Higgs boson pair production at NNLO with top quark mass effects

Journal of High Energy Physics, May 2018

Abstract We consider QCD radiative corrections to Higgs boson pair production through gluon fusion in proton collisions. We combine the exact next-to-leading order (NLO) contribution, which features two-loop virtual amplitudes with the full dependence on the top quark mass M t , with the next-to-next-to-leading order (NNLO) corrections computed in the large-M t approximation. The latter are improved with different reweighting techniques in order to account for finite-M t effects beyond NLO. Our reference NNLO result is obtained by combining one-loop double-real corrections with full M t dependence with suitably reweighted real-virtual and double-virtual contributions evaluated in the large-M t approximation. We present predictions for inclusive cross sections in pp collisions at \( \sqrt{s} \) = 13, 14, 27 and 100 TeV and we discuss their uncertainties due to missing M t effects. Our approximated NNLO corrections increase the NLO result by an amount ranging from +12% at \( \sqrt{s}=13 \) TeV to +7% at \( \sqrt{s}=100 \) TeV, and the residual uncertainty of the inclusive cross section from missing M t effects is estimated to be at the few percent level. Our calculation is fully differential in the Higgs boson pair and the associated jet activity: we also present predictions for various differential distributions at \( \sqrt{s}=14 \) and 100 TeV, and discuss the size of the missing M t effects, which can be larger, especially in the tails of certain observables. Our results represent the most advanced perturbative prediction available to date for this process.

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Higgs boson pair production at NNLO with top quark mass effects

Received: March Higgs boson pair production at NNLO with top quark mass e ects M. Grazzini 0 1 3 6 7 G. Heinrich 0 1 3 4 7 S. Jones 0 1 3 4 7 S. Kallweit 0 1 3 5 7 M. Kerner 0 1 3 4 7 J.M. Lindert 0 1 2 3 7 J. Mazzitelli 0 1 3 6 7 0 CH-1211 Geneva 23 , Switzerland 1 Fohringer Ring 6 , 80805 Munchen , Germany 2 Institute for Particle Physics Phenomenology, Durham University 3 Winterthurerstrasse 190 , CH-8057 Zurich , Switzerland 4 Max Planck Institute for Physics 5 TH Division, Physics Department , CERN 6 Physik-Institut, Universitat Zurich 7 Durham DH1 3LE , U.K We consider QCD radiative corrections to Higgs boson pair production through gluon fusion in proton collisions. We combine the exact next-to-leading order (NLO) contribution, which features two-loop virtual amplitudes with the full dependence on the top quark mass Mt, with the next-to-next-to-leading order (NNLO) corrections computed in the large-Mt approximation. The latter are improved with di erent reweighting techniques in order to account for nite-Mt e ects beyond NLO. Our reference NNLO result is obtained by combining one-loop double-real corrections with full Mt dependence with suitably imation. We present predictions for inclusive cross sections in pp collisions at p reweighted real-virtual and double-virtual contributions evaluated in the large-Mt approx27 and 100 TeV and we discuss their uncertainties due to missing Mt e ects. Our approximated NNLO corrections increase the NLO result by an amount ranging from +12% at NLO Computations; QCD Phenomenology - s = 13 TeV to +7% at p s = 100 TeV, and the residual uncertainty of the inclusive cross section from missing Mt e ects is estimated to be at the few percent level. Our calculation is fully di erential in the Higgs boson pair and the associated jet activity: we also present predictions for various di erential distributions at p s = 14 and 100 TeV, and discuss the size of the missing Mt e ects, which can be larger, especially in the tails of certain observables. Our results represent the most advanced perturbative prediction available to date for this process. 1 Introduction 2 Details on the method and approximations boson production [5{10] or electroweak precision observables [11, 12]. For the gg ! hh production channel, the leading order (LO) calculation was performed some time ago in refs. [13{15]. The next-to-leading-order (NLO) corrections with full top quark mass (Mt) dependence, involving two-loop diagrams with several mass scales, became available only recently [16, 17], and have been supplemented by soft-gluon resummation at small transverse momenta of the Higgs boson pair [18] and parton shower e ects [19, 20]. In the Mt ! 