The three-loop unpolarized and polarized non-singlet anomalous dimensions from off shell operator matrix elements

We calculate the unpolarized and polarized three--loop anomalous dimensions and splitting functions $P_{\rm NS}^+, P_{\rm NS}^-$ and $P_{\rm NS}^{\rm s}$ in QCD in the $\overline{\sf MS}$ scheme by using the traditional method of space--like off shell massless operator matrix elements. This is a gauge--dependent framework. For the first time we also calculate the three--loop anomalous dimensions $P_{\rm NS}^{\rm \pm tr}$ for transversity directly. We compare our results to the literature.


Introduction
The anomalous dimensions of local quark and gluon operators determine the scaling violations of the deep-inelastic scattering structure functions [1,2] by the scale evolution of the parton densities and are therefore instrumental in the measurement of the strong coupling constant a(M 2 Z ) = α s (M 2 Z )/(4π) [3] for this inclusive precision data. They have been calculated to 3-loop order both in the unpolarized and polarized case [4][5][6][7] using the method of on-shell forward Compton amplitudes, in which the scale is set by the virtuality Q 2 = −q 2 of the exchanged current. At four-loop order a series of low moments for the non-singlet anomalous dimensions has been calculated in Refs. [8] and at five-loop order in [9]. The O(T F ) contributions at threeloop order have been confirmed by the calculation of massive on-shell operator matrix elements (OMEs) [10][11][12][13][14].
In this paper we are calculating the unpolarized and polarized three-loop anomalous dimensions for the first time using the method of massless off shell OMEs in the flavor non-singlet case, which is the first complete independent recalculation of the results obtained in Ref. [4]. The present calculation requires the knowledge of the corresponding massless off shell OMEs to two-loop order, cf. [32,33,36], up to the terms of O(ε 0 ) in the dimensional parameter ε = D − 4. The off shell OMEs are gauge-dependent quantities. We will calculate the anomalous dimensions and splitting functions: P + NS , P − NS and P s NS . For the first time we also calculate the three-loop anomalous dimension P ±,tr NS for transversity in a direct way. The paper is organized as follows. In Section 2 we derive the structure of the physical part of the flavor non-singlet unrenormalized off shell OMEs to three-loop order. From their pole terms of O(1/ε) one can extract the non-singlet anomalous dimensions. Due to a known Ward identity, cf. e.g. [10,14], the polarized anomalous dimension can be calculated by applying anticommuting γ 5 . We also calculate the polarized OMEs in the Larin scheme [33,37] from which one can determine the Z-factor Z NS 5 (N) of the corresponding finite renormalization to three-loop order. The details of the calculation are described in Section 3. In Section 4 we present the three-loop anomalous dimensions and splitting functions. We compare with results in the literature in Section 5 and Section 6 contains the conclusions. In an appendix we briefly summarize the transition from the Larin to the MS scheme for the polarized anomalous dimension in the vector case.
Here S is the symmetry operator, λ r a SU(N F ) flavor matrix and D µ = ∂ µ + ig s t a A a µ the covariant derivative, with A a µ the gluon field, ψ the quark field, t a the generators of SU(N C ), and g s = √ 4πα s . The Feynman rules of QCD are given in [38] and for the local operators in [14,39]. The local operator in the case of transversity is given by where σ µν = (i/2)[γ µ γ ν − γ ν γ µ ].
The operator matrix elements have the representation Here ∆ denotes a light-like vector, ∆.∆ = 0. The following projectors are applied to separate the physical (phys) contribution and the one vanishing by the equation of motion (EOM), which does not hold in the off shell case, A NS,EOM qq = 1 4(∆.p) N tr ∆ /Â NS qq .
