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\title{$b$-Hadrons: Mixing and Lifetimes with a Lattice Perspective}
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\author{JM Flynn\address{School of Physics \& Astronomy\\
University of Southampton, Highfield, Southampton SO17 1BJ, UK}}
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\runtitle{$b$-Hadrons: Mixing and Lifetimes}
\runauthor{JM Flynn}
\begin{document}
\begin{abstract}
I review theoretical calculations of the nonperturbative parameters
needed to describe $B$-meson mixing and $b$-hadron lifetime ratios and
lifetime differences. I take a lattice QCD perspective and close with
some comments on the current status of lattice calculations.
\end{abstract}
\maketitle
\section{MIXING AND DECAY}
The effective Hamiltonian and physical states for the $\ket B$,
$\ket{\bar B}$ system are
\begin{gather*}
\begin{pmatrix}
M_{11} & M_{12} \\ M^*_{12} & M_{11}
\end{pmatrix}
- \frac i2
\begin{pmatrix}\Gamma_{11} & \Gamma_{12} \\ \Gamma^*_{12} & \Gamma_{11}
\end{pmatrix}\\
\ket{B_H} = p\ket B + q\ket{\bar B},\qquad
\ket{B_L} = p\ket B - q\ket{\bar B}
\end{gather*}
where $|p|^2+|q|^2=1$. The off-diagonal, $\Delta B=2$, entries can be
probed by measuring:
\[
\begin{aligned}
\mbox{mass difference}&& \Delta m = M_H-M_L \approx 2|M_{12}|\\
\mbox{width difference}&& \Delta\Gamma = \Gamma_L - \Gamma_H
\approx 2 |\Gamma_{12}|\cos\phi\\
\mbox{CP asymmetry}&& \left|\frac qp \right| -1 \approx
-\frac12 \mathrm{Im}\, \frac{\Gamma_{12}}{M_{12}}
\end{aligned}
\]
In the Standard Model (SM) the phase $\phi$ is expected to be small
for $\Delta\Gamma_s$, while $\Delta\Gamma_d$ is negligible.
\section{MIXING}
In the SM, $\Delta m_q$ for a neutral $B_q$-meson containing light
quark $q$, with mass $M_{B_q}$, is governed by the matrix element,
\[
\langle \bar B_q | (\bar b q)_{V-A}(\bar b q)_{V-A}| B_q\rangle =
\frac83 B_{B_q} f_{B_q}^2 M_{B_q}
\]
where $f_{B_q}$ is the meson decay constant and $B_{B_q}$ would be $1$
if the vacuum insertion approximation were correct. Knowledge of
$f_{B_q}\sqrt{B_{B_q}}$ for $q=d,s$ is needed to use experimental
$B_q$ mixing information to constrain the CKM unitarity triangle.
From the point of view of lattice calculations, the quantities with
the least-correlated errors are
\[
f_{B_s}\sqrt{B_{B_s}} \qquad\mbox{and}\qquad
\xi\equiv\frac{f_{B_s}\sqrt{B_{B_s}}}{f_{B_d}\sqrt{B_{B_d}}}.
\]
The ratio $\xi$ is most sensitive to chiral extrapolation errors,
while $f_{B_s}\sqrt{B_{B_s}}$ is more sensitive to the remaining
lattice systematics. Figure~\ref{fig:phirat} shows the ratio
$\phi(B_s)/\phi(B_q) \equiv
f_{B_s}\sqrt{m_{B_S}}/f_{B_q}\sqrt{m_{B_q}}$ using HPQCD lattice
results, taken from~\cite{Wingate:2006ie}. This ratio, and $\xi$ to
which it is closely related, depends quite strongly on the light quark
mass. To get the physical result one has to reach the blue line on the
left of the plot. Impressive progress has been made using staggered
fermions to push down the mass of the light dynamical quarks in
simulations. Figure~\ref{fig:fbs-history} (updated from the one shown
by Hashimoto at ICHEP04~\cite{Hashimoto:2004hn}) shows a history of
lattice results for $f_{B_s}$, culminating with the most recent
$2{+}1$-flavour dynamical simulations using staggered fermions. There
is a discernible increase in the value of $f_{B_s}$ in the unquenched
results.
