]>NUPHB13450S0550-3213(15)00256-410.1016/j.nuclphysb.2015.07.016The AuthorsHigh Energy Physics – PhenomenologyFig. 1Diagrammatical representation of the correlation function Πμ(n⋅p,n¯⋅p) at tree level.Fig. 2Diagrammatical representation of the correlation function Πμ(n⋅p,n¯⋅p) at O(αs).Fig. 3One-loop diagrams for the B-meson DA Φbu¯αβ(ω′) defined in (12).Fig. 4Four different models of ϕB+(ω,μ0) (left plot) and ϕB−(ω,μ0) (right plot). A reference value of ω0(μ0)=350 MeV is taken for all the models. Solid (red), dotted (blue), dashed (green) and dot-dashed (black) curves correspond to ϕB,I±, ϕB,II±, ϕB,III± and ϕB,IV±, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)Fig. 5The shape of fBπ+(q2) with the value at zero momentum transfer fixed to the prediction from the LCSR with pion DAs, from which the parameter ω0(1 GeV) is determined for a given model of ϕB±(ω,μ0). Solid (blue), solid (black), dotted (black), dashed (black) and dot-dashed (black) curves are obtained from the pion LCSR and from the ones with the B-meson DAs ϕB,I±, ϕB,II±, ϕB,III± and ϕB,IV±, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)Fig. 6Dependence of the form factor fBπ+(0) computed from the NLL resummation improved sum rules (78) on the Borel parameter (left panel) and on the effective threshold (right panel). The solid, dashed and dot-dashed curves correspond to s0=0.70 GeV2,0.65 GeV2,0.75 GeV2 (left panel) and M2=1.25 GeV2,1.0 GeV2,1.25 GeV2 (right panel), respectively.Fig. 7Hard-collinear scale dependence of the form factor fBπ+(0) (left panel) and q2 dependence of the NLO radiative correction to fBπ+(q2) with both hard and hard-collinear scales varied in the allowed regions as explained in the text (right panel).Fig. 8Left: The pion energy dependence of the ratio R1(Eπ). The black curves correspond to the sum rule predictions with the Borel mass taken as 1.25 GeV (solid), 1.0 GeV (dashed) and 1.5 GeV (dot-dashed). The two green curves illustrate a pure 1/Eπ and a pure 1/Eπ2 dependence. Right: The heavy-quark mass dependence of the quantity R2(mQ). The three curves are predicted from the sum rules with the Borel mass varied between 1.0 GeV and 1.5 GeV around the default value 1.25 GeV. Note that in both plots we take the pion decay constant fπ=130.41 MeV [40] instead of using the two-point QCD sum rules in Appendix C as done in the remainder of this paper. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)Fig. 9q2 dependence of the vector form factor fBπ+(q2) (left) and of the scalar form factor fBπ0(q2) (right). The pink (solid) and the blue (solid) curves are computed from the LCSR with B-meson DAs and with pion DAs, respectively, and the shaded regions indicate the estimated uncertainties. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)Fig. 10q2 dependence of the re-scaled form factor (1−q2/mB⁎2)fBπ+(q2) predicted from the sum rules with B-meson DAs and the parameter ω0(1 GeV) determined by matching the Lattice point at q2=17.34 GeV2 [55]. The Lattice data are taken from Fermilab/MILC [55] (pink band), HPQCD [56] (blue squares), RBC/UKQCD [57] (green triangles). The blue curve is again obtained from the sum rules with pion DAs [48] with central inputs. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)Fig. 11Top: The normalized differential q2 distribution of B→πμνμ computed from (103) with the form factor fBπ+(q2) predicted from the sum rules with B-meson DAs and fitted to the z-parametrization (red band), and that predicted from the sum rules with pion DAs and z-parametrization (blue band). The experimental data bins are taken from [58] (brown spade suits), [59] (green five-pointed stars), [60] (purple squares), [61] (orange triangles), [62] (magenta full circles). Bottom: The normalized differential distribution of B→πτντ. The red and blue bands are obtained with the form factors computed from the B-meson LCSR and from the pion LCSR, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)Fig. 12Top: Contributions of B-meson three-particle DAs to the correlation function Πμ at tree level (the diagram (a)) and at one-loop order (typical diagrams displayed in (b), (c) and (d)). Bottom: Typical diagrams for renormalization of two-parton (the diagram (e)) and three-parton (the diagram (f)) DAs at O(gs3). The black blob in the diagram (f) indicates the external gluon field in the string operator defining three particle DAs of the B-meson.Table 1Fitted values of the form factor fBπ+(q2) at zero momentum transfer and of the slop parameters b1, b˜1 entering the z expansions (100) and (101). The notation “default” means that all the parameters are taken as the central values in the numerical evaluation. Note that the central value of fBπ+(0) is taken from [48] to determine ω0(1 GeV) from the matching condition, whose variations induce the combined uncertainty estimated in [48] by construction. Negligible uncertainties induced by variations of the remaining parameters are not shown but are taken into account in the combined uncertainty.ParameterDefaultω0σB(1)μμh1(2){M2,s0}{M‾2,s‾0}ϕB±(ω)

fBπ+(0)0.281+0.027−0.029+0.008−0.008−0.031+0.015−0.004+0.005−0.014+0.008−0.007+0.012–

b1−3.92+0.09−0.10+0.03−0.03−0.00+0.06+0.06−0.01−0.09+0.08–−0.95+0.14

b˜1−5.37+0.12−0.13+0.03−0.03−0.41+0.21−0.00+0.05−0.12+0.11–−1.15+0.17

QCD corrections to B→π form factors from light-cone sum rulesYu-MingWanga⁎yuming.wang@tum.deYue-LongShenbaPhysik Department T31, Technische Universität München, James-Franck-Straße 1, D-85748 Garching, GermanyPhysik Department T31Technische Universität MünchenJames-Franck-Straße 1GarchingD-85748GermanybCollege of Information Science and Engineering, Ocean University of China, Qingdao, Shandong 266100, PR ChinaCollege of Information Science and EngineeringOcean University of ChinaQingdaoShandong266100PR China⁎Corresponding author.Editor: Tommy OhlssonAbstractWe compute perturbative corrections to B→π form factors from QCD light-cone sum rules with B-meson distribution amplitudes. Applying the method of regions we demonstrate factorization of the vacuum-to-B-meson correlation function defined with an interpolating current for pion, at one-loop level, explicitly in the heavy quark limit. The short-distance functions in the factorization formulae of the correlation function involves both hard and hard-collinear scales; and these functions can be further factorized into hard coefficients by integrating out the hard fluctuations and jet functions encoding the hard-collinear information. Resummation of large logarithms in the short-distance functions is then achieved via the standard renormalization-group approach. We further show that structures of the factorization formulae for fBπ+(q2) and fBπ0(q2) at large hadronic recoil from QCD light-cone sum rules match that derived in QCD factorization. In particular, we perform an exploratory phenomenological analysis of B→π form factors, paying attention to various sources of perturbative and systematic uncertainties, and extract |Vub|=(3.05−0.38+0.54|th.±0.09|exp.)×10−3 with the inverse moment of the B-meson distribution amplitude ϕB+(ω) determined by reproducing fBπ+(q2=0) obtained from the light-cone sum rules with π distribution amplitudes. Furthermore, we present the invariant-mass distributions of the lepton pair for B→πℓνℓ (ℓ=μ,τ) in the whole kinematic region. Finally, we discuss non-valence Fock state contributions to the B→π form factors fBπ+(q2) and fBπ0(q2) in brief.1IntroductionMaking every endeavor to achieve precision determinations of heavy-to-light transition form factors is of utmost importance to, on the one hand, test the CKM sector of the Standard Model, and on the other side to sharpen our knowledge towards diverse facets of the theory of strong interaction (QCD). We are continually surprised by complexities and subtleties of factorization properties and heavy quark expansions of even the simplest B→π form factors in the context of both QCD factorization and QCD sum rules on the light-cone (LCSR), not to mention more sophisticated B→ρ,K⁎ form factors with an unstable particle in the final states. The purposes of this paper are to pursue an endeavor to understand factorization structures of B→π form factors from the LCSR with B-meson distribution amplitudes (DAs) at O(αs) in QCD [1,2] (see also [3,4] for an alternative formulation in the framework of soft-collinear effective theory (SCET)); and to provide a complementary approach to anatomize the topical |Vub| tension arising from the mismatch in exclusive and inclusive determinations.Constructions of the LCSR with B-meson DAs are accomplished by introducing the B-meson-to-vacuum correlation function, demonstrating factorization of the considered correlator in the proper kinematic regime, and applying the parton–hadron duality ansatz in the light-meson channel. It is evident that proof of QCD factorization for the correlation function defined with an on-shell B-meson state at next-to-leading order (NLO) constitutes a primary task in such program, in addition to further refinements of the duality relation. Inspecting the tree-level contribution to the correlation function shows that three different momentum modes with the scaling behaviors(1)Pμ≡(n⋅P,n¯⋅P,P⊥),Ph,μ∼O(1,1,1),Phc,μ∼O(1,λ,λ1/2),Ps,μ∼O(λ,λ,λ), appear in the problem under consideration, where nμ and n¯μ are light-cone vectors, satisfying n2=n¯2=0 and n⋅n¯=2, and are chosen such that the four-momentum of the fast-moving pion state has a large component n⋅p of order mb. Ph,μ, Phc,μ and Ps,μ corresponding to the four-momentum of the external b-quark, of the interpolating current of pion and of the light-spectator quark, will be called hard, hard-collinear and soft modes hereafter. The transfer momentum qμ of the weak current u¯Γb can correspond to either a hard mode or a hard-collinear mode dependent on the kinematic region; a unified description for the purpose of demonstrating factorization of the correlation function at NLO can be achieved by focusing on the kinematic variable n⋅p. The heavy-quark expansion parameter λ scales as Λ/mb where Λ is a hadronic scale of order ΛQCD. It is well known that computing multi-scale amplitudes at loop level can be facilitated by applying the method of regions [5] in dimensional regularization, which has been extensively used for evaluating multi-loop integrals in heavy quarkonium decays, top-quark pair production near threshold, Higgs production at hadron colliders and last but not least B-meson decays. More importantly, we also benefit from a separation of dynamics at distinct energy scales allowing for resummation of large logarithms in the resulting matching coefficients and non-perturbative distribution functions with the standard renormalization-group (RG) approach in the momentum space. It is then our favored strategy to establish a factorization