]>NUPHB116210116210S05503213(23)00139610.1016/j.nuclphysb.2023.116210The Author(s)High Energy Physics – PhenomenologyFig. 1Matrix element 〈O(p4)u¯(p1)d¯(p2)s¯(p3)〉 of a threequark baryonic operator in momentum space, where we omit all spinor and color indices. The momentum p4 = −(p1 + p2 + p3) is the one coming into the operator.Fig. 1Table 1Number of form factors for different numbers of loops in d and four dimensions.Table 1# of loops012
N (in d dimensions)167581
N (in 4 dimensions)164247
Table 2Symmetric structures ordered according to their signatures and values of [p], numbers # of entities for given signature and value of [p], and total numbers ## of entities for given signature.Table 2signature[p]tensor structure###
0000Γ0 ⊗ Γ0 ⊗ Γ011

2220Γμ1μ2⊗Γμ2μ3⊗Γμ3μ11
22261/(−μ2)3Γpp⊗Γpp⊗Γpp2728
Table 3Nonsymmetric structures. The meaning of the columns is the same as in Table 2.Table 3signature[p]tensor structure###
20021/(−μ2)1Γpp⊗Γ0⊗Γ033

2200Γμ1μ2⊗Γμ1μ2⊗Γ01
22021/(−μ2)1Γpμ1⊗Γpμ1⊗Γ09
22041/(−μ2)2Γpp⊗Γpp⊗Γ0919

22221/(−μ2)1Γpμ1⊗Γpμ2⊗Γμ1μ29
22221/(−μ2)1Γpp⊗Γμ1μ2⊗Γμ1μ23
22241/(−μ2)2Γpp⊗Γpμ1⊗Γpμ12739

40221/(−μ2)1Γppμ1μ2⊗Γ0⊗Γμ1μ23
40241/(−μ2)2Γpppμ1⊗Γ0⊗Γpμ136

42021/(−μ2)1Γppμ1μ2⊗Γμ1μ2⊗Γ03
42041/(−μ2)2Γpppμ1⊗Γpμ1⊗Γ036

4400Γμ1μ2μ3μ4⊗Γμ1μ2μ3μ4⊗Γ01
44021/(−μ2)1Γpμ1μ2μ3⊗Γpμ1μ2μ3⊗Γ09
44041/(−μ2)2Γppμ1μ2⊗Γppμ1μ2⊗Γ09
44061/(−μ2)3Γpppμ1⊗Γpppμ1⊗Γ0120

4220Γμ1μ2μ3μ4⊗Γμ1μ2⊗Γμ3μ41
42221/(−μ2)1Γpμ1μ2μ3⊗Γpμ1⊗Γμ2μ39
42221/(−μ2)1Γpμ1μ2μ3⊗Γμ2μ3⊗Γpμ19
42221/(−μ2)1Γppμ1μ2⊗Γμ2μ3⊗Γμ3μ13
42241/(−μ2)2Γppμ1μ2⊗Γpμ1⊗Γpμ227
42241/(−μ2)2Γppμ1μ2⊗Γpp⊗Γμ1μ29
42241/(−μ2)2Γppμ1μ2⊗Γμ1μ2⊗Γpp9
42241/(−μ2)2Γpppμ1⊗Γpμ2⊗Γμ1μ23
42241/(−μ2)2Γpppμ1⊗Γμ1μ2⊗Γpμ23
42261/(−μ2)3Γpppμ1⊗Γpp⊗Γpμ19
42261/(−μ2)3Γpppμ1⊗Γpμ1⊗Γpp991
Renormalization of threequark operators at two loops in the RI′/SMOM schemeBernd A.Kniehla⁎kniehl@desy.deOleg L.VeretinbaII. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, GermanyII. Institut für Theoretische PhysikUniversität HamburgLuruper Chaussee 149Hamburg22761GermanyII. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, GermanybInstitut für Theoretische Physik, Universität Regensburg, Universitätsstraße 31, 93040 Regensburg, GermanyInstitut für Theoretische PhysikUniversität RegensburgUniversitätsstraße 31Regensburg93040GermanyInstitut für Theoretische Physik, Universität Regensburg, Universitätsstraße 31, 93040 Regensburg, Germany⁎Corresponding author.Editor: HongJian HeAbstractWe consider the renormalization of the threequark operators without derivatives at nexttonexttoleading order in QCD perturbation theory at the symmetric subtraction point. This allows us to obtain conversion factors between the MS‾ scheme and the regularization invariant symmetric MOM (RI/SMOM, RI′/SMOM) schemes. The results are presented both analytically in Rξ gauge in terms of a set of master integrals and numerically in Landau gauge. They can be used to reduce the errors in determinations of baryonic distribution amplitudes in lattice QCD simulations.Data availabilityNo data was used for the research described in the article.1IntroductionLight cone distribution amplitudes (DAs) play an important rôle in the analysis of hard exclusive reactions involving large momentum transfer from the initial to the final state. The cases of baryon asymptotic states have been considered already long ago [1–3].The theoretical description of DAs is based on the relation of their moments to matrix elements of local operators. Such