Evolution equation for the B-meson distribution amplitudein the heavy-quark effective theory in coordinate space

# Evolution equation for the B-meson distribution amplitude in the heavy-quark effective theory in coordinate space

## Abstract

The -meson distribution amplitude (DA) is defined as the matrix element of a quark-antiquark bilocal light-cone operator in the heavy-quark effective theory, corresponding to a long-distance component in the factorization formula for exclusive -meson decays. The evolution equation for the -meson DA is governed by the cusp anomalous dimension as well as the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi-type anomalous dimension, and these anomalous dimensions give the “quasilocal” kernel in the coordinate-space representation. We show that this evolution equation can be solved analytically in the coordinate-space, accomplishing the relevant Sudakov resummation at the next-to-leading logarithmic accuracy. The quasilocal nature leads to a quite simple form of our solution which determines the -meson DA with a quark-antiquark light-cone separation in terms of the DA at a lower renormalization scale with smaller interquark separations (). This formula allows us to present rigorous calculation of the -meson DA at the factorization scale for less than  GeV, using the recently obtained operator product expansion of the DA as the input at  GeV. We also derive the master formula, which reexpresses the integrals of the DA at for the factorization formula by the compact integrals of the DA at  GeV.

###### pacs:
12.38.Cy, 12.39.Hg, 12.39.St, 13.25.Hw
1

## I Introduction

The -meson light-cone distribution amplitude (LCDA) is one of the important ingredients of the QCD factorization formula for exclusive decays Beneke:2000ry (); Bauer:2001cu (); Li:2003yj () and has recently attracted much attention due to its central role for the analysis of the experimental data, e.g., hadronic and radiative -decay data Antonelli:2009ws (). The -meson LCDA appears in the factorization formula for hard spectator interaction amplitudes, where a large momentum is transferred to the spectator light-quark via gluon exchange Beneke:2000ry (); Bauer:2001cu (); Li:2003yj (); Antonelli:2009ws (); Korchemsky:1999qb (); BBNL (); BBNL2 (); Beneke:2000wa (); Beneke:2005vv (); Bell08 (), and represents the nonperturbative matrix element that describes the leading amplitude to have the valence quark and antiquark with a light-like separation inside the meson Szczepaniak:1990dt (). Grozin and Neubert Grozin:1997pq () studied constraints on the -meson LCDA from the equations of motion, heavy-quark symmetry and the renormalization group (RG), and they gave the first quantitative estimate of the LCDA using QCD sum rules with the leading perturbative and nonperturbative effects taken into account. The light-cone QCD sum rules for the -decay form factors were also used Ball:2003fq (); Khodjamirian:2005ea () to estimate the first inverse moment foot1 () of the LCDA, which participates in the corresponding factorization formula. The Grozin-Neubert’s QCD sum rule calculation was extended by Braun, Ivanov and Korchemsky Braun:2003wx () including the perturbative and nonperturbative corrections, and the importance of the NLO perturbative corrections was emphasized. Indeed, the true non-analytic behavior of the -meson LCDA associated with the “cusp singularities” Korchemsky:1987wg () is only revealed at this level including the radiative corrections Lange:2003ff (), and it is this behavior that renders the (non-negative) moments of the LCDA divergent, even after renormalization Grozin:1997pq () (see Ball:2008fw () for a similar behavior in three-quark LCDAs for the baryon). Introducing the regularization for the moments with an additional momentum cutoff, Lee and Neubert Lee:2005gza () evaluated the first two moments for a large value of the cutoff in terms of the operator product expansion (OPE) with the NLO perturbative corrections, as well as the power corrections which are generated by the local operators of dimension 4, and they used the results as constrains on model building of the -meson LCDA.