In the polarized case the operator (1) In the case of transversity we consider the unrenormalized Green's function [40] G ij,NS,tr where i, j are external color indices and the coefficients c k | k=1...5 denote other OMEs than those we are going to deal with. Since the non-singlet anomalous dimensions receive only contributions from the unrenormalized OMEÂ NS,(5),phys qq we will consider only this operator matrix element in the following. In Mellin N space it has the representation with the spherical factor where γ E is the Euler-Mascheroni number andâ the bare coupling constant. The free gluon propagator is given by 1 which defines the gauge parameter in the Rξ gauge. The renormalization of the massive off shell non-singlet OMEs encounters the renormalization of the coupling constant and the gauge parameter, as well as that of the local operator. In the following we will deviate from Refs. [32,33] and perform the renormalization of the coupling constant and the gauge parameter and use the resulting expression,Ã, at µ 2 = −p 2 to extract the anomalous dimensions. In the unrenormalized OME obtained in the diagrammatic calculation the coupling constant and the gauge parameter are renormalized before comparing toÃ in Eq. (25). The unrenormalized coupling is given bŷ where a denotes the renormalized strong coupling constant. The expansion coefficients of the QCD β-function are given by [41] The bare gauge parameterξ is renormalized bŷ where Z 3 is the Z-factor of the gluon propagator, cf. [42][43][44][45], with The color factors are C F = (N 2 C − 1)/(2N C ), C A = N C , T F = 1/2 for SU(N C ) and N C = 3 for QCD; N F denotes the number of massless quark flavors. 1 Note a typo in [32], Eq. (2.6). 2 Note a typographical error in [32], Eq. (2.13) and [33], Eq. (2.14).
In Mellin N space the Z-factor of a local non-singlet operator reads [39] In (23) the terms γ (k),NS qq , k = 0, 1, 2, . . . denote the expansion coefficients of the anomalous dimension The partly renormalized OME,Ã NS,phys qq , reads The expansion coefficients a NS,(i,j) qq are in general gauge dependent. The renormalized OMEs are given by expanded to O(a 3 ) and setting S ε = 1. The anomalous dimensions are iteratively extracted form the 1/ε pole terms and the other expansion coefficients a NS,(i,j) qq are given in Ref. [36]. Eq. (25) is understood to hold both for the unpolarized as well as the polarized case, by relabeling the corresponding quantities to f → ∆f . Similar expressions hold for transversity. From them we will determine γ (2) NS and ∆γ (2) NS in both cases. The further three-loop non-singlet anomalous dimensions γ (2),s NS can be derived from other quarkonic diagrams at three-loop order. 3 Because γ (2),s NS occurs for the first time at the three-loop loop level, there is no renormalization of the OME The other expansion coefficients occurring in (25) potentially coming from lower orders in the coupling a do all vanish in this case. The anomalous dimension γ (2)s NS is formally obtained as the O(1/ε) pole term of the pure-singlet OME by considering in the unpolarized case the analytic continuation from odd values of N. The d abc d abc terms in the non-singlet + contributions vanish. One considers the contributions ∝ d abc d abc of this OME for the odd moments. In this way γ (2),s NS corresponds to the non-singlet combination γ (2),s qq − γ (2),s qq . In deep-inelastic scattering one may form up to three different combinations of quark distributions in the unpolarized and polarized case and analogously for (q k ,q k ) → (∆q k , ∆q k ). Here i, k denote the different flavors. These combinations can be obtained by combining the scattering cross sections for different neutral and charged current exchanges off proton and neutron targets. 4 The corresponding anomalous dimensions ruling the evolution of these non-singlet distributions are γ + NS , γ − NS and γ v NS = γ − NS + γ s NS . In the polarized case mostly pure virtual photon exchange has been studied experimentally, which is described by the structure functions g 1,2 (x, Q 2 ). Their non-singlet contributions evolve with ∆γ + NS . The following relations hold ∆γ

