UNCERTAINTIES ON THE CP PHASE-ALPHA DUE TO PENGUIN DIAGRAMS

A major problem in the determination of the the CP angle alpha, that s hould be measured through modes of the type B-d, ($) over bar B-d --> pi pi,..., is the uncertainty coming from Penguin diagrams. We conside r the different ground state modes pi pi, pi rho, rho rho, and, assumi ng the FSI phases to be negligible, we investigate the amount of uncer tainty coming from Penguins that can be parametrized by a dilution fac tor D and an angle shift Delta alpha. The parameter D is either 1 or v ery close to 1 in all these modes, and it can be measured independentl y, 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 rh o rho than for pi pi, and we plot their allowed region as a function o f ct itself. Moreover, we show that most of the modes contribute to th e 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 sta tistical 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.