matrix elements involve longdistance dynamics and, thus, cannot be accessed via perturbation theory alone.First estimates of the lower moments of the baryon DAs have been obtained more than 30 years ago using QCD sum rules [4–8]. An alternative way to access the moments is to calculate them from first principles using lattice QCD. Such studies for the nucleon DAs have a long history [9–12]. More recently, this analysis has been extended to include the full SU(3) octet of baryons [13].To renormalize the matrix elements on the lattice, the RI′/SMOM scheme [14] was used in Ref. [13]. However, in order to embed lattice estimations of hadronic matrix elements into the complex of other studies and to assure comparability, it is necessary to present the result in the widely used MS‾ scheme. Since the RI′/SMOM prescription can be used in both perturbative and nonperturbative calculations, the conversion from the RI′/SMOM to the MS‾ scheme can be evaluated perturbatively as a power series in the strongcoupling constant αs(μ) at some typical mass scale μ of the order of a few GeV.The letter “S” in RI′/SMOM stands for symmetric configuration of kinematics. For the bilinear quark operator, this implies that the virtualities of the momenta of the quarks and the operator itself are all taken at the same euclidean point μ2. Such analyses including operators with zero, one, and two derivatives, were done at one loop in Refs. [14,15] and at two loops in Refs. [16–20]. These calculations were performed fully analytically. In our previous works [21,22], we have evaluated the matching constants for the bilinear quark operators with up to two derivatives and up to three loops with RI′/SMOM subtraction numerically. Our results for the mass operator [21] have been confirmed later in Ref. [23]. Similar results in the RI′/MOM scheme may be found in Refs. [16,24].By contrast, baryonic operators consist of three quark fields, which we may call u, d, s, the actual flavor structure being irrelevant for the following discussion, and a number of covariant derivatives Dμ:ϵijk(Dμ1…Dμlu)ξ1i(Dμl+1…Dμl+md)ξ2j(Dμl+m+1…Dμl+m+ns)ξ3k, where i,j,k are color indices, μi are Lorentz indices, and ξi are spinor indices. The matrix element of such an operator is shown schematically in Fig. 1.The simplest baryonic operators, without derivatives, were studied in perturbative QCD a long time ago. In Refs. [25,26], the anomalous dimensions of the octet baryonic currents were evaluated at two loops in the MS‾ scheme working in Feynman gauge. In Ref. [27], the renormalization of the threequark operators with open indices, which are considered also here, was performed at two loops in Feynman gauge as well. In Ref. [28], this was extended to arbitrary gauge, and the anomalous dimensions of the baryonic operators were provided at three loops, thus lifting the results of Refs. [25–27] by one loop.In addition, in Ref. [28], twoloop matrix elements were evaluated at the RI′/MOM subtraction point with zeromomentum operator insertion. Specifically, the kinematics of the MOM scheme adopted in Ref. [28] implies that p12=p22=p32=μ2 for some euclidean point μ2, while p4=0. However, it has been argued that the presence of zero momentum can generate additional sensitivity to infrared dynamics, which aggravates lattice QCD analyses [14]. This problem may be avoided by selecting a more symmetric kinematic configuration of the baryonic current