Correlators of vector, tensor, and scalar composite vertices of order $O(\alpha_s^2\beta_0)$

We present analytical results for massless correlators of two vector, tensor, and scalar composite vertices with the Bjorken fractions $x$ and $y$ of order $\alpha_s^2 \beta_0$ of QCD. The structure of these correlators $\Pi^\text{V,T,S}(x,y; p^2)$ and properties of its main elements are discussed in detail. Special attention is paid to verifying the results and comparing them with known particular cases. We apply the correlators to evaluate radiative corrections to the distribution amplitudes of light mesons within the QCD sum rules.


D Borel transform 23
E Distribution amplitudes of twist 2 and 3 for π and ρ mesons 24

Introduction
In this paper, we investigate massless two-point correlators of composite vertices that "live" on the light cone. The local composite vertices presented below emerge in QCD due to applying the "factorization procedure" (or operator product expansion, OPE) to the amplitudes of hard inclusive and exclusive processes. A well-known example of composite vertices arises from the collinear factorization of "handbag" diagrams in deep inelastic scattering. Another example related to exclusive processes is given by the V (q 1 )V (q 2 )A(p) triangle diagram (V and A are the standard vector and axial fermion currents) with hard momentum transfers −q 2 1 , −q 2 2 ≫ p 2 = (q 1 + q 2 ) 2 . A two-point correlator with one composite vertex appears here as a result of factoring out V V subgraphs -the "hard subgraphs" of the diagram (figure 1). A correlator of two composite vertices originates from the factorization of a box diagram if we "contract" its hard subgraphs including the side edges of the diagram at large values of transferred t. Such a two-point correlator is a universal object that determines the asymptotic behavior of the initial amplitude with respect to a hard momentum (i.e., in the leading twist). Correlators like this describe the perturbative content of the hadron distribution amplitudes (DAs) -universal hadron characteristics in the collinear approximation, which are ordered by their twist. Besides, these two-vertex correlators are important to investigate the conformal properties of composite vertices under renormalization [1]. Figure 1. Leading-twist factorization of a three-point function V V A into a convolution ⊗ of a hard four-point function and soft two-point one involving a non-local composite operator, which is denoted by the vertex ⊗.
Let us consider some of the simplest composite bilinear fermion currents involving the N th derivatives of a quark field, Jμ X (η; N ) ≡d(η)Γμ X (iñ∇) N u(η), X = S, P, V, A, T, (1.1) where η is a space-time point, ∇ µ = ∂ µ −igt a A a µ is the covariant derivative,ñ µ is a light-like vector,ñ 2 = 0, and Γμ X is a combination of the Dirac matrices, optionally carrying a string of the Lorentz indicesμ. In particular, we are interested in the (pseudo)scalar, X = S and P, vector V, axial A, and tensor T currents with, respectively, 1 Our goal in this work is to calculate two-point massless correlators containing the composite vertices (see [2]), e.g., the tensor-tensor T T correlator that depends on the longitudinal momentum fractions -the Bjorken variables 0 x, y 1 [3].
Here and in what follows, we underline the arguments of the images of the Mellin transform, i.e. our notation for the Mellin transform is Note that the scalar SS and pseudoscalar P P correlators agree in the massless limit as well as a pair of axial AA and vector V V ones: Π S (x, y; p 2 ) = Π P (x, y; p 2 ), Π V (x, y; p 2 ) = Π A (x, y; p 2 ). (1.5) The (x, y)-representation allows us to obtain any kind of composite vertices by means of convolutions ϕ(x) ⊗ Π(x, y; p 2 ) ⊗ φ(y), 2 where the functions ϕ and φ replace monomials in the corresponding composite vertices. Moreover, the calculation becomes much easier if we apply the inverse Mellin transforms to the composite vertices, from the very beginning [4,5]. The Feynman rules for the vertices Jμ X (η; x) are presented in appendix A. In what follows, we will deal with the Π X (x, y; p 2 ) correlators of x and y-vertices of different γ-matrix structures, X = S (P), V (A), T. The key technical element necessary for our calculation -the "kite" two-loop scalar integral -was evaluated in [5].
In the calculation, we use the BPHZ R-operation in the MS renormalization scheme (for dimensional regularization with D = 4 − 2ε). Along with Π X (x, y; p 2 ), we consider its Mellin moments which are important for various applications. The correlators calculated in this work are also important as perturbative ingredients in evaluating meson DAs within the QCD sum rule (QCD SR) approach. In this approach, the correlators are usually Borel transformed, which implies that only terms containing logarithms of p 2 , external momentum squared, contribute to QCD SR, while the finite parts of the correlators do not survive the Borel transform. Hence, in this paper, we are mostly interested in the log-parts of the correlators.
The paper is organized as follows. In section 2, we discuss the results of 2-loop calculations for the correlators V V , T T , and SS . We consider some checks on these results as well as their relation to the perturbative content of the corresponding DAs. The log-part of the results has a direct physical meaning, while more lengthy nonlogarithmic parts are less interesting in the scope of this paper, see the discussion in [6], and are reserved for appendix B and .m files appended to the arXiv submission. In section 3, we present the 3-loop expressions for the same correlators of order O(β 0 a 2 s ). We discuss their general structure in detail and pay attention to checking their correctness. To this end, we extract some special cases of the Mellin moments (1.6) from the results of [2] and compare them with the ones obtained by us. As an immediate application, we use Π X (x, y; p 2 ) to estimate the impact of radiative corrections on different meson distribution amplitudes in section 4. For all cases, the radiative contributions to the DAs look significant and should be taken into account in future estimations. In section 5, we formulate our conclusions. Some important technical details and part of the results are given in five appendices.
2 Correlators V V , T T , S S at NLO In pQCD, the p 2 -dependence of the correlators manifests itself through the logarithm L = ln −p 2 /µ 2 = ln P 2 /µ 2 , except for the case of S S ( P P ) containing, also, a common factor of P 2 = −p 2 (see the definition in section 2.3): The generalized one-loop ERBL evolution kernels a s C F V X are important and natural elements in the calculations of the corresponding Π X i [7]. These kernels are generated by all subgraphs with a composite vertex that are contracted to be substituted by counterterms as required by the BPHZ R-operation. Therefore, in our results, all the leading-log terms Π X i,i+1 , counterterm contributions, and some other parts of the correlators are proportional to the kernels and their generalizations, see below. We shall start with the vector-vector correlator and the corresponding V V kernel.

