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Approximate four-loop QCD corrections to the Higgs-boson production cross section

We study the soft and collinear (SV) contributions to inclusive Higgs-boson production in gluon-gluon fusion at four loops. Using recent progress for the quark and gluon form factors and Mellin moments of splitting functions, we are able to complete the soft-gluon enhanced contributions exactly in the limit of a large number of colours, and to a sufficiently accurate numerical accuracy for QCD. The four-loop SV contributions increase the QCD cross section at 14 TeV by 2.7% and 0.2% for the standard choices mu_R=m_H and mu_R=m_H/2 of the renormalization scale, and reduce the scale uncertainty to below +-3%. As by-products, we derive the complete delta(1-x) term for the gluon-gluon splitting function at four loops and its purely Abelian contributions at five loops, and provide a numerical result for the single pole of the four-loop gluon form factor in dimensional regularization. Finally we present the closely related fourth-order coefficients D_4 for the soft-gluon exponentiation of Higgs-boson and Drell-Yan lepton-pair production.

The production of the Standard Model (SM) Higgs boson in proton-proton collisions and its subsequent decay are flagship measurements in run 2 of the Large Hadron Collider (LHC) [1,2]. The main production mechanism for pp → H + X is the gluon-gluon fusion (ggF) process. The corresponding inclusive cross section serves as a benchmark for the achieved accuracy, both in the LHC experiments and for theoretical research. The radiative corrections in Quantum Chromodynamics (QCD) for the ggF process are large and have motivated significant efforts to improve the precision of the predictions. The QCD corrections are currently known to the next-to-next-to-nextto-leading order (N 3 LO) in the effective theory for a large top-quark mass, m t m H [3,4], and to next-to-next-to-leading order (NNLO) in the full theory for Higgs-boson masses m H < ∼ 2 m t [5][6][7]. Near the production threshold, for z = m 2 H /ŝ close to unity, where m H is the Higgs mass and √ŝ the partonic center-of-mass energy, the QCD corrections to the ggF process are dominated by the well-known large logarithmic corrections. At n-th order they appear in the partonic cross section in the MS scheme as plus-distributions D k = [(1−z) −1 ln k (1−z)] + with 2n − 1 ≥ k ≥ 0, while the virtual contributions lead to δ(1−z) terms. In Mellin N-space, where N is the conjugate variable of z, the threshold logarithms read ln k N with 2n ≥ k ≥ 1, and the virtual contributions lead to a constant in N. The soft-virtual (SV) approximation to the partonic ggF cross section in N-space yields reliable predictions for the total Higgs production cross section, as has been demonstrated with comparisons to exact fixed-order results up to NNLO, see, e.g., Refs. [8,9]. In addition, Mellin N-space lends itself to an all-order exponentiation of threshold contributions up to next-to-next-to-next-to-leading logarithmic (N 3 LL) accuracy and beyond [10,11].
These facts motivate the derivation of approximate QCD corrections to the ggF process at four loops in the effective theory, which can be achieved thanks to recent progress in the computation of QCD corrections for related quantities at the four-loop level. This comprises results for specific colour contributions, including quartic group invariants, and the planar limit of quark and gluon form factors [12][13][14][15][16][17][18], correlators of Wilson lines [19][20][21], splitting functions for the evolution of parton distributions (PDFs) [22][23][24][25], and, related, the knowledge of a low number of Mellin moments for the structure functions in deep-inelastic scattering (DIS), see Ref. [26].
Taken together, this knowledge enables us to determine precise numerical results for the complete SV approximation of the ggF process at four loops as well as partial information on terms suppressed by a power 1/N in Mellin N-space using physical evolution kernels at the same order [9]. The results are used to provide new predictions for the ggF cross section at the collision energy of 14 TeV, as planned for run 3 of the LHC. We also present the corresponding expression for the Drell-Yan (DY) process, pp → γ * + X, which is closely related to the ggF process in the threshold limit, and the N 4 LL soft-gluon exponentiation coefficient D 4 for both processes. As by-products, we obtain a complete result for the so-called gluon virtual anomalous dimension, i.e. the δ(1−z) terms of the gluon-gluon splitting function at four loops in QCD, together with partial information at five loops, and we derive a numerical result for the single pole 1/ε of the dimensionally regulated gluon form factor at four loops.