matrix elements, where all four virtualities pi2 are taken to coincide at some euclidean point μ2. This leaves residual freedom of how to fix the scalar products pi⋅pj with i≠j. The most symmetric setting would be pi⋅pj=−μ2/3 for i≠j. However, this choice of kinematics turns out to be technically inconvenient in lattice QCD analyses. A recent such analysis [13] used the following kinematics:(1)p12=p22=p32=p42=μ2,p1⋅p2=p3⋅p1=−μ22,p2⋅p3=0, where μ2 is the euclidean subtraction point of the SMOM scheme. To allow for direct comparisons of our results with those of Ref. [13], we adopt Eq. (1) in this paper.An auxiliary analysis, based on differential equations, has revealed that even in the most symmetric kinematics, let alone the kinematics of Eq. (1), the analytic results beyond one loop cannot be expressed in terms of polylogarithms, but include a more complicated class of special functions, involving elliptic structures. In this work, we present our twoloop results in numerical form. We perform the renormalization of the threequark operators for RI′/SMOM kinematics, which allows for the lowest moments of the baryonic DAs to be converted between the RI′/SMOM and MS‾ schemes.It is well known that the MS‾ renormalization prescription is ambiguous for operators with more than one open spinor chain. This happens because there are more tensor structures in d dimensions than in four. In particular, there are operators in d dimensions that have no counterparts in four dimensions. These operators vanish as d→4 and are traditionally called evanescent operators. One cannot simply ignore these structures, since they mix with the physical operators under renormalization [29,30], thus leading to finite contributions.To take these contributions properly into account, we use the renormalization scheme proposed in Ref. [27]. The idea is to express all ddimensional operators in terms of ddimensional tensor structures built from antisymmetrized products of n gamma matrices, Γn=(1/n!)γ[μ1…γμn]. All structures involving Γn with n>4 are evanescent and vanish for d=4. We first renormalize the coefficients in front of these tensor structures and then take the limit d→4. Since the γ matrix structures are now convoluted with renormalized, finite quantities, we can safely put Γn=0 for n>4.This scheme has another very useful property: we completely avoid any problems with γ5 in dimensional regularization. However, the disadvantage of this scheme is that we are confronted with a large number of different spin tensor structures.This paper is organized as follows. In Section 2, we introduce our notations and definitions. In Section 3, we discuss the tensor decomposition and the renormalization procedure. In Section 4, we present sample results, while our complete results are provided in ancillary files submitted to the ArXiv along with this paper. In Section 5, we present our conclusions. In Appendix A, we expose the relevant spin tensor structures. In Appendix B, we list useful fourdimensional identities for the spin tensor structures.2Basic setupThe basic object for the threequark operators without derivatives located at the origin is the amputated fourpoint function,(2)Hβ1β2β3,α1α2α3(p1,p2,p3)=−∫d4x1d4x2d4x3ei(p1⋅x1+p2⋅x2+p3⋅x3)ϵb1b2b3ϵa1a2a3×〈uβ1b1(0)dβ2b2(0)sβ3b3(0)u¯α1′a1(x1)d¯α2′a2(x2)s¯α3′a3(x3)〉×G2−1(p1)α1′α1G2−1(p2)α2′α2G2−1(p3)α3′α3, where all quantities are to be understood as Euclidean. The quark