Description

The $B^-$ and $\bar{B}^0$ meson lifetimes are measured using $\bar{B}\rightarrow \ell^- \bar{\nu} D^0 X$ decays, where the $D^0$ meson is reconstructed in the $K^-\pi^+$ decay mode. The data are collected with 8 GeV single lepton triggers (both muon and electron) in CDF Run II. The corresponding integrated luminosity used for this analysis is about 260 pb$^{-1}$, collected through January 2004. After the lepton+$D^0$ reconstruction, we separate the lepton+$D^{*+}$ candidates according to $D^{*+}\rightarrow D^0 \pi_s^+$ decay signal. After the separation, the lepton+$D^0$ ($D^{*+}$ excluded) and lepton+$D^{*+}$ samples are dominated by the $B^-$ and $\bar{B}^0$ mesons respectively. There is a small cross talk, it is estimated using monte carlo simulation. Thus we extract the $B^-$ and $\bar{B}^0$ meson lifetimes by fitting the lepton+$D^0$ ($D^{*+}$ excluded) and lepton+$D^{*+}$ samples simultaneously. The mass plots for lepton+$D^0$ ($D^{*+}$ excluded) and lepton+$D^{*+}$ signals are shown in Figure 1. The event yield and signal fractions are summarized in Section 2.1.

From the lepton+$D^0$ pairs, we reconstruct the $B$ decay vertices and measure $B$ decay length $L_B$. A schematic view of the reconstruction is shown in Figure 2. The decay length $L_B$ is calculated as the projection of $\vec{V}_B$ on the direction of $\ell^-D^0$ transverse momentum,

$$L_B = \vec{V}_B \cdot \frac{\vec{P}_{T}(\ell^-D^0)}{\vert\vec{P}_{T}(\ell^-D^0)\vert},$$

where the $\vec{V}_B$ is the vector from the primary vertex to the $B$ decay vertex. From $L_B$ and transverse momentum of the $B$ meson, proper decay length $ct$ is defined as follows.

$$ct(B) = L_B \frac{M_B}{p_T(B)},$$

where the $M_B$ is the $B$ meson mass (5.279 GeV/$c^2$). However, since the $B$ meson is partially reconstructed, we can only measure the the transverse momentum of the lepton+$D^0$ system, not the $B$ meson. So the pseudo-proper decay length $ct^*$ is also defined as,

$$ct^*(B) = L_B \frac{M_B}{p_T(\ell^-D^0)} = ct \frac{1}{K}$$

where the $K$ is a correction factor for the missing momentum of the $B$ meson, defined as

$$K \equiv \frac{p_T(\ell^-D^0)}{p_T(B)}$$

The K factor distributions are calculated from the monte carlo simulation, and involved into probability density function for the lifetime fit.

Then we extract the $B^-$ and $\bar{B}^0$ meson lifetimes by fitting the lepton+$D^0$ ($D^{*+}$ excluded) and the lepton+$D^{*+}$ samples simultaneously. The likelihood function for the semileptonic $B$ decay signal is given by,

$${\cal F}_B(ct^*,\sigma_{ct^*}) = \frac{K}{c\tau} \int D(K)\ ... ...t(-\frac{ct^*}{c\tau}K\right) \otimes Gaus(ct^*;s\sigma_{ct^*})$$

$ct^*,\sigma_{ct^*}$	...	Pseudo-proper decay length of the $B$ meson, and its resolution.
$c\tau$	...	$B$ meson lifetime.
$D(K)$	...	$K$ factor distribution, which correct the missing momentum of the
		$B$ meson. It is obtained from monte carlo simulation.
$s$	...	Resolution scale factor.

Secondly we define the probability density function as follows.

$$\begin{eqnarray*} P(ct^*,\sigma_{ct^*}) &=& f_{\mbox{\normalsize sig}} \left\{(1... ...ze sl}} & = & g_-{\cal F}_{B^-} + (1-g_-){\cal F}_{\bar{B}^0}\\ \end{eqnarray*}$$

$f_{\mbox{\normalsize sig}}$	...	Fraction of $D^0\ (D^{*+})$ events
${\cal F}_{\mbox{\scriptsize BG}}(ct^*)$	...	Combinatorial background function
$f_c$, ${\cal F}_c(ct^*)$	...	Prompt charm background function
$f_b$, ${\cal F}_b(ct^*)$	...	Bottom background function
$g_-$	...	fraction of $B^-$ component

Then the following combined likelihood is formed.

$$L = \prod_i P_i,$$

where the product is taken over all the events in signal region of the each sample. We extract the $B^-$ and $\bar{B}^0$ lifetimes by minimizing negative log likelihood $- l = - \log{L}$. The obtained $B^-$ and $\bar{B}^0$ meson lifetimes and their ratio are listed in Section 2.3.

The systematic uncertainties are mainly from the estimation of sample composition, and treatment of prompt charm background component. Missing momentum correction gives some systematics to the $B^-$ and $\bar{B}^0$ meson lifetimes.