# B⟶P,V Form Factors with the B-Meson Light-Cone Sum Rules

Authors: Yue-Long Shen; Yan-Bing Wei
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

In this review, we discuss the calculation of theB⟶P,V form factors within the framework of the light-cone sum rules with the light-cone distribution amplitudes of the B-meson. A detailed introduction to the definition, scale evolution, and phenomenological models of the B-meson distribution amplitudes is presented. We show two equivalent approaches of calculating the next-to-leading order QCD corrections to the sum rules for the form factors, i.e., the method of regions and the step-by-step matching in the soft-collinear effective theory. The power suppressed corrections to the B⟶P,V form factors especially the contributions from the higher-twist B-meson distribution amplitudes are displayed. We also present numerical results of the form factors including both the QCD and the power corrections, and phenomenological applications of the predicted form factors such as the determination of the CKM matrix element Vub.


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## Body

## 1. Introduction

The decays ofBq mesons (q=d,s) have been playing a crucial role in the determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements as well as the understanding of the QCD dynamics in the heavy-light meson system. The heavy-to-light transition form factors are essential ingredients in the semileptonic decays B⟶Mℓν (M stands for vector (V) or pseudoscalar (P) meson), in the flavor-changing-neutral-current (FCNC) processes B⟶Mℓ+ℓ−νν¯ and B⟶Vγ, and in the nonleptonic B-meson decays. In the small recoil region, the heavy-to-light form factors can be determined from the experiments or calculated by the nonperturbative approach, among which the lattice QCD simulation which is based on the first principle of QCD is regarded to give the most reliable predictions. At small hadronic recoil, the lattice QCD calculations of B⟶π,K, Bs⟶K form factors have been performed [1–3] using the gauge-field ensembles with (2+1)-flavor lattice configurations. In addition, the Flavor Lattice Average Group (FLAG) has given the results of these form factors with an extrapolation to the whole kinematic region from the small hadronic recoil region of the light meson [4]. The unquenched lattice QCD calculations of B⟶K∗ form factors have been performed [5, 6] by employing the gauge-field ensembles with an improved staggered quark action from the MILC Collaboration [7].Due to the limited computing capability, the lattice simulation cannot be applied to the large recoil region directly. In the framework of the QCD factorization, the heavy-to-light form factors at large recoil contain both the soft contribution satisfying the large-recoil symmetry relations and the hard spectator scattering effect violating the symmetry relations at leading power inΛ/mQ [8]. The soft-collinear effective theory (SCET) provides a more transparent insight on the factorization property of heavy-to-light form factors by integrating out the hard and hard-collinear fluctuations step by step. Implementing the first-step matching procedure for the QCD current ψ¯ΓiQ will give rise to the so-called A0-type and B1-type SCETI operators [9–11], both of which can contribute to heavy-to-light form factors at leading power in Λ/mQ. The matrix elements of the A0-type operator are nonfactorizable due to the emergence of endpoint divergences in the convolution integrals of the jet functions from the matching between SCETI and SCETII and the light-cone distribution amplitudes (LCDAs). By contrast, the matrix elements of the B1-type SCETI operators can be further factorized into convolutions of the jet functions and the LCDAs [11]. Since the latter one is suppressed by the strong coupling constant, the heavy-to-light form factors are dominated by the soft form factor in the QCD factorization. An alternative approach to compute the heavy-to-light form factors is based upon the transverse-momentum-dependent (TMD) factorization for hard processes, where the on-shell Sudakov form factor [12] arises from the resummation of large logarithmic terms that can effectively suppress the region with small transverse momentum [13]. In the TMD factorization approach (also called the PQCD approach), the endpoint singularity will be regularized by the transverse momentum; then, the form factors are perturbative calculable. The B⟶π form factors within the PQCD approach have been pushed to Oαs for twist-2 [14, 15] and twist-3 [16] contributions of pion LCDAs. However, the infrared subtractions beyond the leading order in αs [17] are much more complex than that in the QCD factorization and a complete understanding of the TMD factorization for exclusive processes with large momentum transfer has not been achieved to date on the conceptual side.The lattice QCD simulation for the heavy-to-light form factors is valid in the small recoil region; its predictions need to be extrapolated to the whole kinematic region from the small hadronic recoil of the light meson through special models. The extrapolation-model dependence will produce unavoidable uncertainties in the determination of observations, especially at large recoil. Therefore, to obtain theq2 (invariant mass of lepton pair in the semileptonic decays) dependence of the form factors with high accuracy, it is important to compute the form factors in the large hadronic recoil region directly. The light-cone sum rule (LCSR) approach, which is a combination of the SVZ sum rules with the QCD theory of hard exclusive processes, provides an appropriate method to evaluate the form factors at large recoil. According to the correlation functions employed in the calculation, two different frameworks of LCSR, i.e., LCSR with the light-meson LCDAs [18–25] and LCSR with the B-meson LCDAs (we will also call this the B-meson LCSR) [26–29], have been established. The advantage of LCSR with light-meson LCDAs is that it can be applied to a larger region of the square of the momentum transfer and also the LCDAs of the light meson are better determined than that of the B-meson. Meanwhile, the B-meson LCSR has its unique advantage that the input LCDAs are universal for all the B⟶M form factors, and the theoretical uncertainty can be sizeably reduced in the calculation of ratios of the form factors. The LCSR for the SCETI matrix elements entering the QCD factorization formulae of heavy-to-light B-meson decay form factors, which is also called the SCET improved sum rules, has been achieved [30, 31] employing the vacuum-to-B-meson correlation functions. To improve the accuracy of the B⟶PP=π,K,D form factors, the next-to-leading order (NLO) corrections to the correlation functions in the strong coupling αs have been performed in [32–35], where the factorization formulae for the correlation functions were established with the diagrammatic approach and the strategy of regions [36, 37]. With a similar method, the baryonic transition Λb⟶Λ form factors have also been investigated [38]. The B⟶V form factors have also been evaluated with the SCET improved sum rules, and the NLO correction to the hard-collinear function is performed [39]. Except for the QCD corrections, the power suppressed contributions also play an important role in the precise calculation of the heavy-to-light form factors. Since it is a notoriously difficult task to perform a complete power expansion on a physical amplitude, one has to deal with some special kinds of power corrections with respect to the leading power contribution so far. The power suppressed contributions from the high-twist LCDAs of B-meson to the B⟶P and B⟶V form factors were calculated in [35, 39, 40].The fundamental nonperturbative inputs in theB-meson LCSR are the LCDAs of the B-meson. Since they are nonperturbative in nature, the hard effect must be integrated out in the definition of the LCDAs. Therefore, one has to employ the effective bottom quark field in the heavy quark effective theory (HQET) to construct the matrix elements of the nonlocal operators [41]. As a soft objective, the B-meson LCDAs do not have a definite twist in principle, while when entering a process with large momentum transfer and thus only one of the light-cone components of the momentum of the soft light quark would be picked up, the twist of the B-meson LCDAs makes sense. The scale dependence of the LCDAs of the B-meson can be obtained by the renormalization group (RG) equation approach, and the RG equation for the leading-twist LCDA has been derived at two-loop level [42, 43], and the one-loop level evolution equations for higher-twist LCDAs are also known [44]. The evolution behavior of the LCDAs can provide constraints on the model of the LCDAs; in addition, it also plays an important role in proving the factorization-scale independence of the factorization formula.The plan of this review is as follows. In Section2, we will discuss the LCDAs of the B-meson, including the definition, the evolution behavior, and the phenomenological models of the leading-twist and higher-twist LCDAs. In Sections 3 and 4, we will introduce the LCSR with B-meson LCDAs, emphasizing the two equivalent methods to evaluate the QCD corrections to the correlation functions, i.e., the method of regions and the matching between the SCETI and SCETII. In Section 5, we will discuss the power suppressed contributions to the heavy-to-light form factors with LCSR, concentrating on the contributions from high-twist B-meson LCDAs. In Section 6, we will present some numerical results of the form factors and the phenomenological applications. We summarize in the last section.


## 2. The Light-Cone Distribution Amplitudes of theB-Meson

### 2.1. Definition of the Leading-TwistB-Meson LCDAs

TheB-meson LCDAs are among the most important ingredients of the QCD factorization formula for exclusive B decays. Before talking about the heavy-to-light form factors, we would like to introduce the LCDAs of the B-meson. The two-particle LCDAs of the B-meson in the HQET can be obtained from the coordinate-space matrix elements [41]:
(1)0q¯βxx,0hvα0B¯v=−if~BμmB41+υ22Φ+t,x2+Φ−t,x2−Φ+t,x2txαβ,where t=v·x. The LCDAs Φ±t,x2 can be expanded around x2=0. In the limit x2⟶0, we assume t⟶z=n·x/2; then, the B-meson LCDAs in the momentum space are defined through the Fourier transformation:
(2)ϕ±ω=∫dz2πeiωzΦ±z,Φ±z≡Φ±z,0.For convenience, we introduce two light-like vectorsn and n¯ satisfying n·n¯=2,n2=n¯2=0. Any four-vector kμ can be expressed as kμ=n·kn¯μ/2+n¯·knμ/2+k⊥μ.When the LCDAs of theB-meson are applied in the calculation of B-meson decay processes, the momentum space projector is usually required. To obtain the projector, we adopt the reference frame satisfying v=n+n¯/2 and assume n¯·x≪x⊥≪n·x, thus x≃zn¯+x⊥, then
(3)Φ+t,x2+Φ−t,x2−Φ+t,x2tx≃Φ+zn+Φ−zn¯+Φ−z−Φ+zzx⊥.After the Fourier transform, the projector ofB-meson LCDAs is obtained as [8]
(4)MBω=−if~BμmB41+v2ϕ+ωn+ϕ−ωn¯−∫0ωdηϕ−η−ϕ+ηγ⊥μ∂∂k⊥μγ5,with k is the momentum of the antilight quark in the B-meson.The LCDAsΦ± defined above do not contribute at the same power in the QCD factorizations of the heavy hadron decay processes, since they do not have the same collinear twist (in the following, we call “twist” for short). The twist t and conformal spin j of the light quark and gluon fields are given by the usual expressions [45]:
(5)t=d−s,j=12d+s,where d is the canonical dimension and s is the spin projection on the light cone. The twist and conformal spin are closely related to the collinear subgroup of the conformal group. The collinear subgroup of the conformal group is locally equivalent to the SL2,R group. The subgroup contains four generators, namely, P+,M−+,D, and K−, where P+ and M−+ are the projections of the generator of the Poincare group pμ,Mνμ on the light cone, D is the generator of dilatation, and K− is the generator of the special conformal transformation along the light cone [45]. Specifically, if we write the light-like vector x=αn, then the dilatation indicates the transform α⟶α′=λα, and for the special conformal transformation,
(6)α⟶α′=aα+bcα+d,ad−bc=1.The four generators of the collinear subgroup of the conformal group can be rearranged to form the algebra ofSL2,R, i.e.,
(7)J+=J1+iJ2=−iP+,J−=J1−iJ2=i2K−,J0=i2D+M∓,E=i2D−M−+,with
(8)J0,J∓=∓J∓,J−,J+=−2J0.For the quantized fieldΦα, we have
(9)J+,Φα=−∂αΦα,J−,Φα=α2∂α+2jαΦα,J0,Φα=α∂α+jΦα,E,Φα=12ℓ−sΦα.For a spinor fieldΨ, the projectors Ψ+=nn¯/4Ψ and Ψ−=n¯n/4Ψ have definite twist, namely, tΨ±=±1. In an appropriate reference frame, Ψ± only have two nonzero components; thus, it is more convenient to define these fields with two-component spinors. Any light-like vector can be represented by a product of two spinors. One can write
(10)nαα̇=nμσαα̇μ=λαλ¯α̇,n¯αα̇=n¯μσαα̇μ=μαμ¯α̇,where for the auxiliary λ and μ spinors, λ¯=λ†, μ¯=μ† which satisfy λμ=λαμα=2, μ¯λ¯=μ¯α̇λ¯α̇=2. The “+” and “–” fields are defined as
(11)χ+=λαψα,ψ¯+=λ¯α̇ψα̇,f++=λαλβfαβ,f±=λαμβfαβ,f¯++=λ¯α̇λ¯β̇f¯α̇β̇,etc. The Dirac (antiquark) spinors
(12)q=ψαχ¯β̇,q¯=χβ,ψ¯α̇are written in terms of the following two component fields:
(13)λμχα=μαχ+−λαχ−,μ¯λ¯ψ¯α̇=μ¯α̇ψ¯+−λ¯α̇ψ¯−.The large component of the heavy quark field in the HQET satisfiesvhv=hv; then,
(14)h+=−h¯−,h−=h¯+.The gluon strength tensorFμν can be decomposed as
(15)Fαβ,α̇β̇=σαα̇μσββ̇νFμν=2ϵα̇β̇fαβ−ϵαβf¯α̇β̇,iF~αβ,α̇β̇=σαα̇μσββ̇νiF~μν=2ϵα̇β̇fαβ+ϵαβf¯α̇β̇.Here,fαβ and f¯α̇β̇ are chiral and antichiral symmetric tensors, f∗=f¯, which belong to 1,0 and 0,1 representations of the Lorentz group, respectively. It is easy to see that the twist of the field ψ+ and χ+ is 1, the twist of ψ− and χ− is 2, and we assign the twist of h± to be 1. Employing the relevant operators with definite twist in spinor notation, one obtains
(16)f~BμmBΦ+z=i0ψ¯+zh+0−χ+zh¯+0B¯v,f~BμmBΦ−z=i0ψ¯−zh−0−χ−zh¯−0B¯v.


### 2.2. Evolution of the Leading-TwistB-Meson LCDA

At leading power, onlyϕ+ω is relevant in the factorization formula of various B-meson decay processes, and the RG equation of ϕ+ω up to the leading-logarithmic (LL) accuracy is the well-known Lange-Neubert (LN) equation [42], which reads
(17)ddlnμϕ+ω,μ=−∫0∞dω′Γ+ω,ω′,μϕ+ω′,μ,where μ is the renormalization scale. The anomalous dimensions are
(18)Γ+ω,ω′,μ=Γcusplnμω+γ+δω−ω′+ωΓcuspΓω,ω′,Γcusp0=4CF,Γcusp1=4CF673−π2−109nf,γ+0=−2CF,Γω,ω′=−θω′−ωω′ω′−ω+θω−ω′ωω−ω′+,∫0∞dyfx,y+gy=∫0∞dyfx,ygy−gx,where the anomalous dimensions are expanded in the same way as
(19)γ+=∑n=0γ+nan+1,witha=αs4π.Since the evolution equation of the leading-twistB-meson LCDA is the integrodifferential equation, it is difficult to obtain the solution directly. A commonly used method is to simplify the evolution equation by an integral transformation. There exist several kinds of integral transformations which are helpful to work out the solution of the evolution equation. It was found that the evolution kernel is diagonalized when it is transformed into the so-called “dual” space [46]. The leading-twist LCDA in the dual space can be obtained by
(20)ρ+ω′,μ=∫0∞dωωωω′J12ωω′ϕ+ω,μ,which satisfies an ordinary differential equation:
(21)ddlnμρ+ω′,μ=−Γcusplnμe−2γEω′+γ+ρ+ω′,μ.The evolution equation can also be calculated in the position space, where it takes the form [47]
(22)ddlnμΦ+z,μ=−Γcusplnizμ~+γ~+Φ+z,μ−Γcusp∫01duu¯uΦ+z,μ−Φ+u¯z,μ,where μ~=μeγE, u¯=1−u, and γ~+0=2CF. This equation can be related to the LN equation by a Fourier transform. Performing the Mellin transformation to the evolution equation in the position space [47, 48]:
(23)φ+j,μ=12πi∫−i0−i∞dzzizμ~−jΦ+z,μ,the evolution equation can also be diagonalized and easily solved. The different kinds of integral transform mentioned above are equivalent, and the LCDAs ϕ+ω, Φ+z, φ+j, and ρ+ω′ are different expressions of an identical objective. Because the momentum space and the position space are related through a standard Fourier transformation, we are able to derive
(24)φ+j,μ=Γ−j2πi∫0∞dωωμ~jϕ+ω,μ,φ+j,μ=μ~2πiΓ2+j∫0∞dω′ω′ρ+ω′,μμ~ω′−1−j.At the one-loop level, the most convenient method is to work in the dual space since the Bessel function is the eigenfunction of the LN kernel, which is confirmed in [49, 50]. The LN kernel can be expressed as a logarithm of the generator of special conformal transformations along the light cone. When the eigenfunction of the generator is transformed to the momentum space, it is simply the Bessel function in (20).The two-loop level anomalous dimension of theB-meson LCDA was first calculated in the coordinate space in [43]
(25)ddlnμΦ+z,μ=−Γcusplnizμ~+γ~+Φ+z,μ−Γcusp∫01duu¯u1+ahuΦ+z,μ−Φ+u¯z,μ,where
(26)γ~+1=CF4CF2+π2−6ζ3+6629−356π2−18ζ3−nf8027−19π2,Γcusp2=CF1470−536π23+44π45+264ζ3+nf−12769+80π29−2083ζ3−1627nf2,hu=lnu¯11−23nf+2CFlnu¯−1+u¯u¯lnu−32.This equation has also been transformed into the momentum space in [51], resulting in the two-loop level LN equation (17) with the integral kernel modified as
(27)Γ+ω,ω′,μ=Γcusplnμω+γ+δω−ω′+ωΓcuspΓω,ω′+γ^+ω,ω′.The anomalous dimensions up to two loops are(28)γ+=γ~+−Γcusp1−aβ01−π26−CF3−π26,γ^+ω,ω′=−a4CFωθω′−ωω′ω′−ωhωω′.The advantage of solving the evolution equation at the two-loop level in the dual space does not hold since the two-loop evolution kernel is not diagonal in this space. On the contrary, the elegant form of the RG equation in the Mellin space is maintained:(29)ddlnμ+V+j,αsφ+j,μ=0,with
(30)V+j,αs=j−γ~++Γcuspψj+2−ψ2+ϑj,where
(31)ϑj=aβ0−3CFψ′j+2−ψ′2+2CF1j+13+ψ′j+2ψj+2−ψ1+ψ′j+1ψj+1−ψ1−π26.The solution in the Mellin space is then obtained directly [48]
(32)φ+jμ,αsμ,μ=φ+jμ0,αsμ0,μ0exp−∫μ0μdssV+js,αss.In a recent paper [52], an alternative approach to solving the evolution equation at two-loop level was proposed. The essential idea of this approach is to perform a Laplace transformation to the B-meson LCDA:
(33)ϕ~+η,μ=∫0∞dωωωω¯−ηϕ+ω,μ,where ω¯ is a fixed reference scale, which can be used to eliminate the logarithmic moment σ1 in the factorization formula of B⟶γℓν¯ℓ. We note that the LCDA φ+j is related to ϕ~+η through [53]
(34)φ+j,μ=Γ−j2πieγEμω¯eγEμj+1ϕ~+−j−1,μ.Then, one could derive the RG equation forϕ~+ and solve the evolution equation directly [52, 53].


