NLO QCD corrections to Higgs boson plus jet production with full top-quark mass dependence

We present the next-to-leading order QCD corrections to the production of a Higgs boson in association with one jet at the LHC including the full top-quark mass dependence. The mass of the bottom quark is neglected. The two-loop integrals appearing in the virtual contribution are calculated numerically using the method of Sector Decomposition. We study the Higgs boson transverse momentum distribution, focusing on the high $p_{t,\mathrm{H}}$ region, where the top-quark loop is resolved. We find that the next-to-leading order QCD corrections are large but that the ratio of the next-to-leading order to leading order result is similar to that obtained by computing in the limit of large top-quark mass.


INTRODUCTION
During Run I and early Run II of the LHC great progress has been made in establishing many of the properties of the Higgs boson particle discovered in 2012. Already the spin and CP properties are well constrained and its couplings to the Standard Model (SM) weak vector bosons and heavier fermions (top quarks and tau leptons) have been measured [1]. So far the measured properties are consistent with the predictions of the SM.
Among the different channels for the production of a SM Higgs boson, gluon fusion, which we consider here, is the mechanism yielding the largest contribution. At lowest order in perturbation theory this process is mediated by a closed loop of heavy quarks and Next-to-Leading Order (NLO) QCD corrections therefore require the computation of two-loop contributions. A Higgs Effective Field Theory (HEFT) was derived long ago [2], in which the heavy quarks are integrated out and the Higgs boson couples directly to the gluons. This allows the computations to be simplified considerably.
One interesting regime to consider is that of Higgs boson production with a transverse momentum p t,H of the order of the top-quark mass, m T , or larger. Here the top quark loop is resolved and it becomes possible to disentangle the SM contribution from effects of New Physics. However, in this regime finite top-quark mass effects are not negligible and the effective theory approximation becomes increasingly poor. In other words, events in which the Higgs boson is recoiling against one or more jets acquiring a large transverse momentum do not fall into the validity range of the effective field theory description. It is thus important to go beyond the HEFT approximation and include finite top-quark mass effects to obtain reliable predictions in this kinematical range.
Within the HEFT approximation corrections to inclusive Higgs boson production are known to N 3 LO QCD accuracy [3], whilst the fully differential corrections for H + 1 jet production are known to NNLO QCD accuracy [4][5][6][7]. Finite quark mass corrections to H + 1 jet production have been known at LO for a long time [8,9] and LO results are also known for the higher multiplicity processes H+2jets [10,11] and recently H+3jets [12,13]. The HEFT results for H + 1 jet production have also been supplemented by an expansion in 1/m 2 T at NLO QCD accuracy [14,15] and combined with the exact Born and real corrections [16]. They were also included in multipurpose Monte Carlo generators to produce merged samples matched to parton showers [17][18][19] and used to improve the Higgs NNLO QCD transverse momentum distributions in the HEFT above the top-quark mass threshold [20].
One of the first major steps towards the computation of the full two-loop NLO QCD virtual corrections was made in Ref. [21], where the planar master integrals were computed analytically in the Euclidean region and shown to contain elliptic integrals. At the same time an expansion valid in the limit of small bottom-quark mass allowed insight to be gained into the NLO QCD effects due to nearly massless quarks [22][23][24][25]. Very recently a NLO QCD result expanded in the regime where the Higgs boson and top-quark masses are small, relevant for the description of the Higgs boson transverse momentum distribution at large p t,H 400 GeV, was also studied [26,27].
On the experimental side, recently the CMS collaboration has considered events where the Higgs boson transverse momentum p t,H is larger than 450 GeV [28]; a feat made possible through the use of boosted techniques [29], which allow the Higgs to be identified through its decay to bottom quarks.
In this letter we present the first NLO QCD computation of Higgs boson production in association with one jet retaining the full top-quark mass dependence. In the following sections we present the computational setup used for this calculation and selected phenomenological results.

COMPUTATIONAL SETUP
We compute using conventional dimensional regularization (CDR) with d = 4 − 2 . The top-quark mass is renormalized in the on-shell scheme and the QCD coupling and gluon wave-function in the MS scheme with n f = 5 light quarks, with the top quark loops subtracted at zero momentum. The top-quark mass renormalization is performed by inserting the mass counterterm into the heavy quark propagators. Alternatively, the mass renormalization can be calculated by taking the derivative of the one-loop amplitude with respect to m T . We have used both methods as a cross-check.
The sampling of the phase-space generator has been adapted to generate a nearly uniform distribution of points in p t,H .

