$B\to V\gamma$ at NLO from QCD Factorization\footnote{ Invited talks given at the Workshop on Exclusive B Decays, 20--22 July 2001, Regensburg and at the 9th International Symposium on Heavy Flavor Physics, 10--13 September 2001, Caltech, Pasadena. Based on work together with Gerhard Buchalla \cite{BB}}
Stefan W. Bosch
We discuss the exclusive radiative $B$-meson decays $B\to K^*\gamma$ and $B\to\rho\gamma$ in a model-independent manner. The analysis is based on the heavy-quark limit of QCD. This allows a factorization of perturbatively calculable contributions to the $B\to V\gamma$ matrix elements from non-perturbative form factors and universal light-cone distribution amplitudes. These results allow us to compute {exclusive\/} $b\to s(d) \gamma$ decays systematically beyond the leading logarithmic approximation. We present results for these decays complete to next-to-leading order in QCD and to leading order in the heavy-quark limit. Phenomenological implications for various observables of interest are discussed, including direct CP violation and isospin breaking effects.
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\begin{document}
\begin{flushright}
MPI-PHT-2001-38\\
September 2001
\end{flushright}
%\vspace{-0.5cm}
\title{$B\to V\gamma$ at NLO from QCD Factorization\footnote{
Invited talks given at the Workshop on Exclusive B Decays, 20--22 July 2001, Regensburg and at the 9th International Symposium on Heavy Flavor Physics, 10--13 September 2001, Caltech, Pasadena.
Based on work together with Gerhard Buchalla \cite{BB}}
}

\classification{classification}
\keywords{keywords}

\author{Stefan W. Bosch}{
  address={Max-Planck-Institut f\"ur Physik,
% Werner-Heisenberg-Institut, 
F\"ohringer Ring 6, D-80805 Munich, Germany},
  email={bosch@mppmu.mpg.de}
}

%\iftrue
%\author{Gerhard Buchalla}{
%  address={Theory Division, CERN, CH-1211 Geneva 23, Switzerland},
%  email={gerhard.buchalla@cern.ch},
%}
%
%\fi

\copyrightyear  {2001}



\begin{abstract}
We discuss the exclusive radiative $B$-meson decays $B\to K^*\gamma$ and $B\to\rho\gamma$ in a model-independent manner. The analysis is based on the heavy-quark limit of QCD. This allows a factorization of perturbatively calculable contributions to the $B\to V\gamma$ matrix elements from non-perturbative form factors and universal light-cone distribution amplitudes. These results allow us to compute {\it exclusive\/} $b\to s(d) \gamma$ decays systematically beyond the leading logarithmic approximation. We present results for these decays complete to next-to-leading order in QCD and to leading order in the heavy-quark limit. Phenomenological implications for various observables of interest are discussed, including direct CP violation and isospin breaking effects.
\end{abstract}


\date{\today}


\maketitle

\section{Introduction}

The radiative transitions $b\to s(d)\gamma$ are among the most valuable probes of flavour physics. Although they are rare decays the Cabibbo-favoured $b\to s\gamma$ modes are experimentally accessible already at present. The inclusive branching fraction has been measured to be
\begin{equation}\label{bsgamex}
B(B\to X_s\gamma)=(3.23\pm 0.42)\cdot 10^{-4}
\end{equation}
combining the results of \cite{CHENHON,BAR,ABENAK}.
The branching ratios for the exclusive channels have been determined by CLEO \cite{COA}, and more recently also by BaBar \cite{SPARYD} and BELLE \cite{NAK} to give the averaged values:
\begin{eqnarray}\label{b0kgamex}
B(B^0\to K^{*0}\gamma)=(4.51\pm 0.54)\cdot 10^{-5}\\
\label{bpkgamex}B(B^+\to K^{*+}\gamma)=(3.86\pm 0.59)\cdot 10^{-5}.
\end{eqnarray}
On the theoretical side, the flavour-changing neutral current (FCNC)
reactions $b\to s(d)\gamma$ are characterized by their high sensitivity
to New Physics and by the particularly large impact of short-distance
QCD corrections \cite{BBM,AGM,BMMP,CFMRS}.
Considerable efforts have therefore been devoted to achieve a full
calculation of the inclusive decay $b\to s\gamma$ at next-to-leading
order (NLO) in renormalization group (RG) improved perturbation
theory \cite{AY,GHW,CMM} (see \cite{MG} for recent reviews).

Whereas the inclusive mode can be computed perturbatively, using
the fact that the $b$-quark mass is large and employing the heavy-quark
expansion, the treatment of the exclusive channel $B\to K^*\gamma$
is in general more complicated. In this case bound state effects
are essential and need to be described by nonperturbative 
hadronic quantities (form factors).
The basic mechanisms at next-to-leading order were already
discussed previously for the $B\to V\gamma$ amplitudes \cite{AAWGSW}.
However, hadronic models were used to evaluate the various
contributions, which did not allow a clear separation
of short- and long-distance dynamics and a clean distinction
of model-dependent and model-independent features.

In this talk we present the results of \cite{BB} where a systematic analysis of the exclusive radiative decays $B\to V\gamma$ ($V=K^*$, $\rho$) in QCD, based
on the heavy quark limit $m_b\gg\Lambda_{QCD}$, was performed. We quote factorization formulas for the evaluation of the relevant hadronic matrix elements of local operators in the weak Hamiltonian. A similar subject was treated in \cite{BFS,AP}.
Factorization holds in QCD to leading power in the heavy quark limit.
This result relies on arguments similar to those used previously
to demonstrate QCD factorization for hadronic two-body modes
of the type $B\to\pi\pi$ \cite{BBNS1,BBNS2}. 

This framework allows us to separate perturbatively calculable
contributions from the nonperturbative form factors and universal meson
light-cone distribution amplitudes (LCDA) in a systematic way. 
This includes the treatment of loop effects from light quarks,
in particular up and charm.
Such loop effects are straightforwardly included for the inclusive
decays $b\to s(d)\gamma$. For the exclusive modes, however,
the effects from virtual charm and up quarks have so far been
considered to be uncalculable ``long-distance'' contributions
and have never been treated in a model independent fashion.

Finally, power counting in $\Lambda_{QCD}/m_b$  implies a hierarchy
among the possible mechanisms for $B\to V\gamma$ transitions. This
allows us to identify leading and subleading contributions.
For example, weak annihilation contributes only at subleading
power in the heavy quark limit.

Within this approach, higher order QCD corrections can be consistently
taken into account. We give the $B\to V\gamma$ decay amplitudes
at next-to-leading order (NLO). Furthermore numerical values for CP-asymmetries and isospin-violating ratios are given. After including NLO corrections the largest uncertainties still come from the $B\to V$ form factors, which are at present known only with limited precision ($\sim \pm 15\%$), mostly from QCD sum rule calculations \cite{BB98}. The situation should improve in the future with the help of both lattice QCD \cite{FS} and analytical methods based on the heavy-quark and large-energy limits \cite{CLOPR,BF,BH}.

%Despite the more complicated theoretical situation of the exclusive channels in comparison to the inclusive decays, the goal of obtaining a better understanding of the exclusive modes is very well motivated. This is because the exclusive decays are easier to investigate experimentally, especially in the difficult environment of hadron machines as the Fermilab Tevatron or the LHC at CERN. In any case the systematic uncertainties, both experimental and theoretical, are very different for $B\to V\gamma$ and $b\to s(d)\gamma$. A careful study of the exclusive modes can therefore yield valuable complementary information in testing the Standard Model.

%The remainder of this paper is organized as follows. In section \ref{sec:basics} we collect basic expressions and present the structure of the QCD factorization formulas for $B\to V\gamma$ matrix elements in general terms. Sections \ref{sec:BKgam} and \ref{sec:Brhogam} give a detailed discussion of $B\to K^*\gamma$ and $B\to\rho\gamma$, respectively. Phenomenological applications are studied in section \ref{sec:phen}. Finally, section \ref{sec:concl} contains our conclusions.


\section{Basic Formulas}
%\label{sec:basics}

The effective Hamiltonian for $b\to s\gamma$ transitions reads
\begin{equation}\label{heff}
{\cal H}_{eff}=\frac{G_F}{\sqrt{2}}\sum_{p=u,c}\lambda_p^{(s)}
\left[ C_1 Q^p_1 + C_2 Q^p_2 +\sum_{i=3,\ldots ,8} C_i Q_i\right]
\end{equation}
where $\lambda_p^{(s)}=V^*_{ps}V_{pb}$. The relevant operators are given by
\begin{eqnarray}\label{q1def}
Q^p_1 &=& (\bar sp)_{V-A}(\bar pb)_{V-A} \\
Q^p_2 &=& (\bar s_i p_j)_{V-A}(\bar p_j b_i)_{V-A} \\
%Q_3 &=& (\bar sb)_{V-A} \sum_q (\bar qq)_{V-A} \\
%Q_4 &=& (\bar s_i b_j)_{V-A} \sum_q (\bar q_j q_i)_{V-A} \\
%Q_5 &=& (\bar sb)_{V-A} \sum_q (\bar qq)_{V+A} \\
%Q_6 &=& (\bar s_i b_j)_{V-A} \sum_q (\bar q_j q_i)_{V+A} \\
\label{q7def}
Q_7 &=& \frac{e}{8\pi^2}m_b\, 
        \bar s_i\sigma^{\mu\nu}(1+\gamma_5)b_i\, F_{\mu\nu}\\
\label{q8def}
Q_8 &=& \frac{g}{8\pi^2}m_b\, 
        \bar s_i\sigma^{\mu\nu}(1+\gamma_5)T^a_{ij} b_j\, G^a_{\mu\nu}
\end{eqnarray}
%The most important operators are $Q^p_{1,2}$ and the magnetic penguin
%operator $Q_7$.
%Note that the numbering of $Q^p_{1,2}$ is reversed with respect to the convention of \cite{BBL}. 
%i.e. $C_1^{(0)}(M_W)=1$ and $C_2^{(0)}(M_W)=0$.
%The sign conventions for the electromagnetic and 
%strong couplings correspond to the covariant derivative
%$D_\mu=\partial_\mu +ie Q_f A_\mu + i g T^a A^a_\mu$. With these
%definitions the coefficients $C_{7,8}$ are negative in the
%Standard Model, which is the choice generally adopted
%in the literature. 
%Throughout this work 
Note that the numbering of $Q^p_{1,2}$ is reversed with respect to the convention of \cite{BBL}. We neglect the contribution from the QCD penguin operators $Q_{3\ldots 6}$, which enter at ${\cal O}(\alpha_s)$ and are further suppressed by very small Wilson coefficients. The effective Hamiltonian for $b\to d\gamma$ is obtained from
(\ref{heff}--\ref{q8def}) by the replacement $s\to d$. 
%The Wilson coefficients $C_i$ in (\ref{heff}) are known at
%next-to-leading order \cite{CMM}.

