Moments $n=2$ and $n=3$ of the Wilson twist-two operators at three loops in the RI${}'$/SMOM scheme

We study the renormalization of the matrix elements of the twist-two non-singlet bilinear quark operators, contributing to the $n=2$ and $n=3$ moments of the structure functions, at next-to-next-to-next-to-leading order in QCD perturbation theory at the symmetric subtraction point. This allows us to obtain conversion factors between the $\overline{\rm MS}$ scheme and the regularization-invariant symmetric MOM (RI/SMOM, RI${}'$/SMOM) schemes. The obtained results can be used to reduce errors in determinations of moments of structure functions from lattice QCD simulations. The results are given in Landau gauge.


Introduction
The great success of QCD in the description of the structure of hadrons relies on the principle of factorization. Phenomenologically it is possible to access this problem only in some particular kinematical conditions, as provided, for instance, in experiments like deep-inelastic scattering, vector boson or heavymeson production, Drell-Yan process and others.
At the operator level, the most significant contributions in hard processes arise from operators of twist two. In particular, in case of the non-siglet distributions, bilinear quark operators play a crucial rôle. Such operators, contributing to the nth moment of a distribution, are given by symmetric traceless combinations, like where the symbol S denotes total symmetrization over indices µ 1 , . . . , µ n (including the factor 1/n!) and subtraction of all possible traces over pairs of indices.
Since the matrix elements of the operators in Eq. (1) are of nonpertubative nature, they can be accessed only by experiments, QCD sum rules, or lattice-QCD simulations. The most important examples of recent lattice studies include determinations of low moments of light-cone distribution amplitudes of mesons (see, e.g., Refs. [12,13,14,15]) and low moments of the proton PDFs and GPDs (see, e.g., Refs. [16,17,18,19,20]).
To renormalize the matrix elements of the operators in Eq. (1) on the lattice, the regularization-invariant momentum-subtraction (RI/MOM) scheme and its modification, the RI /MOM scheme, have been developed [21,22]. Improved variants include the RI/SMOM and RI /SMOM schemes [23,24], which differ in the way three-point functions are treated. In the RI/MOM and RI /MOM schemes, the subtraction is done at vanishing operator momenta, which potentially generates additional sensitivity to short-distance effects in this channel. On the other hand, in the RI/SMOM and RI /SMOM schemes, the subtraction of three-point functions is performed at the symmetric Euclidean point, −µ 2 , by setting where the four-momenta p and q are as depicted in Fig. 1. Thus, there is no channel with exceptional momenta in this scheme. The next step after the nonpertubative renormalization is the perturbative convertion of the results from one of the above schemes into the modified minimal-subtraction (MS) scheme of dimensional regularization, which serves as the worldwide standard in perturbative QCD calculations. Choosing the parameter −µ 2 to be of the order of a few GeV, such a convertion can be done perturbatively order by order in the expansion in the strong-coupling constant α s (−µ 2 ). This matches the lattice simulations with the high-energy behavior determined by conventional perturbation theory in the continuum using the MS scheme.
The RI/SMOM to MS conversion functions of non-singlet bilinear quark operators without derivatives have been considered in Refs. [23,25] at one loop p+q p q and in Refs. [24,26,27] at two loops. In our previous paper [28], we extended this analysis to the three-loop order numerically. Our three-loop result for the (pseudo)scalar current has been confirmed by an analytical calculation [29] in terms of constants constructed earlier in Ref. [30]. The corresponding conversions for the n = 2, 3 moments of the bilinear quark operators of twist two with one or two covariant derivatives have been considered in Refs. [25,27,31,32] at the one-and two-loop orders.
In this paper, we extend this analysis to the three-loop order. We concentrate on the cases of n = 2 and n = 3 and study the relevant operators at the symmetric kinematical point up to three loops. This paper is organized as follows. In Section 2, we introduce notations and the definitions. In Sections 3 and 4, we present our three-loop results for n = 2 and n = 3 moments, respectively. In Section 5, we conclude with a summary.

