Likelihood fit

The relative $B_{d,s}\rightarrow h^{+}h^{'-}$ fractions are measured by an unbinned maximum likelihood fit which combines mass, kinematics and $dE/dx$ information. The discriminating variables used in the fit are:

• $M_{\pi\pi}$ which is the invariant mass of the track pair computed by assigning the pion mass to both tracks;
• $\alpha = (1-\frac{p_1}{p_2}) \cdot q_{1}$ which is the charge-signed momentum imbalance. $p_1$ ($p_2$) is the modulus of the smaller (larger) momentum of the two tracks and $q_1$ is the charge of the track with smaller momentum. The $M_{\pi\pi}$ dependence on $\alpha$ is computed analytically, the formulas relative to the six signals $B_d\rightarrow\pi^+\pi^-$, $B_d\rightarrow K^+\pi^-$, $\overline{B}_d\rightarrow K^-\pi^+$, $B_s\rightarrow K^-\pi^+$, $\overline{B}_s\rightarrow K^+\pi^-$, $B_s\rightarrow K^+K^-$ show that these two variables provide some separation between particles and antiparticles (see Tab.  and top six plots of Fig. );
• ID1 (ID2) which contain the $dE/dx$ information for the track with smaller (higher) modulus of the momentum.

The Likelihood function is $\mathcal{L}=\prod_{i=1}^{nev}\mathcal{L}_i$. The index $i$ runs over the events and the Likelihood of the $i^{th}$ event is:

$$\mathcal{L}_i = b\cdot \mathcal{L}^{bckg} + (1-b)\cdot \mathcal{L}^{sign}$$ (1)

The index $sig$ ($bckg$) labels the likelihood term of signal (background), $b$ is the background fraction.

• $\mathcal{L}^{sig} = \sum_{j} f_j\cdot \mathcal{L}^{kin}_j\cdot \mathcal{L}^{PID}_j$ is the signal likelihood. The index $j$ runs over the six $B_{d,s}\rightarrow h^+h^{'-}$ signal modes shown in Tab.  and the parameters $f_j$ are their relative fractions determined by the fit. $\mathcal{L}^{kin}_j$ and $\mathcal{L}^{PID}_j$ are the factors of the likelihood that use the information derived respectively from the kinematics and from particle identification.
  • The kinematics term is:
    $$\begin{eqnarray*}\mathcal{L}_j^{kin}= pdf^{kin}_j(M_{\pi\pi},\alpha ; \sigma,\m... ...}{2}}\left(\frac{M_{\pi\pi}-\mathcal{M}(\alpha)}{\sigma}\right)^2\end{eqnarray*}$$
    
    
    
    
    
    where $P(\alpha)$ is the distribution function of the variable $\alpha$ of signal events which is extracted from Monte Carlo (see bottom six plots of Fig. ). The $pdf$ of the $M_{\pi\pi}$ variable is a Gaussian of width $\sigma$ and centered on $\mathcal{M}(\alpha)$. $\sigma$ is the mass resolution of each mode and is rescaled to the data from Monte Carlo. $\mathcal{M}(\alpha)$ are the analytic functions defined in Tab.  that determine the expected value of $M_{\pi\pi}$ as a function of the momentum imbalance ($\alpha$).
  • The term (relative to the $B_{d,s}\rightarrow h^{'}h$ decay) which uses the particle identification information is
    $$\begin{eqnarray*}\mathcal{L}^{PID}_j&=& pdf^{PID}_{h^{'}h}(\textsf{ID(1), ID(2)... ...2} \left(\frac{\mathsf{ID}(2)-PID_h(2)}{\sigma_h^*(2)}\right)^2}\end{eqnarray*}$$
    
    
    
    
    
