Systematic Uncertainties

Systematic uncertainties in this analysis come from several sources. The contribution due to the background model is estimated by varying the background parameterization which assumes the $e^{c_0+c_1\cdot M_{\pi\pi}}+c_2$ shape for the central value. We have varied the background model by assuming polynomials of second or third degree and repeating the fit. The quoted systematics is the largest variation with respect to the central value. The likelihood fit takes the mass resolutions of the several $B^0\rightarrow h^{+}h^{'-}$ signals as input parameters and these parameters are estimated from Monte Carlo. CDF II Monte Carlo shows some difficulty in reproducing the widths of several two-body decays, like $D^0\rightarrow h^{+}h^{'-}$, $J/\psi \rightarrow \mu^+\mu^-$ and $\Upsilon \rightarrow \mu^+\mu^-$. In order to estimate the systematics deriving from the assumption of the Monte Carlo value of the mass resolution, we have varied the parameter of an amount consistent with the discrepancy observed in data. Several sources of systematics are due to the particle identification information provided by the $dE/dx$ measured in the drift chamber. A correlation between the measured $dE/dx$ of the two tracks used to reconstruct the $D^0\rightarrow h^{+}h^{'-}$ has been measured and a similar, although smaller correlation, since the accessible phase space to the decay is larger, is expected in the $B^0\rightarrow h^{+}h^{'-}$ signal. This correlation can be modeled as a common mode of the measured $dE/dx$ values of the two tracks belonging to the decays. The parameters varied to estimate the systematics are the width of the distribution of the common mode and the shape of the distribution itself. The minimum and maximum values of the width are estimated from the $D^0\rightarrow h^{+}h^{'-}$ sample and as extreme cases of possible distributions we have chosen a Gaussian and a pair of delta distributions posed at symmetric values and at a distance to produce a distribution with the same width as the Gaussian. We also evaluated the effect due to small non Gaussian tails observed in the distributions of the $dE/dx$ pulls and to small shifts of the mean value observed in the pull distribution of low momentum positive tracks. The $\mathcal{M}(\alpha)$ functions are parameterized with the values of the $B_d$ and $B_s$ meson masses measured by CDF II. We have estimated the systematics associated to this parameterization by repeating the fit using mass values varied by $\pm 1 \sigma$ of the CDF II statistical error. There are contributions to the systematics due the uncertainties on the $B_d$ and $B_s$ lifetimes and from the unknown value of $\Delta\Gamma_s/\Gamma_s$ which affect the estimate of the relative efficiency corrections. For the uncertainty on the $B_d$ lifetime we have used the PDG2002 value. The $B_s\rightarrow K^+K^-$ decay is expected to be almost purely CP even state and consequently to be the state with short lifetime. The Standard Model estimates $\Delta\Gamma_s/\Gamma_s = 0.12 \pm 0.06$ and the $\pm 0.06$ uncertainty is propagated to the efficiency correction factors. The effect due to the limited statistics of the Monte Carlo samples used to estimate the relative efficiency corrections is estimated by propagating the statistical error of the correction factors to the quoted measurements. The statistical error of the $B$ isolation efficiency introduces a systematics only in the measurements involving relative branching fractions of $B_s$ to $B_d$ decays. The CDF II detector shows a small tracking asymmetry between positive and negative tracks, which has been used to correct the measurement of $A_{\mathsf{CP}}$. The uncertainty on the measurement of this asymmetry ($\pm25$ %) is propagated to the final measurement. The CDF II Monte Carlo does not reproduce accurately the efficiency of the Level 1 track processor which shows a small tracking efficiency difference between kaons and pions. This difference has been measured from data and used to determine the relative efficiency corrections. The statistical error on this measurement is propagated to the final results as a systematic. The Monte Carlo samples used to estimate the relative efficiency corrections have been generated assuming the transverse momentum distribution of a mixture of $b$ hadrons measured at CDF II. The theory expects slightly different distributions for $B_d$ and $B_s$. The systematic error due to this approximation has been estimated by recalculating the efficiency corrections varying the transverse momentum distributions of 1 % [3]. A summary of all the systematic errors is reported in Tab. .

Table: Summary of the systematic uncertainties.

source	$\frac{f_s}{f_d}\cdot\frac{BR(B_s\rightarrow KK)}{BR(B_d\rightarrow K\pi)}$	$A_{\mathsf{CP}}(B_d\rightarrow K\pi)$	$\frac{BR(B_d\rightarrow \pi\pi)}{BR(B_d\rightarrow K\pi)}$	$\frac{f_d}{f_s}\cdot\frac{BR(B_d\rightarrow \pi\pi)}{BR(B_s\rightarrow KK)}$
background model	$^{+0.005}_{-0.005}$	$^{+0.002}_{-0.002}$	$^{+0.003}_{-0.003}$	$^{+0.000}_{-0.000}$
mass resolution	$^{+0.001}_{-0.004}$	$^{+0.001}_{-0.001}$	$^{+0.001}_{-0.002}$	$^{+0.001}_{-0.001}$
$dE/dx$ correlation: RMS(s)	$^{+0.043}_{-0.031}$	$^{+0.002}_{-0.002}$	$^{+0.034}_{-0.025}$	$^{+0.029}_{-0.017}$
$dE/dx$ correlation: $pdf$(s)	$^{+0.002}_{-0.002}$	$^{+0.002}_{-0.002}$	$^{+0.000}_{-0.000}$	$^{+0.002}_{-0.002}$
$dE/dx$ tail	$^{+0.056}_{-0.056}$	$^{+0.003}_{-0.003}$	$^{+0.020}_{-0.020}$	$^{+0.017}_{-0.017}$
$dE/dx$ shift	$^{+0.001}_{-0.002}$	$^{+0.001}_{-0.001}$	$^{+0.001}_{-0.003}$	$^{+0.017}_{-0.005}$
input masses	$^{+0.027}_{-0.028}$	$^{+0.003}_{-0.003}$	$^{+0.009}_{-0.010}$	$^{+0.009}_{-0.010}$
$B_d$, $B_s$ lifetime	$^{+0.004}_{-0.004}$	-	-	$^{+0.004}_{-0.004}$
$\Delta\Gamma_s/\Gamma_s$ Standard Model	$^{+0.007}_{-0.006}$	-	-	$^{+0.006}_{-0.006}$
MC statistics	$^{+0.004}_{-0.004}$	$^{+0.001}_{-0.001}$	$^{+0.003}_{-0.003}$	$^{+0.006}_{-0.006}$
isolation efficiency	$^{+0.051}_{-0.051}$	-	-	$^{+0.050}_{-0.050}$
charge asymmetry	-	$^{+0.002}_{-0.002}$	-	-
XFT-bias correction	$^{+0.010}_{-0.007}$	-	$^{+0.004}_{-0.004}$	$^{+0.015}_{-0.010}$
$p_T(B)$ spectrum	$^{+0.007}_{-0.007}$	-	-	$^{+0.007}_{-0.007}$
TOTAL	$\pm 0.09$	$\pm 0.01$	$\pm 0.04$	$\pm 0.07$