The most difficult step in computing the decay amplitudes is the
evaluation of the hadronic matrix elements of the operators in 
(\ref{heff}). A systematic treatment can be given in the heavy-quark
limit. In this case the following factorization formula is valid 
\begin{equation}\label{fform}
\langle V\gamma(\epsilon)|Q_i|\bar B\rangle =
\left[ F^{B\to V}(0)\, T^I_{i} +
\int^1_0 d\xi\, dv\, T^{II}_i(\xi,v)\, \Phi_B(\xi)\, \Phi_V(v)\right]
\cdot\epsilon
\end{equation}
where $\epsilon$ is the photon polarization 4-vector.
Here $F^{B\to V}$ is a $B\to V$ transition form factor,
and $\Phi_B$, $\Phi_V$ are leading twist light-cone distribution amplitudes
of the $B$ meson and the vector meson $V$, respectively.
These quantities are universal, nonperturbative objects. They
describe the long-distance dynamics of the matrix elements, which
is factorized from the perturbative, short-distance interactions
expressed in the hard-scattering kernels $T^I_{i}$ and $T^{II}_i$.
%The type-I contributions are the hard vertex corrections where the light quark in the $B$-meson is a real spectator whereas the spectator participates in the hard scattering for the type-II contributions.
The QCD factorization formula (\ref{fform}) holds up to
corrections of relative order $\Lambda_{QCD}/m_b$.

In the leading logarithmic approximation
(LO) and to leading power in the heavy-quark limit, $Q_7$ gives the only contribution to the amplitude of $\bar B\to V\gamma$ and the factorization formula (\ref{fform}) is trivial.
The matrix element is simply expressed in terms of the standard
form factor, $T^I_{7}$ is a purely kinematical function and
the spectator term $T^{II}_7$ is absent. An illustration is given
in Fig. \ref{fig:q7}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
%   \vspace{-3cm}
%   \epsfysize=4cm
%   \epsfxsize=6cm
   \includegraphics[height=2.5cm]{bvgfig1}
%   \vspace*{-15.5cm}
\caption{Contribution of the magnetic penguin operator $Q_7$
described by $B\to V$ form factors. All possible gluon exchanges 
between the quark lines are included in the form factors and have not
been drawn explicitly. \label{fig:q7}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The matrix element reads
\begin{equation}\label{q7f1f2}
\langle V(k,\eta) \gamma(q,\epsilon)|Q_7|\bar B\rangle =
-\frac{e}{2\pi^2}m_b\, c_V F_V \left[ 
\varepsilon^{\mu\nu\lambda\rho}\epsilon_\mu\eta_\nu k_\lambda q_\rho +
i (\epsilon\cdot\eta\, k\cdot q-\epsilon\cdot k\, \eta\cdot q)\right]
\end{equation}
where $c_V=1$ for $V=K^*$, $\rho^-$ and $c_V=1/\sqrt{2}$ for $V=\rho^0$.
The $\bar B\to V$ form factor $F_V$ is evaluated at momentum transfer
$q^2=0$. Our phase conventions coincide with those of \cite{BB98,ABS}.
%In particular we have $F_V>0$, $\varepsilon^{0123}=-1$, and the
%phases of $V$ (with flavour content $\bar qq'$) and $\bar B$ are such that
%\begin{eqnarray}\label{fbfv}
%\langle V(k,\eta)|\bar q\sigma_{\mu\nu}q'|0\rangle &=&
%-i(\eta_\mu k_\nu-\eta_\nu k_\mu)f^\perp_V \\
%\langle 0|\bar u\gamma_\mu\gamma_5 b|\bar B(p)\rangle &=& +i f_B p_\mu
%\end{eqnarray}
%with positive $f_B$, $f^\perp_V$.

The matrix elements of $Q_1$ and $Q_8$ start contributing at ${\cal O}(\alpha_s)$. In this case the factorization formula becomes nontrivial. The diagrams for the hard-scattering kernels $T^I_{i}$ are shown in Fig. \ref{fig:qit1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
%   \vspace{-3cm}
%   \epsfysize=4cm
%   \epsfxsize=12cm
   \includegraphics[height=3cm]{bvgfig2}
%   \vspace*{-15.5cm}
\caption{${\cal O}(\alpha_s)$ contribution to the hard-scattering kernels $T^I_1$ from four-quark operators $Q_1$. The crosses indicate the places where the emitted photon can be attached. \label{fig:qit1}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for $Q_1$ and in Fig. \ref{fig:q8t1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
%   \vspace{-3cm}
%   \epsfysize=2.5cm
%   \epsfxsize=12cm
   \includegraphics[height=1.875cm]{bvgfig4}
%   \vspace*{-15.5cm}
\caption{${\cal O}(\alpha_s)$ contribution to the hard-scattering 
kernels $T^I_{8}$ from chromomagnetic penguin operator $Q_8$.
\label{fig:q8t1}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for $Q_8$. These diagrams were computed in \cite{GHW} to get the virtual corrections to the inclusive matrix elements of $Q_1$ and $Q_8$. In our case they determine the kernels $T^I_{1}$ and $T^I_{8}$.
As required for the consistency of the factorization formula these corrections must be dominated by hard scales of order $m_b$ and hence must be infrared finite. This is indeed the case. Re-interpreted as the perturbative hard-scattering kernels for the exclusive process, the results from \cite{GHW} imply
\begin{equation}\label{q1me1}
\langle Q_{1,8}\rangle^I=\langle Q_7\rangle 
\frac{\alpha_s C_F}{4\pi} G_{1,8}
\end{equation}
where $C_F=(N^2-1)/(2N)$, with $N=3$ the number of colours, and
\begin{eqnarray}\label{G1}
G_1(s_c) &=& -\frac{104}{27}\ln\frac{\mu}{m_b}+ g_1(s_c) \\
\label{G8}
G_8 &=& \frac{8}{3}\ln\frac{\mu}{m_b} + g_8
\end{eqnarray}
%The logarithms of the renormalization scale $\mu$ cancel the corresponding scale-dependence of the Wilson coefficients.
The finite part $g_1(s_c)$ of the $Q_1$-contribution depends via $s_c=\frac{m^2_c}{m^2_b}$ on the mass of the quark running in the loop in Fig. \ref{fig:qit1}.
%\begin{eqnarray}\label{g1s}
%g_1(s) &=& -\frac{833}{162}-\frac{20i\pi}{27} 
%            +\frac{8\pi^2}{9} s^{3/2} \nonumber\\
%        && {}+\frac{2}{9} \bigg[ 48+30i\pi-5\pi^2-2i\pi^3 -36\zeta (3) 
%          +\left( 36+6i\pi-9\pi^2\right)\ln s \nonumber\\
%        && \qquad\;\;\, +\left( 3+6i\pi\right) \ln^2\! s+\ln^3\! s \bigg] s 
%           \nonumber\\
%        && {}+\frac{2}{9} \bigg[ 18+2\pi^2 -2i\pi^3 
%         +\left( 12-6\pi^2 \right)\ln s +6i\pi\ln^2\! s+\ln^3\! s\bigg] s^2 
%           \nonumber\\
%        && {}+\frac{1}{27} \bigg[ -9+112 i\pi-14\pi^2
%          +\left(182-48i\pi\right)\ln s-126\ln^2\! s\bigg] s^3 \\
%\label{g8}
%g_8 &=& \frac{11}{3}-\frac{2\pi^2}{9}+\frac{2i\pi}{3}
%\end{eqnarray}
%Here we denote
%\begin{equation}\label{sdef}
%s_c=\frac{m^2_c}{m^2_b}
%\end{equation}

We now turn to the mechanism where the spectator participates
in the hard scattering.