flavors are called u, d, and s, but the only essential feature is that they are all different. All masses are supposed to vanish. αi and βj are spinor indices, ak and bl are color indices in the fundamental representation, and pm are external momenta. The matrix element of the threequark operator is shown schematically in Fig. 1.The twopoint function G2(p) required for the amputation of the external legs is defined by(3)δa′aG2(p)α′α=∫d4xeip⋅x〈uα′a′(0)u¯αa(x)〉.Our goal is to evaluate the matrix element (2) in the kinematics defined by Eq. (1) at the twoloop order.3Tensor decomposition and projectionAs was already mentioned in the Introduction, we renormalize Eq. (2) without contracting the spinor indices and projecting on some particular baryonic currents. For this purpose, let us decompose the tensor in Eq. (2) as(4)Hβ1β2β3,α1α2α3(p1,p2,p3)=∑n=1NTn,β1β2β3,α1α2α3(p1,p2,p3)fn({pi⋅pj}), where Tn are spin tensor structures and fn are scalar form factors. The explicit construction of these structures is discussed in Appendix A. The form factors fn generally depend on six kinematic invariants, p12, p22, p32, p1⋅p2, p2⋅p3, and p3⋅p1. In the following discussion, we omit spinor indices and arguments, and simply write(5)H=∑n=1NTnfn. The upper limit N of summation in Eqs. (4) and (5) is the number of the linearly independent spin tensor structures. It depends on the number of loops. We also have to distinguish between the decompositions in d and four dimensions. In d dimensions, the number of independent structures is larger, owing to the presence of evanescent operators. The values N of independent form factors through two loops are given in Table 1.Let us introduce the following notation. If Xβ1β2β3,α1α2α3 is an object with six spinor indices, we denote by tr3(X) the trace over three pairs of indices, i.e.,(6)tr3(X)=∑α1,α2,α3=14Xα1α2α3,α1α2α3. Using this definition, we can introduce the symmetric N×N matrix(7)Mkn=tr3(TkTn), where Tj are the spin tensor structures from Eqs. (4) and (5). Then, the projectors on the form factors fj take the form(8)Pl=∑k=1NMlk−1Tk, where M−1 is the inverse matrix, and we obviously have(9)fl=tr3(PlH). The use of Eqs. (8) and (9) for unrenormalized amplitudes is delicate within dimensional regularization, since the projectors Pl depend nontrivially on the dimension d. A better way is to first renormalize the amplitude H in Eq. (2) and then to use the projectors in four dimensions. In order to achieve this, we construct N scalar amplitudes ak as(10)ak=tr3(TkH),k=1,…,N. After renormalization of all amplitudes ak in the MS‾ scheme, the form factors can be obtained as(11)fl=∑k=1NMlk−1ak, where M−1 is now taken in four dimensions. In this limit, all elements of M−1 are just rational numbers.However, in four dimensions, we cannot apply the formula in Eq. (11) directly, since the determinant of the matrix Mlk is then zero. This may be understood from Table 1 by observing that the number of independent structures in four dimensions is less than in d dimensions. In this case, we need to solve for the unknown f→ the system (in matrix notation)(12)Mf→=a→, where f→=(f1,…,fN)T, etc.The system (12) is overdetermined, but consistent by construction. We find the solution for f→ in the form(13)f→=f→0+∑j=0Nd−N4Cjy→j, where f→0 is some particular solution of system (12), the vectors y→j form a basis of the Nd−N4=334 dimensional null space of the matrix M, and Cj are arbitrary