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\title{B hadron lifetime measurements at the Tevatron experiments}

\author{E. De La Cruz Burelo
\thanks{Mailbox E26610, CERN CH-1211 Gen\'eve 23, Switzerland. Email: {\em Eduard.Burelo@cern.ch} .}
\address{Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109, USA.}\\
(On behalf of the CDF and D0 collaborations)
}
       
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\runtitle{$B$ hadron lifetimes at the Tevatron}
\runauthor{E. De La Cruz Burelo}

\begin{document}

\begin{abstract}
We present the recent $B$ hadron lifetime measurements in the CDF and D0 collaborations at the Tevatron collider.
D0 reports a new world-best measurement of the $B_{s}$ lifetime in semileptonic decays, and CDF presents the most
precise $\Lambda_{b}$ lifetime measurement for a single experiment, and which precision is at the level of the 
uncertainty in the world average of the $\Lambda_{b}$ lifetime.

\vspace{1pc}
\end{abstract}

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\section{INTRODUCTION}

The measurement of $B$ hadron lifetimes is a important part of the rich
B physics program at the Tevatron experiments, CDF\cite{CDF} and D0\cite{D0}.
The lifetime of $B$ hadrons provide information on the $b$ quark decay, and
what is the role that play the lighter quarks which along with the $b$-quark form the $B$ hadron.
Also, $b$-quark decays are related to the $V_{ub}$ and $V_{cb}$ elements 
of the CKM matrix.

In this note we present the lifetime measurement of the $B_{s}$ in the semileptonic
decays $B_{s}\rightarrow D_{s}l\nu X$. These decays are 50\% CP-even and 50\% CP-odd
at time t=0, and impose an extra constraint to the $\Delta\Gamma_{s}/\Gamma_{s}$
measurement~\cite{dgamma}, and then to new physics in the $B_{s}$-$\overline{B}_{s}$ mixing sector.
We also present the recent $\Lambda_{b}$ lifetime measurement from CDF. 
The reported earlier difference in the measured $\Lambda_{b}$ lifetime
and the prediction created a great deal of interest and it was named as
the {\em $\Lambda_{b}$ lifetime puzzle}\cite{LbLifetimePuzzle}. 
This new measurement from CDF
is at the precision level of the current world average, however it is higher than previous 
measurements, situating the $\Lambda_{b}$-baryon decay at the same level of the $B$-meson decays.

\section{LIFETIME MEASUREMENT TECHNIQUES}

The lifetime measurements presented here are based on the determination of the distance $L$
from the decay vertex of the $B$ hadron to the primary interaction vertex. The proper decay length
$\lambda=\vec{L}\cdot\vec{P}_{T}(B) M_{B} / P_{T}(B)= L_{xy}/(\beta\gamma)_{T}$ is
determined on an event-per-event basis, and its distribution $F(\lambda)$ is related to the lifetime
of the $B$ hadron by:
\begin{equation}\label{pdf}
F(\lambda)=\int_{0}^{\infty}R(\lambda - \lambda')\frac{1}{c\tau_{B}}\exp\left(\frac{-\lambda'}{c\tau_{B}}\right)d\lambda'
\end{equation}
where $R(\lambda-\lambda')$ is the detector resolution.
In semileptonic decays, where the momentum $\vec{P}(B)$ of the $B$ hadron is not fully determined,
the momentum $\vec{P}(semi)$ of the semi-reconstructed $B$ hadron decay is used instead, and 
a correction factor $K=P_{T}(semi)/P_{T}(B)$ is introduced to account for the missing part of the momentum. 
The K factor
is extracted from Monte Carlo(MC) simulation. Then, Eq.~\ref{pdf} is rewritten using the pseudo-proper 
decay length (PPDL)
$\lambda=\vec{L}\cdot\vec{P}_{T}(semi) M_{B} / P_{T}(semi)$, in the form:
\begin{eqnarray}\label{pdf2}
G(\lambda)=\int dK H(K)
\left[
\int_{0}^{\infty}R(\lambda - \lambda')\right.\nonumber\\
\left.
\frac{K}{c\tau_{B}}\exp\left(\frac{-K\lambda'}{c\tau_{B}}\right)d\lambda'
\right]
\end{eqnarray}
where $H(K)$ is the distribution of the $K$ factor.