### 2.3. Higher-TwistB-Meson LCDAs

Power corrections to theB-meson decay processes are of great importance, and higher-twist B-meson LCDAs provide one kind of important power suppressed contributions. The LCDA ϕ− defined in the previous subsection is of twist-3, and it is suppressed due to different components of the quark field in the definition of LCDAs. Besides, the additional gluon or quark fields will also give rise to higher-twist LCDAs. Compared with two-particle LCDAs, the three-particle quark-gluon LCDAs are more numerous. There exist eight independent Lorentz structures [54], and they can be defined as
(35)0q¯n¯z1n¯z1,n¯z2gGμνn¯z2n¯z2,0Γhv0B¯v==12f~BμmBTrγ5ΓP+vμγν−vνγμΨA−ΨV−iσμνΨV−n¯μvν−n¯νvμXA+n¯μγν−n¯νγμW+YA−iϵμναβn¯αvβγ5X~A+iϵμναβn¯αγβγ5Y~A−n¯μvν−n¯νvμn¯W+n¯μγν−n¯νγμn¯Zz1,z2;μ,where the totally antisymmetric tensor ε0123=1, the covariant derivative is defined as Dμ=∂μ−igAμ, and the dual gluon strength tensor is G~μν=1/2ϵμναβGαβ. The momentum space distributions are defined through Fourier transformations:
(36)ΨAz1,z2=∫0∞dω1∫0∞dω2e−iω1z1−iω2z2ψAω1,ω2,and similarly for the other functions. The LCDAs defined above do not have definite twist. In order to construct the LCDAs with definite twist, one can take advantage of the two-component spinors as follows:
(37)2f~BμmBΦ3z1,z2;μ=−0χ+z1f¯++z2h+0+ψ¯+z1f++z2h¯+0B¯v,2f~BμmBΦ4z1,z2;μ=0χ−z1f++z2h−0+ψ¯−z1f¯++z2h¯−0B¯v,f~BμmBΨ4+Ψ~4z1,z2;μ=−0χ+z1f+−z2h−0+ψ¯+z1f¯+−z2h¯−0B¯v,f~BμmBΨ4−Ψ~4z1,z2;μ=−0χ+z1f¯+−z2h−0+ψ¯+z1f+−z2h¯−0B¯v,2f~BμmBΦ5z1,z2;μ=0χ+z1f−−z2h+0+ψ¯+z1f¯−−z2h¯+0B¯v,f~BμmBΨ5+Ψ~5z1,z2;μ=0χ−z1f+−z2h+0+ψ¯−z1f¯+−z2h¯+0B¯v,f~BμmBΨ5−Ψ~5z1,z2;μ=0χ−z1f¯+−z2h+0+ψ¯−z1f+−z2h¯+0B¯v,2f~BμmBΦ6z1,z2;μ=0χ−z1f¯−−z2h−0+ψ¯−z1f−−z2h¯−0B¯v,where fαβ and f¯α̇β̇ are chiral and antichiral symmetric tensors, and the twist of their light-cone components satisfies: tf++=tf¯++=1, tf+−=tf¯+−=2, and tf−−=tf¯−−=3. This eight invariant function is related to the B-meson higher-twist LCDAs:
(38)Φ3=ΨA−ΨV,Φ4=ΨA+ΨV,Ψ4=ΨA+XV,Ψ~4=ΦA−X~A,Ψ5=−ΨA+XA−2YA,Φ5=ΨA+ΨV+2YA−2Y~A+2W,Ψ~5=−ΨV−X~A+2Y~A,Φ6=ΨA−ΨV+2YA+2Y~A+2W−4Z.Except for the higher Fock state, the higher twist also arises from the nonvanishing parton transverse momenta (or virtuality). The twist-4 and twist-5 two-particleB-meson LCDAs can be defined as
(39)0q¯xΓx,0hv0B¯v=−i2f~BμmBTrγ5ΓP+∫0∞dωe−iωvxϕ+ω+x2g+ω+i4f~BμmBTrγ5ΓP+x1vx∫0∞dωe−iωvxϕ+−ϕ−ω+x2g+−g−ω,where we have to assume x2≪1/ΛQCD2; thus, (39) can be understood as a light-cone expansion to the tree-level accuracy. This definition contains the constraints
(40)∫0∞dωϕ+ω−ϕ−ω=0,∫0∞dωg+ω−g−ω=0.From QCD equation of motion (EOM), one can derive the following relations among the LCDAs [55] (the last two relations in (41a) follow from the expressions given in [55] by simple algebra).
(41a)zddz+1Φ−z=Φ+z+2z2∫01uduΦ3z,uz,2z2G+z=−zddz−12+izΛ¯Φ+z−12Φ−z−z2∫01u¯duΨ4z,uz≡2z2G^+z−z2∫01u¯duΨ4z,uz,2z2G−z=−zddz−12+izΛ¯Φ−z−12Φ+z−z2∫01u¯duΨ5z,uz≡2z2G^−z−z2∫01u¯duΨ5z,uz,Φ−z=zddz+1+2izΛ¯Φ+z+2z2∫01duuΦ4z,uz+Ψ4z,uz,where
(42)G±z,μ=∫0∞dωe−iωzg±ω,μ,Λ¯=mB−mb.With the above relations, one can calculate the LCDAsG± with the leading-twist and three-particle higher-twist LCDAs.


### 2.4. The Phenomenological Models

Different from the LCDAs of light mesons, which can be expanded in terms of Gagenbauer polynomials and the corresponding Gagenbauer moments can be calculated by Lattice or QCD sum rules since they are determined by the matrix elements of local operators, the LCDAs of theB meson are more difficult to be modeled. The evolution of the B-meson can provide some model-independent constraints to the behavior of the leading-twist LCDA of the B-meson. In the large ω region, the operator product expansion (OPE) can be employed to explore the model-independent properties of the LCDA. The method is to calculate the first several moments of the distribution amplitude, derive its asymptotic behavior, and study its properties under RG evolution; then, the constraints on the LCDAs can be found. The result indicates that at large ω, the LCDA falls off faster than 1/ω. In the low ω region, the behavior of the LCDA cannot be constrained by the perturbative QCD and only can be modeled with a nonperturbative method. In practice, the B-meson LCDA is usually applied to the decay processes of the B meson; therefore, the spectator is regarded as a soft quark, and low ω region behavior is more important. Hereafter, we introduce some commonly used models.The asymptotic behavior of the LCDAs at the small quark and glue momenta is relative to the conformal spins of the quark and gluon field:(43)ϕω1,ω2~ω12j1−1ω22j2−1,ϕ∈ϕ3,ϕ4,ψ4⋯.This relation can be obtained from the correlation function of the light-ray operators and suitable local current [56]. Several models for the two-particle and three-particle LCDAs to the twist-four accuracy have been given with a more general ansatz in [57], such as the exponential model, the free parton model, and the local duality model. Similarly, we can obtain all the LCDA models in accord with the correct low-momentum behavior [56] and EOM constrains (tree level):
(44)ϕ+ω=ωfω,ϕ−ω=∫ω∞dρfρ+16ϰλE2−λH2ω2f′ω+4ωfω−2∫ω∞dρfρ,g−ω=14∫ω∞dxx−ωϕ+ω−ϕ−ω−2Λ¯−xϕ−ω−12∫0ωdω1∫ω−ω1∞dω21ω21−ω−ω1ω2ψ5ω1,ω2,ϕ3ω1,ω2=−12ϰλE2−λH2ω1ω22f′ω1+ω2,ϕ4ω1,ω2=12ϰλE2+λH2ω22fω1+ω2,ψ4ω1,ω2=ϰλE2ω1ω2fω1+ω2,ϕ5ω1,ω2=ϰλE2+λH2ω1∫ω1+ω2∞dωfω,ψ5ω1,ω2=ϰλE2ω2∫ω1+ω2∞dωfω,ψ~5ω1,ω2=ϰλH2ω2∫ω1+ω2∞dωfω,ϕ6ω1,ω2=ϰλE2−λH2∫ω1+ω2∞dω∫ω∞dω′fω′,where the normalization constant
(45)ϰ−1=12∫0∞ω3fωdω.It is clear that this model is in agreement with the local duality model and the exponential model in [35]. The parameters λE and λH in HQET are defined as the hadronic matrix element for both the light and the heavy quarks with EOM constraints:
(46)0q¯gsGμνΓhvB¯v=−f~BmB6Tr1+v2iσμνλH2+vμγν−vνγμλH2−λE2γ5Γ.The renormalization scale dependence ofλE and λH can be obtained by solving the RG equation at the one-loop order:
(47)λE2μλH2μ=V^αsμαsμ0γi0/2β0diagV∧−1λE2μ0λH2μ0.Here, we employ the three-parameter model ofϕ+ω,μ0=ωf3pω,μ0 where
(48)f3pω,μ0=ΓβΓα1ω02e−ω/ω0Uβ−α,3−α,ωω0.The three parametersα,β, and ω0 determine the model of B-meson LCDAs at the initial scale μ=1.0 GeV. The values of the logarithmic moments λB, σ^1, and σ^2 are needed, which are defined as [57]
(49)σ^nμ=∫0∞dωλBωlnnλBe−γEωϕ+ω,μ,where λB is defined by σ^0=1. In the terms of the three parameters ω0,α,andβ, we obtain with a short calculation
(50)λB=α−1β−1ω0,σ^1=ψβ−1−ψα−1+lnα−1β−1,σ^2=π26+ψβ−1−ψα−1+lnα−1β−12−ψ′β−1−ψ′α−1.


## 2.1. Definition of the Leading-TwistB-Meson LCDAs

TheB-meson LCDAs are among the most important ingredients of the QCD factorization formula for exclusive B decays. Before talking about the heavy-to-light form factors, we would like to introduce the LCDAs of the B-meson. The two-particle LCDAs of the B-meson in the HQET can be obtained from the coordinate-space matrix elements [41]:
(1)0q¯βxx,0hvα0B¯v=−if~BμmB41+υ22Φ+t,x2+Φ−t,x2−Φ+t,x2txαβ,where t=v·x. The LCDAs Φ±t,x2 can be expanded around x2=0. In the limit x2⟶0, we assume t⟶z=n·x/2; then, the B-meson LCDAs in the momentum space are defined through the Fourier transformation:
(2)ϕ±ω=∫dz2πeiωzΦ±z,Φ±z≡Φ±z,0.For convenience, we introduce two light-like vectorsn and n¯ satisfying n·n¯=2,n2=n¯2=0. Any four-vector kμ can be expressed as kμ=n·kn¯μ/2+n¯·knμ/2+k⊥μ.When the LCDAs of theB-meson are applied in the calculation of B-meson decay processes, the momentum space projector is usually required. To obtain the projector, we adopt the reference frame satisfying v=n+n¯/2 and assume n¯·x≪x⊥≪n·x, thus x≃zn¯+x⊥, then
(3)Φ+t,x2+Φ−t,x2−Φ+t,x2tx≃Φ+zn+Φ−zn¯+Φ−z−Φ+zzx⊥.After the Fourier transform, the projector ofB-meson LCDAs is obtained as [8]
(4)MBω=−if~BμmB41+v2ϕ+ωn+ϕ−ωn¯−∫0ωdηϕ−η−ϕ+ηγ⊥μ∂∂k⊥μγ5,with k is the momentum of the antilight quark in the B-meson.The LCDAsΦ± defined above do not contribute at the same power in the QCD factorizations of the heavy hadron decay processes, since they do not have the same collinear twist (in the following, we call “twist” for short). The twist t and conformal spin j of the light quark and gluon fields are given by the usual expressions [45]:
(5)t=d−s,j=12d+s,where d is the canonical dimension and s is the spin projection on the light cone. The twist and conformal spin are closely related to the collinear subgroup of the conformal group. The collinear subgroup of the conformal group is locally equivalent to the SL2,R group. The subgroup contains four generators, namely, P+,M−+,D, and K−, where P+ and M−+ are the projections of the generator of the Poincare group pμ,Mνμ on the light cone, D is the generator of dilatation, and K− is the generator of the special conformal transformation along the light cone [45]. Specifically, if we write the light-like vector x=αn, then the dilatation indicates the transform α⟶α′=λα, and for the special conformal transformation,
(6)α⟶α′=aα+bcα+d,ad−bc=1.The four generators of the collinear subgroup of the conformal group can be rearranged to form the algebra ofSL2,R, i.e.,
(7)J+=J1+iJ2=−iP+,J−=J1−iJ2=i2K−,J0=i2D+M∓,E=i2D−M−+,with
(8)J0,J∓=∓J∓,J−,J+=−2J0.For the quantized fieldΦα, we have
(9)J+,Φα=−∂αΦα,J−,Φα=α2∂α+2jαΦα,J0,Φα=α∂α+jΦα,E,Φα=12ℓ−sΦα.For a spinor fieldΨ, the projectors Ψ+=nn¯/4Ψ and Ψ−=n¯n/4Ψ have definite twist, namely, tΨ±=±1. In an appropriate reference frame, Ψ± only have two nonzero components; thus, it is more convenient to define these fields with two-component spinors. Any light-like vector can be represented by a product of two spinors. One can write
(10)nαα̇=nμσαα̇μ=λαλ¯α̇,n¯αα̇=n¯μσαα̇μ=μαμ¯α̇,where for the auxiliary λ and μ spinors, λ¯=λ†, μ¯=μ† which satisfy λμ=λαμα=2, μ¯λ¯=μ¯α̇λ¯α̇=2. The “+” and “–” fields are defined as
(11)χ+=λαψα,ψ¯+=λ¯α̇ψα̇,f++=λαλβfαβ,f±=λαμβfαβ,f¯++=λ¯α̇λ¯β̇f¯α̇β̇,etc. The Dirac (antiquark) spinors
(12)q=ψαχ¯β̇,q¯=χβ,ψ¯α̇are written in terms of the following two component fields:
(13)λμχα=μαχ+−λαχ−,μ¯λ¯ψ¯α̇=μ¯α̇ψ¯+−λ¯α̇ψ¯−.The large component of the heavy quark field in the HQET satisfiesvhv=hv; then,
(14)h+=−h¯−,h−=h¯+.The gluon strength tensorFμν can be decomposed as
(15)Fαβ,α̇β̇=σαα̇μσββ̇νFμν=2ϵα̇β̇fαβ−ϵαβf¯α̇β̇,iF~αβ,α̇β̇=σαα̇μσββ̇νiF~μν=2ϵα̇β̇fαβ+ϵαβf¯α̇β̇.Here,fαβ and f¯α̇β̇ are chiral and antichiral symmetric tensors, f∗=f¯, which belong to 1,0 and 0,1 representations of the Lorentz group, respectively. It is easy to see that the twist of the field ψ+ and χ+ is 1, the twist of ψ− and χ− is 2, and we assign the twist of h± to be 1. Employing the relevant operators with definite twist in spinor notation, one obtains
(16)f~BμmBΦ+z=i0ψ¯+zh+0−χ+zh¯+0B¯v,f~BμmBΦ−z=i0ψ¯−zh−0−χ−zh¯−0B¯v.


## 2.2. Evolution of the Leading-TwistB-Meson LCDA

At leading power, onlyϕ+ω is relevant in the factorization formula of various B-meson decay processes, and the RG equation of ϕ+ω up to the leading-logarithmic (LL) accuracy is the well-known Lange-Neubert (LN) equation [42], which reads
(17)ddlnμϕ+ω,μ=−∫0∞dω′Γ+ω,ω′,μϕ+ω′,μ,where μ is the renormalization scale. The anomalous dimensions are
(18)Γ+ω,ω′,μ=Γcusplnμω+γ+δω−ω′+ωΓcuspΓω,ω′,Γcusp0=4CF,Γcusp1=4CF673−π2−109nf,γ+0=−2CF,Γω,ω′=−θω′−ωω′ω′−ω+θω−ω′ωω−ω′+,∫0∞dyfx,y+gy=∫0∞dyfx,ygy−gx,where the anomalous dimensions are expanded in the same way as
(19)γ+=∑n=0γ+nan+1,witha=αs4π.Since the evolution equation of the leading-twistB-meson LCDA is the integrodifferential equation, it is difficult to obtain the solution directly. A commonly used method is to simplify the evolution equation by an integral transformation. There exist several kinds of integral transformations which are helpful to work out the solution of the evolution equation. It was found that the evolution kernel is diagonalized when it is transformed into the so-called “dual” space [46]. The leading-twist LCDA in the dual space can be obtained by
(20)ρ+ω′,μ=∫0∞dωωωω′J12ωω′ϕ+ω,μ,which satisfies an ordinary differential equation:
(21)ddlnμρ+ω′,μ=−Γcusplnμe−2γEω′+γ+ρ+ω′,μ.The evolution equation can also be calculated in the position space, where it takes the form [47]
(22)ddlnμΦ+z,μ=−Γcusplnizμ~+γ~+Φ+z,μ−Γcusp∫01duu¯uΦ+z,μ−Φ+u¯z,μ,where μ~=μeγE, u¯=1−u, and γ~+0=2CF. This equation can be related to the LN equation by a Fourier transform. Performing the Mellin transformation to the evolution equation in the position space [47, 48]:
(23)φ+j,μ=12πi∫−i0−i∞dzzizμ~−jΦ+z,μ,the evolution equation can also be diagonalized and easily solved. The different kinds of integral transform mentioned above are equivalent, and the LCDAs ϕ+ω, Φ+z, φ+j, and ρ+ω′ are different expressions of an identical objective. Because the momentum space and the position space are related through a standard Fourier transformation, we are able to derive
(24)φ+j,μ=Γ−j2πi∫0∞dωωμ~jϕ+ω,μ,φ+j,μ=μ~2πiΓ2+j∫0∞dω′ω′ρ+ω′,μμ~ω′−1−j.At the one-loop level, the most convenient method is to work in the dual space since the Bessel function is the eigenfunction of the LN kernel, which is confirmed in [49, 50]. The LN kernel can be expressed as a logarithm of the generator of special conformal transformations along the light cone. When the eigenfunction of the generator is transformed to the momentum space, it is simply the Bessel function in (20).The two-loop level anomalous dimension of theB-meson LCDA was first calculated in the coordinate space in [43]
(25)ddlnμΦ+z,μ=−Γcusplnizμ~+γ~+Φ+z,μ−Γcusp∫01duu¯u1+ahuΦ+z,μ−Φ+u¯z,μ,where
(26)γ~+1=CF4CF2+π2−6ζ3+6629−356π2−18ζ3−nf8027−19π2,Γcusp2=CF1470−536π23+44π45+264ζ3+nf−12769+80π29−2083ζ3−1627nf2,hu=lnu¯11−23nf+2CFlnu¯−1+u¯u¯lnu−32.This equation has also been transformed into the momentum space in [51], resulting in the two-loop level LN equation (17) with the integral kernel modified as
(27)Γ+ω,ω′,μ=Γcusplnμω+γ+δω−ω′+ωΓcuspΓω,ω′+γ^+ω,ω′.The anomalous dimensions up to two loops are(28)γ+=γ~+−Γcusp1−aβ01−π26−CF3−π26,γ^+ω,ω′=−a4CFωθω′−ωω′ω′−ωhωω′.The advantage of solving the evolution equation at the two-loop level in the dual space does not hold since the two-loop evolution kernel is not diagonal in this space. On the contrary, the elegant form of the RG equation in the Mellin space is maintained:(29)ddlnμ+V+j,αsφ+j,μ=0,with
(30)V+j,αs=j−γ~++Γcuspψj+2−ψ2+ϑj,where
(31)ϑj=aβ0−3CFψ′j+2−ψ′2+2CF1j+13+ψ′j+2ψj+2−ψ1+ψ′j+1ψj+1−ψ1−π26.The solution in the Mellin space is then obtained directly [48]
(32)φ+jμ,αsμ,μ=φ+jμ0,αsμ0,μ0exp−∫μ0μdssV+js,αss.In a recent paper [52], an alternative approach to solving the evolution equation at two-loop level was proposed. The essential idea of this approach is to perform a Laplace transformation to the B-meson LCDA:
(33)ϕ~+η,μ=∫0∞dωωωω¯−ηϕ+ω,μ,where ω¯ is a fixed reference scale, which can be used to eliminate the logarithmic moment σ1 in the factorization formula of B⟶γℓν¯ℓ. We note that the LCDA φ+j is related to ϕ~+η through [53]
(34)φ+j,μ=Γ−j2πieγEμω¯eγEμj+1ϕ~+−j−1,μ.Then, one could derive the RG equation forϕ~+ and solve the evolution equation directly [52, 53].


## 2.3. Higher-TwistB-Meson LCDAs

Power corrections to theB-meson decay processes are of great importance, and higher-twist B-meson LCDAs provide one kind of important power suppressed contributions. The LCDA ϕ− defined in the previous subsection is of twist-3, and it is suppressed due to different components of the quark field in the definition of LCDAs. Besides, the additional gluon or quark fields will also give rise to higher-twist LCDAs. Compared with two-particle LCDAs, the three-particle quark-gluon LCDAs are more numerous. There exist eight independent Lorentz structures [54], and they can be defined as
(35)0q¯n¯z1n¯z1,n¯z2gGμνn¯z2n¯z2,0Γhv0B¯v==12f~BμmBTrγ5ΓP+vμγν−vνγμΨA−ΨV−iσμνΨV−n¯μvν−n¯νvμXA+n¯μγν−n¯νγμW+YA−iϵμναβn¯αvβγ5X~A+iϵμναβn¯αγβγ5Y~A−n¯μvν−n¯νvμn¯W+n¯μγν−n¯νγμn¯Zz1,z2;μ,where the totally antisymmetric tensor ε0123=1, the covariant derivative is defined as Dμ=∂μ−igAμ, and the dual gluon strength tensor is G~μν=1/2ϵμναβGαβ. The momentum space distributions are defined through Fourier transformations:
(36)ΨAz1,z2=∫0∞dω1∫0∞dω2e−iω1z1−iω2z2ψAω1,ω2,and similarly for the other functions. The LCDAs defined above do not have definite twist. In order to construct the LCDAs with definite twist, one can take advantage of the two-component spinors as follows:
(37)2f~BμmBΦ3z1,z2;μ=−0χ+z1f¯++z2h+0+ψ¯+z1f++z2h¯+0B¯v,2f~BμmBΦ4z1,z2;μ=0χ−z1f++z2h−0+ψ¯−z1f¯++z2h¯−0B¯v,f~BμmBΨ4+Ψ~4z1,z2;μ=−0χ+z1f+−z2h−0+ψ¯+z1f¯+−z2h¯−0B¯v,f~BμmBΨ4−Ψ~4z1,z2;μ=−0χ+z1f¯+−z2h−0+ψ¯+z1f+−z2h¯−0B¯v,2f~BμmBΦ5z1,z2;μ=0χ+z1f−−z2h+0+ψ¯+z1f¯−−z2h¯+0B¯v,f~BμmBΨ5+Ψ~5z1,z2;μ=0χ−z1f+−z2h+0+ψ¯−z1f¯+−z2h¯+0B¯v,f~BμmBΨ5−Ψ~5z1,z2;μ=0χ−z1f¯+−z2h+0+ψ¯−z1f+−z2h¯+0B¯v,2f~BμmBΦ6z1,z2;μ=0χ−z1f¯−−z2h−0+ψ¯−z1f−−z2h¯−0B¯v,where fαβ and f¯α̇β̇ are chiral and antichiral symmetric tensors, and the twist of their light-cone components satisfies: tf++=tf¯++=1, tf+−=tf¯+−=2, and tf−−=tf¯−−=3. This eight invariant function is related to the B-meson higher-twist LCDAs:
(38)Φ3=ΨA−ΨV,Φ4=ΨA+ΨV,Ψ4=ΨA+XV,Ψ~4=ΦA−X~A,Ψ5=−ΨA+XA−2YA,Φ5=ΨA+ΨV+2YA−2Y~A+2W,Ψ~5=−ΨV−X~A+2Y~A,Φ6=ΨA−ΨV+2YA+2Y~A+2W−4Z.Except for the higher Fock state, the higher twist also arises from the nonvanishing parton transverse momenta (or virtuality). The twist-4 and twist-5 two-particleB-meson LCDAs can be defined as
(39)0q¯xΓx,0hv0B¯v=−i2f~BμmBTrγ5ΓP+∫0∞dωe−iωvxϕ+ω+x2g+ω+i4f~BμmBTrγ5ΓP+x1vx∫0∞dωe−iωvxϕ+−ϕ−ω+x2g+−g−ω,where we have to assume x2≪1/ΛQCD2; thus, (39) can be understood as a light-cone expansion to the tree-level accuracy. This definition contains the constraints
(40)∫0∞dωϕ+ω−ϕ−ω=0,∫0∞dωg+ω−g−ω=0.From QCD equation of motion (EOM), one can derive the following relations among the LCDAs [55] (the last two relations in (41a) follow from the expressions given in [55] by simple algebra).
(41a)zddz+1Φ−z=Φ+z+2z2∫01uduΦ3z,uz,2z2G+z=−zddz−12+izΛ¯Φ+z−12Φ−z−z2∫01u¯duΨ4z,uz≡2z2G^+z−z2∫01u¯duΨ4z,uz,2z2G−z=−zddz−12+izΛ¯Φ−z−12Φ+z−z2∫01u¯duΨ5z,uz≡2z2G^−z−z2∫01u¯duΨ5z,uz,Φ−z=zddz+1+2izΛ¯Φ+z+2z2∫01duuΦ4z,uz+Ψ4z,uz,where
(42)G±z,μ=∫0∞dωe−iωzg±ω,μ,Λ¯=mB−mb.With the above relations, one can calculate the LCDAsG± with the leading-twist and three-particle higher-twist LCDAs.


