Constraints on the strong phases in B →ππ and B →πK decays

The nonleptonic decays B→ππ and B→π K received much attention lately. The decay B→π K is suppressed in the Standard Model due to the Wilson coefficients and the Cabibbo-Kobayashi-Maskawa matrix elements, and a significant experimental rate of this process would be an interesting signal of New Physics. The hadronic decays of B into light pseudoscalar mesons are also good candidates for providing information on the weak angles α, β and γ of the unitarity triangle, which parametrize CP violation in the Standard Model. The time dependent CP asymmetry in B0 →π+π- was proposed as one of the ways of extracting the angle α, while the processes B0 →π+ K- and B+ →π0 K+ were considered in recent strategies for determining the angle γ. However, the unknown strong phases of the amplitudes of these processes seriously affect the accuracy of the determination. In this work we obtain theoretical constraints on the strong phases using a dispersive formalism recently proposed, based on the analytic continuation in the mass squared of one final particle of the decay B → P1P2. The main result is a dispersion relation from which we write the amplitude of the decay B0 →π+ π- as a diagram sums in terms of complex amplitudes corresponding to the tree, penguin, exchange and penguin annihilation diagrams, respectively. We indicated explicitly the weak phases β and γ which, according to recent works, are restricted to the ranges 0.17≤ β ≤ 0.52 and 0.349 ≤γ ≤ 2.79. As intermediate states we keep π+ π- giving the elastic channel, as well as two meson states responsible for the soft inelastic scattering, e.g., π0 π0, K+ K-, K0 K0 bar, π0 η8 and η8 η8. By SU(3) flavour symmetry the amplitudes of these processes are written in terms of the same diagram amplitudes. Assuming in addition that the dominant amplitudes are AT and AP as suggested by the naive quark-diagram approximation, we obtain the relative strong phase δ = δP - δT as a function of the weak angles γ and β. A more complete analysis including the exchange and penguin annihilation amplitudes is in progress. In the case of B →π K decays, a detailed investigation exploiting the properties of the weak Hamiltonian and isospin symmetry was performed recently, providing a convenient parametrization of the amplitudes, where the only free parameters are the strong phases. On the other hand, a two channel Regge model for the strong scattering π K →π K, including both t-channel and u-channel Regge poles was discussed recently. With this input, the dispersion relation will provide information about the strong phases of the weak amplitudes of B →π K decays. (authors)