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%% section 5 Results and conclusions [slacpub7144005 in slacpub7144005: slacpub7144006]
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\section{\usemenu{slacpub7144::context::slacpub7144005}{Results and conclusions}}\label{section::slacpub7144005}
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We have calculated the virtual
corrections to $b \to s \gamma$ coming from the operators
$O_2$, $O_7$ and $O_8$. The contributions from the other
fourFermi operators
in eq. (\docLink{slacpub7144001.tcx}[operators]{1.2}) which are given by the analogous diagrams
as shown in Figs. (\docLink{slacpub7144002.tcx}[fig:1]{1})  (\docLink{slacpub7144002.tcx}[fig:4]{4}) were
neglected, because they either
vanish ($O_1$) or have Wilson coefficients
which are about fifty times
smaller than that of $O_2$ while their matrix elements
can be enhanced at most by color factors.
However, we did include the nonvanishing
diagrams of $O_5$ and $O_6$
where the gluon connects the external
quark lines and the photon is radiated from the charm quark
because these corrections are
automatically considered
when $C_7^{eff}$ (defined in eq.(\docLink{slacpub7144001.tcx}[C78eff]{1.3}))
is used instead of $C_7$.
As discussed in section three, some of the Bremsstrahlung
corrections to the operator $O_7$ have been transferred into
the matrix element for $b \to s \gamma$ in order
to present results which are free from
infrared and collinear singularities.
The sum of the various contributions derived
in the previous sections yields the amplitude $A(b \to s \g)$
for $b \to s \gamma$. The result can be presented in
a convenient way, following the treatment of
Buras et al. \cite{6}, where the general structure
of the nexttoleading order result is discussed in detail.
We write
\be
\label{amplitudevirt}
A(b \to s \g) = \frac{4 G_F \l_t}{\sqrt{2}} \, \hat{D} \,
\bra s \gO_7(\mu)b \ket _{tree}
\ee
with
$\hat{D}$
\be
\label{dhat}
\hat{D} = C_7^{eff}(\mu) + \frac{\a_s(\mu)}{4\p} \sum_{i}
\left(
C_i^{(0)eff}(\mu) \ell_i \log \frac{m_b}{\mu} +
C_i^{(0)eff} r_i
\right) \quad ,
\ee
and where the quantities $\ell_i$ and $r_i$ are given
for $i=2,7,8$ in sections 2, 3 and 4, respectively.
For the full nexttoleading logarithmic result one would need
the first term on the rhs of eq. (\docLink{slacpub7144005.tcx}[dhat]{5.2}) ,$C_7^{eff}(\mu)$,
at nexttoleading logarithmic
precision. In contrast,
it is consistent
to use the leading logarithmic values
for the other Wilson coefficients in eq. (\docLink{slacpub7144005.tcx}[dhat]{5.2}).
As the nexttoleading coefficient $C_7^{eff}$ is not known
yet, we replace it by its
leading logarithmic value $C_7^{(0)eff}$
in the numerical investigations.
The notation $\bra s \gO_7(\mu)b \ket _{tree}$ in
eq. (\docLink{slacpub7144005.tcx}[amplitudevirt]{5.1}) indicates that the explicit
factor $m_b$
in the operator $O_7$ is the running mass taken
at the scale $\mu$.
As the relevant scale for a $b$ quark decay is expected to
be $\mu \sim m_b$, we expand the matrix elements of the
operators around
$\mu=m_b$ up to order $O(\a_s)$.
Thus we arrive at
\be
\label{amplitudevirtuell}
A(b \to s \g) = \frac{4 G_F \l_t}{\sqrt{2}} \, D \,
\bra s \gO_7(m_b)b \ket _{tree}
\ee
with
$D$
\be
\label{d}
D = C_7^{eff}(\mu) + \frac{\a_s(m_b)}{4\p} \sum_{i} \left(
C_i^{(0)eff}(\mu) \gamma_{i7}^{(0)eff} \log \frac{m_b}{\mu} +
C_i^{(0)eff} r_i
\right) \quad ,
\ee
with the quantities $\gamma_{i7}^{(0)eff}$
\be
\gamma_{i7}^{(0)eff} = \ell_i + 8 \delta_{i7}
\ee
being just the entries of the (effective) leading order anomalous
dimension matrix \cite{6}.
As also pointed out in this reference,
the explicit logarithms of the form
$\a_s(m_b) \log(m_b/\mu)$
in eq. (\docLink{slacpub7144005.tcx}[d]{5.4})
should be cancelled by the $\mu$dependence of $C_7^{(0)eff}(\mu)$.
This is the crucial point why the scale dependence
is reduced significantly as we will see later.
\footnote{As we neglect the virtual correction of $O_3$$O_6$,
there is a small leftover
$\mu$ dependence, of course.}
From $A(b \to s \g)$ in eq.
(\docLink{slacpub7144005.tcx}[amplitudevirtuell]{5.3}) we obtain
the decay width $\G^{virt}$ to be
\be
\label{widthvirt}
\G^{virt} = \frac{m_{b,pole}^5 \, G_F^2 \l_t^2 \a_{em}}{32 \p^4}
\, F \, D^2 \quad ,
\ee
where we discard term of $O(\a_s^2)$
in $D^2$.