Another feature peculiar to the -meson LCDA is that it involves a complicated mixture of the multiparticle Fock states of higher-twist nature through nonperturbative quark-gluon interactions, as demonstrated using the equations of motion and heavy-quark symmetry Grozin:1997pq (); KKQT (); GW (). A first systematic treatment of the mixing of the multiparticle states, disentangling the singularities from the radiative corrections, has recently been accomplished by the present authors, and the -meson LCDA is obtained in a form of the OPE as the short-distance expansion for the quark-antiquark light-cone separation, with the subleading and subsubleading power corrections, generated by the local operators of dimension and 5, respectively, and the NLO corrections for the corresponding Wilson coefficients Kawamura:2008vq (). This OPE enables us to evaluate the -meson LCDA for interquark distances with , where is the renormalization scale of the LCDA, in a rigorous way in terms of three nonperturbative parameters in the heavy-quark effective theory (HQET), one of which is the usual mass difference between the -meson and -quark, , associated with matrix elements of dimension-4 operators, and the other two are the novel HQET parameters Grozin:1997pq (); KKQT (); GW () associated with matrix elements of the quark-antiquark-gluon three-body operators of dimension 5. Note that the range of where the OPE is directly applicable becomes wider for the smaller value of the scale , as : choosing  GeV, corresponding to typical hadronic scale, the model-independent result for interquark distances  GeV has been obtained from the OPE and this result has also been used to constrain the behavior of the LCDA at large distances  GeV Kawamura:2008vq (). Those results of the -meson LCDA at  GeV have to be evolved to the factorization scale of order that corresponds to the characteristic “hard-collinear” scale for hard spectator scattering in exclusive decays Beneke:2000ry (); Bauer:2001cu (); Li:2003yj (), when we substitute the LCDA into the relevant factorization formula.

For this purpose, in principle, we can utilize the analytic solution for the evolution equation of the -meson LCDA obtained in Lange:2003ff (); Lee:2005gza (). However, the corresponding solution is directly applicable when the LCDA is given in the momentum representation, which we find inconvenient in our case: the Fourier transformation of the above OPE-based results to the momentum space mixes up the model-independent behavior for  GeV with the behavior for  GeV which relies on a certain model for the large behavior. On the other hand, it has been noted that the relevant evolution kernel embodies the particularly simple geometrical structure in the coordinate-space representation Braun:2003wx (). These facts motivate us to treat the evolution of the -meson LCDA in an unconventional way, working in the coordinate-space representation. We are able to find the analytic solution for the corresponding evolution equation, and demonstrate that the solution determines the -meson LCDA in terms of the LCDA at a lower scale with smaller interquark separations and thus preserves the boundary at  GeV between the model-independent and -dependent behaviors of our LCDA, even after evolving from  GeV to . We emphasize that such simple RG structure of the -meson LCDA can be manifested only in the coordinate space. Furthermore, as we shall demonstrate, it is this simple structure that enables us to derive the master formula, by which the relevant integrals of the LCDA at the scale , arising in the factorization formula for the exclusive -meson decays, can be reexpressed by the compact integrals of the LCDA at the scale  GeV. Therefore, we believe that the coordinate-space approach for the RG evolution of the -meson LCDA deserves detailed discussions in the present paper. We also show that our solution can be organized so as to include the Sudakov resummation to the next-to-leading logarithmic (NLL) accuracy, taking into account the effects of the anomalous dimension at the two-loop level which is associated with the cusp singularity. We present the first rigorous result of the -meson LCDA at the relevant factorization scale for  GeV. Combining with the results for the long-distance behavior, we discuss an estimate for the inverse moments of the LCDA at .

The paper is organized as follows. Sec. II is mainly introductory; we give the operator definition of the -meson LCDA, explain the result for its renormalization in the coordinate space, and derive the corresponding RG evolution equation. We demonstrate in Sec. III that, as the solution of this equation, we can obtain the new coordinate-space representation for the evolution of the -meson LCDA, which manifests the simple RG structure, and also organize the result so as to include the Sudakov resummation at the NLL-level. In Sec. IV, we use our coordinate-space representation of the evolution to derive a compact and closed formula for the inverse moments of the LCDA in terms of the certain integrals of the LCDA at a lower scale . Application of our results to calculate the evolution of the OPE-based form of the -meson LCDA is presented in Sec. V, and we discuss an estimate for the inverse moments of the LCDA. Sec. VI is reserved for conclusions.