V V (x, y) correlator
Evaluating the correlator Π V(A) (x, y; p 2 ), where the current J X (η; x) is defined by eqs. (1.1) and (1.2), it is convinient and natural to express the result in terms of some "building blocks" [4] -the LO function d(y; ε) (which is proportional to the one-loop correlator) and, starting from NLO, the generalized kernels V a (x, y; ε) and V b (x, y; ε): 3 where the part V b of the complete kernel absorbs the contributions with a gluon leg (or a renormalon chain) attached to the composite vertex, while V a corresponds to all other topologies contributing to the one-loop kernel. The part V a and V b of the complete kernel enter in Π V i,j in different ways. Here, a s C F V (0) + is the one-loop ERBL kernel, which describes the ERBL evolution of the DAs of the longitudinally polarized vector (ρ) and pseudoscalar (π) mesons (see appendix E). The plus-distribution form of the V V kernels is the general property for any number of loops -it is the consequence of the vector (axial) current conservation, its anomalous dimension being γ(0) ∼ 1 0 dx V (x, y) + = 0. Therefore, the kernel can be written as Higher derivatives of V a,b (x, y; ε) and d(y; ε) with respect to ε proliferate in expressions for higher orders in a s [10].
The LO V V correlator can be written as in terms of the derivatives of the one-loop function d(y; ε).
The NLO V V correlator (figure 2) obtained in an arbitrary covariant gauge reads The quantitiesḢ a ,ḣ a ,ḣ b are the symmetric functions presented in appendix B together with the nonlogarithmic termΠ V 1,0 (B.1), which, as far as we know, has never been calculated before. The plus distributions for a function f (x, y) are defined as The expressions in eqs. (2.5b) and (2.5c) coincide with the ones obtained in [7]. The 0th moment Π V 1,1 (x, 0) was evaluated in [7,11,12] and a few first two-fold Mellin moments of the complete correlator Π V 1(2) (N , M ) were computed in [2]. We will come back to that in section 3.1.1 to verify our results.
2. Due to the vector-current conservation, the one-fold 0th moments of the leading-log terms vanish, It should be stressed that the identity Π V n,n+1 (x, 0) = 0 originates from the vector-current conservation and (x ↔ y) permutation symmetry rather than particular properties of a specific calculation; therefore, it holds true not only in NLO, but in any higher loop orders as well.
The zeroth moment of the correlator, is the source of perturbative contributions to the QCD sum rules for the meson DAs ϕ M with appropriate meson quantum numbers [7]. We will discuss it in more details at the beginning of section 4 and mention here only that the Borel transformed correlator Π V (x; P 2 ) determines ∆ϕ M -perturbative part of the DA ϕ M for the leading twist of π mesons and longitudinally polarized vector mesons such as ρ . Indeed, applying the Borel transformB (M 2 ) to Π V (x; P 2 ), 4 we arrive at the well-known NLO expression [12] ∆ϕ where M 2 is the Borel parameter. The radiative content of the π and ρ meson DAs of twist-2 will be considered further in section 4.1.

