pm0.30)\times10^{-4},
\label{bsgsmvalue:ref}
\end{eqnarray}
it has become clear that 
little room is now 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 $\Lambda$ at which new
physics appears. In the case of low--energy supersymmetry (SUSY),
 $\Lambda$ is set
by the SUSY--breaking scale $\msusy$, which is expected to remain
within ${\cal O}(1\tev)$ on the grounds of naturalness.

The SM contribution to $\bsgamma$ involves a one--loop exchange of the
top quark and the $W^{-}$. % (and the unphysical state $\phi^-$). 
In 2--Higgs doublet models (2HDM), 
there is, in addition, a constructive contribution from the charged Higgs
$H^{-}$. %lr replacing the $W$--boson.
%***
A complete NLO analysis  in the SM has been performed in
several stages and recently completed in~\cite{bcmu02} and, likewise,
in the 2HDM~\cite{bg98,crs98,cdgg98}.
%The remaining uncertainty in Eq.~\ref{bsgsmvalue:ref} comes mostly
%from perturbative and non--perturbative effects related to the charm
%quark~\cite{bcmu02}.
%lr A complete NLO analysis
%lr has also been performed in the 2HDM~\cite{bg98,crs98,cdgg98}.

In SUSY, additional one--loop diagrams come from
%superpartners of the SM fields, and can be comparable in size. The
an exchange of the charged Higgs--top $H^{-}$--$t$ and the
chargino--stop $\charginom$--$\wt{t}$.  The latter contributes
constructively (destructively), depending on whether the
Higgs/higgsino mass parameter $\mu>0$ or $<0$, %is negative or
 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 
%lr come from two sources: they 
are
enhanced by factors involving large $\tanb=\langle
H^0_2\rangle/\langle H^0_1\rangle$ [the ratio of the neutral Higgs 
VEVs] and $\log\left(\msusy/\mw\right)$.

The good agreement between the
ranges~(\ref{bsgexptvalue:ref}) and (\ref{bsgsmvalue:ref}) has also been used
to derive stringent constraints on the mass spectra of the
superpartners in specific popular models, like the minimal
supersymmetric SM (MSSM), or the constrained MSSM (CMSSM).  For
example, in the CMSSM, the case of $\mu<0$ has been shown to be ruled
out, except for the very large common gaugino $\mhalf$ and scalar $\mzero$
masses ($\sim$ few~$\tev$), where SUSY contributions become tiny. 
%lr 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 the neutralino as a dark matter
weakly interacting massive particle (WIMP),
and in other nonaccelerator 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
$\sigma^{SI}_p$ that would otherwise be accessible to today's
experimental sensitivity~\cite{knrr1}.  
%lr Generally, it has been concluded that 
%lr %lr suppressing SUSY contributions to $\bsgamma$ can be achieved 
%lr ** one can suppress SUSY contributions to $\bsgamma$ 
%lr by taking larger $\msusy$, and therefore typically larger masses of
%lr superpartners.

The above conclusions have often been reached by assuming, sometimes
implicitly, that the mixings among the mass eigenstates of
squarks 
% *** supersymmetric partners of the quarks 
closely resemble the Cabibbo--Kobayashi--Maskawa (CKM)
structure of the quark sector. This scenario, often called minimal
flavor violation (MFV), 
is perhaps the simplest way
% *** 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
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 mixings with 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{bghw99,ekrww02}.
The question thus arises
about the robustness of constraints on SUSY derived in 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~\cite{bcmu02,bg98,crs98,cdgg98}, we introduce NLO QCD
resummation above $\mw$ and threshold corrections from the heavy
%lr gluino field %*** by taking into account all the sources of flavor
%lr mixings in the down--squark sector.
gluino field $\gluino$ %*** by taking into account all the sources of flavor
in the presence of all mixings in the down--squark 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{bghw99}, 
% in which case the
In contrast to MFV~\cite{cgnw00,dgg00}, most
analyses of the GFM have been done with only LO matching
conditions (MCs) for SUSY contributions~\cite{bghw99}, in which case 
%*** 
NLO effects become an issue. In order to be able to compare
with the SM+SUSY contributions to $\brbsgamma$ in MFV and in LO, 
we calculate all $\tanb$--enhanced NLO contributions in the framework of GFM
and show that they lead to dramatic implications.
%lr  for SUSY.  
%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
%$\brbsgamma$.
%
%*** NEW, KEEP? *** We find that even small departures from the MFV scenario often lead to
%considerably relaxed constraints on the gaugino mass and that even the
%case of $\mu<0$ becomes again readily allowed.  Furthermore, we
%demonstrate that, at the level of NLO, some constraints on FV terms
%become considerably weaker than in the LO case, especially in the case
%of large $\tanb$ and $\mu>0$.  We identify the underlying ``focusing
%effect'' which generally reduces the gluino NLO contributions to
%$\brbsgamma$.
%
%We will first summarize our 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 Procedure}.---
%\
The contributions to $\bsgamma$ from all states that are much heavier
than $\mb$ are described by the effective Lagrangian~\cite{gm01}
%\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 matching conditions
% by comparing Greens' functions of
between an effective and an underlying theory.
% in the limit of external momenta much smaller than the masses of the
% decoupling heavy states.
One evaluates the $C_{i}^{(\prime)}$s
% ***Wilson coefficients 
at $\mub\sim \mb$ 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 MCs %lr 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 with MCs at $\mub$~\cite{ghw96}.

New physics effects usually appear in the Wilson coefficients
$C_{7,8}$ and $C_{7,8}^{\prime}$
%, which are associated with *** 
of the 
%lr $\Delta B=1$ 
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 $\ms/\mb$. 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{bghw99}.
%The limit of the contribution to $\bsgamma$ from SUSY coming only
%from $C_{7,8}^{\prime}$ has been considered in~\cite{ekrww02}.