formula of the considered correlation function at leading power in Λ/mb and at O(αs) using the method of regions.The fundamental non-perturbative inputs entering LCSR discussed in this paper are the B-meson DAs defined by hadronic matrix elements of HQET string operators, which also serve as essential ingredients for the theoretical description of many other exclusive B-meson decays, e.g., the radiative leptonic B→γℓνℓ decays. It will be shown that the constructed B-meson LCSR for B→π form factors are not only sensitive to the inverse moment of ϕB+(ω,μ), i.e., λB(μ), but also dependent heavily on small ω behaviors of the B-meson DAs (see also [4]). We are therefore provided with a golden opportunity to probe more actuate images of the B meson in terms of the elementary constituents (quarks and gluons), anticipating precision measurements of differential q2 distributions of B→πℓνl at high luminosity experiments and alternative (refined) determinations of |Vub| exclusively (for instance, the leptonic B→τντ decay). We should also mention that understanding renormalization properties of the B-meson DAs and perturbative QCD constraints of ϕB±(ω,μ) at high ω are also of conceptual interests for many reasons.As diverse techniques for computing B→π form factors have been developed so far and theory predictions are continuously refined with yet higher precision, several comments on the state-of-art of QCD calculations might be meaningful. •The up-to-date calculations of B→π form factors from the LCSR with pion DAs are restricted to NLO corrections to twist-2 and twist-3 terms [6–8] where asymptotic expressions of twist-3 DAs were taken to demonstrate factorization of the relevant correlation functions without bothering about mixing of the two- and three-particle DAs under renormalization. In addition, next-to-next-to-leading-order (NNLO) perturbative corrections to the twist-2 part induced by the running QCD coupling were fulfilled recently in [9]. Such computations should however be taken cum grano salis, because the large-β0 approximation generally overestimates the complete perturbative corrections strongly. Further improvements of the pion LCSR, including complete NLO calculations of the twist-3 terms beyond the asymptotic limit and detailed analysis of the sub-leading power corrections from twist-5 and 6 parts, are highly desirable.•The industries of investigating heavy-to-light B-meson form factors in QCD factorization were initiated in [10] where O(αs) corrections were found to be dominated by the spectator-scattering terms suffering sizeable uncertainty from λB(μ). Perturbative corrections to hard matching coefficients were carried out at one loop [11,12] for A-type currents and [12,13] for B-type currents, and at two loops [14–18] for A-type currents. The jet functions from integrating out dynamics of the hard-collinear fluctuation were accomplished at one-loop level [13,19,20]. One should however keep in mind that hadronic matrix elements of A-type currents (up to perturbatively calculable contributions dependent on the factorization schemes) cannot be further factorized in SCET(c,s) [21] and must be taken as fundamental inputs from other approaches.•Yet another approach to compute B→π form factors is based upon transverse-momentum-dependent (TMD) QCD factorization for hard processes developed from the theory of on-shell Sudakov form factor [22] and the asymptotic behavior of elastic hadron–hadron scattering at high energy [23] with the underlying physical principle that the elastic scattering of an isolated parton suffers a strong suppression at high energy from radiative QCD corrections. Recently, computations of B→π form factors with TMD factorization approach have been pushed to O(αs) for twist-2 [24,25] and twist-3 [26] contributions of pion DAs. However, one needs to be aware of the fact that TMD factorization of hard exclusive processes becomes extraordinarily delicate due to complex infrared subtractions beyond the leading order in αs [27] and a complete understanding of TMD factorization for exclusive processes with large momentum transfer has not been achieved to date on the conceptual side.The remainder of this paper is structured as follows. In Section 2 we briefly review the method of the LCSR with B-meson DAs by illustrating the tree-level calculation of B→π form factors. To facilitate proof of QCD factorization for the considered correlation function at NLO we recapitalize basic of the diagrammatical factorization approach at tree level as an instructive example. We then generalize factorization proof of the correction function to the one-loop order in Section 3 by showing a complete cancellation of soft contributions to the one-loop QCD diagrams and infrared subtractions determined by convolutions of the one-loop partonic DAs of the B-meson and the tree-level hard-scattering kernel, at leading power in Λ/mb. Hard functions and jet functions entering factorization formulae of the correlation function are simultaneously obtained by computing the relevant one-loop integrals with the method of regions. Next-to-leading-logarithmic (NLL) resummation of the hard coefficient functions is performed by virtue of the RG approach and a detailed comparison of the obtained perturbative matching coefficients with the equivalent expressions computed in SCET is also presented in Section 3. We further derive NLL resummation improved LCSR for B→π form factors in Section 4, which constitute the main results of this paper. Phenomenological applications of the new sum rules are explored in Section 5, including determinations of the q2 shapes of B→π form factors, extractions of the CKM matrix element |Vub| and predictions of the normalized q2 distributions in B→πℓνℓ. In Section 6 we turn to discuss the impact of three-particle B-meson DAs on B→π form factors, which is still the missing ingredient of our calculations. The concluding discussions are presented in Section 7. Appendix A contains some useful expressions of one-loop integrals after expanding integrands with the method of regions. Spectral representations of the convolution integrals for constructing the LCSR with B-meson DAs and two-point QCD sum rules for the decay constants of the B-meson and the pion are collected in Appendices B and C.2Recapitulation of the LCSR methodWe construct LCSR of the form factors fBπ+(q2) and fBπ0(q2) with the correlation function(2)Πμ(n⋅p,n¯⋅p)=∫d4x eip⋅x〈0|T{d¯(x)n̸γ5u(x),u¯(0)γμb(0)}|B¯(p+q)〉=Π(n⋅p,n¯⋅p)nμ+Π˜(n⋅p,n¯⋅p)n¯μ, defined with a pion interpolating current carrying a four-momentum pμ and a weak b→u transition current. We work in the rest frame of the B-meson with the velocity vector satisfying n⋅v=n¯⋅v=1 and v⊥=0. For definiteness, we adopt the following conventions(3)n⋅p≃mB2+mπ2−q2mB=2Eπ,n¯⋅p∼O(ΛQCD). The correlation function Πμ(n⋅p,n¯⋅p) can be computed from light-cone operator–product–expansion (OPE) at n¯⋅p<0. Evaluating the diagram in Fig. 1 yields(4)Π˜(n⋅p,n¯⋅p)=f˜B(μ)mB∫0∞dω′ϕB−(ω′)ω′−n¯⋅p−i0+O(αs),Π(n⋅p,n¯⋅p)=O(αs). The B-meson distribution amplitude (DA) ϕB−(ω′) is defined as [28](5)〈0|d¯β(τn¯)[τn¯,0]bα(0)|B¯(p+q)〉=−if˜B(μ)mB4{1+v̸2[2ϕ˜B+(τ)+(ϕ˜B−(τ)−ϕ˜B+(τ))n̸]γ5}αβ, where the light-cone Wilson line is given by(6)[τn¯,0]=P{Exp[igs∫0τdλn¯⋅A(λn¯)]}, with the convention of the covariant derivative in QCD as Dμ=∂μ−igsTaAμa, and the Fourier transformations of ϕ˜B±(τ) lead to(7)ϕB±(ω′)=∫−∞+∞dτ2πeiω′τϕ˜B±(τ−i0). One then can construct the light-cone projector in momentum space [4](8)Mβα=−if˜B(μ)mB4×{1+v̸2[ϕB+(ω′)n̸+ϕB−(ω′)n̸¯−2ω′D−2ϕB−(ω′)γ⊥ρ∂∂k⊥ρ′]γ5}αβ in D dimensions. Here, f˜B(μ) is the B-meson decay constant in the static limit and it can be expressed in terms of the QCD decay constant(9)fB=f˜B(μ)[1+αsCF4π(−3lnμmb−2)]. Note that a single B-meson DA ϕB−(ω′) appears in the tree-level LCSR (4) in contrast to factorization of B→γℓν where only ϕB+(ω′) enters the factorization formulae of the form factors FV,A(Eγ) at leading power in Λ/mb [29]. The discrepancy can be traced back to the longitudinally polarized interpolating current for the pion in the former and to the transversely polarized photon in the latter.Factorization of Πμ(n⋅p,n¯⋅p) at tree level is straightforward due to the absence of infrared (soft) divergences. The hard-collinear fluctuation of the internal u-quark guarantees light-cone expansion of the non-local matrix element defining the B-meson DAs. For the sake of a clear demonstration of factorization of Πμ(n⋅p,n¯⋅p) at one-loop order, we write down the tree-level approximation of the partonic correlation function11Perturbative matching coefficients entering the factorization formulae of Πμ(n⋅p,n¯⋅p) are independent of the external partonic state, and it is a matter of convenience to choose a certain configuration for the practical calculation. More detailed discussions of this point in the context of factorization of B→γℓν can be found in Ref. [30]. (defined as replacing |B¯(p+q)〉 by |b(pB−k)d¯(k)〉 in Eq. (2) with pB≡p+q)(10)Πμ,bd¯(0)(n⋅p,n¯⋅p)=∫dω′Tαβ(0)(n⋅p,n¯⋅p,ω′)Φbd¯(0)αβ(ω′), where the superscript (0) indicates the tree level, the Lorenz index “μ” is suppressed on the right-hand side, the leading-order hard-scattering kernel is given by(11)Tαβ(0)(n⋅p,n¯⋅p,ω′)=i21n¯⋅p−ω′+i0[n̸γ5n̸¯γμ]αβ, and the partonic DA of the B-meson reads(12)Φbd¯αβ(ω′)=∫dτ2πeiω′τ〈0|d¯β(τn¯)[τn¯,0]bα(0)|b(pB−k)d¯(k)〉 with the tree-level contribution(13)Φbd¯(0)αβ(ω′)=δ(n¯⋅k−ω′)d¯β(k)bα(pB−k). It is worthwhile to point out that the variable ω′ is not necessarily to be the same as ω≡n¯⋅k despite the equivalence at tree level. The partonic light-cone projector can be obtained from Eq. (8) via the replacement ϕB±(ω′)→ϕbd¯±(ω′), and we can write down(14)Πμ,bd¯(0)(n⋅p,n¯⋅p)=Πbd¯(0)(n⋅p,n¯⋅p)nμ+Π˜bd¯(0)(n⋅p,n¯⋅p)n¯μ,Π˜bd¯(0)(n⋅p,n¯⋅p)=f˜B(μ)mBϕbd¯−(ω)ω−n¯⋅p−i0,Πbd¯(0)(n⋅p,n¯⋅p)=0.With definitions of the B→π form factors and the pion decay constant(15)〈π(p)|u¯γμb|B¯(pB)〉=fBπ+(q2)[pB+p−mB2−mπ2q2q]μ+fBπ0(q2)mB2−mπ2q2qμ,〈π(p)|d¯n̸γ5u|0〉=−in⋅pfπ, we obtain the hadronic dispersion relation for the correlation function(16)Πμ(n⋅p,n¯⋅p)=fπn⋅pmB2(mπ2−p2){n¯μ[n⋅pmBfBπ+(q2)+fBπ0(q2)]+nμmBn⋅p−mB[n⋅pmBfBπ+(q2)−fBπ0(q2)]}+∫ωs+∞dω′1ω′−n¯⋅p−i0[ρh(ω′,n⋅p)nμ+ρ˜h(ω′,n⋅p)n¯μ], where ωs is the hadronic threshold in the pion channel. Applying the quark–hadron duality ansatz, the integrals over the hadronic spectral densities can be approximated by the integrals over the QCD spectral functions with the threshold parameter reinterpreted as an effective “internal” parameter of the sum rule approach. Then, one can derive the final expressions of the LCSR after implementing the Borel transformation in the variable n¯⋅p→ωM(17)fBπ+(q2)=f˜B(μ)mBfπn⋅pexp[mπ2n⋅pωM]∫0ωsdω′e−ω′/ωMϕB−(ω′)+O(αs),fBπ0(q2)=n⋅pmBfBπ+(q2)+O(αs), which are in agreement with Refs. [2,3].Albeit with the rather simple structures of the tree-level LCSR, some interesting observations can be already made.