constants.After renormalization, we have 581 twoloop form factors fn in four dimensions, 247 of which are linearly independent. We have calculated all of them analytically in Rξ gauge in terms of a set of complicated master integrals, which we have evaluated numerically.Our calculational procedure is similar to Refs. [21,22]. As usual, the evaluation of the Feynman diagrams is organized in two steps: the reduction to master integrals and the evaluation of the latter. After the projection and the evaluation of the color and Dirac traces, we first reduce the large number of Feynman integrals using integrationbyparts (IBP) relations [31] to a small set of master integrals. This is done with the help of the computer package FIRE [32]. Besides the IBP relations, we have additional relations arising from the symmetric kinematics. With these new relations, we can further reduce the number of master integrals. Finally, we can express all the Feynman diagrams in terms of 4 oneloop master integrals, to be evaluated analytically, and 44 twoloop master integrals, to be evaluated numerically. As a byproduct of our analytic twoloop calculation, we reproduce the wellknown renormalization constant of the baryonic operator in the MS‾ scheme at this order [25–28].For the numerical evaluation of the twoloop master integrals, we adopt the method of sector decomposition [33,34], which is based on the analytic resolution of singularities and the successive numerical integration of the parametric integrals by Monte Carlo methods. This is done with the help of the program package FIESTA [35]. At the twoloop level, we have up to sevenfold parametric integrals resulting usually in several hundreds of socalled sector integrals, which are then evaluated numerically using the program library CUBA [36]. With a typical sample of 108 function calls, we achieve a relative accuracy of order 10−6 for individual master integrals. However, due to large cancellations between different terms, the resulting relative accuracy is expected to be worse.4ResultsBecause of their large number, we refrain from listing the renormalized twoloop form factors fn here, but supply them in ancillary files submitted to the ArXiv along with this manuscript. Specifically, we present our analytic results in Rξ gauge in the form of Eq. (13) including explicit expressions with the constants Ci, and our numerical results for Ci=0. To obtain the results in Landau gauge, one sets ξ=0.To illustrate the structure and typical size of the corrections, we present here, in numerical form, the twoloop form factor f1, corresponding to the structure Γ0⊗Γ0⊗Γ0, in Rξ gauge. We have(14)f1f1,Born=1+a3[3−c2+ln2+ξ(9−2c1−c2−ln2)]+a2[10.4515(4)+3.5881(4)ξ+1.4232(2)ξ2−0.68933(2)nf]=1+a(0.6204053307351691+0.5957023845688996ξ)+a2(10.4515(4)+3.5881(4)ξ+1.4232(2)ξ2−0.68933(2)nf), where f1,Born=ϵijkϵijk=6 is the Born result, a=αs/π, nf is the number of light quark flavors, ξ is the gauge parameter, and(15)c1=43Cl2(π3)=2.3439072386894588906...,(16)c2=2Cl2(π2)=1.8319311883544380301..., with(17)Cl2(θ)=−∫0θdϕln(2sinϕ2) being Clausen's integral. The errors quoted in Eq. (14) reflect the uncertainties from the numerical integration of the twoloop master integrals.Having all spin tensor components of the matrix element H in Eq. (2) at our disposal, we are now in a position to build any baryonic current without derivatives. As examples, let us evaluate the matrix elements