## 2.4. The Phenomenological Models

Different from the LCDAs of light mesons, which can be expanded in terms of Gagenbauer polynomials and the corresponding Gagenbauer moments can be calculated by Lattice or QCD sum rules since they are determined by the matrix elements of local operators, the LCDAs of theB meson are more difficult to be modeled. The evolution of the B-meson can provide some model-independent constraints to the behavior of the leading-twist LCDA of the B-meson. In the large ω region, the operator product expansion (OPE) can be employed to explore the model-independent properties of the LCDA. The method is to calculate the first several moments of the distribution amplitude, derive its asymptotic behavior, and study its properties under RG evolution; then, the constraints on the LCDAs can be found. The result indicates that at large ω, the LCDA falls off faster than 1/ω. In the low ω region, the behavior of the LCDA cannot be constrained by the perturbative QCD and only can be modeled with a nonperturbative method. In practice, the B-meson LCDA is usually applied to the decay processes of the B meson; therefore, the spectator is regarded as a soft quark, and low ω region behavior is more important. Hereafter, we introduce some commonly used models.The asymptotic behavior of the LCDAs at the small quark and glue momenta is relative to the conformal spins of the quark and gluon field:(43)ϕω1,ω2~ω12j1−1ω22j2−1,ϕ∈ϕ3,ϕ4,ψ4⋯.This relation can be obtained from the correlation function of the light-ray operators and suitable local current [56]. Several models for the two-particle and three-particle LCDAs to the twist-four accuracy have been given with a more general ansatz in [57], such as the exponential model, the free parton model, and the local duality model. Similarly, we can obtain all the LCDA models in accord with the correct low-momentum behavior [56] and EOM constrains (tree level):
(44)ϕ+ω=ωfω,ϕ−ω=∫ω∞dρfρ+16ϰλE2−λH2ω2f′ω+4ωfω−2∫ω∞dρfρ,g−ω=14∫ω∞dxx−ωϕ+ω−ϕ−ω−2Λ¯−xϕ−ω−12∫0ωdω1∫ω−ω1∞dω21ω21−ω−ω1ω2ψ5ω1,ω2,ϕ3ω1,ω2=−12ϰλE2−λH2ω1ω22f′ω1+ω2,ϕ4ω1,ω2=12ϰλE2+λH2ω22fω1+ω2,ψ4ω1,ω2=ϰλE2ω1ω2fω1+ω2,ϕ5ω1,ω2=ϰλE2+λH2ω1∫ω1+ω2∞dωfω,ψ5ω1,ω2=ϰλE2ω2∫ω1+ω2∞dωfω,ψ~5ω1,ω2=ϰλH2ω2∫ω1+ω2∞dωfω,ϕ6ω1,ω2=ϰλE2−λH2∫ω1+ω2∞dω∫ω∞dω′fω′,where the normalization constant
(45)ϰ−1=12∫0∞ω3fωdω.It is clear that this model is in agreement with the local duality model and the exponential model in [35]. The parameters λE and λH in HQET are defined as the hadronic matrix element for both the light and the heavy quarks with EOM constraints:
(46)0q¯gsGμνΓhvB¯v=−f~BmB6Tr1+v2iσμνλH2+vμγν−vνγμλH2−λE2γ5Γ.The renormalization scale dependence ofλE and λH can be obtained by solving the RG equation at the one-loop order:
(47)λE2μλH2μ=V^αsμαsμ0γi0/2β0diagV∧−1λE2μ0λH2μ0.Here, we employ the three-parameter model ofϕ+ω,μ0=ωf3pω,μ0 where
(48)f3pω,μ0=ΓβΓα1ω02e−ω/ω0Uβ−α,3−α,ωω0.The three parametersα,β, and ω0 determine the model of B-meson LCDAs at the initial scale μ=1.0 GeV. The values of the logarithmic moments λB, σ^1, and σ^2 are needed, which are defined as [57]
(49)σ^nμ=∫0∞dωλBωlnnλBe−γEωϕ+ω,μ,where λB is defined by σ^0=1. In the terms of the three parameters ω0,α,andβ, we obtain with a short calculation
(50)λB=α−1β−1ω0,σ^1=ψβ−1−ψα−1+lnα−1β−1,σ^2=π26+ψβ−1−ψα−1+lnα−1β−12−ψ′β−1−ψ′α−1.


## 3. LCSR withB-Meson LCDAs

The LCSR withB-meson LCDAs is widely employed to calculate the heavy-to-light form factors which receive the well-known end-point divergence in the QCD factorization. Starting from the QCD two-point correlation functions at the Euclidean space, the B-meson LCSR can avoid the end-point divergence with the help of the dispersion relation, the Borel transformation, and the quark-hadron duality. The heavy-to-light form factors can also be calculated with the SCET improved LCSR. In the following, we will calculate the B⟶P form factors and the B⟶V form factors with the B-meson LCSR and the SCET improved LCSR, respectively.


### 3.1. The Heavy-to-Light Form Factors

The form factors for theB-meson decays into a pseudoscalar meson are defined by the matrix elements of the weak transition current sandwiched between the B-meson and a pseudoscalar meson states, i.e.,
(51)Ppq¯γμbB¯pB=fB⟶P+q2pBμ+pμ−mB2−mP2q2qμ+fB⟶P0q2mB2−mP2q2qμ,Ppq¯σμνqνbB¯pB=ifB⟶PTq2mB+mPq2pBμ+pμ−mB2−mP2qμ,where mB and mP are, respectively, the masses of the B meson and the pseudoscalar meson and the momentum transfer q=pB−p. The relevant form factors for B decays into vector mesons are defined as
(52)Vp,ε∗q¯γμbB¯pB=2iVq2mB+mVϵμνρσεν∗pρpB,σ,Vp,ε∗q¯γμγ5bB¯pB=2mVA0q2ε∗·qq2qμ+mB+mVA1q2ε∗μ−ε∗·qq2qμ−A2q2ε∗·qmB+mVpBμ+pμ−mB2−mV2q2qμ,Vp,ε∗∣q¯σμνqνb∣B¯pB=2T1q2εμνρσεν∗pB,ρpσ,Vp,ε∗∣q¯σμνγ5qνb∣B¯pB=−iT2q2mB2−mV2ε∗μ−ε∗·qpBμ+pμ+−iT3q2ε∗·qqμ−q2mB2−mV2pBμ+pμ,where mV (ε) is the mass (polarisation vector) of the vector meson and we use the sign convention ε0123=1. At maximal hadronic recoil q2=0, there exist three relations for the above-mentioned B⟶M form factors in QCD:
(53)fB⟶P+0=fB⟶P00,mB+mV2mVA10−mB−mV2mVA20=A00,T10=T20,which are free of both radiative and power corrections.


### 3.2. TheB⟶P Form Factors with LCSR

In this subsection, we intend to obtain the light-cone sum rules for theB⟶P form factors at the leading power in 1/mb and leading order in αs. In order to obtain the light-cone sum rules for B⟶P form factors, we use the following vacuum-to-B-meson correlation function:
(54)Πμn·p,n¯·p=∫d4xeip·x0Td¯xnγ5qx,q¯0Γμb0B¯pB=Πn·p,n¯·pnμ+Π~n·p,n¯·pn¯μ,Γμ=γμ,ΠTn·p,n¯·pn¯·qnμ−n·qn¯μ,Γμ=σμνqν,where the Lorentz structure Γμ stands for the two different b⟶q weak currents in QCD. In the center-of-mass frame, the B-meson momentum pB=p+q=mBv, where pandq stand for the momentum of the light-meson and the weak current. At large recoil n·p~mb, we work with p2<0 in order that the correlation function can be calculated perturbatively. We take n¯ standing for the direction of p and use the following power counting with the representation of q~n·q,n¯·q,q⊥:
(55)pB~mb1,1,1,p~mb1,λ2,λ,k~mbλ2,λ2,λ2,where the power counting parameter λ=ΛQCD/mb, k is the momentum of the light-quark in the B meson, and mb is the b-quark mass. For the interpolation current, we employ d¯xnγ5qx so that we can obtain the leading power contribution to the form factors directly.To guarantee the validity of the light-cone OPE, one must prove the light-cone dominance of the correlation functions. In the correlation functions, we have assumed that the four-momentump is spacelike, p2<0, and sufficiently large: P2=−p2≫ΛQCD2; in addition, the ratio ξ=2p·k/P2~O1; then, x2~1/P2 must be satisfied under the condition that the exponent eipx does not oscillate strongly. This leads to a constraint on region of the q2 which is accessible to OPE on the light-cone in the B⟶P transitions
(56)0≤q2<mb2−mbP2ΛQCD.The light-cone dominance of the correlation function allows one to contract theq and q¯ fields and use the free-quark propagator Sqx=−i0∣qxq¯0∣0 as a leading-order approximation. Then, the correlation function can be written by
(57)Πμn·p,n¯·p=∫d4xeip·xnγ5SqxΓμαβ0d¯xαhv0βB¯v,where the heavy quark field has been expanded using HQET and only the large component has been kept. If one expands the bilocal operator d¯xαhv0β in the small x region, an infinite series of matrix elements of local operators are required for the vacuum-B-meson amplitude. Instead, one has to retain the matrix element of the nonlocal operator, which introduces the LCDAs of the B-meson. The detailed discussion on the B-meson LCDAs has been done in the previous section.We will use the calculation of the form factorsfB⟶P+ and fB⟶P0 as an illustration of the LCSR approach. With Γμ=γμ, the correlation function is then expressed in terms of the convolution of the hard scattering kernel and the LCDAs of the B meson, and at tree level, the result reads
(58)Π~n·p,n¯·p=f~BμmB∫0∞dωϕ−ωω−n¯·p−i0,Πn·p,n¯·p=0.In order to arrive at the sum rules, one has to express this result in terms of the dispersion integral with respect ton¯·p:
(59)Π~QCDn·p,n¯·p=∫0∞dω′ω′−n¯·pImω′Π~n·p,ω′,and Πn·p,ω′ should also be expressed in the dispersion form. The relative hadronic representation of the vacuum-to-B-meson correlation function and the decay constant of the pseudoscalar meson are given by
(60)ΠμhadPB,p=0d¯nγ5qPpPpq¯γμbB¯pBmP2−p2+continuum=fPmB2mP2/n·p−n¯·pn¯μn·pmBfB⟶P+q2+fB⟶P0q2+nμmBn·p−mBn·pmBfB⟶P+q2−fB⟶P0q2(61)+∫ωsP∞dω′ω′−n¯·p−i0ρnhadn·p,ω′nμ+ρn¯hadn·p,ω′n¯μ,where we have used the definitions of the form factors (51) and the light-meson decay constant
(62)0d¯nγ5qPp=in·pfP.The parameterωsP=s0P/n·p~λ2 corresponds to the hadronic threshold of the light-meson channel. Taking advantage of the parton-hadron duality assumption, one can relate the result from the parton level calculation to the parametrization in the hadronic level:
(63)∫ωsP∞dω′ω′−n¯·pρn¯hadn·p,ω′=∫ωsP∞dω′ω′−n¯·pImω′Π~n·p,ω′.A similar relation also holds forρnhadn·p,ω′ and Πn·p,ω′. Then, we should perform the Borel transformation with respect to the variable n¯·p to both the hadronic and partonic representation of the correlation [58]:
(64)ΠBωM≡BωMΠn¯·p=lim−n¯·p,r⟶∞−n¯·p/r=ωM−n¯·pr+1r!ddn¯·prΠn¯·p.With the above procedures, one can obtain the sum rules for the form factors(65)fB⟶P+q2=f~BμmBfPn·pexpmP2n·pωMP∫0ωsPdω′e−ω′/ωMPϕ−ω′,fB⟶P0q2=n·pmBfB⟶P+q2,where the Borel mass ωMP=MP2/n·p~λ2. At tree level as well as leading power, the scalar correlation functions Πq2 vanishes, which exhibits the large recoil symmetry of the form factors. The tensor form factor can also be calculated in the same method by using the correlation function with Γμ=σμνqν.


### 3.3. TheB⟶V Form Factors with SCET Improved LCSR

In this subsection, we will calculate theB⟶V form factors with the SCET improved LCSR. First, we match QCD onto SCET where the heavy-to-light form factors are given by [9, 11, 59]
(66)q¯Γib0=∫ds^∑jC~ijA0s^OjA0s;0+∫ds^∑jC~ijμA1s^OjA1μs;0+∫ds^1∫ds^2∑jC~ijμB1s^1,s^2OjB1μs1,s2;0+⋯.The hard matching coefficients for both theA0-type and B1-type SCET currents have been computed at one-loop accuracy [60–62]. The seven QCD B⟶V form factors are expressed in terms of the four “effective” form factors in SCET at leading power in the heavy quark expansion [63]:
(67)Vp,ε∗ξ¯Wcγ5hvB¯v=−n·pε∗·vξ∥n·p,Vp,ε∗ξ¯Wcγ5γμ⊥hvB¯v=−n·pεμ∗−ε∗·vn¯μξ⊥n·p,Vp,ε∗ξ¯Wcγ5Wc†iDc⊥WcrnhvB¯v=−n·pmbε∗·v∫01dτeiτn·prΞ∥τ,n·p,Vp,ε∗ξ¯Wcγ5γμ⊥Wc†iDc⊥WcrnhvB¯v=−n·pmbεμ∗−ε∗·vn¯μ∫01dτeiτn·prΞ⊥τ,n·p,where the light-cone Wilson line is introduced to restore the collinear gauge invariance [11, 64]:
(68)Wcx=Pexpigs∫−∞0dsn·Acx+sn.The relation between the QCD form factors and the SCET form factors are [63]
(69)fB⟶Vin·p=CiA0n·pξan·p+∫dτCiB1τ,n·pΞaτ,n·p,a=∥,⊥,where
(70)mBmB+mVVn·p=CVA0n·pmb,μξ⊥n·p+∫01dτCVB1n·pτ¯mb,n·pτmb,μΞ⊥τ,n·p,2mVn·pA0n·p=Cf0A0n·pmb,μξ∥n·p+∫01dτCf0B1n·pτ¯mb,n·pτmb,μΞ∥τ,n·p,mB+mVn·pA1n·p=CVA0n·pmb,μξ⊥n·p+∫01dτCVB1n·pτ¯mb,n·pτmb,μΞ⊥τ,n·p,mB+mVn·pA1n·p−mB−mVmBA2n·p=Cf+A0n·pmb,μξ∥n·p+∫01dτCf+B1n·pτ¯mb,n·pτmb,μΞ∥τ,n·p,T1n·p=CT1A0n·pmb,μξ⊥n·p+∫01dτCT1B1n·pτ¯mb,n·pτmb,μΞ⊥τ,n·p,mBn·pT2n·p=CT1A0n·pmb,μξ⊥n·p+∫01dτCT1B1n·pτ¯mb,n·pτmb,μΞ⊥τ,n·p,mBn·pT2n·p−T3n·p=CfTA0n·pmb,μξ∥n·p+∫01dτCfTB1n·pτ¯mb,n·pτmb,μΞ∥τ,n·p.The coefficient functionsCijA0 and CijμB1 are obtained from the Fourier transformations of the position-space coefficient functions C~ijA0 and C~ijμB1 [11]. It is evident that only five independent combinations of A0- and B1-type SCET operators appear in the factorization formulae for the seven different B⟶V form factors, implying the two additional relations [8, 65]:
(71)mBmB+mVVn·p=mB+mVn·pA1n·p,T1n·p=mBn·pT2n·p,which are fulfilled to all orders in perturbative expansion at leading power in Λ/mb.With the relations between the form factors at hand, we only need to calculate the SCET form factorsξi and Ξi. We will use the calculation of the SCET form factor ξ∥n·p as an example to illustrate the application of SCET improved LCSR. In the SCET sum rules, the correlation function is constructed with fields in the SCET. We start with the the vacuum-to-B-meson correlation function:
(72)Πν,∥p,q=∫d4xeip·x0Tjνx,ξ¯Wc0γ5hv0B¯v,where the local QCD current jν interpolates current for the longitudinal polarization state of the collinear vector meson [10]:
(73)jνx=q¯′xγνqx=jξξ,ν0+jξξ,⊥ν1+jξqs,∥ν2+jξqs,⊥ν2+⋯,where the explicit expressions of the effective currents are given by
(74)jξξ,ν0=ξ¯n2ξn¯ν,jξξ,⊥ν1=ξ¯γν⊥1in·DciDc⊥n2ξ+ξ¯iDc⊥1in·Dcγν⊥n2ξ,jξqs,∥ν2=ξ¯Wcn2Ys†qs+q¯sYsn2Wc†ξn¯ν,jξqs,⊥ν2=ξ¯Wcγ⊥νYs†qs+q¯sYsγ⊥νWc†ξ,where the collinear Wilson line defined in (68) and the following soft Wilson line
(75)Ysx=Pexpigs∫−∞0dsn¯·Asx+sn¯,is introduced to keep the current gauge invariance. It is then straightforward to write down the leading-power contribution to the correlation function at tree level:
(76)Πν,∥p,q=f~BμmB2∫0∞dω1n¯·p−ω+i0ϕ−ω,μn¯νNow, we apply the standard method of sum rules and match the spectral representation of the factorization formula (76) with the corresponding hadronic dispersion relation:
(77)Πν,∥hadp,q=−fV,∥mVmV2/n·p−n¯·p−i0n·p2mV2ξ∥n·p+∫ωsV∞dω′ω′−n¯·p−i0ρ∥hadn·p,ω′n¯ν,to obtain the SCET form factor ξ∥n·p(78)ξ∥n·p=2f~BμfV,∥mBmVn·p2∫0ωsVdω′exp−n·pω′−mV2n·pωMVϕ−ω′,μ.The scale-independent longitudinal decay constant of the vector meson is defined as follows:(79)cVVp,ε∗∣jν∣0=−ifV,∥mVεν∗p.The other SCET form factorsξi and Ξi can be calculated with the same procedure from different SCET correlation functions (72).


## 3.1. The Heavy-to-Light Form Factors

The form factors for theB-meson decays into a pseudoscalar meson are defined by the matrix elements of the weak transition current sandwiched between the B-meson and a pseudoscalar meson states, i.e.,
(51)Ppq¯γμbB¯pB=fB⟶P+q2pBμ+pμ−mB2−mP2q2qμ+fB⟶P0q2mB2−mP2q2qμ,Ppq¯σμνqνbB¯pB=ifB⟶PTq2mB+mPq2pBμ+pμ−mB2−mP2qμ,where mB and mP are, respectively, the masses of the B meson and the pseudoscalar meson and the momentum transfer q=pB−p. The relevant form factors for B decays into vector mesons are defined as
(52)Vp,ε∗q¯γμbB¯pB=2iVq2mB+mVϵμνρσεν∗pρpB,σ,Vp,ε∗q¯γμγ5bB¯pB=2mVA0q2ε∗·qq2qμ+mB+mVA1q2ε∗μ−ε∗·qq2qμ−A2q2ε∗·qmB+mVpBμ+pμ−mB2−mV2q2qμ,Vp,ε∗∣q¯σμνqνb∣B¯pB=2T1q2εμνρσεν∗pB,ρpσ,Vp,ε∗∣q¯σμνγ5qνb∣B¯pB=−iT2q2mB2−mV2ε∗μ−ε∗·qpBμ+pμ+−iT3q2ε∗·qqμ−q2mB2−mV2pBμ+pμ,where mV (ε) is the mass (polarisation vector) of the vector meson and we use the sign convention ε0123=1. At maximal hadronic recoil q2=0, there exist three relations for the above-mentioned B⟶M form factors in QCD:
(53)fB⟶P+0=fB⟶P00,mB+mV2mVA10−mB−mV2mVA20=A00,T10=T20,which are free of both radiative and power corrections.