Setup
To fix the notation, we start from the following expression in Minkowsky coordinate space: where O stands for some bilinear quark operator, ξ, ζ are spinor indices, i, j are color indices in the fundamental representation of the SU (N ) group, S(q) is the quark propagator, and Λ(p, q) is the amputated Green's function, which is shown schematically in Fig. 1.
In the cases n = 2 and n = 3, we can write explicitly for any operators O µν and O µνσ : where g µν is the metric tensor and d = 4 − 2ε is the space-time dimension.
In the definition in Eq. (1), we still have the freedom to define in which directions the covariant derivatives act. Thus, in the case with one derivative, we can define two operators, from which we can construct operators with either sign of charge conjugation (C), where we have omitted the indices µ, ν for the ease of notation. Notice that the operators in Eqs. (8) and (9) The factor 1/2 in Eq. (10) appears because it has been omitted in the definitions of W 2 and ∂W 2 in Refs. [25,27,31]. We should also note that, in these papers, W 2 corresponds to the operator where the covariant derivative acts to the right, while, according to our definitions, the derivative in W 2 acts to the left. Only with such conventions, we find agreement with Ref. [25,27,31]. For the operators with two derivatives, we introduce the following basis of three operators: From these operators, we can define the following combinations with definite C parities: where we again omit the indices µ, ν, σ for simplicity. Operators O 1 and O 2 mix under renormalization, so that the 3 × 3 operator renormalization matrix takes a block diagonal form in this basis, with one block of size 2 × 2 and one of size 1 × 1.
In Refs. [25,27,32], a different triplet of operators, called W 3 , ∂W 3 , and ∂∂W 3 , has been introduced. We can express these in terms of the operators in Eqs. (11)-(13) as Similarly to the previous case, we find that the directions in which the covariant derivatives act in the operator W 3 defined in Refs. [25,27,32] should be flipped. Upon this change, we find agreement with the previous one-and two-loop calculations.
In order to renormalize the above operators, we use appropriate matrices Z of enormalization constants, a 2 × 2 matrix for n = 2 and a 3 × 3 matrix for n = 3. In the MS scheme, we can write where Z i are constant matrices depending on the QCD coupling constant, These matrices can be related to the matrix of anomalous dimensions γ by the following matrix equations: a ∂ a Z 2 = a ∂ a 1 2 where β is QCD β function, ξ is the gauge parameter, and γ 3 is the anomalous dimension associated with the latter [33]. The matrix γ for n = 2 has been evaluated analytically through O(a 3 ) in Ref. [31]. The corresponding matrix for n = 3 can be found in Ref. [32]. 1 Moreover, in Ref. [32], the nondiagonal matrix elements are only given through order O(a 2 ). We evaluate the missing O(a 3 ) contributions numerically for color group SU(3). In the basis (W 3 , ∂W 3 , ∂∂W 3 ), Eq. (2.10) in Ref. [32] should be extended by the following three-loop contributions where n f is the number of light quark flavors.
To represent our results, we adopt the tensor decompositions from Refs. [31,32]. It is convenient to contract the open indices of the operators O µν and O µνσ with the light-cone vector ∆, with ∆ 2 = 0. This automatically takes into account the symmetry and the tracelessness of the operators. Specifically, we write Here is the fully antisymmetric combination of the Dirac γ matrices, and we use the short-hand notation Γ 3,∆pq = Γ 3,µνσ ∆ µ p ν q σ . With the definitions in Eqs. (25) and (26), the definitions of the formactors F 1 , . . . , F 10 and F 1 , . . . , F 14 coincide with those in Ref. [31] and [32], respectively. We refrain from describing our calculation because it is similar to the one in Ref. [28], where details may be found, and mere list our results, which we do for the n = 2 case in Section 3 and for the n = 3 case in Section 4.

Numerical results for n = 2 moment
Here, we present the numerical results for the formfactors F L j and F R j of the n = 2 moment at three loops in the MS scheme. For F L j , we have Via crossing symmetry in the decomposition in Eq. (25), we obtain for Comparing with the previous calculations by Gracey [25,27,31], we find agreement by verifying the relations for j = 1, . . . , 10 through the two-loop order.

Numerical results for n = 3 moment
Here, we present the numerical results for the formfactors F LL j , F RR j , and F LR j of the n = 3 moment at three loops in the MS scheme. For F LL j , we have Comparing with the previous calculations by Gracey [25,27,32], we find agreement by verifying the relations F LL for j = 1, . . . , 14 through the two-loop order.

Conclusion
In this paper, we have calculated the n = 2 and n = 3 moments of the twisttwo non-singlet bilinear quark operators in SMOM kinematics at three loops in QCD. This allows us to match, with unprecedented precision, lattice QCD simulations of these quantities to their high-energy behaviors in the continuum limit as determined from perturbative QCD calculations in the MS scheme. We have presented the relevant conversion factors between the RI/SMOM and MS schemes in numerical form, ready to use by the lattice community. The threeloop corrections are comparable in size to the two-loop contributions available from Refs. [25,27,31,32], which we were able to reproduce after clarifying some issues with the definitions.