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%% section* Appendix C: Comment on $\tau_{B^+}/\tau_{B_d}$ [slacpub7165005u3 in slacpub7165005u3: slacpub7165005u4]
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\section*{\usemenu{slacpub7165::context::slacpub7165005u3}{Appendix C: Comment on $\tau_{B^+}/\tau_{B_d}$}}\label{section::slacpub7165005u3}
\label{bpbd}
Some of the issues in the calculation of lifetime differences
among $B_s$ and $B_d$ mesons that we have discussed in
this paper are also relevant for the prediction of
$\tau_{B^+}/\tau_{B_d}$. We will therefore take the opportunity
to also have a brief look at the question of the $B^+B_d$
lifetime difference.
In the literature this quantity has been estimated to be
\cite{3}
\begin{equation}\label{tbpd}
\frac{\tau_{B^+}}{\tau_{B_d}}\simeq 1+0.05\cdot
\frac{f^2_B}{(200\,{\rm MeV})^2},
\end{equation}
\noindent predicting the $B^+$ lifetime to exceed $\tau_{B_d}$ by
several percent. In the following we would like to reexamine
this estimate, emphasizing the theoretical uncertainties that are
involved in its derivation.
Assuming isospin symmetry, the mechanisms that produce a difference
in $\tau_{B^+}$ and $\tau_{B_d}$ first enter at the level of
dimension six operators, or equivalently at ${\cal O}(1/m^3_b)$,
in the heavy quark expansion \cite{3}. These effects are
weak annihilation for the $B_d$ and Pauli interference in the
case of $B^+$. As we have seen in section \docLink{slacpub7165004.tcx}[bsbd]{4}, the
weak annihilation contribution to $\tau_{B_d}$ is very small
and we shall neglect it. In this approximation the difference
between $\tau_{B^+}$ and $\tau_{B_d}$ arises only through
Pauli interference and one may write
\begin{equation}\label{tbpi}
\frac{\tau_{B^+}}{\tau_{B_d}}=1+24\pi^2 B(B\to Xe\nu)\,
\frac{f^2_B}{m^2_b} V^2_{ud} \frac{(1z)^2}{g(z)\,\tilde{\eta}_{QCD}}
\left[(C^2_C^2_+)B^{(u)}_1\frac{1}{N_c}
(C^2_++C^2_)B^{(u)}_2\right],
\end{equation}
\noindent where
\begin{eqnarray}\label{bu12}
\langle B^+(\bar b_iu_i)_{VA}(\bar u_jb_j)_{VA}B^+\rangle
&=& f^2_{B} m^2_b B^{(u)}_1 \nonumber \\
\langle B^+(\bar b_iu_j)_{VA}(\bar u_jb_i)_{VA}B^+\rangle
&=& \frac{1}{N_c} f^2_{B} m^2_b B^{(u)}_2
\end{eqnarray}
\noindent define the bag parameters $B^{(u)}_{1,2}$.
The Wilson coefficients $C_\pm$ have been given
in (\docLink{slacpub7165002.tcx}[c21pm]{11}).
With $m_b=4.8\,$GeV, $m_c=1.4\,$GeV, $\Lambda_{LO}=0.2\,$GeV
and taking $B^{(u)}_{1,2}=1$, $f_B=0.2$\,GeV, one finds
$\tau_{B^+}/\tau_{B_d}=1.02$, indicating a slightly longer
lifetime for $B^+$ than for $B_d$. This number can however not
be viewed as a very accurate prediction. In fact, the two
contributions proportional to $B^{(u)}_1$ and $B^{(u)}_2$
in (\docLink{slacpub7165005u3.tcx}[tbpi]{77}) enter with different sign. This leads to a
partial cancellation that has the tendency to make the result
unstable. For instance, allowing the unphysical scale
$\mu={\cal O}(m_b)$ in the coefficients $C_\pm$ to vary from
$m_b/2$ to $2m_b$ gives a range of $1.00  1.06$ for the
$B^+$ to $B_d$ lifetime ratio. Switching off short distance
QCD corrections completely ($C_\pm\to 1$), the hierarchy of
lifetimes would even be reversed to $\tau_{B^+}/\tau_{B_d}=0.95$,
which is another aspect of the large sensitivity to QCD effects.
An alternative way of estimating the present uncertainty is to
allow a variation in the bag parameters (keeping $\mu=m_b$ fixed).
A range of $B^{(u)}_{1,2}=1.0\pm 0.3$ is certainly conceivable,
considering the uncertainties in the nonperturbative dynamics
and from the scale and scheme dependence in the
longdistance to shortdistance matching. Assuming this,
we obtain for $f_B=0.2$\,GeV, $\tau_{B^+}/\tau_{B_d}=1.02\pm 0.04$.
A combination of both variations, of scale and bag parameters,
would even allow us to obtain a lifetime difference of up to $20\%$,
$\tau_{B^+}/\tau_{B_d}\sim 1.2$. Although we consider this case highly
unlikely the point to note is that a lifetime that large could be
tolerated by QCD as well as equal
lifetimes, or even a marginally shorter lifetime for the $B^+$.
A decisive improvement of this situation could only be achieved
by a reliable lattice calculation of $B^{(u)}_{1,2}$ in
conjunction with a nexttoleading order computation of
shortdistance QCD corrections to ensure a proper matching
in renormalization scheme and scale between Wilson coefficients
and hadronic matrix elements. Alternatively one could use the
present measurement $\tau_{B^+}/\tau_{B_d}=1.06\pm 0.04$ \cite{26}
to constrain the bag parameters. At present such constraints
appear to be of limited use, because of the large renormalization
scale dependence of Pauli interference at leading order.
Similar conclusions have been reached in the recent paper
by Neubert and Sachrajda \cite{36}.
The authors of \cite{3} have modeled the bag parameters
in their estimate of $\tau_{B^+}/\tau_{B_d}$ by factorizing
at a low scale $\mu_h< m_b$ and explicitly including the
leading logarithms of HQET. This yields
\begin{equation}\label{buhl}
B^{(u)}_1(m_b)=\frac{8}{9}
\left[\frac{\alpha_s(m_b)}{\alpha_s(\mu_h)}\right]^{3/50}+
\frac{1}{9}
\left[\frac{\alpha_s(m_b)}{\alpha_s(\mu_h)}\right]^{12/25}
\qquad
B^{(u)}_2(m_b)=
\left[\frac{\alpha_s(m_b)}{\alpha_s(\mu_h)}\right]^{12/25} .
\end{equation}
\noindent Taking $\mu_h=1$\,GeV this gives $B^{(u)}_1(m_b)=1.01$,
$B^{(u)}_2(m_b)=0.72$ and $\tau_{B^+}/\tau_{B_d}=1.04$
(for $f_B=0.2$\,GeV), favoring $\tau_{B^+}>\tau_{B_d}$.
However, as discussed at the end of section \docLink{slacpub7165002.tcx}[basic]{2}, the
quantitative reliability of an estimate based on hybrid logarithms
is not entirely clear.
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