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

TL;DR

This paper calculates the five-loop contributions to non-singlet anomalous dimensions in QCD, providing crucial data for high-precision evolution of quark distributions and advancing the understanding of perturbative QCD at N^4LO.

Contribution

It presents the first N^4LO calculations of non-singlet anomalous dimensions at N=2 and 3 in QCD, extending perturbative accuracy to five loops.

Findings

N^4LO coefficients are larger than lower-order predictions.

Including N^4LO stabilizes perturbative expansions for alpha_s < 0.3.

Results enable more precise evolution of non-singlet quark distributions.

Abstract

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.

Equations

$$\gamma_{\,\rm kk}(N)\;=\;A_{\rm k}\,\ln\widetilde{N}\,-\,B_{\rm k}\,+\,C_{\rm k}\>N^{\,-1}\ln\widetilde{N}\,-\,\big{(}D_{\rm k}-\mbox{\footnotesize{$\displaystyle\frac{1}{2}$}}\>A_{\rm k}\big{)}\,N^{-1}\,+\,{\cal O}\big{(}N^{\,-2}\ln^{\,n}\widetilde{N}\big{)}$$

$$q_{ab}^{\pm }=q_a\pm {\bar{q}}_a-(q_b\pm {\bar{q}}_b),q_{\mathrm{v}}={\sum }_{a=1}^{n_f}(q_a-{\bar{q}}_a)$$

$$\gamma_{\,\rm ns}^{\;\rm a}(N)\;=\;-P_{\rm ns}^{\>\rm a}(N)\;=\;\sum_{n=0}\gamma_{\,\rm ns}^{\,(n)\,\rm a}(N)\>{a_{\rm s}^{\,n+1}}\quad\mbox{ with }\quad a_{\rm s}\;=\;\mbox{\footnotesize{$\displaystyle\frac{\alpha_{\rm s}(\mu_{\>\!\!f}^{\,2})}{4\pi}$}}\>\>.$$

$${\gamma }_{\mathrm{ns}}^{(4)+}(N=2)=$$

$${\gamma }_{\mathrm{ns}}^{(4)-}(N=3)=$$

$${\gamma }_{\mathrm{ns}}^{(4)\mathrm{v}}(N=3)={\gamma }_{\mathrm{ns}}^{(4)-}(N=3)$$

$${\gamma }_{\mathrm{ns}}^{+}(N=2,n_f=0)$$

$${\gamma }_{\mathrm{ns}}^{+}(N=2,n_f=3)$$

$${\gamma }_{\mathrm{ns}}^{+}(N=2,n_f=4)$$

$${\gamma }_{\mathrm{ns}}^{+}(N=2,n_f=5)$$

$${\gamma }_{\mathrm{ns}}^{-}(N=3,n_f=0)$$

$${\gamma }_{\mathrm{ns}}^{-}(N=3,n_f=3)$$

$${\gamma }_{\mathrm{ns}}^{-}(N=3,n_f=4)$$

$${\gamma }_{\mathrm{ns}}^{-}(N=3,n_f=5)$$

$$2.3617$$

$$4.520$$

$$2.0954$$

$$3.954$$

$${\gamma }_{\mathrm{ns}}^{(0)}(L)$$

$${\gamma }_{\mathrm{ns}}^{(1)}(L)$$

$${\gamma }_{\mathrm{ns}}^{(2)}(L)$$

$${\gamma }_{\mathrm{ns}}^{(3)}(L)$$

$${\gamma }_{\mathrm{ns}}^{(4)}(L)$$

$$A_5=(1.7\pm 0.5,1.1\pm 0.5,0.7\pm 0.5)\cdot 10^5\text{mbox}for{n}_f=3,4,5.$$

$$A_{\mathrm{q}}(n_f=3)$$

$$A_{\mathrm{q}}(n_f=4)$$

$$A_{\mathrm{q}}(n_f=5)$$

$$C_{\mathrm{q}}=(A_{\mathrm{q}}{)}^2,D_{\mathrm{q}}=A_{\mathrm{q}}\cdot (B_{\mathrm{q}}-\beta (a_{\mathrm{s}})/a_{\mathrm{s}}),$$

$${\gamma }_{\mathrm{ns}}(N,a_{\mathrm{s}})\cdot \left({\gamma }_{\mathrm{ns}}(N,a_{\mathrm{s}})+N-\beta (a_{\mathrm{s}})/a_{\mathrm{s}}\right)=O(1).$$

$$A_{5,\mathrm{L}}=(1.5\pm 0.25,0.8\pm 0.2,0.4\pm 0.1)\cdot 10^5\text{mbox}for{n}_f=3,4,5$$

$$A_{\mathrm{q},\mathrm{L}}(n_f=3)$$

$$A_{\mathrm{q},\mathrm{L}}(n_f=4)$$

$$A_{\mathrm{q},\mathrm{L}}(n_f=5)$$

Full text

Nikhef 2018-066 December 2018

DESY 18–227

LTH 1191

**Five-loop contributions to low-${\bm{N}}$

non-singlet anomalous dimensions in QCD

** F. Herzog${}^{\,a}$, S. Moch${}^{\,b}$, B. Ruijl${}^{\,c}$, T. Ueda${}^{\,d}$, J.A.M. Vermaseren${}^{\,a}$ and A. Vogt${}^{\,e}$