## 3.2. TheB⟶P Form Factors with LCSR

In this subsection, we intend to obtain the light-cone sum rules for theB⟶P form factors at the leading power in 1/mb and leading order in αs. In order to obtain the light-cone sum rules for B⟶P form factors, we use the following vacuum-to-B-meson correlation function:
(54)Πμn·p,n¯·p=∫d4xeip·x0Td¯xnγ5qx,q¯0Γμb0B¯pB=Πn·p,n¯·pnμ+Π~n·p,n¯·pn¯μ,Γμ=γμ,ΠTn·p,n¯·pn¯·qnμ−n·qn¯μ,Γμ=σμνqν,where the Lorentz structure Γμ stands for the two different b⟶q weak currents in QCD. In the center-of-mass frame, the B-meson momentum pB=p+q=mBv, where pandq stand for the momentum of the light-meson and the weak current. At large recoil n·p~mb, we work with p2<0 in order that the correlation function can be calculated perturbatively. We take n¯ standing for the direction of p and use the following power counting with the representation of q~n·q,n¯·q,q⊥:
(55)pB~mb1,1,1,p~mb1,λ2,λ,k~mbλ2,λ2,λ2,where the power counting parameter λ=ΛQCD/mb, k is the momentum of the light-quark in the B meson, and mb is the b-quark mass. For the interpolation current, we employ d¯xnγ5qx so that we can obtain the leading power contribution to the form factors directly.To guarantee the validity of the light-cone OPE, one must prove the light-cone dominance of the correlation functions. In the correlation functions, we have assumed that the four-momentump is spacelike, p2<0, and sufficiently large: P2=−p2≫ΛQCD2; in addition, the ratio ξ=2p·k/P2~O1; then, x2~1/P2 must be satisfied under the condition that the exponent eipx does not oscillate strongly. This leads to a constraint on region of the q2 which is accessible to OPE on the light-cone in the B⟶P transitions
(56)0≤q2<mb2−mbP2ΛQCD.The light-cone dominance of the correlation function allows one to contract theq and q¯ fields and use the free-quark propagator Sqx=−i0∣qxq¯0∣0 as a leading-order approximation. Then, the correlation function can be written by
(57)Πμn·p,n¯·p=∫d4xeip·xnγ5SqxΓμαβ0d¯xαhv0βB¯v,where the heavy quark field has been expanded using HQET and only the large component has been kept. If one expands the bilocal operator d¯xαhv0β in the small x region, an infinite series of matrix elements of local operators are required for the vacuum-B-meson amplitude. Instead, one has to retain the matrix element of the nonlocal operator, which introduces the LCDAs of the B-meson. The detailed discussion on the B-meson LCDAs has been done in the previous section.We will use the calculation of the form factorsfB⟶P+ and fB⟶P0 as an illustration of the LCSR approach. With Γμ=γμ, the correlation function is then expressed in terms of the convolution of the hard scattering kernel and the LCDAs of the B meson, and at tree level, the result reads
(58)Π~n·p,n¯·p=f~BμmB∫0∞dωϕ−ωω−n¯·p−i0,Πn·p,n¯·p=0.In order to arrive at the sum rules, one has to express this result in terms of the dispersion integral with respect ton¯·p:
(59)Π~QCDn·p,n¯·p=∫0∞dω′ω′−n¯·pImω′Π~n·p,ω′,and Πn·p,ω′ should also be expressed in the dispersion form. The relative hadronic representation of the vacuum-to-B-meson correlation function and the decay constant of the pseudoscalar meson are given by
(60)ΠμhadPB,p=0d¯nγ5qPpPpq¯γμbB¯pBmP2−p2+continuum=fPmB2mP2/n·p−n¯·pn¯μn·pmBfB⟶P+q2+fB⟶P0q2+nμmBn·p−mBn·pmBfB⟶P+q2−fB⟶P0q2(61)+∫ωsP∞dω′ω′−n¯·p−i0ρnhadn·p,ω′nμ+ρn¯hadn·p,ω′n¯μ,where we have used the definitions of the form factors (51) and the light-meson decay constant
(62)0d¯nγ5qPp=in·pfP.The parameterωsP=s0P/n·p~λ2 corresponds to the hadronic threshold of the light-meson channel. Taking advantage of the parton-hadron duality assumption, one can relate the result from the parton level calculation to the parametrization in the hadronic level:
(63)∫ωsP∞dω′ω′−n¯·pρn¯hadn·p,ω′=∫ωsP∞dω′ω′−n¯·pImω′Π~n·p,ω′.A similar relation also holds forρnhadn·p,ω′ and Πn·p,ω′. Then, we should perform the Borel transformation with respect to the variable n¯·p to both the hadronic and partonic representation of the correlation [58]:
(64)ΠBωM≡BωMΠn¯·p=lim−n¯·p,r⟶∞−n¯·p/r=ωM−n¯·pr+1r!ddn¯·prΠn¯·p.With the above procedures, one can obtain the sum rules for the form factors(65)fB⟶P+q2=f~BμmBfPn·pexpmP2n·pωMP∫0ωsPdω′e−ω′/ωMPϕ−ω′,fB⟶P0q2=n·pmBfB⟶P+q2,where the Borel mass ωMP=MP2/n·p~λ2. At tree level as well as leading power, the scalar correlation functions Πq2 vanishes, which exhibits the large recoil symmetry of the form factors. The tensor form factor can also be calculated in the same method by using the correlation function with Γμ=σμνqν.


## 3.3. TheB⟶V Form Factors with SCET Improved LCSR

In this subsection, we will calculate theB⟶V form factors with the SCET improved LCSR. First, we match QCD onto SCET where the heavy-to-light form factors are given by [9, 11, 59]
(66)q¯Γib0=∫ds^∑jC~ijA0s^OjA0s;0+∫ds^∑jC~ijμA1s^OjA1μs;0+∫ds^1∫ds^2∑jC~ijμB1s^1,s^2OjB1μs1,s2;0+⋯.The hard matching coefficients for both theA0-type and B1-type SCET currents have been computed at one-loop accuracy [60–62]. The seven QCD B⟶V form factors are expressed in terms of the four “effective” form factors in SCET at leading power in the heavy quark expansion [63]:
(67)Vp,ε∗ξ¯Wcγ5hvB¯v=−n·pε∗·vξ∥n·p,Vp,ε∗ξ¯Wcγ5γμ⊥hvB¯v=−n·pεμ∗−ε∗·vn¯μξ⊥n·p,Vp,ε∗ξ¯Wcγ5Wc†iDc⊥WcrnhvB¯v=−n·pmbε∗·v∫01dτeiτn·prΞ∥τ,n·p,Vp,ε∗ξ¯Wcγ5γμ⊥Wc†iDc⊥WcrnhvB¯v=−n·pmbεμ∗−ε∗·vn¯μ∫01dτeiτn·prΞ⊥τ,n·p,where the light-cone Wilson line is introduced to restore the collinear gauge invariance [11, 64]:
(68)Wcx=Pexpigs∫−∞0dsn·Acx+sn.The relation between the QCD form factors and the SCET form factors are [63]
(69)fB⟶Vin·p=CiA0n·pξan·p+∫dτCiB1τ,n·pΞaτ,n·p,a=∥,⊥,where
(70)mBmB+mVVn·p=CVA0n·pmb,μξ⊥n·p+∫01dτCVB1n·pτ¯mb,n·pτmb,μΞ⊥τ,n·p,2mVn·pA0n·p=Cf0A0n·pmb,μξ∥n·p+∫01dτCf0B1n·pτ¯mb,n·pτmb,μΞ∥τ,n·p,mB+mVn·pA1n·p=CVA0n·pmb,μξ⊥n·p+∫01dτCVB1n·pτ¯mb,n·pτmb,μΞ⊥τ,n·p,mB+mVn·pA1n·p−mB−mVmBA2n·p=Cf+A0n·pmb,μξ∥n·p+∫01dτCf+B1n·pτ¯mb,n·pτmb,μΞ∥τ,n·p,T1n·p=CT1A0n·pmb,μξ⊥n·p+∫01dτCT1B1n·pτ¯mb,n·pτmb,μΞ⊥τ,n·p,mBn·pT2n·p=CT1A0n·pmb,μξ⊥n·p+∫01dτCT1B1n·pτ¯mb,n·pτmb,μΞ⊥τ,n·p,mBn·pT2n·p−T3n·p=CfTA0n·pmb,μξ∥n·p+∫01dτCfTB1n·pτ¯mb,n·pτmb,μΞ∥τ,n·p.The coefficient functionsCijA0 and CijμB1 are obtained from the Fourier transformations of the position-space coefficient functions C~ijA0 and C~ijμB1 [11]. It is evident that only five independent combinations of A0- and B1-type SCET operators appear in the factorization formulae for the seven different B⟶V form factors, implying the two additional relations [8, 65]:
(71)mBmB+mVVn·p=mB+mVn·pA1n·p,T1n·p=mBn·pT2n·p,which are fulfilled to all orders in perturbative expansion at leading power in Λ/mb.With the relations between the form factors at hand, we only need to calculate the SCET form factorsξi and Ξi. We will use the calculation of the SCET form factor ξ∥n·p as an example to illustrate the application of SCET improved LCSR. In the SCET sum rules, the correlation function is constructed with fields in the SCET. We start with the the vacuum-to-B-meson correlation function:
(72)Πν,∥p,q=∫d4xeip·x0Tjνx,ξ¯Wc0γ5hv0B¯v,where the local QCD current jν interpolates current for the longitudinal polarization state of the collinear vector meson [10]:
(73)jνx=q¯′xγνqx=jξξ,ν0+jξξ,⊥ν1+jξqs,∥ν2+jξqs,⊥ν2+⋯,where the explicit expressions of the effective currents are given by
(74)jξξ,ν0=ξ¯n2ξn¯ν,jξξ,⊥ν1=ξ¯γν⊥1in·DciDc⊥n2ξ+ξ¯iDc⊥1in·Dcγν⊥n2ξ,jξqs,∥ν2=ξ¯Wcn2Ys†qs+q¯sYsn2Wc†ξn¯ν,jξqs,⊥ν2=ξ¯Wcγ⊥νYs†qs+q¯sYsγ⊥νWc†ξ,where the collinear Wilson line defined in (68) and the following soft Wilson line
(75)Ysx=Pexpigs∫−∞0dsn¯·Asx+sn¯,is introduced to keep the current gauge invariance. It is then straightforward to write down the leading-power contribution to the correlation function at tree level:
(76)Πν,∥p,q=f~BμmB2∫0∞dω1n¯·p−ω+i0ϕ−ω,μn¯νNow, we apply the standard method of sum rules and match the spectral representation of the factorization formula (76) with the corresponding hadronic dispersion relation:
(77)Πν,∥hadp,q=−fV,∥mVmV2/n·p−n¯·p−i0n·p2mV2ξ∥n·p+∫ωsV∞dω′ω′−n¯·p−i0ρ∥hadn·p,ω′n¯ν,to obtain the SCET form factor ξ∥n·p(78)ξ∥n·p=2f~BμfV,∥mBmVn·p2∫0ωsVdω′exp−n·pω′−mV2n·pωMVϕ−ω′,μ.The scale-independent longitudinal decay constant of the vector meson is defined as follows:(79)cVVp,ε∗∣jν∣0=−ifV,∥mVεν∗p.The other SCET form factorsξi and Ξi can be calculated with the same procedure from different SCET correlation functions (72).


## 4. QCD Corrections to theB⟶M Form Factors

In this section, we will show the calculation of the QCD corrections to theB⟶M form factors. First, we calculate the NLO correction to the B⟶P form factors with traditional LCSR with B-meson LCDAs. Then, we provide the NLO computation of the B⟶V form factors with SCET improved LCSR.


### 4.1. TheB⟶P Form Factors with B-Meson LCSR

The objective of this section is to establish the factorization formulae forΠμn·p,n¯·p in (54) at the one-loop level. In the previous section, we have seen that the correlation function can be factorized into the hard scattering part and the LCDAs of the B meson. Since the B-meson LCDA is nonperturbative, what we need is to find the QCD correction to the hard scattering part. The correlator Πμ,bd¯ with free partonic states can be expanded as
(80)Πμ,bd¯=Πμ0+Πμ1+⋯=Φbq¯⊗T=Φbq¯0⊗T0+Φbq¯0⊗T1+Φbq¯1⊗T0+⋯,where ⊗ denotes the convolution in the variable ω′, and T1 is the one-loop level hard scattering kernel; it is then determined by the matching condition:
(81)Φbq¯0⊗T1=Πμ1−Φbq¯1⊗T0,where the second term serves as the infrared (soft) subtraction. We will demonstrate that the soft divergence will be completely absorbed into the B-meson LCDA and that there is no leading contribution to the correlation function from the collinear region (with the momentum scaling lμ~1,λ4,λ2), which confirms the factorization at one-loop level. As a result, the hard-scattering kernel T can be contributed only from hard and/or hard-collinear regions at leading power in Λ/mb. We will evaluate the master formula of T1 in (81) diagram by diagram. There exist two typical perturbative scales, namely, the hard scale mb and the hard-collinear scale mbΛ; therefore, the hard scattering part can be further factorized into the hard function and the jet function. It is more convenient to apply the method of regions [36] to compute the loop integrals in order to obtain the hard coefficient function (C) and the jet function (J) simultaneously. 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,bq¯ without introducing an additional “analytical” regulator. In the following, we will calculate the contribution in Figure 1(a) in detail and present the final results for the other diagram directly.Figure 1
Diagrammatic representation of the correlation functionΠμn·p,n¯·p at Oαs. Figure 1 is reproduced from [32].
(a)(b)(c)(d)


#### 4.1.1. Weak Vertex Diagram

The contribution toΠμ1 from the QCD correction to the weak vertex (Figure 1(a)) is
(82)Πμ,weak1=gs2CF2n¯·p−ω∫dDl2πD1p−k+l2+i0mbv+l2−mb2+i0l2+i0d¯knγ5n¯γρp−k+lγμmbv+l+mbγρhv,where D=4−2ε. For the following scalar integral:
(83)I1=∫dl1p−k+l2+i0mbv+l2−mb2+i0l2+i0,the power counting of I1 is I1~λ0 for the three leading regions (the hard, hard-collinear, and soft regions); thus, only the leading-power contributions of the numerator in (82) need to be kept for a given region. We define the integration measure as
(84)dl≡4π2iμ2eγE4πεdDl2πD.Inserting the partonic light-cone projector yields the hard contribution ofΠμ,weak1 at leading power
(85)Πμ,weak1,h=igs2CFf~BμmBϕbq¯−ωn¯·p−ω∫dDl2πD1l2+n·pn¯·l+i0l2+2mbv·l+i0l2+i0×n¯μ2mbn·p+l+D−2l⊥2−nμD−2n¯·l2,where the superscript “h” denotes the hard contribution and we adopt the conventions
(86)l⊥2≡g⊥μνlμlν,g⊥μν≡gμν−nμn¯ν2−nνn¯μ2.The loop integrals can be evaluated directly, and we obtain(87)Πμ,weak1,h=αsCF4πf~BμmB%∫0∞dωϕbq¯−ωn¯·p−ωn¯μ1∫2+1∫2lnμn·p+1+2ln2μn·p+2lnμmb−ln2r−2Li2−r¯r+2−rr−1lnr+π212+3+nμ1r−11+rr¯lnr,with r=n·p/mb and r¯=1−r.Along the same vein, one can identify the hard-collinear contribution ofΠμ,weak1 at leading power
(88)Πμ,weak1,hc=igs2CFf~BμmBϕbq¯−ωn¯·p−ω∫dDl2πD2mbn·p+ln·p+ln¯·p−k+l+l⊥2+i0mbn·l+i0l2+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 yields
(89)Πμ,weak1,hc=αsCF4πf~BμmB∫0∞dωϕbq¯−ωω−n¯·pn¯μ2∫2+2∫lnμ2n·pω−n¯·p+1+ln2μ2n·pω−n¯·p+2lnμ2n·pω−n¯·p−π26+4.Applying the method of regions we extract the soft contribution ofΠμ,weak1(90)Πμ,weak1,s=gs2CF2n¯·p−ω∫0∞dωn¯·p−ω∫dDl2πD1n¯·p−k+l+i0v·l+i0l2+i0d¯knγ5n¯γμhv=αsCF4πf~BμmB∫0∞dωϕbd¯−ωn¯·p−ωn¯μ1∫2+2∫lnμω−n¯·p+2ln2μω−n¯·p+3π24,where the superscript “s” represents the soft contribution.Now, we compute the corresponding infrared subtraction termΦbd¯,a1⊗T0 as displayed in Figure 2(a). With the Wilson-line Feynman rules, we obtain
(91)Φbq¯,aαβ,1ω,ω′=igs2CF∫dDl2πDδω′−ω−n¯·l−δω′−ωn¯·l+i0v·l+i0l2+i0d¯kαhvβ,from which we can derive the soft subtraction term
(92)Φbq¯,a1⊗T0=gs2CF2n¯·p−ω∫dDl2πDd¯knγ5n¯γμhvn¯·p−k+l+i0v·l+i0l2+i0.Figure 2
One-loop diagrams for theB-meson LCDA. The figure is reproduced from [32].
(a)(b)(c)We then conclude that(93)Πμ,weak1,s=Φbq¯,a1⊗T0,at leading power in Λ/mb, which is an essential point to prove factorization of the correlation function Πμ.


#### 4.1.2. Interpolation Current Vertex Diagram

Now, we turn to compute the QCD correction to the interpolation current vertex (Figure1(b)). The soft region contribution will generate a scaleless integral which vanishes in dimensional regularization, and the hard region induces a power-suppression factor λ from the spinor structure. Therefore, the contribution from the hard-collinear region will be the same as the full QCD result. It is then straightforward to obtain the contribution of the current vertex:
(94)Πμ,P1=Πμ,P1,hc=αsCF4πf~BμmB%∫0∞dω1n¯·p−ωnμϕbq¯+ωn¯·p−ωωlnn¯·p−ωn¯·p+n¯μϕbq¯−ω1∫+ln−μ2p22n¯·pωlnn¯·p−ωn¯·p+1n¯·pωlnn¯·p−ωn¯·plnn¯·p−ωn¯·p+2ωn¯·p−4+4.The soft contribution ofΠμ,P1 vanishes in dimensional regularization; it can be proved that the precise cancelation of Πμ,P1,s and Φbq¯,b1⊗T0 from Figure 2(b) is independent of regularization schemes.


#### 4.1.3. Wave Function Renormalization

The self-energy correction to the intermediate quark propagator (Figure1(c)) is free of soft and collinear divergences and a straightforward calculation gives
(95)Πμ,wfc1=αsCF4πf~BμmB∫0∞dωϕbq¯−ωn¯·p−ωn¯μ1∫+lnμ2n·pω−n¯·p+1.For the external quark fields, 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.,(96)Πμ,dwf1−Φbq¯,dwf1⊗T0=0.The wave function renormalization of theb-quark is actually the match coefficient between the full QCD and HQET:
(97)Πμ,bwf1−Φbq¯,bwf1⊗T0=−αsCF8π3∫+3lnμ2mb2+4Πμ0.