% In SUSY, *** there is a new mass scale $\msusy$, 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}.
% 
% *** NEW 
% 
In SUSY, the new mass scale $\msusy$ is set by $\mgluino$ and the
squark mass $\msquark$, which tend to be heavier than the other
states. (We assume no large mass splittings among the squarks.) It is
reasonable to expect a hierarchy $\mususy\gg\muw$ just like one has
$\muw\gg\mub$. In order for QCD corrections beyond the LO to match
the underlying SUSY theory and the effective theory at $\muw$, we
extend the treatment of~\cite{cdgg98susy,dgg00} to the case of GFM.
The usual procedure, which we follow here, is to assume that
$\mgluino\sim\msquark\sim\mususy$, while all the other SUSY states
play a role at $\muw$~\cite{dgg00}.  We thus do not resum the
logarithms of $m_H/\muw$ and $m_{\wt{q}(\wt{g})}/\mususy$.  The
resummation of QCD correction between $\mususy$ and $\muw$ is
performed with the NLO anomalous dimensions with six quark
flavors~\cite{misiakmunz95}.
% and validity of our results are up to them.  
We compute two--loop gluon corrections to the $H^{-}$ contributions to
$C_{4,7,8}$ at $\muw$, and those from the $\charginom$, $\neut$ and
$\gluino$ fields to $C_{7,8}^{(\prime)}$ at $\mususy$ as
in~\cite{bmu00}.  
%We have verified that their contributions to
%$C_{1-6}^{(\prime)}$ are numerically negligible 
We have verified that their contributions to $C_{1-6}^{(\prime)}$ and
their mixing to $C_{7,8}^{(\prime)}$ are numerically negligible when using
available five--flavor anomalous dimensions~\cite{cmm98}.
%%in the case with 5--flavor anomalous dimensions~\cite{cmm98}.
%This is in contrast to the SM where a tree--level
%contribution from $W^{-}$ to $C_2$ plays a dominant role 
%through the mixing with these operators.  
We also neglect operators outside of the SM basis.  At LO
they are numerically subdominant~\cite{bghw99},
% are also beyond the scope of this paper.   
and we could not identify any enhancement
mechanism beyond the LO.  At the NLO--level, SUSY QCD contributions come
from ${\cal O}(\alpha_s)$ corrections involving the gluino.  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 with 
GFM in the down--squark sector~\cite{or2}.

For the fields at $\mususy$, this effective approach cannot be
applied.  Instead of calculating a full set of two--loop diagrams (a
rather formidable task), we include finite threshold corrections to
Yukawa matrices, which become important at large
$\tanb$~\cite{bottomcorr,cgnw00,dgg00}.  As described in detail
in~\cite{or2}, we first integrate out the gluino and the squarks to calculate
effective quark mass matrices in the super-CKM (SCKM) basis. By identifying these with
physical ones, we extract the Yukawa couplings that appear in the higgsino--quark--squark
vertices and in the F--term contributions to squark mass matrices, and
further perturbatively include ``$\tan\beta$
resummation''~\cite{cgnw00,dgg00}, generalized to GFM~\cite{or2} for
self--consistency.
%lr To achieve self--consistency and to perturbatively include the effect of
%lr ``$\tan\beta$ resummation''~\cite{cgnw00,dgg00}, we use numerical
%lr iteration first identifing the ``bare'' quark matrices with physical
%lr ones and repeating the above steps with the resultant mass matrices
%lr until they converge~\cite{or2}.
%They are identified as
%counter terms for the Yukawa couplings in view of perturbative
%renormalization and inserted in the higgsino--quark--squark
%vertices~\cite{cgnw00,dgg00} and in the F--term contrubutions to the squark
%mass matrices.  
%lr Again, it is crucial to include the effect of GFM.  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{0.15cm}
%\noindent
{\it 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
nonzero (and likewise for the up--type squarks).
It is natural
to break it into four $3\times3$ submatrices of the $LL$, $LR$, $RL$
and $RR$ sectors. For the 2nd to 3rd generation mixing 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{\mb(A_d)_{32}^*}{\sqrt{(m_D^2)_{33}(m_Q^2)_{22}}},
% \eeq
%lr $\deltadll= (m_Q^2)_{23}/{\sqrt{(m_Q^2)_{22} (m_Q^2)_{33}}}$ and
%lr $\deltadlr= {v_d (A_d)_{23}^*}/{\sqrt{(m_D^2)_{33}(m_Q^2)_{22}}}$,
%lr %$\deltadlr= {\mb(A_d)_{32}^*}/{\sqrt{(m_D^2)_{33}(m_Q^2)_{22}}}$,
$\deltadll=
%\left
(m^{2}_{d,LL}
%\right
)_{23}/{\sqrt{
%\left
(m^{2}_{d,LL}
%\right
)_{22}
%\left
(m^{2}_{d,LL}
%\right
)_{33}}}$ and 
%lr $\deltadlr= {v_d \left(A_d\right)_{23}^{\ast}}
$\deltadlr= 
%\left
(m^{2}_{d,LR}
%\right
)_{23}
  /{\sqrt{
%\left
(m^{2}_{d,LL}
%\right
)_{33}
%\left
(m^{2}_{d,RR}
%\right
)_{22}}}$,
%$\deltadlr= {\mb(A_d)_{32}^*}/{\sqrt{(m_D^2)_{33}(m_Q^2)_{22}}}$,
and analogously for $\deltadrl$ and $\deltadrr$, where
$m^{2}_{d,LL(RR,LR)}$ are, respectively, the $3\times3$ soft mass matrices
$m^2_{Q(D)}$ and the soft
trilinear term matrix $A_d^{\ast}$ rotated
into the SCKM basis~\cite{or2}.

In the case  
% *** framework 
of GFM, these mixings induce new 1--loop
contributions to $\bsgamma$, with one internal line involving
$\gluino$ (or $\neut$) and the other from 
%superpartner of the bottom ***
$\sbottom$ turning into 
% ***
%lr one of the strange quark 
$\sstrange$
due to the 2nd to 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 the less accurate mass--insertion approximation.