•Since the B-meson DA ϕB+(ω′) does not enter the factorization formulae of Πμ(n⋅p,n¯⋅p) at tree level and ϕB±(ω′) do not mix under renormalization at one loop in the massless light-quark limit, the convolution integrals of ϕB+(ω′) entering the contributions of the one-loop diagrams of Πμ(n⋅p,n¯⋅p) in QCD must be infrared finite at O(αs) to guarantee the validity of QCD factorization of Πμ(n⋅p,n¯⋅p).•Since only a single invariant function Π˜(n⋅p,n¯⋅p) survives at tree level, one concludes that the one-loop contributions to Π(n⋅p,n¯⋅p) in QCD must be infrared finite due to the vanishing infrared (soft) subtraction at O(αs), provided that factorization of Πμ(n⋅p,n¯⋅p) holds.•The Borel mass ωM and the threshold parameter ωs enter into the LCSR from the dispersive analysis with respect to the variable n¯⋅p, indicating that one needs to identify ωM=M2/n⋅p and ωs=s0/n⋅p with (M2,s0) from the dispersive construction of the LCSR in the variable p2. From the scaling M2∼s0∼Λ2, one then finds the power counting of fBπ+ and fBπ0 as ∼(Λ/mb)3/2 at tree level, consistent with the observations of [10,21].3Factorization of the correlation function at O(αs)The objective of this section is to establish the factorization formulae for Πμ(n⋅p,n¯⋅p) in QCD at one-loop level. We adopt the diagrammatic factorization method expanding the correlator Πμ,bd¯, the short-distance function T and the partonic DA of the B meson Φbd¯ in perturbation theory. Schematically,(18)Πμ,bd¯=Πμ,bd¯(0)+Πμ,bd¯(1)+…=Φbd¯⊗T=Φbd¯(0)⊗T(0)+[Φbd¯(0)⊗T(1)+Φbd¯(1)⊗T(0)]+…, where ⊗ denotes the convolution in the variable ω′ defined in Eq. (12), and the superscripts indicate the order of αs. The hard-scattering kernel at O(αs) is then determined by the matching condition(19)Φbd¯(0)⊗T(1)=Πμ,bd¯(1)−Φbd¯(1)⊗T(0), where the second term serves as the infrared (soft) subtraction. One crucial point in the proof of factorization of Πμ,bd¯ is to demonstrate that the hard-scattering kernel T can be contributed only from hard and/or hard-collinear regions at leading power in Λ/mb, due to a complete cancellation of the soft contribution to Πμ,bd¯(1) and Φbd¯(1)⊗T(0). In addition, since B-meson DAs can only collect the soft QCD dynamics of Πμ,bd¯, we must show that there is no leading contribution to the correlation function from the collinear region (with the momentum scaling lμ∼(1,λ2,λ)) at leading power.Following Ref. [30], we will evaluate the master formula of T(1) in Eq. (19) diagram by diagram. However, we will apply the method of regions [5] to compute the loop integrals in order to obtain the hard coefficient function (C) and the jet function (J) simultaneously. To establish the factorization formula(20)Πμ,bd¯=Φbd¯⊗T=C⋅J⊗Φbd¯, C and J must be well defined in dimensional regularization. This guarantees that we can adopt dimensional regularization to evaluate the leading-power contributions of Πμ,bd¯ without introducing an additional “analytical” regulator. The strategies of our calculations are as follows: (i) Identify leading regions of the scalar integral for each diagram; (ii) Simplify the Dirac algebra in the numerator for a given leading region and evaluate the relevant integrals using the method of regions; (iii) Evaluate the hard and hard-collinear contributions with the light-cone projector of the B meson in momentum space; (iv) Show the equivalence of the soft subtraction term and the correlation function in the soft region; (v) Add up the contributions from the hard and hard-collinear regions separately.3.1Weak vertex diagramThe contribution to Πμ(1) from the QCD correction to the weak vertex (the diagram in Fig. 2(a)) is(21)Πμ,weak(1)=gs2CF2(n¯⋅p−ω)∫dDl(2π)D1[(p−k+l)2+i0][(mbv+l)2−mb2+i0][l2+i0]×d¯(k)n̸γ5n̸¯γρ(p̸−k̸+l̸)γμ(mbv̸+l̸+mb)γρb(v), where the label “bd¯” of the partonic correlation function Πμ,bd¯ will be suppressed from now on and D=4−2ϵ. Since the perturbative matching coefficients are insensitive to infrared physics, we thus assign the external momenta mbv to the bottom quark and k (with k2=0) to the light quark. In accordance with the scaling behaviors(22)n⋅p∼mb,n¯⋅p∼Λ,kμ∼Λ, we identify the leading-power contributions of the scalar integral(23)I1=∫[dl]1[(p−k+l)2+i0][(mbv+l)2−mb2+i0][l2+i0] from the hard, hard-collinear and soft regions and the power counting I1∼λ0 implies that only the leading-power contributions of the numerator in Eq. (21) need to be kept for a given region taking into account the power counting of the tree-level contribution in Eq. (11). We define the integration measure as(24)[dl]≡(4π)2i(μ2eγE4π)ϵdDl(2π)D.Inserting the partonic light-cone projector yields the hard contribution of Πμ,weak(1) at leading power(25)Πμ,weak(1),h=igs2CFf˜B(μ)mBϕbd¯−(ω)n¯⋅p−ω∫dDl(2π)D1[l2+n⋅pn¯⋅l+i0][l2+2mbv⋅l+i0][l2+i0]×{n¯μ[2mbn⋅(p+l)+(D−2)l⊥2]−nμ(D−2)(n¯⋅l)2}, where the superscript “h” denotes the hard contribution and we adopt the conventions(26)l⊥2≡g⊥μνlμlν,g⊥μν≡gμν−nμn¯ν2−nνn¯μ2. Using the results of loop integrals provided in Appendix A, we obtain(27)Πμ,weak(1),h=αsCF4πf˜B(μ)mBϕbd¯−(ω)n¯⋅p−ω{n¯μ[1ϵ2+1ϵ(2lnμn⋅p+1)+2ln2μn⋅p+2lnμmb−ln2r−2Li2(−r¯r)+2−rr−1lnr+π212+3]+nμ[1r−1(1+rr¯lnr)]}, with r=n⋅p/mb and r¯=1−r.Along the same vein, one can identify the hard-collinear contribution of Πμ,weak(1) at leading power(28)Πμ,weak(1),hc=igs2CFf˜B(μ)mBϕbd¯−(ω)n¯⋅p−ω∫dDl(2π)D2mbn⋅(p+l)[n⋅(p+l)n¯⋅(p−k+l)+l⊥2+i0][mbn⋅l+i0][l2+i0], where the superscript “hc” indicates the hard-collinear contribution and the propagators have been expanded systematically in the hard-collinear region. Evaluating the integrals with the relations collected in Appendix A yields(29)Πμ,weak(1),hc=αsCF4πf˜B(μ)mBϕbd¯−(ω)ω−n¯⋅pn¯μ[2ϵ2+2ϵ(lnμ2n⋅p(ω−n¯⋅p)+1)+ln2μ2n⋅p(ω−n¯⋅p)+2lnμ2n⋅p(ω−n¯⋅p)−π26+4].Applying the method of regions we extract the soft contribution of Πμ,weak(1)(30)Πμ,weak(1),s=gs2CF2(n¯⋅p−ω)∫dDl(2π)D1[n¯⋅(p−k+l)+i0][v⋅l+i0][l2+i0]d¯(k)n̸γ5n̸¯γμb(pb)=αsCF4πf˜B(μ)mBϕbd¯−(ω)n¯⋅p−ωn¯μ×[1ϵ2+2ϵlnμω−n¯⋅p+2ln2μω−n¯⋅p+3π24], where the superscript “s” represents the soft contribution.Now, we compute the corresponding infrared subtraction term Φbd¯,a(1)⊗T(0) as displayed in Fig. 3(a). With the Wilson-line Feynman rules, we obtain(31)Φbd¯,aαβ,(1)(ω,ω′)=igs2CF∫dDl(2π)D1[n¯⋅l+i0][v⋅l+i0][l2+i0]×[δ(ω′−ω−n¯⋅l)−δ(ω′−ω)][d¯(k)]α[b(v)]β, from which we can derive the soft subtraction term(32)Φbd¯,a(1)⊗T(0)=gs2CF2(n¯⋅p−ω)∫dDl(2π)D1[n¯⋅(p−k+l)+i0][v⋅l+i0][l2+i0]d¯(k)n̸γ5n̸¯γμb(v), where the tree-level hard kernel in Eq. (11) is used. We then conclude that(33)Πμ,weak(1),s=Φbd¯,a(1)⊗T(0) at leading power in Λ/mb, which is an essential point to prove factorization of the correlation function Πμ.3.2Pion vertex diagramNow we turn to compute the QCD correction to the pion vertex (the diagram in Fig. 2b)(34)Πμ,pion(1)=−gs2CFn⋅p(n¯⋅p−ω)∫dDl(2π)D1[(p−l)2+i0][(l−k)2+i0][l2+i0]d¯(k)γρl̸n̸γ5(p̸−l̸)γρ(p̸−k̸)γμb(v). One can identify the leading-power contributions of the scalar integral(35)I2=∫[dl]1[(p−l)2+i0][(l−k)2+i0][l2+i0] from the hard-collinear and soft regions, which have the scaling behaviors(36)I2hc∼I2s∼λ−1, by virtue of the power counting analysis. It is evident that the pion vertex correction would give rise to the power enhanced effect in relative to the tree-level contribution of Eq. (11), provided that no additional suppression factors come from the spinor structure. Closer inspection shows that expanding the integrand of I2 in the soft region will generate a scaleless integral which vanishes in dimensional regularization. For the hard-collinear loop momentum, the spinor structure is reduced to(37)d¯(k)[...]b(v)=d¯(k)γ5[2(p̸−l̸)n̸l̸+(D−4)l̸n̸(p̸−l̸)](p̸−k̸)b(v) which indeed induces a power-suppression factor λ. It turns out to be less transparent to extract the leading-power contribution in the hard-collinear region with the insertion of the B-meson light-cone projector. Instead, we first compute the loop integral of Eq. (34) exactly without resorting to the method of regions; then we express Πμ,pion(1) in terms of the partonic DAs by inserting the momentum-space projector.Employing the expressions of loop integrals in Appendix A we find(38)Πμ,pion(1)=Πμ,pion(1),hc=αsCF4πf˜B(μ)mB1n¯⋅p−ω{nμϕbd¯+(ω)[n¯⋅p−ωωlnn¯⋅p−ωn¯⋅p]+n¯μϕbd¯−(ω)[(1ϵ+ln(−μ2p2))(2n¯⋅pωlnn¯⋅p−ωn¯⋅p+1)−n¯⋅pωlnn¯⋅p−ωn¯⋅p(lnn¯⋅p−ωn¯⋅p+2ωn¯⋅p−4)+4]}. While the soft contribution of Πμ,pion(1) vanishes in dimensional regularization, it remains to demonstrate that the precise cancellation of Πμ,pion(1),s and Φbd¯,b⊗T(0) is independent of regularization schemes. Applying the method of regions yields(39)Πμ,pion(1),s=−gs2CF2(n¯⋅p−ω)∫dDl(2π)D1[n¯⋅(p−l)+i0][(l−k)2+i0][l2+i0]d¯(k)n̸¯l̸n̸γ5n̸¯γμb(v). The corresponding contribution to the partonic DA (the diagram in Fig. 3b) is given by(40)Φbd¯,bαβ,(1)(ω,ω′)=igs2CF∫dDl(2π)D1[n¯⋅l+i0][(k+l)+i0][l2+i0]×[δ(ω′−ω−n¯⋅l)−δ(ω′−ω)][d¯(k)n̸¯(k̸+l̸)]α[b(v)]β. One then deduces the soft subtraction term(41)Φbd¯,b(1)⊗T(0)=−gs2CF2(n¯⋅p−ω)∫dDl(2π)D1[n¯⋅(p−k−l)+i0][(k+l)2+i0][l2+i0]d¯(k)n̸¯(k̸+l̸)n̸γ5n̸¯γμb(v), which coincides with Πμ,pion(1),s exactly after the shift of the loop momentum l→l−k.3.3Wave function renormalizationThe self-energy correction to the intermediate quark propagator (the diagram in Fig. 2c) can be written as(42)Πμ,wfc(1)=gs2CF(n⋅p)2(n¯⋅p−ω)2∫dDl(2π)D1[(p−k+l)2+i0][l2+i0]d¯(k)n̸γ5(p̸−k̸)γρ(p̸−k̸+l̸)γρ(p̸−k̸)γμb(v). Apparently, Πμ,wfc(1) is free of soft and collinear divergences and a straightforward calculation gives(43)Πμ,wfc(1)=αsCF4πf˜B(μ)mBϕbd¯−(ω)n¯⋅p−ωn¯μ[1ϵ+lnμ2n⋅p(ω−n¯⋅p)+1].Now we evaluate the perturbative matching coefficient from the wave function renormalization of the external quark fields. It is evident that the wave function renormalization of a massless quark does not contribute to the matching coefficient when dimensional regularization is applied to regularize both ultraviolet and infrared divergences, i.e.,(44)Πμ,dwf(1)−Φbd¯,dwf(1)⊗T(0)=0. The wave function renormalization of the b-quark in QCD gives(45)Πμ,bwf(1)=−αsCF8π[3ϵ+3lnμ2mb2+4]Πμ(0), with Πμ(0) displayed in Eq. (14). The wave function renormalization of the b-quark in HQET is(46)Φbd¯,bwf(1)⊗T(0)=0, due to the scaleless integral, we then find(47)Πμ,bwf(1)−Φbd¯,bwf(1)⊗T(0)=−αsCF8π[3ϵ+3lnμ2mb2+4]Πμ(0).3.4Box diagramThe one-loop contribution to Πμ from the box diagram is given by(48)Πμ,box(1)=gs2CF×∫dDl(2π)D−1[(mbv+l)2−mb2+i0][(p−k+l)2+i0][(k−l)2+i0][l2+i0]d¯(k)γρ(k̸−l̸)n̸γ5(p̸−k̸+l̸)γμ(mbv̸+l̸+mb)γρb(v). This is the only diagram at one-loop level without a hard-collinear propagator outside of the loop, hence we must identify the enhancement factor mb/Λ from the corresponding scalar integral so that it can give rise to the leading-power contribution compared to the tree-level amplitude in Eq. (14). With the scaling behaviors of the external momenta, one can establish the scaling