NLO Higgs+jet at large transverse momenta including top quark mass effects

We present a next-to-leading order calculation of H+jet in gluon fusion including the effect of a finite top quark mass $m_t$ at large transverse momenta. Using the recently published two-loop amplitudes in the high energy expansion and our previous setup that includes finite $m_t$ effects in a low energy expansion, we are able to obtain $m_t$-finite results for transverse momenta below 225 GeV and above 500 GeV with negligible remaining top quark mass uncertainty. The only remaining region that has to rely on the common leading order rescaling approach is the threshold region $\sqrt{\hat s}\simeq 2m_t$. We demonstrate that this rescaling provides an excellent approximation in the high $p_T$ region. Our calculation settles the issue of top quark mass effects at large transverse momenta. It is implemented in the parton level Monte Carlo code MCFM and is publicly available immediately in version 8.2.


Introduction
With the Higgs boson discovery at the Large Hadron Collider (LHC) [1,2] setting a milestone for physics research, the hunt for signals beyond those described by the Standard Model (SM) has been more active than ever. Early Higgs studies during Run I, limited by statistics and energy, probed rather inclusive properties, and no significant deviations from the SM have been found [1][2][3][4][5][6][7]. Differential Higgs measurements [8][9][10][11][12][13][14] testing the SM were limited by statistics rather than theory predictions. Recent experimental analyses consider the Higgs boson in a highly boosted regime with transverse momenta (p T ) of 450 GeV to 1 TeV [15][16][17]. Clearly there is an evolving need for most precise predictions in differential quantities and in the high energy regime that needs to be filled.
Unfortunately an effective field theory (EFT) description that integrates out the top quark as a heavy particle has to be used at higher perturbative orders to approximate the complicated massive loop integrals. The operator used in the infinite top quark mass EFT is the same operator that constitutes the leading EFT operator to search for new physics. Thus the only reliable way to directly disentangle SM gluon fusion from heavy BSM loop contributions requires computing the full top quark mass dependence at large energies. It is this region, in which finite top quark mass effects are unconstrained at NLO, that motivates our study. 1 When it comes to precision SM predictions to constrain BSM physics, not just the size of the perturbative corrections to Higgs production are of importance, which are well described by the EFT, but also the shape of transverse momentum distributions, for example. In this study we can also evaluate whether top quark mass effects distort the shape of the Higgs high transverse momentum distribution compared to previous approximations.
While gluon fusion induced Higgs production mediated through a massive top quark loop was calculated at LO a long time ago [33] 2 , the difficulty is considerably amplified at NLO and in Higgs production with a jet, where massive two-loop amplitudes have to be calculated. An efficient analytical evaluation of these integrals for Higgs+jet is currently not within reach [35,36]. Fortunately the EFT approach turns out to provide an excellent approximation of perturbative corrections even for differential Higgs+jet quantities [37,38]. This was shown using a low energy expansion below the top quark threshold p T 2m T , and as such can not constrain the effects of a finite top quark mass for large energies. Going beyond estimations of top quark mass effects, studies with direct predictions including them were performed [39][40][41][42], but were always limited by low energy approximations. And only the EFT approach, which reduces the needed number of loops to evaluate by one, allowed for an evaluation of Higgs+jet at NNLO [43][44][45][46][47]. Residual perturbative uncertainties estimated by renormalization and factorization scale variation are estimated to be about 10% at NNLO, cutting in half the estimated NLO uncertainty.
Studies targeting top quark mass effects specifically in the large transverse momentum region of Higgs production were performed using resummation [48], and by using a factorization of the mass scale from p T in the high p T limit [49]. In the latter case the developed formalism was only applied to the subprocess qq at leading order in 1/p T and leading order in α s . The former study only improves the NLO calculation through a rescaling k-factor and not directly, as their high energy approximation was shown to deviate from the exact result at LO. Their suggested approach is to rescale exact LO results by a k-factor obtained in their high energy approximation. A different approach is to match different hard-jet multiplicities and parton showers [26,41]. 1 While finalizing our manuscript we became aware of a study similar to ours in ref. [31]. Additionally, results based on a fully numerical evaluation of the two-loop integrals with full top quark mass dependence have been presented in ref. [32]. We still believe that our study and implementation are useful for large transverse momenta, where the difference to the full calculation should be negligible and the full calculation seems to be numerically challenging for large transverse momenta. 2 For a recent overview of Higgs production and decay cross sections we refer to ref. [34].
It is the goal of this study to extend our previous setup [42], publicly available in MCFM [50][51][52], and which provides finite m t effects in the region p t 2m t , to predictions with a finite m t for p T 2m t . We also study the validity of the common LO rescaling approach in the region of high p T . To do this we implement the recently published two-loop amplitudes in the high energy expansion [53].