\vspace{0.15cm}
%\noindent
{\it Focusing effect}.---  We find that, relative to the LO MC 
%lr matching condition 
at $\muw$, NLO corrections generally reduce SUSY
contributions (mostly from the gluino) to $\brbsgamma$. As a result,
constraints on $\mgluino$
% *** the mass of the gluino, 
% *** and related SUSY parameters (like the chargino and neutralino soft mass parameters), 
and related SUSY parameters (like the $\chargino$ and $\neut$ soft mass parameters), 
and on the FV couplings become considerably relaxed. This {\em focusing effect}
becomes particularly strong at large $\tanb$. This is illustrated in
Fig.~\ref{brmhalft40ldfmup:fig}. 
We present ranges of the $\brbsgamma$ as a function of 
the common squark mass, $m_{\widetilde{q}}\,\equiv \,\sqrt{(m^2_{Q,U,D})_{ii}}$
%$\mhalf\simeq 0.35 \mgluino$ 
in the case of MFV ($\deltadlr=0$) and for
$\deltadlr=\pm 0.02$. We scan over the ranges of
$1<m^2_{\widetilde{g}}/m^2_{\widetilde{q}}<2$
% $\mzero<1\tev$
and
apply collider bounds on superpartner masses. The brown (darker) bands
are obtained by using our expressions for dominant NLO--level
contributions, while the green (light) bands correspond to applying the
approximation of the LO matching at $\muw$. 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.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
\begin{figure}[b!]
\vspace*{-0.2in}
%\vspace*{-0.4in}
%\hspace*{-.70in}
\begin{center}
\begin{minipage}{3in} %{3.5in}
\centerline{
%\psfig{figure=or1-brmhalf-t45-dlr-mup.eps, angle=0,height=2.8in}
\resizebox*{2.5in}{2.3in}{\includegraphics{or1-brmhalf-t45-dlr-mup.eps}}
}
% \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
(color online).
 We plot $\brbsgamma$
\vs\
$m_{\widetilde{q}}$ 
%$\mhalf$ in the Constrained MSSM 
for $\deltadlr=\pm0.02$ and $0$ (MFV). 
For each case we show the bands obtained by varying
$m^2_{\widetilde{g}}/m^2_{\widetilde{q}}$
% $\mzero$ 
in the approximation of using only LO MCs (green/light grey) and of
including NLO corrections (brown/dark grey). 
%Remaining parameters are fixed as $A_u=-m_{\widetilde{q}}$,
% $M_{1,2}=m_{\widetilde{q}}$ and $m_{H^-}=0.5\,m_{\widetilde{q}}$.
The SM value and $2\sigma$~C.L.
experimental limits are also marked.
}
}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Focusing comes from two sources. First, RGE evolution between
$\mususy$ and $\muw$ generally reduces SUSY contributions. For example, for
$\msusy\sim1\tev$, 
one finds $\alpha_s(\mususy)/\alpha_s(\muw)\simeq 0.8$, 
$C_7(\muw)\simeq 0.8 C_7(\mususy) +0.06
C_8(\mususy)$ and $C_8(\muw)\simeq 0.8 C_8(\mususy)$.  
%lr **ko to lr* 
%lr This correction gives as the same order of reduction for gluino
%lr contributions to $C_{7,8}$ as their own running and can not be ignored.
%lr  ***
(The NLO QCD matching condition at $\mususy$ causes some additional
reduction.)  Second, at $\mu>0$ we find a remarkable tendency of
``alignment'' between the effective $b$--quark mass
%gluino correction to
%$\mb$ (and $\ms$)~\cite{hrs}
and the corresponding penguin operator at large $\tanb$.  This reduces
the gluino contribution to $C_{7,8}$ and, as a 
result, the 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
effective quark mass matrices through gluino--squark loops.  In
order to rediagonalize them, one also rotates the Yukawa couplings
that appear in the higgsino vertices and squark mass matrices, thus inducing 
${\cal O}(\alpha_s)$ corrections to them in SCKM basis.  The resultant
$C_{7,8}$ is also ``rotated''.
%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 $C_{7,8}$ exhibits a strong correlation with
%the initial gluino contribution to $C_{7,8}$ 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 $\mu>0$.  
For example, $\deltadlr$ (or
$\deltadll$) induces a correction  
proportional to 
$\delta m_{bs}^*/\cos \beta$
%$-\Delta m_{bs}^*/\cos \beta$
in the higgsino vertex.
This generates a diagram with an exchange of $\widetilde c_L$ and
$\chi^-$, and another one with the $\widetilde b_R$--$\widetilde s_L$
(with mass insertion 
$\delta m_{bs}^*\mu \tan \beta$
%$-\Delta m_{bs}^*\mu \tan \beta$
) and the gluino
in the loop, as shown in Fig.~\ref{fdiags:fig}. Both are proportional
to
$\tan \beta$ 
%$\tan^2 \beta$
and the rotation occurs in the direction that
reduces the LO effect at $\mu>0$. 
% and reduce the LO 
%effect in $C_{7,8}$ at $\mu>0$.  
%More details will be provided in~\cite{or2}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\begin{figure}[b!]
%\vspace{-0.35in} %{-0.3in}	%{-0.25in}
%	\hspace*{-.70in}
\begin{center}
\begin{minipage}{9.2cm} %{8.6cm}
\centerline{
%\hspace*{-.2cm}\psfig{figure=or1-fdiag1-v5.eps, angle=0,width=4.6cm} %4.3cm}
%\hspace*{-.4cm}\psfig{figure=or1-fdiag2-v5.eps, angle=0,width=4.6cm}
\resizebox*{4.6cm}{!}{\includegraphics{or1-fdiag1-v5.eps}}
\resizebox*{4.6cm}{!}{\includegraphics{or1-fdiag2-v5.eps}}
%\centerline{\hspace*{-.8cm}\psfig{figure=or1-fdiag1.eps, angle=0,width=4.6cm}
%            \hspace*{-.4cm}\psfig{figure=or1-fdiag2.eps, angle=0,width=4.6cm}
}
\end{minipage}
\caption{\label{fdiags:fig} {\small Examples of $\tanb$--enhancement in GFM.
}}
\end{center}
\end{figure}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% \begin{figure}
% \begin{fmffile}{fdc7correct1}
% \begin{fmfchar*}(50,40)
%    \fmfright{v4,sl,v0}
%    \fmfleft{br}
%    \fmflabel{$s_L$}{sl}
%    \fmflabel{$b_R$}{br}
%    \fmfv{d.shape=square,d.size=8thin,d.filled=shaded,label=$\Delta
%     m_{bs}^*\langle H_1 \rangle^{-1}$,l.angle=-90,l.dist=15}{v1}
%    \fmfv{d.shape=cross,d.size=8thin,label=$\mu$,l.angle=90}{mu}
%    \fmfv{d.shape=cross,d.size=8thin,label=$\langle H_2\rangle$,l.angle=90}{h2}
%    \fmfv{d.shape=cross,d.size=8thin,label=$M_2$,l.angle=90}{m2}
%    \fmf{fermion}{v1,br}
%    \fmf{fermion}{v2,sl}
%    \fmf{plain,label=$\widetilde{\chi}$,l.side=right,tension=0}{v1,v2}
%    \fmf{dashes,label=$\widetilde{c}_L$,l.side=left,left,tension=0}{v1,v2}
%    \fmf{phantom}{v1,mu,h2,m2,v2}
%    \fmf{phantom}{br,v3,sl}
%    \fmf{photon}{v3,v4}
%    \fmfdot{v2}
% \end{fmfchar*}
% \end{fmffile}
% \end{figure}
% 
% \begin{figure}
% \begin{fmffile}{fdc7correct2}
% \begin{fmfchar*}(50,40)
%    \fmfright{v4,sl,v0}
%    \fmfleft{br}
%    \fmftop{top}
%    \fmflabel{$s_L$}{sl}
%    \fmflabel{$b_R$}{br}
%    \fmf{fermion}{v1,br}
%    \fmf{fermion}{v2,sl}
%    \fmf{plain}{v1,mu,h2,m2,v2}
%    \fmf{gluon,label=$\widetilde{g}$,l.side=left,tension=0}{v1,v2}
%    \fmffreeze
%    \fmf{dashes,l.side=left,left,tenstion=0,tag=1}{v1,v2}
%    \fmfipath{p[]}
%    \fmfiset{p1}{vpath1(__v1,__v2)}
%    \fmfiv{d.shape=square,d.size=8thin,d.filled=shaded,label=$\mu\Delta
%     m_{bs}^*\tan\beta$,l.angle=90,l.dist=15}
%    {point length(p1)/2 of p1}
%    \fmfiv{label=$\widetilde{b}_R$,l.angle=135}
%    {point length(p1)/4 of p1}
%    \fmfiv{label=$\widetilde{s}_L$,l.angle=45}
%    {point 3length(p1)/4 of p1}
%    \fmf{phantom}{br,v3,sl}
%    \fmf{photon}{v3,v4}
%    \fmfdot{v1,v2}
% \end{fmfchar*}
% \end{fmffile}
% \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.
The first diagram can be even larger than the LO gluino effect
if the ``diagonal part'' of the chargino penguin %lr (which is hidden at LO) 
%lr ***lr: what do you mean by hidden??? ***
%lr ***ko to lr* 
%lr With $\deltadrl$, there's no chargino contribution to $C_{7,8}^\prime$
%lr at LO, even how much its contribution to
%lr  the diagonal (chromo-)magnetic operator is large.
%lr I mean this 'hidden' large contiribution can be transfered to
%lr  $C_{7,8}$ with the small NLO effect to Yukawa coupling,
%lr  giving same order of contribution as LO gluino contribution. 
%lr *** 
is arbitrarily large.
For more details see~\cite{or2}.
At $\mu<0$, the rotation described above occurs in the direction
which increases the LO effect.
This competes with the reduction caused by the RGE evolution.
%For $\mu<0$, the corrections described above add
%constructively and cause a miss--alignment instead.
%This competes with the reduction caused by the RGE evolution.
As $\mususy$ increases, the RGE effect increases, while the quark mass
correction decreases.  At $\mususy \sim 1$ TeV, the two effects have
similar size for both signs of $\mu$.