Born and Real radiation
The computation of everything but the virtual amplitudes is performed within the POWHEG-BOX-V2 framework [30], taking advantage of the existing HJ generator [31] for HEFT, in which the Born and real radiation amplitudes are computed using Madgraph4 [32,33] and the virtual amplitudes are taken from MCFM [34]. The subtraction of the infra-red divergences is performed using FKS [35].
We supplemented this code with the analytical Born amplitudes with full top-quark mass dependence from Ref. [9], whereas the one-loop real radiation contribution was generated with GoSam [36,37] using the BLHA [38,39] interface developed in Ref. [40]. For the purpose of this computation GoSam has been improved such that it now automatically switches to quadruple precision in regions where the amplitude becomes unstable due to one of the final state partons becoming soft or collinear to another parton. The amplitudes generated by GoSam are computed at run time with Ninja [41][42][43] using the quadninja feature. The scalar one-loop integrals are computed with the OneLoop [44] integral library. As a consistency check the virtual amplitudes in HEFT were cross checked with GoSam, whereas the real radiation amplitudes in the full theory were compared against OpenLoops [45].

The virtual amplitude
The Lorentz structure of the H → qqg and H → ggg partonic amplitudes can be decomposed, after imposing parity conservation, transversality of the gluon polarization vectors and the Ward identity, in terms of 2 and 4 tensor structures respectively, see for example Ref. [46]. This decomposition is not unique. For the H → ggg amplitude we follow Ref. [47] and choose to decompose it such that three of the form factors (which multiply the tensor structures) are related by cyclic permutations of the external gluon momenta whilst the fourth is itself invariant under such permutations. For the H → qqg am-plitude our decomposition is chosen such that the form factors are related by interchanging the external quark and anti-quark momenta. We separately compute all form factors and use these symmetries as a cross-check of our result.
In order to compute the amplitudes we closely follow the method of Refs. [48,49]. We construct projection operators for each of the form factors and contract them with the amplitude omitting external spinors and polarization vectors. This procedure allows us to write the amplitude in terms of scalar integrals.
The Feynman diagrams contributing to the two-loop virtual amplitude are generated using Qgraf [50] and further processed using Reduze [51], Ginac [52], Fermat [53], and Mathematica. We cross checked the amplitudes with expressions obtained from a two-loop extension of GoSam, which uses Qgraf and form [54,55]. The integrals appearing in the amplitude are reduced to master integrals using a customized version of the program Reduze. To simplify the numerical evaluation we choose a quasi-finite basis of master integrals [56]. The resulting integrals are calculated numerically using SecDec [57,58]. For the numerical integration we use a quasi-Monte Carlo method based on a rank-one lattice rule [59][60][61]. Neglecting crossings we evaluate a total of 102 planar and 18 non-planar two-loop integrals.
The Higgs boson mass is set to m H = 125 GeV and the top-quark mass is chosen such that m 2 H /m 2 T = 12/23, which means that m T = 173.055 GeV. Fixing the ratio of the Higgs boson to top-quark mass allows us to reduce by one the number of independent scales appearing in the two-loop virtual amplitudes. This simplifies the integral reduction and the form of the resulting reduced amplitude.
We subtract the infra-red and collinear poles of the virtual amplitude to obtain the finite part of the virtual amplitude V fin as required in the POWHEG-BOX-V2 framework [62]. The IR subtraction procedure requires the one-loop amplitudes up to order 2 , which we compute numerically using the same procedure as for the two-loop amplitudes.

