A ug 2 00 0 QCD Factorization For B Decays To Two Light Pseudoscalars Including Chirally Enhanced Corrections ∗

In this letter, we examine QCD factorization including chirally enhanced corrections. To avoid the logarithmically divergent integrals, we describe the emitted meson with twist-2 and twist-3 wave functions instead of two collinear massless on-shell quarks. Our results are encouraging: the QCD coefficients a6 and a8 at next-to-leading order of αs are infrared finite and no logarithmically divergent integrals exist, which means that we can include the most important 1/mb corrections consistently in QCD factorization. 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 Last year, Beneke, Buchalla, Neubert and Sachrajda (BBNS) [1] proposed a promising method: under the assumption that the emitted light meson is described by two collinear massless on-shell quarks, they find that 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 [2]. In the heavy quark limit, they show that the hadronic matrix elements can be expressed as [1] 〈M1M2|Q|B〉 = 〈M2|J1|0〉〈M1|J2|B〉 · [1 + Σrnα s +O(ΛQCD/mb)]. (1) 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 [1], 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. But unfortunately, Beneke et al. [1,3] have shown that a6, a8 involve logarithmically divergent integrals which violate factorization when including next-to-leading corrections. This makes us very uncomfortable. In this letter, we try to solve this problem. Chirally enhanced corrections arise from twist-3 light-cone distribution amplitudes. For example, π meson twist-3 distribution amplitude is γ5μπφ 0 π(x) where μπ = m2π mu+md , and φπ(x) is twist-3 wave function. In the framework of BBNS method, when considering the contributions of Fig.(f), there would involve an integral: 1 ∫ 0 dx 0 π(x) x , because φπ(x) do not fall off fast enough at the end point, the above integral is logarithmically divergent. However, we notice that the Dirac structure of this twist-3 distribution amplitude is indeed in contradiction with the assumption of BBNS method where the emitted meson is described by two collinear massless on-shell quarks. We shall show this below: for definiteness, we take the emitted meson as π(dū) and the corresponding momentum of valence quarks are pd = x∗pπ and pū = (1− x) ∗ pπ, massless on-shell condition means that / pd(dū) = / pū(dū) = (dū) / pū = (dū) / pd = 0. (2) Thus two collinear massless on-shell spinors dū can only lead to Dirac structure of / pπ or γ5 / pπ. So the assumption of two collinear massless on-shell quarks is only consistent with leading twist wave function of π meson, γ5 / pπφπ(x), but unsuitable to be applied in calculating chirally enhanced corrections because twist-3 distribution amplitudes conflict with Eq.(2). Thus we hope that logarithmically divergent integrals would disappear in a6 and a8 when using twist-3 distributions. Of course, we shall demonstrate that the infrared divergences from the ”vertex correction” diagrams cancel after summing over Fig.(a)-(d) when using twist-3 light-cone distribution amplitudes. The remaining finite terms are also different from the results calculated by BBNS method. We have noticed that in Ref [4], the authors have used twist-3 distribution amplitude to calculate the strong penguin corrections (Fig.(e)-(f)) and find that the logarithmically divergent integrals disappear in a6 and a8. 2 However, they did not calculate ”vertex correction” diagrams and also did not show the infrared safety of a6 and a8 at the next-to-leading order of αs. In addition chirally enhanced corrections from hard spectator scattering diagrams are not considered too in their paper. In the following, we take leading twist and twist-3 wave functions to describe the emitted light meson, and we will show the infrared safety under this approach and compare our results with BBNS’s. The |∆B| = 1 effective Hamiltonian is given by [5] Heff = GF √ 2 [