\vspace{0.15cm}
%\noindent
{\it Relaxation of constraints on SUSY and FV terms}.---
%In the GFM scenario, the focusing effect leads to striking
%implications for SUSY.
%% *** contributions to $\brbsgamma$.
%We will illustrate this again in 
%the case of the CMSSM.
The constraints obtained in the MFV scenario~\cite{cgnw00,dgg00}
are no longer valid with GFM, and their robustness is of particular
phenomenological interest.
We illustrate this by including the above large NLO--level corrections.
In the three panels of
Fig.~\ref{dellrvsmhalf:fig} we show the contours of $\brbsgamma$
 in the plane of
$m_{\widetilde{q}}$
%$\mhalf$
and $\deltadlr$ (left panel) and
$\deltadll$ (middle and right panels) for 
$\tan\beta=45$, $m^2_{\widetilde{g}}/m^2_{\widetilde{q}}=2$,
 $A_u=-m_{\widetilde{q}}$, and $\mu>0$
%$\tanb=40$, $\mzero=500\gev$,
%$A_0=0$ and $\mu<0$ ($\mu>0$). 
(left and right panels) and $\mu<0$ (middle panel).
The light green (light grey) and dark yellow (dark grey) bands
agree with the experimental range~(\ref{bsgexptvalue:ref}) at the
$1\sigma$ ($2\sigma$)~C.L. Larger departures are denoted as
``excluded''. In the case of $\deltadlr$, in MFV one finds
$m_{\widetilde{q}}\gsim 800\gev$
 at the $2\sigma$~C.L. for $\mu>0$.
%$\mgluino\gsim600\gev$ ($\mhalf\gsim200\gev$)
% at the $1\sigma$~CL for
%$\mu>0$.  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure*}[t]
\begin{figure*}[t!]
\onecolumngrid
\vspace*{-0.2in} %{-0.3in}	%{-0.25in}
%	\hspace*{-.70in}
\begin{center}
%
%%\begin{minipage}{18.4cm}%{9.2cm} %{8.6cm}
%\centerline{
%%\hspace*{-.2cm}\psfig{figure=or1-t45-dlr-mun.eps, angle=0,width=4.6cm}
%\hspace*{-.3cm}\psfig{figure=or1-t45-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-t45-dll-mun.eps, angle=0,width=4.6cm}
% %4.3cm}
%\hspace*{-.3cm}\psfig{figure=or1-t45-dll-mup.eps, angle=0,width=4.6cm}
%	    %4.3cm}
%}
%%\end{minipage}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\resizebox{5.5cm}{5.5cm}{\includegraphics{or1-t45-dlr-mup.eps}}
\resizebox{5.5cm}{5.5cm}{\includegraphics{or1-t45-dll-mun.eps}}
\resizebox{5.5cm}{5.5cm}{\includegraphics{or1-t45-dll-mup.eps}}
%\begin{widetext}
\caption{\label{dellrvsmhalf:fig} {\small
(color online). We plot contours of $\brbsgamma$
in the plane of $m_{\widetilde{q}}$
 and $\deltadlr$ (left panel) and $\deltadll$ (
middle and right panels).
% for $\mu<0$ (left side) and $\mu>0$ (right side)  in the
%Constrained MSSM. 
In the left and right panels $\mu>0$ and in the middle one $\mu<0$.
Regions (in grey)
beyond the $1\sigma$ (green/light grey) and $2\sigma$~C.L. (dark yellow/grey)
in agreement with
experiment [Eq.~(\ref{bsgexptvalue:ref})] are marked ``excluded''.
}}
\end{center}
\end{figure*}
%\vspace*{-0.2in}
\twocolumngrid
%\vspace{-0.1in}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\noindent
%
In contrast, even relatively small nonzero values of
$|\deltadlr|\simeq -0.04$ 
%$|\deltadlr|\simeq0.01$ 
remove the constraint from $\bsgamma$ altogether.  The effect is even
more striking for $\delta^d_{LL}$ and $\mu<0$ where relatively 
light squark becomes again
allowed.  We have checked numerically that, at $\mu>0$, constraints on
FV terms at NLO become also considerably relaxed.  
For example, in Fig.~\ref{dellrvsmhalf:fig} at
$m_{\widetilde{q}}=1000\gev$
%$\mhalf=500\gev$
 and $\mu>0$,
we find
$\deltadlr(NLO)/\deltadlr(LO)\simeq7$. %lr ~\cite{or2}.
%$\deltadlr(NLO)/\deltadlr(LO)\simeq7.5$~\cite{or2}.
%
Similar GFM deconstraining can also be obtained by introducing
nonzero $\deltadll$s instead, for both $\mu>0$ and $\mu<0$
but with a milder NLO effect.
It is interesting to note the existence of two branches of allowed
solutions in Fig.~\ref{dellrvsmhalf:fig}.  By
simultaneously allowing for more than one $\delta^d$ to become
nonzero, one can relax the constraint from $\bsgamma$ even further,
even at LO~\cite{bghw99}. In addition, in general FV contributions can
also come from the up--squark sector~\cite{up-mixing}. We have checked
that this effect is numerically less important.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Finally, we must reemphasize that in our analysis we assume
$\msusy/\mw$ larger than a few and compute only the leading NLO
effects, which are enhanced by $\msusy/\mw$, large $\tanb$, or flavor
mixing. We also remain in the SM basis of operators, instead of
considering the full operator basis of the MSSM.  Despite this, we do
believe that the results presented here should remain as the dominant
effect in a more complete study.