%
\begin{figure}
\includegraphics[width=\hsize]{wingate-phiratio.eps}
\caption{Chiral extrapolation of $\phi(B_s)/\phi(B_q) \equiv
f_{B_s}\sqrt{m_{B_S}}/f_{B_s}\sqrt{m_{B_S}}$ using HPQCD lattice
results~\cite{Wingate:2006ie}.}
\label{fig:phirat}
\end{figure}
%
\begin{figure}
\includegraphics[width=\hsize,bb=79 432 379 771]{fbs-history.ps}
\caption{Lattice results for $f_{B_s}$, updated
from~\cite{Hashimoto:2004hn} (see~\cite{Hashimoto:2004hn} for
the $N_f=0,2$ references). The grey band is the average,
$f_{B_s}=230\pm30\mev$ given in~\cite{Hashimoto:2004hn}.}
\label{fig:fbs-history}
\end{figure}
%
Results for $B_{B_q}$ are not yet available from staggered fermions,
so rather than combine results from different formalisms, I prefer to
quote the averages given by Hashimoto~\cite{Hashimoto:2004hn}:
\[
\begin{aligned}
f_{B_s} &= 230(30)\mev,\\
f_{B_s}\sqrt{B_{B_s}} &= 262(35)\mev,\\
\xi &=1.23(6).
\end{aligned}
\]
However, the combination \emph{is} done
in~\cite{Wingate:2006ie,Okamoto:2005zg,Mackenzie:2006un} with the
result that $f_{B_s}$ and $f_{B_s}\sqrt{B_{B_s}}$ go up by about
$30\mev$, while the central value of $\xi$ is not much affected, but
the quoted error is less: $\xi = 1.21({}^{+5}_{-4})$.
\section{LIFETIME RATIOS}
The lifetime of a hadron $H_b$ containing a $b$-quark is calculated
from
\[
\Gamma(H_b) = \frac1{m_{H_b}}
\mathop{\mathrm{Im}} \langle H_b|\mathcal{T}|H_b\rangle
\]
where
\[
\mathcal{T} = i \int \mathrm{d}^4x \mathrm{T}
\{ H^{|\Delta B=1|}(x)H^{|\Delta B=1|}(0) \}
\]
is a non-local product of two $|\Delta B{=}1|$ effective Hamiltonians.
$H^{|\Delta B=1|}$ is known to NNLO~\cite{Gorbahn:2004my}. The large
energy release in a $b$ decay allows an operator product expansion
(OPE) of $\mathcal{T}$ as a series of local operators of increasing
dimension and increasing inverse powers of $m_b$, with calculable
coefficients (containing the CKM factors). This \emph{heavy quark
expansion} leads to an expression for the lifetime of the form
\[
\Gamma(H_b) = \sum_k
\frac{c_k(\mu)\langle H_b|O_k(\mu)|H_b\rangle}{m_b^k}.
\]
The dependence of the coefficients $c_k$ on the renormalisation scale
$\mu$ cancels that of the $\Delta B=0$ operators $O_k$ to give a
scale-independent physical result.
The operators occurring at the first few orders in $1/m_b$ (and
leading order in QCD) are~\cite{Neubert:1996we}:
\[
\def\arraystretch{1.10}
\begin{tabular}{>{$}r<{$}l}
O(1) & $\bar b b$\\
O(1/m_b) & no contribution\\
O(1/m_b^2) & $\bar b g_s \sigma\cdot G b$, chromomagnetic operator\\
O(1/m_b^3) & $\bar b\Gamma q \,\bar q\Gamma b$, $4$-quark
operators, $\Delta B{=}0$
\end{tabular}
\]
Their matrix elements are further expanded using
heavy quark effective theory, leading to
\[
\begin{aligned}
\langle H_b|\bar b b|H_b\rangle &= 1-\frac{\mu_\pi^2
-\mu_G^2}{2m_b^2}+O(1/m_b^3)\\
\langle H_b|\bar b g_s \sigma{\cdot}G b|H_b\rangle &=
2\mu_G^2 +O(1/m_b)
\end{aligned}
\]
The quantities $\mu_\pi$ and $\mu_G$ (which depend on the hadron,
$H_b$) can be determined from masses and mass-splittings. The ``$1$''
appearing in the matrix element of $\bar b b$ is a universal term,
corresponding to the decay of a free $b$ quark.