The effective coupling of the Higgs boson to partons is described by the Lagrangian where υ 246 GeV is the Higgs vacuum expectation value in the SM and G a µν denotes the gluon field strength tensor. The inclusive hadronic cross section for Higgs-boson production at a centerof-mass energy E cm = √ S is given in standard QCD factorization by where τ = m 2 H /S, and µ F , µ R are the mass-factorization and renormalization scales, and f a/h (x, µ 2 F ) the PDFs of the proton. The expansion in the strong coupling α s of the large-m t effective vertex for the Higgs coupling to gluons is included in σ H 0 , viz where n A = (n 2 c − 1) denotes the dimension of the adjoint representation of the SU(n c ) gauge group, and the matching coefficient  [38] and that of D 1 is fixed by the results of Ref. [39] and has been given in Eq. (13) of Ref. [40]. An approximate result for the D 0 term has been provided before in Eqs. (2.13) and (2.14) of Ref. [9].
Here we present a new result for the latter coefficient for a general gauge group. The relevant Casimir invariants for SU(n c ) are C A = n c , C F = (n 2 c − 1)/(2n c ) and With the recent progress at four loops on the pole structure of the QCD form factors, on splitting functions and on Mellin moments for DIS structure functions, and following the same procedure as employed for DIS structure functions in Ref. [11], the D 0 term in c H, Here the term f is related to the eikonal anomalous dimension of the four-loop quark form factor, i.e., to the single pole in dimensional regularization, cf. Ref. [11]. The expressions b denote the four-loop coefficients of the respective colour factor in the quark virtual anomalous dimension B q , i.e.,the coefficient of δ(1−z) in the quark-quark splitting function P qq . At n-th order, expanding in powers of a s ≡ α s (µ 2 R )/(4π) analogous to Eq. (4), the flavor-diagonal splitting functions P ii admit the large-z expansion [41,42] as where the coefficients A i n are the well-known lightlike cusp anomalous dimensions. The quantities f are not known analytically. They drop out in the large-n c limit of Eq. (6). Precise numerical estimates, i.e., b .780 ± 0.005 have been given in Ref. [11] together While the large-n c limit of Eq. (6) is exact, the above general expression uses one assumption on the relation of quark and gluon form factors F q and F g in QCD in dimensional regularization, which, in the normalization of Eq. (4), admit the perturbative expansion as see e.g. Refs. [11,43] for the higher orders.
The logarithms of F q and F g contain double and single poles in ε, the former being controlled by their respective cusp anomalous dimensions, A q and A g , cf. Eq. (7), which exhibit generalized Casimir scaling through four loops, see, e.g. [18,24]. The single poles on the other hand, which are proportional to functions G p n (ε) at n-th order, are controlled by the collinear anomalous dimensions and can be converted to appropriate eikonal (Wilson line) quantities, after separation of the virtual anomalous dimensions B q and B g , cf. Eq. (7), and terms proportional to the QCD β-function.
In detail (see, e.g. [11,43]), the functions G p n (ε) satisfy the following relations to five loops where the functions f p 0n (ε) at n loops are polynomials in ε and β n are the coefficients of the QCD β-function normalized as in Eq. (4), i.e., β(a s ) = −β 0 a 2 s − . . . with β 0 = 11/3 C A − 2/3 n f . The eikonal anomalous dimensions f q and f g of these Wilson line quantities for quarks and gluons exhibit the same maximal non-Abelian colour structure as the cusp anomalous dimensions, a fact verified explicitly at lower fixed orders [43,44] and generalized in Ref. [45]. Hence we assume here that also the expressions for f q and f g are related by generalized Casimir scaling at four loops (and beyond), in complete analogy to the cusp anomalous dimensions, A q and A g , see also the recent work [46]. This leads immediately to the expression for the full colour dependence of the four-loop gluon virtual anomalous dimension B g 4 as where the n 3 f -dependent terms agree with Ref. [22]. In addition, we have checked that a numerical fit for the d abcd F d abcd A term in the gluon-gluon splitting function [24] to the known Mellin moments nicely confirms the value quoted in Eq. (10). Altogether, we take this as strong indications on the correctness of the assumption made in the derivation of Eqs. (6) and (10). The remaining unknowns can be determined numerically as b We also note that the purely Abelian (QED) contributions in B g 4 coincide with the respective terms in the four-loop QCD β-function [47,48]. This concerns the colour factors n f C 3 and is a direct consequence of the generalized Casimir scaling of f q and f g , which implies that f ). This reproduces a pattern already observed for B g n up to third order, n ≤ 3, see [49], for all terms n k f C n−k  With the help of Eqs. (9) and (10) we obtain the single pole in ε in the gluon form factor F g 4 at four loops as where all n f -dependent contributions have been given analytically in Ref. [18].
The observed relation between the gluon virtual anomalous dimension B g n and the corresponding coefficient of the β-function for purely Abelian terms leads to new predictions at five loops. Using the expression for the β-function for a general gauge group at five loops [50][51][52] (the QCD result was obtained before in Ref. [53]), one deduces for the splitting function P gg in Eq. (7) .