*a**Nikhef Theory Group

Science Park 105, 1098 XG Amsterdam, The Netherlands*

*b**II. Institute for Theoretical Physics, Hamburg University

Luruper Chaussee 149, D-22761 Hamburg, Germany

*c**Institute for Theoretical Physics, ETH Zürich

Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland

d* Department of Materials and Life Science, Seikei University

3-3-1 Kichijoji Kitamachi, Musashino-shi, Tokyo 180-8633, Japan

*e**Department of Mathematical Sciences, University of Liverpool

Liverpool L69 3BX, United Kingdom

Abstract

We present the first calculations of next-to-next-to-next-to-next-to-leading order (N4LO) 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 $\alpha_{\rm s}$. These results set the scale for the N4LO 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 N4LO 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 $\alpha_{\rm s}<0.3$.

The anomalous dimensions $\gamma_{\,\rm ik}(N)$ 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 $P_{\,\rm ik}(x)$ 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 $A_{\rm k}$ of the leading large-$N$ term of the diagonal ($\rm i=k$) anomalous dimensions in the standard $\overline{\mbox{MS}}$ scheme [1, 2, 3]

[TABLE]

(with $\ln\widetilde{N}=\ln N+\gamma_{\>\!e}$, where $\gamma_{\>\!e}$ 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, N2LO), 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\raisebox{-1.99168pt}{;\stackrel{{\scriptstyle<}}{{{\scriptstyle\sim}}}>}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 (N4LO) twist-2 anomalous dimensions in QCD and its generalization to a general gauge group. Specifically, we have computed $\gamma_{\,\rm ns}^{\>+}(N\!=\!2)$ and $\gamma_{\,\rm ns}^{\;k}(N\!=\!3)$ for $k=-,\rm v$. The superscripts refer to the combinations of quark densities

[TABLE]

and ${n_{\!f}}$ represents the number of effectively massless flavours. These results set the scale for the N4LO corrections to the evolution of the non-singlet quark distributions outside the small-$x$ region. In particular they allow, together with $\gamma_{\,\rm ns}^{\,-,\rm v}(N\!=\!1)=0$ 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 $\gamma_{\,\rm ns}^{\,-,\rm v}(N=3)$ require an effort comparable to that for the N4LO corrections for Higgs decay to hadrons in the heavy top-quark limit [12], the hardest calculation performed before with the R${}^{\ast\!}$ 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

[TABLE]

Here and in eqs. (4) – (S0.Ex53) below we identify the renormalization scale $\mu_{r}$ with the factorization scale $\mu_{\>\!\!f}$. The expansion of $\gamma_{\,\rm ns}^{\>+}(N=2)$ to the fourth order in $a_{\rm s}$ and the 4-loop contribution to $\gamma_{\,\rm ns}^{\>-,\rm v}(N=3)$ 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

[TABLE]

[TABLE]

and

[TABLE]

Here $N_{A}$ and $N_{F}$ are the dimensions of the adjoint and fermion representation, with $N_{A}=8$ and $N_{F}=3$ in QCD, where the quadratic, cubic and quartic group invariants take the values $C_{A}=3$, $C_{F}=4/3$, $d_{abc}d^{\,abc}=5/6$ and $d^{\,(4)}_{AA}\equiv d_{A}^{\,abcd}d_{A}^{\,abcd}=135$, $d^{\,(4)}_{FA}=15/2$, $d^{\,(4)}_{FF}=5/12$, see ref. [28].

The terms with even-$n$ values of Riemann’s $\zeta$-function in eqs. (4) – (6) provide a partial check that was not yet known at the time of ref. [12]. Consistent with the ‘no-$\pi^{2}\!$ theorem’ for Euclidean physical quantities [29, 30, 31, 32], the ${\zeta_{6}}$ terms cancel when the $\overline{\mbox{MS}}$ 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 ${\zeta_{4}}$ terms are removed by an additional transformation to a renormalization scheme in which the N4LO beta function [11, 37, 38, 39] does not include ${\zeta_{4}}$-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) with the lower-order results leads to the numerical QCD expansions

[TABLE]

and

[TABLE]

in powers of the strong coupling constant $\alpha_{\rm s}$. Here we have included ${n_{\!f}}=0$ besides the physically relevant values, since it provides useful information about the behaviour of the perturbation series. The new N4LO coefficients in eqs. (S0.Ex49) and (S0.Ex53) are larger than one may have expected from the N2LO and N3LO contributions.