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


\vspace{0.15cm}
%\noindent
{\it Conclusions and outlook}.---  
Contributions from new physics to rare processes like $\bsgamma$,
do not necessarily have to be suppressed by the largeness of the
related effective mass scale beyond the SM. Instead, this can be caused by new effects
beyond the LO. In the case of SUSY with GFM, at the
NLO level we have pointed out the existence of one such suppressing effect, which
we call focusing.
%lr , and identified its origin in the alignment mechanism
%lr which enhances NLO effect, together with the RG running.
%lr renomalization group running. 
%Its effect is often that of considerably reducing the 
%% *** 
%magnitude of the SUSY contribution to $\brbsgamma$ relative to the LO.
%*** and to the MFV scenario.  
Despite this, $\bsgamma$ still maintains strong
sensitivity to even small deviations from MFV
%Its strong sensitivity to even small deviations from MFV 
(although less so than at LO)  of
especially the flavor mixing terms $\deltadll$ and $\deltadlr$. This
puts into question the robustness of the commonly assumed constraints
on $\msusy$ in MFV.
%lr $m_{\widetilde{q}}$ and also on other SUSY parameters.  
It appears that the alignment mechanism is generic to
processes involving chirality flip. We are now
exploring its role in other FCNC processes.
%, 
%like $\mu \to e
%\gamma$ or $B \to X_s ll$, for which stringent constraints on ``new
%physics'' effects either exist or will soon become available.

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgments We thank P.~Gambino, T.~Hurth,
A.~Masiero, M.~Misiak and L.~Silvestrini for helpful comments.
%
% *** quote hurth review **
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
\vspace*{-0.1in}
%
\bibitem{bbmr90}
%\bibitem{Bertolini:1990if}
S.~Bertolini, \etal,
%	F.~Borzumati, A.~Masiero and G.~Ridolfi,
%``Effects Of Supergravity Induced Electroweak Breaking On Rare B Decays And Mixings,''
Nucl. Phys. {\bf B353}, 591 (1991).
%%CITATION = NUPHA,B353,591;%%

\bibitem{exp}
%%\bibitem{Chen:2001fj}
%S.~Chen, {\it et al.}  [CLEO Collaboration],
%%``Branching fraction and photon energy spectrum for b $\to$ s gamma,''
%Phys. Rev. Lett.  {\bf 87}, 251807 (2001);
%%[arXiv:hep-ex/0108032].
%%%CITATION = HEP-EX 0108032;%%
%%\bibitem{Barate:1998vz}
%R.~Barate, {\it et al.}  [ALEPH Collaboration],
%%``A measurement of the inclusive b $\to$ s gamma branching ratio,''
%Phys. Lett. {\bf B429}, 169 (1998);
%%%CITATION = PHLTA,B429,169;%%
%%\bibitem{Abe:2001hk}
%K.~Abe, {\it et al.}  [Belle Collaboration],
%%``A measurement of the branching fraction for the inclusive B $\to$ X/s  gamma decays with Belle,''
%Phys. Lett. {\bf B511}, 151 (2001);
%%[arXiv:hep-ex/0103042].
%%%CITATION = HEP-EX 0103042;%%
%%C.~Jessop, {\it et al.} [BaBar Collaboration],
%%talk at ICHEP-02, Amsterdam, July, 2002.
%B.Aubert {\it et al.} [BaBar Collaboration], hep-ex/0207074; hep-ex/0207076.
C.~Jessop, Report No. SLAC-PUB-9610 and references therein.

\bibitem{gm01}
%\bibitem{Gambino:2001ew}
P.~Gambino and M.~Misiak,
%``Quark mass effects in anti-B $\to$ X/s gamma,''
Nucl. Phys. {\bf B611}, 338 (2001).
%[arXiv:hep-ph/0104034].
%%CITATION = HEP-PH 0104034;%%

\bibitem{bcmu02}
%\bibitem{Buras:2002tp}
A.J.~Buras,  \etal,
%	A.~Czarnecki, M.~Misiak and J.~Urban,
%``Completing the NLO QCD calculation of anti-B $\to$ X/s gamma,''
Nucl. Phys. {\bf B631}, 219 (2002)
%[arXiv:hep-ph/0203135].
%%CITATION = HEP-PH 0203135;%%
and references therein.%



%	\bibitem{bcmu01}
%\bibitem{Buras:2001mq}
%	A.~J.~Buras, A.~Czarnecki, M.~Misiak and J.~Urban,
%``Two-loop matrix element of the current-current operator in the decay b  $\to$ X/s gamma,''
%	Nucl.\ Phys.\ B {\bf 611}, 488 (2001)
%[arXiv:hep-ph/0105160].
%%CITATION = HEP-PH 0105160;%%
%	and reference there in.

%\bibitem{2hdnlo}
\bibitem{bg98}
%\bibitem{Borzumati:1998tg}
F.M.~Borzumati and C.~Greub,
%``2HDMs predictions for anti-B $\to$ X/s gamma in NLO {QCD},''
Phys. Rev. {\bf D58}, 074004 (1998).
%[arXiv:hep-ph/9802391].
%%CITATION = HEP-PH 9802391;%%
%

\bibitem{crs98}
%\bibitem{Ciafaloni:1997un}
P.~Ciafaloni,  \etal,
%	A.~Romanino and A.~Strumia,
%``Two-loop QCD corrections to charged-Higgs-mediated b $\to$ s gamma  decay,''
Nucl. Phys. {\bf B524}, 361 (1998).
%[arXiv:hep-ph/9710312].
%%CITATION = HEP-PH 9710312;%%

\bibitem{cdgg98}
%\bibitem{Ciuchini:1997xe}
M.~Ciuchini,  \etal,
%	G.~Degrassi, P.~Gambino and G.~F.~Giudice,
%``Next-to-leading QCD corrections to B $\to$ X/s gamma: Standard model and  two-Higgs doublet model,''
Nucl. Phys. {\bf B527}, 21 (1998).
%[arXiv:hep-ph/9710335].
%%CITATION = HEP-PH 9710335;%%

\bibitem{cdgg98susy}
%\bibitem{Ciuchini:1998xy}
M.~Ciuchini,  \etal,
%	G.~Degrassi, P.~Gambino and G.~F.~Giudice,
%``Next-to-leading {QCD} corrections to B $\to$ X/s gamma in supersymmetry,''
Nucl. Phys. {\bf B534}, 3 (1998).
%[arXiv:hep-ph/9806308].
%%CITATION = HEP-PH 9806308;%%

\bibitem{bmu00}
%\bibitem{Bobeth:1999ww}
C.~Bobeth,  \etal,
%	M.~Misiak and J.~Urban,
%``Matching conditions for b $\to$ s gamma and b $\to$ s gluon in extensions  of the standard model,''
Nucl. Phys. {\bf B567}, 153 (2000).
%[arXiv:hep-ph/9904413].
%%CITATION = HEP-PH 9904413;%%