## Ii Definition and renormalization in the coordinate space

The leading quark-antiquark component of the -meson LCDA is defined as the vacuum-to-meson matrix element in the HQET Grozin:1997pq ():

 (1)

where is the light-antiquark field, is the effective heavy-quark field, and these fields form a gauge-invariant bilocal operator linked by a light-like Wilson line,

 [tn,0]=Pexp[ig∫t0dλ n⋅A(λn)] , (2)

with as the light-like vector, and , and denoting the 4-velocity of the meson. The bilocal operator is renormalized at the scale and, here and below, refers to the renormalization scale. In the definition (1),

 (3)

denotes the -meson decay constant in the HQET Neubert:1994mb () and in the RHS is the LCDA in the momentum representation where denotes the light-cone “”-component of the momentum carried by the light antiquark.

The renormalization of the bilocal operator of (1) was studied in Grozin:1997pq (); Lange:2003ff (), calculating the UV divergence in the one-loop diagrams of Fig. 1 in the momentum space (see also Grozin:2005iz (); DescotesGenon:2009hk ()). The calculation of those diagrams has also been carried out in the coordinate space Braun:2003wx (); Kawamura:2008vq (), and the result yields the renormalization of the bilocal operator in the coordinate-space representation as (unless otherwise indicated, )

 Θbare(t) =Θren(t,μ)+αsCF2π{(−12ε2−Lε+14ε)Θren(t,μ) (4) +1ε∫10dz(z1−z)+Θren(zt,μ)} ,

connecting the bare and renormalized operators by the “-dependent” renormalization constant in dimensions and Feynman gauge, where , and

 L=ln[i(t−i0)μeγE] , (5)

with the Euler constant and the “” prescription coming from the position of the pole in the relevant propagators in the coordinate space. The plus-distribution is defined, as usual, as

 ∫10dz(z1−z)+f(z)≡∫10dzz[f(z)−f(1)]1−z , (6)

for a smooth test function . In the one-loop contributions in (4), the first two terms, the double-pole term and the single-pole term involving , manifest the cusp singularity Korchemsky:1987wg () in Fig. 1 (a), i.e., the singularity in the radiative correction around the cusp (at the origin) in the Wilson line,

 [tn,0][0,−∞v] , (7)

which is contained in (1), using the relation . The last term in (4) comes from Fig. 1 (b), accompanying the plus-distribution characteristic of the loop integral associated with the massless degrees of freedom only, while the remaining one-loop term comes from the contribution of the renormalization constants of the two quark fields, and . We note that Fig. 1 (c) is UV-finite in the Feynman gauge foot2 () and does not contribute to (4).

The RG invariance to the one-loop accuracy based on (4) implies that the -meson LCDA (1) obeys the evolution equation in the coordinate space,

 μddμ~ϕ+(t,μ)=−[Γcusp(αs)L+γF(αs)]~ϕ+(t,μ)+∫10dzK(z,αs)~ϕ+(zt,μ) , (8)

with the one-loop RG functions

 Γcusp(αs) =Γ(1)cuspαs4π ,Γ(1)cusp=4CF , (9) γF(αs) =γ(1)Fαs4π ,γ(1)F=2CF , (10)

and

 K(z,αs)=K(1)(z)αs4π ,K(1)(z)=4CF(z1−z)+ . (11)

Here, corresponds to the anomalous dimension of the Wilson line with a cusp, (7), and coincides with the LO term of the universal cusp anomalous dimension of Wilson loops with light-like segments Korchemsky:1987wg (); we obtain the finite result (9), because the contribution from the double-pole term of (4) is canceled by the contribution generated by taking the derivative of in the next term with respect to . of (10) represents the anomalous dimension from the above-mentioned contribution of the renormalization constants of the two quark fields, combined with the so-called hybrid anomalous dimension of heavy-light currents in the HQET Neubert:1994mb (), which governs the scale dependence of the decay constant (3) as, at one-loop accuracy,

 μddμF(μ)=3CFαs4πF(μ) . (12)

of (11) comes from the last term of (4) and represents the -dependent anomalous dimension associated with the massless degrees of freedom only. We note the remarkable property in (8) that the evolution kernel in the RHS, composed of (9)-(11), is quasilocal, such that the evolution mixes the LCDA with itself and with the LCDA associated with smaller light-cone separation (). This is due to the similar structure appearing in the renormalization in the coordinate space, (4), and reflects Braun:2003wx () the fact that the cusp renormalization induced by Fig. 1 (a) is multiplicative in the coordinate space Korchemsky:1987wg () while Fig. 1 (b) gives the contribution identical to the similar correction to the light-quark-antiquark bilocal operators, which embodies simple geometrical structure in the coordinate space so as to mix operators associated with smaller “size” Balitsky:1987bk ().

