Violation in Mixing

188 Analysis of the time-dependent �È -violating asymmetry in � � � � decays
The kinematics of two-body decay are taken into account in the toy Monte Carlo generator:
¯ generate a � meson in the § �Ë frame with momentum randomly selected from a Gaussian
distribution (� � Å�Î� , � � Å�Î� ) and polar angle selected from a Ó× �
distribution,
¯ decay the � candidate to two tracks isotropically,
¯ boost to the laboratory frame,
¯ use the resulting track momenta and polar angles to compute the expected value of � and ¡�.
This “generation” procedure faithfully reproduces the (� �� ) and ¡��� correlations observed in Table
7-14. In the case the data ¡� resolution ( � Å�Î) is used instead of the Monte Carlo value ( � Å�Î)
the correlation between ¡� and � is reduced.
7.8.2 Effect of floating yields in the �È fit
The rms of the ¡Ø distribution for �� events is greater than the corresponding rms for continuum ÕÕ
events. It is therefore expected that adding this variable in the likelihood function will improve the statistical
separation between signal and background. To see how much we gain in the branching fraction analysis
by floating yields in the �È fit, we generated 685 toy experiments and fit each one with and without
the ¡Ø PDF in the likelihood. Figure 7-14 shows the difference in the fitted error on Æ�� and the two
asymmetry parameters. The error on �� improves by � , while the asymmetry errors increase only
slightly. The conclusion is that fitting simultaneously for yields and asymmetries optimizes the branching
fraction measurement and leads to a more accurate asymmetry measurement (since the uncertainty on the
yield is included directly in the fit error).
7.8.3 Monte Carlo fits
Since we expect � � signal �� and Ã� events in fb , an important consistency check on the ¡Ø
resolution function, and in the fit mechanism itself, is to fit for the � lifetime and mixing frequency in
the � � sample. Table 7-15 shows the results of several test fits on Monte Carlo samples. Fitting for
the lifetime and ¡Ñ�� in pure signal, or in a mix of �� and Ã� events in the proper ratio, returns correct
values for both parameters. We have also tried mixing the correct signal yield into the ��� fb continuum
Monte Carlo sample and find consistent values of � and ¡Ñ�� . Finally, fitting for Ë�� and ��� returns the
correct values in high statistics signal Monte Carlo and consistent values in the sample with background.
To check that the fit errors on the �È asymmetries in simulated Monte Carlo samples are consistent with
what we estimate in toy Monte Carlo, we take the same sample of ��� fb continuum Monte Carlo and
add in ten different sets of (Æ��,ÆÃ�) with exact values determined by Poisson statistics. Table 7-16
summarizes the results of this test. The average error on �� and ��� are � and ���, respectively.
Scaled to fb we would predict an expected error of ��� on �� and ���, in excellent agreement with
the toy Monte Carlo prediction (Fig. 7-12).
MARCELLA BONA