    This is the product of two Gaussians in the variable $\textsf{ID}$. The definition of $\mathsf{ID}(track)$ and $\sigma^*(track)$ is easily derived in terms of the more popular pulls. $\mathsf{ID} = \frac{\frac{dE}{dx}_{meas}-\frac{dE}{dx}_{exp-\pi}} {\frac{dE}{dx}_{exp-K}-\frac{dE}{dx}_{exp-\pi}}$ where $dE/dx_{meas}$ is the measured value of the $dE/dx$, $dE/dx_{exp-K}$ is the expected value of $dE/dx$ in kaon hypothesis and $dE/dx_{exp-\pi}$ is the expected value of $dE/dx$ in pion hypothesis. The relations between pulls, $\mathsf{ID}$ and $\sigma^*$ are $pull_{\pi} = \frac{\mathsf{ID}}{\sigma^*}$ and $pull_{K} = \frac{\mathsf{ID}-1}{\sigma^*}$, where $\sigma^* =\frac{{\sigma_{dE/dx}}}{\frac{dE}{dx}_{exp-K}-\frac{dE}{dx}_{exp-\pi}}$ with $\sigma_{dE/dx}$ being the track-dependent resolution on the $dE/dx$ measurement. $PID_{h(h^{'})}$ is the $\mathsf{ID}$ expectation value and depends on the mass assumption (0 for a pion, 1 for a kaon).
• $\mathcal{L}^{bckg} = \mathcal{L}^{kin}_{bckg} \cdot \mathcal{L}^{PID}_{bckg}$ is the background likelihood function.
  • The kinematics term is:
    $$\begin{eqnarray*}\mathcal{L}^{kin}_{bckg} = pdf^{kin}_{bckg}(M_{\pi\pi}, \alpha... ...a) \cdot \frac{1}{Norm}\cdot (e^{(c_0+c_1\cdot M_{\pi\pi}}+c_2) \end{eqnarray*}$$
    
    
    
    
    
    where $P^{'}(\alpha)$ is the distribution function of the variable $\alpha$ of background events which is extracted from data using the $B_{d,s}\rightarrow h^{+}h^{'-}$ sidebands ( $M_{\pi\pi} \in[4.600, 5.125]\cup[5.400, 5.800]$ GeV/c$^2$, see Fig. , top). The $M_{\pi\pi}$ profile of background is determined by the parameters $c_i$ that are left floating in the fit.
  • The PID term is:
    $$\begin{eqnarray*}\mathcal{L}_{bckg}^{PID} &=& pdf^{PID}(\textsf{ID(1), ID(2)},... ...t(\frac{\mathsf{ID}(2)-PID_K(2)}{\sigma_K^*(2)}\right)^2} \right]\end{eqnarray*}$$
    
    
    
    
    
    where holds the assumption that the background is composed of pions and kaons only and $f_{\pi}$ is the pion fraction determined by the fit separately in the three mass regions reported in Tab.  top right.

The PID Likelihood functions for both signal and background reported here are a simplified version of those actually used in the fit. The fit includes the track-by-track $dE/dx$ correlations (measured in an independent data sample) by using the full joint $pdf(\mathsf{ID}(1),\mathsf{ID}(2))$.

Table: Left table: $\mathcal{M}(\alpha)$ for $\alpha <0$ (i.e. the negative particle carries smaller momentum). Right table: $\alpha >0$ (i.e. the positive particle carries larger momentum).

mode	$\mathcal{M}^2(\alpha)= \mathcal{M}^2(\alpha<0)$
$B_d\rightarrow\pi^+\pi^-$	$M^2_{B^0_d}$
$B^0_d \rightarrow \pi^- K^+$	$M^2_{B^0_d}+(2+\alpha)(m^2_{\pi}-m^2_K)$
$\overline{B}^0_d\rightarrow K^-\pi^+$	$M^2_{B^0_d} +(1+\frac{1}{1+\alpha})(m^2_{\pi}-m^2_K)$
$\overline{B}^0_s\rightarrow \pi^- K^+$	$M^2_{B^0_s} +(2+\alpha)(m^2_{\pi}-m^2_K)$
$B^0_s \rightarrow K^- \pi^+$	$M^2_{B^0_s} +(1+\frac{1}{1+\alpha})(m^2_{\pi}-m^2_K)$
$B_s\rightarrow K^+K^-$	$M^2_{B^0_s} +(3+\alpha+\frac{1}{1+\alpha})(m^2_{\pi}-m^2_K)$

mode	$\mathcal{M}^2(\alpha)= \mathcal{M}^2(\alpha > 0)$
$B_d\rightarrow\pi^+\pi^-$	$M^2_{B^0_d}$
$\overline{B}^0_d \rightarrow \pi^+ K^-$	$M^2_{B^0_d} +(2-\alpha)(m^2_{\pi}-m^2_K)$
$B^0_d\rightarrow K^+\pi^-$	$M^2_{B^0_d} +(1+\frac{1}{1-\alpha})(m^2_{\pi}-m^2_K)$
$B^0_s\rightarrow \pi^+ K^-$	$M^2_{B^0_s}+(2-\alpha)(m^2_{\pi}-m^2_K)$
$\overline{B}^0_s \rightarrow K^+ \pi^-$	$M^2_{B^0_s} +(1+\frac{1}{1-\alpha})(m^2_{\pi}-m^2_K)$
$B_s\rightarrow K^+K^-$	$M^2_{B^0_s} +(3-\alpha+\frac{1}{1-\alpha})(m^2_{\pi}-m^2_K)$