Details of the calculation
The Feynman diagrams for the massless off shell OMEs are generated by QGRAF [39,49] and the Dirac and Lorentz algebra is performed by FORM [50]. The color algebra is performed by using Color [51]. The local operators are resummed into propagators by observing the current crossing relations, cf. [1,46], as has been described in Ref. [36], in the corresponding OMEsÂ NS,(5) qq for even or odd moments, which will depend on the resummation variable t quadratically only. To calculate the anomalous dimension γ (2),s NS we resum first, using the variable t itself. In the flavor non-singlet case 684 irreducible diagrams contribute. The reducible diagrams are accounted for by wave-function renormalization [43][44][45], decorating the OMEs at lower order in the coupling constant [32,33,36]. The different local operator insertions are resummed using generating functions of the type where t denotes an auxiliary parameter for the resummation of the formal Taylor series, see [52]. Eq. (33) implements the corresponding current crossing relations in the unpolarized (+) and the polarized case (−) [1,46], which is not just a formality. Only the moments contributing to the respective cases exist. 5 In the calculation of the one-and two-loop contributions we also used the package Eval-uateMultiSums [55] and also applied LiteRed [56] for some checks, cf. [36]. The irreducible three-loop diagrams are reduced to 252 master integrals using the code Crusher [57] by applying the integration-by-parts relations [58,59]. Relations between a small number of t-dependent master integrals are difficult to prove analytically for general values of D. However, they can be proven for the whole finite range of Mellin N and ε used in the present analysis by the method of arbitrary large moments [60]. For the calculation of the necessary initial values for the difference equations we use the results given in [59,61].
The method of arbitrary large moments implemented within the package SolveCoupledSystem [62] is also used to generate a large number of moments for the massless OMEs. By using the method of guessing [63,64] and its implementation in Sage [65,66] we determine the difference equations, which correspond to the different color and multiple zeta value factors [67]. To calculate γ (2),± NS we generate 3000 even resp. odd moments and for γ (2),s NS 500 moments. It turns out that the determination of the largest recurrence requires 1537 moments for γ In the present calculation we kept only one power in the gauge parameterξ to check the renormalization, which has been sufficient to compute the non-singlet anomalous dimensions. In calculating the complete OMEs, no gauge-dependent contribution can be neglected.
The anomalous dimensions, γ NS , can be expressed by harmonic sums [71,72] Their Mellin inversion to the splitting functions P qq (z) can be performed using routines of the packages HarmonicSums and is expressed in terms of harmonic polylogarithms [73] given by with the alphabet of letters In z-space one usually distinguishes three contributions to the individual splitting functions, because of their different treatment in Mellin convolutions, where P δ (z) = p 0 δ(1 − z), P reg (z) is a regular function in z ∈ [0, 1] and P plu (z) denotes the remaining genuine +-distribution, the Mellin transformation of which is given by We will use this representation in Section 4.

The anomalous dimensions and splitting functions
In the following we use the minimal representations in terms of the contributing harmonic sums and harmonic polylogarithms by applying the algebraic relations between the harmonic sums and the harmonic polylogarithms [79]. 26 harmonic sums up to weight w = 5 contribute. Both the anomalous dimensions in the vector case, γ Later the respective analytic continuations from N ∈ N → C proceeds from the even or the odd integers [78]. We will therefore refer to the complete expressions, respectively, as long as they are written in terms of harmonic sums. Considering their Mellin inversion to z space allows then to consider the respective difference term, since the corresponding expression is free of N.
We obtain the following expressions for the non-singlet anomalous dimensions in Mellin N space, using the shorthand notation S a (N) ≡ S a . In the vector case they are given by In the case of transversity we obtain γ (2),tr,+ NS We have calculated the transversity anomalous dimensions for the first time directly and without any assumptions. The polynomials in Eqs. (40)(41)(42)(43) read Finally, we turn to the non-singlet anomalous dimension γ (2),s NS for which we obtain with d abc d abc /N C = 40/9 for N C = 3 in QCD and the polynomials Note that γ (2),s NS has no pole at N = 1, but vanishes. The splitting functions in z space are given by For transversity the contributions to the splitting function P NS,+,tr 2 read P (2),+,tr,δ NS