The factor $F$ in eq. (\docLink{slacpub7144005.tcx}[widthvirt]{5.6}) is
\be
F = \left( \frac{m_b(\mu=m_b)}{m_{b,pole}} \right)^2 =
1 \frac{8}{3} \, \frac{\a_s(m_b)}{\p} \quad .
\ee
To obtain the inclusive rate for $B \to X_s \g$
consistently at the nexttoleading order level,
we have to take into account all the Bremsstrahlung contributions.
They have been calculated by
Ali and Greub for the operators
$O_2$ and $O_7$ some time ago \cite{3}, while the complete set has been
worked out only recently
\cite{4,14,15}.
Here, we neglect the small
contribution
of the operators $O_3$  $O_6$ as we did for the virtual corrections,
i.e., we only consider $O_2$, $O_7$ and $O_8$.
The corresponding Bremsstrahlung formulae are collected
in Appendix B.
In order to arrive at the branching ratio $\mbox{BR}(b \to s \g (g))$,
we divide, as usual, the decay width $\G(B \to X_s \g) =
\G^{virt} + \G^{brems}$
by the theoretical expression for the
semileptonic decay width $\G_{sl}$ and
multiply this ratio with the measured semileptonic branching ratio
$\mbox{BR}_{sl} = (10.4 \pm 0.4)\%$ \cite{27}, i.e.,
\be
\label{brformel}
\mbox{BR}(b \to s \g (g)) = \frac{\G}{\G_{sl}} \,
\mbox{BR}_{sl} \quad .
\ee
\begin{figure}[htb]
\vspace{0.10in}
\centerline{
\epsfig{file=mtspecllog.ps,height=2.5in,angle=0,clip=}
}
\vspace{0.08in}
\caption[]{Branching ratio for $b \to s \g$
based on the leading logarithmic formula in eq.
(\docLink{slacpub7144005.tcx}[leadinglog]{5.12}).
The upper (lower) solid curve
is for $\mu=m_b/2$ ($\mu = 2 m_b$). The dotted curves show the
CLEO bounds \cite{2}.
\label{fig:mtspecllog}}
\end{figure}
\begin{figure}[htb]
\vspace{0.10in}
\centerline{
\epsfig{file=mtspecali.ps,height=2.5in,angle=0,clip=}
}
\vspace{0.08in}
\caption[]{Branching ratio for $b \to s \g (g)$ neglecting
the virtual corrections of $O_2$ and $O_8$
calculated in the present
paper.
The upper (lower) solid curve
is for $\mu=m_b/2$ ($\mu = 2 m_b$).
The dotted curves show the CLEO bounds \cite{2}.
\label{fig:mtspecali}}
\end{figure}
\begin{figure}[htb]
\vspace{0.10in}
\centerline{
\epsfig{file=mtspecfinal.ps,height=2.5in,angle=0,clip=}
}
\vspace{0.08in}
\caption[]{Branching ratio for $b \to s \g (g)$
based on the complete
formulae presented in this section. The upper (lower) solid curve
is for $\mu=m_b/2$ ($\mu = 2 m_b$). The dotted curves show the CLEO
bounds \cite{2}.
\label{fig:mtspecfinal}}
\end{figure}
The semileptonic decay width is
\be
\label{semileptonic}
\G_{sl} = \frac{G_F^2 \, m_{b,pole}^5 \, V_{cb}^2}{192 \p^3} \,
g(m_c/m_b) \, \left( 1 
\frac{2 \a_s(m_b)}{3 \p} f(m_c/m_b) \right) \quad ,
\ee
where the phase space function $g(u)$ is defined as
\be
\label{gz}
g(u) = 1  8 u^2 + 8 u^6  u^8  24 u^4 \log u \quad ,
\ee
and an approximate analytic form
for the radiative correction function
$f(u)$ has been found
in \cite{28} to be
\be
\label{fz}
f(u) = \left( \p^2  \frac{31}{4} \right) \, (1u)^2 + \frac{3}{2}
\quad .
\ee
In Figs. \docLink{slacpub7144005.tcx}[fig:mtspecllog]{10}\docLink{slacpub7144005.tcx}[fig:mtspecfinal]{12} we compare the
available leadinglog results and our new results
for the inclusive branching ratio for $B \to X_s \g$
as a function of the top quark mass.
A rather crucial parameter is the ratio $m_c/m_b$;
it enters both,
$b \to s \g$ through the
virtual corrections of $O_2$ and
the semileptonic decay width through phase space.
It can be written as $m_c/m_b=1(m_bm_c)/m_b$.
While the mass difference $m_b  m_c$
is determined quite precisely through the $1/m_Q$
expansion \cite{29} or from the semileptonic
$b \to c$ spectrum; we use $m_b  m_c=3.40$ GeV \cite{30},
the $b$quark mass is not precisely known.
Using for the $b$ quark pole
mass $m_{b,pole}=4.8 \pm 0.15$ GeV one arrives at $m_c/m_b=0.29 \pm 0.02$.