\bibitem{cgnw00}
%\bibitem{Carena:2000uj}
M.~Carena,  \etal,
%	D.~Garcia, U.~Nierste and C.~E.~Wagner,
%``b $\to$ s gamma and supersymmetry with large tan(beta),''
Phys. Lett. {\bf B499}, 141 (2001).
%[arXiv:hep-ph/0010003].
%%CITATION = HEP-PH 0010003;%%

\bibitem{dgg00}
%\bibitem{Degrassi:2000qf}
G.~Degrassi,  \etal,
%	P.~Gambino and G.~F.~Giudice,
%``B $\to$ X/s gamma in supersymmetry: Large contributions beyond the leading order,''
J. High Energy Phys. {\bf 0012} (2000) 009
%JHEP {\bf 0012}, 009 (2000).
%[arXiv:hep-ph/0009337].
%%CITATION = HEP-PH 0009337;%%


\bibitem{cmssmbsgamma}
See, \eg, J.~Ellis, \etal,
%T. Falk, G. Ganis, K.A. Olive, M. Srednicki,
Phys. Lett. {\bf B510}, 236 (2001);
%hep-ph/0102098
% The CMSSM Parameter Space at Large tan beta
L.~Roszkowski, \etal,
J. High Energy Phys. {\bf 0108} (2001) 024.
%{\bf JHEP 0108}, 024 (2001).
% Roberto Ruiz de Austri, Takeshi Nihei
% hep-ph/0106334
% New Cosmological and Experimental Constraints on the CMSSM


\bibitem{knrr1} Y.G.~Kim, \etal,
J. High Energy Phys. {\bf 0212} (2002) 034.

\bibitem{agis02}
%\bibitem{D'Ambrosio:2002ex}
G.~D'Ambrosio,  \etal,
%	G.~F.~Giudice, G.~Isidori and A.~Strumia,
%``Minimal flavour violation: An effective field theory approach,''
hep-ph/0207036.
%%CITATION = HEP-PH 0207036;%%


\bibitem{bghw99} %gluino}
%\bibitem{Borzumati:1999qt}
F.M.~Borzumati,  \etal,
%	C.~Greub, T.~Hurth and D.~Wyler,
%``Gluino contribution to radiative B decays: Organization of QCD  corrections and leading order results,''
Phys. Rev. {\bf D62}, 075005 (2000);
%[arXiv:hep-ph/9911245].
%%CITATION = HEP-PH 9911245;%%
%\bibitem{Besmer:2001cj}
T.~Besmer,  \etal,
%	C.~Greub and T.~Hurth,
%``Bounds on flavor violating parameters in supersymmetry,''
Nucl. Phys. {\bf B609}, 359 (2001).
%[arXiv:hep-ph/0105292].
%%CITATION = HEP-PH 0105292;%%

\bibitem{ekrww02}
%\bibitem{Everett:2001yy}
L.~Everett,  \etal,
%	G.~L.~Kane, S.~Rigolin, L.~T.~Wang and T.~T.~Wang,
%``Alternative approach to b $\to$ s gamma in the uMSSM,''
J. High Energy Phys. {\bf 0201} (2002) 022
%JHEP {\bf 0201}, 022 (2002).
%[arXiv:hep-ph/0112126].
%%CITATION = HEP-PH 0112126;%%

%\bibitem{gsw90}
%%\bibitem{Grinstein:tj}
%B.~Grinstein,  \etal,
%%	R.~P.~Springer and M.~B.~Wise,
%%``Strong Interaction Effects In Weak Radiative Anti-B Meson Decay,''
%Nucl.\ Phys.\ B {\bf 339}, 269 (1990).
%%%CITATION = NUPHA,B339,269;%%

\bibitem{bmmp94}
%\bibitem{Buras:xp}
A.J.~Buras,  \etal,
%	M.~Misiak, M.~Munz and S.~Pokorski,
%``Theoretical Uncertainties And Phenomenological Aspects Of B $\to$ X(S) Gamma Decay,''
Nucl. Phys. {\bf B424}, 374 (1994).
%(hep-ph/9311345).
%%CITATION = HEP-PH 9311345;%%

\bibitem{NLOmatchingSM}
%\bibitem{Adel:1993ah}
K.~Adel and Y.P.~Yao,
%``Exact alpha-s calculation of b $\to$ s + gamma b $\to$ s + g,''
Phys. Rev. {\bf D49}, 4945 (1994);
%[arXiv:hep-ph/9308349].
%%CITATION = HEP-PH 9308349;%%
%\bibitem{Greub:1997hf}
C.~Greub and T.~Hurth,
%``Two-loop matching of the dipole operators for b $\to$ s gamma and b $\to$  s g,''
Phys. Rev. {\bf D56}, 2934 (1997);
%[arXiv:hep-ph/9703349].
%%CITATION = HEP-PH 9703349;%%
%\bibitem{Buras:1997xf}
A.J.~Buras,  \etal,
%	A.~Kwiatkowski and N.~Pott,
%``Next-to-leading order matching for the magnetic photon penguin operator  in the B $\to$ X/s gamma decay,''
Nucl. Phys. {\bf B517}, 353 (1998).
%[arXiv:hep-ph/9710336].
%%CITATION = HEP-PH 9710336;%%

\bibitem{ghw96}
C.~Greub, \etal,
Phys. Rev. {\bf D54}, 3350 (1996); Phys. Lett. {\bf B380}, 385 (1996). 

\bibitem{misiakmunz95}
%\bibitem{Misiak:1994zw}
M.~Misiak and M.~M{\" u}nz,
%``Two loop mixing of dimension five flavor changing operators,''
Phys. Lett. {\bf B344}, 308 (1995).
%[arXiv:hep-ph/9409454].
%%CITATION = HEP-PH 9409454;%%


\bibitem{cmm98}
%\bibitem{Chetyrkin:1996vx}
K.G.~Chetyrkin, \etal,
%	M.~Misiak and M.~Munz,
%``Weak radiative B-meson decay beyond leading logarithms,''
Phys. Lett. {\bf B400}, 206 (1997);{\bf 425}, 414(E)
%[Erratum-ibid. {\bf B425}, 414 (1998)]%.
%[arXiv:hep-ph/9612313].
%%CITATION = HEP-PH 9612313;%%
; P.~Gambino, \etal, Nucl. Phys. {\bf B673}, 238 (2003).


%\bibitem{Anlauf:1994df}
%H.~Anlauf,
%%``On the QCD corrections to b $\to$ s gamma in supersymmetric models,''
%Nucl.\ Phys.\ B {\bf 430}, 245 (1994)
%[arXiv:hep-ph/9406286].
%%%CITATION = HEP-PH 9406286;%%

\bibitem{or2}
K.~Okumura and L.~Roszkowski, 
J. High Energy Phys. {\bf 0310} (2003) 024.
%JHEP {\bf 0310}, 024 (2003).

