Description

The $\bar{B}_s^0$ meson lifetime is measured using the partially reconstructed semileptonic decay $\bar{B}_s^0 \rightarrow \ell^- \bar{\nu}D_s^+ X$, where $\ell^-$ is a lepton ($e^-$ or $\mu^-$) and the $D_s^+$ meson is reconstructed with the $D_s^+ \rightarrow \phi \pi^+$, $\phi \rightarrow K^+K^-$ decay mode. The data are collected with the CDF detector in Tevatron Run II with 8 GeV single lepton triggers (muons and electrons). The corresponding integrated luminosity used for this analysis is about 360 pb$^{-1}$, collected through August 2004.

The method is very similar to previous CDF measurements [1]. We reconstruct the $D^+_s$ meson produced near the lepton candidates. The $D_s^+$ mass distribution for the combined lepton sample is shown in Figure 1. The event yield and signal fraction are summarized in Section 2.1.

From the $\ell ^-D_s^+$ candidates, the $\bar{B}_s^0$ meson decay vertex is reconstructed and the decay length is measured. The reconstruction is performed in the plane perpendicular to the beam axis ($z$ axis). A schematic view of the decay and its reconstruction is displayed in Figure 2. The $\bar{B}_s^0$ meson decay length, $L_{B}$, is defined as the distance between the primary vertex and the $B^0_s$ decay vertex, projected onto the direction of the lepton-$D^+_s$ system, $\vec{p}_T(\ell^-D_s^+)$ :

$$L_B \equiv \vec{V}_B \cdot \frac{\vec{p}_T(\ell^-D_s^+)}{\vert\vec{p}_T(\ell^-D_s^+)\vert} .$$

Here $\vec{V}_B$ is the vector connecting the primary vertex and the $\bar{B}_s^0$ decay vertex, and the symbol ``$\cdot$'' denotes the inner product of the two 2-dimensional vectors.

To measure the lifetime of the $B^0_s$ meson, we need to know its proper time. The proper decay length $ct(B_s^0)$ is related with the decay length $L_B$ by

$$ct(\bar{B}_s^0) = \frac{L_B}{\beta\gamma} = L_B\frac{M_{\bar{B}_s^0}}{p_T(\bar{B}_s^0)}$$

where $M_{\bar{B}_s^0}$ is the $\bar{B}_s^0$ meson mass, and $\beta$ and $\gamma$ are the Lorentz factors. To calculate the proper decay length, we need to know the momentum of the $B^0_s$ meson for each event. Although we can not reconstruct fully the momentum of the $\bar{B}_s^0$ meson in the semileptonic decays, we can use the momentum of the $\ell^- D^+_s$ system, $p_T(\ell^-D_s^+)$, as a good approximation. We define the ``pseudo-proper decay length'' $ct^*$ as follows:

$$ct^*(\bar{B}_s^0) \equiv L_B \, \frac{M_{\bar{B}_s^0}}{p_T(\ell^-D_s^+)}.$$

To relate $ct^*$ and $ct$, we introduce the factor $K$, which is the ratio of the transverse momenta of the $\ell ^-D_s^+$ system and the $\bar{B}_s^0$ meson:

$$ct^*(\bar{B}_s^0) = ct \, \frac{p_T(\ell^-D_s^+)}{p_T(\bar{B... ...x{ with } K \equiv \frac{p_T(\ell^-D_s^+)}{p_T(\bar{B}_s^0)} .$$

We obtain the distribution of the ratio $K$ using a Monte Carlo simulation of signal events. Figure 3 show the $K$ factor distributions for the muon and electron datasets. They differ slightly because of implicit isolation requirements imposed by electron identification cuts. We use the distributions when we extract the $B^0_s$ meson lifetime from the observed distribution of $ct^*$.

We extract the $\bar{B}_s^0$ meson lifetime using the maximum likelihood method. The probability density function of the variable $x \equiv c t^*$ for the signal, the semileptonic $\bar{B}_s^0$ decay, can be expressed as follows:

$${\cal F}_{\bar{B}_s^0}(x;\sigma_{ct^*};c\tau,s) = \int dk ... ...c { ct^* } { c\tau/K } \right ) \otimes G( s \sigma_{ct^*}) ,$$

where $H(K)$ denotes the normalized $K$ factor distribution, $G(\sigma)$ is a Gaussian distribution with the width $\sigma$. $\sigma_{ct^*}$ is the calculated resolution in $ct^*$ for the event, and $s$ is the scale factor in the resolution, introduced to account for possible incompleteness in $\sigma_{ct^*}$ calculation.

The full probability density function, including the background contributions, is given by

$${\cal F}(x ) %%;\sigma_{ct^*}) = f_{\mbox{\normalsize sig}}... ...\normalsize sig}}) \, {\cal F}_{\mbox{\scriptsize BG}} (x) ,$$

where ${\cal F}_{\mbox{\scriptsize BG}} (x)$ represents the combinatorial background events under the $D^+_s$ meson mass peak, and its shape is determined using events in the sideband of the $D^+_s$ mass distribution. The term ${\cal F}_{\rm sig} (x)$ corresponds to the real $D^+_s$ meson decays. Among them there are events which are not originating from the semileptonic decays of the $B^0_s$ meson, such as promptly produced $D^+_s$ mesons, and $\ell^- D^+_s$ pairs produced in decays of the non-strange $B$ mesons. Therefore, the ``signal'' function ${\cal F}_{\rm sig}$ consists of three terms,

$${\cal F}_{\rm sig} (x ) = (1-f_c) \, [ \, ( 1 -f_b) \, {... ...r{B}_s^0} + f_b \, {\cal F}_b \, ] \, + f_c \, {\cal F}_c ,$$

the true signal, the bottom background, and the prompt charm background. The fraction of the prompt charm background is determined using real data, and its $ct^*$ distribution is mostly prompt. The amount and the $ct^*$ distribution of the bottom background are estimated using Monte Carlo calculations, based on measured branching fractions and $b$-quark fragmentation fractions.

The parameter to be determined by the fit is the $B^0_s$ meson lifetime $c\tau$. We maximize the joint likelihood, defined for the muon and electron datasets.

$${\cal L}_{\rm combined} \equiv \prod_{\ell} \prod_i {\cal F} (x_i ) , \ \ \ \ x_i \equiv ct^*_i ,$$

where $i$ is the event index, $ct^*_i$ is the pseudo-proper decay length measured for event $i$, and the product is taken over all events in the signal region in the sample, and $\ell$ denotes the lepton sample (muon or electron).

The decay length distribution and the fit result are displayed in Figure 4, and the extracted $\bar{B}_s^0$ meson lifetime is given in Section 2.4. The systematic uncertainties in the $\tau(\bar{B}_s^0)$ measurement are summarized in Section 2.2.

We also measure the ratio of the $\bar{B}_s^0$ meson lifetime to the $\bar{B}^0$ meson lifetime. We use a previous CDF measurement of the $\bar{B}^0$ meson lifetime [2], which is obtained also using the semileptonic decays and thus similar final states. The lifetime ratio $\tau(\bar{B}_s^0)/\tau(\bar{B}^0)$ is given in Section 2.4, and its systematic uncertainties are summarized in Section 2.3.