%\section{NLO type II}
The non-vanishing contributions to
$T^{II}_i$ are shown in Fig.~\ref{fig:qit2}. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
%   \vspace{-3cm}
%   \epsfysize=4cm
%   \epsfxsize=15cm
   \includegraphics[height=3cm]{bvgfig3}
%   \vspace*{-15.5cm}
\caption{${\cal O}(\alpha_s)$ and leading power contribution to the hard-scattering kernels $T^{II}_i$ from four-quark operators $Q_i$ (left) and from $Q_8$. \label{fig:qit2}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To find the correction for $\langle Q_1\rangle$ we compute the first diagram in Fig. \ref{fig:qit2}
%, using the
%light-cone projectors in (\ref{phi12def}) and (\ref{phiperpdef}).
We obtain
\begin{equation}\label{q1me2}
\langle Q_1\rangle^{II}=\langle Q_7\rangle 
\frac{\alpha_s(\mu_h) C_F}{4\pi} H_1(s_c)
\end{equation}
with
\begin{equation}\label{h1s}
H_1(s)=-\frac{2\pi^2}{3 N}\frac{f_B f^\perp_V}{F_V m^2_B}
\int^1_0 d\xi\frac{\Phi_{B1}(\xi)}{\xi}\int^1_0 dv\, h(\bar v,s)
\Phi_\perp(v)
\end{equation}
The first negative moment of the light-cone distribution amplitude $\Phi_{B1}(\xi)$ can be pa\-ra\-me\-trized by $m_B/\lambda_B$ where $\lambda_B={\cal O}(\Lambda_{QCD})$. The hard-scattering kernel $h(\bar v,s_c)$ is displayed in Fig. \ref{fig:hcvbar}. It is real for $\bar v\leq 4s$ and develops an imaginary part for the light-cone momentum fraction of the anti-quark in the vector meson $\bar v > 4s$.
%\begin{equation}\label{hus}
%h(u,s)=\frac{4s}{u^2}\left[ 
%L_2\!\left(\frac{2}{1-\sqrt{\frac{u-4s+i\varepsilon}{u}}}\right)+
%L_2\!\left(\frac{2}{1+\sqrt{\frac{u-4s+i\varepsilon}{u}}}\right)\right]
%-\frac{2}{u}
%\end{equation}
%$L_2$ is the dilogarithmic function
%\begin{equation}\label{dilog}
%L_2(x)=-\int^x_0 dt\frac{\ln(1-t)}{t}
%\end{equation}
%The function $h(u,s)$ is real for $u\leq 4s$ and develops an
%imaginary part for $u > 4s$. At small values of $u$ it has the
%expansion
%\begin{equation}\label{husexp}
%h(u,s)=\frac{1}{6s}+\frac{1}{45 s^2}u+
%{\cal O}\left(\frac{u^2}{s^3}\right)
%\end{equation}
%It is also regular for $s\to 0$
%\begin{equation}\label{hus0}
%h(u,0)=-\frac{2}{u}
%\end{equation}
%The function $h(\bar v,s_c)$ is displayed in Fig. \ref{fig:hcvbar}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
%   \vspace{-3cm}
%   \epsfysize=10cm
%   \epsfxsize=12cm
   \includegraphics[height=6cm]{hcvbar}
%   \vspace*{-15.5cm}
\caption{The hard-scattering kernel $h(\bar v,s_c)$ as a function
of $\bar v$. \label{fig:hcvbar}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The correction to $\langle Q_8\rangle$ from the hard spectator
interaction comes from the second diagram in Fig. \ref{fig:qit2}.
One finds 
\begin{equation}\label{q8me2}
\langle Q_8\rangle^{II}=\langle Q_7\rangle 
\frac{\alpha_s(\mu_h) C_F}{4\pi} H_8
\end{equation}
where
\begin{equation}\label{h8}
H_8=\frac{4\pi^2}{3 N}\frac{f_B f^\perp_V}{F_V m^2_B}
\int^1_0 d\xi\frac{\Phi_{B1}(\xi)}{\xi}
\int^1_0 dv\frac{\Phi_\perp(v)}{v}
\end{equation}
Here the LCDA $\Phi_\perp(v)$ is of leading power for a transversly polarized vector meson. The last convolution integral in (\ref{h8}) can be performed explicitely and leads to a combination of Gegenbauer moments of $\Phi_\perp$ \cite{BB96,BB98}.
%The gluons in Fig. \ref{fig:qit2} transfer a momentum of order 
%$\mu_h \sim \sqrt{\Lambda_{QCD} m_b}$. 
%Therefore we set $\alpha_s=\alpha_s(\mu_h)$ in (\ref{q1me2}) and
%(\ref{q8me2}). For our numerical analysis we shall use 
%$\mu_h=\sqrt{\Lambda_h \mu}$ with $\Lambda_h=0.5$ GeV and 
%$\mu={\cal O}(m_b)$. 
%
%
%
%The diagrams in Figs. \ref{fig:qit1} -- \ref{fig:qit2} represent
%the complete set of contributions from $Q_{1,2}$ and $Q_8$
%to leading order in the heavy-quark limit and at ${\cal O}(\alpha_s)$.
%The first term in the factorization formula (\ref{fform}) is
%given by the form factor from (\ref{q7f1f2}) and the hard-scattering
%kernels $T^I_{i}$. The latter are calculable functions of
%$m_q/m_b$, where $m_q$ is the mass of the internal quarks in the loop
%diagram. For the second term in (\ref{fform}), where the spectator
%quark is involved, we need light-cone distribution amplitudes
%(wave functions) for $B$ mesons and light vector mesons.
%In the case of the $B$ meson we have to leading power \cite{BBNS2,BF}
%\begin{equation}\label{phi12def}
%\langle 0|b(0) \bar u(z)|\bar B(p)\rangle =
%\frac{i f_B}{4}(\not\! p + m_b)\gamma_5\,
%\int^1_0 d\xi\, e^{-i\xi p_+ z_-}[\Phi_{B1}(\xi)+\not\! n \Phi_{B2}(\xi)]
%\end{equation}
%with $n=(1,0,0,-1)$, chosen to be parallel to the 4-momentum of
%the vector meson. The functions $\Phi_{1,2}(\xi)$ describe
%the distribution of light-cone momentum fraction $\xi=l_+/p_+$
%of the spectator quark with momentum $l$ inside the $B$ meson.
%Here light-cone components of four-vectors $v$ are defined by
%\begin{equation}\label{vlcc}
%v_\pm =\frac{v^0\pm v^3}{\sqrt{2}}
%\end{equation}
%The wave functions are highly asymmetric with 
%$\xi={\cal O}(\Lambda_{QCD}/m_b)$. They are normalized as
%\begin{equation}\label{phi12norm}
%\int^1_0 d\xi\, \Phi_{B1}(\xi)=1 \qquad
%\int^1_0 d\xi\, \Phi_{B2}(\xi)=0
%\end{equation}
%
%In $\bar B\to V\gamma$ decays the vector meson is transversely
%polarized. The leading-twist and leading-power distribution amplitude 
%for light vector mesons with transverse polarization, $\Phi_\perp$, 
%is defined by
%\begin{equation}\label{phiperpdef}
%\langle V(k,\eta)|q'(z)\bar q(0)|0\rangle =\frac{i f^\perp_V}{4}
%\sigma^{\mu\nu}\eta_\mu k_\nu \int^1_0 dv\, e^{i\bar v k\cdot z}
%\Phi_\perp(v)
%\end{equation}
%Here and in the following we use the short-hand notation
%\begin{equation}
%\bar v \equiv 1-v
%\end{equation}
%for light-cone variables.
%
%The light-cone wave function $\Phi_\perp$ has an expansion in terms
%of Gegenbauer polynomials $C^{3/2}_n(2v-1)$
%\begin{equation}\label{phiperpgbp}
%\Phi_\perp(v)=6v(1-v)\left[ 1+\sum^\infty_{n=1}
%\alpha^\perp_n(\mu) C^{3/2}_n(2v-1)\right]
%\end{equation}
%where $C^{3/2}_1(x)=3x$, $C^{3/2}_2(x)=\frac{3}{2}(5x^2-1)$, etc.
%The Gegenbauer moments $\alpha^\perp_n(\mu)$ are multiplicatively
%renormalized. They vanish logarithmically as the scale $\mu\to\infty$.
%In this limit $\Phi_\perp$ reduces to its asymptotic form
%$\Phi_\perp(v)=6v\bar v$, which often is a reasonable first approximation.
%The remaining leading-twist light-cone wave functions for light
%vector mesons, $\Phi_\parallel$, $g_\perp^{(v)}$ and
%$g_\perp^{(a)}$ \cite{BB98,BB96}, do not contribute at leading power
%if the mesons are transversely polarized.
%
%The form factor $F_V$, $f_B/\lambda_B$, and the light-cone wave function
%$f^\perp_V \Phi_\perp(v)$ are the nonperturbative
%quantities required to describe $\bar B\to V\gamma$ at next-to-leading
%order in QCD. ($\Phi_{B2}(\xi)$ does not contribute.)

There are further mechanisms that can in principle contribute
to $\bar B\to V\gamma$ decays.
One possibility is weak annihilation.
%, depicted in Fig. \ref{fig:ann}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[t]
%%   \vspace{-3cm}
%%   \epsfysize=4cm
%%   \epsfxsize=6cm
%   \includegraphics[height=4cm]{bvgfig5}
%%   \vspace*{-15.5cm}
%\caption{Annihilation contribution to $\bar B\to V\gamma$ decay.
%The dominant mechanism is the radiation of the photon from
%the light quark in the $B$ meson, as shown. This amplitude is
%suppressed by one power of $\Lambda_{QCD}/m_b$, but it is still
%calculable in QCD factorization. Radiation of the photon from the
%remaining three quark lines is suppressed by $(\Lambda_{QCD}/m_b)^2$
%for operators $Q_{1,2}$. \label{fig:ann}}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this case the leading-power projection onto the meson $V$ vanishes because the trace over an odd number of Dirac matrices is zero. A non-vanishing result arises from the subleading-power projections onto $\Phi_\parallel$ and $g_\perp$.
%\begin{equation}\label{fvmveta}
%\langle V(k,\eta)| \bar q\gamma_\nu q' |0\rangle = f_V m_V \eta_\nu
%\end{equation}
%which, however, is suppressed by one power of $\Lambda_{QCD}/m_b$
%compared to (\ref{fbfv}), (\ref{phiperpdef}) for transverse
%polarization $\eta_\nu$ ($f_V$, $f^\perp_V$, $m_V\sim \Lambda_{QCD}$,
%$k\sim m_b$).
%This results in a corresponding power suppression of weak annihilation.
%The dominant contribution comes from the diagram shown in
%Fig. \ref{fig:ann}. Here the photon is emitted from the light-quark
%constituent in the $B$ meson, which leads to a quark propagator
%scaling as $1/\Lambda_{QCD}$. This is in contrast to the
%remaining three possible diagrams with a quark propagator $\sim 1/m_b$.
%The latter contributions are therefore even stronger suppressed,
%5$\sim (\Lambda_{QCD}/m_b)^2$ relative to the leading $\bar B\to V\gamma$
%amplitude.

Despite its power suppression, the dominant annihilation amplitude
can be computed within QCD factorization. This is because the
colour-transparency argument applies to the emitted, highly energetic
vector meson in the heavy-quark limit \cite{BBNS2,GP}.
%A similar observation was already made in \cite{GP}.
%On the other hand, we disagree with the claim made in this paper
%that the ${\cal O}(\alpha_s)$ correction to Fig. \ref{fig:ann}
%would vanish identically in the chiral limit and to leading-twist
%order. This claim is based on the observation that the projector in
%(\ref{phiperpdef}) gives zero when applied to a current with an odd 
%number of Dirac matrices. However, by the same argument, the diagram
%in Fig. \ref{fig:ann} would vanish even at leading order
%in $\alpha_s$, which is not the case.
%The proper treatment
%of the ${\cal O}(\alpha_s)$ correction should employ the subleading-power 
%projections related to the wave functions
%$\Phi_\parallel$, $g_\perp^{(v)}$, $g_\perp^{(a)}$ \cite{BB98,BB96},
%corresponding to the use of (\ref{fvmveta}) at ${\cal O}(\alpha^0_s)$.
%This correction has not yet been computed, but we see no a-priori
%reason to expect a vanishing result.

Since weak annihilation is a power correction, we will content 
ourselves with the lowest order result (${\cal O}(\alpha^0_s)$)
for our estimates below. In particular, we shall include the 
annihilation effects from operators $Q_{1,2}$ to estimate
isospin-breaking corrections in $B\to\rho\gamma$ decays.
The reason for including this class of power corrections is that they
come with a numerical enhancement from the large Wilson
coefficients $C_{1,2}$ ($C_1\approx 3|C_7|$) and are not CKM suppressed.
Instead, a CKM suppression of annihilation effects occurs for
$B\to K^*\gamma$ and these contributions are thus very small in this case.

