UNCERTAINTIES ON THE CP PHASE-ALPHA DUE TO PENGUIN DIAGRAMS

A major problem in the determination of the the CP angle alpha, that should be measured through modes of the type B-d, ($) over bar B-d --> pi pi,..., is the uncertainty coming from Penguin diagrams. We consider the different ground state modes pi pi, pi rho, rho rho, 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 Delta alpha. 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 Delta alpha is much smaller for the modes rho pi and rho rho than for pi pi, and we plot their allowed region as a function of ct itself. Moreover, we show that most of the modes contribute to the asymmetry with the same sign, and define for their sum an effective D-eff and an effective Delta alpha(eff), an average of Delta alpha for the different modes. It turns out that D-eff is of the order of 0.9, Delta alpha(eff) is between 5% and 10% and, relative to pi pi, the statistical gain for the sum is of about a factor 10. Finally, we compute the ratios K pi/pi pi,... that test the strength of the Penguins and depend on the CP angles, as emphasized by Silva and Wolfenstein and by Deandrea et al.