P
(2),+,tr,reg NS in the vector and transversity cases. Here we used the shorthand notation H a (z) ≡ H a . 21 harmonic polylogarithms of up to weight w = 4 are contributing. 18 harmonic polylogarithms of up to weight w = 4 are contributing to the difference terms. The difference terms P ( In N space the leading term for N → ∞ is ∝ ln(N)/N 2 . Finally, the splitting function P

Comparison to the literature
We confirm the results for the non-singlet case for γ (2),s NS in Ref. [4] where the on-shell forward Compton amplitude has been used for the calculation. The contributions ∝ T F have already been calculated independently as a by-product of the massive on-shell operator matrix elements in Ref. [10]. We also agree with the fixed moments, which were calculated in Refs. [39,[80][81][82][83] and the prediction of the leading N F terms for P (2),+ NS and P (2),− NS computed in [84].
Furthermore, we derive the small z limit of the splitting functions, given by The leading small z terms for P (2),+ NS and P (2),− NS agree with the prediction in Ref. [85] after correcting some misprints there [86], see also [87]. Numerically the leading contributions are not dominant but they are significantly reduced by subleading corrections, cf. [86]. For N C = 3 and N F = 3 one obtains P (2) There are no predictions from genuine small z calculations for subleading terms. Also the small z behaviour of P (2),s NS has not been predicted. The splitting functions in the case of transversity do not contain logarithmically enhanced terms in the small z region to three-loop order, but approach the following constants lim z→0 P (2),tr,+ NS lim z→0 P (2),tr,− NS For transversity we agree with the moments ∝ T F calculated in [40] and the corresponding complete N and z-space expressions given in [10]. In [88] the moments 1 and 3-8 of the transversity anomalous dimension have been computed, to which we agree 6 , as well as to the result given in the attachment to [89]. There the anomalous dimensions have has been obtained from 15 moments, under certain special assumptions on their mathematical structure. 7 We also agree to the 16th moment of the transversity anomalous dimension calculated in [90].

Conclusions
We have calculated the three-loop non-singlet anomalous dimensions γ (2),s NS in Quantum Chromodynamics for unpolarized and polarized deep-inelastic scattering. The method used in this first complete recalculation of the former results in [4,89] has been the traditional one, cf. [15,17], of massless off shell operator matrix elements, unlike the on-shell Compton amplitude at virtuality Q 2 in [4]. The present method requests to obtain the anomalous dimensions in a gauge-dependent framework. We confirm results given in the literature, also on partial results both in the unpolarized and polarized case. The former three-loop calculations have been performed using gauge-invariant quantities. For the non-singlet anomalous dimensions a finite renormalization can be avoided in the polarized case, due to a known Ward identity and all the results are obtained in the MS scheme directly. The present calculation has been performed fully automatically in all its parts using a chain of dedicated codes from diagram generation to the final results. The three-loop anomalous dimensions have a comparatively simple mathematical structure, since they can be expressed in harmonic sums only [71,72]. We remark that also the three-loop unpolarized and polarized singlet anomalous dimensions (∆)γ (2) PS and (∆)γ (2) qg have been recalculated in complete form using the framework of massive on-shell OMEs in [12][13][14]. The flavor non-singlet anomalous dimensions play a particular role in the associated scheme-invariant evolution equations for non-singlet structure functions [91], allowing for a direct measurement of the strong coupling constant.

A Relation between the Larin and the MS scheme
The known Ward identity in the non-singlet case allows to derive the transformation between the Larin scheme and the MS scheme directly. We calculated the anomalous dimension ∆γ NS,− with P 63 = 2N 4 + N 3 + 8N 2 + 5N + 2, 6 In the last term of the 1st moment a factor N 2 F is missing. 7 There is a sign error in the term ∝ C 2 F T F N F in Eq. (A.15) of [89]. P 64 = 103N 4 + 140N 3 + 58N 2 + 21N + 36. (131)