%Finally, Fig. \ref{fig:subl} displays the remaining diagrams
%with a power suppression. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[t]
%%   \vspace{-3cm}
%%   \epsfysize=4cm
%%   \epsfxsize=15cm
%   \includegraphics[height=4cm]{bvgfig6}
%%   \vspace*{-15.5cm}
%\caption{Other contributions that are power-suppressed in the
%heavy-quark limit. \label{fig:subl}}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The penguin-annihilation diagrams (left)
%are power suppressed in a way similar to the ordinary annihilation
%topologies. The spectator diagrams from $Q_8$ (right) lead to
%amplitudes that are either manifestly power-suppressed
%(photon emitted from the quark line in the upper right),
%or that are superficially of leading power, but vanish when the
%leading-order projections are performed (photon emitted from
%either of the quarks forming the $B$ meson).
%Recall, however, that photon emission from the light-quark line
%from $Q_8$ is a leading-power effect (see Fig. \ref{fig:qit2}).
% 
%
%
%
%\section{$B\to K^*\gamma$}
%%\label{sec:BKgam}
%
%In the case of $B\to K^*\gamma$ the component of the
%Hamiltonian (\ref{heff}) proportional to $\lambda_u$ is strongly
%CKM suppressed ($|\lambda_u/\lambda_c|\approx 0.02$) and has
%only a minor impact on the decay rate. It is essentially
%negligible, but will be included later on for completeness.
%We note that to ${\cal O}(\alpha_s)$ the matrix element of $Q_2$
%is zero because of its colour structure.
%The amplitude for $\bar B\to K^*\gamma$ then reads
%($\langle Q_i\rangle\equiv\langle K^*\gamma|Q_i|\bar B\rangle$)
%\begin{equation}\label{abkgam718}
%A(\bar B\to K^*\gamma)=\frac{G_F}{\sqrt{2}}\lambda_c^{(s)}
%\left(C_7\langle Q_7\rangle +C_1\langle Q^c_1\rangle + 
%C_8\langle Q_8\rangle \right)
%\end{equation}
%The leading-order matrix element $\langle Q_7\rangle$ is given in
%(\ref{q7f1f2}) with $V\equiv K^*$. At subleading order in $\alpha_s$
%the matrix elements $\langle Q_{1,8}\rangle$ need to be computed
%from the diagrams in Figs. \ref{fig:qit1} -- \ref{fig:qit2}.
%The result for the diagrams in Figs. \ref{fig:qit1} and \ref{fig:q8t1},
%which enter the hard-scattering kernels $T^I_{1}$, $T^I_{8}$,
%can be infered from \cite{GHW}. In these papers the diagrams were
%computed to obtain the matrix elements for the inclusive mode
%$b\to s\gamma$ at next-to-leading order. In this context
%Figs. \ref{fig:qit1} and \ref{fig:q8t1} represented the virtual
%corrections to the inclusive matrix elements of $Q_1$ and $Q_8$. 
%
%
%
\section{Results}
Finally, we can combine these results and write, adding the up- and charm- quark contribution,
\begin{equation}\label{abvgam}
A(\bar B\to V\gamma)=\frac{G_F}{\sqrt{2}}
\left[\sum_{p=u,c} \lambda_p^{(s)}\, a^p_7(V\gamma)\right] 
\langle V\gamma|Q_7|\bar B\rangle
\end{equation}
where, at NLO, the factorization coefficients $a^p_7(V\gamma)$ are given as
\begin{eqnarray}\label{a7vgam}
a^p_7(V\gamma) = C_7 &+& \frac{\alpha_s(\mu) C_F}{4\pi}
\left( C_1(\mu) G_1(s_p)+ C_8(\mu) G_8\right) \nonumber \\  
  &+& \frac{\alpha_s(\mu_h) C_F}{4\pi} 
  \left( C_1(\mu_h) H_1(s_p)+ C_8(\mu_h) H_8\right)
\end{eqnarray}
Here the NLO expression for $C_7$ \cite{CMM} has to be used, while the leading
order values are sufficient for $C_1$ and $C_8$. 
%The explicit formulas for the Wilson coefficients can be found in \cite{CMM}.
Numerically we obtain for central values of all input parameters, at $\mu=m_b$, and displaying separately the size of the various correction terms:
\begin{eqnarray}\label{numa7cK}
a_7^c(K^* \gamma) &=& \begin{array}[t]{cccc}
-0.3221 & +0.0113 & -0.0820-0.0147i & -0.0144-0.0109i\\
C_7^{LO} & \Delta C_7^{NLO} & T^I_{1,8}\mbox{-contribution} & 
  T^{II}_{1,8}\mbox{-contribution}\\
\end{array} \nonumber\\
&=& -0.4072 -0.0256i.
\end{eqnarray}
We note a sizable enhancement of the leading order value, dominated by the 
$T^I$-type correction.
%This feature was already observed in the context of 
%the inclusive case in \cite{GHW}. 
A complex phase is generated at NLO, 
where the $T^I$-corrections and the hard-spectator interactions ($T^{II}$) 
yield comparable effects.

The net enhancement of $a_7$ at NLO leads to a corresponding enhancement 
of the branching ratios, for fixed value of the form factor. This is 
illustrated in Fig. \ref{fig:bkrhomu}, where we show the residual scale 
dependence for $B(\bar{B}\to \bar{K}^{*0}\gamma)$ and $B(B^-\to\rho^-\gamma)$ 
at leading and next-to-leading order.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
%\epsfig{figure=bk0barmu.eps,width=7.2cm,height=8.0cm}
%\epsfig{figure=brhomu.eps,width=7.2cm,height=8.0cm}
\includegraphics[height=8cm]{bk0barmu}\includegraphics[height=8cm]{brhomu}
\caption{Dependence of the branching fractions $B(\bar{B}^0 \to \bar{K}^{*0} \gamma)$ and $B(B^- \to \rho^- \gamma)$ on the renormalization scale $\mu$. The dotted line shows the LO, the dash-dotted line the NLO result including type-I corrections only and the solid line shows the complete NLO result.}
\label{fig:bkrhomu}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As expected, the inclusion of the hard-vertex corrections ($T^I$) reduces the scale dependence coming from the Wilson coefficients. The scale dependence of the complete NLO result is ``deteriorated'' because the type-II contributions appear at ${\cal O}(\alpha_s)$ for the first time and therefore introduce a completely new scale dependence.


%The scale dependence of the matrix element of $Q_7$ is reflected in the 
%running of the product of $b$-quark mass and form factor which is 
%explicitely given as
%\begin{equation}\label{runmbff}
%\left( m_b \cdot F_V \right)[\mu] = \left( m_b \cdot F_V \right)[m_b]
%\left(1-\frac{\alpha_s(\mu)}{4\pi} 8C_F \ln\frac{\mu}{m_b}\right)
%\end{equation}
%This dependence on the scale $\mu$ has to be taken into account
%when the residual scale dependence of physical quantities
%is investigated.
%
%From the amplitude in (\ref{abkgam}) the branching ratio is obtained as
%\begin{equation}\label{brbkgam}
%B(\bar B\to K^*\gamma)=\tau_B\frac{G^2_F \alpha m^3_B m^2_b}{32\pi^4}
%\left(1-\frac{m^2_{K^*}}{m^2_B}\right)^3 
%|\sum_p \lambda_p^{(s)}\, a^p_7(K^*\gamma)|^2\, |F_{K^*}|^2
%\end{equation}



%\section{$B\to\rho\gamma$}
%\label{sec:Brhogam}
%
For the decay $\bar B\to\rho\gamma$ both sectors of the effective 
Hamiltonian have the same order of magnitude.
%and have to be included. 
%It is straightforward to translate the expressions from the previous 
%section to this case. 
%The amplitude can be written as
%\begin{equation}\label{abrgam}
%A(\bar B\to \rho\gamma)=\frac{G_F}{\sqrt{2}}
%\left[\sum_p \lambda_p^{(d)}\, a^p_7(\rho\gamma) \right]
%\langle \rho\gamma|Q_7|\bar B\rangle
%\end{equation}
%where $a^p_7(V\gamma)$ is given in (\ref{a7vgam}).
%The branching fraction becomes
%\begin{equation}\label{brbrgam}
%B(\bar B\to \rho\gamma)=\tau_B\frac{G^2_F \alpha m^3_B m^2_b}{32\pi^4}
%\left(1-\frac{m^2_\rho}{m^2_B}\right)^3 
%|\sum_p \lambda_p^{(d)}\, a^p_7(\rho\gamma)|^2\, c^2_\rho |F_\rho|^2
%\end{equation}
The amplitude for the CP-conjugated mode $B\to\rho\gamma$ 
is obtained by replacing $\lambda_p^{(d)} \to \lambda_p^{(d)*}$. 
We may then consider the CP asymmetry
\begin{equation}\label{acpbrgdef}
{\cal A}_{CP}(\rho\gamma)=
\frac{\Gamma(B\to\rho\gamma)-\Gamma(\bar B\to\rho\gamma)}{
      \Gamma(B\to\rho\gamma)+\Gamma(\bar B\to\rho\gamma)}
\end{equation}