of(49)I4=∫[dl]1[(mbv+l)2−mb2+i0][(p−k+l)2+i0][(k−l)2+i0][l2+i0] as λ−1 (λ−2) in the hard-collinear and semi-hard (soft) regions.22No power enhanced factor can be induced for I4 in other regions by the power counting analysis, which are therefore irrelevant here. It is straightforward to verify that the semi-hard contribution will be reduced to a scaleless integral, since there is no external semi-hard mode in the box diagram. We are only left with the hard-collinear and soft regions at leading power in Λ/mb. The term (k̸−l̸) in the spinor structure will give a suppression factor λ in the soft region so that both the hard-collinear and the soft contributions are of the same power. One might be curious about the observation that the box diagram contributes to the jet function entering the factorization formulae of the B-meson-to-vacuum correlation function Πμ at one-loop level while the hard-collinear contribution of the box diagram vanishes in the radiative leptonic decay B→γℓν [30,31]. The crucial discrepancy attributes to the longitudinally polarized pion interpolating current in the former and the transversely polarized photon in the latter. As a consequence, one is not able to pick up the large components of two intermediate up-quark propagators(50)(k̸−l̸)ϵ̸γ⁎(p̸−k̸+l̸) simultaneously in the case of B→γℓν, while this is possible in the contribution of the box diagram for Πμ as indicated in Eq. (48).Evaluating the hard-collinear contribution of Πμ,box(1) with the partonic momentum-space projector yields(51)Πμ,box(1),hc=igs2CFf˜B(μ)mBmbn¯μ∫dDl(2π)D[(2−D)n⋅lϕbd¯+(ω)+2mbϕbd¯−(ω)]×n⋅(p+l)[n⋅(p+l)n¯⋅(p−k+l)+l⊥2+i0][n⋅ln¯(l−k)+l⊥2+i0][l2+i0]. Using the expressions of loop integrals collected in Appendix A we obtain(52)Πμ,box(1),hc=αsCF4πf˜B(μ)mBωn¯μ{ϕbd¯+(ω)[rln(1+η)]−2ϕbd¯−(ω)ln(1+η)×[1ϵ+lnμ2n⋅p(ω−n¯⋅p)+12ln(1+η)+1]}, with η=−ω/n¯⋅p.Extracting the soft contribution of Πμ,box(1) with the method of regions gives(53)Πμ,box(1),s=−gs2CF2∫dDl(2π)D1[v⋅l+i0][n¯⋅(p−k+l)+i0][(k−l)2+i0][l2+i0]d¯(k)v̸(k̸−l̸)n̸γ5n̸¯γμb(v). Now we compute the corresponding NLO contribution to the partonic DA (the diagram in Fig. 3c)(54)Φbd¯,cαβ,(1)(ω,ω′)=−igs2CF∫dDl(2π)D1[(l−k)2+i0][v⋅l+i0][l2+i0]×δ(ω′−ω+n¯⋅l)[d¯(k)v̸(l̸−k̸)]α[b(v)]β, from which one can deduce the soft subtraction term(55)Φbd¯,c(1)⊗T(0)=gs2CF2∫dDl(2π)D1[v⋅l+i0][n¯⋅(p−k+l)+i0][(l−k)2+i0][l2+i0]d¯(k)v̸(l̸−k̸)n̸γ5n̸¯γμb(v), which cancels out the soft contribution of the correlation function Πμ,box(1),s completely. The absence of such soft contribution to the perturbative matching coefficient is particularly important for the box diagram, since the relevant loop integrals in the soft region depend on two components of the soft spectator momentum n¯⋅k and v⋅k, and the light-cone OPE fails in the soft region.33The bottom and down quarks entering the B-meson state is not light-cone separated for the soft exchanged gluon in Fig. 2d, therefore one is not allowed to use B-meson DAs to absorb the long-distance physics (i.e., non-perturbative QCD dynamics). The construction of QCD factorization itself requires decoupling of soft contributions from perturbative fluctuations in general.3.5The hard-scattering kernel at O(αs)The one-loop hard-scattering kernel of the correlation function Πμ can be readily computed from the matching condition in Eq. (19) by collecting different pieces together(56)Φbd¯(0)⊗T(1)=[Πμ,weak(1)+Πμ,pion(1)+Πμ,wfc(1)+Πμ,box(1)+Πμ,bwf(1)+Πμ,dwf(1)]−[Φbd¯,a(1)+Φbd¯,b(1)+Φbd¯,c(1)+Φbd¯,bwf(1)+Φbd¯,dwf(1)]⊗T(0)=[Πμ,weak(1),h+(Πμ,bwf(1)−Φbd¯,bwf(1))]+[Πμ,weak(1),hc+Πμ,pion(1),hc+Πμ,wfc(1),hc+Πμ,box(1),hc], where the terms in the first and second square brackets of the second equality correspond to the hard matching coefficients and the jet functions at O(αs). Finally, one can derive the factorization formulae of Π and Π˜ defined in Eq. (2)(57)Π=f˜B(μ)mB∑k=±C(k)(n⋅p,μ)∫0∞dωω−n¯⋅p J(k)(μ2n⋅pω,ωn¯⋅p)ϕB(k)(ω,μ),Π˜=f˜B(μ)mB∑k=±C˜(k)(n⋅p,μ)∫0∞dωω−n¯⋅p J˜(k)(μ2n⋅pω,ωn¯⋅p)ϕB(k)(ω,μ), at leading power in Λ/mb, where we keep the factorization-scale dependence explicitly, the hard coefficient functions are given by(58)C(+)=C˜(+)=1,C(−)=αsCF4π1r¯[rr¯lnr+1],C˜(−)=1−αsCF4π[2ln2μn⋅p+5lnμmb−ln2r−2Li2(−r¯r)+2−rr−1lnr+π212+5], and the jet functions are(59)J(+)=1rJ˜(+)=αsCF4π(1−n¯⋅pω)ln(1−ωn¯⋅p),J(−)=1,J˜(−)=1+αsCF4π[ln2μ2n⋅p(ω−n¯⋅p)−2lnn¯⋅p−ωn¯⋅plnμ2n⋅p(ω−n¯⋅p)−ln2n¯⋅p−ωn¯⋅p−(1+2n¯⋅pω)lnn¯⋅p−ωn¯⋅p−π26−1].Now, we verify the factorization-scale independence of Π and Π˜ as a consequence of QCD factorization by construction. Note that the correlation function Πμ is defined by the conserved currents in QCD, hence the ultraviolet renormalization-scale dependence of Πμ is determined by the renormalization constant of the strong coupling constant αs and no additional QCD operator renormalization (ultraviolet subtraction) is needed in obtaining the renormalized hard coefficients and jet functions. It is straightforward to write down the following evolution equations(60)ddlnμC˜(−)(n⋅p,μ)=−αsCF4π[Γcusp(0)lnμn⋅p+5]C˜(−)(n⋅p,μ),(61)ddlnμJ˜(−)(μ2n⋅pω,ωn¯⋅p)=αsCF4π[Γcusp(0)lnμ2n⋅pω]J˜(−)(μ2n⋅pω,ωn¯⋅p)+αsCF4π∫0∞dω′ωΓ(ω,ω′,μ)J˜(−)(μ2n⋅pω′,ω′n¯⋅p),(62)ddlnμ[f˜B(μ)ϕB−(ω,μ)]=−αsCF4π[Γcusp(0)lnμω−5][f˜B(μ)ϕB−(ω,μ)]−αsCF4π∫0∞dω′ωΓ(ω,ω′,μ)[f˜B(μ)ϕB−(ω′,μ)], where the function Γ is given by [32](63)Γ(ω,ω′,μ)=−Γcusp(0)θ(ω′−ω)ωω′−Γcusp(0)[θ(ω′−ω)ω′(ω′−ω)+θ(ω−ω′)ω(ω−ω′)]⊕ at one-loop order, with the ⊕ function defined as(64)∫0∞dω′[f(ω,ω′)]⊕g(ω′)=∫0∞dω′f(ω,ω′)[g(ω′)−g(ω)], and Γcusp(0)=4 determined by the geometry of Wilson lines. The renormalization kernel of ϕB−(ω,μ) at one-loop level was first computed in [32] and then confirmed in [33]. We also mention in passing that the RG equations of both the B-meson DAs and the jet functions take a particularly simple form in the “dual” momentum space where the Lange–Neubert kernel [34] at one loop is diagonalized. More details can be found in Ref. [35] (see also [36]) and we will not pursue the discussions along this line further. With the evolution equations displayed above, it is evident that(65)ddlnμ[Π(n⋅p,n¯⋅p),Π˜(n⋅p,n¯⋅p)]=O(αs2).Inspection of Eqs. (58), (59) and (9) indicates that one cannot avoid the parametrically large logarithms of order ln(mb/ΛQCD) in the hard functions, the jet functions, f˜B(μ) and the B-meson DAs concurrently, by choosing a common value of μ. Resummation of these logarithms to all orders of αs can be achieved by solving the three RG equations shown above. Since the hadronic scale entering the initial conditions of the B-meson DAs ϕB±(ω,μ0), μ0≃1 GeV, is quite close to the hard-collinear scale μhc≃mbΛQCD≈1.5 GeV, we will not sum logarithms of μhc/μ0 due to the minor evolution effect [29]. Because the hard scale μh1∼n⋅p in the hard function C˜(−)(n⋅p,μ) differs from the one μh2∼mb in f˜B(μ), the resulting evolution functions due to running of the renormalization scale from μh1 (μh2) to μhc in C˜(−) (n⋅p,μ) (f˜B(μ)) are(66)C˜(−)(n⋅p,μ)=U1(n⋅p,μh1,μ)C˜(−)(n⋅p,μh1),f˜B(μ)=U2(μh2,μ)f˜B(μh2). To achieve NLL resummation of large logarithms in the hard coefficient C˜(−) we need to generalize the RG equation (60) to(67)ddlnμC˜(−)(n⋅p,μ)=[−Γcusp(αs)lnμn⋅p+γ(αs)]C˜(−)(n⋅p,μ), where the cusp anomalous dimension, γ(αs) and the QCD β-function are expanded as(68)Γcusp(αs)=αsCF4π[Γcusp(0)+(αs4π)Γcusp(1)+(αs4π)2Γcusp(2)+…],γ(αs)=αsCF4π[γ(0)+(αs4π)γ(1)+…],β(αs)=−8π[(αs4π)2β0+(αs4π)3β1+(αs4π)4β2+…]. The cusp anomalous dimension at the three-loop order and the remanning anomalous dimension γ(αs) determining renormalization of the SCET heavy-to-light current at two loops will enter U1(n⋅p,μh1,μ) at NLL accuracy. The manifest expressions of Γcusp(i), γ(i) and βi can be found in [29] and references therein,44Note that there is a factor CF difference of our conventions of Γcusp(i) and γ(i) compared with [29]. the evolution function U1(n⋅p,μh1,μ) can be read from Eq. (A.3) in [29] with the replacement rules Eγ→n⋅p/2 and μh→μh1. The three-loop evolution of the strong coupling αs in the MS‾ scheme(69)αs(μ)=2πβ0{1−β12β02ln(2L)L+β124β04L2[(ln(2L)−12)2+β2β0β12−54]},L=ln(μΛQCD(nf)) is used with ΛQCD(4)=229 MeV.The RG equation of f˜B(μ) at the two-loop order is given by(70)ddlnμf˜B(μ)=γ˜(αs)f˜B(μ), with(71)γ˜(αs)=αsCF4π[γ˜(0)+(αs4π)γ˜(1)+…],γ˜(0)=3,γ˜(1)=1276+14π29−53nf, where nf=4 is the number of light quark flavors. Solving this RG equation yields(72)U2(μh2,μ)=Exp[∫αs(μh2)αs(μ)dαsγ˜(αs)β(αs)]=z−γ˜02β0CF[1+αs(μh2)CF4π(γ˜(1)2β0−γ˜(0)β12β02)(1−z)+O(αs2)], with z=αs(μ)/αs(μh2).The final factorization formulae of Π and Π˜ with RG improvement at NLL accuracy can be written as(73)Π=mB[U2(μh2,μ)f˜B(μh2)]∫0∞dωω−n¯⋅p J(+)(μ2n⋅pω,ωn¯⋅p)ϕB(+)(ω,μ)+mB[U2(μh2,μ)f˜B(μh2)]C(−)(n⋅p,μ)∫0∞dωω−n¯⋅p ϕB(−)(ω,μ),Π˜=mB[U2(μh2,μ)f˜B(μh2)]∫0∞dωω−n¯⋅p J˜(+)(μ2n⋅pω,ωn¯⋅p)ϕB(+)(ω,μ)+mB[U1(n⋅p,μh1,μ)U2(μh2,μ)][f˜B(μh2)C˜(−)(n⋅p,μh1)]×∫0∞dωω−n¯⋅p J˜(−)(μ2n⋅pω,ωn¯⋅p)ϕB(−)(ω,μ), where μ should be taken as a hard-collinear scale of order mbΛ.3.6Comparison with previous approachesThe aim of this subsection is to develop a better understanding of the factorization structures of Π and Π˜ obtained above. Inspecting Eq. (56) shows that the hard-scale fluctuation of the correlation function Πμ(n⋅p,n¯⋅p) comes solely from the contributions of the weak vertex diagram and the b-quark wave function renormalization. This demonstrates that the hard matching coefficients C(−) and C˜(−) can be also extracted from the one-loop hard matching coefficients of the QCD current q¯γμb in SCET [11](74)q¯γμb→[C4n¯μ+C5vμ]ξ¯n¯WhcYs†bv+…, where Whc and Ys† denote the hard-collinear and soft Wilson lines, the ellipses represent terms with different Dirac structures and sub-leading power contributions. Inserting (57) into (2) and comparing with (74) gives55C(−) and C˜(−) correspond to the hard matching coefficients of A-type SCET currents. This can be understood from the fact that factorization of the associated SCET matrix elements involve the same DA ϕB(−)(ω) as in the tree-level approximation. C(+) and C˜(+) are the hard matching coefficients of B-type SCET currents whose matrix elements start at the first order of αs, therefore only the tree-level contributions of C(+) and C˜(+) enter the factorization formulae of the correlation function Πμ(n⋅p,n¯⋅p) at one loop.