It is straightforward to perform the Fourier transformation of (8) to the momentum space and derive the evolution equation for of (1), using

 12π∫∞−∞dteiωtL~ϕ+(t,μ) (13) =i2π∫∞0dω′(1ω′−ω−i0lnω′−ω−i0μ−1ω′−ω+i0lnω′−ω+i0μ)ϕ+(ω′,μ) =−ϕ+(ω,μ)lnωμ−∫∞0dω′θ(ω−ω′)ω−ω′[ϕ+(ω′,μ)−ϕ+(ω,μ)] ,

and

 12π∫∞−∞dteiωt∫10dz(z1−z)+~ϕ+(zt,μ) (14) =ϕ+(ω,μ)+∫∞0dω′ωω′θ(ω′−ω)ω′−ω[ϕ+(ω′,μ)−ϕ+(ω,μ)] ,

and, indeed, the result reproduces the evolution equation obtained through the renormalization of the bilocal operator of (1) in the momentum space Lange:2003ff (); Grozin:2005iz (); DescotesGenon:2009hk (). We note that the momentum representation of the kernel in (14) coincides with (a part of) the Brodsky-Lepage kernel for the pion LCDA Lepage:1979zb () and physically represents a Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) splitting function that vanishes for . On the other hand, a local contribution in (8), associated with the cusp anomalous dimension , yields the new evolution kernel for in the RHS of (13), so that the evolution in the momentum space mixes the LCDA with over the entire region,  Lange:2003ff (); Grozin:2005iz (); Lee:2005gza (). We also note that we cannot derive the moment-space representation of the evolution equation (8) in a usual way as in the case of the LCDAs for the light mesons Braun:1990iv (), because the presence of the logarithm (5) prevents us from performing the Taylor expansion of (8) about ; indeed, (4) shows that the renormalized LCDA is non-analytic at (see also the discussion in Braun:2003wx (); Kawamura:2008vq ()). Thus, the evolution equation for the -meson LCDA manifests simple geometrical structure as the quasilocality of the kernel only in the coordinate-space representation (8).

One may anticipate that the evolution equation (8) would hold to all orders in perturbation theory by taking into account the higher-loop terms in the RG functions (9)-(11). This is correct, at least, for a particular class of higher-loop corrections associated with the universal cusp anomalous dimension of Wilson loops. For example, when we take into account the diagrams that correspond to the two-loop corrections to the relevant Wilson line (7), of (9) gets modified into Korchemsky:1987wg ()

 Γcusp(αs)=Γ(1)cuspαs4π+Γ(2)cusp(αs4π)2 , (15)
 (16)

where and denotes the number of active flavors. Actually, it is not known at present whether the effects of all the other two-loop corrections to the bilocal operator in (1) can be absorbed into the remaining two RG functions in (8), and , as their two-loop terms. Still, the evolution equation (8), with (10), (11) and the two-loop cusp anomalous dimension (15) taken into account, is useful for resumming the Sudakov logarithms to a consistent accuracy, as we will demonstrate in the next section.

## Iii Analytic solution in the coordinate space

The LO solution for the evolution equation of the -meson LCDA was obtained in the momentum representation in Lange:2003ff (), and the result determines of (1) as the convolution of at a lower scale and the (complicated) evolution operator, over the entire range of (see (79) in Appendix A). Its Fourier transformation in principle gives the solution for our evolution equation, (8)-(11), in the coordinate space, but, in practice, we find it more useful to solve (8) directly: mathematically, (8) is an integro-differential equation of similar type as the corresponding equation in the momentum space and has simpler structure for the kernel of integral operator than the latter case, as noted in Sec. II. This would imply that the strategy devised in Lange:2003ff () to solve the evolution equation for the latter case should also allow us to solve (8), possibly with simpler manipulations. Moreover, intermediate steps of those manipulations reveal peculiar structures behind a rather simple final form of our solution, (35) below.