#### 4.1.4. Box Diagram

The one-loop contribution toΠμ from the box diagram (Figure 1(d)) receives leading-power contribution from both the hard-collinear and the soft regions. Evaluating the hard-collinear contribution of Πμ,box1 yields
(98)Πμ,box1,hc=αsCF4πf~Bμ∫0∞dωmBωn¯μϕbq¯+ωrln1+η−2ϕbq¯−ωln1+η×1∫+lnμ2n·pω−n¯·p+12ln1+η+1,with η=−ω/n¯·p.Extracting the soft contribution ofΠμ,box1 with the method of regions gives
(99)Πμ,box1,s=−gs2CF2∫dDl2πDd¯kvk−lnγ5n¯γμhvv·l+i0n¯·p−k+l+i0k−l2+i0l2+i0.Now, we compute the corresponding NLO contribution to the partonic LCDA (Figure2(c)):
(100)Φbq¯,cαβ,1ω,ω′=−igs2CF∫dDl2πDδω′−ω+n¯·ll−k2+i0v·l+i0l2+i0d¯kvl−kαhvβ,from which one can deduce the soft subtraction term
(101)Φbq¯,c1⊗T0=gs2CF2∫dDl2πDd¯kvl−knγ5n¯γμhvv·l+i0n¯·p−k+l+i0l−k2+i0l2+i0,which cancels out the soft contribution of the correlation function Πμ,box1,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.After the results of all the diagrams obtained, the one-loop hard-scattering kernel of the correlation functionΠμ can be readily computed from the matching condition in (81) by collecting different pieces together
(102)Φbq¯0⊗T1=Πμ,weak1+Πμ,P1+Πμ,wfc1+Πμ,box1+Πμ,bwf1+Πμ,dwf1−Φbq¯,a1+Φbq¯,b1+Φbq¯,c1+Φbq¯,bwf1+Φbq¯,dwf1⊗T0=Πμ,weak1,h+Πμ,bwf1−Φbq¯,bwf1+Πμ,weak1,hc+Πμ,P1,hc+Πμ,wfc1,hc+Πμ,box1,hc,where the one-loop level hard function and the jet function can be extracted from the first and second square brackets of the second equality, respectively. Finally, the factorization formulae of Π and Π~ are given by
(103)Π=f~BμmB∑k=±Ckn·p,μ∫0∞dωω−n¯·pJkμ2n·pω,ωn¯·pϕkω,μ,Π~=f~BμmB∑k=±C~kn·p,μ∫0∞dωω−n¯·pJ~kμ2n·pω,ωn¯·pϕkω,μ,at leading power in Λ/mb, where we keep the factorization-scale dependence explicitly, the hard coefficient functions are given by
(104)C+=C~+=1,C−=aCF1r¯rr¯lnr+1,C~−=1−aCF2ln2μn·p+5lnμmb−ln2r−2Li2−r¯r+2−rr−1lnr+π212+5,and the jet functions are
(105)J+=1rJ~+=aCF1−n¯·pωln1−ωn¯·p,J−=1,J~−=1+aCFln2μ2n·pω−n¯·p−2lnn¯·p−ωn¯·plnμ2n·pω−n¯·p−ln2n¯·p−ωn¯·p−1+2n¯·pωlnn¯·p−ωn¯·p−π26−1.From the one-loop calculation, one can obtain the anomalous dimensions of the hard function, the jet function, and the soft contribution; then, the factorization-scale independence ofΠ and Π~ can be demonstrated explicitly. It is straightforward to write down the following evolution equations:
(106)ddlnμC~−n·p,μ=−aCFΓcusp0lnμn·p+5C~−n·p,μ,(107)ddlnμJ~−μ2n·pω,ωn¯·p=aCFΓcusp0lnμ2n·pωJ~−μ2n·pω,ωn¯·p+aCF∫0∞dω′ωΓω,ω′,μJ~−μ2n·pω′,ω′n¯·p,(108)ddlnμf~Bμϕ−ω,μ=−aCFΓcusp0lnμω−5f~Bμϕ−ω,μ−aCF∫0∞dω′ωΓω,ω′,μf~Bμϕ−ω′,μ,where the evolution kernel in the last equation is actually the LN kernel (except for the anomalous dimension of the decay constant f~B) for ϕ−, and Γ has been given in the previous section. The renormalization kernel of ϕ−ω,μ at one-loop level was first computed in [66] and then confirmed in [67]. With the evolution equations displayed above, it is evident that
(109)ddlnμΠn·p,n¯·p,Π~n·p,n¯·p=Oαs2.One cannot avoid the large logarithms of orderlnmb/Λ in the hard functions, the jet functions, f~Bμ, and the B-meson LCDAs concurrently by choosing a common value of μ, and the large logarithms have to be resummed by solving the three RG equations shown above. The solution to the RG equation of the LCDA has been discussed detailed in the previous section; in addition, the hadronic scale entering the initial conditions of the B-meson LCDAs μ0≈1.0 GeV is quite close to the hard-collinear scale μhc≃mbΛ≈1.5GeV; it is not important phenomenologically to sum logarithms of μhc/μ0 [68]. To achieve NLL resummation of large logarithms in the hard coefficient C~−, we need to generalize the RG equation (106) to
(110)ddlnμC~−n·p,μ=−Γcuspαslnμn·p+γαsC~−n·p,μ,ddlnμf~Bμ=γhlαsf~Bμ.The cusp anomalous dimension at the three-loop order and the remaining anomalous dimensionγαs determining renormalization of the SCET heavy-to-light current at two loops will enter U1n·p,μh1,μ at NLL accuracy. The manifest expressions of γi, γhli, and βi can be found in [68] and references therein (note that there is a factor CF difference of our conventions of Γcuspi and γi compared with [68]); the evolution function U1n·p,μh1,μ can be read from (A.3) in [68] with the replacement rules Eγ⟶n·p/2 and μh⟶μh1. Because the hard scale μh1~n·p in the hard function C~−n·p,μ differs from the one μh2~mb in f~Bμ, the resulting evolution functions due to running of the renormalization scale from μh1(μh2) to μhc in C~−n·p,μ (f~Bμ) are
(111)C~−n·p,μ=U1n·p,μh1,μC~−n·p,μh1,f~Bμ=U2μh2,μf~Bμh2.The final factorization formulae ofΠ and Π~ with RG improvement at NLL accuracy can be written as
(112)Π=mBU2μh2,μf~Bμh2∫0∞dωω−n¯·pJ+μ2n·pω,ωn¯·pϕ+ω,μ+mBU2μh2,μf~Bμh2C−n·p,μ∫0∞dωω−n¯·pϕ−ω,μ,Π~=mBU2μh2,μf~Bμh2∫0∞dωω−n¯·pJ~+μ2n·pω,ωn¯·pϕ+ω,μ+mBU1n·p,μh1,μU2μh2,μf~Bμh2C~−n·p,μh1×∫0∞dωω−n¯·pJ~−μ2n·pω,ωn¯·pϕ−ω,μ,where μ should be taken as a hard-collinear scale of order mbΛ.Since the one-loop partonic level result of the correlation function has been obtained, we are ready to construct the sum rules offB⟶P+q2 and fB⟶P0q2 including the radiative corrections at Oαs. To achieve this target, one must express the one-loop correlation function in terms of the dispersion integral. Using the quark-hadron duality assumption and the Borel transform, we obtain the form factors for the B⟶π transition:
(113)fπe−mπ2/n·pωMπn·pmBfB⟶π+q2,fB⟶π0q2=U2μh2,μf~Bμh2∫0ωsπdω′e−ω′/ωMπ×rϕB,eff+ω′,μ+U1n·p,μh1,μC~−n·p,μh1ϕB,eff−ω′,μ±n·p−mBmBϕB,eff+ω′,μ+C−n·p,μϕ−ω′,μ,where the functions ϕB,eff±ω′,μ are defined as
(114)ϕB,eff+ω′,μ=aCF∫ω′∞dωωϕ+ω,μ,ϕB,eff−ω′,μ=ϕ−ω′,μ+aCF∫0ω′dω2ω−ω′lnμ2n·pω′−2lnω′−ωω′⊕ϕ−ω,μ−∫ω′∞dωln2μ2n·pω′−2lnμ2n·pω′+3lnω−ω′ω′+2lnωω′+π26−1dϕ−ω,μdω.The otherB⟶P form factors could be calculated in the same way.


### 4.2. TheB⟶V Form Factors with SCET Improved LCSR

In this subsection, we will calculate the NLO corrections to the SCET form factorsξi and Ξi. The computation of the form factor ξ∥ will be presented in detail; the calculation of the other form factors, which can be derived in a similar procedure, will be neglected.We start with the SCET vacuum-to-B-meson correlation function (72):
(115)Πν,∥p,q=∫d4xeip·x0Tjνx,ξ¯Wc0γ5hv0B¯v.It is straightforward to write down the leading-power contribution to the correlation function:(116)Πν,∥p,q=∫d4xeip·x0Tjξqs,∥ν2x,ξ¯Wc0γ5hv0B¯v+∫d4xeip·x∫d4y0Tjξξ,ν0x,iLξqs2y,ξ¯Wc0γ5hv0B¯v+∫d4xeip·x∫d4y∫d4z0Tjξξ,ν0x,iLξqs1y,iLξm1z,ξ¯Wc0γ5hv0B¯v≡Πν,∥Ap,q+Πν,∥Bp,q+Πν,∥Cp,q,where the third term Πν,∥C takes into account the light-quark mass effect. The multipole expanded SCET Lagrangian up to the Oλ2 accuracy [64] has been derived with the position-space formalism [10]:
(117)Lξ0=ξ¯in¯⋅D+iD⊥cn2ξ,Lξm1=mξ¯iD⊥c,1in⋅Dcn2ξ,Lξm2=−m2ξ¯1in⋅Dcn2ξ,Lξqs1=q¯sWc†iD⊥c,ξ−ξ¯iD⟵⊥cWcqs,Lξqs2=q¯sWc†in¯⋅D+iD⊥c,1in⋅DciD⊥cn2ξ−ξ¯n2in¯⋅D⟵+iD⟵⊥c,1in⋅D⟵ciD⟵⊥cWcqs+q¯sD⟵sμx⊥μWc†iD⊥cξ−ξ¯iD⟵⊥cWcx⊥μDsμqs.In the following, we aim at calculating the jet function in the following factorization formulaΠν,∥i as defined in (116) onto SCETII:
(118)Πν,∥ip,q=f~BμmB2∑m=±∫0∞dωJ∥,miμ2n·pω,ωn¯·pϕmω,μn¯ν,i=A,B,C,at the one-loop accuracy.Firstly, we calculate the jet functionJ,mA entering the SCET factorization formula (118) by investigating the following matrix element:
(119)F∥A=∫d4xeip⋅x0Tq¯sxYsn2Wc†ξx,ξ¯Wc0γ5hv0q¯skhv.At tree level, we have(120)F∥,LOA=−in¯·p−ω′+i0∗O∥,−ω,ω′0,where the light-cone matrix element O∥,−ω,ω′ is defined as
(121)O,−ω,ω′=0O,−ω′q¯skhv=q¯skn2γ5hvδω−ω′+Oαs,and the HQET operator O∥,∓ω′ in the momentum space reads
(122)O,∓ω′=12π∫dteitω′q¯sYstn¯γ5n2,n¯2Ys†hv0.Remembering the definition of the LCDAϕ−, we can obtain the jet functions at tree level as follows:
(123)J∥,−A,0=1n¯·p−ω′+i0,J∥,+A,0=0.We proceed to determine the NLO contribution to the jet functionJ,±A; the corresponding one-loop SCETI diagrams are presented in Figure 3 with the subleading power SCET Feynman rules collected in [69]. The self-energy correction to the hard-collinear quark propagator displayed in Figure 3(a) can be readily written as [38]
(124)F∥,NLOA,a=−αsCF4π1ε+lnμ2n·pω−n¯·p−i0+1F∥,LOA.Figure 3
Diagrammatic representation of the vacuum-to-B-meson correlation function Πν,∥Ap,q defined with the A0-type SCET operator OA0=ξ¯Wcγ5hv and the power suppressed interpolating current jξqs,∥ν2 at one loop.
(a)(b)(c)(d)Diagram (c) of Figure3 yields vanishing contribution due to n2=0 where n is from the Wilson line. One can further verify that the hard-collinear corrections displayed in the diagrams (b) and (d) of Figure 3 give rise to the identical results:
(125)F∥,NLOA,b=F∥,NLOA,d=−2gs2CFn¯⋅p−ωq¯skn2γ5hv×∫dDl2πDn⋅p+ln⋅p+ln¯.p−k+l+l⊥2+i0n⋅l+i0l2+i0,which can be evaluated straightforwardly with dimensional regularization scheme
(126)F,NLOA,b=F,NLOA,d=αsCF2π1ε2+1εlnμ2n·pω−n¯·p+1+12ln2μ2n·pω−n¯·p+lnμ2n·pω−n¯·p−π212+2F∥,LOA.Adding up different pieces together leads to the jet functions at the one-loop accuracy:(127)J,−A=J,−A,01+aCF4ε2+1ε4lnμ2n·pω−n¯·p+3+2ln2μ2n·pω−n¯·p+3lnμ2n·pω−n¯·p−π23+7,J,+A=0,which are in precise agreement with the results presented in [31].The jet functionJ,±B and J,±C appear only at loop level, and they can be determined with a similar method with J,±A. J,±B and J,±C can be obtained through diagrams in Figure 4 and the diagram in Figure 5 with the SCET Feynman rules, respectively. We collect the corresponding jet functions here:
(128)J,−B=aCFJ,−A,0−2ε2+1ε−2lnμ2n·pω−n¯·p−2ln1+η−3−ln2μ2n·pω−n¯·p+lnμ2n·pω−n¯·p−2ln1+η−3−ln21+η+2η−1ln1+η+π26−8,J,+B=0,J,+C=−aCFJ,−A,0mωlnn¯·p−ωn¯·p,J,−C=0.Figure 4
Diagrammatic representation of the vacuum-to-B-meson correlation function Πν,Bp,q defined with the A0-type SCET operator O∥A0=ξ¯Wcγ5hv, the leading power interpolating current jξξ,ν0, and the subleading power SCET Lagrangian Lξqs2.
(a)(b)Figure 5
Diagrammatic representation of the vacuum-to-B-meson correlation function Πν,Bp,q defined with the A0-type SCET operator OA0=ξ¯Wcγ5hv, the leading power interpolating current jξξ,ν0 and the subleading power SCET Lagrangians Lξqs1 and Lξm1.Plugging the obtained jet functions into the factorization formula (118) and employing the decomposition of Πν, defined in (116) yields
(129)Πν,p,q=f~BμmB2∫0∞dωn¯·p−ω+i01+aCFJ^,−A0μ2n·pω,ωn¯·pϕ−ω,μ+aCFJ^,+mμ2n·pω,ωn¯·pϕ+ω,μn¯ν,where the normalized one-loop jet functions J^,−A0 and J^,+m read
(130)J^,−A0=ln2μ2n·pω−n¯·p−2lnμ2n·pω−n¯·pln1+η−ln21+η+2η−1ln1+η−π26−1,J^,+m=−mωlnn¯·p−ωn¯·p.To construct the SCET sum rules for the effective form factorξ∥n·p, we write the correlation function as a dispersion integral:
(131)Πν,p,q=−f~BμmB2∫0∞dω′ω′−n¯·p−i0ϕB,eff−ω′,μ+ϕB,m+ω′,μn¯ν,where the effective B-meson “distribution amplitudes” are identical to that used in the B⟶P form factors. After applying the NLL resummation to the decay constants f~B, we derive
(132)ξ∥n·p=2U2μh2,μf~Bμh2fV,mBmVn·p2×∫0ωsVdω′exp−n·pω′−mV2n·pωMVϕB,eff−ω′,μ+ϕB,m+ω′,μ.The calculations of the otherξ⊥, Ξ∥, and Ξ⊥ are similar but lengthy, see [39] for the details. We will not show the results of these three form factors here.


## 4.1. TheB⟶P Form Factors with B-Meson LCSR

The objective of this section is to establish the factorization formulae forΠμn·p,n¯·p in (54) at the one-loop level. In the previous section, we have seen that the correlation function can be factorized into the hard scattering part and the LCDAs of the B meson. Since the B-meson LCDA is nonperturbative, what we need is to find the QCD correction to the hard scattering part. The correlator Πμ,bd¯ with free partonic states can be expanded as
(80)Πμ,bd¯=Πμ0+Πμ1+⋯=Φbq¯⊗T=Φbq¯0⊗T0+Φbq¯0⊗T1+Φbq¯1⊗T0+⋯,where ⊗ denotes the convolution in the variable ω′, and T1 is the one-loop level hard scattering kernel; it is then determined by the matching condition:
(81)Φbq¯0⊗T1=Πμ1−Φbq¯1⊗T0,where the second term serves as the infrared (soft) subtraction. We will demonstrate that the soft divergence will be completely absorbed into the B-meson LCDA and that there is no leading contribution to the correlation function from the collinear region (with the momentum scaling lμ~1,λ4,λ2), which confirms the factorization at one-loop level. As a result, the hard-scattering kernel T can be contributed only from hard and/or hard-collinear regions at leading power in Λ/mb. We will evaluate the master formula of T1 in (81) diagram by diagram. There exist two typical perturbative scales, namely, the hard scale mb and the hard-collinear scale mbΛ; therefore, the hard scattering part can be further factorized into the hard function and the jet function. It is more convenient to apply the method of regions [36] to compute the loop integrals in order to obtain the hard coefficient function (C) and the jet function (J) simultaneously. 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,bq¯ without introducing an additional “analytical” regulator. In the following, we will calculate the contribution in Figure 1(a) in detail and present the final results for the other diagram directly.Figure 1
Diagrammatic representation of the correlation functionΠμn·p,n¯·p at Oαs. Figure 1 is reproduced from [32].
(a)(b)(c)(d)


### 4.1.1. Weak Vertex Diagram

The contribution toΠμ1 from the QCD correction to the weak vertex (Figure 1(a)) is
(82)Πμ,weak1=gs2CF2n¯·p−ω∫dDl2πD1p−k+l2+i0mbv+l2−mb2+i0l2+i0d¯knγ5n¯γρp−k+lγμmbv+l+mbγρhv,where D=4−2ε. For the following scalar integral:
(83)I1=∫dl1p−k+l2+i0mbv+l2−mb2+i0l2+i0,the power counting of I1 is I1~λ0 for the three leading regions (the hard, hard-collinear, and soft regions); thus, only the leading-power contributions of the numerator in (82) need to be kept for a given region. We define the integration measure as
(84)dl≡4π2iμ2eγE4πεdDl2πD.Inserting the partonic light-cone projector yields the hard contribution ofΠμ,weak1 at leading power
(85)Πμ,weak1,h=igs2CFf~BμmBϕbq¯−ωn¯·p−ω∫dDl2πD1l2+n·pn¯·l+i0l2+2mbv·l+i0l2+i0×n¯μ2mbn·p+l+D−2l⊥2−nμD−2n¯·l2,where the superscript “h” denotes the hard contribution and we adopt the conventions
(86)l⊥2≡g⊥μνlμlν,g⊥μν≡gμν−nμn¯ν2−nνn¯μ2.The loop integrals can be evaluated directly, and we obtain(87)Πμ,weak1,h=αsCF4πf~BμmB%∫0∞dωϕbq¯−ωn¯·p−ωn¯μ1∫2+1∫2lnμn·p+1+2ln2μn·p+2lnμmb−ln2r−2Li2−r¯r+2−rr−1lnr+π212+3+nμ1r−11+rr¯lnr,with r=n·p/mb and r¯=1−r.Along the same vein, one can identify the hard-collinear contribution ofΠμ,weak1 at leading power
(88)Πμ,weak1,hc=igs2CFf~BμmBϕbq¯−ωn¯·p−ω∫dDl2πD2mbn·p+ln·p+ln¯·p−k+l+l⊥2+i0mbn·l+i0l2+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 yields
(89)Πμ,weak1,hc=αsCF4πf~BμmB∫0∞dωϕbq¯−ωω−n¯·pn¯μ2∫2+2∫lnμ2n·pω−n¯·p+1+ln2μ2n·pω−n¯·p+2lnμ2n·pω−n¯·p−π26+4.Applying the method of regions we extract the soft contribution ofΠμ,weak1(90)Πμ,weak1,s=gs2CF2n¯·p−ω∫0∞dωn¯·p−ω∫dDl2πD1n¯·p−k+l+i0v·l+i0l2+i0d¯knγ5n¯γμhv=αsCF4πf~BμmB∫0∞dωϕbd¯−ωn¯·p−ωn¯μ1∫2+2∫lnμω−n¯·p+2ln2μω−n¯·p+3π24,where the superscript “s” represents the soft contribution.Now, we compute the corresponding infrared subtraction termΦbd¯,a1⊗T0 as displayed in Figure 2(a). With the Wilson-line Feynman rules, we obtain
(91)Φbq¯,aαβ,1ω,ω′=igs2CF∫dDl2πDδω′−ω−n¯·l−δω′−ωn¯·l+i0v·l+i0l2+i0d¯kαhvβ,from which we can derive the soft subtraction term
(92)Φbq¯,a1⊗T0=gs2CF2n¯·p−ω∫dDl2πDd¯knγ5n¯γμhvn¯·p−k+l+i0v·l+i0l2+i0.Figure 2
One-loop diagrams for theB-meson LCDA. The figure is reproduced from [32].
(a)(b)(c)We then conclude that(93)Πμ,weak1,s=Φbq¯,a1⊗T0,at leading power in Λ/mb, which is an essential point to prove factorization of the correlation function Πμ.


### 4.1.2. Interpolation Current Vertex Diagram

Now, we turn to compute the QCD correction to the interpolation current vertex (Figure1(b)). The soft region contribution will generate a scaleless integral which vanishes in dimensional regularization, and the hard region induces a power-suppression factor λ from the spinor structure. Therefore, the contribution from the hard-collinear region will be the same as the full QCD result. It is then straightforward to obtain the contribution of the current vertex:
(94)Πμ,P1=Πμ,P1,hc=αsCF4πf~BμmB%∫0∞dω1n¯·p−ωnμϕbq¯+ωn¯·p−ωωlnn¯·p−ωn¯·p+n¯μϕbq¯−ω1∫+ln−μ2p22n¯·pωlnn¯·p−ωn¯·p+1n¯·pωlnn¯·p−ωn¯·plnn¯·p−ωn¯·p+2ωn¯·p−4+4.The soft contribution ofΠμ,P1 vanishes in dimensional regularization; it can be proved that the precise cancelation of Πμ,P1,s and Φbq¯,b1⊗T0 from Figure 2(b) is independent of regularization schemes.