A background contribution is added to the probability density function in Eq. \ref{pdf} or \ref{pdf2},
and then fitted to the $\lambda$ distribution extracted from data. 
The background contribution can be modeled from data by using the sidebands to the $B$ hadron signal, or
from MC simulation. 

\section{$B_{s}$ LIFETIME IN SEMILEPTONIC DECAYS}

The D0 collaboration measured the $B_{s}$ lifetime in the semileptonic decays 
$B_{s}\rightarrow D_{s}^{-}\mu^{+}\nu X$, where the $D_{s}^{-}$ decays to $\phi\pi^{-}$, 
and $\phi\rightarrow K^{+}K^{-}$.
From a sample of integrated luminosity of 400 pb$^{-1}$, D0 reconstructed
5176 $\pm$ 242 (stat.) $\pm$ 314 (syst.) $B_{s}$ candidates.

The lifetime was measured using an unbinned maximum likelihood fit to
the pseudo-proper decay length distribution. The $B_{s}$ signal was modeled 
by Eq.~\ref{pdf2}, with the K factor extracted from MC. The background contributions 
from $B$ hadron decays, such as 
$\bar{B}^{0}\rightarrow D_{s}^{(*)-}D^{(*)+}X$, $B^{-}\rightarrow D_{s}^{(*)-}\bar{D}^{(*)0}X$,
and $\bar{B}_{s}\rightarrow D_{s}^{(*)-}D^{(*)}X$, were taking into account by including similar components
to those in the signal, but using fixed lifetimes according to the world average values. The weight and K factor of each
component were determined from MC. A $c\bar{c}$ background was included in the fit as a Gaussian with fixed
parameters as found in MC. Finally, the combinatorial background was parametrized using a Gaussian for zero-lived events,
plus several exponential decays as found in the sideband events to the $B_{s}$ signal. 
Figure~\ref{fig::BsM} shows the
invariant mass distribution of the $\phi\pi^{+}$ candidates, and Fig.~\ref{fig::BsLF} shows
the PPDL distribution of the $D_{s}^{-}\mu^{+}$ candidates. 
The $B_{s}$ lifetime was measured\cite{BsLifetimeD0} to be: 
\begin{equation}\label{BsD0ct}
\tau(B_{s})=1.398 \pm 0.044(\mbox{stat.}) ^{+0.028}_{-0.025}(\mbox{syst.}) \mbox{ ps} 
\end{equation} 
\begin{figure}
\includegraphics[scale=0.4]{B06CF01.eps}
\caption{\label{fig::BsM} Invariant mass distribution of $\phi\pi^{-}$ candidates. The open squares show the 
distribution from same sign combination of $\mu$ and $D_{s}$ candidates. 
The peak around 1.96 GeV/c$^{2}$ is due to $B_{s}$ semileptonic decays, and 
the second peak is from the Cabibbo-suppressed decay $D^{-}\rightarrow\phi\pi^{-}$ which main contribution 
was found to come from $B^{0}\rightarrow D^{-}\mu^{+}\nu X$ decays.}
\end{figure}
\begin{figure}
\includegraphics[scale=0.4]{B06CF02.eps}
\caption{\label{fig::BsLF} Pseudo-proper decay length distribution 
of $D_{s}^{-}\mu^{+}$ candidates from the D0 experiment. The dotted curve represents the combinatorial background 
and the filled area represents the $B_{s}$ signal.}
\end{figure}
This measurement is in good agreement with previous experiments, as well with the current world average lifetime
for all flavor-specific $B_{s}$ decays, and its precision is better than the current world average 
of the $B_{s}$ lifetime, when measurements from semileptonic and hadronic decays are combined\cite{PDG}. 
The major contribution to the systematic uncertainties comes from the modeling of the combinatoric background.

Similarly, the CDF collaboration measured the $B_{s}$ lifetime in the decays $B_{s}\rightarrow D_{s}l\nu X$, where
$l$ is a lepton ($e$ or $\mu$) . This measurement was performed in a data sample of integrated luminosity 
of 360 pb$^{-1}$. The extracted $B_{s}$ lifetime was\cite{BsLifetimeCDF}:
\begin{equation}
\tau(B_{s})=1.381 \pm 0.055(\mbox{stat.}) ^{+0.052}_{-0.046}(\mbox{syst.}) \mbox{ ps} 
\end{equation} 
This measurement in very consistent with the D0 measurement reported in Eq.~\ref{BsD0ct}.

