$B\to \pi \pi$ Form Factors from Light-Cone Sum Rules with $B$-meson Distribution Amplitudes

Abstract

We study $B\to \pi \pi$ form factors using QCD light-cone sum rules with $B$-meson distribution amplitudes. These form factors describe the semileptonic decay $B\to \pi \pi \ell {\bar{\nu }}_{\ell }$, and constitute an essential input in $B\to \pi \pi {\ell }^{+}{\ell }^{-}$ and $B\to \pi \pi \pi$ 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 ${\bar{B}}^0\to {\pi }^{+}{\pi }^0$ form factors with the time-like 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 $B\to \rho$ form factors in the limit of $\rho$-dominance with zero width, thus providing a systematics for so-far-unaccounted corrections to $B\to \rho$ transitions. Using data for the pion vector form factor, we estimate finite width-effects and the contribution of excited $\rho$-resonances to the $B\to \pi \pi$ form factors. We find that these contributions amount up to $\sim 20\%$ in the small dipion mass region where they can be effectively regarded as a nonresonant ($P$-wave) background to the $B\to \rho$ transition.

Comments: 28 pages, 3 figures. A few comments added. Version published in JHEP