\bibitem{bottomcorr}
%\bibitem{Rattazzi:1996fb}
R.~Rattazzi and U.~Sarid,
%``Large tan(beta) in gauge-mediated SUSY-breaking models,''
Nucl. Phys. {\bf B501}, 297 (1997);
%[arXiv:hep-ph/9612464].
%%CITATION = HEP-PH 9612464;%%
%\bibitem{Baer:1997jq}
H.~Baer, \etal,
%	M.~Brhlik, D.~Castano and X.~Tata,
%``b $\to$ s gamma constraints on the minimal supergravity model with large  tan(beta),''
Phys. Rev. {\bf D58}, 015007 (1998);
%[arXiv:hep-ph/9712305].
%%CITATION = HEP-PH 9712305;%%
%\bibitem{Blazek:1997cs}
T.~Blazek and S.~Raby,
%``b $\to$ s gamma with large tan(beta) in MSSM analysis constrained
%by a  realistic SO(10) model,''
Phys. Rev. {\bf D59}, 095002 (1999).
%[arXiv:hep-ph/9712257].
%%CITATION = HEP-PH 9712257;%%

%	\bibitem{ggms96}
%\bibitem{Gabbiani:1996hi}
%	F.~Gabbiani,  \etal,
%	E.~Gabrielli, A.~Masiero and L.~Silvestrini,
%``A complete analysis of FCNC and CP constraints in general SUSY extensions of the standard model,''
%	Nucl. Phys. {\bf B477}, 321 (1996).
%[arXiv:hep-ph/9604387].
%%CITATION = HEP-PH 9604387;%%

% \bibitem{mpr97}
% %\bibitem{Misiak:1997ei}
% See, \eg, M.~Misiak,  \etal,
% %	S.~Pokorski and J.~Rosiek,
% %``Supersymmetry and FCNC effects,''
% Adv.\ Ser.\ Direct.\ High Energy Phys.\  {\bf 15}, 795 (1998),
% (hep-ph/9703442).
%[arXiv:hep-ph/9703442].
%%CITATION = HEP-PH 9703442;%%

%\bibitem{hrs}
%%\bibitem{Hall:1993gn}
%L.J.~Hall,  \etal,
%%	R.~Rattazzi and U.~Sarid,
%%``The Top quark mass in supersymmetric SO(10) unification,''
%Phys. Rev. {\bf D50}, 7048 (1994).
%%[arXiv:hep-ph/9306309].
%%%CITATION = HEP-PH 9306309;%%

\bibitem{up-mixing}
%\bibitem{Ali:2001ej}
A.~Ali and E.~Lunghi,
%``Extended minimal flavor violating MSSM and implications for B physics,''
Eur. Phys. J. {\bf C21}, 683 (2001).
%[arXiv:hep-ph/0105200].
%%CITATION = HEP-PH 0105200;%%

\end{thebibliography}


\end{document}












hep-ph0208101.extracted_bib
\begin{thebibliography}{99}
\vspace*{-0.1in}
%
\bibitem{bbmr90}
%\bibitem{Bertolini:1990if}
S.~Bertolini, \etal,
%	F.~Borzumati, A.~Masiero and G.~Ridolfi,
%``Effects Of Supergravity Induced Electroweak Breaking On Rare B Decays And Mixings,''
Nucl. Phys. {\bf B353}, 591 (1991).
%%CITATION = NUPHA,B353,591;%%

\bibitem{exp}
%%\bibitem{Chen:2001fj}
%S.~Chen, {\it et al.}  [CLEO Collaboration],
%%``Branching fraction and photon energy spectrum for b $\to$ s gamma,''
%Phys. Rev. Lett.  {\bf 87}, 251807 (2001);
%%[arXiv:hep-ex/0108032].
%%%CITATION = HEP-EX 0108032;%%
%%\bibitem{Barate:1998vz}
%R.~Barate, {\it et al.}  [ALEPH Collaboration],
%%``A measurement of the inclusive b $\to$ s gamma branching ratio,''
%Phys. Lett. {\bf B429}, 169 (1998);
%%%CITATION = PHLTA,B429,169;%%
%%\bibitem{Abe:2001hk}
%K.~Abe, {\it et al.}  [Belle Collaboration],
%%``A measurement of the branching fraction for the inclusive B $\to$ X/s  gamma decays with Belle,''
%Phys. Lett. {\bf B511}, 151 (2001);
%%[arXiv:hep-ex/0103042].
%%%CITATION = HEP-EX 0103042;%%
%%C.~Jessop, {\it et al.} [BaBar Collaboration],
%%talk at ICHEP-02, Amsterdam, July, 2002.
%B.Aubert {\it et al.} [BaBar Collaboration], hep-ex/0207074; hep-ex/0207076.
C.~Jessop, Report No. SLAC-PUB-9610 and references therein.

\bibitem{gm01}
%\bibitem{Gambino:2001ew}
P.~Gambino and M.~Misiak,
%``Quark mass effects in anti-B $\to$ X/s gamma,''
Nucl. Phys. {\bf B611}, 338 (2001).
%[arXiv:hep-ph/0104034].
%%CITATION = HEP-PH 0104034;%%

\bibitem{bcmu02}
%\bibitem{Buras:2002tp}
A.J.~Buras,  \etal,
%	A.~Czarnecki, M.~Misiak and J.~Urban,
%``Completing the NLO QCD calculation of anti-B $\to$ X/s gamma,''
Nucl. Phys. {\bf B631}, 219 (2002)
%[arXiv:hep-ph/0203135].
%%CITATION = HEP-PH 0203135;%%
and references therein.%



%	\bibitem{bcmu01}
%\bibitem{Buras:2001mq}
%	A.~J.~Buras, A.~Czarnecki, M.~Misiak and J.~Urban,
%``Two-loop matrix element of the current-current operator in the decay b  $\to$ X/s gamma,''
%	Nucl.\ Phys.\ B {\bf 611}, 488 (2001)
%[arXiv:hep-ph/0105160].
%%CITATION = HEP-PH 0105160;%%
%	and reference there in.

%\bibitem{2hdnlo}
\bibitem{bg98}
%\bibitem{Borzumati:1998tg}
F.M.~Borzumati and C.~Greub,
%``2HDMs predictions for anti-B $\to$ X/s gamma in NLO {QCD},''
Phys. Rev. {\bf D58}, 074004 (1998).
%[arXiv:hep-ph/9802391].
%%CITATION = HEP-PH 9802391;%%
%

\bibitem{crs98}
%\bibitem{Ciafaloni:1997un}
P.~Ciafaloni,  \etal,
%	A.~Romanino and A.~Strumia,
%``Two-loop QCD corrections to charged-Higgs-mediated b $\to$ s gamma  decay,''
Nucl. Phys. {\bf B524}, 361 (1998).
%[arXiv:hep-ph/9710312].
%%CITATION = HEP-PH 9710312;%%

\bibitem{cdgg98}
%\bibitem{Ciuchini:1997xe}
M.~Ciuchini,  \etal,
%	G.~Degrassi, P.~Gambino and G.~F.~Giudice,
%``Next-to-leading QCD corrections to B $\to$ X/s gamma: Standard model and  two-Higgs doublet model,''
Nucl. Phys. {\bf B527}, 21 (1998).
%[arXiv:hep-ph/9710335].
%%CITATION = HEP-PH 9710335;%%

