Four-loop splitting functions in QCD – The gluon-to-quark case –

We have computed the even-N moments N ≤ 20 of the gluon-to-quark splitting function P qg at the fourth order of perturbative QCD via the renormalization of off-shell operator matrix elements. Our results, derived analytically for a general gauge group, agree with all results obtained for this function so far, in particular with the lowest ﬁve moments obtained via physical cross sections. Using our new moments and all available endpoint constraints, we construct approximations for the four-loop P qg ( x ) that should be sufﬁcient for a wide range of collider-physics applications. The N 3 LO corrections resulting from these and the corresponding quark-quark splitting functions lead to a marked improvement of the perturbative accuracy for the scale derivative of the singlet quark distribution, with effects of 1% or less at x > ∼ 10 − 4 at a standard reference scale with α s = 0 . 2.

High-energy particle physics is entering an era of precision measurements.Systematic errors at the ATLAS and CMS experiments at the Large Hadron Collider (LHC) will shrink towards the percent level [1,2].Statistical uncertainties of many observables will be drastically reduced by increasing the amount of collected data by a factor of about 20 during the high-luminosity phase of the LHC [3].Furthermore, new measurements of deep-inelastic scattering (DIS) with percent-level precision are expected at the forthcoming Electron-Ion Collider (EIC) [4].
In order to match this accuracy in theoretical predictions of hard LHC processes and DIS, it is mandatory to compute radiative corrections up to the next-to-next-to-next-to-leading order (N 3 LO) of perturbative QCD [5].The partonic cross sections for some key processes at the LHC [6][7][8][9][10] and for the structure functions F 1,2,3 in inclusive DIS [11][12][13] are already available at this accuracy.The N 3 LO (four-loop) contributions to the splitting functions governing the scale evolution of the parton distribution functions (PDFs) are needed to complete these results.
Approximate results for the corresponding four-loop quark-quark splitting functions, which should be sufficient for most collider-physics applications, have been obtained for the non-singlet cases in refs.[14,15] and, recently, for the pure-singlet (ps) case in ref. [16].In this letter, we continue to address the scale dependence of the (unpolarized) flavour-singlet PDFs, d d ln µ 2 q s g = P qq P qg P gq P gg ⊗ q s g .
Here q s = ∑ n f i=1 (q i + qi ) and g are the singlet quark and gluon distributions, with n f the number of light flavours, and ⊗ denoting the Mellin convolution in the momentum variable x.
The splitting functions P ij , and the corresponding anomalous dimensions γ ij related to their even-N Mellin moments by a conventional sign, can be expanded in powers of a s = α s (µ 2 )/(4π), The next-to-next-to leading order (NNLO, N 2 LO) n = 2 terms in eq. ( 2) were computed almost 20 years ago [17,18].Over the years these were checked bit by bit in various calculations, recently a complete re-calculation was performed using the operator-product expansion (OPE) [19].
Here we derive, in the same approach, the first 10 even-N values of the anomalous dimensions γ (3) qg , thus completing the upper row of the N 3 LO matrix in eq. ( 1) to N = 20.We then combine these results with large-x and small-x constraints to provide approximate expressions for P (3) qg (x).The quantities γ ij are the anomalous dimensions of the gauge invariant twist-two operators where ψ is the quark field, F µν is the gluon field strength tensor, and D µ = ∂ µ −ig A µ is the covariant derivative with the coupling g, where g 2 /(4π) = α s .The curly brackets indicate symmetrization in the indices µ 1 . . .µ N .For brevity, we have not written out the additional terms that render these operators traceless.
O q,g renormalize multiplicatively with a matrix of renormalization constants Z ij .As already pointed out in ref. [20], O q and O g mix with a set of gauge-variant operators, known as aliens.The basis of aliens was determined in ref. [21] at two loops.The general theory on the renormalization of gauge-invariant operators was then developed in refs.[22][23][24][25][26].Recently the basis of alien operators up to four loops was constructed explicitly for any fixed moment N [27].This basis is consistent with the counterterms computed at two and three loops for all values of N in refs.[21,28,29] and [19], respectively.For the case at hand the renormalization proceeds along the lines of these references, but since it is technically more involved we defer the details to a later publication.
The anomalous dimensions γ ij in eq. ( 2) are obtained from the renormalization constants Z ij .They are determined by requiring the finiteness of the renormalized operator matrix elements (OMEs) A ij = j(p)|O i | j(p) .These are two-point Green functions of the operator O i with offshell external states j of momentum p, where j can be a quark (q), gluon (g) or ghost (c).
The Feynman diagrams for the OMEs have been generated using QGRAF [30] and then processed, see ref. [31], by a FORM [32][33][34] program which collects self-energy insertions, determines the colour factors [35] and classifies the topologies according to the conventions of the FORCER program [36].An optimized in-house version of this program has been employed to perform the integral reduction for fixed even values of N in 4 − 2ε dimensions.Diagrams with the same colour factor and topology have been merged into meta-diagrams for computational efficiency.
In this manner, we were able to compute the physical OMEs A qg for the gluon-to-quark splitting function to four loops for N ≤ 20.The other physical OMEs A ps , A gq and A gg are needed at three loops for the determination of γ qg , and the OMEs with the alien operators inserted into a gluon two-point function, A Ag , are required only to two loops.These computations yield the following results for the N 3 LO contributions to the anomalous dimensions γ qg in eq. ( 2) for QCD, i.e., the gauge group SU(n c = 3): The corresponding exact results, in terms of fractions and the values ζ 3 , ζ 4 and ζ 5 of the Riemann ζ-function, are given for a general compact simple gauge group in app.A, eqs.(A.3) -(A.12).