(75)C(−)=12C5,C˜(−)=C4+12C5. The explicit expressions of C4 and C5 can be found in [11,12](76)C4=1−αsCF4π[2ln2μmb−(4lnr−5)lnμmb+2ln2r+2Li2(1−r)+π212+(r2r¯2−2)lnr+r1−r+6],(77)C5=2r+2rr¯2lnr, from which one can readily verify the relations in Eq. (75). Conceptually, this is just an example to show that perturbative coefficient functions entering QCD factorization formulae are independent of the external partonic configurations used in the matching procedure.The jet functions J˜(±) also confront with the earlier calculations in [4] with SCET Feynman rules. It is a straightforward task to show that J˜(−) coincides with (2.23) in [4] while J˜(+) (J(+)) is in agreement with (3.9) of [4]. A final remark is devoted to J(−). Because the corresponding hard coefficient C(−) starts at O(αs), only the tree-level jet function J(−) enters the one-loop factorization of Πμ.4The LCSR for B→π form factors at O(αs)Now, we are ready to construct the sum rules of fBπ+(q2) and fBπ0(q2) including the radiative corrections at O(αs). Following the prescriptions to construct the tree-level sum rules in Section 2 and expressing the correlation function Πμ in a dispersion form with the relations in Appendix B, we obtain(78)fπe−mπ2/(n⋅pωM){n⋅pmBfBπ+(q2),fBπ0(q2)}=[U2(μh2,μ)f˜B(μh2)]∫0ωsdω′e−ω′/ωM[rϕB,eff+(ω′,μ)+[U1(n⋅p,μh1,μ)C˜(−)(n⋅p,μh1)]ϕB,eff−(ω′,μ)±n⋅p−mBmB(ϕB,eff+(ω′,μ)+C(−)(n⋅p,μ)ϕB−(ω′,μ))], where the functions ϕB,eff±(ω′,μ) are defined as(79)ϕB,eff+(ω′,μ)=αsCF4π∫ω′∞dωωϕB+(ω,μ),(80)ϕB,eff−(ω′,μ)=ϕB−(ω′,μ)+αsCF4π{∫0ω′dω[2ω−ω′(lnμ2n⋅pω′−2lnω′−ωω′)]⊕×ϕB−(ω,μ)−∫ω′∞dω[ln2μ2n⋅pω′−(2lnμ2n⋅pω′+3)lnω−ω′ω′+2lnωω′+π26−1]dϕB−(ω,μ)dω}.Several comments on the structures of the sum rules are in order.•The symmetry-breaking effects of the form-factor relation (17) can be immediately read from the last line of (78). The first term comes from the hard-collinear fluctuation and the corresponding integral is infrared finite in the heavy quark limit. One can readily confirm that this term gives an identical result of the spectator-interaction induced symmetry-breaking correction shown in Eq. (56) of [10] in the leading approximation, provided that the tree-level sum rules of fπ in Appendix C and the asymptotic expression of the twist-2 pion DA are implemented [3]. The second term corresponds to the symmetry-breaking effect induced by the hard fluctuation and it also coincides with the second term in the bracket of Eq. (30) in [10].•The scaling behavior of ω′ in (78) is ω′∼Λ2/mb due to the bounds of the integration, while the power counting of ω in (80) is O(Λ) determined by the canonical behaviors of the B-meson DAs ϕB±(ω,μ). It is then evident that ln[(ω−ω′)/ω′] and ln(ω/ω′) appeared in ϕB,eff−(ω′,μ) are counted as ln(mb/Λ) in the heavy quark limit. Such large logarithms are identified as the end-point divergences in QCD factorization approach (see also the discussions in [4]). However, we should also keep in mind that the NLL resummation improved hard coefficient [U1(n⋅p,μh1,μ)C˜(−)(n⋅p,μh1)] vanishes in the heavy quark limit.5Numerical analysisIn this section we aim at exploring phenomenological implications of the sum rules for fBπ+,0(q2) in Eq. (78) including the shapes of the two form factors, the normalized q2 spectra of B→πℓν for ℓ=μ,τ as well as the determinations of the CKM matrix element |Vub|. We will first discuss the theory inputs (the B-meson DAs, the “internal” sum rule parameters, the decay constants of the B-meson and pion, etc.) entering the sum rule analysis, compute the form factors at zero momentum transfer, and then predict the shapes of fBπ+,0(q2) in the small q2 region and extrapolate the sum rule computations to the full kinematic region with the z-series parametrization.5.1Theory input parametersThe B-meson DAs serve as fundamental ingredients for the LCSR of the B→π form factors fBπ+,0(q2). Albeit with the encouraging progresses in understanding their properties at large ω in perturbative QCD [37,38], our knowledge of the behaviors of ϕB±(ω,μ) at small ω is still rather limited due to the poor understanding of non-perturbative QCD dynamics (see [39] for discussions in the context of the QCD sum rule method). To achieve a better understanding of the model dependence of ϕB±(ω,μ) in the sum rule analysis, we consider the following four different parameterizations for the shapes of the B-meson DA ϕB+(ω,μ0):(81)ϕB,I+(ω,μ0)=ωω02e−ω/ω0,ϕB,II+(ω,μ0)=14πω0kk2+1[1k2+1−2(σB(1)−1)π2lnk],k=ω1 GeV,ϕB,III+(ω,μ0)=2ω2ω0ω12e−(ω/ω1)2,ω1=2ω0π,ϕB,IV+(ω,μ0)=ωω0ω2ω2−ωω(2ω2−ω)θ(ω2−ω),ω2=4ω04−π. ϕB,I+(ω,μ0) was originally proposed in [28] inspired by a tree-level QCD sum rule analysis. ϕB,II+(ω,μ0) suggested in [39] was motivated from the QCD sum rule calculations at O(αs) with the parameter σB(1) defined as(82)σB(n)(μ)=λB(μ)∫0∞dωωlnnμωϕB+(ω,μ),λB−1(μ)=∫0∞dωωϕB+(ω,μ). ϕB,III+(ω,μ0) and ϕB,IV+(ω,μ0) are deduced from the two models of ϕB−(ω,μ0) [4] with the Wandzura–Wilczek approximation (i.e., neglecting contributions of B-meson three-particle DAs) to maximize the model dependence of ϕB±(ω,μ0) in theory predictions, because these two models result in the same value of λB as ϕB,I+(ω,μ0) while the derivative dϕB+(ω,μ0)/dω at ω=0 takes extreme values 0 and ∞. The corresponding expression of ϕB−(ω,μ0) for each model is determined by the equation-of-motion constraint in the absence of contributions from three-particle DAs [10](83)ϕB−(ω,μ0)=∫01dξξϕB+(ωξ,μ0). We emphasize that the above models can only provide a reasonable description of ϕB±(ω,μ0) at small ω due to the radiative tail developed from QCD corrections (except the second model) and the mismatch of large ω behaviors predicted from the perturbative QCD analysis [38]. Nevertheless, the dominant contributions of fBπ+,0(q2) in the LCSR (78) come from the small ω region due to the strong suppression of ϕB±(ω,μ0) at large ω. This is also an essential prerequisite to validate QCD factorization of the correlation function Πμ whose qualifications rely on the power counting scheme ω∼Λ by construction.As a default value, we take the factorization scale μ=1.5 GeV with a variation between 1.0 GeV and 2.0 GeV for the estimate of theory uncertainty. The scale dependence of λB−1(μ) and of σB(1)(μ) are governed by the following evolution equations [32,35](84)ddlnμλB−1(μ)=−λB−1(μ)[Γcusp(αs)σB(1)(μ)+γ+(αs)],ddlnμ[σB(1)(μ)]=1+Γcusp(αs)[(σB(1)(μ))2−σB(2)(μ)], at O(αs), where the anomalous dimension γ+(αs) is(85)γ+(αs)=αsCF4π[γ+(0)+(αs4π)γ+(1)+…],γ+(0)=−2. Solving these equations yields(86)λB(μ0)λB(μ)=1+αs(μ0)CF4πlnμμ0[2−2lnμμ0−4σB(1)(μ0)]+O(αs2),(87)σB(1)(μ)=σB(1)(μ0)+lnμμ0(1+αs(μ0)CFπ[(σB(1)(μ0))2−σB(2)(μ0)])+O(αs2), where we need the evolution equation of σB(2)(μ) [32](88)ddlnμ[σB(2)(μ)]=2σB(1)(μ)+Γcusp(αs)[σB(1)(μ)σB(2)(μ)−σB(3)(μ)+4ζ3σB(0)(μ)]+O(αs2) to derive the second relation (87) with ζ3 being the Riemann zeta function. As mentioned before we are not aiming at the resummation of ln(μ/μ0) here. The two logarithmic moments will be taken as σB(1)(1 GeV)=1.4±0.4 [39] and σB(2)(1 GeV)=3±2 [29]. The determination of λB(μ0), which constitutes the most important theory uncertainty in the B-meson LCSR approach, will be discussed later. Note also that we will presume the validity of the parameterizations of ϕB±(ω,μ0) in (81) at a “hard-collinear” scale of order 1.5 GeV to avoid a complicated RG evolution of ϕB±(ω,μ) in the momentum space. We will first determine λB(μ0) at a “hard-collinear” scale and then convert it to λB(1 GeV), using the relation in (86), for a comparison of values determined in other approaches. To illustrate the features of four models displayed in (81), numerical examples for the small ω behaviors of ϕB±(ω,μ0) at μ0=1.5 GeV are plotted in Fig. 4 with a reference value of ω0(μ0)=350 MeV, where σB(1)(μ0) is evaluated from σB(1)(1 GeV) with the relation in (87).Now we turn to discuss the determinations of the Borel parameter ωM and the effective threshold ωs. We first recall the power counting(89)ωs∼ωM∼Λ2/mb, in addition to which the following requirements•The continuum contributions in the dispersion integrals of Π and Π˜ need to be less than 50%.•The sum rules for fBπ+,0(q2) are insensitive to the variation of the Borel mass ωM. For definiteness, we impose the constraint proposed in [3](90)∂lnfBπ+,0∂lnωM≤35%.•The effective threshold needs to be close to that determined from the two-point correlation function with pion interpolating currents:(91)s0≃4π2fπ2, indicated by the parton–hadron duality. are implemented to determine these “internal” sum rule parameters. Proceeding with the above-mentioned procedure yields(92)M2≡n⋅pωM=(1.25±0.25) GeV2,s0≡n⋅pωs=(0.70±0.05) GeV2, in agreement with the intervals in [2].The static decay constant f˜B(μ) entering the sum rules (78) will be traded into the QCD decay constant fB with the relation (9), which is evaluated from the two-point QCD sum rules at O(αs) as presented in Appendix C. The Borel parameter and the effective duality threshold are taken as M‾2=5.0±1.0 GeV2 and s¯0=35.6−0.9+2.1 GeV2 [8]. The pion decay constant fπ determined from the sum of branching ratios of π−→μν¯ and π−→μν¯γ is fπ=(130.41±0.03±0.02) MeV [40]. To reduce the theory uncertainties induced by the “internal” sum rule parameters we will instead use the two-point sum rules of fπ presented in Appendix C for the numerical analysis. We will return to this point later on.A “reasonable” choice of the factorization scale is μ=1.5 GeV with the variation in the interval 1 GeV≤μ≤2 GeV and the hard scales μh1 and μh2 will be set to be equal and varied in [mb/2,2mb] around the default value mb. Following [41], we adopt the bottom-quark mass in the MS‾ scheme m¯b(m¯b)=(4.16±0.03) GeV taken from [42] with a doubled uncertainly, which is still in agreement with the most recent determinations from the non-relativistic sum rules at next-to-next-to-next-to-leading order (NNNLO) [43] and from the relativistic sum rules at O(αs3) [44].5.2Numerical results of the form factors fBπ+,0(q2)Now we are in a position to discuss the inverse moment λB(1 GeV) whose determination is also of central importance in the theoretical description of the radiative leptonic B-meson decays as well as the semi-leptonic and charmless hadronic B decays. Unfortunately, the favored values of λB(1 GeV) implied by the hadronic B-decay data in QCD factorization [45] are not supported by the NLO QCD sum rule calculation [39] (see also [46] for a discussion). Recent searches of the radiative leptonic B→ℓνγ (ℓ=e,μ) decays from the Belle Collaboration [47] only set a boundary λB(1 GeV)>238 GeV.66We were informed by M. Beneke that a slightly different constraint λB(1 GeV)>217 GeV is obtained with the formulae presented in [29].Given the poor knowledge of λB(1 GeV) we will attempt to determine this parameter by matching the B-meson LCSR of fBπ+(q2) at zero momentum transfer to a given input value computed from a different method. Taking fBπ+(0)=0.28±0.03 [48] evaluated from the LCSR with pion DAs (see [49] for a recent update with somewhat larger values) and proceeding with the matching procedure yields(93)ω0(1 GeV)=354−30+38 MeV,(Model-I)ω0(1 GeV)=368−32+42 MeV,(Model-II)ω0(1 GeV)=389−28+35 MeV,(Model-III)ω0(1 GeV)=303−26+35 MeV,(Model-IV) where the four models correspond to that shown in (81). It is evident that the extracted values of ω0(1 GeV) are sensitive to the specific models of ϕB±(ω,μ0) entering the LCSR of fBπ+,0(q2) in (78), because these sum rules cannot be controlled