It is interesting in this context to consider the effect of the quartic group invariants. For example, the ${n_{\!f}}=0$ coefficients in eqs. (S0.Ex49) and (S0.Ex53) at N3LO and N4LO can be decomposed as

[TABLE]

and

[TABLE]

with $d^{\,(4)}_{FA}/{n_{c}}=2.5$ 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 $\gamma_{\,\rm ns}$ at N4LO for the first time, the series would look much more benign with consecutive ratios of 1.4 – 1.6 between the N4LO, N3LO, N2LO and NLO coefficients. This sizeable $d^{\,(4)}_{AA}$ contribution ($\,\sim{n_{c}^{\,2}}+36\,$) also implies that the leading large-${n_{c}}$ contribution provides a less good approximation at N4LO, at least for low $N$, than at the previous orders.

The generalization of the expansion coefficients in eq. (3) to $L\equiv\ln(\mu_{r}^{2}/\mu_{\>\!\!f}^{\,2})\neq 0$ is given by [36]

[TABLE]

to N4LO, where we have suppressed the superscript ‘a’ of eq. (3). $\beta_{0,1,2,3}$ are the $\overline{\mbox{MS}}$ coefficients of the beta function up to N3LO [44, 45] with $\beta_{0}=11-2/3\>{n_{\!f}}$, $\beta_{1}=102-38/3\>{n_{\!f}}\,$ etc in QCD.

The numerical impact of the higher-order contributions to the anomalous dimensions $\gamma_{\,\rm ns}^{\,\pm}$ on the evolution of the $N=2$ and $N=3$ moments of the respective parton distributions (2) are illustrated in fig. 1. At $\alpha_{\rm s}(\mu_{\>\!\!f}^{\,2})=0.2$ and ${n_{\!f}}=4$, the N4LO corrections are about 0.15% at the default choice $\mu_{r}=\mu_{\>\!\!f}$ of the renormalization scale, roughly half the size of their N3LO counterparts. Varying $\mu_{r}$ up and down by a factor of 2 one arrives at a band with a full width of about 0.7%. The N3LO and N4LO corrections are about twice as large at a lower scale with ${n_{\!f}}=3$ and $\alpha_{\rm s}(\mu_{\>\!\!f}^{\,2})=0.25$.

In order to assess the implications of the above results beyond $N=2$ and $N=3$, and in particular for the five-loop cusp anomalous dimension, it is useful to consider the $N$-dependence of $\gamma_{\,\rm ns}^{\,\pm}(N)$ at lower orders and the large-${n_{c}}$ limit. In the left part of fig. 2, moments (3) of the NLO, N2LO and N3LO splitting functions $P_{\rm ns}^{\,\pm}(x)$ and their common large-$n_{c}$ (Ln${}_{\,\rm c}$) limit are displayed for ${n_{\!f}}=3$ in a manner that facilitates a direct comparison with the size of the corresponding cusp anomalous dimensions, defined by $A_{\rm q}=A_{1}{a_{\rm s}^{\,}}+A_{2\,}{a_{\rm s}^{\,2}}+\ldots\,$, in eq. (1).

At all orders known so far, $\gamma_{\,\rm ns}^{\,(n)\,-}(N)/\ln N$ 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

[TABLE]

Together with the lower-order results [4, 8] these lead to the QCD expansions

[TABLE]

for the physically relevant values of ${n_{\!f}}$. Here and in fig. 2 also the N3LO results are approximate; their uncertainties are however irrelevant and amount to $2\cdot\!10^{\,-4}$ for the coefficients in eq. (S0.Ex63).

A more direct determination is possible for the leading large-${n_{c}}$ contribution of $A_{5}$. In this limit $\gamma_{\,\rm ns}^{\,+}=\gamma_{\,\rm ns}^{\,-}$, 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 $C_{\rm q}$ and $D_{\rm q}$ in eq. (1) can be predicted from known coefficients using [3, 8]

[TABLE]

where $\beta(a_{\rm s})=-\beta_{0}\,{a_{\rm s}^{\,2}}-\beta_{1}\,{a_{\rm s}^{\,3}}-\ldots\;$. Finally the coefficients of all Ln${}_{\,\rm c}$ small-$x$ logarithms at five loops, $\ln^{\,\ell\!}x$ with $\ell=1,\ldots,8$, can be predicted from the results of ref. [8] by solving [46]

[TABLE]

Together, these endpoint constraints imply that the function $P_{\,\rm ns}^{\,(4)}(x)$ is known in the large-${n_{c}}$ limit up to the large-$x$ coefficients $A_{5}$ and $B_{5}$ of $1/{(1\!-\!x)}_{+}$ and $\delta{(1\!-\!x)}$, respectively, and a smooth function that approaches a constant for $x\rightarrow 0$ and vanishes for $x\rightarrow 1$.

