Meson from Correlations

The meson proper lifetime distribution has been examined
for a lifetime difference between the two **CP** eigenstates of the
meson, and .

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Measurements of the individual lifetimes of **B** mesons provide a
well defined test of theoretical models of heavy quark decays. The lifetimes
of the and meson are expected to be approximately equal,
while the lifetime is predicted to be about
greater than the neutral **B** mesons [1].
CDF has already published a lifetime measurement of the
meson [2] with Run Ia data, using both
decays and decays.

The two mass eigenstates of the meson, and
(**L =** `light', **H =** `heavy') are related to the
flavour eigenstates by

We further define

where and denote the mass and decay width of and . In the CKM Model the ratio depends only on QCD corrections [3]. Thus a measurement of would imply a determination of and a way to infer the mixing parameter of the meson, as well as a measurement of the ratio of CKM matrix elements .

Since the decay is expected to be an equal mixture of and , one way to measure is to describe the proper lifetime distribution from correlations by a functional form of

The meson decays to
X, and the is reconstructed through the decay
, with .
We use oppositely charged pairs of tracks, each assigned the **K** mass,
in order to reconstruct the . If the invariant mass of
the pair of tracks corresponds to the mass, we
then add a third track which is assigned the mass to reconstruct
the .

The mass spectrum for right-sign events is shown in Figure 1 with the fit result overlaid. We find a signal of 254 21 events in the peak and also see evidence of the Cabibbo-suppressed decay . The shaded histogram shows the mass spectrum for wrong-sign events. As expected, there is no evidence for any enhancement in the wrong-sign mass spectrum.

Having found the decay vertex and reconstructing the of the , an intersection with the lepton in the transverse plane is used to define the decay vertex. The decay length of the in the transverse plane is then defined as the distance between the primary and vertices. Since the is not fully reconstructed, we use the of the system to approximate (). We find from Monte Carlo that the system carries a mean of 85.8% of the momentum of the . We include this correction in the fit to the lifetime distributions.

We use an unbinned maximum log-likelihood method to extract the lifetime. The probability distribution of the signal consists of an exponential function, convoluted with the correction factor distribution, convoluted with a gaussian resolution function. The background probability distribution has three components. There is a gaussian for the prompt component, and two exponentials, for the positive and negative lifetime backgrounds, each convoluted with a gaussian.

We also include contributions from real physics backgrounds in our fit. We include the decays and , with the decaying semi-leptonically, and decays with one decaying semi-leptonically. We estimate the contribution of these processes to our signal to be but include their contributions in our lifetime fit.

Table 1 shows the contributions to the systematic error on our lifetime measurement. The main contribution comes from the shape of the background, which is modelled by the lifetime distribution of the right-sign sidebands and wrong-sign events. Adding the individual contributions in quadrature, our total systematic error is 13 m.

**Table 1:** Systematic errors in the measurement of the lifetime.

Using the values quoted above for the physics background contributions, we find

Figure 2(a) shows the proper decay length distribution for the signal sample with the result of the fit overlaid. The shaded curve shows the sum of the background probability function over the events in the signal sample. Figure 2(b) shows the corresponding distribution for the background sample with the result of the fit overlaid.

We also fit for the lifetime of the meson. We find

where the error given is statistical only. This compares well with the current world average = 140 m [4]. Figure 3(a) and Figure 3(b) show the distributions for the signal and background samples, respectively, with the results of the fit overlaid.

In an attempt to fit the meson lifetime distribution for two lifetime components, we try to fit our signal to a function of the form

where is fixed to the lifetime found in fitting the sample for a single lifetime.

We find

and conclude that we have no sensitivity to determine a lifetime difference with the statistics of our current data sample.

**1**- I.I. Bigi, in
*B Decays*, revised 2nd edition, edited by S. Stone; World Scientific, Singapore, 1994. **2**- F. Abe et al. (CDF),
Phys.Rev.Lett.
**74**(1995) 4988. **3**- A. Datta, E.A. Paschos, and U. Türke,
Phys.Lett.
**B196**(1987) 382;

I. Dunietz, Ann.Phys.**184**(1988) 350. **4**- Particle Data Group,
Phys.Rev.
**D50**(1994) 1173.

CDF