%One finds
%\begin{equation}\label{acpbrgam}
%{\cal A}_{CP}(\rho\gamma)=
%\frac{2\, {\rm Im}\lambda_u^{(d)*} \lambda_c^{(d)}\, {\rm Im}a^u_7 a^{c*}_7}{
% |\lambda_u^{(d)}|^2 |a^u_7|^2 + |\lambda_c^{(d)}|^2 |a^c_7|^2 + 
%  2 {\rm Re}\lambda_u^{(d)*} \lambda_c^{(d)}\, {\rm Re}a^{u*}_7 a^c_7}
%\end{equation}
%
%which is non-vanishing at ${\cal O}(\alpha_s)$ only. 
%Expanding ${\cal A}_{CP}$ in $\alpha_s$ and using the improved Wolfenstein 
%parametrization for the CKM elements we obtain
%\begin{eqnarray}\label{acpbrgamsimp}
%{\cal A}_{CP}(\rho\gamma) &=&
%\frac{2\,{\rm Im}\lambda^{(d)*}_u\lambda^{(d)}_c}{|\lambda^{(d)}_t|^2}
%\, 
%\frac{{\rm Im}a^{u*}_7 a^c_7}{|C_7|^2} \\
%&=& -\frac{2\bar{\eta}}{(1-\bar\rho)^2+\bar{\eta}^2}
%\frac{\alpha_s C_F}{4\pi}\, \frac{C_1}{C_7}\,
%{\rm Im}\left(G_1(s_c)+H_1(s_c)-G_1(0)-H_1(0)\right)\nonumber
%\end{eqnarray}
%
%As we have discussed above, weak annihilation from the
%leading operators $Q_{1,2}$ contributes to the $\bar B\to\rho\gamma$ 
%amplitude only at ${\cal O}(\Lambda_{QCD}/m_b)$, but is enhanced
%by large Wilson coefficients. These effects can be calculated in
%QCD factorization. To lowest order in $\alpha_s$ we obtain
%from the diagram in Fig. \ref{fig:ann}, using the projection
%in (\ref{fvmveta})
%\begin{eqnarray}\label{aannrho}
%A_{ann}(B^-\to\rho^-\gamma) &=& 
%\frac{G_F}{\sqrt{2}}\lambda^{(d)}_u\, a_1 b_u \,
%\langle\rho^-\gamma|Q_7|B^-\rangle \\
%A_{ann}(\bar B^0\to\rho^0\gamma) &=& 
%\frac{G_F}{\sqrt{2}}\lambda^{(d)}_u\, a_2 b_d \,
%\langle\rho^0\gamma|Q_7|\bar B^0\rangle 
%\end{eqnarray}
%Here
%\begin{equation}\label{ac12}
%a_{1,2}=C_{1,2}+\frac{1}{N} C_{2,1}
%\end{equation}
%(evaluated in leading logarithmic approximation) and
%\begin{equation}\label{bubd}
%b_u=\frac{2\pi^2 Q_u f_B f_\rho m_\rho}{F_\rho m^2_B \lambda_B}
%\ \ \  b_d=\frac{1}{2}b_u
%\end{equation}
%Recalling that $F_\rho\sim m^{-3/2}_b$, $f_B\sim m^{-1/2}_b$
%in the heavy-quark limit, we note that $b_u\sim \Lambda_{QCD}/m_b$.
%This shows explicitly the power suppression of weak annihilation. 
%The ratio of $b_d$ to $b_u$ is $-Q_d/Q_u=1/2$, where the minus sign
%comes from the relative sign between the up-quark and down-quark 
%components of the $\rho^0$ wave function (only the up-quarks
%produce the $\rho^0$ in the annihilation process, while only the
%down-quarks are relevant in $\langle Q_7\rangle$).
%
%The annihilation components are included in the decay
%amplitudes by substituting
%\begin{eqnarray}\label{a7ba}
%a^u_7 &\to& a^u_7 + b_u a_1 \ \ \ \ {\rm for}\ \ \ \ B^-\to\rho^-\gamma \\
%a^u_7 &\to& a^u_7 + b_d a_2 \ \ \ \ {\rm for}\ \ \ \ \bar B^0\to\rho^0\gamma 
%\end{eqnarray}
%
%The annihilation contribution for $B^-\to K^{*-}\gamma$ is similar
%to the one for $B^-\to\rho^-\gamma$, with obvious replacements
%$\rho\to K^*$ in $b_u$. Its impact is very small for $B^-\to K^{*-}\gamma$ 
%because of the strong CKM suppression of the up-quark sector.
%It will be included in our numerical analysis for definiteness.
%Within our approximations weak annihilation gives no contribution
%to $\bar B^0\to \bar K^{*0}\gamma$.
%
%
%\section{Phenomenology}
%%\label{sec:phen}
%
%In this section we present numerical results for various
%observables of interest for the phenomenology of 
%$B\to V\gamma$ decays, based on the NLO QCD expressions derived
%in this paper.
%Our choice of input parameters is summarized in Table \ref{tab:input}.
%Our default choice for the CKM angle $\gamma$ is $58$ deg.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%% table of input parameters   %%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{table}[p]
%\caption[]{Summary of input parameters.}
%\label{tab:input}
%\renewcommand{\arraystretch}{1.2}
%%\begin{center}
%\begin{tabular*}{140mm}{@{\extracolsep\fill}|c|c|c|c|c|c|}
%\hline\hline
%\multicolumn{6}{|c|}{CKM parameters and coupling constants}\\
%\hline
%$V_{us}$ & $V_{cb}$ & $\left|V_{ub}/V_{cb}\right|$ & 
%$\Lambda_{\overline{MS}}^{(5)}$ & $\alpha$ & $G_F$\\
%\hline
%0.22 & 0.041 & $0.085 \pm 0.025$ & 225 MeV & 1/137 & $1.166 \times 10^{-5} 
%\mbox{GeV}^{-2}$\\
%\hline
%\end{tabular*}
%
%\begin{tabular*}{140mm}{@{\extracolsep\fill}|c|c|c|c|c|}
%\hline
%\multicolumn{5}{|c|}{Parameters related to the $B$ mesons}\\
%\hline
%$m_B$ & $f_B$ & $\lambda_B$ & $\tau_{B^+}$ & $\tau_{B^0}$\\
%\hline
%5.28 GeV & 180 MeV & $(350 \pm 150)$ MeV & 1.65 ps & 1.56 ps\\
%\hline
%\end{tabular*}
%
%\begin{tabular*}{140mm}{@{\extracolsep\fill}|c|c|c|c|c|c|}
%\hline
%\multicolumn{6}{|c|}{Parameters related to the $K^*$ meson \cite{BB98}}\\
%\hline
%$F_{K^*}$ & $f_{K^*}^\perp$ & $m_{K^*}$ & $\alpha_1^{K^*}$ & 
%$\alpha_2^{K^*}$ & $f_{K^*}$ \\
%\hline
%$0.38 \pm 0.06$ & 185 MeV & 894 MeV & 0.2 & 0.04 & 230 MeV\\
%\hline\hline
%\multicolumn{6}{|c|}{Parameters related to the $\rho$ meson \cite{BB98}}\\
%\hline
%$F_\rho$ & $f_\rho^\perp$ & $m_\rho$ & $\alpha_1^\rho$  & $\alpha_2^\rho$ & 
%$f_\rho$\\
%\hline
%$0.29 \pm 0.04$ & 160 MeV & 770 MeV & 0 & 0.2 & 200 MeV\\
%\hline
%\end{tabular*}
%
%\begin{tabular*}{140mm}{@{\extracolsep\fill}|c|c|c|c|}
%\hline
%\multicolumn{4}{|c|}{Quark and W-boson masses}\\
%\hline
%$m_b(m_b)$ & $m_c(m_b)$ & $m_{t,\mbox{pole}}$ & $M_W$\\
%\hline
%4.2 GeV & $(1.3 \pm 0.2)$ GeV & 174 GeV & 80.4 GeV\\
%\hline\hline
%\end{tabular*}
%%\end{center}
%\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%We begin with the numerical result for the NLO QCD coefficient 
%$a_7^c(K^*\gamma)$ as a typical example.
%\begin{table}[p]
%\label{tab:output}
%\caption[]{Predictions for branching ratios and CP asymmetries with the 
%errors from the individual input uncertainties.}
%\renewcommand{\arraystretch}{1.1}
%%\begin{center}
%\begin{tabular}{|l|c|c|c|}
%\hline\hline
%   & $B(\bar{B}^0 \to \bar{K}^{*0} \gamma)[10^{-5}]$    
%   & 
%     $B(B^- \to \rho^- \gamma)[10^{-6}]$        & 
%${\cal A}_{CP}(\rho^\pm\gamma)[\%]$\\
%\hline\hline
%central                         & 7.09                                  
%  & 1.58                                        & 9.89\\
%\hline\hline
%$F_{K^*}$                       & +2.32/$-$1.99                         
%  &   --    &  --  \\
%\hline
%$F_{\rho}$                      &   --                                     
%  & +0.42/$-$0.37                               & +0.12/$-$0.13\\
%\hline
%$\mu$                           & +0.67/$-$1.00                         
%  & +0.07/$-$0.19                               & +5.25/$-$2.40\\
%\hline
%$\lambda_B$                     & +0.38/$-$0.15                         
%  & +0.26/$-$0.09                               & +0.35/$-$0.28\\
%\hline
%$m_c$                           & +0.38/$-$0.43                         
%  & +0.12/$-$0.12                               & +1.40/$-$1.52\\
%\hline
%$\left|V_{ub}/V_{cb}\right|$    & +0.05/$-$0.05                         
%  & +0.15/$-$0.13                               & +3.48/$-$3.30\\
%%\hline
%%$\gamma$                        & +0.11/$-$0.45                         
%%  & +2.01/$-$0.41                               & +0.01/$-$9.89\\
%\hline\hline
%\end{tabular}
%%\end{center}
%\end{table}
%
%
%
%
%The sensitivity of the $\bar{B}\to \bar{K}^{*0}\gamma$ and 
%$B^-\to\rho^-\gamma$ branching ratios, and of the CP asymmetry 
%${\cal A}_{CP}(\rho\gamma)$ to variations in the relevant input parameters 
%are summarized in Table 2.
%The uncertainty of the branching fractions is 
%currently dominated by the form factors $F_{K^*}$, $F_\rho$. The values for 
%$B(B^-\to K^{*-}\gamma)$ are very close to those for 
%$B(\bar{B}\to \bar{K}^{*0}\gamma)$ with the main shift coming
%from the different lifetimes. For central input parameters we find
%$B(B^-\to K^{*-}\gamma)=7.45\cdot 10^{-5}$ compared to
%$B(\bar{B}\to \bar{K}^{*0}\gamma)=7.09\cdot 10^{-5}$.
%The CP asymmetry ${\cal A}_{CP}(K^*\gamma)$ is typically
%$-0.5\%$.
%
%Taking the sizable uncertainties into 
%account, the results for $B\to K^*\gamma$ in Table 2 are compatible with the 
%experimental measurements in (\ref{b0kgamex}) and (\ref{bpkgamex}), 
%even though the central theoretical values appear to be somewhat high.
%
%
It is substantial for the $\rho\gamma$ modes and much less dependent on the form factors. Here the largest theoretical uncertainty comes from the scale dependence. This is to be expected because the direct CP asymmetry is proportional to the perturbative strong phase difference, which arises at ${\cal O}(\alpha_s)$ for the first time. Unknown power corrections could have some impact on the prediction.