by the inverse moment λB(1 GeV) of the DA ϕB+(ω,μ0) to a good approximation and the precise shapes of B-meson DAs at small ω are in demand for the sum rule analysis [4]. In other words,(94)∫0ωsdω′e−ω′/ωMϕB−(ω,μ0)≃ϕB−(ω=0,μ0)∫0ωsdω′e−ω′/ωM should not be taken seriously as one would expect at first sight. Mathematically, the precision of such approximation depends on the fluctuant rapidity of ϕB−(ω,μ0) at small ω. A similar observation was already made by inspecting the LCSR with pion DAs in the heavy quark limit [3], where the knowledge of the two lowest-order Gegenbauer moments is not sufficient to determine the key non-perturbative object ϕπ′(1) which is highly dependent on the exact form of ϕπ(u). We stress that the quantity λB(μ0) itself is well defined at the operator level and is independent of the specific models of ϕB+(ω,μ0). A precision determination of λB(μ0) by other means (e.g., Lattice QCD simulation) would be of great value to discriminate certain models of the B-meson DAs.To reduce the sizeable uncertainty from modeling the B-meson DAs, we will merely aim at predicting the shape of fBπ+(q2) which is insensitive to the precise behaviors of ϕB±(ω,μ0) at small ω, as displayed in Fig. 5, due to a large cancellation of the model dependence in the form-factor ratio fBπ+(q2)/fBπ+(0). We also find that the results of fBπ+(q2) evaluated from different models of ϕB±(ω,μ0) are systematically lower than that obtained from the LCSR with pion DAs confirming an earlier observation from the tree-level calculations [2]. The underlying mechanism responsible for such discrepancy might be due to the yet unaccounted sub-leading power corrections and/or the different ansatz of the parton–hadron duality in the constructions of sum rules, and we will return to this point later on. Hereafter, we will take ϕB,I±(ω,μ0) as the default model to study the implications of the sum rules in (78) and the systemic uncertainty from the model dependence of the B-meson DAs will be included in the final predictions of the two form factors fBπ+,0(q2).To demonstrate the stability of the LCSR predictions we show the dependencies of fBπ+(q2) on the “internal” sum rule parameters M2 and s0 in Fig. 6 where the two plots on the top are obtained from NLL resummation improved sum rules (78) with fπ extracted from the experimental data as explained before; while the two-point QCD sum rules of fπ are substituted in the LCSR to produce the two plots on the bottom. One can readily find that the systematic uncertainties induced by the Borel parameter and the effective threshold are significantly reduced in the latter case, albeit with the absence of a model-independent justification of correlating the “internal” parameters in the two types of sum rules.Now we come to investigate the factorization-scale dependence of the NLL and the leading-logarithmic (LL) resummation improved LCSR for fBπ+,0(q2), where the LL predictions can be achieved by employing the cusp anomalous dimension at O(αs2) as well as γ(αs) and γ˜(αs) at the one-loop order in the evolution functions U1(n⋅p,μh1,μ) and U2(μh2,μ) of (78). Fig. 7 shows that the scale dependence of the NLL predictions is not significantly reduced compared to the LL approximation for the hard-collinear scale varied in the interval [1.0,2.0] GeV and the discrepancy of the scale dependency for the NLL and LL predictions will be more visible for a somewhat “unrealistic” hard-collinear scale μ<1.0 GeV which is therefore excluded in the plot. The dominant radiative effect arises from the NLO QCD corrections to perturbative matching coefficients instead of resummation of the parametrically large logarithms in the heavy quark limit. However, the resummation improvement stabilizes the factorization-scale dependence in the allowed region and strengthens the predictive power of the LCSR method. One can also find that the NLO QCD correction is stable against the momentum-transfer dependence of fBπ+(q2) in contrast to the case of B→γℓν [29].Understanding the pion energy and the heavy quark mass dependencies of the form factors fBπ+,0(q2) are of both theoretical and phenomenological interest in that different competing mechanisms appear in the theory description of heavy-to-light form factors in the large recoil region and a better control of the form factor shapes can be achieved by incorporating the energy-scaling laws and the Lattice (sum-rule) calculations of form factors at high (low) q2. In accordance with the factorization formulae [20](95)fBπi(Eπ)=Ci(Eπ)ξπ(Eπ)+∫dτCi(B1)(Eπ,τ)Ξa(τ,Eπ),Ξa(τ,Eπ)=∫0∞dω∫01duJ‖(τ,u,ω)f˜B(μ)ϕB+(ω,μ)fπϕπ(u,μ), one can readily deduce that both terms in the first line of (95) scale as 1/Eπ2 in the large energy limit and as (Λ/mb)3/2 in the heavy quark limit [10,50]. It is our objective to verify such scaling behaviors from the NLL resummation improved sum rules (78). In doing so we define the following two ratios [4](96)R1(Eπ)≡fBπ+(Eπ)fBπ+(mB/2),R2(mQ)≡mQf˜B(μ)mBf˜Q(μ)fQπ+(mQ/2)fBπ+(mB/2), where the argument of the form factor refers to n⋅p/2 different from that (q2) used in the remaining of this paper, the pre-factors in the definition of R2(mQ) is introduced to achieve a simple scaling R2(mQ)→1 in the heavy quark limit. The expression of fQπ+(n⋅p/2) can be obtained from Eq. (78) via the replacement (mb,mB)→(mQ,mQ). One should also keep in mind that the scalings of the “internal” sum rule parameters shown in (89) need to be respected when deriving the power-counting laws of the large energy and the heavy quark mass dependencies. We present the sum rule predictions for the two ratios R1(Eπ) and R2(mQ) in Fig. 8, where we observe that the yielding energy dependence is indeed close to the 1/Eπ2 behavior and the heavy-quark mass scaling is also justified from the LCSR with B-meson DAs. However, the sum rule results become more and more instable at mQ>2mB where the Borel parameter dependence is not under control any more as displayed in Fig. 8, and one can also find that the continuum effect dominates over the ground state contribution in the dispersion integral of the correlation function Πμ (see also the discussions in [4]).One more comment concerns the ratio R2(mQ) which allows to estimate D→π form factors from the corresponding B-meson cases in the leading-power approximation. However, this statement needs to be taken with a grain of salt in reality in view of the sizeable power correction in the decay-constant ratio fB/fD which is determined as(97)fBfD=[mDmB]1/2[αs(mc)αs(mb)]γ˜02β0CF{1+[αs(mb)−αs(mc)]CF4π×[−2+(γ˜(1)2β0−γ˜(0)β12β02)]}≃0.69, significantly lower than the QCD sum rule prediction 0.93≤fB/fD≤1.19 [51] and the Lattice QCD result computed from fB=(190.5±4.2) MeV and fD=(209.3±3.3) MeV with Nf=2+1 [52].To validate the light-cone expansion of the correlation function Πμ in the region |n¯⋅p|∼O(Λ) we need to keep the photon energy as a hard scale, above the practical value of a hard-collinear scale ∼1.5 GeV, then the LCSR with B-meson DAs can be trusted at q2≤qmax2=8 GeV2 (see [2] for more detailed discussions) on the conservative side. To extrapolate the computed form factors from the LCSR method at large recoil toward large momentum transfer q2 we apply the z-series parametrization based upon the analytical and asymptotic properties of the form factors, where the entire cut q2-plane is mapped onto the unit disk |z(q2,t0)|<1 via the conformal transformation(98)z(q2,t0)=t+−q2−t+−t0t+−q2+t+−t0, where t+=(mB+mπ)2 denotes the threshold of continuum states in the B⁎(1−) meson channel. The free parameter t0∈(−∞,t+) determines the value of q2 mapped onto the origin in the z plane and can be adjusted to minimize the z interval from mapping the LCSR region qmin2≤q2≤qmax2. For definiteness, we follow [48](99)t0=t+2−t+−t−t+−qmin2, with qmin2=−6.0 GeV2 and t−≡(mB−mπ)2, and we also refer to [48,53] and the references therein for more discussions on different versions of the z-parametrization and to [49] for a new implementation of the unitary bounds for the vector B→π form factor.Employing the z-series expansion and taking into account the threshold t+ behavior implies the following parametrization of the vector form factor [48](100)fBπ+(q2)=fBπ+(0)1−q2/mB⁎2{1+∑k=1N−1bk(z(q2,t0)k−z(0,t0)k−(−1)N−kkN[z(q2,t0)N−z(0,t0)N])}, where the expansion coefficients bk can be determined by matching the computed fBπ+(q2) at low q2 onto Eq. (100) and we truncate the z-series at N=2 in the practical calculation. One can keep more terms of the z expansion in the fitting program to quantify the systematic uncertainty induced by the truncation, however, one could also run the risk of increasingly unconstrained fit when introducing too many parameters [54], and we will leave a refined statistic analysis for the future. Along this line, one can further parameterize the scalar form factor as(101)fBπ0(q2)=fBπ0(0){1+∑k=1Nb˜k(z(q2,t0)k−z(0,t0)k)}, where the pole factor is removed because the lowest scalar B(0+) meson is located above the continuum cut t+, and the series is truncated at N=1 with fBπ0(0)=fBπ+(0) by definition. We also implement the unitary bound constraints on the coefficients of bk and b˜k in the fitting program, which are however too weak to take effect for the truncation at N=2 for fBπ+(q2) and at N=1 for fBπ0(q2).Fig. 9 shows the q2 dependence of the two form factors fBπ+,0(q2) computed from the LCSR with B-meson DAs at q2<8 GeV2 with an extrapolation to q2=12 GeV2 (pink band) using the z expansion, and theoretical predictions from the LCSR with pion DAs [48] without any extrapolation at q2<12 GeV2 (blue band) are also presented for a comparison. It is evident that the predict shape of fBπ0(q2) is in good agreement with that computed from the sum rules with pion DAs while a similar comparison for the vector form factor fBπ+(q2) reveals perceptible discrepancies in particular at high q2 as already observed before.As the first attempt to understand this issue it would be interesting to inspect influence of the matching condition of ω0(1 GeV), described before Eq. (93), on the final predictions of the form-factor shapes. Taking fBπ+(17.34 GeV2)=0.94−0.07+0.06 from Fermilab/MILC Collaborations [55] as an input and proceeding with the matching procedure we obtain ω0(1 GeV)=525±29 MeV for the default model of ϕB±(ω,μ0), which is significantly larger than the determinations displayed in (93). The resulting shape of the re-scaled form factor (1−q2/mB⁎2)fBπ+(q2) is presented in Fig. 10 where the Lattice data from HPQCD Collaboration [56], RBC/UKQCD Collaborations [57] and Fermilab/MILC Collaborations [55] are also displayed for a comparison. One can readily observe that the higher q2 shape of fBπ+(q2) predicted by Fermilab/MILC [55] lies in between that obtained from the LCSR with B-meson DAs and the one with pion DAs. In fact, the recent Lattice calculations [55] (see Fig. 24 there) already revealed a faster growing form factor fBπ+(q2) in the momentum transfer squared compared to that computed from the LCSR with pion DAs. We should stress that the new matching procedure discussed here needs to be interpreted more carefully, because extrapolating the sum rule computations toward large momentum transfer with the z expansion is also implemented for the sake of determining ω0(1GeV) from the Lattice input at a high q2=17.34 GeV2.We present the fitted values of fBπ+(0) and of the slop parameters b1 and b˜1 in Table 1 where breakdown of the numerically important uncertainties is also shown. A few comments on the numerical results obtained above are in order. •The very limited information of ϕB±(ω) (indicated by the variations of ω0 and by the model dependence of ϕB±(ω) in the table) remains the most significant source of theory uncertainties.