of the baryonic currents previously considered in Refs. [25,26,28],(18)(O1)α=ϵijkuαi[(uj)TCdk],(19)(O2)α=ϵijkγ5uαi[(uj)TCγ5dk], where C is the unitary, antisymmetric matrix, with C−1=C†=−C⁎, of charge conjugation,(20)CγμC−1=−γμT,(21)Cγ5C−1=γ5T. The matrix γ5 in Eqs. (19) and (21) is taken to be fourdimensional, since we are working here with finite renormalized quantities.The matrix elements of the baryonic currents O1 and O2 in Eqs. (18) and (19) may be decomposed as(22)〈(O1,2)δ(p4)u¯α(p1)u¯β(p2)d¯γ(p3)〉=∑n(Γαδ⊗Γβγ)nfnO1,2, with some scalar form factors fnO1,2 being linear combinations of fl as defined in Eq. (9). It is actually sufficient to consider O1, since the expression for O2 may be obtained by multiplying that for O1 with γ5⊗γ5. Exploiting the fourdimensional identities listed in Appendix B, we may reduce the number n of form factors in Eq. (22) down to 32, some of which are zero.Omitting spinor indices, we thus obtain(23)O1=I⊗I[1+a(2.00280429+0.549441049ξ)+a2(26.486(1)+3.03333(7)ξ+1.3064(3)ξ2−2.79941(6)nf)]+(I⊗σp1p2+I⊗σp3p1)[−a×0.52086828+a2(−10.85904(3)−0.03004(3)ξ−0.12631(9)ξ2+0.75237(3)nf)]+I⊗σp2p3[a(1.38629436+0.462098120ξ)+a2(34.345(1)+5.98739(6)ξ+1.95648(2)ξ2−2.4645(1)nf)]+γ5⊗γ5a2(−1.59871(1)−0.02545(2)ξ)+(γ5⊗γ5σp1p2+γ5⊗γ5σp3p1)a2(0.14070(2)+0.22190(3)ξ)+γ5⊗γ5σp2p3a2(0.19179(7)+0.03528(2)ξ)+(σp1p2⊗I−σp3p1⊗I)[−a×0.52086828+a2(12.9088(4)+0.6163(4)ξ+0.1125(1)ξ2−0.752365(4)nf)]+(γ5σp1p2⊗γ5−γ5σp3p1⊗γ5)a2(−0.3571(1)−0.02859(4)ξ)+(σp1p2⊗σp1p2−σp3p1⊗σp3p1)[a(0.076423831−0.076423831ξ)+a2(0.6196(9)−0.652(2)ξ−0.196(7)ξ2−0.084915(3)nf)]+(σp2p3⊗σp1p2−σp2p3⊗σp3p1)a2(−0.0041(4)−0.0301(1)ξ)+(σp1p2⊗σp3p1−σp3p1⊗σp1p2)a2(−0.5875(8)−0.1445(5)ξ+0.0346(1)ξ2)+(σp1p2⊗σp2p3−σp3p1⊗σp2p3)a2(1.1520(6)+0.7480(8)ξ+0.1593(1)ξ2)+σp2p3⊗σp2p3a2(2.95198(6)−6.2222(3)ξ)+(σμp2⊗σμp1−σμp3⊗σμp1)[a×0.111111111+a2(2.6434(7)−0.06095(1)+0.0387(1)ξ)]+(σμp1⊗σμp2−σμp1⊗σμp3)[a(0.37154525+0.26043414ξ)+a2(9.6843(8)+3.0743(2)ξ+0.9510(1)ξ2−0.697562(2)nf)]+(σμp2⊗σμp3−σμp3⊗σμp2)a2(1.3642(2)+0.5083(1)ξ)+(σμp2⊗σμp2−σμp3⊗σμp3)[a×0.22222222+a2(2.9803(8)+0.1734(1)ξ−0.208701(1)nf)]+σμp1⊗σμp1a2×4.2853(1).5ConclusionIn this work, we have established a framework for the evaluation of the corrections to the baryonic current without derivatives through the twoloop order. The main difficulty in the study of the baryonic operators is the presence of evanescent operators that mix under renormalization with the physical operators. This leads to a large mixing matrix and the necessity for finite renormalizations. On the other hand, if we use the openindices approach, there is no ambiguity in the interpretation within the MS‾ scheme. Exploiting this observation, we have evaluated all the form factors appearing through two loops and presented them in a numerical form that is ready for use in lattice QCD simulations.CRediT authorship contribution statementBernd Kniehl: research, manuscript preparation. Oleg Veretin: research, manuscript preparation. Declaration of Competing InterestThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.AcknowledgementsWe are grateful to Vladimir M. Braun, Meinulf Göckeler, and Alexander N. Manashov for fruitful discussions. O.L.V. is grateful to the University of Hamburg for the warm hospitality. This work was supported in part by the German Research Foundation DFG through Research Unit FOR 2926 “Next Generation Perturbative QCD for Hadron Structure: Preparing for the ElectronIon Collider” with Grant No. KN 365/131.Appendix ASpin tensor structuresIn this Section, we explicitly enumerate all linearly independent spin tensor structures Tn through two loops in d dimensions. All tensors Tn are represented as tensor products of three Dirac structures, as(24)Tα1α2α3,β1β2β3=Γα1β1⊗Γα2β2⊗Γα3β3. The building blocks Γ are antisymmetric products of Dirac γ matrices,(25)Γ0=1,(26)Γμ1μ2=12!