Five-loop contributions to low-N non-singlet anomalous dimensions in QCD

We present the first calculations of next-to-next-to-next-to-next-to-leading order (N^4LO) contributions to anomalous dimensions of spin-N twist-2 operators in perturbative QCD. Specifically, we have obtained the respective non-singlet quark-quark anomalous dimensions at N=2 and N=3 to the fifth order in the strong coupling alpha_s. These results set the scale for the N^4LO contributions to the evolution of the non-singlet quark distributions of hadrons outside the small-x region, and facilitate a first approximate determination of the five-loop cusp anomalous dimension. While the N^4LO coefficients are larger than expected from the lower-order results, their inclusion stabilizes the perturbative expansions for three or more light flavours at a sub-percent accuracy for alpha_s<0.3.

The anomalous dimensions γ ik (N) of spin-N twist-2 operators are important quantities in perturbative QCD. They are closely related, by an integer-N Mellin transform, to the splitting functions P ik (x) that govern the scale dependence (evolution) of the parton densities of hadrons, and are hence directly relevant to the analysis of hard processes at the LHC. The coefficients A k of the leading large-N term of the diagonal (i = k) anomalous dimensions in the standard MS scheme [1][2][3] (with ln N = ln N + γ e , where γ e is the Euler-Mascheroni constant) are identical to the (lightlike) cusp anomalous dimensions [1], and thus relevant well beyond the evolution of parton distributions.
At present, the splitting functions are fully known to three loops (next-to-next-to-leading order, N 2 LO), see refs. [4,5] for the main unpolarized case and refs. [6,7] for the helicity-dependent case. At four loops, the non-singlet quark-quark splitting functions have been determined analytically in the limit of a large number of colours n c ; the remaining terms are known with a high numerical accuracy except at momentum fractions x < ∼ 10 −2 [8]. The less advanced present status in the flavour-singlet sector beyond the leading large-n f terms [9] has been summarized in ref. [10].
In this letter, we report on the first complete calculations of five-loop (N 4 LO) twist-2 anomalous dimensions in QCD and its generalization to a general gauge group. Specifically, we have computed γ + ns (N =2) and γ k ns (N =3) for k = −, v. The superscripts refer to the combinations of quark densities and n f represents the number of effectively massless flavours. These results set the scale for the N 4 LO corrections to the evolution of the non-singlet quark distributions outside the small-x region. In particular they allow, together with γ −,v ns (N = 1) = 0 and specific properties in the large-n c limit, see below, first serious (if unavoidably rough) estimates of the five-loop cusp anomalous dimension.
In terms of operator definitions and renormalization, the present calculation is a direct generalization of ref. [8]. The computation of the required five-loop self-energy integrals is performed as in refs. [11,12], i.e., we employ a recent implementation [13,14] on the local R * operation [15][16][17] to reduce these to four-loop integrals that can be evaluated by the FORCER program [18]. All our symbolic manipulations are carried out using the latest version [19] of FORM [20,21]. The five-loop computation of γ −,v ns (N = 3) require an effort comparable to that for the N 4 LO corrections for Higgs decay to hadrons in the heavy top-quark limit [12], the hardest calculation performed before with the R * program of ref. [13]. A full extension to higher N is not realistic with the present setup.
Our notation for the twist-2 anomalous dimensions and their perturbative expansion is ( Here and in eqs. (4) -(8) below we identify the renormalization scale µ r with the factorization scale µ f . The expansion of γ + ns (N = 2) to the fourth order in a s and the 4-loop contribution to γ −,v ns (N = 3) have been written down in eqs. (B.1), (B.9) and (B.16) of ref. [8], see also refs. [22][23][24][25]. The lower orders of the latter can be found in appendix C of ref. [26] where, however, the normalization of the group factor d abc d abc is larger by a factor of 16; see the discussion below eq. (30) in ref. [7].
Our new (except for the C F n 4 f terms which are identical to those obtained already in ref. [27]) five-loop contributions to these anomalous dimensions read FF = 5/12, see ref. [28]. The terms with even-n values of Riemann's ζ-function in eqs. (4) -(6) provide a partial check that was not yet known at the time of ref. [12]. Consistent with the 'no-π 2 theorem' for Euclidean physical quantities [29][30][31][32], the ζ 6 terms cancel when the MS anomalous dimensions are combined with the corresponding coefficient functions [33][34][35] to physical evolution kernels for the structure functions F 2 at N = 2 and F 3 at N = 3 in deep-inelastic scattering (the required transformation can by found in eqs. (2.7) -(2.9) of ref. [36]). The ζ 4 terms are removed by an additional transformation to a renormalization scheme in which the N 4 LO beta function [11,[37][38][39] does not include ζ 4 -terms, such as the MINIMOM scheme in the Landau gauge [40][41][42] or the scheme introduced in ref. [43].
Combining eqs. (4) and (5)  It is interesting in this context to consider the effect of the quartic group invariants. For example, the n f = 0 coefficients in eqs. (7) and (8) with d FA /n c = 2.5 and d (4) AA /N a = 16.875, see, e.g., appendix C of ref. [42]. Without the rather large contributions of d (4) AA , which enters γ ns at N 4 LO for the first time, the series would look much more benign with consecutive ratios of 1.4 -1.6 between the N 4 LO, N 3 LO, N 2 LO and NLO coefficients. This sizeable d (4) AA contribution ( ∼ n 2 c + 36 ) also implies that the leading large-n c contribution provides a less good approximation at N 4 LO, at least for low N, than at the previous orders.
