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S-, P- and D-wave final state interactions and CP violation in B+- --> pi+- pi-+ pi+- decays

We study CP violation and the contribution of the strong pion-pion interactions in the three-body B+- --> pi+- pi-+ pi+- decays within a quasi two-body QCD factorization approach. The short distance interaction amplitude is calculated in the next-to-leading order in the strong coupling constant with vertex and penguin corrections. The meson-meson final state interactions are described by pion non-strange scalar and vector form factors for the S and P waves and by a relativistic Breit-Wigner formula for the D wave. The pion scalar form factor is calculated from a unitary relativistic coupled-channel model including pi pi, Kbar K and effective (2pi)(2pi) interactions. The pion vector form factor results from a Belle Collaboration analysis of tau- --> pi- pi0 nu_tau data. The recent B+- --> pi+- pi-+ pi+- BABAR Collaboration data are fitted with our model using only three parameters for the S wave, one for the P wave and one for the D wave. We find not only a sizable contribution of the S wave just above the pi pi threshold but also under the rho(770) peak a significant interference, mainly between the S and P waves. For the B to f_2(1270) transition form factor, we predict F^{Bf_2}(m_π^2)=0.098+-0.007. Our model yields a unified unitary description of the contribution of the three scalar resonances f_0(600), f_0(980) and f_0(1400) in terms of the pion non-strange scalar form factor.