First of all, we demonstrate that the strategy of Lange:2003ff () is applicable to (8) and allows us to construct its general solution which is exact even when the higher-loop terms in the RG functions , , are taken into account. For this purpose, we further put forward the above-mentioned similarity between (8) and the corresponding integro-differential equation of Lange:2003ff () in the momentum space, by performing the analytic continuation for (8) as . Then the evolution equation (8) becomes the integro-differential equation for the -meson LCDA at imaginary light-cone separation, , as

 μddμ~ϕ+(−iτ,μ)=−[Γcusp(αs)ln(τμeγE)+γF(αs)]~ϕ+(−iτ,μ)+∫10dzK(z,αs)~ϕ+(−izτ,μ) . (17)

We recall that the kernel corresponds to the coordinate-space representation of a DGLAP-type splitting function and thus can be diagonalized in the moment space. In the coordinate-space language Balitsky:1987bk (), the corresponding moment is given as

 K(j,αs)=∫10dzzjK(z,αs)=K(1)(j)αs4π+⋯ , (18)

where (11) gives the coefficient for the order term as

 K(1)(j)=4CF∫10dzzj(z1−z)+=−4CF[ψ(j+2)+γE−1] , (19)

with being the di-gamma function, and the ellipses in (18) stand for the (presently unknown) terms of order and higher. As mentioned below (14), however, the usual moment is not useful for treating (17): the presence of logarithm in the RHS suggests that the values for the moment will be modified under the variation of the scale . The authors in Lange:2003ff () demonstrated that taking into account the corresponding “evolution” of the moment indeed enables them to construct the general solution of the corresponding integro-differential equation in the momentum space (see also Grozin:2005iz ()). Thus, we take the ansatz,

 ~ϕ+(−iτ,μ)=12πi∫c+i∞c−i∞dj(τμ0eγE)j−ξ(μ,μ0)φ(j,μ) , (20)

with a real constant , and we determine and such that (20) obeys (17). This ansatz has the form similar to the inverse Mellin transformation to construct the solution for the DGLAP-type evolution equation in the coordinate-space language (see Balitsky:1987bk ()), except for the contribution of , which describes the evolution of the power of from a certain (low) scale to the scale . We assume , without loss of generality, and multiplied by is put in the integrand of (20) for convenience. Substituting (20) into (17), we obtain

 μddμφ(j,μ) =[−Γcusp(αs)ln(τμeγE)−γF(αs)+K(j−ξ(μ,μ0), αs) (21) + μdξ(μ,μ0)dμln(τμ0eγE)]φ(j,μ) .

Because the RHS of this equation should be independent of , obeys

 μdξ(μ,μ0)dμ=Γcusp(αs) . (22)

This shows that is independent of and is integrated to give, introducing the function, ,

 ξ(μ,μ0)=∫αs(μ)αs(μ0)Γcusp(α)β(α)dα≡Ξ(αs(μ),αs(μ0)) . (23)

Now (21) reduces to

 μddμφ(j,μ)=[−Γcusp(αs)lnμμ0−γF(αs)+K(j−ξ(μ,μ0), αs)]φ(j,μ) , (24)

and this simple differential equation is immediately solved to give

 φ(j,μ)=exp[V(μ,μ0)+W(μ,μ0,j)]φ(j,μ0) , (25)

where

 V(μ,μ0) =−∫αs(μ)αs(μ0)dαβ(α)[Γcusp(α)∫ααs(μ0)dα′β(α′)+γF(α)] , (26) W(μ,μ0,j) =∫αs(μ)αs(μ0)dαβ(α)K(j−Ξ(α,αs(μ0)), α) , (27)

and should be expressed by the Mellin transform of the initial condition, , from (20) (see (29) below). Substituting these results into (20), we obtain