Predictions for the purely Abelian part of the gluon form factor at five loops are possible, e.g., for the n f C 4 F in F g 5 , which can be read off from Eq. (12) using G g 5 in Eq. (9), while finite terms of F g at lower orders are needed for other colour structures.
Beyond the SV approximation, predictions for the ggF cross section are possible using physical evolution kernels [9,54,55]. In z-space, this concerns terms enhanced as ln k (1−z) with 2n − 1 ≥ k ≥ 1 at n-th order, or equivalently power suppressed contributions (ln k N)/N in Mellin N-space. Such next-to-leading power threshold effects have also been studied in Refs. [56][57][58][59][60][61][62][63]. At N 4 LO, these subleading terms in c H,(4) gg (z) can be obtained from the physical evolution kernel K gg , which one can define by re-expressing Eq. (2) as dimensionless 'structure functions' F ab , i.e., The kernel K gg and its perturbative expansion for a scale choice µ F = m H are then given in terms of the splitting function P gg , the β-function and the gluon coefficient function c H gg by where ⊗ denotes the usual Mellin convolution and β(a s ) the β-function as defined below Eq. (9).
The key feature of the kernel K gg is its single-logarithmic enhancement. In z-space, this implies at n-th order that all terms ln k (1−z) with 2n − 1 ≥ k ≥ n + 1 have to cancel in Eq. (13) to all orders in (1−z), which leads to predictions for coefficient function c H gg , cf. Ref. [9]. In Mellin N-space the leading large-N logarithms of the sub-dominant N −1 contributions in K gg take the simple form, where the first three lines follow from the known coefficient functions c H,(n) gg up to N 3 LO. The expression for K (4) gg contains an unknown coefficient ξ A β 0 ln 6 (1−z) 6 We note that the pattern of the ratios of the lower order coefficients is 112/32 = 7/2 and (896/3)/112 = 8/3.
A generalization of this pattern leads to an estimate of ξ H = 670 ± 300 with a conservative numerical uncertainty, which has a sub-percent effect on the N −1 ln 4 N coefficient in Eq. (16) below.
where c H,(4) gg (z)| SV denotes the z-space SV approximation at N 4 LO as discussion above.
We are now ready to assemble the results of Refs. [9,38,40] and Eqs. (6) and (14)  The N 4 LO coefficient function in the SV approximation, together with the sub-dominant N −1 contributions in Mellin N-space, can be expected to provide a reliable approximation of the exact result. As pointed out earlier, the exact Mellin N-space result at lower orders resides inside the band spanned by the SV and SV+N −1 terms at moderately large N. This is shown in Fig. 1 (left) at N 3 LO (for corresponding NLO and NNLO plots see Fig. 1 of Ref. [9]). The exact coefficient functions differ from the approximation based on the SV+N −1 terms by 0.44% at NLO, 0.83% at N 2 LO and 1.15% at N 3 LO at N = 12. For smaller N values the difference between the exact results and the approximation based on the SV+N −1 terms is larger, however they always remain inside the SV and SV+N −1 band. At N 4 LO, see Fig. 1 (right), the SV approximation of Eq. (16) is shown including the N-independent constant g 04 and the known 1/N terms as specified in Eq. (16).
The predictions for the ggF cross sections at the collision energy of 14 TeV use a Higgs mass m H = 125 GeV, an on-shell top quark mass m t = 172.5 GeV, n f = 5 active quark flavors and the PDF sets ABMP16 [64] and MMHT2014 [65] using the lhapdf [66] interface. The PDF sets and as well as the value of the strong coupling constant α s corresponding to the respective PDF set are taken order-independent at NNLO throughout. The prefactor C(µ 2 R ) in Eq. (3) is improved with the full top-mass dependence of the Born cross section. The results up to N 3 LO are computed with the program iHixs [67] which directly provides the cross sections in this rescaled effective field theory.  Fig. 2 (left). For the scale choice µ R = m H /2, which is closer to the point of minimal sensitivity, the effect of higher order corrections is indeed small, leading to a K-factor close to unity, cf. also Ref. [9]. At the scale µ R = m H a K-factor of 1.027 is obtained for the LHC at 14 TeV.
In Fig. 2 (right) we show the dependence of the cross section for the E CM = 14 TeV on the renormalization scale µ R . The µ R dependence indeed decreases order by order in perturbation theory up to N 4 LO. In the range m H /4 ≤ µ R ≤ 2m H the effect of the µ R variation decreases drastically from ±27% at NLO and ±14% at NNLO to ±5% at N 3 LO, while it amounts to less than ±3% at N 4 LO. Also here the factorization scale is kept fixed, at µ F = m H , since beyond NNLO only flavour non-singlet results have published for the QCD splitting functions P ik (z) [22,23,68], and PDFs fits have been limited to NNLO so far.