PHENOMENOLOGY
In this section we present results for H + 1 jet production at the LHC at a center-of-mass energy of 13 TeV. Jets were clustered using the anti-kt jet algorithm implemented in FastJet [63-65] with a radial distance of R = 0.4 and requiring a minimum transverse momentum of p t,j > 30 GeV. We used the PDF4LHC15 nlo 30 pdfas [66][67][68][69] set interfaced through LHAPDF [70] for both LO and NLO predictions, and fixed the default value of factorization and renormaliza-tion scales µ F and µ R to H T /2, defined as where the sum runs over all final state partons i. This scale is known to give a good convergence of the perturbative expansion and stable differential K-factors (ratio of NLO to LO predictions) in the effective theory [71].
To estimate the theoretical uncertainty we vary independently µ F and µ R by factors of 0.5 and 2, and exclude the opposite variations. The total uncertainty is taken to be the envelope of this 7-point variation.
To better highlight the differences arising from the twoloop massive contributions, we compare the new results with full top-quark mass dependence, which we label as "full theory result" or simply "full" in the following, to two different approximations. In addition to predictions in the effective theory, which are referred to as HEFT in the following, we show results in which everything but the virtual amplitudes is computed with full top-quark mass dependence. In this latter case only the virtual contribution is computed in the effective field theory and reweighted by the full theory Born amplitude for each phase space point. Following Ref. [72] we call this prediction "approximated full theory" and label it as FT approx from now on.
We start by presenting the total cross sections, which are reported in Table I. For comparison we present results also for the HEFT and FT approx approximations. Together with the prediction obtained with the central scale defined according to Eq. (1) we show the upper and lower values obtained by varying the scales. While at LO the top-quark mass effects lead to an increase of 4.3%, at NLO this is enhanced to 4.9% when compared to the HEFT approximation. Comparing the full theory result with the FT approx result we obtain an increase of 1%. It is important to keep in mind that when taking into account massive bottom-quark loop contributions, the interference effects are sizable and cancel to a large extent the increase in the total cross section observed here between the HEFT and the full theory results (see e.g. the results in Ref. [13]). At LO the bottom-quark mass effects are of the order of 2% or smaller above the top quark threshold. Considering more differential observables, it is well known that very significant effects due to resolving the top-quark loop are displayed by the Higgs boson transverse momentum distribution, which is softened for larger values of p t,H by the full top-quark mass dependence. By considering the high energy limit of a point-like gluongluon Higgs interaction and one mediated via a quark loop it is possible to derive the scaling of the squared transverse momentum distribution dσ/dp 2 t,H [73,74], which drops as (p 2 t,H ) −1 in the effective theory, and goes instead as (p 2 t,H ) −2 in the full theory. This fact was shown to hold numerically at LO for up to three jets in Ref. [13]. It is interesting to verify this also after NLO QCD corrections are applied. To do so, in Figure 1 we show the transverse momentum spectrum of the Higgs boson at LO and NLO in the HEFT approximation and with the full top-quark mass dependence.
In the upper panel we display each differential distribution with the theory uncertainty band originating from scale variation. To highlight the different scaling in p t,H , in the middle panel we normalize all the distributions to the LO curve in the effective theory. It is thus possible to see that for low transverse momenta the full theory predictions overshoot slightly the effective theory ones. For p t,H > 200 GeV the two predictions start deviating more substantially. At LO the two uncertainty bands do not overlap any more above 400 GeV, whereas at NLO this happens already around 340 GeV due to reduction of the uncertainty at this order. The logarithmic scale also allows to see that the relative scaling behavior within the two theory descriptions is preserved between LO and NLO. The curves in the lowest panel of Figure 1 show the differential K-factor in HEFT and in the full theory. In both cases above 150 GeV they become very stable and amount to about 1.85 and 1.95 respectively. Thus the NLO corrections are large also in the full theory. This broadly agrees with the conclusions of Ref. [27], where the expanded result showed a similar enhancement of the K-factor by about 6% in the tail compared to the HEFT.
To conclude this section we compare the new predictions for the Higgs boson transverse momentum with the one in FT approx . At LO the two predictions are identical by construction, it is however interesting to check how good FT approx can reproduce the full theory results. In the main panel of Figure 2 we plot the three curves. To highlight better the differences among the two predictions, in the middle panel we normalize the distributions to the LO prediction. This allows us to compare the two differential K-factors, which behave very similarly over the full kinematical range. In order to quantify the difference between the two predictions, in the lower panel we display the ratio of the Full NLO curve to the FT approx NLO curve showing an increase at the 1-2% level of the Full NLO compared to the FT approx prediction.

CONCLUSIONS AND OUTLOOK
In this letter we have presented for the first time NLO QCD corrections to Higgs boson plus jet production retaining the full top-quark mass dependence. We observe that the size of the NLO corrections is large but, for our choice of the renormalization and factorization scale, the K-factor is approximately constant above the top-quark threshold. Compared to FT approx predictions, the full two-loop contribution enhances the NLO predictions by about 1% at the level of the total cross section, with nearly no dependence on the transverse momentum of the Higgs boson. Despite a completely different p t scaling, the K-factors in the HEFT and in the full theory behave in a very similar way.
The result removes the theoretical uncertainty on differential H + 1 jet distributions due to the unknown top mass corrections at NLO in QCD. Besides the transverse momentum distribution, shown here, this calculation enables accurate predictions to be made also for other observables where the top-quark mass effects may play a significant role.
As the experimental precision at the LHC improves in the coming years, this result aids the study of the Higgs boson properties also in boosted regimes. Providing a more accurate theoretical description of the Higgs boson production at large transverse momentum will be helpful not only for unraveling the details of the electroweak symmetry breaking mechanism, but also in the search for indirect signs of new physics.