In the plots the central values for $m_{b,pole}$
and $m_c/m_b$ have been used. Moreover, we put $V_{cb}=V_{ts}$ and $V_{tb}=1$
and also use the central value for the measured semileptonic branching
ratio $\mbox{BR}_{sl}=10.4\%$ in eq. (\docLink{slacpub7144005.tcx}[brformel]{5.8}).
In all three plots the horizontal dotted
curves show the CLEO $1 \sigma$limits for the branching ratio
$\mbox{BR}(B \to X_s \g)$ \cite{2}.
In Fig. \docLink{slacpub7144005.tcx}[fig:mtspecllog]{10} we show the
leading logarithmic
result for the branching ratio of $b \to s \g$,
based on the formula
\be
\label{leadinglog}
\mbox{BR}(b \to s \g)^{leading} =
\frac{6 \a_{em}}{\p g(m_c/m_b)} \, C_7^{(0)eff}^2 \,
\mbox{BR}_{sl} \quad .
\ee
Similarly, Fig. \docLink{slacpub7144005.tcx}[fig:mtspecali]{11} exhibits
the results also
taking into
account the Bremsstrahlung corrections and the virtual
corrections to $O_7$ without including the virtual corrections of $O_2$
and $O_8$. We can
reproduce this result by
putting $\ell_2=r_2=\ell_8=r_8=0$ in our formulae.
As noticed in the literature \cite{3,15},
the $\mu$ dependence in this case is even larger
than in the leading
logarithmic result shown in Fig. \docLink{slacpub7144005.tcx}[fig:mtspecllog]{10}.
Also, the experimentally allowed region is shown.
The theoretical results shown in Figs. \docLink{slacpub7144005.tcx}[fig:mtspecllog]{10}
and
\docLink{slacpub7144005.tcx}[fig:mtspecali]{11}
allowed for a reasonable
prediction of the branching ratio within a large error
which was essentially determined by the $\mu$ dependence.
As we see, they do agree well with experiment but within
large uncertainties.
In Fig. \docLink{slacpub7144005.tcx}[fig:mtspecfinal]{12}, finally, we
give the branching ratio
based on formula (\docLink{slacpub7144005.tcx}[brformel]{5.8}) which
includes all virtual corrections calculated
in the present paper.
Because all the logarithms
of the form $\a_s(m_b) \log(m_b/\mu)$ cancel as discussed
above, the $\mu$ dependence is significantly reduced in our improved
calculation (Fig. \docLink{slacpub7144005.tcx}[fig:mtspecfinal]{12}); the bands of
scale uncertainty are rather narrow.
To illustrate the remaining renormalization scale dependence,
we present two 'scenarios'
which differ by higher order contributions. First, we take the
explicit $\alpha_s$ factors in eqs. (\docLink{slacpub7144005.tcx}[d]{5.4})
and (\docLink{slacpub7144005.tcx}[semileptonic]{5.9}) at $\mu=m_b$ as indicated in these
formulae; varying $\mu$ between $(m_b/2)$ and $(2 \, m_b)$
in formula (\docLink{slacpub7144005.tcx}[d]{5.4}) leads to the
dashdotted curves in Fig. \docLink{slacpub7144005.tcx}[fig:mtspecfinal]{12}.
Second, we evaluate the explicit $\alpha_s$ in eqs. (\docLink{slacpub7144005.tcx}[d]{5.4})
and (\docLink{slacpub7144005.tcx}[semileptonic]{5.9}) at the (variable) scale $\mu$.
Varying again the scale $\mu$ between $(m_b/2)$ and $(2\,m_b)$
yields the solid lines in Fig. \docLink{slacpub7144005.tcx}[fig:mtspecfinal]{12}.
In both scenarios
the upper (lower) curve corresponds to $\mu=m_b/2$ ($\mu=2m_b$).
We mention that the $\mu$band is larger in the second scenario
and it is therefore safer to use this band to
obtain a feeling for the remaining scale uncertainties.
While this result shows that the theoretical accuracy can
be strongly improved by the nexttoleading calculations,
it would be premature to extract a prediction
for the branching ratio from Fig. \docLink{slacpub7144005.tcx}[fig:mtspecfinal]{12}
(with obviously a small error) and claim, for instance,
a discrepancy with experiment. This only will become
possible (with high theoretical precision)
if also $C_7^{eff}$ is known to nexttoleading
logarithmic precision. This additional effect will , essentially,
shift the narrow bands in Fig. \docLink{slacpub7144005.tcx}[fig:mtspecfinal]{12}, without
broadening them substantially.
The drastic reduction of the theoretical
uncertainties shows that significant experimental
improvements are
necessary to extract the important information
available in these decays.
\vspace{2cm}
{\bf Acknowledgements}
We thank A. Ali, M. Beneke, R. Blankenbecler,
S. Brodsky, J. Hewett, M. Lautenbacher, M. Peskin,
J. Soares
and M. Worah for discussions.
We are particularly indebted to M. Misiak and L. Reina
for many useful comments. One of us (C.G.) would like to thank the
Institute for Theoretical Physics in Z\"urich
for the kind hospitality.
\newpage
\centerline{\Large {\bf Appendix}}
\appendix
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