\section{$\Lambda_{b}$ LIFETIME}

The $\Lambda_{b}$ lifetime has been measured by the CDF collaboration
in the exclusive decay channel
$\Lambda_{b}\rightarrow J/\psi\Lambda$, where the
$J/\psi$ decays to $\mu^{+}\mu^{-}$ and the $\Lambda$ to $p \pi$.
From a data sample of integrated luminosity of 1 fb$^{-1}$,
CDF reconstructed 538 $\pm$ 38 $\Lambda_{b}$ candidates. This is the largest collected
sample of fully reconstructed $\Lambda_{b}$ baryons. 

The lifetime was extracted using an unbinned maximum likelihood fit to mass,
proper decay length, and their errors distributions.
The $\lambda$ distribution for signal was modeled by $F(\lambda)$ in Eq.~\ref{pdf}, and the background
was described with a negative and two positive exponential decays accounting for mis-measured decay vertices
and background from other heavy-flavor decays. The mass distribution was modeled as the sum of a Gaussian
signal and linear background. Figure~\ref{fig::CDFLb} shows the 
%mass and 
proper decay length %distributions
distribution
with the projection of the fit result superimposed.
The $\Lambda_{b}$ lifetime was found to be\cite{LbLifetimeCDF}:
\begin{equation}
\tau(\Lambda_{b})=1.593^{+0.083}_{-0.078}(\mbox{stat.})\pm0.033(\mbox{syst.})\mbox{ ps}
\end{equation}
This measurement is comparable in precision to the world average, however, it is
3.1 standard deviations higher. Figure
\ref{Lbctaus} shows this new measurement compared to the previous measurements and to the world average.
\begin{figure}
\includegraphics[scale=0.39]{Lb_ctau_Fit.eps}
\caption{\label{fig::CDFLb} Proper decyay length distribution of the $\Lambda_{b}$ 
  candidates from CDF experiment.}
\end{figure}
\begin{figure}
\includegraphics[scale=0.39]{Lambdab_Lifetime_Measurements_1fb.eps}
\caption{\label{Lbctaus} Comparison between the new $\Lambda_{b}$ lifetime measurement from CDF and those
  reported in the PDG.}
\end{figure}
The main contribution to the systematics uncertainties comes from the resolution model in Eq.~\ref{pdf}.

\section{OTHER $B$ HADRON LIFETIME MEASUREMENTS}

The CDF and D0 collaborations have measured the lifetime of $B$ hadrons in different decay modes. Many of these
measurements are
very competitives at the world level. However
due to the lack of space, it is not possible to cover all of them here. Those important measurements
which have not been 
reported in other contributions to these proceedings are presented in Table~\ref{table:1},
and the reference are given in~\cite{AllLifetimes}. Some important $B$ hadron lifetime measurements presented 
in other contributions are: the first report on the $B_{c}$ meson lifetime, which was part of 
the special contribution on the
$B_{c}$ properties~\cite{Bc}; 
and the $\Delta\Gamma_{s}/\Gamma_{s}$ measurement in the $B_{s}\rightarrow J/\psi\phi$ 
decay and in the CP-even (to within 5\%) decay mode $B_{s}\rightarrow K^{+} K^{-}$ presented by K. Yip\cite{dgamma}.  
\begin{table}[htb]
\caption{$B$ hadron lifetime measurements which are not presented in this note.}
\label{table:1}
\begin{tabular}{c}
\hline
B lifetimes in hadronic decays (CDF)\\
\hline
$\tau(B^{+})$ = 1.66$\pm$0.03(stat)$\pm$0.01(syst.) ps.\\
$\tau(B^{0})$ = 1.51$\pm$0.02(stat)$\pm$0.01(syst.) ps.\\
$\tau(B_{s})$ = 1.60$\pm$0.10(stat)$\pm$0.02(syst.) ps.\\
\hline
B lifetimes in semileptonic decays (CDF)\\
\hline
$\tau(B^{+})$ = 1.653$\pm$0.029(stat)$^{+0.033}_{-0.031}$(syst.) ps.\\
$\tau(B^{0})$ = 1.473$\pm$0.036(stat)$\pm$ 0.054(syst.) ps.\\
\hline
Direct lifetime ratio (D0)\\
\hline
$\frac{\tau(B^{+})}{\tau(B^{0})}$ = 1.080$\pm$0.016(stat)$\pm$0.014(syst.)\\
\hline
\end{tabular}
\end{table}