Since the $O(1/m_b^2)$ terms are not large, any substantial deviations
from equality of the hadron lifetimes should come from the
$O(1/m_b^3)$ terms. In particular, there are \emph{spectator effects}
where the matrix elements first involve the light ``spectator'' quarks
in $H_b$. These arise from $1$-loop terms in the HQE, whereas the
$O(1/m_b,1/m_b^2)$ contributions come from $2$-loop terms (see
figure~\ref{fig:ope-hqe}), and thus have a relative loop-factor
enhancement of $16\pi^2$. QCD corrections to the Wilson coefficients
of the HQE have been computed~\cite{Franco:2002fc,Beneke:2002rj},
which bring in penguin operators at $O(1/m_b^3)$.
\begin{figure}
\psset{unit=\hsize,arrowinset=0}
\begin{pspicture}(0,0)(1,0.748)
%\psframe(0,0)(1,0.748)
\rput[bl](0,0){\includegraphics[width=\hsize]{hqe.eps}}
\rput(0.62,0.6){$O(1)$}
\rput(0.62,0.32){$O(1/m_b^2)$}
\rput(0.62,0.042){$O(1/m_b^3)$}
\rput(0.06,0.69){$b$}
\rput(0.06,0.41){$b$}
\rput(0.06,0.173){$b$}
\rput(0.42,0.69){$b$}
\rput(0.42,0.41){$b$}
\rput(0.42,0.173){$b$}
\rput(0.82,0.69){$b$}
\rput(0.82,0.41){$b$}
\rput(0.84,0.178){$b$}
\rput(0.94,0.69){$b$}
\rput(0.94,0.41){$b$}
\rput(0.92,0.178){$b$}
\rput(0.06,0.017){$q$}
\rput(0.42,0.017){$q$}
\rput(0.84,0.0168){$q$}
\rput(0.92,0.0168){$q$}
\end{pspicture}
\caption{Generating local operators in the HQE.}
\label{fig:ope-hqe}
\end{figure}
The matrix elements of the $1/m_b^3$ operators have been calculated in
quenched lattice
QCD~\cite{DiPierro:1998ty,DiPierro:1999tb,Becirevic:2001fy}. These
calculations have not included the penguin contributions and so-called
``eye'' diagrams (for which a power-divergent subtraction has to be
controlled). The missing parts are expected to be small
$\mathrm{SU}(2)$ and $\mathrm{SU}(3)$ breaking effects for
$\tau(B^+,B_s)/\tau(B^0)$, but could be important for
$\tau(\Lambda_b)/\tau(B^0)$. The leading $O(1/m_b^4)$ spectator
contributions have also been
analysed~\cite{Gabbiani:2003pq,Gabbiani:2004tp}, leading to eight new
dimension-$7$ $4$-quark operators, whose matrix elements were
evaluated from vacuum insertion for $B$-mesons, or using a
quark-diquark model for $b$-baryons.
Analyses of the lifetime ratios incorporating all these pieces are
compared to experiment in figure~\ref{fig:lifetime-ratios}.
The experimental picture for $b$-hadron lifetime ratios is in flux. At
FPCP2006, Van Kooten~\cite{VanKooten:2006tr} updated the average for
$\tau(B_s)/\tau(B^0)$ compared to the HFAG~\cite{hfag:2006bi} summary
of early 2006, while at this meeting, a new measurement by CDF of
$\tau(\Lambda_b)/\tau(B^0)$ was presented~\cite{bureloBEACH2006}, with
value $1$ within errors. These numbers are shown in the figure.
The theoretical analyses, labelled T05~\cite{Tarantino:2005zi} and
GOP~\cite{Gabbiani:2004tp} agree very well with each other. In the
theoretical calculations, it is very hard to get a significant
deviation from $1$ for the ratio $\tau(B_s)/\tau(B^0)$, so that this
could become a problem in comparison to experiment. On the other hand,
the relevant matrix elements are large enough to accomodate a
substantial deviation of $\tau(\Lambda_b)/\tau(B^0)$ from $1$. If the
recent CDF result is correct, it will be essential to revisit the
lattice calculations.
