An analysis of the time-dependent CP asymmetryin $B \to \pi \pi$ decays in the standard model

Abstract

Measurements of the time-dependent CP-asymmetry in the decay \(B^0_d (t)\to \pi^ + \pi^-\) and its charge conjugate by the BELLE and BABAR collaborations currently yield \(C_{\pi \pi}^{ + -} = -0.46 \pm 0.13\) and \(S_{\pi \pi}^{ + -} = -0.74 \pm 0.16\), characterizing the direct and mixing-induced CP-asymmetries, respectively. We study the implication of these measurements on the CKM phenomenology taking into account the available information in the quark mixing sector. Our analysis leads to the results that the ratio |Pc/Tc| involving the QCD-penguin and tree amplitudes and the related strong phase difference \(\delta_c = \delta_c^P - \delta_c^T\) in the \(B^0_d/\bar{B}_d^0 \to \pi^ + \pi^-\) decays are quite substantial. Using the isospin symmetry to constrain |Pc/Tc| and \(\cos (2\theta)\), where \(2 \theta\) parameterizes the penguin-induced contribution, we present a fit of the current data including the measurements of \(S_{\pi \pi}^{ + -}\) and \(C_{\pi \pi}^{ + -}\). Our best-fits yield \(\alpha = 92^\circ\), \(\beta = 24^\circ\), \(\gamma = 64^\circ\), \(\vert P_c/T_{\mathrm c}\vert = 0.77\), and \(\delta_c = -43^\circ\). At 68% C.L., the ranges are \(81^\circ \leq \alpha \leq 103^\circ\), \(21.9^\circ \leq \beta \leq 25.5^\circ\), \(54^\circ \leq \gamma \leq 75^\circ\), \(0.43 \leq \vert P_c/T_{\mathrm c}\vert \leq 1.35\) and \(-64^\circ \leq \delta_c \leq -29^\circ\). Currently en vogue dynamical approaches to estimate the hadronic matrix elements in \(B \to \pi \pi\) decays do not provide a good fit of the current data.