CP Violation and Strong Pion-Pion Interactions in the Weak $${B^\pm \to \pi^\pm \pi^\mp \pi^\pm}$$ Decays

Abstract

An analysis of $${B^\pm \to \pi^\pm \pi^\mp \pi^\pm}$$ decay data in a quasi two-body B ± → π ± [π + π −] S, P, D QCD factorization framework is performed. The short distance amplitudes are calculated including next-to-leading order in α s vertex and penguin corrections. The long distance amplitudes due to the [π + π −] S, P, D final state interactions are described by the pion non-strange scalar and vector form factors for the [π + π −] S S-wave state and [π + π −] P P-wave state, respectively and by a relativistic Breit–Wigner formula for the[π + π −] D D-wave state. We achieve a good fit of the data with only three parameters for the S wave, one for the P wave and one for the D wave. Our model gives a unified unitary description of the three scalar resonances, f 0(600), f 0(980) and f 0(1400) in terms of the pion scalar form factor. We predict for the B to f 2(1270) transition form factor, $${F^{Bf_2}(m_\pi^2)=0.098\pm0.007}$$ .