 ~ϕ+(−iτ,μ)=eV(μ,μ0)(τμ0eγE)−ξ∫∞0dτ′τ′~ϕ+(−iτ′,μ0)∫c+i∞c−i∞dj2πi(ττ′)jeW(μ,μ0,j) . (28)

Here and below, , unless otherwise indicated. The formula (28) in principle gives the solution for (17), which is exact even when the higher-order terms in , , are taken into account. However, this solution has been obtained by assuming tacitly that of (20), expressed by the Mellin transform of as

 φ(j,μ)=∫∞0dττ(τμ0eγE)−j(τμ0eγE)ξ~ϕ+(−iτ,μ) , (29)

is a regular function in a certain “band” of region in the complex plane, and that the constant in (28) is chosen such that the integration contour in this formula is contained within this band. Now we consider the condition for the convergence of the integral in (29), which in turn determines this band, as well as the range where (28) is applicable: the short-distance behavior of as in the integrand of (29) can be determined Lee:2005gza (); Kawamura:2008vq () by perturbation theory, as a constant modulo (see (65) below), so that (29) is convergent for the integration region when . On the other hand, studies of the IR structure of the DA indicate or more strongly suppressed as  Grozin:1997pq (); KKQT (), so that the integral in (29) is convergent as when . These considerations show that (29) gives a regular function for the band with in the complex plane, and the constant in (20), (28) should be chosen as

 ξ−2

Because of (23) grows from 0, as increases from (see (9), (15), and (16)), only for the values of scales and satisfying

 ξ=ξ(μ,μ0)<2 , (31)

can be chosen as a fixed constant and thus the solution (28) describes the exact evolution of the meson LCDA from to . (Note that the condition for the convergence of the convolution integrals of (28) and the corresponding hard part in the QCD factorization formula for exclusive decays eventually requires (60) below.)

To proceed further, we change the integration variable in (27) from to . Defining such that , we obtain

 W(μ,μ0,j)=∫ξ0dxK(j−x, αx)Γcusp(αx)=∫ξ0dx K(1)(j−x)Γ(1)cusp+⋯ , (32)

where the ellipses stand for the NLO or higher contributions that involve the two- or higher-loop anomalous dimensions. Using (9) and (19), one finds

 eW(μ,μ0,j)=e(1−γE)ξ Γ(j+2−ξ)Γ(j+2) , (33)

up to the corrections of the two-loop level. In the complex plane, (33) has poles at with , which are all located in the left of the integration contour in (28) with (30): by the theorem of residues, these poles give rise to nonzero contribution to the integral for , while the integral vanishes for . Evaluation of those pole contributions yields

 ∫c+i∞c−i∞dj2πi(ττ′)jΓ(j+2−ξ)Γ(j+2) =θ(τ−τ′)∞∑n=0(−1)nn!Γ(ξ−n)(ττ′)ξ−2−n (34) =θ(τ−τ′)(τ′/τ)2−ξ(1−τ′/τ)1−ξΓ(ξ) .

Substituting this result into (28) and changing the integration variable from to , we obtain

 ~ϕ+(−iτ,μ)=eV(μ,μ0)(τμ0eγE)−ξe(1−γE)ξΓ(ξ)∫10dz(z1−z)1−ξ~ϕ+(−iτz,μ0) , (35)

which is exact up to the NLO corrections mentioned in (32). The contribution of the kernel (11) in perturbation theory receives the RG improvement in (35) as with the modified power , where of (23) is induced by the cusp anomalous dimension. For the case with , we have the singular behavior as , but this eventually gives the finite contribution to the RHS of (35), combined with the behavior of the gamma function, . This also shows that the RHS of (35) reduces to when , i.e., when and (see (23), (26)), as it should be. In (35), it is straightforward to perform the analytic continuation from the imaginary light-cone separation to the real one, as , and the resulting solution for the evolution equation (8) embodies a quite simple structure to determine the -meson LCDA with a quark-antiquark light-cone separation in terms of the LCDA at a lower renormalization scale with smaller interquark separations. The Fourier transformation of this result is calculated in Appendix A, and the obtained momentum representation (79) reproduces the result of Lange:2003ff (); Lee:2005gza () derived in the momentum space; in particular, the factor in (35), which is non-analytic at , produces the radiative tail as for large in (79), which renders all non-negative moments of the LCDA, with , divergent, irrespective of the initial behavior,  Lange:2003ff (). We also emphasize that our result (35) has a much simpler structure than (79); i.e., the most compact expression possible for calculating the evolution of the -meson LCDA under changes of the renormalization scale is provided by our coordinate-space result (35).

