0 QCD Factorization For B Decays To Two Light Pseudoscalars Including Chirally Enhanced Corrections ∗

Since b quark mass is not asymptotically large, chirally enhanced corrections which arise from twist-3 wave functions may be important in B decays. We thus evaluate the hadronic matrix elements with the emitted meson described by leading twist and twist-3 distribution amplitudes Φp(x). After summing over the four ”vertex correction” diagrams, we obtain the results with infrared finiteness which shows that chirally enhanced corrections arise from Φp(x) can be consistently included in QCD factorization. We also briefly discuss the contributions from ”hard spectator” diagrams. PACS numbers 13.25.Hw 12.38.Bx Supported in part by National Natural Science Foundation of China and State Commission of Science and Technology of China Email: [email protected], [email protected], [email protected] Mailing address 1 It is well known that two-body, non-leptonic charmless B decays are crucial for extracting CKM matrix elements. However, due to our ignorance on how to calculate hadronic matrix elements, we conventionally resort to Bjorken’s color transparency argument [1] which lead to ”naive factorization assumption”, 〈M1M2|Q|B〉 = 〈M2|J1|0〉〈M1|J2|B〉, (1) This assumption makes the hadronic matrix elements scale-independent. Noting that Wilson Coefficients are schemeand scale-dependent, the theoretical calculations on the branching ratios would then be schemeand scale-dependent which is unacceptable. To save factorization hypothesis, a phenomenological parameter Neff is introduced [4] which is commonly called generalized factorization. However this approach is not satisfactory yet because in principle Neff is process dependent. Recently, Beneke, Buchalla, Neubert and Sachrajda [2,3] proposed a promising method: in the heavy quark limit, they show that the emitted light meson can be described with leading twist-2 distribution amplitude, the infrared divergences of the hard-scattering amplitudes are canceled after summing over the four ”vertex correction” diagrams (Fig.(a)-(d)), which is a one-loop demonstration of Bjorken’s color transparency argument [1]. In the heavy quark limit, they show that the hadronic matrix elements can be expressed as [2] 〈M1M2|Q|B〉 = 〈M2|J1|0〉〈M1|J2|B〉 · [1 + Σrnα s +O(ΛQCD/mb)]. (2) If power corrections in 1/mb can be safely neglected, then everything is perfect. At the zero order of αs, it would come back to ”naive factorization”, and at the higher order of αs, the corrections can be systematically calculated in Perturbative QCD, which means that the decay amplitudes of B meson can be computed from first principles, and the necessary input are heavy-to-light form factors and light-cone distribution amplitudes. But in the real world, bottom quark mass is not asymptotically large (but 4.8 GeV), and numerically power suppression may fail in some cases. An obvious and possibly the most important case is chirally enhanced power corrections. As pointed out in ref [2], numerically the enhanced factor rχ = 2m2π mb(mu+md) ≃ 1.18 which makes the power suppression completely fail. This parameter is multiplied by a6 and a8, where a6 is very important numerically in penguindominated B decays. So an evaluation of the hadronic matrix elements including chirally enhanced corrections may be phenomenologically or numerically important. In this letter, we shall examine this problem in some detail. Chirally enhanced corrections arise from twist-3 light-cone distribution amplitudes, generally called Φp(x) and Φσ(x). For light pseudoscalar mesons, they are defined as [6] 〈P (p)|q̄(y)iγ5q(x)|0〉 = fpμp ∫ 1 0 du e ′′φp(u), (3) 〈P (p)|q̄(y)σμνγ5q(x)|0〉 = ifpμp(p′μzν − p′νzμ) ∫ 1 0 du e i(up′·y+ūp′·x)φσ(u) 6 , (4) where μp = M p mu+md , z = y−x. We notice that in Ref. [2] color transparency is demonstrated in one-loop level in the heavy quark limit. If we want to include chirally enhanced corrections consistently, we should describe the emitted light meson with leading twist-2 and twist-3 distribution amplitudes, which means that we should show the infrared finiteness using 2 twist-3 distribution amplitudes after summing over the ”vertex correction” diagrams. In this paper, we shall restrict ourselves to Φp(x) while postpone the discussion of Φσ(x) to ref [7] because of the complicated derivation in proving the infrared finiteness of the ”vertex correction” diagrams using Φσ(x). We notice that in Ref [8], the authors have used twist-3 distribution amplitude Φp(x) to calculate the strong penguin corrections (Fig.(e)-(f)). The difference of our work from ref [8] is that we calculate ”vertex correction” diagrams and show the infrared finiteness of a6 and a8 at the order of αs. In the following, we take leading twist and twist-3 wave functions Φp(x) to describe the emitted light meson, and we will show the infrared finiteness under this approach. The |∆B| = 1 effective Hamiltonian is given by [9] Heff = GF √ 2 [