Under these circumstances, the three available $N$-values are sufficient, just, for a first approximate reconstruction of $P_{\,\rm ns}^{\,(4)}(x)$: a sufficient number, here 20, of one-parameter smooth functions are chosen, and $A_{5}$, $B_{5}$ and this parameter are determined from the available three moments for each of these choices. The ensuing spread of the values of $P_{\,\rm ns}^{\,(4)}(x)$ indicates the remaining uncertainty of this function. The results are shown in the right part of 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 N3LO, where a comparison with the exact results is possible.

In this manner we arrive at the five-loop Ln${}_{\,\rm c}$ cusp anomalous dimensions

[TABLE]

and $A_{5,\,\rm L}=(4.7\pm 0.6)\cdot 10^{\,5}$ for ${n_{\!f}}=0$. Together with the lower-order results, which are here known exactly to N3LO [8, 49, 50], these lead to the numerical expansions

[TABLE]

These results differ from eq. (S0.Ex63) only from the N2LO contributions which include a (small) term of the form ${C^{\,2}_{F}}\>\!{n_{\!f}}$. The largest part to the more sizeable N3LO difference is due to the (negative) $d^{\,(4)\!}_{FA}/{n_{c}}$ contribution. A difference between the N4LO QCD and Ln${}_{\,\rm c}$ results as shown by the central values in eqs. (S0.Ex63) and (S0.Ex65) would not be surprising in view of eqs. (S0.Ex56) and (S0.Ex57). However, the present uncertainties preclude any conclusions even about the sign of large-${n_{c}}$ suppressed contributions.

Up to N2LO, the gluon cusp anomalous dimension is related to its quark counterpart by a simple ‘Casimir scaling’, $A_{\rm g}/A_{\rm q}\,=\,C_{A}/C_{F}\,=\,2.25$ in QCD. This feature is broken at N3LO by the contributions of the quartic group invariants [8, 51, 52], but appears to persist in a generalized form to N4LO [53, 54] that includes the above $C_{A}/C_{F}$ relation for the Ln${}_{\,\rm c}$ contributions. Assuming that the latter feature holds also at five loops, eq. (S0.Ex65) also provides a first result for the five-loop gluon cusp anomalous dimension $A_{\rm g,L}$. If a numerical estimate at N4LO were required of $A_{\rm g}$ in QCD, we would recommend, for the time being, to use the last column in the main bracket of eq. (S0.Ex65) with the errors enhanced to $\pm 0.6$ (twice the offset between the corresponding Ln${}_{\,\rm c}$ and full QCD coefficients of $A_{\rm g}$ at N3LO) together with the N3LO 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 [22, 23, 24, 25, 8] of the anomalous dimensions $\gamma_{\,\rm ns}(N)$ of the lowest-$N$ non-singlet twist-2 operators to the fifth order in the strong coupling constant $\alpha_{\rm s}$. While the coefficients of ${\alpha_{\rm s}^{\,5}}$ are larger than expected from the lower-order results, these N4LO corrections stabilize the numerical results at a sub-percent level; a 1% correction is reached only at $\alpha_{\rm s}=0.3$ for ${n_{\!f}}=3$.

At least up to N3LO, the anomalous dimensions $\gamma_{\,\rm ns}(N)$ can be written as $f(N)\ln N$, where the functions $f$ depend rather weakly on $N$ at $N\geq 3$. Assuming that this feature also holds at the present order, our results at $N=2$ and $N=3$ set the scale for the N4LO 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 $\gamma_{\,\rm ns}^{\,(4)}(N)/\ln N$ , 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, 46, 8].

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.

Acknowledgements

This work has been supported by the Advanced Grant 320651, HEPGAME, of the *European Research Council * (ERC) and by the Deutsche Forschungsgemeinschaft (DFG) under grant number MO 1801/2-1. The research of F. H. is supported by the Vidi grant 680-47-551 of the Netherlands Organisation for Scientific Research (NWO). For a part of our computations we have used the ulgqcd cluster in Liverpool which was funded by the UK Science & Technology Facilities Council (STFC) grant ST/H008837/1.