\section{SUMMARY}

The CDF and D0 collaborations at the Tevatron collider have measured the $B_{s}$ lifetime in
semileptonic decays. Both measurements are very consistent between them, and the new measurement from D0 have
a better precision than the current world average. This new measurement impossed 
indirectly an extra constraint to the search for new physics in the $B_{s}$-$\overline{B}_{s}$ mixing sector.

The CDF experiment reported a new measurement of the $\Lambda_{b}$ lifetime, which precision is comparable 
to the world average, however, it is 3.1 standard deviations 
highter than the world average value. Recently, in the time of the preparation of this summary, 
a preliminary measurement 
was released by the D0 experiment\cite{NewLbLifetimeD0} in the same decay channel 
$\Lambda_{b}\rightarrow J/\psi\Lambda$. This new measurement is consistent with the world average and 
with the previous D0 measurement\cite{LbLifetimeD0} in the same decay channel, suggesting a
shorter lifetime nature of the $\Lambda_{b}$  lifetime than what was found by CDF. However,
both measurements are consistent within two standard deviations, and the theoretical
prediction~\cite{Tarantino} is consistent with both of them.
More statistics is needed to settle the issue of the $\Lambda_{b}$ lifetime.

Many other measurements of $B$ hadron lifetimes in different decay modes have been performed by the 
CDF and D0 experiments. The first lifetime measurements of the $B_{c}$ meson have confirmed
its expected short-lived nature. Also, the $\Delta\Gamma_{s}/\Gamma_{s}$ measurement in
the $B_{s}\rightarrow J/\psi\phi$ and $B_{s}\rightarrow K^{+} K^{-}$
has been found to be consistent
with the Standard Model predictions. With the increase of the collected data, more precise measurements
will come, and the $B$-baryon lifetime sector will be an unexplored area accessible to the Tevatron experiments.  

%\section{ACKNOWLEDGMENTS}

%The author wishes to thank the organizers of the BEACH 2006 conference for a very interesting program
%and all the attentions provided during the conference. 
%This work has been performed under the auspices
%of the US Department of Energy.

\begin{thebibliography}{15}

\bibitem{CDF} CDF Collaboration, F. Abe {\em et al.}, Nucl. Instrum. Methods Phys. Res. A {\bf 271}, 388 (1998);
  S. Donati, these proceedings.

\bibitem{D0} D0 Collaboration, V. Abazov {\em et al.}, ``The Upgraded D0 Detector'', submitted to 
  Nucl. Instrum. Methods Phys. Res. A; Fermilab-Pub-05/341-E; physics/0507191; L. Welty, these
  proceedings.
 
\bibitem{dgamma} R. Van Kooten, talk at the Flavour Physics and CP Violation Conference, Vancouver 
  2006, hep-exp/0606005; K. Yip, these proceedings.

\bibitem{LbLifetimePuzzle} F. Gabbiani {\em et al.} Phys. Rev. Lett. {\bf D} 68, (2003) 114006-1-4.

\bibitem{BsLifetimeD0} To appear in Phys. Rev. Lett.; Fermilab-Pub-06/085-E; hep-ex/0604046.

\bibitem{PDG} S. Eidelman {\em et al.}, Phys. Lett. {\bf B} 592, 1 (2004).

\bibitem{BsLifetimeCDF} CDF note 7757.

\bibitem{LbLifetimeCDF} CDF  note 8268.

\bibitem{NewLbLifetimeD0} D0 note 5179-Conf.

\bibitem{LbLifetimeD0} V.M. Abazov {\em et al.}, Phys. Rev. Lett. {\bf 94}, 102001 (2005).

\bibitem{Tarantino} C. Tarantino, Nucl. Phys. B (Proc. Suppl.) 156 (2006), 33-37.

\bibitem{AllLifetimes}  CDF note 7514; CDF note 7386; Phys. Rev. Lett. {\bf 94}, 182001 (2005).

\bibitem{Bc} M.A. Rahaman, these proceedings.

\end{thebibliography}


\end{document}