1 limit, also called Higgs E ective Field Theory (HEFT) approxima tion, point-like e ective couplings of gluons to Higgs bosons arise. In this limit, the NLO corrections were rst calculated in ref. [21] and rescaled by a factor BFT=BHEFT, where BFT denotes the LO one-loop matrix element squared in the full theory. This procedure is often called \Born-improved HEFT" approximation. In refs. [4, 22] an approximation for Higgs boson pair production at NLO, labelled \FTapprox", was introduced, in which the real radiation matrix elements contain the full { 1 { top quark mass dependence, while the virtual part is calculated at NLO in the HEFT approximation and rescaled at the event level by the re-weighting factor BFT=BHEFT. At the inlusive cross section level this approximation suggests at the LHC a correction with respect to the \Born-improved HEFT" approximation of about 10%, close to the corresponding correction of 14% later obtained in the full NLO calculation [16, 17]. The next-to-next-to-leading-order (NNLO) QCD corrections in the HEFT approximation have been computed in refs. [23{26], where ref. [26] provides fully di erential results. The NNLO HEFT results for the total cross section have been supplemented by an expansion in 1=Mt2 in ref. [27]. Approximations for the top-quark mass dependence of the two-loop amplitudes in the NLO calculation have been studied in ref. [28] via a Pade ansatz. Soft gluon resummation has been performed at NLO+NNLL in ref. [29] and at NNLO+NNLL in ref. [30]. The NNLO+NNLL HEFT results lead to K-factors of about 1.2 relative to the Born-improved NLO HEFT result. In ref. [31], the recommended value for the total gg ! hh cross section was based on the NNLO+NNLL HEFT results [30], corrected by a factor t accounting for top quark mass e ects, extracted from ref. [16]. However, this procedure is somewhat ad hoc, and not viable to study kinematical distributions. In order to account for the NNLO K-factor in the HEFT calculation as well as for the correct description of the tt threshold and the high-energy tails of the distributions, where the top quark loops are resolved, a rst attempt to combine the two calculations has been made in ref. [17], where the full NLO result for a particular distribution was reweighted by the NNLO K-factor obtained from ref. [26] on a bin-by-bin basis. However, this procedure, called \NLO-improved NNLO" has its drawbacks, as it needs to be repeated for each observable (and binning) under consideration. The aim of this paper is to study alternative methods to combine the two results, i.e. to incorporate top quark mass e ects in the calculation of the production of Higgs boson pairs at NNLO. One of the studied approximations comprises exact top-quark mass dependence up to NLO and also exact top quark mass dependence in the double-real emission contributions to the NNLO cross section at di erential level. The results of this approximation can be regarded as the most advanced prediction currently available for Higgs boson pair production in gluon fusion. This work is organized as follows: in section 2 we describe the technical details of our calculation, and present the di erent approximations we will consider to incorporate mass e ects in the NNLO contribution. In section 3 we present our numerical predictions, both for the total cross section and di erential distributions. Finally, in section 4 we summarise our results. 2 Details on the method and approximations We start by presenting the di erent technical ingredients entering our computation, as well as the de nition of the various approximate ways to include mass e ects in the NNLO calculation introduced and used in this work. Finally, we also discuss the numerical stability of our predictions. { 2 { HJEP05(218)9 Our calculation is based on the publicly available computational framework Matrix [32], which allows the user to perform fully di erential NNLO calculations for a wide class of processes at hadron colliders. For the purpose of the present work, the public version of the code has been extended, based on the calculation of ref. [26], to include the production of a pair of Higgs bosons via gluon fusion. For the calculation of the NNLO corrections the code implements the qT -subtraction formalism [33], in which the genuine NNLO singularities, located where the transverse momentum of the Higgs boson pair, pT;hh qT , vanishes, are explicitly separated from the NLO-like singularities in the hh + jet contribution. The qT where in particular the contribution d NhhL+Ojet can be evaluated using any available NLO subtraction procedure to handle and cancel the corresponding infrared (IR) divergencies.1 The remaining qT ! 0 divergence is canceled by the process-independent counterterm d NCNTLO. The process-dependence of the hard-collinear coe cient HNNLO enters only via hh the NNLO (HEFT) two-loop virtual corrections [23] through an appropriate subtraction procedure [37]. The di erence in the square bracket of eq. (2.1) is nite when qT ! 0, but each of the terms exhibits a logarithmic divergence. Therefore, a technical cut, rcut, needs to be introduced on qT =Q, where the scale Q is chosen to be the invariant mass of the nal-state system. More details about the rcut ! 