The generalization of the expansion coefficients in eq. (3) to L ≡ ln(µ 2 r /µ 2 f ) = 0 is given by [36] γ (0) ns + 3β 0 L γ (2) ns + 2β 1 L + 3β 2 0 L 2 γ (1) ns ,  At all orders known so far, γ (n) − ns (N)/ ln N at N = 3 deviate from A n+1 by less than 8% for n f = 3. The relative deviations are even smaller at n f = 0, but larger at larger n f due to cancellations between the n f -dependent and n f -independent contributions. However, the corresponding absolute deviations at n f = 4 and n f = 5 are comparable to those at n f = 3.
These results suggest that our above five-loop results can be used for a first estimate of the 5-loop cusp anomalous dimension. The situation is complicated somewhat by the large low-N contribution of the new colour structure d (4) AA /N A which may or may not persist to the large-N limit. Treating the size of this contribution as an additional uncertainty, we arrive at the predictions Together with the lower-order results [4,8] these lead to the QCD expansions A more direct determination is possible for the leading large-n c contribution of A 5 . In this limit γ + ns = γ − ns , thus the results at N = 1, N = 2 and N = 3 refer to the same function. Furthermore, as already noted below eq. (3.11) in ref. [8], the large-n c five-loop coefficients of C q and D q in eq. (1) can be predicted from known coefficients using [3,8] where β(a s ) = −β 0 a 2 s − β 1 a 3 s − . . . . Finally the coefficients of all Ln c small-x logarithms at five loops, ln ℓ x with ℓ = 1, . . ., 8, can be predicted from the results of ref. [8] by solving [46] γ ns (N, a s ) · ( γ ns (N, a s ) + N − β(a s )/a s ) = O(1) .
Together, these endpoint constraints imply that the function P (4) ns (x) is known in the large-n c limit up to the large-x coefficients A 5 and B 5 of 1/(1−x) + and δ(1−x), respectively, and a smooth function that approaches a constant for x → 0 and vanishes for x → 1.
Under these circumstances, the three available N-values are sufficient, just, for a first approximate reconstruction of P   fig. 2 for n f = 3 quark flavours. Corresponding procedures (all of which are, of course, mathematically non-rigorous) have been successfully employed to three-loop and four-loop quantities in the past, usually with (many) more calculated moments but weaker endpoint constraints, see, e.g., refs. [47,48] and ref. [8]. We have checked the above setup by applying it at N 3 LO, where a comparison with the exact results is possible.
In this manner we arrive at the five-loop Ln c cusp anomalous dimensions A 5, L = (1.5 ± 0.25 , 0.8 ± 0.2 , 0.4 ± 0.1) · 10 5 for n f = 3 , 4 , 5 (16) and A 5, L = (4.7 ± 0.6) · 10 5 for n f = 0. Together with the lower-order results, which are here known exactly to N 3 LO [8,49,50], these lead to the numerical expansions These results differ from eq. (13) only from the N 2 LO contributions which include a (small) term of the form C 2 F n f . The largest part to the more sizeable N 3 LO difference is due to the (negative) d FA /n c contribution. A difference between the N 4 LO QCD and Ln c results as shown by the central values in eqs. (13) and (17) would not be surprising in view of eqs. (9) and (10). However, the present uncertainties preclude any conclusions even about the sign of large-n c suppressed contributions.
Up to N 2 LO, the gluon cusp anomalous dimension is related to its quark counterpart by a simple 'Casimir scaling', A g /A q = C A /C F = 2.25 in QCD. This feature is broken at N 3 LO by the contributions of the quartic group invariants [8,51,52], but appears to persist in a generalized form to N 4 LO [53,54] that includes the above C A /C F relation for the Ln c contributions. Assuming that the latter feature holds also at five loops, eq. (17) also provides a first result for the five-loop gluon cusp anomalous dimension A g,L . If a numerical estimate at N 4 LO were required of A g in QCD, we would recommend, for the time being, to use the last column in the main bracket of eq. (17) with the errors enhanced to ±0.6 (twice the offset between the corresponding Ln c and full QCD coefficients of A g at N 3 LO) together with the N 3 LO results in eq. (4.4) of ref. [54].
To summarize, we have employed the implementation [13] of the local R * operation and the FORCER program [18] for the parametric reduction of massless self-energy integrals to extend previous calculations [8,[22][23][24][25] of the anomalous dimensions γ ns (N) of the lowest-N non-singlet twist-2 operators to the fifth order in the strong coupling constant α s . While the coefficients of α 5 s are larger than expected from the lower-order results, these N 4 LO corrections stabilize the numerical results at a sub-percent level; a 1% correction is reached only at α s = 0.3 for n f = 3.
At least up to N 3 LO, the anomalous dimensions γ ns (N) can be written as f (N) ln N, where the functions f depend rather weakly on N at N ≥ 3. Assuming that this feature also holds at the present order, our results at N = 2 and N = 3 set the scale for the N 4 LO corrections to the evolution of the non-singlet quark distributions outside the small-x region. Accordingly, we have provided first rough estimates of the large-N limit of γ (4) ns (N)/ ln N , the five-loop quark cusp anomalous dimension A 5 , for the physically relevant number of light flavours n f = 3, 4 and 5 in QCD. A more direct approximate determination of A 5 has been presented in the limit of a large number of colours n c , where the present lack of higher-N results is compensated, to a just sufficient extent, by constraints on the small-x and large-x limits of the corresponding non-singlet splitting functions [3,8,46]. A FORM file with our results in eqs. (4) - (6) and the corresponding lower-order coefficients can be obtained from the preprint server https://arXiv.org by downloading the source of this article. It is also available from the authors upon request.