T T (x, y) correlator
Let us recall the definition of the tensor-tensor correlator, 10) and the components of the corresponding one-loop ERBL kernels [13], As in the case of one-loop vector kernel, V T b designates contributions from the composite vertices with a gluon leg, while V T a correspond to all others. In the tensor case, however, the part V T a comes solely from the quark-propagator radiative corrections, which makes it trivially "diagonal". We write it explicitly in what follows. The T T correlator at NLO can be written in terms of the tensor kernels, the derivativeẆ b introduced in the previous subsection, and the one-loop functions d andḋ of eq. (2.3d): and the variable z is the conformal ratio [5,14]. The nonlogarithmic partΠ T 1,0 is presented in (B.2) of appendix B. All the calculated parts ofΠ T 1 agree with the two-fold 0th moment Π T 1 (0, 0) computed in [2]. After applying the Borel transform to it, the correlator Π T (x; P 2 ) ≡ Π T (x, 0; P 2 ) constitutes the perturbative part ∆ϕ M ⊥ of the twist-2 DA ϕ M ⊥ describing the transversely polarized vector mesons such as the ρ ⊥ meson [11]: 14) The ∆ϕ depends on the logarithm of the Borel parameter, L B = ln M 2 µ 2 e −γ E , since the tensor current is not conserved, seeΠ T 1,2 in (2.13b). The above expression for ∆ϕ was first derived in [11].

S S (x, y) correlator
The scalar-scalar correlator is defined as 15) and the components of the ERBL one-loop kernel corresponding to the scalar composite vertex are In contrast to the vector kernel, the total scalar ERBL kernel is already symmetric by itself, V . It is diagonalized in the basis of the Gegenbauer polynomials C (1/2) n (y −ȳ). The eigenfunctions and corresponding eigenvalues of The one-loop scalar-scalar correlator (prior to expanding it in ε) is proportional to the function its first Taylor coefficients being = ln 2 (ȳy).
(2.19) The components of the expansion (2.1a) for the correlator Π S (x, y; p 2 ) can be naturally expressed using the functions in eqs. (2.16)-(2.19): The momentsΠ S i,j (0, 0) for all the terms in eq. (2.21) coincide with the results in [2]. In contrast to Π V and Π T cases, the correlator Π S (x; P 2 ) ≡ Π S (x, 0; P 2 ) might be related to the pion DA of twist 3, ϕ p 3;π (x), see appendix E and, e.g. [15,16]. Below, we present B (M 2 ) P 2 Π S 0+1 (x; P 2 ) -a possible source of perturbative contribution ∆ϕ p 3;π to ϕ p 3;π : In order a 2 s , the coefficients Π X 2,3 at L 3 , the highest power of L in this order, are yielded by contracting to points all subgraphs of the diagrams involved, and so they are formed by the one-loop renormalization of the coupling a s (i.e. β 0 ) and composite vertex (i.e. V . Notice that the former term is proportional to β 0 , while the latter is not. The same pattern can be observed in all coefficients Π X 2,j , which is an evident example of the β-expansion representation, see e.g. [17]: In this paper, we calculate Π X 2[β],j -the β 0 parts of the N 2 LO correlators. These pieces might be expected to dominate in this order because of the relatively large value of β 0 . In the vector case, harbingers of this dominance can be seen in the lowest Mellin moments of the correlator (see section 4). It should also be noted that to obtain the β 0 parts of the three-loop correlators, it suffices to compute only two-loop-like topologiesthe NLO diagrams modified with two-point one-loop quark insertions in gluon lines. Then the entire β 0 part can be restored unambiguously via a replacement n f → − 3 2 β 0 .
We start with our results for the V V correlator that is important for applications and passes the most comprehensive independent test presented in section 3.1.1 below. Then we turn to the T T and S S correlators.
Explicit expressions for the β 0 piece Π V 2[β] of the vector-vector correlator at N 2 LO are given by the following formulae: As it is expected, the leading-log term Π V 2,3 (x, y) is proportional to a plus-distribution prescribed by the vector-current conservation, which means that Π V 2,3 (x, 0) = 0. In addition, the leading-log term at this order is diagonalized by the same set of the Gegenbauer polynomials {C (3/2) n (y −ȳ)} as at order a s .