\bibitem{cdgg98susy}
%\bibitem{Ciuchini:1998xy}
M.~Ciuchini,  \etal,
%	G.~Degrassi, P.~Gambino and G.~F.~Giudice,
%``Next-to-leading {QCD} corrections to B $\to$ X/s gamma in supersymmetry,''
Nucl. Phys. {\bf B534}, 3 (1998).
%[arXiv:hep-ph/9806308].
%%CITATION = HEP-PH 9806308;%%

\bibitem{bmu00}
%\bibitem{Bobeth:1999ww}
C.~Bobeth,  \etal,
%	M.~Misiak and J.~Urban,
%``Matching conditions for b $\to$ s gamma and b $\to$ s gluon in extensions  of the standard model,''
Nucl. Phys. {\bf B567}, 153 (2000).
%[arXiv:hep-ph/9904413].
%%CITATION = HEP-PH 9904413;%%

\bibitem{cgnw00}
%\bibitem{Carena:2000uj}
M.~Carena,  \etal,
%	D.~Garcia, U.~Nierste and C.~E.~Wagner,
%``b $\to$ s gamma and supersymmetry with large tan(beta),''
Phys. Lett. {\bf B499}, 141 (2001).
%[arXiv:hep-ph/0010003].
%%CITATION = HEP-PH 0010003;%%

\bibitem{dgg00}
%\bibitem{Degrassi:2000qf}
G.~Degrassi,  \etal,
%	P.~Gambino and G.~F.~Giudice,
%``B $\to$ X/s gamma in supersymmetry: Large contributions beyond the leading order,''
J. High Energy Phys. {\bf 0012} (2000) 009
%JHEP {\bf 0012}, 009 (2000).
%[arXiv:hep-ph/0009337].
%%CITATION = HEP-PH 0009337;%%


\bibitem{cmssmbsgamma}
See, \eg, J.~Ellis, \etal,
%T. Falk, G. Ganis, K.A. Olive, M. Srednicki,
Phys. Lett. {\bf B510}, 236 (2001);
%hep-ph/0102098
% The CMSSM Parameter Space at Large tan beta
L.~Roszkowski, \etal,
J. High Energy Phys. {\bf 0108} (2001) 024.
%{\bf JHEP 0108}, 024 (2001).
% Roberto Ruiz de Austri, Takeshi Nihei
% hep-ph/0106334
% New Cosmological and Experimental Constraints on the CMSSM


\bibitem{knrr1} Y.G.~Kim, \etal,
J. High Energy Phys. {\bf 0212} (2002) 034.

\bibitem{agis02}
%\bibitem{D'Ambrosio:2002ex}
G.~D'Ambrosio,  \etal,
%	G.~F.~Giudice, G.~Isidori and A.~Strumia,
%``Minimal flavour violation: An effective field theory approach,''
hep-ph/0207036.
%%CITATION = HEP-PH 0207036;%%


\bibitem{bghw99} %gluino}
%\bibitem{Borzumati:1999qt}
F.M.~Borzumati,  \etal,
%	C.~Greub, T.~Hurth and D.~Wyler,
%``Gluino contribution to radiative B decays: Organization of QCD  corrections and leading order results,''
Phys. Rev. {\bf D62}, 075005 (2000);
%[arXiv:hep-ph/9911245].
%%CITATION = HEP-PH 9911245;%%
%\bibitem{Besmer:2001cj}
T.~Besmer,  \etal,
%	C.~Greub and T.~Hurth,
%``Bounds on flavor violating parameters in supersymmetry,''
Nucl. Phys. {\bf B609}, 359 (2001).
%[arXiv:hep-ph/0105292].
%%CITATION = HEP-PH 0105292;%%

\bibitem{ekrww02}
%\bibitem{Everett:2001yy}
L.~Everett,  \etal,
%	G.~L.~Kane, S.~Rigolin, L.~T.~Wang and T.~T.~Wang,
%``Alternative approach to b $\to$ s gamma in the uMSSM,''
J. High Energy Phys. {\bf 0201} (2002) 022
%JHEP {\bf 0201}, 022 (2002).
%[arXiv:hep-ph/0112126].
%%CITATION = HEP-PH 0112126;%%

%\bibitem{gsw90}
%%\bibitem{Grinstein:tj}
%B.~Grinstein,  \etal,
%%	R.~P.~Springer and M.~B.~Wise,
%%``Strong Interaction Effects In Weak Radiative Anti-B Meson Decay,''
%Nucl.\ Phys.\ B {\bf 339}, 269 (1990).
%%%CITATION = NUPHA,B339,269;%%

\bibitem{bmmp94}
%\bibitem{Buras:xp}
A.J.~Buras,  \etal,
%	M.~Misiak, M.~Munz and S.~Pokorski,
%``Theoretical Uncertainties And Phenomenological Aspects Of B $\to$ X(S) Gamma Decay,''
Nucl. Phys. {\bf B424}, 374 (1994).
%(hep-ph/9311345).
%%CITATION = HEP-PH 9311345;%%

\bibitem{NLOmatchingSM}
%\bibitem{Adel:1993ah}
K.~Adel and Y.P.~Yao,
%``Exact alpha-s calculation of b $\to$ s + gamma b $\to$ s + g,''
Phys. Rev. {\bf D49}, 4945 (1994);
%[arXiv:hep-ph/9308349].
%%CITATION = HEP-PH 9308349;%%
%\bibitem{Greub:1997hf}
C.~Greub and T.~Hurth,
%``Two-loop matching of the dipole operators for b $\to$ s gamma and b $\to$  s g,''
Phys. Rev. {\bf D56}, 2934 (1997);
%[arXiv:hep-ph/9703349].
%%CITATION = HEP-PH 9703349;%%
%\bibitem{Buras:1997xf}
A.J.~Buras,  \etal,
%	A.~Kwiatkowski and N.~Pott,
%``Next-to-leading order matching for the magnetic photon penguin operator  in the B $\to$ X/s gamma decay,''
Nucl. Phys. {\bf B517}, 353 (1998).
%[arXiv:hep-ph/9710336].
%%CITATION = HEP-PH 9710336;%%

\bibitem{ghw96}
C.~Greub, \etal,
Phys. Rev. {\bf D54}, 3350 (1996); Phys. Lett. {\bf B380}, 385 (1996). 

\bibitem{misiakmunz95}
%\bibitem{Misiak:1994zw}
M.~Misiak and M.~M{\" u}nz,
%``Two loop mixing of dimension five flavor changing operators,''
Phys. Lett. {\bf B344}, 308 (1995).
%[arXiv:hep-ph/9409454].
%%CITATION = HEP-PH 9409454;%%


\bibitem{cmm98}
%\bibitem{Chetyrkin:1996vx}
K.G.~Chetyrkin, \etal,
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\end{thebibliography}