•Comparing the new predictions in Table 1 with that of [48] we notice again the greater slop parameters for both the vector and scalar form factors determined by the LCSR with B-meson DAs.•Since the prediction of fBπ+(0) from the LCSR with pion DAs is taken as an input to determine the inverse moment λB(1 GeV) and resummation of large logarithms in the hard function C˜(−) is implemented in the B-meson LCSR, theory uncertainties of the slop parameters b1 and b˜1 in Table 1 are comparable to that presented in [48] where the scale variation induces sizeable errors. However, one should keep in mind that power suppressed contributions induced by the higher twist pion DAs are taken into account in the traditional LCSR calculations [8]; while power suppressed effects to the B-meson LCSR generated by the sub-leading B-meson DAs and/or the sub-dominant hard scattering kernels are not included in the current analysis. In addition, we do not consider the correlation between the normalization and the slop parameters of the form factors as carried out in the LCSR with pion DAs [49].5.3|Vub| and the normalized q2 distributions of B→πℓνℓThe CKM matrix element |Vub| can be determined from the (partial) branching fraction of B→πℓνℓ(102)dΓdq2(B→πℓνℓ)=GF2|Vub|224π3q4mB2(q2−ml2)2|p→π|[(1+ml22q2)mB2|p→π|2|fBπ+(q2)|2+3ml28q2(mB2−mπ2)2|fBπ0(q2)|2], where |p→π| is the magnitude of the pion three-momentum in the B-meson rest frame, and in the massless lepton limit the above equation can be reduced to(103)dΓdq2(B→πμνμ)=GF2|Vub|224π3|p→π|3|fBπ+(q2)|2. Following [48] we define the following quantity(104)Δζ(0,q02)=GF224π3∫0q02dq2|p→π|3|fBπ+(q2)|2, which allows a straightforward extraction of |Vub| when compared to experimental measurements for the partial branching ratio of B→πμνμ integrated over the same kinematic region. Implementing the computed form factor fBπ+(q2) from the sum rules with B-meson DAs and performing the extrapolation to q2=12 GeV2 with the z-series parametrization yield(105)Δζ(0,12 GeV2)=5.89|ω0−1.10+1.12|σB(1)−0.29+0.30|μ−1.22+0.60|μh1(2)−0.21+0.21|M,s0−0.53+0.34|M‾,s‾0−0.25+0.52ps−1=5.89−1.82+1.63 ps−1, where the negligibly small uncertainties from variations of the remaining parameters are not presented but are included in the final combined uncertainty.Employing experimental measurements of the integrated branching ratio(106)ΔBR(0,q02)=|Vub|2Δζ(0,q02) of the semi-leptonic B¯0→π+μνμ decay [58,59]:(107)ΔBR(0,12 GeV2)=(0.83±0.03±0.04)×10−4,[BaBar 2012]ΔBR(0,12 GeV2)=(0.808±0.062)×10−4,[Belle 2013] and taking the mean lifetime τB0=(1.519±0.005)ps [40] we obtain(108)|Vub|=(3.05−0.38+0.54|th.±0.09|exp.)×10−3, where the reduction of |Vub| compared to [48] is attributed to the rapidly increasing form factor fBπ+(q2), with respect to q2, computed from the sum rules with B-meson DAs, and the diminishing ΔBR(0,12 GeV2) from the new measurements [58,59] in relative to the previous BaBar measurements [60,61]; the theoretical uncertainty is from the computation of Δζ(0,12 GeV2) as displayed in (105).Now we turn to compute the normalized differential q2 distributions of B→πℓνℓ using the form factors obtained with the B-meson LCSR and extrapolated with the z-series parametrization. Our predictions of the normalized q2 distribution are plotted in Fig. 11 where the available data from BaBar and Belle Collaborations are also shown for a comparison. We observe a reasonable agreement of our predictions for the q2 distribution of B→πμνμ and the new Belle and BaBar data points [58,59], but a poor agreement when confronted with the previous measurements [60–62] in particular in the low q2 region. It is evident that the theory uncertainty of the normalized differential distribution of B→πμνμ is somewhat smaller than that of the form factors shown in Fig. 9 because of the partial cancellation in the ratio of the differential and the total branching ratio with respect to the variations of theory inputs. The q2 shape of the normalized distribution for B→πμνμ is also confronted with the prediction from the pion LCSR in [48]. As a by-product, we further plot the normalized differential distribution of B→πτντ in Fig. 11, which provides an independent way to extract |Vub| with the aid of future measurements at the Belle-II experiment.6Three-particle DAs of the B mesonWe have not touched three-parton Fock-state contributions to the form factors fBπ+,0(q2) in the context of the LCSR with B-meson DAs. This topical problem has triggered “sophisticated” discussions in the literature from different perspectives, see [21,63–66] for an incomplete list. We will first make some general comments on non-valence Fock state contributions to B→π form factors, and then discuss how B-meson three-particle DAs could contribute to the sum rules presented in this work briefly. •The representation of the heavy-to-light currents in the context of SCET(c,s) indicates that three-parton Fock-state contributions already appear at leading power in Λ/mb [21] and these contributions preserve the large-recoil symmetry relations at leading power albeit with the emergence of endpoint divergences [21,63]. This observation was confirmed independently by QCD sum rule calculations of B→π form factors with pion DAs [64].•The tree-level contribution of three-particle DAs in B→π form factors is of minor importance numerically (at percent level) confirmed by two different types of sum rules with B-meson DAs [2] and with pion DAs [8], respectively. The insignificant tree-level effect can be understood transparently from the sum rules with pion DAs, where the collinear gluon emission from the b-quark propagator yields power suppression in Λ/mb. However, this power-suppression mechanism will be removed at O(αs), because the radiative gluon can be emitted from the (hard)-collinear light-quark propagators in the evaluation of the corresponding correlation function at NLO (for a concrete example, see [64]). A complete calculation of three-parton Fock-state contributions to B→π form factors is unfortunately not available in the framework of both sum rule approaches at present. In the following we will sketch this absorbing and challenging calculation in the context of the LCSR with B-meson DAs.•B-meson three-particle DAs could manifest themselves in the NLO sum rules in a variety of ways. First, these contributions are essential to compensate the factorization-scale dependence of ϕB−(ω,μ) entering the factorization formulae of Π(n⋅p,n¯⋅p) and Π˜(n⋅p,n¯⋅p) at O(gs3) recalling the evolution equation [33](109)ddlnμϕB−(ω,μ)=−αsCF4π{[Γcusp(0)lnμω−2]ϕB−(ω,μ)+∫0∞dω′ωΓ(ω,ω′,μ)ϕB−(ω,μ)+∫0∞dω′∫0∞dξ′γ−,3(1)(ω,ω′,ξ′,μ)[ΨA−ΨV](ω′,ξ′,μ)}, where the mixing term is solely governed by the light degrees of freedom in the composite operator [67]. A sample diagram is shown in Fig. 12(b) whose soft divergences can be reproduced by adding up amplitudes of the two effective diagrams displayed in Figs. 12(e) and 12(f) convoluted with the corresponding tree-level hard-scattering kernel. Second, B-meson three-particle DAs can induce leading-power contributions without recourse to the above-mentioned mixing pattern and one also expects that the soft subtraction is not needed here due to power suppression of the tree-level contribution from three-particle DAs. One should notice that renormalization of B-meson three-particle DAs will not generate the inverse mixing into two-particle DAs, at least, at O(αs), while a similar statement holds to all orders of αs for pion DAs due to conformal symmetry [68]. Third, three-particle DAs can revise the Wandzura–Wilczek relation (83) which needs to be generalized into [35](110)ωϕB−(ω)−∫0ωdη[ϕB−(η)−ϕB+(η)]=2∫0ωdη∫ω−η∞dξξ∂∂ξ[ΨA(η,ξ)−ΨV(η,ξ)].7Conclusions and discussionWe have carried out, for the first time, perturbative corrections to B→π form factors from the QCD LCSR with B-meson DAs proposed in [1,2] where the sum rules for heavy-to-light form factors were established at tree level including contributions from both two-particle and three-particle DAs. We placed particular emphasis on the demonstration of factorization of the vacuum-to-B-meson correlation function Πμ(n⋅p,n¯⋅p) at O(αs) taking advantage of the method of regions which allows a transparent separation of different leading regions with the aid of the power counting scheme. Precise cancellation of the soft contribution to the correlation function Πμ and the infrared subtraction was perspicuously shown at the diagrammatic level. The short-distance function obtained with integrating out the hard-scale fluctuation receives the contribution from the weak-vertex diagram solely because the loop integrals from the remaining diagrams do not involve any external invariant of order 1. The resulting hard coefficients coincide with the corresponding matching coefficients of the vector QCD weak current in SCETI indicating that perturbative coefficients in the OPE are independent of the external partonic configuration chosen in the matching procedure as expected. The computed jet functions from integrating out dynamics of the hard-collinear scale are also in agreement with the expressions derived from the SCET Feynman rules [3]. We further verified factorization-scale independence of the correlation function at O(αs) employing evolution equations of the hard function C˜(−), the jet function J˜(−) and the B-meson DA ϕB−(ω,μ), then summed up the large logarithms due to the appearance of distinct energy scales by the standard RG approach in the momentum space. We left out resummation of the parametrically large logarithms of μhc/μ0 due to the insignificance numerically. However, there is no difficulty to achieve this resummation whenever such theory precision is in demand and an elegant way to perform resummation of large logarithms in the presence of the cusp anomalous dimension in the evolution equations is to work in the “dual” momentum space where the Lange–Neubert kernel of the