Mellin moments of V V (x, y) as a check of the correlator
Vetting our calculation of the correlator V V (x, y), we must compare its lowest Mellin moments with the results of refs. [2,6]. In doing so, we find the following linear combinations of the moments to agree with the previous calculations: [2] is an immediate consequence of the symmetry Π V (x, 0; p 2 ) = Π V (x, 0; p 2 ). As it is seen from eq. (2.7), the moments Π V (n, 0) do not contain the highest possible power of L allowed at a given order of perturbation theory. This is also confirmed in [2]. Finally, it is important to note that the Π V 2,2 (n, 0) and ζ 3 part of Π V 2,1 (n, 0) in the complete calculation in [2] are proportional to β 0 for n = 0, 1, 2. This might hint at the dominance of the β 0 contribution evaluated here, which is discussed in section 4.1 in connection with the meson DAs.

T T (x, y) correlator
The expansion eq. (3.1) for the tensor-tensor correlator reads where all elements of the notation in the above formulae are defined in eqs. Check of the moments of T T (x, y). Integrating eqs. (2.13) and (3.5) over x and y, we can get the twofold zeroth moment Π T (0, 0) which was also obtained in ref. [2] (see section 4.3 therein). The moment we calculated coincides with the one in [2]. In addition, the moment Π T (1, 0) can be extracted from the results listed in section 4.8 of [2]. It is precisely one-half less than the two-fold zeroth moment, Π T (1, 0) = 1 2 Π T (0, 0), which is a corollary of mirror symmetry of the one-fold zeroth moment Π T (x, 0) = Π T (x, 0).

S S (x, y) correlator
The expansion for the scalar-scalar correlator reads Here, d S ,ḋ S , andd S were defined by eqs. (2.19) and we have also introduced the generalized "scalar" kernelsṼ S in analogy with the definitions in eqs. (2.3) for the vector casẽ and In eqs. (3.6) as throughout this paper, dots over functions without arguments designate the coefficients of the corresponding Taylor series in ε, e.g.
S (x, y). Check of the moments of S S (x, y). If we evaluate the double zeroth moment Π S (0, 0) integrating the correlator Π S (x, y) over x and y, the result coincides with the calculation of ref. [2] (see section 4.1 therein).

Radiative content of meson DAs within QCD sum rules
In this section, we apply our results for the correlators to the description of exclusive hard hadron processes in terms of DAs. Technically, these DAs are linked to the moments Π X (x, 0; P 2 ) and Π X (a, 0; P 2 ), see the definitions in (1.6). These moments are obtained from the correlators of two composite vertices, Π X (x, y; P 2 ), presented in sections 2 and 3. The expressions for the moments were given in eqs. (2.9) and (2.14) to two-loop order. Here, we write down the final results up to order β 0 a 2 s and focus on the perturbative content of the DAs to only estimate its effect, while a full-fledged analysis of the DA properties in QCD SR will be given elsewhere.
Let us recall some elements of the Borel SR approach that is used to determine meson DAs. This kind of SR is based on the dispersion relation for the one-fold correlator Π X (x, P 2 ) ≡ Π X (x, 0; P 2 ): Im Π X (x, s) s + P 2 ds + "subtractions", (4.1) where Π X is constructed with a current J X that has a nonvanishing projection on a meson state M described by the corresponding DA, see the discussion and definitions in appendix E. The subtractions in the r.h.s. of the relation above can be polynomials in P 2 .
To reinforce the contribution of the lowest-state meson in the r.h.s. and to improve the convergence in the l.h.s., one usually applies the Borel transformB (M 2 ) , to both sides of (4.1), which leads tô The Borel transform "kills" all polynomials in P 2 in the r.h.s. saving only logarithmic terms L n , n 1 in the l.h.s. of (4.1). Under this transform, any powers L n , n ∈ N turn into a polynomial in L B = ln M 2 µ 2 e −γ E , see eq. (D.3) for the general case. To transform the correlators at N 2 LO, we need the following special cases: Finally, it is instructive to note a useful and general property of the moments Π X (N , 0; P 2 ). All these moments (with N being a natural number) correspond to local vertices. They do not contain terms proportional to π 2 in agreement with Kotikov's and Baikov's conclusions [18,19]. At the same time, the inverse moment Π X (−1, 0; P 2 ) contains the π 2 -term because the moment does not correspond to a local operator.