We note that the first two factors in (35), given by

 eV(μ,μ0)(τμ0eγE)−ξ=eV(μ,μ0)−ξln(τμ0eγE) , (36)

are unaffected by the above manipulations (33), (34), which are valid up to the NLO corrections, and thus (36) gives the exact result even when the higher-order terms in and are taken into account for (23) and (26). Indeed, substituting those definitions of and , we may reexpress the exponent of the RHS in (36) as

 V(μ,μ0)−ξln(τμ0eγE)=−∫μμ0dμ′μ′[Γcusp(αs(μ′))ln(τμ′eγE)+γF(αs(μ′))] , (37)

which shows that (36) corresponds to the general solution of the evolution equation (17) with the contribution of the kernel omitted foot3 (). We now derive the explicit form of (23) and (26) arising in our solution (28), (35): substituting (9), (10), (15) and the usual perturbative expansion for the function,

 β(αs) =μdαsdμ=−2αs∞∑n=0βn(αs4π)n+1 , β0 =113CG−23Nf ,    β1=343C2G−103CGNf−2CFNf , ⋯ , (38)

a straightforward calculation gives

 ξ(μ,μ0)=Γ(1)cusp2β0⎧⎨⎩lnαs(μ0)αs(μ)+αs(μ0)−αs(μ)4π⎛⎝Γ(2)cuspΓ(1)cusp−β1β0⎞⎠⎫⎬⎭+⋯ , (39)

and

 V(μ,μ0)=Γ(1)cusp4β20{4παs(μ0)(1+lnαs(μ0)αs(μ))−4παs(μ)}−γ(1)F2β0lnαs(μ0)αs(μ) (40) +Γ(1)cusp4β20⎧⎨⎩β12β0ln2αs(μ0)αs(μ)+⎛⎝Γ(2)cuspΓ(1)cusp−β1β0⎞⎠(αs(μ0)−αs(μ)αs(μ0)−lnαs(μ0)αs(μ))⎫⎬⎭+⋯ ,

where the ellipses stand for the terms that are down by compared with the preceding terms and receive the contributions due to higher loops, e.g., the three-loop cusp anomalous dimension , the two-loop local anomalous dimension , etc. If we substitute only the one-loop terms of these results, the first term of (39) and the first line of (40), into (35), we obtain the explicit analytic form of the solution for the evolution equation (8), exact at the one-loop level with (9)-(11).

The definition (26) shows that involves the contribution associated with the first term in the RHS of (24), i.e., the cusp anomalous dimension accompanying . As a result, in (40), the contributions associated with the cusp anomalous dimension are enhanced by the factor induced by this logarithm, compared to the contribution from the second term of (26) with the local anomalous dimension (10): the leading term is given by the one-loop cusp anomalous dimension , while the one-loop local anomalous dimension contributes to the next-to-leading term, i.e., at the same level as the two-loop cusp anomalous dimension. Here, the contribution due to corresponds to the one-loop level in the usual RG-improved perturbation theory, and thus the treatment at this level has to be complemented with the two-loop contributions associated with the cusp anomalous dimension, the second line of (40). This pattern is characteristic of the Sudakov-type large logarithmic effects induced by the cusp anomalous dimension. This fact also requires us to reorganize our result (35) as well as (40) according to consistent order counting of those logarithmic contributions. This can be achieved by introducing

 χ=β0αs(μ)4πlnμ2μ20 , (41)

and by following the standard procedure used in the soft gluon resummation formalism in QCD Catani:2003zt (): we organize (40) by a systematic large logarithmic expansion, where is formally considered of order unity and the small expansion parameter is , leading to

 lnαs(μ0)αs(μ)=−ln(1−χ)−αs(μ)4πβ1β0ln(1−χ)1−χ+O(α2s) . (42)

