$B$ to $π$ Form Factors in collinear factorization approach

Abstract

The form factors for semi-leptonic B decays, B → πlν l , are calculated under collinear factorization approach. The end-point divergences are regularized by a ξ-regularization, where ξ means the collinear fraction of the spectator anti-quark of the B meson. The form factors are calculated up-to O(α s /m B ). The complete O(1/m B ) contributions from the π meson are calculated explicitly by a collinear expansion method. A well-defined power expansion scheme is built such that the O(1/m B ) contributions are about 30% of the leading order contributions. A small value F + (0) = 0.164 is found. This confirms the SCET result F + (0) = 0.17 from B → ππ decays. The form factors are calculated for 0 ≤ q 2 ≤ 16 GeV 2 , where q 2 is the invariant mass of the lepton pair in B → πlν l . An extrapolation of the form factors to q 2 > 16 GeV 2 is made to obtain |V ub | −2 ¡ 26.42 GeV 2 0 dq 2 (dΓ/dq 2 ) th = (5.71 ± 0.91)ps −1 . We determine |V ub | = (3.95 ± 0.13 exp ± 0.32 th ) × 10 −3 from the world averaged branching ratio Br( B → πlν l ) = (1.36 ± 0.09) × 10 −4 .