The Unitarity Triangle through $B_d\to \rho^{\pm} \pi^{\mp}$ decays

We analyze the impact of the CP-violating observables in the $B_d\to \rho^{\pm} \pi^{\mp}$ system, combined with the precise measurement of $\sin2\beta$, in the extraction of the CKM matrix. We explore two strategies for determining the Unitarity Triangle in these modes. Computing the penguin parameters $(r_{\pm}, \phi_{\pm})$ and the ratio of two trees $(r_{t}, \phi_{t})$ within QCD factorization yields a precise determination of $(\bar\rho, \bar\eta)$, reflected by a weak dependence on $\phi_{\pm}$, which is shown to be a second order effect, as in the $B_d \to \pi^+ \pi^-$ system. Moreover, we find that the dependence on penguin amplitudes $r_{\pm}$ in $B_d\to \rho^{\pm} \pi^{\mp}$ is less pronounced than in the $B_d \to \pi^+ \pi^-$ case, since penguin contributions $r_{\pm}\approx r_{\pi\pi}/3$, implying an important simplification in our analysis. Independent experimental tests of the factorization framework are proposed and discussed, using $B_d \to \pi^+ \pi^-$ and $B_d \to \rho^{\pm} \rho^{\mp}$ modes.