%
\begin{figure}
\definecolor{dkred}{rgb}{0.8,0,0}
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\begin{pspicture}(0.5,-1.8)(1.1,17)
%\psframe(0.5,-1.8)(1.1,17)
\rput[t](0.8,-1.1){$0.8$}
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\psset{linecolor=dkred,fillcolor=dkred}
\result{15}{HFAG}{1.076}{0.008}(1.076,1.068,1.084)
\psset{linecolor=blue,fillcolor=blue}
\result{14}{T05}{1.06}{0.02}(1.06,1.04,1.08)
\result{13}{GOP}{1.06}{0.02}(1.06,1.04,1.08)
\rput[l](0.5,11.2){$\tau(B_s)/\tau(B^0)$}
\psset{linecolor=dkred,fillcolor=dkred}
\result{10}{HFAG}{0.914}{0.030}(0.914,0.884,0.944)
\result9{VK}{0.957}{0.020}(0.957,0.937,0.977)
\psset{linecolor=blue,fillcolor=blue}
\result8{T05}{1.00}{0.01}(1.0,0.99,1.01)
\result7{GOP}{1.00}{0.01}(1.0,0.99,1.01)
\rput[l](0.5,5.2){$\tau(\Lambda_b)/\tau(B^0)$}
\psset{linecolor=dkred,fillcolor=dkred}
\result4{HFAG}{0.844}{0.043}(0.844,0.801,0.887)
\result3{CDF}{1.037}{0.058}(1.037,0.979,1.095)
\psset{linecolor=blue,fillcolor=blue}
\result2{T05}{0.88}{0.05}(0.88,0.83,0.93)
\result1{GOP}{0.86}{0.05}(0.86,0.81,0.91)
\end{pspicture}
\caption{Lifetime ratios of $b$-flavoured hadrons (red: experiment,
blue: theory). Experimental numbers are from
HFAG~\cite{hfag:2006bi}, Van Kooten~\cite{VanKooten:2006tr} and
CDF~\cite{bureloBEACH2006}. T05 is the theoretical analysis
of~\cite{Tarantino:2005zi}, updating~\cite{Franco:2002fc}, while GOP
is from~\cite{Gabbiani:2004tp}.}
\label{fig:lifetime-ratios}
\end{figure}
\section{WIDTH DIFFERENCES}
The decay width difference between $B_q$ and $\bar B_q$ depends on the
off-diagonal $\Delta B=2$ matrix element
\[
\Delta\Gamma_q = - \frac1{m_{B_q}}
\langle \bar B_q | \mathcal{T} | B_q \rangle.
\]
Once again the heavy quark expansion is used to organise this into a
series of operator matrix elements with operators of increasing
dimension, accompanying increasing inverse powers of $m_b$, and
calculable coefficients:
\def\Lm#1{\bigg(\frac\Lambda{m_b}\bigg)^#1}
\def\G#1#2{\Gamma_{#2}^{#1}}
\[
\begin{split}
\Delta\Gamma &= \Lm3
\left(\G03 + \frac{\alpha_s}{4\pi}\G13+\cdots\right) \\
& \quad + \Lm4 (\G04+\cdots)
+ \Lm5 (\G05+\cdots) + \cdots
\end{split}
\]
The coefficient in the leading $\G03$ term was found long ago, while
the QCD corrections to it,
$\G13$~\cite{Beneke:1998sy,Ciuchini:2003ww}, and the $\G04$
piece~\cite{Ciuchini:2003ww,Beneke:1996gn,Dighe:2002em} have been
evaluated more recently. This year $\G05$ has also been
considered~\cite{Lenz2006}.
The leading contribution, at $O(1/m_b^3)$, involves two dimension-$6$
operators, one of which is the same as appears in the expression for
$\Delta m$. They are parameterised by $\langle\bar B_q |
O_i|B_q\rangle = \text{const}\times f_{B_q}^2 B_i$. The parameters
$B_{1,2}$ have been evaluated by several lattice
simulations~\cite{Hashimoto:2000eh,Hashimoto:2001zq,Becirevic:2000sj,Gimenez:2000en,Gimenez:2000jj,Becirevic:2001xt,Aoki:2002bh}
and are not expected to be significantly different in unquenched
calculations~\cite{Yamada:2001xp,Aoki:2003xb}. Four more dimension-$7$
operators appear at order $O(1/m_b^4)$: two of these are related to
operators in the set of $\Delta B=2$ operators whose matrix elements
were calculated on the lattice in~\cite{Becirevic:2001xt}; the others
are estimated by vacuum insertion. Putting together all the
ingredients~\cite{Tarantino:2005zi,Ciuchini:2003ww} shows that the QCD
corrections in $\G13$ and the $1/m_b$ corrections in $\G04$ are
important. The size and same sign of these corrections led Lenz and
Nierste~\cite{Lenz2006} to consider $1/m_b^2$ corrections which turn
out to be small; they also changed the operator basis to make the
coefficient of the operator responsible for the mass difference
dominant, which reduces the uncertainty from the QCD and $1/m_b$
corrections.