A further interesting observable is the charge averaged isospin breaking ratio
\begin{equation}
\label{isobreak}
\Delta(\rho\gamma)=\frac{\Gamma(B^+\to\rho^+\gamma)}{4\Gamma(B^0\to\rho^0\gamma)}+\frac{\Gamma(B^-\to\rho^-\gamma)}{4\Gamma(\bar{B}^0\to\rho^0\gamma)}-1
\end{equation}
Within our approximations, isospin breaking is generated by weak annihilation. Isospin breaking was already discussed in \cite{AHL}, partially including NLO corrections.
%Our results, including the power-suppressed  annihilation effects at leading order in QCD, together with the complete NLO expressions at leading power, are displayed in Fig. \ref{fig:acpiso}.

$B\to\rho\gamma$ also depends sensitively on fundamental CKM parameters, such as $\left|V_{ub}/V_{cb}\right|$ and $\gamma$, and can thus in principle serve to constrain the latter quantities once measurements become available. This is further illustrated in Fig. \ref{fig:acpiso}, where the dependence on $\gamma$ is shown for ${\cal A}_{CP}(\rho\gamma)$ and $\Delta(\rho\gamma)$, respectively.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[t]
%%   \vspace{-3cm}
%%   \epsfysize=10cm
%%   \epsfxsize=10cm
%   \includegraphics[height=10cm]{brhogamma}
%%   \vspace*{-15.5cm}
%\caption{The branching fraction $B(B^- \to \rho^- \gamma)$ as a function
%of the CKM angle $\gamma$ at leading and next-to-leading order. 
%\label{fig:brhogamma}}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
%   \vspace{-3cm}
%   \epsfysize=10cm
%   \epsfxsize=12cm
   \includegraphics[height=6cm]{cpasymrho}\includegraphics[height=6cm]{isodeltarho}
%   \vspace*{-15.5cm}
\caption{Left: The CP asymmetry ${\cal A}_{CP}(\rho\gamma)$ as a function of the CKM angle $\gamma$ for three values of the renormalization scale $\mu=m_b/2$, $m_b$ and $2 m_b$. Right: The isospin-breaking asymmetry $\Delta(\rho\gamma)$ as a function of the CKM angle $\gamma$ at leading and next-to-leading order.\label{fig:acpiso}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{figure}[t]
%%   \vspace{-3cm}
%%   \epsfysize=10cm
%%   \epsfxsize=12cm
%   \includegraphics[height=7cm]{isodeltarho}
%%   \vspace*{-15.5cm}
%\caption{The isospin-breaking asymmetry $\Delta(\rho\gamma)$ 
%as a function of the CKM angle $\gamma$ at leading and
%next-to-leading order. \label{fig:isodeltarho}}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We remark that our sign of $\Delta(\rho\gamma)$ differs from the one found in \cite{AHL}.

Another application of our results concerns an estimate of U-spin breaking effects in $B\to V\gamma$ decays \cite{GRG,HM,RF}. Let us state here only that U-spin breaking effects can be sizeable. For further details we refer to \cite{BB}.
%U-spin symmetry, the symmetry of strong interactions under exchange
%of $d$ and $s$ quarks, has been advocated as a tool to control
%hadronic uncertainties in tests of the Standard Model \cite{GRG,HM,RF}.
%Defining
%\begin{eqnarray}\label{dbbkr}
%\Delta B(B\to K^*\gamma) &=& 
%     B(B^+\to K^{*+}\gamma) - B(B^-\to K^{*-}\gamma) \\
%\Delta B(B\to \rho\gamma) &=& 
%     B(B^+\to \rho^{+}\gamma) - B(B^-\to \rho^{-}\gamma) 
%\end{eqnarray}
%the quantity
%\begin{equation}\label{dbb}
%\Delta B(B\to K^*\gamma)+\Delta B(B\to \rho\gamma) \equiv 0
%\end{equation}
%in the limit of U-spin symmetry and within the Standard Model.
%This has been discussed in \cite{GRG} and was 
%considered in more detail in \cite{HM}.
%Using our expressions and central values for all
%parameters we find
%\begin{eqnarray}\label{dbbkrnum}
%\Delta B(B\to K^*\gamma) &=& -7\cdot 10^{-7} \\
%\Delta B(B\to \rho\gamma) &=& 4\cdot 10^{-7} 
%\end{eqnarray}
%where we have chosen the CKM angle $\gamma=\pi/2$, which
%maximises the effects. The two quantities indeed have opposite
%signs, but their sum only partly cancels, leaving a U-spin
%breaking remainder of $-3\cdot 10^{-7}$. This effect is
%almost entirely due to the difference $(F_{K^*}-F_\rho)$.
%For form-factor values different from those in Table 1,
%the U-spin breaking effect would approximately be rescaled 
%proportional to $(F_{K^*}-F_\rho)$.
%For our choice the sum of the two asymmetries in (\ref{dbb})
%is of the same order of magnitude as the individual asymmetries.
%This example quantifies the limitations of the relation
%(\ref{dbb}) as a Standard Model test. 


\section{Conclusions}
%\label{sec:concl}

In this talk we have presented a systematic and
model-independent framework for the exclusive radiative
decays $B\to V\gamma$ based on the heavy-quark limit.
This allowed us to compute the decay amplitudes for these modes
consistently at next-to-leading order in QCD.

An important conceptual aspect of this analysis is the interpretation
of loop contributions with charm and up quarks, which come from
leading operators in the effective weak Hamiltonian.
We have argued that these effects are calculable in terms of
perturbative hard-scattering functions and universal meson
light-cone distribution amplitudes. They are ${\cal O}(\alpha_s)$
corrections, but are leading power contributions in the
framework of QCD factorization. This picture is in contrast to the
common notion that considers charm and up-quark loop effects as
generic, uncalculable long-distance contributions.
Non-factorizable long-distance corrections may still exist, but
they are power-suppressed.

Another important feature of the NLO calculation are the strong interaction phases, which are calculable at leading power. They play a crucial role for CP violating observables. We have seen that weak-annihilation amplitudes are power-suppressed, but can be numerically important for $B\to\rho\gamma$ because they enter with large coefficients. These effects also turn out to be calculable and were included in our phenomenological discussion at leading order in QCD.

Finally, a numerical analysis leads to our predictions for the branching ratios $B(\bar B^0\to\bar K^{*0}\gamma)\sim 7.09\cdot 10^{-5}$ and $B(B^-\to\rho^-\gamma)\sim 1.58\cdot 10^{-6}$, and for the CP asymmetries ${\cal A}_{CP}(K^*\gamma)\sim -0.5\%$ and ${\cal A}_{CP}(\rho\gamma)\sim 10\%$ for central values of the input parameters. The uncertainties for the branching ratios are of ${\cal O}(30\ldots 35\%)$ where the form factors $F_{K^*}$ and $F_\rho$ taken from \cite{BB98} clearly dominate this uncertainty. This situation however can be systematically improved. In particular, our approach allows for a consistent perturbative matching of the nonperturbative form factor to the short-distance part of the amplitude.
Power corrections are not expected to be unreasonably large because with the annihilation contribution we have included the probably numerically largest of these corrections. Furthermore there are no chirally enhanced power corrections in the vector meson wave functions.
%Our formalism can also be applied to other radiative rare $B$ decays,
%such as $B_s\to V\gamma$, $V=\phi$, $K^*$, or $B_d\to\omega\gamma$
%(see e.g. \cite{GP} for a discussion of these modes). 
%
%
%The improved theoretical understanding of $B\to V\gamma$ decay
%observables streng\-thens the motivation for still more detailed
%experimental investigations, which will contribute
%significantly to our knowledge of the flavour sector.
 

\begin{theacknowledgments}
I would like to thank the organizers of 
both the Regensburg workshop and
Heavy Flavors 9 for inviting me to such interesting symposia.
%Especially the HF9 Local Organizing Committee did an incredible job coping with the trouble caused by the terrible events on 11th of September.
It is a great pleasure to thank Gerhard Buchalla for the extremely enjoyable collaboration. For the work presented here, I also gratefully acknowledge financial support from the Studienstiftung des deutschen Volkes and thank the CERN Theory Division for the kind hospitality.
\end{theacknowledgments}

\begin{thebibliography}{99}

\bibitem{BB}
%\cite{Bosch:2001gv}
S.~W.~Bosch and G.~Buchalla,
%``The radiative decays B $\to$ V gamma at next-to-leading order in QCD,''
hep-ph/0106081.
%%CITATION = HEP-PH 0106081;%%


\bibitem{CHENHON}
S.~Chen {\it et al.}  [CLEO Collaboration],
%``Branching fraction and photon energy spectrum for b $\to$ s gamma,''
hep-ex/0108032;\\
%%CITATION = HEP-EX 0108032;%%
K. Honscheid, this conference.

\bibitem{BAR}
R.~Barate {\it et al.}  [ALEPH Collaboration],
%``A measurement of the inclusive b --> s gamma branching ratio,''
Phys.\ Lett.\ B {\bf 429} (1998) 169.
%%CITATION = PHLTA,B429,169;%%


\bibitem{ABENAK}
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.\ B {\bf 511}, 151 (2001)
[hep-ex/0103042];\\
%%CITATION = HEP-EX 0103042;%%
M.~Nakao [Belle collaboration], this conference.


\bibitem{COA}
T.~E.~Coan {\it et al.}  [CLEO Collaboration],
%``Study of exclusive radiative B meson decays,''
Phys.\ Rev.\ Lett.\ {\bf 84} (2000) 5283.
%[hep-ex/9912057].
%%CITATION = HEP-EX 9912057;%%


\bibitem{SPARYD}
%\cite{Spanier:2001cc}
S.~Spanier  [BaBar Collaboration],
%``Rare B decays with the BaBar detector,''
SLAC-PUB-8987
{\it Presented at the Kaon 2001 Conference, Pisa, Italy, June 12 - June 17, 2001};\\
A.~Ryd [BaBar collaboration], this conference.


\bibitem{NAK}
M.~Nakao [Belle collaboration], this conference.


\bibitem{BBM}
S.~Bertolini, F.~Borzumati and A.~Masiero,
%``QCD Enhancement Of Radiative B Decays,''
Phys.\ Rev.\ Lett.\ {\bf 59} (1987) 180.
%%CITATION = PRLTA,59,180;%%

\bibitem{AGM}
A.~Ali, C.~Greub and T.~Mannel,
%``Rare B decays in the Standard Model,''
DESY-93-016, {\it To be publ. in Proc. of ECFA Workshop on the 
Physics of a B Meson Factory, Eds. R. Aleksan, A. Ali, 1993}.