Our results for N ≤ 10 agree with refs.[37,38], where those moments were computed, as the three-loop all-N expressions in refs.[17,18], via structure functions in inclusive DIS, a route that is conceptionally simpler but far more demanding in terms of the integral reductions.The coefficients of n 3 f in eq. ( 4) agree with the all-N results in eq.(3.12) of ref. [39].All-N expressions for the anomalous dimensions include Riemann-ζ values, harmonic sums [40] and simple denominators D k a ≡ (N + a) −k .The latter frequently arise in the combinations Besides the leading large-n f contribution, all-N expressions for γ qg have been obtained until now only for the ζ 4 part, in eq. ( 10) of ref. [41], and the ζ 5 coefficients of the quartic group invariants [42].Using our results to N = 20, we have now been able to determine the ζ 5 coefficients for all colour factors, and to extend the all-N results for the quartic group invariants to the ζ 3 terms.
The latter results, as in ref. [42] using the short-hand d qg (N) RR /n a = 256 7/12 where the argument N of the harmonic sums S ... has been suppressed.The complete ζ 5 part reads γ qg (N) Analytic expressions in x-space are, for now, available only for the leading large-n f part of P qg , see eq. (4.22) of ref. [39].The above partial N-space results proportional to Riemann-ζ values do not translate to x-space Riemann-ζ expressions, since additional terms with ζ n are generated by the inverse Mellin transformation.Similarly, it is not possible to read off coefficients of ζ n in the large-N limit from eqs. ( 7) - (9).In particular, as γ qg vanishes for N → ∞, terms of the form ζ 5 (N − 1) need to be compensated by other contributions that develop ζ 5 terms in this limit, as in the functions g i (N) in eqs.(3.18) -(3.20) of ref. [11].
For the time being, only approximations can be provided for the N 3 LO splitting function 1) and ( 2), based on the moments (4) and all known results for the large-x and small-x limits.The large-x expansion of P (n) qg (x) is given by The coefficients C qg 3,ℓ,p have been predicted for ℓ = 0, 1, 2 in ref. [43]; the results for p = 0 have been confirmed and extended to all higher orders n in refs.[44,45].The small-x expansion reads The coefficients E qg n,1 of the leading 1/x logarithms are known [46], as well as those of the highest three sub-dominant x 0 double logarithms, F n,ℓ , for ℓ = 0, 1, 2 at n = 3, 4 [47].
Our procedure for the construction of the approximations is analogous to that for the puresinglet case in ref. [16].The situation is less favourable here due to the presence of three logarithmically enhanced unknown p = 0 terms in eq. ( 10).This is reflected in a larger uncertainty of the crucial x −1 ln x coefficient E qg 3,2 in eq.(11).Instead of a 'direct fit' of this term, as shown in fig. 1 of ref. [16] for P (3) ps (x), we have first determined a conservative range for this parameter, then constructed 80 approximations for the two boundaries of this range, and finally selected representatives P (3) qg, A (x) and P (3) qg, B (x) that provide the error bands for n f = 3, 4, 5 light flavours.Using the abbreviations x 1 = 1−x, L 1 = ln(1−x) and L 0 = ln x, the chosen approximations, shown in red in fig. 1 for n f = 4, are 14) with the known endpoint contributions [43][44][45][46][47] p where all coefficients have been rounded to eight significant figures.
These error bands also lead to the following predictions for the numerical values of γ The resulting perturbative expansion of P qg (x, α s ) to N 3 LO is illustrated in the left panel of fig. 2 for n f = 4 at a standard reference point α s (µ 2 0 ) = 0.2 corresponding to a scale in the range µ 2 0 ≃ 25 . . .50 GeV 2 .The remaining uncertainty due to the approximate nature of P qg (x) is completely unproblematic down to x ≃ 3 • 10 −3 , but reaches about ±10% at x = 10 −4 .This does not mean, however, that the effect of P qg on the scale dependence of the singlet quark PDF q s is that uncertain.As shown in the right panel of fig.2, the uncertainty of the convolution amounts to only about 1% or less even down to x = 10 −5 for α s = 0.2 and the sufficiently realistic (order-independent) reference gluon distribution [18] xg(x, µ Note that in eq. ( 17) P qg at y > ∼ x is multiplied by the small g at x < ∼ 1, while P qg at large y is combined with the much larger small-x gluon PDF.In view of these results we can conclude that the present results for P qg should be sufficient for a wide range of phenomenological applications, even if our error bands were to somewhat underestimate its remaining uncertainty at x < ∼ 10 −3 .13) for the four-loop contribution.Right: The resulting N 2 LO and N 3 LO convolutions (17) with the reference gluon distribution (18), normalized to the NLO result.
Combining our above results with those of refs.[14,16], we are now able to evaluate the N 3 LO contributions to the scale derivative (1) of the quark PDF q s at the chosen reference point.Complementing, as already in ref. [18], eq. ( 18) by the relative size of the N 2 LO and N 3 LO contributions to dq s /d ln µ 2 is shown in the left part of fig. 3. We see that the N 3 LO contributions are much smaller than their N 2 LO counterparts.They exceed 1% only at x < 10 −4 , and even at x = 10 −5 amount only to about (2 ± 1)%.
Up to now, we have identified the renormalization scale µ r with the factorization scale µ f ≡ µ.The expansion in eq. ( 2) is readily extended to µ r = µ f , see, e.g., eq.(2.9) of ref. [48] for the expression to order α 5 s .The scale stability of qs ≡ d ln q s /d ln µ 2 is illustrated in the right part of fig. 3 by the quantity for the conventional range λ = 1/4 . . . 4. Also here we see a clear improvement by including the N 3 LO terms, to uncertainties below 2% at x > ∼ 10 −4 and 1% at x > ∼ 10 −2 .The renormalization-scale uncertainties of these results, as estimated using eq.(20).Note that dq s /d ln µ 2 changes sign close to x = 0.1, which leads to (irrelevant) singularities in both the relative corrections and the relative scale uncertainty (20).
To summarize, we have computed the even moments N ≤ 20 of the four-loop (N 3 LO) gluon-toquark splitting function P (3) qg (x) in the framework of the operator-product expansion.We have used these results, together with the known constraints for x → 1 and x → 0, to provide approximations for P (3) qg (x) that should be sufficient for most phenomenological applications.Further incremental improvements can be obtained by extending the computations beyond N = 20.A larger reduction of the remaining uncertainties at small x would be achieved by the determination of the hitherto unknown coefficient of the next-to-leading x −1 ln x small-x contribution to P