The uncertainty in the predicted ggF cross sections due to the truncation of the perturbation series is now, at N 4 LO, smaller than that due to the use of different sets of PDFs and corresponding different values of the strong coupling constant α s . For √ S = 14 TeV, m H = 125 GeV, the central scale µ R = m H , and including the PDF uncertainties at N 3 LO, one obtains where the spread in predictions is due to different values of the strong coupling constant at NNLO corresponding to the different PDF sets used, i.e., α s (M Z ) = 0.1147 for ABMP16 and α s (M Z ) = 0.1180 for MMHT2014, and due to different gluon PDFs in the relevant kinematic range. These are consequences of different choices for the theoretical framework and assumptions on parameters used in the respective global fits, see Ref. [69], which lead to systematic shifts that are often significantly larger than the PDF and α s (M Z ) uncertainties associated to individual PDF sets.
Due to the universality of threshold dynamics for colourless final states in hadronic collisions, relevant formulae for the Drell-Yan process, pp → γ * + X, can be easily obtained from the above considerations, using Eq. (2) with the replacement Here α is the fine-structure constant of QED and Q 2 the virtuality of the produced photon γ * . The coefficient functions c DY ab enjoy a perturbative expansive analogous to Eq. (4) with the leading order normalization c DY,(0) ab = δ aq δ bq δ(1−z). The coefficients of D k for 7 ≥ k ≥ 2 of the four-loop term c DY,(4) qq can be found in Eq. (6) of Ref. [38] and that of D 1 in Eq. (14) of Ref. [40]. We are now in the position to present the four-loop D 0 term for the DY process. It is given by where the quartic Casimirs are normalized by the dimension of the fundamental representation of the SU(n c ) gauge group, n F = n c . As Eq. (6), this result is exact in the large-n c limit and has an amply sufficient numerical accuracy for all phenomenological applications in QCD.
The next-to-leading power threshold terms for the DY process can also be derived with the help of the corresponding physical evolution kernel K qq , which exhibits the same simple form in Mellin N-space as Eq. (14) for the leading large-N logarithms of the N −1 contributions with the obvious replacement C A → C F , see Ref. [9] for further details.
Finally, by combining our new result (6) with Eq. (2.13) in Ref. [9], and proceeding analogously with its Drell-Yan counterpart (19), we can derive the four-loop coefficient D 4 for the soft-gluon exponentiation of inclusive Higgs production via ggF and DY lepton-pair production. The two results are, as expected, related by generalized Casimir scaling [24] which reduces to standard 'numerical' C A /C F lower-order Casimir scaling in the their exact large-n c limit: Using recent progress on related fourth-order quantities, we have been able to determine the final soft-gluon enhanced contribution (6) to the N 4 LO coefficient function for inclusive Higgsboson production in gluon-gluon fusion in the heavy-top limit, and the corresponding result (19) for the Drell-Yan process pp → γ * + X. These results also fix the respective N 4 LL coefficients D 4 for the soft-gluon exponentiation (20) which are related by the same fourth-order generalization of the well-known Casimir scaling observed before in the cusp anomalous dimensions, now completely known at this order [21]. Our results are exact in the limit of a large number of flavours n c . Their colour-factor decomposition in full QCD involves a few quantities which are known only numerically at this point. The resulting uncertainties are practically negligible as can be seen from the ln N coefficient in Eq. (16) above.
We have employed the latter Mellin N-space results to add the N 4 LO soft + virtual (SV) corrections to the known complete N 3 LO results [3,4] for the LHC at 14 TeV. With the effect of the only uncomputed quantity, the soft-gluon coefficient g 04 for this process, being well below 1%, we find that the cross sections are enhanced by 2.7% for the scale choice µ R = m H , while the results are almost unchanged for µ R = 0.5 m H . It should be noted that these values refer to the not entirely realistic case of an order-independent α s -value and PDFs at µ = m H . The renormalizationscale variation, estimated using the interval 0.25m H ≤ µ R ≤ 2m H , is reduced from about 5% at N 3 LO to less than 3% at N 4 LO. Based on similar calculations at lower orders, we definitely expect that difference between the present N-space SV approximation and the complete N 4 LO coefficient function will amount to well below 1% of the total cross section.
As by-products of our analysis, we have derived the expression (10) for the four-loop gluon virtual anomalous dimension (and determined the corresponding purely Abelian contributions at five loops), and provided a sub-percent accurate value (11) for the hitherto unknown 1/ε coefficient of the matter-independent contribution to the four-loop gluon form factor for which the n f -terms have been recently computed in Ref. [18].