For $\Delta\Gamma_q/\Gamma_q$, two ways to quote a result are
\begin{itemize}
\item $\Delta\Gamma_q/\Gamma_q =
\Delta\Gamma_{q,\,\mathrm{theo}} \tau_\mathrm{expt}$\\
Pro: independent of new physics in mixing\\
Con: depends on $f_{B_q}^2$
\item $\Delta\Gamma_q/\Gamma_q =
(\Delta\Gamma_q/\Delta m_q)_\mathrm{theo} \Delta
m_{q,\,\mathrm{expt}} \tau_\mathrm{expt}$\\
Pro: theoretically clean\\
Con: might depend on new physics in $\Delta m_q$
\end{itemize}
Using the first method, Lenz and Nierste find~\cite{Lenz2006}
\[
\Delta\Gamma_s/\Gamma_s = 0.158^{+0.046}_{-0.051},
\]
where they have taken $f_{B_s}=245\mev$. Both methods agree for
$f_{B_s} = 221\mev$. Table~\ref{tab:lifediff} gives a comparison of
theoretical and experimental results for both
$\Delta\Gamma_s/\Gamma_s$ and $\Delta\Gamma_d/\Gamma_d$, showing good
consistency.
\begin{table}
\caption{Lifetime differences of neutral $B$ mesons. Experimental
numbers are from Van~Kooten~\cite{VanKooten:2006tr} for $B_s$ and
HFAG~\cite{hfag:2006bi} for $B_d$.}
%\begin{center}
\begin{tabular}{@{}>{$\displaystyle}r<{$}>{$}c<{$}>{$}c<{$}@{}}
\hline
& \text{Expt} & \text{Theory~\cite{Lenz2006}}\\
\hline
\Delta\Gamma_s/\Gamma_s & 0.14\pm0.06 & 0.16\pm0.05\\
\Delta\Gamma_d/\Gamma_d & 0.009\pm0.037 & 0.003\pm0.001\\
\hline
\end{tabular}
%\end{center}
\label{tab:lifediff}
\end{table}
\section{CP VIOLATION PARAMETERS}
For the CP-violation parameters $|q/p|-1$, the QCD corrections at
leading order in $1/m_b$ and the $1/m_b$ corrections have been
calculated~\cite{Ciuchini:2003ww,Beneke:2003az}. The results of two
theoretical analyses, BBLN~\cite{Beneke:2003az} and
CFLMT~\cite{Tarantino:2005zi,Ciuchini:2003ww} are given in
table~\ref{tab:cpviol}.
\begin{table}
\caption{CP violation parameters for neutral $B$ mesons from two
theoretical analyses.}
%\begin{center}
\begin{tabular}{@{}>{$\displaystyle}l<{$}@{\hspace{1.4ex}}>{$}c<{$}@{\hspace{1ex}}>{$}c<{$}@{}}
\hline
& \text{BBLN~\cite{Beneke:2003az}} &
\text{CFLMT~\cite{Tarantino:2005zi,Ciuchini:2003ww}}\\
\hline
\vrule height3.5ex depth2.2ex width0pt
\left|\frac qp\right|_s \!\!\!\!-\! 1 &
-(1.1\pm0.2)\times 10^{-5} & -(1.28\pm0.28){\times} 10^{-5}\\
\vrule height4ex depth3ex width0pt
\left|\frac qp\right|_d \!\!\!\!-\! 1 &
(2.5\pm0.6)\times 10^{-4} & (2.96\pm0.67){\times} 10^{-4}\\
\hline
\end{tabular}
%\end{center}
\label{tab:cpviol}
\end{table}
\section{PERSPECTIVE AND CONCLUSIONS}
In today's lattice simulations, the \emph{quenched} approximation, in
which vacuum polarisation effects are neglected, has now largely been
removed. Dynamical simulations have been done for meson decay
constants, but for the majority of the four-quark operator matrix
elements discussed here, we await updated unquenched calculations.