Substituting this expansion, (40) is recast into

 V(μ,μ0)=4παs(μ)h(0)(χ)+h(1)(χ) , (43)

up to the corrections of , with

 h(0)(χ) =Γ(1)cusp4β20[(χ−1)ln(1−χ)−χ] , (44) h(1)(χ) =Γ(1)cusp4β20⎡⎣−β12β0ln2(1−χ)+⎛⎝Γ(2)cuspΓ(1)cusp−β1β0⎞⎠(ln(1−χ)+χ)⎤⎦+γ(1)F2β0ln(1−χ) . (45)

In (43), the first and the second terms, and , collect the terms and , respectively, with , corresponding to the LL and NLL contributions, while the corrections omitted from (43) correspond to the NNLL or higher level. Thus, the first factor in (35) is the exponentiation of the logarithmic terms with , playing analogous role as the Sudakov form factor in the soft gluon resummation in QCD Catani:2003zt (). It is straightforward to see that this factor with only the LL term, , retained in the exponent (43) corresponds to the double leading logarithmic approximation summing up the towers of logarithms , and with the exponent (43) at the NLL accuracy resums the first three towers of logarithms, with , , and , exactly, to all orders in .

The logarithmic expansion (42) can be also applied to (39), yielding

 ξ(μ,μ0)=−Γ(1)cusp2β0ln(1−χ) , (46)

up to the corrections of . This result does not receive the logarithmic enhancement but obeys order counting similar as the contribution from the second term in (26) due to the local anomalous dimension , as apparent comparing (23) with (26). Thus, the substitution of (46) into (35) produces the NLL-level contributions while the omitted contributions correspond to the NNLL or higher level, using the order counting similar as in (43). Therefore, our solution (35) with (43)-(46) substituted embodies the evolution of the -meson LCDA, accomplishing the relevant Sudakov resummation, and is exact up to the corrections of the NNLL-level. We note that controlling the NNLL-level effects completely requires to take into account the three-loop cusp anomalous dimension , as well as the local anomalous dimension and DGLAP-type splitting function at the two-loop level, and , in (28) with (26), (27).

Before ending this section, we mention that the factor accompanying in the RHS of (36) could produce an additional logarithmic enhancement. Also, in the integrand of (35), the behavior as and could receive another logarithmic enhancement. These facts suggest that the higher-order terms associated with the similar types of logarithms could be relevant if we intend to determine the precise shape of the LCDA at the “edge”. However, we do not go into the details of such higher-order effects here: systematic treatment of those higher-order effects would require an approach, which is beyond the scope of this work based on the evolution equation for the renormalization scale . Furthermore, it is the integrals of the LCDA over , like (48) below, that is eventually relevant to exclusive decays, and the above types of higher-order logarithmic effects at the edge region should play minor roles on the value of those convergent integrals, where the only relevant logarithm is treated in this section.

## Iv Master formula for the inverse moments of the LCDA

The -meson LCDA (1) with participates in the QCD factorization formula for exclusive decays Beneke:2000ry (); Beneke:2000wa (); Beneke:2005vv (); Bell08 () through the inverse moments,

 λ−1B(μ)≡∫∞0dωωϕ+(ω,μ) ,σn(μ)≡λB(μ)∫∞0dωωϕ+(ω,μ)lnnμω . (47)

Here, appears as the convolution with the hard part in the LO for the hard spectator amplitudes, while the calculation of the NLO effects for the hard spectator amplitudes requires also the logarithmic moments with . We introduce the logarithmic moments in the coordinate space,

 Rn(μ)≡∫∞0dτ~ϕ+(−iτ,μ)lnn(τμeγE) , (48)

which are related to (47) as

 λ−1B(μ)=R0(μ) ,σ1(μ)=λB(μ)R1(μ) ,σ2(μ)=λB(μ)R2(μ)−π26 ,⋯ . (49)

These relations can be obtained, e.g., by considering the generating function for the inverse moments (47),

 ∫∞0dωω(μω)sϕ+(ω,μ)=∞∑n=0σn(μ)λB(μ)snn! , (50)

and relating this to the generating function for of (