\bibitem{BMMP}
A.~J.~Buras, M.~Misiak, M.~M\"u{}nz and S.~Pokorski,
%``Theoretical uncertainties and phenomenological aspects of 
% B $\to$ X(s) gamma decay,''
Nucl.\ Phys.\ B {\bf 424} (1994) 374.
%[hep-ph/9311345].
%%CITATION = HEP-PH 9311345;%%

\bibitem{CFMRS}
M.~Ciuchini {\it et al.},
%``Scheme independence of the effective Hamiltonian for 
% b $\to$ s gamma and b $\to$ s g decays,''
Phys.\ Lett.\ B {\bf 316} (1993) 127;\\
%[hep-ph/9307364].
%%CITATION = HEP-PH 9307364;%%
M.~Ciuchini, E.~Franco, L.~Reina and L.~Silvestrini,
%``Leading order QCD corrections to b $\to$ s gamma and 
% b $\to$ s g decays in three regularization schemes,''
Nucl.\ Phys.\ B {\bf 421} (1994) 41.
%[hep-ph/9311357].
%%CITATION = HEP-PH 9311357;%%




\bibitem{AY}
K.~Adel and Y.~Yao,
%``Exact alpha-s calculation of b $\to$ s + gamma b $\to$ s + g,''
Phys.\ Rev.\ D {\bf 49} (1994) 4945;\\
%[hep-ph/9308349].
%%CITATION = HEP-PH 9308349;%%
C.~Greub and T.~Hurth,
%``Two-loop matching of the dipole operators 
% for b --> s gamma and b -->  s g,''
Phys.\ Rev.\ D {\bf 56} (1997) 2934;\\
%[hep-ph/9703349].
%%CITATION = HEP-PH 9703349;%%
A.~J.~Buras, A.~Kwiatkowski and N.~Pott,
%``Next-to-leading order matching 
% for the magnetic photon penguin operator  in the B --> X/s gamma decay,''
Nucl.\ Phys.\ B {\bf 517} (1998) 353.
%[hep-ph/9710336].
%%CITATION = HEP-PH 9710336;%%

\bibitem{GHW}
C.~Greub, T.~Hurth and D.~Wyler,
%``Virtual $O(\a_s)$ corrections to the inclusive decay $b \to s \gamma$,''
Phys.\ Rev.\ D {\bf 54} (1996) 3350;\\
%[hep-ph/9603404].
%%CITATION = HEP-PH 9603404;%%
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,''
hep-ph/0105160.
%%CITATION = HEP-PH 0105160;%%



\bibitem{CMM}
K.~Chetyrkin, M.~Misiak and M.~M\"unz,
%``Weak radiative B-meson decay beyond leading logarithms,''
Phys.\ Lett.\ B {\bf 400} (1997) 206;\\
%[hep-ph/9612313].
%%CITATION = HEP-PH 9612313;%%
Erratum-ibid. B {\bf 425} (1998) 414.

\bibitem{MG}
M.~Misiak,
%``Theory of radiative B decays,''
hep-ph/0002007;\\
%%CITATION = HEP-PH 0002007;%%
C.~Greub,
%``Theory of rare B meson decays,''
hep-ph/9911348.
%%CITATION = HEP-PH 9911348;%%


\bibitem{AAWGSW}
H.~H.~Asatrian, H.~M.~Asatrian and D.~Wyler,
%``NLL corrections for B meson radiative exclusive decays,''
Phys.\ Lett.\ B {\bf 470} (1999) 223;\\
%[hep-ph/9905412].
%%CITATION = HEP-PH 9905412;%%
C.~Greub, H.~Simma and D.~Wyler,
%``Branching ratio and direct CP violating rate asymmetry of the rare
%decays B $\to$ K* gamma and B $\to$ rho gamma,''
Nucl.\ Phys.\ B {\bf 434} (1995) 39
[Erratum-ibid.\ B {\bf 444} (1995) 447].
%[hep-ph/9406421].
%%CITATION = HEP-PH 9406421;%%


\bibitem{BFS}
%\cite{Beneke:2001at}
M.~Beneke, T.~Feldmann and D.~Seidel,
%``Systematic approach to exclusive B $\to$ V l+ l-, V gamma decays,''
hep-ph/0106067.
%%CITATION = HEP-PH 0106067;%%


\bibitem{AP}
A.~Ali and A.~Y.~Parkhomenko,
%``Branching ratios for B $\to$ rho gamma decays in 
%next-to-leading order in  alpha(s) including hard spectator corrections,''
hep-ph/0105302.
%%CITATION = HEP-PH 0105302;%%


\bibitem{BBNS1}
M.~Beneke, G.~Buchalla, M.~Neubert and C.~T.~Sachrajda,
%``{QCD} factorization for B --> pi pi decays: 
% Strong phases and CP  violation in the heavy quark limit,''
Phys.\ Rev.\ Lett.\ {\bf 83} (1999) 1914.
%[hep-ph/9905312].
%%CITATION = HEP-PH 9905312;%%

\bibitem{BBNS2}
M.~Beneke, G.~Buchalla, M.~Neubert and C.~T.~Sachrajda,
%``QCD factorization for exclusive, non-leptonic B meson decays: 
% General  arguments and the case of heavy-light final states,''
Nucl.\ Phys.\ B {\bf 591} (2000) 313.
%[hep-ph/0006124].
%%CITATION = HEP-PH 0006124;%%


\bibitem{BB98}
P.~Ball and V.~M.~Braun,
%``Exclusive semileptonic and rare B meson decays in {QCD},''
Phys.\ Rev.\ D {\bf 58} (1998) 094016.
%[hep-ph/9805422].
%%CITATION = HEP-PH 9805422;%%


\bibitem{FS}
J.~M.~Flynn and C.~T.~Sachrajda,
%``Heavy quark physics from lattice QCD,''
hep-lat/9710057.
%%CITATION = HEP-LAT 9710057;%%


\bibitem{CLOPR}
J.~Charles {\it et al.},
%``Heavy-to-light form factors in the heavy mass to 
% large energy limit of  {QCD},''
Phys.\ Rev.\ D {\bf 60} (1999) 014001.
%[hep-ph/9812358].
%%CITATION = HEP-PH 9812358;%%


\bibitem{BF}
M.~Beneke and T.~Feldmann,
%``Symmetry-breaking corrections to heavy-to-light B meson 
% form factors at  large recoil,''
Nucl.\ Phys.\ B {\bf 592} (2001) 3.
%[hep-ph/0008255].
%%CITATION = HEP-PH 0008255;%%


\bibitem{BH}
G.~Burdman and G.~Hiller,
%``Semileptonic form-factors from B $\to$ K* gamma decays in the large  energy limit,''
Phys.\ Rev.\ D {\bf 63}, 113008 (2001)
[hep-ph/0011266].
%%CITATION = HEP-PH 0011266;%%


\bibitem{BBL}
G.~Buchalla, A.~J.~Buras and M.~E.~Lautenbacher,
%``Weak Decays Beyond Leading Logarithms,''
Rev.\ Mod.\ Phys.\ {\bf 68} (1996) 1125.
%[hep-ph/9512380].
%%CITATION = HEP-PH 9512380;%%



\bibitem{ABS}
A.~Ali, V.~M.~Braun and H.~Simma,
%``Exclusive radiative B decays in the light cone QCD sum rule approach,''
Z.\ Phys.\ C {\bf 63} (1994) 437.
%[hep-ph/9401277].
%%CITATION = HEP-PH 9401277;%%


\bibitem{BB96}
P.~Ball and V.~M.~Braun,
%``The $\rho$ Meson Light-Cone Distribution Amplitudes 
%of Leading Twist Revisited,''
Phys.\ Rev.\ D {\bf 54} (1996) 2182.
%[hep-ph/9602323].
%%CITATION = HEP-PH 9602323;%%


\bibitem{GP}
B.~Grinstein and D.~Pirjol,
%``Long-distance effects in B $\to$ V gamma radiative weak decays,''
Phys.\ Rev.\ D {\bf 62} (2000) 093002.
%[hep-ph/0002216].
%%CITATION = HEP-PH 0002216;%%

\bibitem{AHL}
A.~Ali, L.~T.~Handoko and D.~London,
%``Supersymmetric effects on isospin symmetry breaking and direct CP
%violation in B $\to$ rho gamma,''
hep-ph/0006175.
%%CITATION = HEP-PH 0006175;%%


\bibitem{GRG}
M.~Gronau and J.~L.~Rosner,
%``U-spin symmetry in doubly Cabibbo-suppressed charmed meson decays,''
Phys.\ Lett.\ B {\bf 500} (2001) 247;\\
%[hep-ph/0010237].
%%CITATION = HEP-PH 0010237;%%
M.~Gronau,
%``U-spin symmetry in charmless B decays,''
Phys.\ Lett.\ B {\bf 492} (2000) 297.
%[hep-ph/0008292].
%%CITATION = HEP-PH 0008292;%%


\bibitem{HM}
T.~Hurth and T.~Mannel,
%``CP asymmetries in b $\to$ (s/d) transitions as a test of CKM CP  violation,''
Phys.\ Lett.\ B {\bf 511}, 196 (2001)
[hep-ph/0103331].
%%CITATION = HEP-PH 0103331;%%


\bibitem{RF}
R.~Fleischer,
%``New strategies to extract beta and gamma from B/d $\to$ pi+ pi- 
%and  B/s $\to$ K+ K-,''
Phys.\ Lett.\ B {\bf 459} (1999) 306.
%[hep-ph/9903456].
%%CITATION = HEP-PH 9903456;%%







\end{thebibliography}
 
\vfill\eject


\end{document}


hep-ph0109248.extracted_bib
\begin{thebibliography}{99}

\bibitem{BB}
%\cite{Bosch:2001gv}
S.~W.~Bosch and G.~Buchalla,
%``The radiative decays B $\to$ V gamma at next-to-leading order in QCD,''
hep-ph/0106081.
%%CITATION = HEP-PH 0106081;%%


\bibitem{CHENHON}
S.~Chen {\it et al.}  [CLEO Collaboration],
%``Branching fraction and photon energy spectrum for b $\to$ s gamma,''
hep-ex/0108032;\\
%%CITATION = HEP-EX 0108032;%%
K. Honscheid, this conference.

\bibitem{BAR}
R.~Barate {\it et al.}  [ALEPH Collaboration],
%``A measurement of the inclusive b --> s gamma branching ratio,''
Phys.\ Lett.\ B {\bf 429} (1998) 169.
%%CITATION = PHLTA,B429,169;%%


\bibitem{ABENAK}
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.\ B {\bf 511}, 151 (2001)
[hep-ex/0103042];\\
%%CITATION = HEP-EX 0103042;%%
M.~Nakao [Belle collaboration], this conference.


