Sudakov Resummation for Subleading SCET Currents and Heavy-to-Light Form Factors

Hill, Richard J.; Becher, Thomas; Lee, Seung Joon; Neubert, Matthias

doi:10.1088/1126-6708/2004/07/081

Cited by 119 publications

(280 citation statements)

References 49 publications

(147 reference statements)

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“…The next-to-leading order terms have been computed recently and were found to be small [5]. Specifically, for the case of light pseudoscalar mesons and an asymptotic light-cone distribution amplitude Φ π (x) = 6x(1 − x), one finds that the convolution integrals over the jet function give rise to the series i ), and .…”

Section: Validity Of Perturbation Theory At the Hard-collinear Smentioning

confidence: 96%

“…This fact is important for phenomenology when one opts to treat the hard-collinear scale as non-perturbative, but does not by itself represent a conceptual difference between the formulae given in [1,2] and [3]. Furthermore, the usefulness of the universality of J is limited to the approxima- tion where one neglects radiative corrections to the hardscattering kernels C II , since only then does the function ζ Bπ J reduce to a single number [4,5]. In other words, a phenomenological treatment of the hard-collinear scale as non-perturbative relies on the approximation that the kernels are restricted to their tree-level approximations, whereas one of the key features of QCD factorization (as opposed to naive factorization) is that one can consistently include radiative corrections.…”

Section: Equivalence Of the Scet And Qcd Factorization Formulaementioning

confidence: 99%

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

et al. 2005

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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...

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Section: Validity Of Perturbation Theory At the Hard-collinear Smentioning

confidence: 96%

Section: Equivalence Of the Scet And Qcd Factorization Formulaementioning

confidence: 99%

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

et al. 2005

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“…The scalar coefficient is not independent but can be related to the vector coefficients by means of the equations of motion, yielding [13] …”

Section: Matching Coefficientsmentioning

confidence: 99%

Heavy-to-light currents at NNLO in SCET and semi-inclusive decay

et al. 2011

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We perform the two-loop matching calculation for heavy-to-light currents from QCD onto soft-collinear effective theory for the complete set of Dirac structures. The newly obtained matching coefficients enter several phenomenological applications, of which we discuss heavy-to-light form factor ratios and exclusive radiative decays, as well as the semi-inclusive decayB → X s ℓ + ℓ − . For this decay, we observe a significant shift of the forward-backward asymmetry zero and find q 2 0 = (3.34 +0.22 −0.25 ) GeV 2 for an invariant mass cut m cut X = 2.0 GeV.

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“…(4) these logarithms have been resumed in Ref. [18], and the running for the other part was studied in [14]. It has been argued [14] that the α s ( √ E π Λ) is absent in ζ Bπ , however this relies on the conjecture that diagrams containing a soft-collinear messenger mode [15] in the theory below the scale √ E π Λ cancels all endpoint singularities.…”

Section: Formal Comparisonmentioning

confidence: 99%

Differences between soft collinear effective theory and QCD factorization forB→ππdecays

et al. 2005

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We give a detailed description of the differences between the factorization and results derived from SCET and QCDF for decays B → M1M2. This serves as a reply to the comment about our work B → M1M2: Factorization, charming penguins, strong phases, and polarization [1] made by the authors in [2]. We disagree with their criticisms.In [1] we derived a factorization formula for exclusive B decays to two light mesons using the soft collinear effective theory (SCET) [3]. Recently, Beneke, Buchalla, Neubert and Sachrajda posted a comment about our work [2], and compared it with their QCDF (QCD factorization) approach [4]. In this paper we compare results and reply to their comments [2].For easy reference, we summarize a few points made in Ref.[1] that disagree with Ref.[4]. We found that: i) a proper separation of scales Q 2 ≫ E π Λ ≫ Λ 2 in the factorization theorem are different in SCET and QCDF, and can be regarded as a formal disagreement if desired (Q = m b , E π ); ii) only a subset of α s (m b ) corrections are currently known, so results for these corrections are formally incomplete; iii) certain amplitudes are sensitive to the treatment of m c , with parametrically large ∼ v contributions from cc in the NRQCD region where v is the velocity power counting parameter; iv) current B → ππ data analyzed at LO in SCET supports values for the parameters ζ Bπ ∼ ζ Bπ J , in disagreement with numerical inputs adopted in QCDF. We also emphasized that Λ/m b power corrections need to be of natural size to support model independent phenomenology and verify that the expansion converges, concepts which are sometimes relaxed in QCDF phenomenological analyses. If power corrections change the LO values of ζ Bπ and ζ Bπ J substantially then this would indicate that the power expansion is not converging and a different expansion would be needed if model independent results are desired.We take this opportunity to also comment on results from Ref.[1] where we found agreement with points made in Ref. [4]. The original starting idea is the same, that factorization theorems for these decays should be derived by making a systematic expansion of QCD in Λ QCD /m b , where the terms in this expansion are model independent and unique. Earlier discussion of QCD based factorization methods for nonleptonic decays can be found in [5,6]. We agree on the scaling in Λ/m b for the LO amplitudes. There is agreement that input on nonperturbative functions can be obtained from B → π form factors and LO light-cone meson distribution functions φ π (x) and φ B (k + ). We also agree that at LO factorization occurs for amplitudes from light quark penguin loops, as well as tree, and color-suppressed diagrams. Finally, the set of the one-loop hard corrections computed in Ref. It is also worth emphasizing that the scope of our two works was different. In Refs. [4,8] factorization theorems were proposed based on the study of the IR singularities of lowest order diagrams in perturbation theory. Input parameters are taken from QCD sum rules. Certain power suppres...

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Sudakov Resummation for Subleading SCET Currents and Heavy-to-Light Form Factors

Cited by 119 publications

References 49 publications

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

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

Heavy-to-light currents at NNLO in SCET and semi-inclusive decay

Differences between soft collinear effective theory and QCD factorization forB→ππdecays

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