### 4.1.3. Wave Function Renormalization

The self-energy correction to the intermediate quark propagator (Figure1(c)) is free of soft and collinear divergences and a straightforward calculation gives
(95)Πμ,wfc1=αsCF4πf~BμmB∫0∞dωϕbq¯−ωn¯·p−ωn¯μ1∫+lnμ2n·pω−n¯·p+1.For the external quark fields, 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.,(96)Πμ,dwf1−Φbq¯,dwf1⊗T0=0.The wave function renormalization of theb-quark is actually the match coefficient between the full QCD and HQET:
(97)Πμ,bwf1−Φbq¯,bwf1⊗T0=−αsCF8π3∫+3lnμ2mb2+4Πμ0.


### 4.1.4. Box Diagram

The one-loop contribution toΠμ from the box diagram (Figure 1(d)) receives leading-power contribution from both the hard-collinear and the soft regions. Evaluating the hard-collinear contribution of Πμ,box1 yields
(98)Πμ,box1,hc=αsCF4πf~Bμ∫0∞dωmBωn¯μϕbq¯+ωrln1+η−2ϕbq¯−ωln1+η×1∫+lnμ2n·pω−n¯·p+12ln1+η+1,with η=−ω/n¯·p.Extracting the soft contribution ofΠμ,box1 with the method of regions gives
(99)Πμ,box1,s=−gs2CF2∫dDl2πDd¯kvk−lnγ5n¯γμhvv·l+i0n¯·p−k+l+i0k−l2+i0l2+i0.Now, we compute the corresponding NLO contribution to the partonic LCDA (Figure2(c)):
(100)Φbq¯,cαβ,1ω,ω′=−igs2CF∫dDl2πDδω′−ω+n¯·ll−k2+i0v·l+i0l2+i0d¯kvl−kαhvβ,from which one can deduce the soft subtraction term
(101)Φbq¯,c1⊗T0=gs2CF2∫dDl2πDd¯kvl−knγ5n¯γμhvv·l+i0n¯·p−k+l+i0l−k2+i0l2+i0,which cancels out the soft contribution of the correlation function Πμ,box1,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.After the results of all the diagrams obtained, the one-loop hard-scattering kernel of the correlation functionΠμ can be readily computed from the matching condition in (81) by collecting different pieces together
(102)Φbq¯0⊗T1=Πμ,weak1+Πμ,P1+Πμ,wfc1+Πμ,box1+Πμ,bwf1+Πμ,dwf1−Φbq¯,a1+Φbq¯,b1+Φbq¯,c1+Φbq¯,bwf1+Φbq¯,dwf1⊗T0=Πμ,weak1,h+Πμ,bwf1−Φbq¯,bwf1+Πμ,weak1,hc+Πμ,P1,hc+Πμ,wfc1,hc+Πμ,box1,hc,where the one-loop level hard function and the jet function can be extracted from the first and second square brackets of the second equality, respectively. Finally, the factorization formulae of Π and Π~ are given by
(103)Π=f~BμmB∑k=±Ckn·p,μ∫0∞dωω−n¯·pJkμ2n·pω,ωn¯·pϕkω,μ,Π~=f~BμmB∑k=±C~kn·p,μ∫0∞dωω−n¯·pJ~kμ2n·pω,ωn¯·pϕkω,μ,at leading power in Λ/mb, where we keep the factorization-scale dependence explicitly, the hard coefficient functions are given by
(104)C+=C~+=1,C−=aCF1r¯rr¯lnr+1,C~−=1−aCF2ln2μn·p+5lnμmb−ln2r−2Li2−r¯r+2−rr−1lnr+π212+5,and the jet functions are
(105)J+=1rJ~+=aCF1−n¯·pωln1−ωn¯·p,J−=1,J~−=1+aCFln2μ2n·pω−n¯·p−2lnn¯·p−ωn¯·plnμ2n·pω−n¯·p−ln2n¯·p−ωn¯·p−1+2n¯·pωlnn¯·p−ωn¯·p−π26−1.From the one-loop calculation, one can obtain the anomalous dimensions of the hard function, the jet function, and the soft contribution; then, the factorization-scale independence ofΠ and Π~ can be demonstrated explicitly. It is straightforward to write down the following evolution equations:
(106)ddlnμC~−n·p,μ=−aCFΓcusp0lnμn·p+5C~−n·p,μ,(107)ddlnμJ~−μ2n·pω,ωn¯·p=aCFΓcusp0lnμ2n·pωJ~−μ2n·pω,ωn¯·p+aCF∫0∞dω′ωΓω,ω′,μJ~−μ2n·pω′,ω′n¯·p,(108)ddlnμf~Bμϕ−ω,μ=−aCFΓcusp0lnμω−5f~Bμϕ−ω,μ−aCF∫0∞dω′ωΓω,ω′,μf~Bμϕ−ω′,μ,where the evolution kernel in the last equation is actually the LN kernel (except for the anomalous dimension of the decay constant f~B) for ϕ−, and Γ has been given in the previous section. The renormalization kernel of ϕ−ω,μ at one-loop level was first computed in [66] and then confirmed in [67]. With the evolution equations displayed above, it is evident that
(109)ddlnμΠn·p,n¯·p,Π~n·p,n¯·p=Oαs2.One cannot avoid the large logarithms of orderlnmb/Λ in the hard functions, the jet functions, f~Bμ, and the B-meson LCDAs concurrently by choosing a common value of μ, and the large logarithms have to be resummed by solving the three RG equations shown above. The solution to the RG equation of the LCDA has been discussed detailed in the previous section; in addition, the hadronic scale entering the initial conditions of the B-meson LCDAs μ0≈1.0 GeV is quite close to the hard-collinear scale μhc≃mbΛ≈1.5GeV; it is not important phenomenologically to sum logarithms of μhc/μ0 [68]. To achieve NLL resummation of large logarithms in the hard coefficient C~−, we need to generalize the RG equation (106) to
(110)ddlnμC~−n·p,μ=−Γcuspαslnμn·p+γαsC~−n·p,μ,ddlnμf~Bμ=γhlαsf~Bμ.The cusp anomalous dimension at the three-loop order and the remaining anomalous dimensionγαs determining renormalization of the SCET heavy-to-light current at two loops will enter U1n·p,μh1,μ at NLL accuracy. The manifest expressions of γi, γhli, and βi can be found in [68] and references therein (note that there is a factor CF difference of our conventions of Γcuspi and γi compared with [68]); the evolution function U1n·p,μh1,μ can be read from (A.3) in [68] with the replacement rules Eγ⟶n·p/2 and μh⟶μh1. Because the hard scale μh1~n·p in the hard function C~−n·p,μ differs from the one μh2~mb in f~Bμ, the resulting evolution functions due to running of the renormalization scale from μh1(μh2) to μhc in C~−n·p,μ (f~Bμ) are
(111)C~−n·p,μ=U1n·p,μh1,μC~−n·p,μh1,f~Bμ=U2μh2,μf~Bμh2.The final factorization formulae ofΠ and Π~ with RG improvement at NLL accuracy can be written as
(112)Π=mBU2μh2,μf~Bμh2∫0∞dωω−n¯·pJ+μ2n·pω,ωn¯·pϕ+ω,μ+mBU2μh2,μf~Bμh2C−n·p,μ∫0∞dωω−n¯·pϕ−ω,μ,Π~=mBU2μh2,μf~Bμh2∫0∞dωω−n¯·pJ~+μ2n·pω,ωn¯·pϕ+ω,μ+mBU1n·p,μh1,μU2μh2,μf~Bμh2C~−n·p,μh1×∫0∞dωω−n¯·pJ~−μ2n·pω,ωn¯·pϕ−ω,μ,where μ should be taken as a hard-collinear scale of order mbΛ.Since the one-loop partonic level result of the correlation function has been obtained, we are ready to construct the sum rules offB⟶P+q2 and fB⟶P0q2 including the radiative corrections at Oαs. To achieve this target, one must express the one-loop correlation function in terms of the dispersion integral. Using the quark-hadron duality assumption and the Borel transform, we obtain the form factors for the B⟶π transition:
(113)fπe−mπ2/n·pωMπn·pmBfB⟶π+q2,fB⟶π0q2=U2μh2,μf~Bμh2∫0ωsπdω′e−ω′/ωMπ×rϕB,eff+ω′,μ+U1n·p,μh1,μC~−n·p,μh1ϕB,eff−ω′,μ±n·p−mBmBϕB,eff+ω′,μ+C−n·p,μϕ−ω′,μ,where the functions ϕB,eff±ω′,μ are defined as
(114)ϕB,eff+ω′,μ=aCF∫ω′∞dωωϕ+ω,μ,ϕB,eff−ω′,μ=ϕ−ω′,μ+aCF∫0ω′dω2ω−ω′lnμ2n·pω′−2lnω′−ωω′⊕ϕ−ω,μ−∫ω′∞dωln2μ2n·pω′−2lnμ2n·pω′+3lnω−ω′ω′+2lnωω′+π26−1dϕ−ω,μdω.The otherB⟶P form factors could be calculated in the same way.


## 4.1.1. Weak Vertex Diagram

The contribution toΠμ1 from the QCD correction to the weak vertex (Figure 1(a)) is
(82)Πμ,weak1=gs2CF2n¯·p−ω∫dDl2πD1p−k+l2+i0mbv+l2−mb2+i0l2+i0d¯knγ5n¯γρp−k+lγμmbv+l+mbγρhv,where D=4−2ε. For the following scalar integral:
(83)I1=∫dl1p−k+l2+i0mbv+l2−mb2+i0l2+i0,the power counting of I1 is I1~λ0 for the three leading regions (the hard, hard-collinear, and soft regions); thus, only the leading-power contributions of the numerator in (82) need to be kept for a given region. We define the integration measure as
(84)dl≡4π2iμ2eγE4πεdDl2πD.Inserting the partonic light-cone projector yields the hard contribution ofΠμ,weak1 at leading power
(85)Πμ,weak1,h=igs2CFf~BμmBϕbq¯−ωn¯·p−ω∫dDl2πD1l2+n·pn¯·l+i0l2+2mbv·l+i0l2+i0×n¯μ2mbn·p+l+D−2l⊥2−nμD−2n¯·l2,where the superscript “h” denotes the hard contribution and we adopt the conventions
(86)l⊥2≡g⊥μνlμlν,g⊥μν≡gμν−nμn¯ν2−nνn¯μ2.The loop integrals can be evaluated directly, and we obtain(87)Πμ,weak1,h=αsCF4πf~BμmB%∫0∞dωϕbq¯−ωn¯·p−ωn¯μ1∫2+1∫2lnμn·p+1+2ln2μn·p+2lnμmb−ln2r−2Li2−r¯r+2−rr−1lnr+π212+3+nμ1r−11+rr¯lnr,with r=n·p/mb and r¯=1−r.Along the same vein, one can identify the hard-collinear contribution ofΠμ,weak1 at leading power
(88)Πμ,weak1,hc=igs2CFf~BμmBϕbq¯−ωn¯·p−ω∫dDl2πD2mbn·p+ln·p+ln¯·p−k+l+l⊥2+i0mbn·l+i0l2+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 yields
(89)Πμ,weak1,hc=αsCF4πf~BμmB∫0∞dωϕbq¯−ωω−n¯·pn¯μ2∫2+2∫lnμ2n·pω−n¯·p+1+ln2μ2n·pω−n¯·p+2lnμ2n·pω−n¯·p−π26+4.Applying the method of regions we extract the soft contribution ofΠμ,weak1(90)Πμ,weak1,s=gs2CF2n¯·p−ω∫0∞dωn¯·p−ω∫dDl2πD1n¯·p−k+l+i0v·l+i0l2+i0d¯knγ5n¯γμhv=αsCF4πf~BμmB∫0∞dωϕbd¯−ωn¯·p−ωn¯μ1∫2+2∫lnμω−n¯·p+2ln2μω−n¯·p+3π24,where the superscript “s” represents the soft contribution.Now, we compute the corresponding infrared subtraction termΦbd¯,a1⊗T0 as displayed in Figure 2(a). With the Wilson-line Feynman rules, we obtain
(91)Φbq¯,aαβ,1ω,ω′=igs2CF∫dDl2πDδω′−ω−n¯·l−δω′−ωn¯·l+i0v·l+i0l2+i0d¯kαhvβ,from which we can derive the soft subtraction term
(92)Φbq¯,a1⊗T0=gs2CF2n¯·p−ω∫dDl2πDd¯knγ5n¯γμhvn¯·p−k+l+i0v·l+i0l2+i0.Figure 2
One-loop diagrams for theB-meson LCDA. The figure is reproduced from [32].
(a)(b)(c)We then conclude that(93)Πμ,weak1,s=Φbq¯,a1⊗T0,at leading power in Λ/mb, which is an essential point to prove factorization of the correlation function Πμ.


## 4.1.2. Interpolation Current Vertex Diagram

Now, we turn to compute the QCD correction to the interpolation current vertex (Figure1(b)). The soft region contribution will generate a scaleless integral which vanishes in dimensional regularization, and the hard region induces a power-suppression factor λ from the spinor structure. Therefore, the contribution from the hard-collinear region will be the same as the full QCD result. It is then straightforward to obtain the contribution of the current vertex:
(94)Πμ,P1=Πμ,P1,hc=αsCF4πf~BμmB%∫0∞dω1n¯·p−ωnμϕbq¯+ωn¯·p−ωωlnn¯·p−ωn¯·p+n¯μϕbq¯−ω1∫+ln−μ2p22n¯·pωlnn¯·p−ωn¯·p+1n¯·pωlnn¯·p−ωn¯·plnn¯·p−ωn¯·p+2ωn¯·p−4+4.The soft contribution ofΠμ,P1 vanishes in dimensional regularization; it can be proved that the precise cancelation of Πμ,P1,s and Φbq¯,b1⊗T0 from Figure 2(b) is independent of regularization schemes.


## 4.1.3. Wave Function Renormalization

The self-energy correction to the intermediate quark propagator (Figure1(c)) is free of soft and collinear divergences and a straightforward calculation gives
(95)Πμ,wfc1=αsCF4πf~BμmB∫0∞dωϕbq¯−ωn¯·p−ωn¯μ1∫+lnμ2n·pω−n¯·p+1.For the external quark fields, 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.,(96)Πμ,dwf1−Φbq¯,dwf1⊗T0=0.The wave function renormalization of theb-quark is actually the match coefficient between the full QCD and HQET:
(97)Πμ,bwf1−Φbq¯,bwf1⊗T0=−αsCF8π3∫+3lnμ2mb2+4Πμ0.


## 4.1.4. Box Diagram

The one-loop contribution toΠμ from the box diagram (Figure 1(d)) receives leading-power contribution from both the hard-collinear and the soft regions. Evaluating the hard-collinear contribution of Πμ,box1 yields
(98)Πμ,box1,hc=αsCF4πf~Bμ∫0∞dωmBωn¯μϕbq¯+ωrln1+η−2ϕbq¯−ωln1+η×1∫+lnμ2n·pω−n¯·p+12ln1+η+1,with η=−ω/n¯·p.Extracting the soft contribution ofΠμ,box1 with the method of regions gives
(99)Πμ,box1,s=−gs2CF2∫dDl2πDd¯kvk−lnγ5n¯γμhvv·l+i0n¯·p−k+l+i0k−l2+i0l2+i0.Now, we compute the corresponding NLO contribution to the partonic LCDA (Figure2(c)):
(100)Φbq¯,cαβ,1ω,ω′=−igs2CF∫dDl2πDδω′−ω+n¯·ll−k2+i0v·l+i0l2+i0d¯kvl−kαhvβ,from which one can deduce the soft subtraction term
(101)Φbq¯,c1⊗T0=gs2CF2∫dDl2πDd¯kvl−knγ5n¯γμhvv·l+i0n¯·p−k+l+i0l−k2+i0l2+i0,which cancels out the soft contribution of the correlation function Πμ,box1,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.After the results of all the diagrams obtained, the one-loop hard-scattering kernel of the correlation functionΠμ can be readily computed from the matching condition in (81) by collecting different pieces together
(102)Φbq¯0⊗T1=Πμ,weak1+Πμ,P1+Πμ,wfc1+Πμ,box1+Πμ,bwf1+Πμ,dwf1−Φbq¯,a1+Φbq¯,b1+Φbq¯,c1+Φbq¯,bwf1+Φbq¯,dwf1⊗T0=Πμ,weak1,h+Πμ,bwf1−Φbq¯,bwf1+Πμ,weak1,hc+Πμ,P1,hc+Πμ,wfc1,hc+Πμ,box1,hc,where the one-loop level hard function and the jet function can be extracted from the first and second square brackets of the second equality, respectively. Finally, the factorization formulae of Π and Π~ are given by
(103)Π=f~BμmB∑k=±Ckn·p,μ∫0∞dωω−n¯·pJkμ2n·pω,ωn¯·pϕkω,μ,Π~=f~BμmB∑k=±C~kn·p,μ∫0∞dωω−n¯·pJ~kμ2n·pω,ωn¯·pϕkω,μ,at leading power in Λ/mb, where we keep the factorization-scale dependence explicitly, the hard coefficient functions are given by
(104)C+=C~+=1,C−=aCF1r¯rr¯lnr+1,C~−=1−aCF2ln2μn·p+5lnμmb−ln2r−2Li2−r¯r+2−rr−1lnr+π212+5,and the jet functions are
(105)J+=1rJ~+=aCF1−n¯·pωln1−ωn¯·p,J−=1,J~−=1+aCFln2μ2n·pω−n¯·p−2lnn¯·p−ωn¯·plnμ2n·pω−n¯·p−ln2n¯·p−ωn¯·p−1+2n¯·pωlnn¯·p−ωn¯·p−π26−1.From the one-loop calculation, one can obtain the anomalous dimensions of the hard function, the jet function, and the soft contribution; then, the factorization-scale independence ofΠ and Π~ can be demonstrated explicitly. It is straightforward to write down the following evolution equations:
(106)ddlnμC~−n·p,μ=−aCFΓcusp0lnμn·p+5C~−n·p,μ,(107)ddlnμJ~−μ2n·pω,ωn¯·p=aCFΓcusp0lnμ2n·pωJ~−μ2n·pω,ωn¯·p+aCF∫0∞dω′ωΓω,ω′,μJ~−μ2n·pω′,ω′n¯·p,(108)ddlnμf~Bμϕ−ω,μ=−aCFΓcusp0lnμω−5f~Bμϕ−ω,μ−aCF∫0∞dω′ωΓω,ω′,μf~Bμϕ−ω′,μ,where the evolution kernel in the last equation is actually the LN kernel (except for the anomalous dimension of the decay constant f~B) for ϕ−, and Γ has been given in the previous section. The renormalization kernel of ϕ−ω,μ at one-loop level was first computed in [66] and then confirmed in [67]. With the evolution equations displayed above, it is evident that
(109)ddlnμΠn·p,n¯·p,Π~n·p,n¯·p=Oαs2.One cannot avoid the large logarithms of orderlnmb/Λ in the hard functions, the jet functions, f~Bμ, and the B-meson LCDAs concurrently by choosing a common value of μ, and the large logarithms have to be resummed by solving the three RG equations shown above. The solution to the RG equation of the LCDA has been discussed detailed in the previous section; in addition, the hadronic scale entering the initial conditions of the B-meson LCDAs μ0≈1.0 GeV is quite close to the hard-collinear scale μhc≃mbΛ≈1.5GeV; it is not important phenomenologically to sum logarithms of μhc/μ0 [68]. To achieve NLL resummation of large logarithms in the hard coefficient C~−, we need to generalize the RG equation (106) to
(110)ddlnμC~−n·p,μ=−Γcuspαslnμn·p+γαsC~−n·p,μ,ddlnμf~Bμ=γhlαsf~Bμ.The cusp anomalous dimension at the three-loop order and the remaining anomalous dimensionγαs determining renormalization of the SCET heavy-to-light current at two loops will enter U1n·p,μh1,μ at NLL accuracy. The manifest expressions of γi, γhli, and βi can be found in [68] and references therein (note that there is a factor CF difference of our conventions of Γcuspi and γi compared with [68]); the evolution function U1n·p,μh1,μ can be read from (A.3) in [68] with the replacement rules Eγ⟶n·p/2 and μh⟶μh1. Because the hard scale μh1~n·p in the hard function C~−n·p,μ differs from the one μh2~mb in f~Bμ, the resulting evolution functions due to running of the renormalization scale from μh1(μh2) to μhc in C~−n·p,μ (f~Bμ) are
(111)C~−n·p,μ=U1n·p,μh1,μC~−n·p,μh1,f~Bμ=U2μh2,μf~Bμh2.The final factorization formulae ofΠ and Π~ with RG improvement at NLL accuracy can be written as
(112)Π=mBU2μh2,μf~Bμh2∫0∞dωω−n¯·pJ+μ2n·pω,ωn¯·pϕ+ω,μ+mBU2μh2,μf~Bμh2C−n·p,μ∫0∞dωω−n¯·pϕ−ω,μ,Π~=mBU2μh2,μf~Bμh2∫0∞dωω−n¯·pJ~+μ2n·pω,ωn¯·pϕ+ω,μ+mBU1n·p,μh1,μU2μh2,μf~Bμh2C~−n·p,μh1×∫0∞dωω−n¯·pJ~−μ2n·pω,ωn¯·pϕ−ω,μ,where μ should be taken as a hard-collinear scale of order mbΛ.Since the one-loop partonic level result of the correlation function has been obtained, we are ready to construct the sum rules offB⟶P+q2 and fB⟶P0q2 including the radiative corrections at Oαs. To achieve this target, one must express the one-loop correlation function in terms of the dispersion integral. Using the quark-hadron duality assumption and the Borel transform, we obtain the form factors for the B⟶π transition:
(113)fπe−mπ2/n·pωMπn·pmBfB⟶π+q2,fB⟶π0q2=U2μh2,μf~Bμh2∫0ωsπdω′e−ω′/ωMπ×rϕB,eff+ω′,μ+U1n·p,μh1,μC~−n·p,μh1ϕB,eff−ω′,μ±n·p−mBmBϕB,eff+ω′,μ+C−n·p,μϕ−ω′,μ,where the functions ϕB,eff±ω′,μ are defined as
(114)ϕB,eff+ω′,μ=aCF∫ω′∞dωωϕ+ω,μ,ϕB,eff−ω′,μ=ϕ−ω′,μ+aCF∫0ω′dω2ω−ω′lnμ2n·pω′−2lnω′−ωω′⊕ϕ−ω,μ−∫ω′∞dωln2μ2n·pω′−2lnμ2n·pω′+3lnω−ω′ω′+2lnωω′+π26−1dϕ−ω,μdω.The otherB⟶P form factors could be calculated in the same way.


