Leading large-x logarithms of the quark-gluon contributions to inclusive Higgs-boson and lepton-pair production

We present all-order expressions for the leading double-logarithmic threshold contributions to the quark-gluon coefficient functions for inclusive Higgs-boson production in the heavy top-quark limit and for Drell-Yan lepton-pair production. These results have been derived using the structure of the unfactorized cross sections in dimensional regularization and the large-x resummation of the gluon-quark and quark-gluon splitting functions. The resummed coefficient functions, which are identical up to colour factor replacements, are similar to their counterparts in deep-inelastic scattering but slightly more complicated.

The discovery of a particle with a mass of about 125 GeV [1] and properties consistent with those of the standard-model Higgs boson [2] at the LHC has led to increased interest in precision predictions for Higgs production and decay. The main channel for the total production cross section is gluon-gluon fusion via a top quark loop, known at all M H /M top to next-to-leading order (NLO) of perturbative QCD [3,4]. The convergence of the perturbation series is particularly slow in this case, hence calculations are required at, and beyond, the next-to-next-to-leading order (NNLO).
These calculations can be carried out, at a sufficient accuracy [5], for an effective Hgg interaction in the heavy-top limit [6], where G a µν denotes the gluon field strength tensor. The prefactor C H includes all QCD corrections to the top quark loop; it is of first order in the strong coupling constant α s and fully known up to N 3 LO ( α 4 s ) [7], see also Refs. [8]. The NNLO contributions to the total cross sections were computed in this effective theory in Refs. [9][10][11]; a high-accuracy threshold resummation and a first approximation for N 3 LO corrections were subsequently obtained in Refs. [12,13].
Recently a major step has been taken towards deriving the complete N 3 LO corrections: the calculation of the soft-gluon and virtual contributions at this order [14]. This result directly leads to a further improvement in the threshold limit [15][16][17] by fixing the remaining parameter required for a full N 3 LO + next-to-next-to-next-to-leading logarithmic (N 3 LL) accuracy [18] of the soft-gluon exponentiation. The same soft + virtual N 3 LO and resummation accuracy has also been reached for Drell-Yan lepton-pair production pp → ℓ + ℓ − + anything, calculated at NNLO in Refs. [19,20], due to its close similarity with inclusive Higgs-boson production [15,17].
Generally fixed-or all-order results for logarithmically enhanced endpoint contributions, e.g., in the large-x or threshold limit, can provide checks of elaborate Feynman-diagram calculations and estimates of corrections that cannot (yet) be calculated directly. Quite a few studies of the threshold limit have addressed the dominant channels in Higgs and lepton-pair production, i.e., gluon-gluon fusion and quark-antiquark annihilation, respectively. Here we present first all-order results for the sub-dominant quark-gluon contributions to both processes. In particular, we derive the leading large-x logarithms of the coefficient functions c P,qg for P = H and P = DY.
Our derivation starts from the unfactorized partonic cross sections W P, jℓ in which lead to the mass-factorized expressions Here ⊗ abbreviates the Mellin convolutions, and summations over the light quarks and antiquarks and gluons are understood. All charge factors have been suppressed; see, e.g., Appendix A of Ref. [19] for the Drell-Yan process. We use dimensional regularization with D = 4 − 2ε; a tilde marks the D-dimensional counterparts of quantities which are finite for ε = 0. In particular, the coefficient functions in Eq. (2) can be written as for the choice µ r = µ f = M of the renormalization and mass-factorization scales, with M = M H or M = M ℓ + ℓ − , which can by made without loss of information. All factorized expressions refer to the MS scheme; the additional terms defining its difference to MS are suppressed in Eq. (4) and below. The coefficient functions c P,ik in Eq. (3) are obtained from the above by setting ε = 0.
The scale dependence of the factorized parton distributions f i in Eq. (3) is governed by the splitting functions P ik , which are related to the transition functions Z ik in Eq. (2) by (5) can be solved for Z order by order in α s .
The prefactors σ 0,P in Eq. (2) are defined such that the lowest-order contributions to the D-dimensional coefficient functions in Eq. (4) are normalized and independent of ε, i.e., given by We further specify our notation for the coefficient functions and splitting functions by recalling the leading-logarithmic large-x contributions to the NLO quark-gluon coefficient functions: with C F = 4 3 , T f = 1 2 and C A = 3 for QCD. Note that our convention in Eq. (7) differs from the quantities ∆ ik in Refs. [10,11] by a factor of x −1 . On the other hand, our normalization in Eq. (8) is the same as in Ref. [19]. The corresponding NNLO corrections read It is convenient to turn the convolutions above to products by Mellin transforming all quantities, where the parts in curly brackets refer to the case of (1 − x) −1 +-distributions. Here we mainly consider the leading powers of (1−x) in the threshold limit, in particular (1−x) 0 corresponding to N −1 in the large-N limit for the quark-gluon quantities addressed in this letter. Keeping only the leading -and subleading, if ln k N is replaced by ln k N + k γ e ln k−1 N -contributions, the relations between the corresponding expressions in x-space and Mellin-N space read Here and below M = denotes equality under the Mellin transformation (11).
