B->rho pi decays and final state phases

Abstract

Using the isospin analysis, the fact that penguin is pure Delta I=1/2 transition, the unitarity for tree graph and C-invariance of strong interactions, it is shown that delta_{t}=0=tilde{delta}_{t}, r_{-0}=tr_{0-},delta_{-0}-delta_{0-}=pm pi, 2tr_{f}r_{\bar{f}}\cos (δ_{f}-δ_{bar{f}})=(r_{f}^{2}+t^{2}r_{bar{f}}^{2})-r_{-0}^{2}, where delta &#39;s are final state phases and r&#39;s are penguin to tree ratios defined in the text. Using the factorization for tree graph as input and the experimental data, we have obtained the following bounds on r_{f},r_{bar{f}},delta_{f} and delta_{bar{f}}:0.11leq r_{f}leq 0.21,0.18leq r_{bar{f}}leq 0.30; 11^{circ}leq delta_{f}\leq 57^{circ},23^{circ}leq delta_{bar{f}}leq 90^{circ} for the case z_{f,bar{f}}=cos alpha cos delta_{f,bar{f}}<0. For z_{f}<0 and z_{\bar{f}}>0, we obtain the following bounds for r_{bar{f}} and delta_{bar{f}}:0.14leq r_{bar{f}}leq 0.46; 90^{circ}leq delta_{bar{f}}leq 170^{circ}. From experimental data, for the decays B^{-}->rho ^{-}pi ^{0}(rho ^{0}pi ^{-}) we get epsilon_{-0}=0.28pm 0.10,epsilon_{0-}=0.51pm 0.10, frac{A_{CP}^{-0}}{A_{CP}^{0-}}=-0.8\pm 0.1 i.e. A_{CP}^{-0} and A_{CP}^{0-} have opposite sign, where 1+epsilon_{0-},_{0-}=frac{left| T^{-0,0-}+C^{0-,-0}right|}{left| T^{-0,0-}|}. In the naive quark model the above values imply a_{2}/a_{1}=0.39\pm 0.14 and a_{2}/a_{1}=0.37\pm 0.07 consistent with each other.