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, ${F^{Bf_2}(m_\pi^2)=0.098\pm0.007}$ .