## 4.2. TheB⟶V Form Factors with SCET Improved LCSR

In this subsection, we will calculate the NLO corrections to the SCET form factorsξi and Ξi. The computation of the form factor ξ∥ will be presented in detail; the calculation of the other form factors, which can be derived in a similar procedure, will be neglected.We start with the SCET vacuum-to-B-meson correlation function (72):
(115)Πν,∥p,q=∫d4xeip·x0Tjνx,ξ¯Wc0γ5hv0B¯v.It is straightforward to write down the leading-power contribution to the correlation function:(116)Πν,∥p,q=∫d4xeip·x0Tjξqs,∥ν2x,ξ¯Wc0γ5hv0B¯v+∫d4xeip·x∫d4y0Tjξξ,ν0x,iLξqs2y,ξ¯Wc0γ5hv0B¯v+∫d4xeip·x∫d4y∫d4z0Tjξξ,ν0x,iLξqs1y,iLξm1z,ξ¯Wc0γ5hv0B¯v≡Πν,∥Ap,q+Πν,∥Bp,q+Πν,∥Cp,q,where the third term Πν,∥C takes into account the light-quark mass effect. The multipole expanded SCET Lagrangian up to the Oλ2 accuracy [64] has been derived with the position-space formalism [10]:
(117)Lξ0=ξ¯in¯⋅D+iD⊥cn2ξ,Lξm1=mξ¯iD⊥c,1in⋅Dcn2ξ,Lξm2=−m2ξ¯1in⋅Dcn2ξ,Lξqs1=q¯sWc†iD⊥c,ξ−ξ¯iD⟵⊥cWcqs,Lξqs2=q¯sWc†in¯⋅D+iD⊥c,1in⋅DciD⊥cn2ξ−ξ¯n2in¯⋅D⟵+iD⟵⊥c,1in⋅D⟵ciD⟵⊥cWcqs+q¯sD⟵sμx⊥μWc†iD⊥cξ−ξ¯iD⟵⊥cWcx⊥μDsμqs.In the following, we aim at calculating the jet function in the following factorization formulaΠν,∥i as defined in (116) onto SCETII:
(118)Πν,∥ip,q=f~BμmB2∑m=±∫0∞dωJ∥,miμ2n·pω,ωn¯·pϕmω,μn¯ν,i=A,B,C,at the one-loop accuracy.Firstly, we calculate the jet functionJ,mA entering the SCET factorization formula (118) by investigating the following matrix element:
(119)F∥A=∫d4xeip⋅x0Tq¯sxYsn2Wc†ξx,ξ¯Wc0γ5hv0q¯skhv.At tree level, we have(120)F∥,LOA=−in¯·p−ω′+i0∗O∥,−ω,ω′0,where the light-cone matrix element O∥,−ω,ω′ is defined as
(121)O,−ω,ω′=0O,−ω′q¯skhv=q¯skn2γ5hvδω−ω′+Oαs,and the HQET operator O∥,∓ω′ in the momentum space reads
(122)O,∓ω′=12π∫dteitω′q¯sYstn¯γ5n2,n¯2Ys†hv0.Remembering the definition of the LCDAϕ−, we can obtain the jet functions at tree level as follows:
(123)J∥,−A,0=1n¯·p−ω′+i0,J∥,+A,0=0.We proceed to determine the NLO contribution to the jet functionJ,±A; the corresponding one-loop SCETI diagrams are presented in Figure 3 with the subleading power SCET Feynman rules collected in [69]. The self-energy correction to the hard-collinear quark propagator displayed in Figure 3(a) can be readily written as [38]
(124)F∥,NLOA,a=−αsCF4π1ε+lnμ2n·pω−n¯·p−i0+1F∥,LOA.Figure 3
Diagrammatic representation of the vacuum-to-B-meson correlation function Πν,∥Ap,q defined with the A0-type SCET operator OA0=ξ¯Wcγ5hv and the power suppressed interpolating current jξqs,∥ν2 at one loop.
(a)(b)(c)(d)Diagram (c) of Figure3 yields vanishing contribution due to n2=0 where n is from the Wilson line. One can further verify that the hard-collinear corrections displayed in the diagrams (b) and (d) of Figure 3 give rise to the identical results:
(125)F∥,NLOA,b=F∥,NLOA,d=−2gs2CFn¯⋅p−ωq¯skn2γ5hv×∫dDl2πDn⋅p+ln⋅p+ln¯.p−k+l+l⊥2+i0n⋅l+i0l2+i0,which can be evaluated straightforwardly with dimensional regularization scheme
(126)F,NLOA,b=F,NLOA,d=αsCF2π1ε2+1εlnμ2n·pω−n¯·p+1+12ln2μ2n·pω−n¯·p+lnμ2n·pω−n¯·p−π212+2F∥,LOA.Adding up different pieces together leads to the jet functions at the one-loop accuracy:(127)J,−A=J,−A,01+aCF4ε2+1ε4lnμ2n·pω−n¯·p+3+2ln2μ2n·pω−n¯·p+3lnμ2n·pω−n¯·p−π23+7,J,+A=0,which are in precise agreement with the results presented in [31].The jet functionJ,±B and J,±C appear only at loop level, and they can be determined with a similar method with J,±A. J,±B and J,±C can be obtained through diagrams in Figure 4 and the diagram in Figure 5 with the SCET Feynman rules, respectively. We collect the corresponding jet functions here:
(128)J,−B=aCFJ,−A,0−2ε2+1ε−2lnμ2n·pω−n¯·p−2ln1+η−3−ln2μ2n·pω−n¯·p+lnμ2n·pω−n¯·p−2ln1+η−3−ln21+η+2η−1ln1+η+π26−8,J,+B=0,J,+C=−aCFJ,−A,0mωlnn¯·p−ωn¯·p,J,−C=0.Figure 4
Diagrammatic representation of the vacuum-to-B-meson correlation function Πν,Bp,q defined with the A0-type SCET operator O∥A0=ξ¯Wcγ5hv, the leading power interpolating current jξξ,ν0, and the subleading power SCET Lagrangian Lξqs2.
(a)(b)Figure 5
Diagrammatic representation of the vacuum-to-B-meson correlation function Πν,Bp,q defined with the A0-type SCET operator OA0=ξ¯Wcγ5hv, the leading power interpolating current jξξ,ν0 and the subleading power SCET Lagrangians Lξqs1 and Lξm1.Plugging the obtained jet functions into the factorization formula (118) and employing the decomposition of Πν, defined in (116) yields
(129)Πν,p,q=f~BμmB2∫0∞dωn¯·p−ω+i01+aCFJ^,−A0μ2n·pω,ωn¯·pϕ−ω,μ+aCFJ^,+mμ2n·pω,ωn¯·pϕ+ω,μn¯ν,where the normalized one-loop jet functions J^,−A0 and J^,+m read
(130)J^,−A0=ln2μ2n·pω−n¯·p−2lnμ2n·pω−n¯·pln1+η−ln21+η+2η−1ln1+η−π26−1,J^,+m=−mωlnn¯·p−ωn¯·p.To construct the SCET sum rules for the effective form factorξ∥n·p, we write the correlation function as a dispersion integral:
(131)Πν,p,q=−f~BμmB2∫0∞dω′ω′−n¯·p−i0ϕB,eff−ω′,μ+ϕB,m+ω′,μn¯ν,where the effective B-meson “distribution amplitudes” are identical to that used in the B⟶P form factors. After applying the NLL resummation to the decay constants f~B, we derive
(132)ξ∥n·p=2U2μh2,μf~Bμh2fV,mBmVn·p2×∫0ωsVdω′exp−n·pω′−mV2n·pωMVϕB,eff−ω′,μ+ϕB,m+ω′,μ.The calculations of the otherξ⊥, Ξ∥, and Ξ⊥ are similar but lengthy, see [39] for the details. We will not show the results of these three form factors here.


## 5. Power Corrections to theB⟶M Form Factors

In the previous sections, the form factors are calculated at the leading power. The power suppressed contributions are expected to be sizeable due to the finite bottom quark mass. So far, the high twist contribution has been considered in various works [35, 39, 40]. In this section, we will take the scalar and vector B⟶P form factors as an example and investigate the contribution from both the two-particle and three-particle B-meson LCDAs employing a complete parametrization of the corresponding three-particle light-cone matrix element and the EOM constraints of the higher-twist LCDAs presented in [56].To obtain the result from three-particle LCDAs, we make use of the light-cone expansion of the quark propagator in the background gluon field [70]:
(133)0∣Tq¯x,q0∣0⊃igs∫0∞d4k2π4e−ik·x∫01duuxμγνk2−mq2−k+mqσμν2k2−mq22Gμνux,where we only keep the one-gluon part without the covariant derivative of the Gμν terms. The tree-level diagram is displayed in Figure 6, and three-particle higher-twist corrections to the vacuum-to-B-meson correlation function (54) can be derived directly:
(134)Πμ3Pn·p,n¯·p=−f~BμmBn·p∫0∞dω1∫0∞dω2∫01du1n¯·p−ω1−uω22×n¯μρn¯,LP3P+mqn·pρn¯,NLP3Pu,ω1,ω2,μ+nμρn,LP3P+mqn·pρn,NLP3Pu,ω1,ω2,μ,where the light-quark mass-dependent term of the three-particle corrections have been taken into account, which is suppressed by one power of Λ/mb. The explicit expressions of ρi,LP3P and ρi,NLP3P (i=n,n¯) can be found in [35]. The definition of three particle LCDAs in [56] is essential in evaluating these results. Due to nonvanishing quark transverse momentum, the higher-twist two-particle B-meson LCDAs are related expressed in terms of the three-particle configurations with the exact EOM, and they must be taken into account simultaneously for consistency. The calculation of the higher-twist two-particle B-meson LCDAs is quite similar to the leading power calculation.Figure 6
Diagrammatical representation of the three-particle higher-twist corrections to the vacuum-to-B-meson correlation function. The square box indicates the insertion of the weak vertex q¯Γμb, and the waveline represents the interpolating current d¯nγ5q for the light-pseudoscalar meson.Adding up the two-particle and three-particle higher-twist corrections at tree level together and implementing the standard strategy to construct the sum rules for heavy-to-light form factors give rise to the following expressions:(135)fPn·p2exp−mP2n·pωMPfB⟶P+,HTq2+mBn·pfB⟶P0,HTq2=−f~BμmBn·pe−ωsP/ωMPHn¯,LP2PHTωsP,μ+∫0ωsPdω′1ωMPe−ω′/ωMHn¯,LP2PHTω′,μ+∫0ωsPdω1∫ωsP−ω1∞dω2ω2e−ωsP/ωMPHn¯,LP3PHT+mqn·pHn¯,NLP3PHTωsP−ω1ω2,ω1,ω2,μ+∫0ωsPdω′∫0ω′dω1∫ω′−ω1∞dω2ω21ωMPe−ω′/ωMPHn¯,LP3PHT+mqn·pHn¯,NLP3PHTω′−ω1ω2,ω1,ω2,μ,(136)fPn·p2exp−mP2n·pωMPmBn·p−mBfB⟶P+,HTq2−mBn·pfB⟶P0,HTq2=−f~BμmBn·p∫0ωsPdω1∫ωsP−ω1∞dω2ω2e−ωsP/ωMPHn,LP3PHT+mqn·pHn,NLP3PHTωsP−ω1ω2,ω1,ω2,μ+∫0ωsPdω′∫0ω′dω1∫ω′−ω1∞dω2ω21ωMe−ω′/ωMPHn,LP3PHT+mqn·pHn,NLP3PHTω′−ω1ω2,ω1,ω2,μ,where the nonvanishing spectral functions Hi,LP2PHT and Hi,NLP3PHT (i=n,n¯) are given by
(137)Hn¯,LP2PHTω,μ=4g^−ω,μ,Hn,LP3PHTu,ω1,ω2,μ=2u−1ϕ4ω1,ω2,μ,Hn,NLP3PHTu,ω1,ω2,μ=ψ~5ω1,ω2,μ−ψ5ω1,ω2,μ,Hn¯,LP3PHTu,ω1,ω2,μ=ψ~5ω1,ω2,μ−ψ5ω1,ω2,μ,Hn¯,NLP3PHTu,ω1,ω2,μ=2ϕ6ω1,ω2,μ.It is evident that the two-particle higher-twist corrections preserve the large-recoil symmetry relations of theB⟶P form factors and the three-particle higher-twist contributions violate such relations already at tree level.Employing the power counting scheme for the Borel massωMP and the threshold parameter ωsP [32]:
(138)ωsP~ωMP~OΛ2/mb,we can identify the scaling behaviors of the higher-twist corrections to B⟶P form factors
(139)fB⟶P+,HTq2~fB⟶P0,HTq2~Oλ5,in the heavy quark limit, which is suppressed by one power of Λ/mb compared with the leading-twist contribution. Collecting different pieces together, the final expressions for the LCSR of B⟶P form factors at large hadronic recoil can be written as
(140)fB⟶P+q2=fB⟶P+,2PNLLq2+fB⟶P+,2PHTq2+fB⟶P+,3PHTq2,fB⟶P0q2=fB⟶P0,2PNLLq2+fB⟶P0,2PHTq2+fB⟶P0,3PHTq2,where the manifest expressions of fB⟶Pi,2PNLLq2 (i=+,0) include the light-quark mass effect, and the higher-twist corrections fB⟶Pi,2PHTq2 and fB⟶Pi,3PHTq2 can be extracted from (135) and (136).The power suppressed contribution is from much more sources than the high twist contribution considered here. Various kinds of power suppressed contribution to theB⟶γνℓ [57, 71–73] and Bs⟶γγ [53] have been comprehensively studied, among which some kinds of sources can also contribute to the heavy-to-light form factor. One example is from heavy quark expansion, i.e.,
(141)q¯Γμb=e−imbv⋅xq¯Γμ1+iD⟶2mbhv+⋯,where D⟶μ=Dμ−v·Dvμ, and the b-quark mass mb is defined in the pole-mass scheme in HQET. It is clear replacing the first term in the square bracket with the second term yields power suppressed contributions. In addition, the power suppressed local and nonlocal contribution in hard-collinear propagators might also play an important role. A specific calculation of these contributions will be published elsewhere.


## 6. Numerical Results

Having the sum rules for heavy-to-light form factors at hand, we will proceed to perform the numerical analysis. First, we will show the predictions of the form factors with LCSR at large recoil. Then, we will explore some phenomenological implications of the form factors by extrapolating the form factors to the entire physical region with thez-series parametrization.


### 6.1. LCSR Predictions of the Form Factors

Prior to presenting the predictions of the semileptonicB⟶M decay form factors, we need to determine the input parameter used in the numerical analysis. Most importantly, the threshold and the Borel parameters:
(142)s0P,V≡n·pωsP,V,MP,V2≡n·pωMP,V,should be extracted. The threshold parameter, which also enters the QCD sum rules for the light-meson decay constant, could be fixed by the sum rule calculation of the light-meson decay constant [58]. The Borel mass is introduced to suppress the continuum-state contributions; at the same time, the LCSR requires a plain window of the Borel mass inside which the form factor is stable. Then, two requirements are employed to obtain the interval of the Borel parameter [32]:
(i)
The dependence of the form factor on the Borel mass should be small(143)∂fB⟶P,V∂ωMP,V≤0.35(ii)
The continuum contribution in the dispersion integral of the correlation function should be smaller than the light-meson channel contribution(144)∫ωsP,V∞dω′e−ω′/ωMP,VImω′Π,Π~n·p,ω′<∫0ωsP,Vdω′e−ω′/ωMP,VImω′Π,Π~n·p,ω′In Figure7, we show the Borel mass and the threshold parameter dependence of the form factor fB⟶π+. From the left figure, we see that the form factor has a stable window with respect to the Borel mass. For the later convenience, we collect the dominant inputs in Table 1 (see [81–83] for a discussion about the flavor symmetry breaking effects of the decay constants of pseudoscalar mesons).Figure 7
The Borel mass and threshold dependence of the form factorfB⟶π+0.Table 1
The numerical values of the various input parameters employed in the theory predictions of theB⟶M form factors with B-meson LCSR.
ParameterValueRef.ParameterValueRef.ms2GeV93.8±1.5±1.9MeV[74]mbmb4.193−0.035+0.022GeV[75, 76]μ1.5±0.5GeVμh1,μh2,νhmb−mb/2+mbfB192.0±4.3MeV[77]fπ130.2±1.7MeV[74]fK155.6±0.4MeV[74]fρ,213±5MeV[25]fρ,⊥1GeV160±7MeV[25]fω,197±8MeV[25]fω,⊥1GeV148±13MeV[25]fK∗,204±7MeV[25]fK∗,⊥1GeV159±6MeV[25]s0π0.70±0.05GeV2[32, 78]s0K1.05±0.05GeV2[32, 78]s0ρ,1.5±0.1GeV2[27, 79]s0ρ,⊥1.2±0.1GeV2[80]s0ω,s0ρ,∥+mω2−mρ2[27, 79]s0ω,⊥s0ρ,⊥+mω2−mρ2[80]s0K∗,s0ρ,∥+mK∗2−mρ2[27, 79]s0K∗,⊥s0ρ,⊥+mK∗2−mρ2[80]Mπ,K21.25±0.25GeV2[32, 78]Mρ21.5±0.5GeV2[79]Mω2Mρ2+mω2−mρ2[79]MK∗2Mρ2+mK∗2−mρ2[79]Except for these parameters, we still need to determine the inverse moment of theB meson λBμ0. The parameter λBμ0 could be calculated with nonperturbative HQET sum rules [84] and could also be extracted indirectly from B⟶γℓν process [57, 71, 72, 85, 86]. While at present, we are still lack of satisfied constraints of λBμ0. Following the strategy displayed in [32], we will employ a fit approach to determine λBμ0. For the B⟶P form factors, matching the LCSR calculation for the vector B⟶π form factor at q2=0 with the predictions from LCSR with pion LCDAs fB⟶π+q2=0=0.28±0.03 [18, 20, 87] will provide one with a suitable value of λBμ0. While for the B⟶V form factors, λBμ0 could be determined by matching the LCSR prediction of the form factor VB⟶ρq2=0 to that of the improved NLO LCSR with the ρ-meson LCDAs [25]. Performing such matching procedure, we obtain
(145)λBμ0=285−23+27MeV,B⟶P,343−20+22MeV,B⟶V.It is interesting to notice that the extracted values ofλBμ0 for the B⟶V form factors are in a nice agreement with that of the B⟶P form factors and are also consistent with the implications of experimental data for the two-body charmless hadronic B-meson decays from the QCD factorization approach [88]. The determined values of λB are also compatible with the recent extracted values from the B⟶γℓν¯ℓ process [89].Employing the above-determined parameters, one can explore the LCSR predictions of theB⟶M form factors in the large-recoil region. As an illustration, we show the breakdown of different contributions to the form factor fB⟶π+q2 in Figure 8. One can see from the figure that higher-twist corrections to the form factor fB⟶π+q2 are dominated by the two-particle twist-five B-meson LCDA g^−ω,μ, which can shift the leading-power prediction by an amount of approximately 20~30%. Compared to the two-particle higher-twist contribution, the three-particle higher-twist contribution only generates a tiny contribution on the form factor. We further find that the radiative correction can introduce about a O20% reduction of the corresponding LL prediction. We have also verified that such observations also hold true for the other B⟶M form factors at large hadronic recoil.Figure 8
Theq2 dependence of the form factor fB⟶π+q2. The red line represents the total contribution. The black, blue, green, and yellow correspond to the LP contribution at LL, the LP contribution at NLL, the two-particle higher-twist correction, and the three-particle higher-twist correction, respectively.


