B → ππ l ν l decays in a QCD relativistic potential model

In the framework of a QCD relativistic potential model we evaluate the form factors describing the exclusive decay B → ππlν. The calculation is performed in a phase space region far away from the resonances and therefore is complementary to other decay mechanisms where the pions are produced by intermediate particles, e.g. in the chiral approach. We give an estimate of the contribution of the non resonant channel of the order of B(B− → ππlν̄l) ≃ 2.2 × 10−4. In this letter we shall study the B−meson decays B− → ππlν̄l (1) B̄ → ππlν̄l . (2) ¿From the experimental side these decays are interesting in view of the future programs at the B-factories. For example, some of the preliminary studies on the CP violations at these machines [1] have examined the possibility to extract the angle α of the unitarity triangle by the B → ρπ non leptonic decay channel. The non-resonant decay mode B → 3π would be interesting to analyze in this context, as it might provide a significant background to the main decay process. While a calculation from first principles is not available at the moment, a useful approximation might be the factorization approximation and, within this approximation, the decay modes (1) and (2) would provide the crucial hadronic matrix elements needed to compute the relevant amplitudes. In passing we note that there is another channel, i.e. the B− → ππlν̄l decay mode, which will not be examined here because it is less interesting from an experimental point of view. ¿From a theoretical standpoint semileptonic B-meson decays with two hadrons in the final state represent a formidable challenge as they involve hadronic matrix elements of weak currents with three hadrons. They can be studied by pole diagrams, which amounts to a simplification because only two hadrons are involved in the hadronic matrix elements. This is the approach followed in some papers where these decays have been examined in the framework of the chiral perturbation theories for heavy meson decays [2],[3]. This method is based on an effective theory implementing both heavy-quark and chiral symmetry [3], [4], [5], [6] and allows to achieve, for systems comprising both heavy (Q) and light (q) quarks, rigorous results in the combined mQ → ∞, mq → 0 limit. However the range of validity of this approach is limited by the requirement of soft pion momenta. In the soft pion limit the amplitude is dominated by a few tree diagrams with resonances as intermediate states, and some clear predictions can be made, but, at least for B decays, the actual phase space is relatively large and the phenomenological interest of these predictions is modest. The aim of this letter is to examine the decays (1) and (2) in the framework of a QCD relativistic potential model [7] and to extend the kinematical range where theoretical predictions are possible. We shall present a detailed analysis of the four form factors relevant to (1); for reasons of space we shall only give a prediction for the width of the decay (2). We shall not include final state interactions in our calculation since no consistent way to compute them is presently available. It’s clear however that they can modify our numerical calculations [8]. In two recent papers: [9] ,[10] we have presented an analysis of some semileptonic and rare B decays into one light hadron employing the relativistic potential model in an approximation