Much emphasis is now on reducing the masses of the light quarks in
dynamical simulations, to control the chiral extrapolation. The
challenge has been set by those collaborations using improved
staggered quarks, where light quark masses down to around $1/8$ of the
strange mass have been reached. Using staggered fermions requires
using a ``fourth root trick'' to remove unphysical \emph{tastes} (like
extra unwanted flavours). There is no proof that this is correct, but
there is growing circumstantial evidence and no counter example. We
need simulations using alternative fermion formulations, including
overlap or domain-wall quarks (with good chiral symmetry), improved
Wilson and twisted mass quarks in order to gain full confidence.
For heavy quarks a variety of techniques are available, including QCD
with heavy\emph{ish} quarks plus extrapolation,
non-relativistic QCD, discretised static quarks with $1/m_q$
corrections and relativisitic heavy (Fermilab/Tsukuba) quarks. The
results from these different formulations have been consistent to
date.
In conclusion, for neutral meson mixing we await fully unquenched
calculations for the $B_{B_q}$ parameters and confirmation of the
results by more than one group. For lifetime ratios, the situation is
very interesting given the evolution of the experimental results.
Spectator effects can be large enough to explain a substantial
difference between the $\Lambda_b$ and $B^0$ lifetimes: a cancellation
will have to be understood if these lifetimes are measured to be
equal. Theoretical evaluations of $\tau(B_s)/\tau(B^0)$ hardly deviate
from $1$, so there could be a puzzle if the experimental ratio remains
significantly different from $1$. The hadronic uncertainty remains
substantial for the lifetime ratios: updated lattice calculations
would be welcome here.
\medskip
\noindent\textbf{Acknowledgement}: I thank Alexander Lenz, Federico
Mescia, Chris Sachrajda, Achille Stocchi and Cecilia Tarantino for
answering my questions and sharing their expertise. I also thank the
BEACH2006 organisers for arranging a very interesting meeting.
\raggedright
\bibliographystyle{elsevier}
\bibliography{beach}
\end{document}
\[
\begin{aligned}
\frac{\tau(B^+)}{\tau(B^0)} &=
1.076\pm0.008\quad\text{HFAG~\cite{hfag:2006bi}} \\
\frac{\tau(B_s)}{\tau(B^0)} &=
\begin{cases}
0.914\pm0.030&\text{HFAG~\cite{hfag:2006bi}} \\
0.957\pm0.020&\text{Van Kooten~\cite{VanKooten:2006tr}}
\end{cases}\\
\frac{\tau(\Lambda_b)}{\tau(B^0)} &=
\begin{cases}
0.844\pm0.043&\text{HFAG~\cite{hfag:2006bi}} \\
1.037\pm0.058&\text{CDF~\cite{bureloBEACH2006}}
\end{cases}
\end{aligned}
\]
\begin{table}
\caption{Lifetime ratios of $b$-flavoured hadrons. T05 is the
theoretical analysis of~\cite{Tarantino:2005zi},
updating~\cite{Franco:2002fc}, while GOP is
from~\cite{Gabbiani:2004tp}. Experimental numbers are from
HFAG~\cite{hfag:2006bi}, but taking $\tau(B_s)$
from~\cite{VanKooten:2006tr}.}
%\begin{center}
\begin{tabular}{@{}>{$\displaystyle}r<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}@{}}
\hline
& \text{Expt} & \text{T05} & \text{GOP}\\
\hline
\frac{\tau(B^+)}{\tau(B^0)} &
1.076\pm0.008 & 1.06\pm0.02 & 1.06\pm0.02\\
\frac{\tau(B_s)}{\tau(B^0)} &
0.957\pm0.020 & 1.00\pm0.01 & 1.00\pm0.01\\
\frac{\tau(\Lambda_b)}{\tau(B^0)} &
0.844\pm0.043 & 0.88\pm0.05 & 0.86\pm0.05\\
\hline
\end{tabular}
%\end{center}
\label{tab:lifetime-ratios}
\end{table}