\bibitem{COA}
T.~E.~Coan {\it et al.}  [CLEO Collaboration],
%``Study of exclusive radiative B meson decays,''
Phys.\ Rev.\ Lett.\ {\bf 84} (2000) 5283.
%[hep-ex/9912057].
%%CITATION = HEP-EX 9912057;%%


\bibitem{SPARYD}
%\cite{Spanier:2001cc}
S.~Spanier  [BaBar Collaboration],
%``Rare B decays with the BaBar detector,''
SLAC-PUB-8987
{\it Presented at the Kaon 2001 Conference, Pisa, Italy, June 12 - June 17, 2001};\\
A.~Ryd [BaBar collaboration], this conference.


\bibitem{NAK}
M.~Nakao [Belle collaboration], this conference.


\bibitem{BBM}
S.~Bertolini, F.~Borzumati and A.~Masiero,
%``QCD Enhancement Of Radiative B Decays,''
Phys.\ Rev.\ Lett.\ {\bf 59} (1987) 180.
%%CITATION = PRLTA,59,180;%%

\bibitem{AGM}
A.~Ali, C.~Greub and T.~Mannel,
%``Rare B decays in the Standard Model,''
DESY-93-016, {\it To be publ. in Proc. of ECFA Workshop on the 
Physics of a B Meson Factory, Eds. R. Aleksan, A. Ali, 1993}.

\bibitem{BMMP}
A.~J.~Buras, M.~Misiak, M.~M\"u{}nz and S.~Pokorski,
%``Theoretical uncertainties and phenomenological aspects of 
% B $\to$ X(s) gamma decay,''
Nucl.\ Phys.\ B {\bf 424} (1994) 374.
%[hep-ph/9311345].
%%CITATION = HEP-PH 9311345;%%

\bibitem{CFMRS}
M.~Ciuchini {\it et al.},
%``Scheme independence of the effective Hamiltonian for 
% b $\to$ s gamma and b $\to$ s g decays,''
Phys.\ Lett.\ B {\bf 316} (1993) 127;\\
%[hep-ph/9307364].
%%CITATION = HEP-PH 9307364;%%
M.~Ciuchini, E.~Franco, L.~Reina and L.~Silvestrini,
%``Leading order QCD corrections to b $\to$ s gamma and 
% b $\to$ s g decays in three regularization schemes,''
Nucl.\ Phys.\ B {\bf 421} (1994) 41.
%[hep-ph/9311357].
%%CITATION = HEP-PH 9311357;%%




\bibitem{AY}
K.~Adel and Y.~Yao,
%``Exact alpha-s calculation of b $\to$ s + gamma b $\to$ s + g,''
Phys.\ Rev.\ D {\bf 49} (1994) 4945;\\
%[hep-ph/9308349].
%%CITATION = HEP-PH 9308349;%%
C.~Greub and T.~Hurth,
%``Two-loop matching of the dipole operators 
% for b --> s gamma and b -->  s g,''
Phys.\ Rev.\ D {\bf 56} (1997) 2934;\\
%[hep-ph/9703349].
%%CITATION = HEP-PH 9703349;%%
A.~J.~Buras, A.~Kwiatkowski and N.~Pott,
%``Next-to-leading order matching 
% for the magnetic photon penguin operator  in the B --> X/s gamma decay,''
Nucl.\ Phys.\ B {\bf 517} (1998) 353.
%[hep-ph/9710336].
%%CITATION = HEP-PH 9710336;%%

\bibitem{GHW}
C.~Greub, T.~Hurth and D.~Wyler,
%``Virtual $O(\a_s)$ corrections to the inclusive decay $b \to s \gamma$,''
Phys.\ Rev.\ D {\bf 54} (1996) 3350;\\
%[hep-ph/9603404].
%%CITATION = HEP-PH 9603404;%%
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,''
hep-ph/0105160.
%%CITATION = HEP-PH 0105160;%%



\bibitem{CMM}
K.~Chetyrkin, M.~Misiak and M.~M\"unz,
%``Weak radiative B-meson decay beyond leading logarithms,''
Phys.\ Lett.\ B {\bf 400} (1997) 206;\\
%[hep-ph/9612313].
%%CITATION = HEP-PH 9612313;%%
Erratum-ibid. B {\bf 425} (1998) 414.

\bibitem{MG}
M.~Misiak,
%``Theory of radiative B decays,''
hep-ph/0002007;\\
%%CITATION = HEP-PH 0002007;%%
C.~Greub,
%``Theory of rare B meson decays,''
hep-ph/9911348.
%%CITATION = HEP-PH 9911348;%%


\bibitem{AAWGSW}
H.~H.~Asatrian, H.~M.~Asatrian and D.~Wyler,
%``NLL corrections for B meson radiative exclusive decays,''
Phys.\ Lett.\ B {\bf 470} (1999) 223;\\
%[hep-ph/9905412].
%%CITATION = HEP-PH 9905412;%%
C.~Greub, H.~Simma and D.~Wyler,
%``Branching ratio and direct CP violating rate asymmetry of the rare
%decays B $\to$ K* gamma and B $\to$ rho gamma,''
Nucl.\ Phys.\ B {\bf 434} (1995) 39
[Erratum-ibid.\ B {\bf 444} (1995) 447].
%[hep-ph/9406421].
%%CITATION = HEP-PH 9406421;%%


\bibitem{BFS}
%\cite{Beneke:2001at}
M.~Beneke, T.~Feldmann and D.~Seidel,
%``Systematic approach to exclusive B $\to$ V l+ l-, V gamma decays,''
hep-ph/0106067.
%%CITATION = HEP-PH 0106067;%%


\bibitem{AP}
A.~Ali and A.~Y.~Parkhomenko,
%``Branching ratios for B $\to$ rho gamma decays in 
%next-to-leading order in  alpha(s) including hard spectator corrections,''
hep-ph/0105302.
%%CITATION = HEP-PH 0105302;%%


\bibitem{BBNS1}
M.~Beneke, G.~Buchalla, M.~Neubert and C.~T.~Sachrajda,
%``{QCD} factorization for B --> pi pi decays: 
% Strong phases and CP  violation in the heavy quark limit,''
Phys.\ Rev.\ Lett.\ {\bf 83} (1999) 1914.
%[hep-ph/9905312].
%%CITATION = HEP-PH 9905312;%%

\bibitem{BBNS2}
M.~Beneke, G.~Buchalla, M.~Neubert and C.~T.~Sachrajda,
%``QCD factorization for exclusive, non-leptonic B meson decays: 
% General  arguments and the case of heavy-light final states,''
Nucl.\ Phys.\ B {\bf 591} (2000) 313.
%[hep-ph/0006124].
%%CITATION = HEP-PH 0006124;%%


\bibitem{BB98}
P.~Ball and V.~M.~Braun,
%``Exclusive semileptonic and rare B meson decays in {QCD},''
Phys.\ Rev.\ D {\bf 58} (1998) 094016.
%[hep-ph/9805422].
%%CITATION = HEP-PH 9805422;%%


\bibitem{FS}
J.~M.~Flynn and C.~T.~Sachrajda,
%``Heavy quark physics from lattice QCD,''
hep-lat/9710057.
%%CITATION = HEP-LAT 9710057;%%


\bibitem{CLOPR}
J.~Charles {\it et al.},
%``Heavy-to-light form factors in the heavy mass to 
% large energy limit of  {QCD},''
Phys.\ Rev.\ D {\bf 60} (1999) 014001.
%[hep-ph/9812358].
%%CITATION = HEP-PH 9812358;%%


\bibitem{BF}
M.~Beneke and T.~Feldmann,
%``Symmetry-breaking corrections to heavy-to-light B meson 
% form factors at  large recoil,''
Nucl.\ Phys.\ B {\bf 592} (2001) 3.
%[hep-ph/0008255].
%%CITATION = HEP-PH 0008255;%%


\bibitem{BH}
G.~Burdman and G.~Hiller,
%``Semileptonic form-factors from B $\to$ K* gamma decays in the large  energy limit,''
Phys.\ Rev.\ D {\bf 63}, 113008 (2001)
[hep-ph/0011266].
%%CITATION = HEP-PH 0011266;%%


\bibitem{BBL}
G.~Buchalla, A.~J.~Buras and M.~E.~Lautenbacher,
%``Weak Decays Beyond Leading Logarithms,''
Rev.\ Mod.\ Phys.\ {\bf 68} (1996) 1125.
%[hep-ph/9512380].
%%CITATION = HEP-PH 9512380;%%



\bibitem{ABS}
A.~Ali, V.~M.~Braun and H.~Simma,
%``Exclusive radiative B decays in the light cone QCD sum rule approach,''
Z.\ Phys.\ C {\bf 63} (1994) 437.
%[hep-ph/9401277].
%%CITATION = HEP-PH 9401277;%%


\bibitem{BB96}
P.~Ball and V.~M.~Braun,
%``The $\rho$ Meson Light-Cone Distribution Amplitudes 
%of Leading Twist Revisited,''
Phys.\ Rev.\ D {\bf 54} (1996) 2182.
%[hep-ph/9602323].
%%CITATION = HEP-PH 9602323;%%


\bibitem{GP}
B.~Grinstein and D.~Pirjol,
%``Long-distance effects in B $\to$ V gamma radiative weak decays,''
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%[hep-ph/0002216].
%%CITATION = HEP-PH 0002216;%%

\bibitem{AHL}
A.~Ali, L.~T.~Handoko and D.~London,
%``Supersymmetric effects on isospin symmetry breaking and direct CP
%violation in B $\to$ rho gamma,''
hep-ph/0006175.
%%CITATION = HEP-PH 0006175;%%


\bibitem{GRG}
M.~Gronau and J.~L.~Rosner,
%``U-spin symmetry in doubly Cabibbo-suppressed charmed meson decays,''
Phys.\ Lett.\ B {\bf 500} (2001) 247;\\
%[hep-ph/0010237].
%%CITATION = HEP-PH 0010237;%%
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%%CITATION = HEP-PH 0008292;%%


\bibitem{HM}
T.~Hurth and T.~Mannel,
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%%CITATION = HEP-PH 0103331;%%


\bibitem{RF}
R.~Fleischer,
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%%CITATION = HEP-PH 9903456;%%







\end{thebibliography}