Comment on “B→M1M2: Factorization, charming penguins, strong phases, and polarization”

Abstract

We show that the factorization formula for non-leptonic B decays to two light flavor non-singlet mesons derived by Bauer et al. in the context of soft-collinear effective theory is equivalent to the corresponding formula in the QCD factorization approach. The apparent numerical differences in the analysis of B → ππ data performed by these authors, as compared to previous QCD factorization analyses, can largely be attributed to the neglect of known perturbative and power corrections.The extent to which hadronic decays of B mesons to two light hadrons can be computed from first principles in QCD has been the subject of many investigations, following the statement [1,2] that the decay amplitudes factorize in the limit of very large B-meson mass. Taking the final state to consist of two pions, the amplitude can be represented schematically in the formIn this equation, F B→π denotes a physical form factor, and Φ B and Φ π are the leading-twist light-cone distribution amplitudes of the mesons. The quantities T I,II are perturbative hard-scattering kernels involving the two scales m b (hard) and √ m b Λ (hard-collinear), which are linked to the other elements of the formula by convolution integrals (indicated by an asterisk).The authors of [3] suggest a factorization formula (their Eq. (24)) similar to (1) in the framework of soft-collinear effective theory (SCET) and imply that it is conceptually different. They also argue that, even at leading order in the 1/m b expansion, there may be an additional term on the right-hand side of (1), corresponding to longdistance contributions from cc penguins. They do not disprove factorization of charm-penguin loops by providing a counter-example to factorization; rather, they state that they were not able to demonstrate factorization.In this Comment, we explain why the formula given in [3] is identical in content to the QCD factorization formula (1), and why non-factorizable charm-penguin contributions are of higher order in the 1/m b expansion. We also point out that the phenomenological analysis of [3] neglects important perturbative and power corrections which are already known. Once these are included, there is little room for significant additional contributions to the QCD penguin amplitude. I. EQUIVALENCE OF THE SCET AND QCD FACTORIZATION FORMULAEWe first note that the coefficient function of the spectator-scattering term can be represented as a convolution T II = C II ⋆ J of hard and hard-collinear coefficient functions. The formula quoted in [3] follows from (1) by rewritingwith ζ Bπ J defined as J ⋆ Φ B ⋆ Φ π , and C II defined by the decomposition of T II shown above. In addition, in [3] the SCET form factor ζ Bπ rather than the physical QCD form factor F B→π is used. As discussed in [2], this implies another rearrangement of this type.In general SCET provides a powerful tool to simplify factorization proofs (a task that has not yet been completed for the case of B → ππ considered here), but the resulting QCD factorization formulae can also be obtained using traditional...

Power corrections in charmless nonleptonicBdecays: Annihilation is factorizable and real

“…The possibility of large nonperturbative charm loop contributions was first discussed in Refs. [12,13], and the size of these terms remains controversial [25,26]. 3 Recall that to derive the η 6 , we note that λ 8 = η 4 , and changing the scaling λ → η for four collinear quark fields in matching SCET I → SCET II gives the extra η 2 .…”

“…(26) and (44). Other Kπ channels have similar expressions with different Clebsch-Gordan coefficients.…”

NNLO vertex corrections to non-leptonic B decays: Tree amplitudes

“…A successful attempt to gain control over the complicated QCD effects in non-leptonic B decays is QCD factorization (QCDF) [1][2][3][4]. Within this approach hadronic matrix elements are represented as convolutions of perturbative objects (hard-scattering kernels) with non-perturbative quantities (light-cone distribution amplitudes, LCDA).…”

Taming the end-point singularities in heavy-to-light-quark decays

“…(2) are comparable in size. This point differs from the QCD factorization (QCDF) analysis [12], where the hard scattering contribution is very small.…”