γ[μ1γμ2],(27)Γμ1μ2μ3μ4=14!γ[μ1γμ2γμ3γμ4], where 1 is the unit Dirac matrix and square brackets […] denote antisymmetrization. Notice that Dirac structures with odd numbers of Dirac matrices do not appear in our calculation.We also introduce the following notation for the contraction of a vector and a tensor (Schoonship notation):(28)pμΓ…μ…=Γ…p…. Furthermore, we introduce the following wildcards: p can take one of p1,p2,p3, pp can take one of p1p2,p2p3,p3p1, and ppp stands for p1p2p3.For the sake of systematics, we assign to each tensor structure a signature, which is an ordered triplet of the numbers 0, 2, and 4 of γ matrices appearing in each Γ factor, and a number [p] counting the overall appearances of momenta. Furthermore, we distinguish between symmetric and nonsymmetric structures. The symmetric structures do not have copartners arising under the change of order of the Γ factors in the tensor products, while the nonsymmetric ones do. So, the numbers of nonsymmetric structures should be multiplied by 3. In Tables 2 and 3, we systematically list the symmetric and nonsymmetric tensor structures, respectively, and specify the number (#) of entities for each signature and each value of [p]. We also give the total number (##) of entities for each signature.Appendix BFourdimensional identitiesIn d=4 dimensions, all Γ matrices with more than four indices vanish. For Eqs. (26) and (27), we may write(29)Γμ1μ2=σμ1μ2,Γμ1μ2μ3μ4=εμ1μ2μ3μ4γ5, where εμ1μ2μ3μ4 is the totally antisymmetric tensor.For any fourmomenta q1,…,q4, the following identities hold:(30)Γμ1μ2μ3μ4⊗Γμ1μ2μ3μ4=24γ5⊗γ5,Γμ1μ2μ3q1⊗Γμ1μ2μ3q2=6(q1⋅q2)γ5⊗γ5,Γμ1μ2q1q2⊗Γμ1μ2q3q4=2[(q1⋅q3)(q2⋅q4)−(q1⋅q4)(q2⋅q3)]γ5⊗γ5,Γμ1q1q2q3⊗Γμ1q1q2q3=[q12q22q32+2(q1⋅q2)(q2⋅q3)(q3⋅q1)−q12(q2⋅q3)2−q22(q3⋅q1)2−q32(q1⋅q2)2]γ5⊗γ5,Γμ1μ2q1q2⊗Γμ1μ2=−2γ5⊗γ5σq1q2,Γμ1q1q2q3⊗Γμ1q4=−(q1⋅q4)γ5⊗γ5σq2q3−(q2⋅q4)γ5⊗γ5σq3q1−(q3⋅q4)γ5⊗γ5σq1q2.There is one more identity that relates the 19 structures of the form σμν⊗σμν, σμp⊗σμp, and σpp⊗σpp in four dimensions. This relation is quite cumbersome for arbitrary kinematics. We present it here for the particular kinematics of Eq. (1):(31)0=σμν⊗σμν+4(−μ2)2(σp1p2⊗σp1p2+σp2p3⊗σp2p3+σp3p1⊗σp3p1)−2(−μ2)2(σp1p2⊗σp2p3+σp3p1⊗σp2p3+σp2p3⊗σp1p2+σp2p3⊗σp3p1)−2(−μ2)(σμp1⊗σμp2+σμp2⊗σμp1+σμp1⊗σμp3+σμp3⊗σμp1)−1(−μ2)(σμp2⊗σμp3+σμp3⊗σμp2)−3(−μ2)(σμp2⊗σμp2+σμp3⊗σμp3)−4(−μ2)σμp1⊗σμp1.Appendix CSupplementary materialSupplementary material related to this article can be found online at https://doi.org/10.1016/j.nuclphysb.2023.116210. Appendix CSupplementary materialThe following is the Supplementary material related to this article.MMCFile <attachment.tar> includes 14 files including explanation README file. These are the analytical and numerical results for the amplitudes a_j, form factors f_j, and the matrix elements M_ij as discussed in the text.MMCReferences[1]A.V.EfremovA.V.RadyushkinFactorization and asymptotic behaviour of pion form factor in QCDPhys. Lett. B941980245250A. V. Efremov and A. V. Radyushkin. 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