The diagonal splitting function are not logarithmically enhanced at higher orders for the N 0 contributions [21] (nor at N −1 , see Refs. [22,23] ). Hence only their leading-order contributions are relevant here (and at NLL), with The corresponding off-diagonal contributions can be readily read off from Eqs. (7) and (8), These functions do exhibit a double-logarithmic higher-order enhancement, derived in Ref. [24], in terms of the function where B n are the Bernoulli numbers in the standard normalization of Ref. [25], and For the corresponding NLL and NNLL resummations of the splitting functions see Refs. [26,27].
We are now prepared to return to the unfactorized cross sections in Eq. (2). For brevity the following steps are written out only for Higgs-boson production. We have checked that the corresponding relations for the Drell-Yan case can be obtained, as expected from Eqs. For the resummation of the quark-gluon coefficient function c H,qg = c H,qg we need to consider and which provides c H,gg for the right-hand-side of Eq. (19). Other coefficient functions such as c H,qq are not relevant for the leading logarithms in Eq. (19) even at higher orders in N −1 .
At the leading (and next-to-leading) power in N −1 the a n s contributions to the diagonal and off-diagonal transition functions are given by [24] Here additional sign factors have been avoided by using the anomalous dimensions γ defined in Eq. ( and at order N 0 . The difference of Eq. (24) to the corresponding structure function in deep-inelastic scattering (DIS) is the replacement ε → 2 ε in the exponentials due to the different phase space. An extension of Eqs. (21) - (24) to higher logarithmic accuracy is no problem, but not required here.
The right-hand-side of Eq. (19) is thus known at LL accuracy at all powers of α s and ε except for the quark-gluon coefficient function. Hence an all-order result for W H,qg on the left-hand-side corresponding to Eqs. (23) and (24) leads to a LL resummation of c H,qg ; determining this result is the crucial step of our calculations.
Taking into account (1−x) −k ε factors due to real and virtual corrections, cf. the discussion of the phase-space master integrals in Ref. [10], the general form of the a n s contribution to W H,qg is The parameters A (n,ℓ) H,qg combine to the coefficients of the LL contributions a n s ε −2 n+m ln m−1 N in Eqs. (19), which, of course, vanish for 1 ≤ m ≤ n − 1 due to Eqs. (21) and (22). Correspondingly, the quantities B The presence of 2n − 1 terms in the sums (25) represents a crucial difference to W (n) H,gg in the N 0 soft-gluon limit, where only the n even values of ℓ occur [13], and inclusive DIS and semiinclusive e + e − annihilation (SIA), where the corresponding sums run from ℓ = 1 to ℓ = n [26,28]. In those cases, an N n LO calculation leads to a N n LL resummation with a large number of relations to spare. Here, instead, all 2n − 1 terms with negative powers of ε are required to fix the LL coefficients A (n,ℓ) H,qg , i.e., the terms to ε −2 fixed by lower-order contributions together with the ε −1 term provided by the splitting-function resummation (16). Consequently, due to the extra factor of ε, the NLL coefficients B  H,qg , they involve the splitting functions provided by fixed-order calculations at n ≤ 3 and the resummations (15) and (16). Finally entries marked by 'D' are determined, at each order, from the above coefficients via Eq. (25). Checks of this procedure are provided by the a 2 s ε 0 terms of Refs. [9][10][11]19,20], see Eqs. (9) and (10), and the a 2 s ε 1 contributions to Higgs production calculated in Ref. [29].
We have determined the coefficients A which can be cast in a closed, if not very transparent, form in terms of binomial coefficients: with ρ = ℓ − 2m and ⌊a⌋ the largest integer not greater than a. The simplicity of especially the special cases (26) provides some additional insurance against calculational errors. It is interesting to note that not only A (n,3) H,qg , but all odd-ℓ coefficients vanish for C F = C A .
With these results the LL mass-factorization of W H,qg can be performed order by order; it leads to a table of coefficients which has been given to n = 12 in Ref. [30]. Finally this table can be used to find and verify the all-order resummation formula for the quark-gluon coefficient functions, which involves the same ingredients as its counterpart for DIS [24] but is slightly more complicated. The corresponding coefficient function for the Drell-Yan process can be obtained from (28) by C F → T f in the numerator of the prefactor and C A ↔ C F everywhere else, including the argument of the function B 0 . Expansion of Eq. (28) and Mellin inversion yields the explicit third-and fourth-order predictions c (3) LL H,qg (x, a s ) = ln 5 and their obvious analogues for lepton-pair production.
To summarize, we have derived the leading-logarithmic large-x resummation of the quarkgluon coefficient functions for inclusive Higgs-boson and lepton-pair production; our main results are Eq. (28) and its closely related counterpart for the Drell-Yan process. Our calculations have been confined to the leading term in the expansion in powers of (1−x); yet we definitely expect the structure with P (0) ik (x) in Eq. (7) - (10) to occur at all orders. An extension of our results to the next-to-leading double logarithms, α n s ln 2n−2 (1−x), would require additional all-order insight into the corresponding coefficients in the crucial decomposition of the unfactorized partonic cross section (25). One may hope that an extension of Ref. [14] to the complete N 3 LO corrections will soon provide useful information also for the large-x resummation of the quark-gluon channel.