Uncertainties on the CP phase α due to Penguin diagrams

A major problem in the determination of the the CP angle α, that should be measured through modes of the type Bd,Bd → ππ, …, is the uncertainty coming from Penguin diagrams. We consider the different ground state modes ππ, πϱ, ϱϱ, and, assuming the FSI phases to be negligible, we investigate the amount of uncertainty coming from Penguins that can be parametrized by a dilution factor D and an angle shift Δα. The parameter D is either 1 or very close to 1 in all these modes, and it can be measured independently, up to a sign ambiguity, by the t dependence. Assuming factorization, we show that Δα is much smaller for the modes ϱπ and ϱϱ than for ππ, and we plot their allowed region as a function of α itself. Moreover, we show that most of the modes contribute to the asymmetry with the same sign, and define for their sum an effective Deff and an effective Δαeff, an average of Δα for the different modes. It turns out that Deff is of the order of 0.9, Δαeff is between 5% and 10% and, relative to ππ, the statistical gain for the sum is of about a factor 10. Finally, we compute the ratios Kπππ, … that test the strength of the Penguins and depend on the CP angles, as emphasized by Silva and Wolfenstein and by Deandrea et al.