Calculation
Our calculation is based on the existing NLO Higgs+jet setup in MCFM-8.1 [42]. It uses an asymptotic expansion in Λ/(2m t ) only for the finite part of the virtual two-loop amplitudes, but is exact in the top quark mass otherwise. Here Λ is a placeholder for all kinematical scales of the process. For Higgs p T smaller than 225 GeV the asymptotic expansion was shown to be convergent and provides an excellent approximation of the full top quark mass dependence. For energies larger than 300 GeV the expansion breaks down and finite top quark mass effects could become larger than 8%, such that either a full calculation is necessary or another approximation is needed for sufficiently large p T . Here, we fill this gap for the latter case.
We have implemented the one-and two-loop Higgs plus three parton helicity amplitudes in the high energy expansion from ref. [53].
Here m H is the Higgs mass andŝ the partonic center of mass energy. The amplitudes are given as the finite parts after UV renormalization and Catani IR subtraction [54] using d = 4 − 2 Born one-loop amplitudes. We have performed a conversion to the 't Hooft-Veltman scheme for use in MCFM and additionally restored the renormalization scale dependence.
At LO Higgs+jet relies on one-loop amplitudes and is known with the exact top quark mass dependence, which allows us to compare it with the result from the high energy expansion. This gives an estimate on how far we can trust the approximation when using the two-loop amplitudes. Having established trust in the validity of the two-loop amplitudes in the high p T region we can then compare the results with the Born-rescaling approximation. In this rescaling approximation the finite part of the two-loop virtual amplitude is point-wise rescaled by the Born amplitude in the full theory divided by the Born amplitude in the EFT.
For our study we choose a center of mass energy of √ s = 13 TeV and a common renormalization and factorization scale of µ R = µ F = m 2 H + p 2 T , where m H = 125.0 GeV and p T is the Higgs transverse momentum. Although the region of high p T motivates using the six-flavor scheme, no matching parton distribution functions (PDFs) are available. So for consistency we work in the five-flavor scheme with an on-shell top quark mass of m t = 173.2 GeV. We use CT14 PDFs [55] at NLO accuracy for the NLO cross section and at LO accuracy for our LO results. The value of α s is given at the according order by the PDF set. Finally, we use the anti-k T jet algorithm with p T,jet > 30 GeV, |η jet | < 2.4 and R = 0.5.

Results
The first question one has to ask is in how far one can trust the two-loop high energy amplitudes to describe the exact m t dependence. In lack of the m t -exact two-loop amplitudes for comparison, one has to resort to a different method to evaluate this trust. For example, one could observe a convergent behavior of the expansion, but this would require some higher expansion order than is available as we will see below. Instead we can study how well the expansion works at LO. This has some limitations though, as will be discussed below. To study the high energy expansion we consider fig. 1: Shown is the LO Higgs transverse momentum distribution in various approximations normalized to the distribution with exact m t -dependence. The approximations shown are the low energy expansions up to order 1/m k t for k = 0, 2, 4, where k = 0 describes the EFT, as well as the high energy expansion.
For the low energy expansion the convergence is poor and is practically non-existent beyond 100 GeV. At NLO though, using only the expansion in the two-loop amplitude, the region of convergence increases to about 250 GeV as shown in ref. [42]. This can also be seen in fig. 2. A simple explanation is that for two-loop diagrams the topology does not force all center of mass energy to go through the top quark loop, such that ∼ 2m t threshold effects are further washed out.
The amplitudes in the high energy expansion are given up to order κ 1 ≡ (m 2 t /ŝ) 1 . Naively using them for the LO cross section includes partial effects of order κ 2 . This is labeled in the plot as "high, partial m 4 t ". Only including m 2 t terms in the cross section is labeled with "high m 2 t ". By the same argument given above, one would expect the high energy approximation to work better for one-loop diagrams than for two-loop diagrams. Nevertheless the difference between using the full O(κ 1 ) amplitudes and the m t -exact result is less than two percent beyond 500 GeV, which gives motivation to trust that the two-loop high energy amplitudes describe the full top quark mass dependence at a similar level. Considering that the NLO scale uncertainty is about 20% [42] and still about 10% at NNLO [45,47], any remaining top quark mass uncertainty can then be considered negligible. Ideally a more precise estimate could be established by including full O(κ 2 ) and O(κ 3 ) terms for the one-and two-loop amplitudes.
Having shown that the large energy expansion describes the full LO result at percent level accuracy beyond 500 GeV, we expect a similar behavior for the two-loop amplitudes. At NLO the two-loop amplitudes additionally only enter as the virtual corrections and a bulk of the perturbative corrections at large p T comes from the real emission which we include with full m t dependence. The error from using the large energy expansion estimated at LO should thus be conservative. At NLO we show the Higgs p T distribution in fig. 2, where, to emphasize again, only the finite part of the two-loop virtual corrections is not exact in m t and is approximated in different ways. It is obtained using either a 1/m t expansion in the region of small p T , or in the high energy expansion up to order κ 1 for large p T . Additionally, using the rescaling approach as described in section 2 we obtain an approximation that can be used over the whole range of p T . The latter approach was used for example in ref. [41] and shown to agree with the low energy asymptotic expansion at the percent level for p T 225 GeV [42].
Since we are strongly interested in possible shape corrections due to using a finite top quark mass, we present the distribution normalized to the rescaling approximation result. This shows the additional corrections compared to the previous best approximation at high p T in percent. The scale variation uncertainty of about 20 % changes only little with respect to the EFT result and other approximations and can be found for example in our previous study [42].
The low energy expansion indeed extends its convergent behavior to about 250 GeV with corrections of less than a percent compared to the rescaling approach. The high energy expansion, which we believe approximates the full result by better than 2%, is consistent with the corrections at low p T and increases the cross section by about 1 − 2% compared to the rescaled result. It is remarkable that these top quark mass corrections are flat within 1% over the whole range of large p T . In this sense one is free to choose either the rescaling approach or the high energy approximation. Nevertheless the high energy  approximation is a systematic approach, whereas the rescaling approach was done ad hoc without prior validation. We thus recommend to use the high energy approximation for transverse momenta beyond ∼ 500 GeV.

Conclusions
We have presented a NLO calculation of Higgs+jet with negligible remaining top quark mass uncertainty in the region of low transverse momenta p T 225 GeV [42] and, as shown in this analysis, also in the region of large transverse momenta p T 500 GeV. We have demonstrated that the approach of rescaling the finite part of the two-loop virtual amplitude by the m t -exact Born amplitude approximates the m t -exact NLO result by better than a few percent. The elimination of the top-quark mass uncertainty at high p T at NLO now allows one to rescale NNLO results obtained in the EFT by a NLO k-factor NLO(m t )/NLO(EFT). Our implementation is publicly available immediately in