Uncertainties on the CP Phase $\bfalpha$ 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 $B_d, \bar{B}_d \to \pi \pi , \cdots ,$ 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 $\alpha$ 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 $K\pi /\pi \pi , \cdots$ that test the strength of the Penguins and depend on the CP angles, as emphasized by Silva and Wolfenstein and by Deandrea et al.