A Mellin moments of P (3) qg
Here we present the exact results for the four-loop quark-to-gluon anomalous dimensions γ   The expressions (A.3) -(A.7) have also been computed, in a different manner, in refs.[37,38].
The terms proportional to d RR /n a and d RA /n a to N ≤ 16 have been obtained before in ref. [42].The n 3 f contributions are known at all N [39].A FORM file with the results for γ qg (N) at even N ≤ 20, all partial all-N expressions in the main text, and a FORTRAN subroutine of the approximate splitting function P (3) qg (x) have been deposited at the preprint server https://arXiv.org with the sources of this letter.They are also available from the authors upon request.

Figure 1 :Figure 2 :
Figure1: Two sets of 80 trial functions, one for a large and one for a small value of the unknown coefficient of x −1 ln x, for the four-loop (N 3 LO) contribution to the gluon-to-quark splitting function at n f = 4.The two cases selected for eq.(13) are shown by the solid (red) lines.

Figure 3 :
Figure3: Left: The relative N 2 LO and N 3 LO corrections to the scale derivative of the quark PDF q s at the reference point with n f = 4 and α s (µ 2 0 ) = 0.2.Right: The renormalization-scale uncertainties of these results, as estimated using eq.(20).Note that dq s /d ln µ 2 changes sign close to x = 0.1, which leads to (irrelevant) singularities in both the relative corrections and the relative scale uncertainty(20).

( 3 )
qg (N) at even N ≤ 20 for a general compact simple gauge group.The numerical values in QCD, i.e., SU(n c = 3) have been given in eq.(4) above.The quadratic Casimir invariants in SU(n c ) are C A = n c and C F = (n 2 c − 1)/(2n c ).The relevant quartic group invariants arise as products of two symmetrized traces of four generators T a r of the fundamental (R) or adjoint (A) representation, RR /n a = 5/96.γ