B-meson DAs are diagonalized [35].With the resummation improved sum rules (78) at hand, we explored their phenomenological implications on B→π form factors at large hadronic recoil in detail. Due to our poor knowledge of the inverse moment of the B-meson DA ϕB+(ω,μ) we first determine this parameter by matching the B-meson LCSR prediction of fBπ+(q2) at zero momentum transfer to the result obtained from the sum rules with pion DAs at NLO, utilizing four different models of ϕB±(ω,μ) displayed in (81). While these models do not capture the features of large ω behaviors from perturbative QCD analysis, the power counting rule ω∼Λ, thanks to the canonical picture of the B-meson bound state, implemented in the construction of QCD factorization for the correlation function Πμ requires that the dominant contribution in the factorized amplitude must be from the small ω region. We then found that the obtained values of the shape parameter ω0(1 GeV) from the matching procedure are rather sensitive to the shapes of ϕB±(ω,μ) at small ω, pointing to the poor “local” approximation (94) and confirming an earlier observation made in [4]. However, the q2 shape of fBπ+(q2) predicted from LCSR is insensitive to the specific model of the B-meson DAs after determining ω0(1 GeV) from the above-described matching condition. This is not surprising because of a large cancellation of the theory uncertainty in the form-factor ratio fBπ+(q2)/fBπ+(0). Moreover, we showed that the dominating radiative effect originates from the NLO QCD correction instead of the NLL resummation of large logarithms in the heavy quark limit, which does however improve the stability of varying the factorization scale. Proceeding with the resummation improved sum rules (78) in the large recoil region, q2≤8 GeV2, and extrapolating the computed from factors toward large momentum transfer, we found that the obtained scalar form factor fBπ0(q2) is in a reasonable agreement with that from the LCSR with pion DAs at q2≤12 GeV2; while a similar comparison of the vector form factor fBπ+(q2) calculated from two different sum rule approaches reveals noticeable discrepancies particularly at higher q2. We made a first step towards understanding this intriguing problem by determining the parameter ω0(1 GeV) from matching the form factor fBπ+(13.74 GeV2) to the Lattice data from Fermilab/MILC Collaborations. It was then shown that the shape of fBπ+(q2) at high q2 predicted from Lattice QCD simulation lies in between that obtained from the two sum rule approaches. It remains unclear whether the observed discrepancies arise from the sub-leading power contributions to both LCSR and/or from the systematic uncertainties induced by different kinds of parton–hadron duality relations in the constructions of sum rules and/or from the yet unknown leading-power contribution from three-particle DAs in both approaches. As a result of the rapidly growing form factor fBπ+(q2) we obtain lower values of |Vub|=(3.05−0.38+0.54|th.±0.09|exp.)×10−3, in contrast to the predictions of the pion LCSR [48,49], by comparing BaBar and Belle measurements of the integrated branching ratio of B→πμνμ in the region 0<q2<12 GeV2 with the computed quantity Δζ(0,12 GeV2) in (105). The theory uncertainty is dominated by the rather limited information of the B-meson DAs at small ω encoding formidable (non-perturbative) strong-interaction dynamics. Precision measurements of the differential q2 distributions of B→πℓνℓ (ℓ=μ,τ) at the Belle-II experiment might shed light on the promising orientation to resolve the tension of the form factor shapes and subsequently to put meaningful constraints on the small ω behaviors of ϕB±(ω).We further turned to discuss non-valence Fock-state contributions to the form factors fBπ+,0(q2), which are the missing ingredients of our computations, and to illustrate modifications of the sum rule analysis in the presence of three-particle DAs of the B meson. Given the fact that non-valence Fock-state contributions to the form factors fBπ+,0(q2) are either suppressed in powers of Λ/mb at tree level or suppressed by the QCD coupling constant αs at NLO, one may expect that including three-particle DAs of the B meson may generate a modest effect on the form-factor predictions albeit with a high demand of computing their contributions at O(αs) in the conceptual aspect.Developing the LCSR of B→π form factors with B-meson DAs beyond this work can be pursued further by including the sub-leading power contributions of the considered correlation function, which requires better knowledge of the sub-leading DAs of the B meson (e.g., off-light-cone corrections) and demonstrations of factorization for the correlation function at sub-leading power in Λ/mb. Computing yet higher order QCD corrections of the correlation function would be also interesting conceptually, but one would need the two-loop evolution equation of ϕB−(ω,μ), which can be complicated by more involved mixing of string operators under renormalization.77The new technique developed in [69] using exact conformal symmetry of QCD at the critical point would be powerful in this respect.Moreover, we also expect phenomenological extensions of this work to compute NLO QCD corrections of many other hadronic matrix elements from the LCSR with bottom-hadron DAs. First, it is of interest to perform a comprehensive analysis of B→P,V form factors with P=π,K and V=ρ,K⁎ from the B-meson LCSR at O(αs), which serve as fundamental theory inputs for QCD factorization of the electro-weak penguin B→K(⁎)ℓℓ decays and the charmless hadronic B→PP and B→PV decays. In particular, such analysis could be of value to understand the tension of form-factor ratios between the traditional LCSR and QCD factorization firstly observed in [10], keeping in mind that these ratios are less sensitive to the shapes of B-meson DAs at small ω. Second, a straightforward extension of this work to compute form factors describing the exclusive B→D(⁎)τντ decays will deepen our understanding towards the topical R(D(⁎)) puzzles, referring to the 2σ (2.7σ) deviations of the measured ratios of the corresponding branching fractions in muon and tauon channels. Such computations will enable us to pin down perturbative uncertainties in the tree-level predictions of B→D(⁎) form factors [70] and also allow for a better comparison of LCSR and heavy-quark expansion in a different testing ground. A complete discussion of radiative corrections to B→D(⁎) form factors from the B-meson LCSR including a proper treatment of the charm-quark mass will be presented elsewhere. Third, the techniques discussed in this work can be applied to compute Λb→p,Λ transition form factors from the sum rules with the Λb-baryon DAs [71,72], which are of phenomenological interest for an alternative determination of |Vub| exclusively and for a complementary search of physics beyond the Standard Model. However, one should be aware of the fact that constructing baryonic sum rules are more involved than the mesonic counterpart due to various ways to choose baryonic interpolating currents and potential contaminations from negative-parity baryons in the hadronic dispersion relations [73]. To summarize, we foresee straightforward extensions of this work into different directions.AcknowledgementsWe are grateful to M. Beneke, N. Offen and J. Rohrwild for illuminating discussions, to M. Beneke for many valuable comments on the manuscript, and to D.P. Du for providing us with the Lattice data points in [55]. YMW is supported in part by the Gottfried Wilhelm Leibniz programme of the Deutsche Forschungsgemeinschaft (DFG).Appendix ALoop integralsIn this appendix, we collect some useful one-loop integrals in our calculations.(111)I0=∫[dl]1[(p−k+l)2+i0][l2+2mbv⋅l+i0]=1ϵ+2lnμmb+rr¯lnr+2,(112)I1h=∫[dl]1[l2+n⋅pn¯⋅l+i0][l2+2mbv⋅l+i0][l2+i0]=−12mbn⋅p[1ϵ2+2ϵlnμn⋅p+2ln2μn⋅p−ln2r−2Li2(−r¯r)+π212],(113)I1hc=∫dDl(2π)D1[n⋅(p+l)n¯⋅(p−k+l)+l⊥2+i0][n⋅l+i0][l2+i0]=1n⋅p[1ϵ2+1ϵlnμ2n⋅p(ω−n¯⋅p)+12ln2μ2n⋅p(ω−n¯⋅p)−π212],(114)I1μ=∫[dl]lμ[(p−k+l)2+i0][l2+2mbv⋅l+i0][l2+i0]≡C1(p−k)μ+C2mbvμ,(115)C1h=−1mb2r[1ϵ+2lnμmb−r−2r−1lnr+2],(116)C2=C2h=−1mb2r¯lnr,(117)C1hc=C1−C1h=1mb2r[1ϵ+lnμ2n⋅p(ω−n¯⋅p)+2],(118)I1,a=∫[dl]n⋅ln¯⋅l[(p−k+l)2+i0][l2+2mbv⋅l+i0][l2+i0]=12[1ϵ+2lnμmb+rr¯lnr+2],(119)I1,b=∫[dl](n¯⋅l)2[(p−k+l)2+i0][l2+2mbv⋅l+i0][l2+i0]=−12r¯2[rlnr+r¯],(120)I2=∫[dl]lα(p−l)β[(p−l)2+i0][(l−k)2+i0][l2+i0]≡−gαβ2I2,a−1p2[kαkβI2,b−pαpβI2,c−kαpβI2,d+pαkβI2,e],(121)I2,a=12[1ϵ+ln(−μ2p2)−1+ηηln(1+η)+3],(122)I2,b=2η−η2−2ln(1+η)2η3[1ϵ+ln(−μ2p2)−ln(1+η)+3]+η2−ln2(1+η)2η3,(123)I2,c=ln(1+η)2η,(124)I2,d=ln(1+η)−ηη2[1ϵ+ln(−μ2p2)−ln(1+η)+52]+ln2(1+η)2η2,(125)I2,e=η−ln(1+η)2η2,(126)I3=∫[dl]1[(p−k+l)2+i0][l2+i0]=1ϵ+lnμ2n⋅pn¯⋅(k−p)+2,(127)I3μ=∫[dl]lμ[(p−k+l)2+i0][l2+i0]=−I32(p−k)μ,(128)I4,a=∫dDl(2π)Dn⋅(p+l)[n⋅(p+l)n¯⋅(p−k+l)+l⊥2+i0][n⋅ln¯(l−k)+l⊥2+i0][l2+i0]=ln(1+η)ω[1ϵ+lnμ2n⋅p(ω−n¯⋅p)+12ln(1+η)+1],(129)I4,b=∫dDl(2π)Dn⋅ln⋅(p+l)[n⋅(p+l)n¯⋅(p−k+l)+l⊥2+i0][n⋅ln¯(l−k)+l⊥2+i0][l2+i0]=n⋅p2ωln(1+η), with r=n⋅p/mb, r¯=1−r, ω=n¯⋅k and η=−ω/n¯⋅p.Appendix BSpectral representationsWe collect the spectral functions of convolution integrals entering the factorization formulae of Π and Π˜ in (73). These expressions were first derived in [4], we confirmed these spectral functions independently and also verified the corresponding dispersion integrals.(130)1πImω′∫0∞dωωlnω′−ωω′ϕB+(ω,μ)=∫ω′∞dωωϕB+(ω,μ),(131)1πImω′∫0∞dωω−ω′−i0ln2μ2n⋅p(ω−ω′)ϕB−(ω,μ)=∫0∞dω[2θ(ω′−ω)ω−ω′lnμ2n⋅p(ω′−ω)]⊕ϕB−(ω,μ)+[ln2μ2n⋅pω′−π23],(132)1πImω′∫0∞dωω−ω′−i0ln2ω′−ωω′ϕB−(ω,μ)=−∫ω′∞dω[ln2ω−ω′ω′−π23]ddωϕB−(ω,μ),(133)1πImω′∫0∞dωω−ω′−i0lnω′−ωω′lnμ2n⋅p(ω−ω′)ϕB−(ω,μ)=∫0∞dω[θ(ω′−ω)ω−ω′lnω′−ωω′]⊕ϕB−(ω,μ)+12∫ω′∞dω[ln2μ2n⋅p(ω−ω′)−ln2μ2n⋅pω′+π23]ddωϕB−(ω,μ),(134)1πImω′∫0∞dωω−ω′−i0lnω′−ωω′ϕB−(ω,μ)=−∫ω′∞dωlnω−ω′ω′ddωϕB−(ω,μ).Appendix CTwo-point QCD sum rules for fB and fπFor completeness, we collect the two-point sum rules for the B-meson decay constant fB in QCD [51,74] including NLO corrections to the perturbative term and to the D=3 quark condensate part:(135)fB2=emB2/M‾2mb2mB4{∫mb2s¯0dse−s/M‾238π2[(s−mb2)2s+αsCFπρpert(1)(s,mb2)]+e−mb2/M‾2[−mb〈q¯q〉(1+αsCFπρqq¯(1)(s,mb2))−mb〈q¯Gq〉2M‾2(1−mb22M‾2)+112〈αsπGG〉−16π27αs〈q¯q〉2M‾2(1−mb24M‾2−mb412M‾4)]}, where M‾2 and s¯0 are the Borel parameter and the effective threshold, and the relevant NLO spectral functions are given by(136)ρpert(1)(s,mb2)=x¯s2{x¯[4Li2(x)+2lnxlnx¯−(5−2x)lnx¯]+(1−2x)(3−x)lnx+3(1−3x)lnμ2mb2+17−33x2},(137)ρqq¯(1)(s,mb2)=−32[Γ(0,mb2M‾2)emb2/M‾2−(1−mb2M‾2(lnμ2mb2+43))−1], with x=mb2/s, x¯=1−x and the incomplete Γ function defined as(138)Γ(n,x)=∫x∞dttn−1e−t.The two-point QCD sum rules of the pion decay constant fπ including the perturbative term at O(αs) reads [75](139)fπ2=M2[14π2(1−e−s0/M2)(1+αs(M)π)+112M4〈αsπGG〉+176π81αs〈q¯q〉2M6].References[1]A.KhodjamirianT.MannelN.OffenPhys. 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