Fit assuming a non-zero $\Delta \Gamma / \Gamma$ value

In the $B^0_s \bar B^0_s$ meson system, there should exist two eigenstates of the masses and the lifetimes, $B^0_{s,\, {\rm Heavy}}$ and $B^0_{s, \, {\rm Light}}$. The width difference $\Delta \Gamma$ could be sizable, of order 10% in $\Delta \Gamma / \Gamma$, according to theory. Experimentally, CDF finds $\Delta \Gamma / \Gamma = 0.65 ^{ + 0.25} _ {-0.33} \pm 0.01$ using $B^0_s \rightarrow J/\psi \phi$ decays, and the 2005 web update version of PDG [4] quotes a world average of

$$\Delta \Gamma / \Gamma = 0.37 ^{ + 0.17} _ {-0.20}.$$

The results presented thus far have been obtained assuming $\Delta \Gamma / \Gamma=0$. We could, however, fit for the average decay width $\Gamma$ assuming a value of $\Delta \Gamma / \Gamma$. The decay time distribution of the signal is modified to be

$$P(t) = \frac{1}{\tau_L+\tau_S} \left[e^{-t/\tau_L}+e^{-t/\tau_S}\right] ,$$

where $\tau_L$ and $\tau_S$ are the lifetimes of the long-lived and short-lived mesons, $\Gamma_{S,L} \equiv \Gamma \pm \Delta\Gamma/2 = 1 / \tau_{S, \, L}$, $\Gamma \equiv ( \Gamma_S+\Gamma_L) /2$ is the average decay width, and $\Delta\Gamma \equiv \Gamma_S-\Gamma_L$ is the width difference.

We fit for $\Gamma$ with $\Delta \Gamma / \Gamma$ fixed. For $\Delta \Gamma / \Gamma = 0.37$, we find

$$c /\Gamma = 386.7 \pm 16.0\ \mu\mbox{m} ,$$

where the uncertainty is statistical only. From this value and $\Delta \Gamma / \Gamma$, we calculate $\tau_L$ and $\tau_S$. The table below summarizes these numbers, including other values of $\Delta \Gamma / \Gamma$ we have assumed.

Figure 7 shows the fit result for $\Delta \Gamma / \Gamma = 0.37$. The data points are identical to those of Figure 4.

$\Delta \Gamma / \Gamma$	$c\tau_S$ ($\mu$m)	$c\tau_L$ ($\mu$m)	$c/\Gamma$ ($\mu$m)
$0.17$	$376.1 \pm 15.2$	$446.0 \pm 18.0$	$408.1 \pm 16.5$
$0.37$	$326.3 \pm 13.5$	$474.4 \pm 19.6$	$386.7 \pm 16.0$
$0.54$	$281.7 \pm 11.9$	$490.1 \pm 20.7$	$357.8 \pm 15.1$