### 6.2. Phenomenological Applications

It is well-known that the light-cone operator-product expansion of the vacuum-to-B-meson correlation function is only valid in the large-recoil region [27, 32]. One then needs to extrapolate the LCSR predictions of B⟶M form factors at q2≤8GeV2 to the full kinematic region with the z-series expansion. The parameter z corresponds to a map of the entire cut q2-plane onto a unit disk ∣zq2,t0∣≤1:
(146)zq2,t0=t+−q2−t+−t0t+−q2+t+−t0.Here,t+=mB+mM2 is determined by the threshold of the lowest continuum state which can be generated by the weak transition currents in QCD. The auxiliary parameter t0, which determines the q2 point to be mapped onto the origin of the complex z plane, will be further chosen as [18, 35]
(147)t0=mB+mPmB+mP2,B⟶P,t0=mB−mV2,B⟶V.We will use the simplified Bourrely-Caprini-Lellouch (BCL) series expansion for theB⟶M form factors [90] (see [91, 92] for an alternative parametrization and [20] for more discussions for the B⟶π form factors):
(148)fB⟶P+,Tq2=fB⟶P+,T01−q2mBs∗21+∑k=1N−1bk,P+,Tzq2,t0k−z0,t0k−−1N−kkNzq2,t0N−z0,t0N,(149)fB⟶P0q2=fB⟶P001+∑k=1Nbk,P0zq2,t0k−z0,t0k,(150)fB⟶Viq2=fB⟶Vi01−q2mi,pole21+∑k=1Nbk,Vizq2,t0k−z0,t0K.The expansion parameter is small in the entire regionzq2,t02≤0.04; the expansion could be truncated at certain N. We will truncate the z-series at N=2 for the form factors fB⟶P+,T and at N=1 for all the other form factors (see [93] for further discussions on the systematic truncation uncertainties). The adopted values of the various resonance masses from the Particle Data Group (PDG) [74] and from the heavy-hadron chiral perturbation theory [94] are summarized in Table 2.Table 2
The resonance masses with different quantum numbers entering thez-series expansions of the B⟶V form factors (149) where A12=mB+mV/n·pA1−mB−mV/mBA2 and T23=mB/n·pT2−T3.
fB⟶Viq2JPb⟶d (in GeV)b⟶s (in GeV)Vq2, T1q21−5.3255.415A0q20−5.2795.366A1q2, A12q2, T2q2, T23q21+5.7245.829Table3 shows the numerical predictions of the form factors fB⟶π+,0 at q2=0GeV2. In this table, we display the significant uncertainties from different input parameters. One concludes from the table that the parameter λB introduces an about 10% uncertainty to the form factors. The fitted z-expansion parameters are also included in the table.Table 3
Theory predictions for the form factorsfB⟶π+,00 and the corresponding z-expansion shape parameters b1,π+,0 with the dominant uncertainties from variations of different input parameters.
ParametersCentral valueλBσ^1μMπ2s0πfB⟶π+,000.280+0.031−0.030+0.013−0.012−0.032+0.000−0.017+0.012−0.014+0.014b1,π+−2.77−0.02+0.05−0.01+0.02−0.16+0.09−0.03+0.02−0.07+0.07b1,π0−4.88+0.11−0.10+0.04−0.04−0.61+0.17−0.06+0.04−0.11+0.11Having at our disposal the theory predictions forB⟶π form factors, we proceed to explore phenomenological aspects of the semileptonic B⟶πℓν decays, which serves as the golden channel for the determination of CKM matrix element ∣Vub∣ exclusively (see [95] for the future advances of precision measurements of Belle II). It is straightforward to write down the differential decay rate for B⟶πℓν¯ℓ:
(151)dΓB⟶πℓν¯ℓdq2=GF2Vub2192π3mB3λ32mB2,mπ2,q21−ml2q221+ml22q2fB⟶π+q2∣2+3ml2mB2−mπ22λmB2,mπ2,q2ml2+2q2fB⟶π0q22,dΓB⟶Vℓν¯ℓdq2=GF2Vub2192π3mB3q2cV2λ1/2mB2,mV2,q2H0q22+H+q22+H−q22,where λa,b,c=a2+b2+c2−2ab−2ac−2bc and the three helicity amplitudes Hiq2 (i=±,0) can be expressed in terms of the semileptonic B⟶V form factors:
(152)H±q2=mB+mVA1q2∓2mB∣p⟶V∣mB+mV2Vq2,H0q2=mB+mV2mVq2mB2−mV2−q2A1q2−4mB2p⟶V2mB+mV2A2q2,with the momentum ∣p⟶V∣ of the light-vector meson in the B-meson rest frame given by ∣p⟶V∣=12mBλ1/2mB2,mV2,q2.Employing the experimental measurements ofB⟶πℓν¯ℓ [96, 97], and taking advantage of the measurements of the partial branching fractions for B⟶ρℓν¯ℓ [97, 98] and B⟶ωℓν¯ℓ [96, 97], we can derive the following intervals for exclusive ∣Vub∣:
(153)Vubexc.=3.23−0.48+0.66th.−0.11+0.11exp.×10−3,fromB⟶πℓνℓ,Vubexc.=3.05−0.52+0.67th.−0.20+0.19exp.×10−3,fromB⟶ρℓνℓ,Vubexc.=2.54−0.40+0.56th.−0.19+0.18exp.×10−3,fromB⟶ωℓνℓ.The total uncertainties are obtained by adding in quadrature the separate uncertainties from individual variations of all input parameters. Apparently, the extracted values of∣Vub∣ from B⟶Vℓν¯ℓ decay are lower than that from the B⟶πℓν¯ℓ channel. We also note that the obtained values from the B⟶πℓν¯ℓ decay are in good agreements with the results from B⟶Vℓ+ℓ− decays [25]. The values of ∣Vub∣ from B⟶πℓν¯ℓ decay are in agreement with the averaged exclusive determinations presented in PDG [74], while the central values of both determinations of Vub from B⟶Vℓν¯ℓ are somewhat smaller than the corresponding result in PDG [74]. We also observe that the obtained Vub from the B⟶ωℓν¯ℓ process are significantly smaller than that from the exclusive channel B⟶ρℓν¯ℓ as already observed in [97]. All the three exclusively extracted values of Vub from the B⟶Mℓν¯ℓ decays are significantly smaller than the averaged inclusive determinations reported in [74]
(154)Vubinc.=4.49±0.15−0.17+0.16±0.17×10−3.


## 6.1. LCSR Predictions of the Form Factors

Prior to presenting the predictions of the semileptonicB⟶M decay form factors, we need to determine the input parameter used in the numerical analysis. Most importantly, the threshold and the Borel parameters:
(142)s0P,V≡n·pωsP,V,MP,V2≡n·pωMP,V,should be extracted. The threshold parameter, which also enters the QCD sum rules for the light-meson decay constant, could be fixed by the sum rule calculation of the light-meson decay constant [58]. The Borel mass is introduced to suppress the continuum-state contributions; at the same time, the LCSR requires a plain window of the Borel mass inside which the form factor is stable. Then, two requirements are employed to obtain the interval of the Borel parameter [32]:
(i)
The dependence of the form factor on the Borel mass should be small(143)∂fB⟶P,V∂ωMP,V≤0.35(ii)
The continuum contribution in the dispersion integral of the correlation function should be smaller than the light-meson channel contribution(144)∫ωsP,V∞dω′e−ω′/ωMP,VImω′Π,Π~n·p,ω′<∫0ωsP,Vdω′e−ω′/ωMP,VImω′Π,Π~n·p,ω′In Figure7, we show the Borel mass and the threshold parameter dependence of the form factor fB⟶π+. From the left figure, we see that the form factor has a stable window with respect to the Borel mass. For the later convenience, we collect the dominant inputs in Table 1 (see [81–83] for a discussion about the flavor symmetry breaking effects of the decay constants of pseudoscalar mesons).Figure 7
The Borel mass and threshold dependence of the form factorfB⟶π+0.Table 1
The numerical values of the various input parameters employed in the theory predictions of theB⟶M form factors with B-meson LCSR.
ParameterValueRef.ParameterValueRef.ms2GeV93.8±1.5±1.9MeV[74]mbmb4.193−0.035+0.022GeV[75, 76]μ1.5±0.5GeVμh1,μh2,νhmb−mb/2+mbfB192.0±4.3MeV[77]fπ130.2±1.7MeV[74]fK155.6±0.4MeV[74]fρ,213±5MeV[25]fρ,⊥1GeV160±7MeV[25]fω,197±8MeV[25]fω,⊥1GeV148±13MeV[25]fK∗,204±7MeV[25]fK∗,⊥1GeV159±6MeV[25]s0π0.70±0.05GeV2[32, 78]s0K1.05±0.05GeV2[32, 78]s0ρ,1.5±0.1GeV2[27, 79]s0ρ,⊥1.2±0.1GeV2[80]s0ω,s0ρ,∥+mω2−mρ2[27, 79]s0ω,⊥s0ρ,⊥+mω2−mρ2[80]s0K∗,s0ρ,∥+mK∗2−mρ2[27, 79]s0K∗,⊥s0ρ,⊥+mK∗2−mρ2[80]Mπ,K21.25±0.25GeV2[32, 78]Mρ21.5±0.5GeV2[79]Mω2Mρ2+mω2−mρ2[79]MK∗2Mρ2+mK∗2−mρ2[79]Except for these parameters, we still need to determine the inverse moment of theB meson λBμ0. The parameter λBμ0 could be calculated with nonperturbative HQET sum rules [84] and could also be extracted indirectly from B⟶γℓν process [57, 71, 72, 85, 86]. While at present, we are still lack of satisfied constraints of λBμ0. Following the strategy displayed in [32], we will employ a fit approach to determine λBμ0. For the B⟶P form factors, matching the LCSR calculation for the vector B⟶π form factor at q2=0 with the predictions from LCSR with pion LCDAs fB⟶π+q2=0=0.28±0.03 [18, 20, 87] will provide one with a suitable value of λBμ0. While for the B⟶V form factors, λBμ0 could be determined by matching the LCSR prediction of the form factor VB⟶ρq2=0 to that of the improved NLO LCSR with the ρ-meson LCDAs [25]. Performing such matching procedure, we obtain
(145)λBμ0=285−23+27MeV,B⟶P,343−20+22MeV,B⟶V.It is interesting to notice that the extracted values ofλBμ0 for the B⟶V form factors are in a nice agreement with that of the B⟶P form factors and are also consistent with the implications of experimental data for the two-body charmless hadronic B-meson decays from the QCD factorization approach [88]. The determined values of λB are also compatible with the recent extracted values from the B⟶γℓν¯ℓ process [89].Employing the above-determined parameters, one can explore the LCSR predictions of theB⟶M form factors in the large-recoil region. As an illustration, we show the breakdown of different contributions to the form factor fB⟶π+q2 in Figure 8. One can see from the figure that higher-twist corrections to the form factor fB⟶π+q2 are dominated by the two-particle twist-five B-meson LCDA g^−ω,μ, which can shift the leading-power prediction by an amount of approximately 20~30%. Compared to the two-particle higher-twist contribution, the three-particle higher-twist contribution only generates a tiny contribution on the form factor. We further find that the radiative correction can introduce about a O20% reduction of the corresponding LL prediction. We have also verified that such observations also hold true for the other B⟶M form factors at large hadronic recoil.Figure 8
Theq2 dependence of the form factor fB⟶π+q2. The red line represents the total contribution. The black, blue, green, and yellow correspond to the LP contribution at LL, the LP contribution at NLL, the two-particle higher-twist correction, and the three-particle higher-twist correction, respectively.


## 6.2. Phenomenological Applications

It is well-known that the light-cone operator-product expansion of the vacuum-to-B-meson correlation function is only valid in the large-recoil region [27, 32]. One then needs to extrapolate the LCSR predictions of B⟶M form factors at q2≤8GeV2 to the full kinematic region with the z-series expansion. The parameter z corresponds to a map of the entire cut q2-plane onto a unit disk ∣zq2,t0∣≤1:
(146)zq2,t0=t+−q2−t+−t0t+−q2+t+−t0.Here,t+=mB+mM2 is determined by the threshold of the lowest continuum state which can be generated by the weak transition currents in QCD. The auxiliary parameter t0, which determines the q2 point to be mapped onto the origin of the complex z plane, will be further chosen as [18, 35]
(147)t0=mB+mPmB+mP2,B⟶P,t0=mB−mV2,B⟶V.We will use the simplified Bourrely-Caprini-Lellouch (BCL) series expansion for theB⟶M form factors [90] (see [91, 92] for an alternative parametrization and [20] for more discussions for the B⟶π form factors):
(148)fB⟶P+,Tq2=fB⟶P+,T01−q2mBs∗21+∑k=1N−1bk,P+,Tzq2,t0k−z0,t0k−−1N−kkNzq2,t0N−z0,t0N,(149)fB⟶P0q2=fB⟶P001+∑k=1Nbk,P0zq2,t0k−z0,t0k,(150)fB⟶Viq2=fB⟶Vi01−q2mi,pole21+∑k=1Nbk,Vizq2,t0k−z0,t0K.The expansion parameter is small in the entire regionzq2,t02≤0.04; the expansion could be truncated at certain N. We will truncate the z-series at N=2 for the form factors fB⟶P+,T and at N=1 for all the other form factors (see [93] for further discussions on the systematic truncation uncertainties). The adopted values of the various resonance masses from the Particle Data Group (PDG) [74] and from the heavy-hadron chiral perturbation theory [94] are summarized in Table 2.Table 2
The resonance masses with different quantum numbers entering thez-series expansions of the B⟶V form factors (149) where A12=mB+mV/n·pA1−mB−mV/mBA2 and T23=mB/n·pT2−T3.
fB⟶Viq2JPb⟶d (in GeV)b⟶s (in GeV)Vq2, T1q21−5.3255.415A0q20−5.2795.366A1q2, A12q2, T2q2, T23q21+5.7245.829Table3 shows the numerical predictions of the form factors fB⟶π+,0 at q2=0GeV2. In this table, we display the significant uncertainties from different input parameters. One concludes from the table that the parameter λB introduces an about 10% uncertainty to the form factors. The fitted z-expansion parameters are also included in the table.Table 3
Theory predictions for the form factorsfB⟶π+,00 and the corresponding z-expansion shape parameters b1,π+,0 with the dominant uncertainties from variations of different input parameters.
ParametersCentral valueλBσ^1μMπ2s0πfB⟶π+,000.280+0.031−0.030+0.013−0.012−0.032+0.000−0.017+0.012−0.014+0.014b1,π+−2.77−0.02+0.05−0.01+0.02−0.16+0.09−0.03+0.02−0.07+0.07b1,π0−4.88+0.11−0.10+0.04−0.04−0.61+0.17−0.06+0.04−0.11+0.11Having at our disposal the theory predictions forB⟶π form factors, we proceed to explore phenomenological aspects of the semileptonic B⟶πℓν decays, which serves as the golden channel for the determination of CKM matrix element ∣Vub∣ exclusively (see [95] for the future advances of precision measurements of Belle II). It is straightforward to write down the differential decay rate for B⟶πℓν¯ℓ:
(151)dΓB⟶πℓν¯ℓdq2=GF2Vub2192π3mB3λ32mB2,mπ2,q21−ml2q221+ml22q2fB⟶π+q2∣2+3ml2mB2−mπ22λmB2,mπ2,q2ml2+2q2fB⟶π0q22,dΓB⟶Vℓν¯ℓdq2=GF2Vub2192π3mB3q2cV2λ1/2mB2,mV2,q2H0q22+H+q22+H−q22,where λa,b,c=a2+b2+c2−2ab−2ac−2bc and the three helicity amplitudes Hiq2 (i=±,0) can be expressed in terms of the semileptonic B⟶V form factors:
(152)H±q2=mB+mVA1q2∓2mB∣p⟶V∣mB+mV2Vq2,H0q2=mB+mV2mVq2mB2−mV2−q2A1q2−4mB2p⟶V2mB+mV2A2q2,with the momentum ∣p⟶V∣ of the light-vector meson in the B-meson rest frame given by ∣p⟶V∣=12mBλ1/2mB2,mV2,q2.Employing the experimental measurements ofB⟶πℓν¯ℓ [96, 97], and taking advantage of the measurements of the partial branching fractions for B⟶ρℓν¯ℓ [97, 98] and B⟶ωℓν¯ℓ [96, 97], we can derive the following intervals for exclusive ∣Vub∣:
(153)Vubexc.=3.23−0.48+0.66th.−0.11+0.11exp.×10−3,fromB⟶πℓνℓ,Vubexc.=3.05−0.52+0.67th.−0.20+0.19exp.×10−3,fromB⟶ρℓνℓ,Vubexc.=2.54−0.40+0.56th.−0.19+0.18exp.×10−3,fromB⟶ωℓνℓ.The total uncertainties are obtained by adding in quadrature the separate uncertainties from individual variations of all input parameters. Apparently, the extracted values of∣Vub∣ from B⟶Vℓν¯ℓ decay are lower than that from the B⟶πℓν¯ℓ channel. We also note that the obtained values from the B⟶πℓν¯ℓ decay are in good agreements with the results from B⟶Vℓ+ℓ− decays [25]. The values of ∣Vub∣ from B⟶πℓν¯ℓ decay are in agreement with the averaged exclusive determinations presented in PDG [74], while the central values of both determinations of Vub from B⟶Vℓν¯ℓ are somewhat smaller than the corresponding result in PDG [74]. We also observe that the obtained Vub from the B⟶ωℓν¯ℓ process are significantly smaller than that from the exclusive channel B⟶ρℓν¯ℓ as already observed in [97]. All the three exclusively extracted values of Vub from the B⟶Mℓν¯ℓ decays are significantly smaller than the averaged inclusive determinations reported in [74]
(154)Vubinc.=4.49±0.15−0.17+0.16±0.17×10−3.


## 7. Summary

In this review, we have discussed the LCSR withB-meson LCDAs and its application to calculating the B⟶P and B⟶V form factors. The fundamental nonperturbative inputs in this approach are the LCDAs of the B meson, which are defined in terms of nonlocal operators including the effective b-quark field and the soft light parton fields sandwiched between the vacuum and the B-meson state. At the one-loop level, the leading-twist B-meson LCDA satisfies the LN equation, which can be simplified by some kinds of integral transforms. The two-loop level evolution equation of the leading-twist B-meson LCDA has also been obtained. The evolution function is useful in the determination of the models of the LCDAs, especially the behavior at the endpoint. The higher-twist B-meson LCDAs can be introduced by adjusting the spinor structure, or adding an additional gluon field, or taking the higher power contribution from the light-cone expansion. They can be studied using a similar method as the leading-twist one.We introduced the investigation of theB⟶M form factors with the B-meson LCSR in detail. To obtain the sum rules, one must start from the correlation function which is defined as the vacuum-to-B-meson matrix element of the time-ordered product of the weak current and the interpolation current of the light meson. At the partonic level, the correlation function can be factorized into the convolution of the hard function, the jet function, and the LCDAs of the B meson. The short-distance hard function and jet function can be calculated by choosing free partonic external states. To evaluate the NLO corrections to the hard function and the jet function, one might employ the method of regions, i.e., the loop momentum was assigned to be hard, collinear, and soft, respectively. The loop integral was calculated in different regions, and the hard function and jet function can be extracted directly. An alternative approach to obtain the NLO hard function and jet function is to perform the match from QCD first to SCETI and then to SCETII, respectively. The most subtle place in the loop calculation is the infrared subtraction, especially under the condition that the evanescent operators took the place. The convolution of the hard function, the jet function, and the LCDAs is independent of the factorization scale. However, the large logarithmic terms need to be resummed with the RG equation approach. Having the hard function and the jet function at hand, the correlation function can be expressed in terms of the dispersion integral. After this, we employed the quark-hadron duality assumption and the Borel transformation and then obtained the sum rules for the form factors.We have discussed the power suppressed contributions to the form factors. The higher-twist corrections to the form factors from both the two-particle and three-particleB-meson LCDAs were calculated at tree level. For the contribution from three-particle B-meson LCDAs, we employed the quark propagator in the background field. The quark-mass-dependent term in the sum rules with three-particle B-meson LCDAs is power suppressed, which is different from the leading-twist contribution. Except for the high-twist contribution, there exist more sources of power suppressed contributions, such as the power suppressed contribution from heavy quark expansion, the power suppressed contribution from hard-collinear propagators, and the local terms. The investigation of power suppressed contributions is one of the main goals of future studies.After employing appropriate input parameters, we evaluated theB⟶P and B⟶V form factors numerically within the momentum region 0<q2<8GeV2 where the LCSR with B-meson LCSR is applicable. Then, we extrapolated the result to the whole physical region by the z-series expansion. Applying the experimental measurements of the B⟶πℓν¯ℓ process and taking advantage of the measurements of the partial branching fractions for B⟶ρℓν¯ℓ and B⟶ωℓν¯ℓ decays, we obtained intervals for exclusive Vub. The extracted values of Vub from B⟶Vℓν¯ℓ decay are lower than that from the B⟶πℓν¯ℓ channel. The values of Vub from B⟶πℓν¯ℓ decay are in agreement with the averaged exclusive determinations presented in PDG. All of the three exclusively extracted values of ∣Vub∣ from the B⟶Mℓν¯ℓ decays are significantly smaller than the averaged inclusive determinations.



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