B →ππ form factors from light-cone sum rules with B-meson distribution amplitudes

We study B → ππ form factors using QCD light-cone sum rules with B-meson distribution amplitudes. These form factors describe the semileptonic decay B→ π π ν̄ , and constitute an essential input in B → ππℓ + ℓ − and B → πππ decays. We employ the correlation functions where a dipion isospin-one state is interpolated by the vector light-quark current. We obtain sum rules where convolutions of the P -wave B̄⁰→ π⁺π⁰ form factors with the timelike pion vector form factor are related to universal B-meson distribution amplitudes. These sum rules are valid in the kinematic regime where the dipion state has a large energy and a low invariant mass, and reproduce analytically the known light-cone sum rules for B → ρ form factors in the limit of ρ-dominance and zero width, thus providing a systematics for so far unaccounted corrections to B → ρ transitions. Using data for the pion vector form factor, we estimate finite-width effects and the contribution of excited ρ-resonances to the B → ππ form factors. We find that these contributions amount up to ∼ 20% in the small dipion mass region where they can be effectively regarded as a nonresonant (P -wave) background to the B → ρ transition.