An Analysis of the Time-Dependent CP Asymmetry in $B \to \pi \pi$ 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 $|P_c/T_c|$ 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 $|P_c/T_c|$ 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: $\alpha = 92^\circ$, $\beta = 24^\circ$, $\gamma = 64^\circ$, $|P_c/T_c| = 0.77$, and $\delta_c = -43^\circ$. At 68% C.L., the ranges are: $81^\circ \leq \alpha \leq 103^\circ$, $21.9^\circ \leq \beta \leq 25.5^\circ$, $54^\circ \leq \gamma \leq 75^\circ$, $0.43 \leq |P_c/T_c| \leq 1.35$ and $-64^\circ \leq \delta_c \leq -29^\circ$. Currently {\it 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.