Radiative content of twist-2 DAs for π and ρ L mesons
Here, we start with Π V correlator that determines the perturbative part of π and ρ L meson DAs. Integrating eq. (3.3) over y, taking its Borel transform, and combining the result with eq. (2.9), we arrive at ∆ϕ (0+1+2) (1) which has been already presented in the proceedings [20], while the last term ∆φ Phenomenologically important characteristics of ∆ϕ M are its norm and normalized moments defined as  In particular, we are interested in the inverse and second ξ moment, ξ = 2x − 1: 8) where N 0 is the norm (4.6) with the (β 0 ) 0 piece being omitted in order a 2 s since only the β 0 part of the inverse moment has been calculated up to date. The norm (4.6) is essentially the Adler D-function (up to a factor).
In eqs. (4.6) and (4.8), we have extracted the (β 0 ) 0 pieces from the correlators in ref. [2]. It is worth stressing again that all other terms of the norm and ξ 2 moment calculated by us coincide with those that can be extracted from ref. [2].
The ∆ϕ M (A) in (4.4d) makes a minor contribution to the inverse moment with respect to lower orders -compare the third term and the second one in eq. (4.7), their ratio is 0.085 for L B = 0 and α s (µ 2 = 1 GeV 2 ) ≈ 0.494. This β 0 part, however, is known to dominate the norm (4.6) numerically in order a 2 s . 6 It is instructive to verify numerical validity of large-β 0 approximation for the ξ 2 M moment comparing it with the exact expression that can be obtained from the complete calculations in [2], It is easy to see that the β 0 part is dominant in this moment also (at L B ≈ 0). In addition, we can estimate the perturbative QCD contribution to the Gegenbauer moment a 2 , although one should recognize that a significant contribution to a 2 could come from nonperturbative vacuum-condensate interactions that can vary depending on quantum numbers of mesons. The perturbative contribution a r 2 (r stands for "radiative") is proportional exactly to a s (µ 2 ): As we can see, the radiative contribution a r 2 is of the same order of magnitude as the complete a 2 , so that the contribution a r 2 is comparable numerically with the nonperturbative one and, therefore, is important to take it into account.

Radiative content of ρ T -meson twist-2 DAs
From the tensor correlator (3.5) we get a next-order correction to the NLO amplitude (2.14): In comparison with the LO and NLO terms, the β 0 part of the N 2 LO contribution is mainly of the opposite sign and comparable in magnitude with NLO in the middle region of x, see figure 4.
The norm, the inverse and ξ 2 moments of ∆ϕ where N 0 is the norm (4.12b) with the (β 0 ) 0 piece (4.12a) omitted, the latter one can be obtained using the results of ref. [2]. The β 0 part of the norm (4.12b) is larger in magnitude and has the opposite sign in comparison with the sum of non-β 0 terms in (4.12a) at L B = 0. So the β 0 approximation works satisfactorily here, although it is not as reliable in this case as in the vector one.
A significant convexity in the x-behavior of the β 0 part occurs in the middle values of x. The negative NNLO contribution to x −1 M ⊥ in eq. (4.12d) is not strong in comparison with the NLO one. The radiative contribution a ⊥r 2 to a ⊥ρ 2 can be estimated in analogy to eq. (4.10), ≈ 0.05949 . (4.13) The estimate in the r.h.s. of (4.13) is obtained for L B = 0, µ 2 = 1 GeV 2 , α s (µ 2 ) = 0.494. At these conditions, a ⊥r 2 = 0.059 < a ⊥ρ 2 = 0.130 from the lattice results [25] (originally, a ⊥ρ 2 = 0.101 (22) at µ 2 = 4 GeV 2 ). Again, the radiative contribution a ⊥r 2 is large and as important as for the vector (axial) case.

Conclusion
Here, we have calculated the massless correlators Π V,T,S (x, y; p 2 ) of two vector, tensor, and scalar composite vertices with the Bjorken fractions x and y at orders α s and α 2 s β 0 of QCD. These correlators are universal objects appearing as a result of the collinear factorization procedure in hard processes. We have discussed in detail the structure of the correlators and its elements and their relation to generalized ERBL evolution kernels. Moreover, we have verified our results by comparing them with the known particular cases for Mellin moments. These results are used to estimate the impact of the radiative corrections following from 1 0 Π X (x, y; p 2 )dy on distribution amplitudes of different light mesons within QCD sum-rule approach. For all cases, these radiative corrections are significant and should be taken into account in DA calculations.
with the help of the Dirac deltas [10], e.g.
where Θ(R) is equal to 1, where the relations R are satisfied, and 0 elsewhere.