0 extrapolation are provided in section 2.3. At variance with the calculation of ref. [26], which was strictly done within the HEFT, this time all the routines needed to compute the full NLO cross section as well as the di erent NNLO reweightings have been implemented. This includes linking the code to the NLO two-loop virtual corrections obtained via a grid interpolation [19] and to several loop-induced amplitudes provided by the OpenLoops amplitude generator [38]. Within this framework we reproduced the di erential NLO results of refs. [16, 17] at the per mille level. The grid for the NLO virtual two-loop amplitudes is based on the calculation presented in refs. [16, 17], which in turn for the calculation of the two-loop amplitudes relies on an extension of the program GoSam [39, 40] to two loops [41], using also Reduze2 [42], SecDec3 [43] and the Quasi-Monte Carlo technique as described in ref. [44] for the numerical integration. These amplitudes (for xed values of the Higgs boson and top quark masses) are provided in a two-dimensional grid together with an interpolation framework, which allows us to evaluate them at any phase space point without having to perform the computationally costly two-loop integration. For more details, see refs. [19, 45]. All tree and one-loop amplitudes in the HEFT and also all loop-squared amplitudes in the full theory as discussed below are obtained via a process independent interface to OpenLoops [38, 46, 47]. For the latter this comprises loop-squared amplitudes for 1Matrix uses the automated implementation of the Catani-Seymour dipole subtraction method [34, 35] within the Monte Carlo program Munich [36]. { 3 { pp ! hh + 1; 2 jets, that need to be evaluated in IR divergent unresolved limits. In particular the limit qT ! 0 represents a signi cant challenge for the numerical stability of the hh + 2 jets amplitudes in the full theory. Thanks to the employed algorithms the numerical stability is under control, as discussed in detail in section 2.3. A major element of this stability originates from the employed tensor integral reduction library COLLIER [48]. 2.2 Approximations for top-mass e ects at NNLO In the following we present three approximations for the NNLO Higgs boson pair production cross section, which take into account nite top quark mass e ects in di erent ways. In all cases, we always include the full NLO result when computing the NNLO prediction, and only apply the di erent approximations to the O( S4) contribution. NNLONLO-i. The NLO-improved NNLO approximation (NNLONLO-i) has already been presented in ref. [17]. It can be constructed based on an observable-level multiplicative approach. In this approximation, for each bin of each histogram we multiply the full NLO result by the ratio between the HEFT NNLO and NLO predictions for this bin. NNLOB-proj. A di erent approximation can be obtained by reweighting each NNLO event by the ratio of the full and HEFT Born squared amplitudes. We denote this procedure as Born-projected approximation (NNLOB-proj). Of course, in order to do so and due to the di erent multiplicities involved, an appropriate projection to Born-like kinematics is needed; for this purpose we make use of the qT -recoil procedure de ned in ref. [49]. Following this prescription, the momenta of the Higgs bosons remain unchanged, and the new initial-state parton momenta are obtained by absorbing the recoil due to the additional radiation. Speci cally, denoting the momenta of the incoming partons by p1 and p2, and the momentum of the Higgs boson pair system by q, the new momentum to be used for the LO projection k1 (then, k2 = q k1) is given by Q2 2 k1T q p1 k1 = z1 2 q p1 p1 + k1T + z1 Q2 p1 p2 p2 ; k1T k1T = 2 k1T ; (2.2) where z1 = Q2 + 2 qT k1T + q (Q2 + 2 qT k1T )2 and k1T is a two-dimensional vector in the qT plane which needs to ful ll the condition k1T ! 0 when qT ! 0, and we set k1T = qT =2 (and therefore k2T = qT =2). This condition guarantees that the subsequently applied reweighting does not spoil the NNLO qT -cancellation. More details about this procedure can be found in ref. [49]. NNLOFTapprox . The third approximation we consider is constructed to pro t from the fact that the double-real emission contributions to the NNLO cross section require only oneloop amplitudes in the full theory (FT) and can thus be computed by using OpenLoops. Of course, the inclusion of these loop-induced amplitudes needs to be done in such a way { 4 { HJEP05(218)9 that the dipole cancellations in the NLO hh + j calculation and the low-qT cancellation for hh at NNLO are not spoiled. We will de ne our approximation by using the following procedure: working in the HEFT, for each n-loop squared amplitude that needs to be computed for a given partonic (n) subprocess AHEFT(ij ! HH + X), we apply the reweighting R(ij ! HH + X) = AFBuolrln(ij ! HH + X) (0) AHEFT(ij ! HH + X) ; where AFBuolrln stands for the lowest order (loop-induced) squared amplitude for the corresponding partonic subprocess, computed in the full theory.2 We note that, contrary to what happens in the Born-projected approach, here the reweighting is de ned using amplitudes that correspond to the same subprocess under consideration. Therefore, the kinematics is always preserved and there is no need to de ne a Born projection. Moreover, for amplitudes that are of tree-level type in the HEFT (as it is the case for the double-real emission contributions), this reweighting simply implies using the exact loop-induced amplitudes with full top mass dependence. The reweighting procedure de ned by eq. (2.4) agrees at NLO with the so-called FTapprox introduced in ref. [22], therefore we will use the same notation. Given that the performance of the Born-projection and FT approximations was already studied in ref. [17] at NLO, we directly present NNLO predictions in section 3. We point out that, based on the ingredients entering each of the approximations, the NNLOFTapprox is expected to be the most advanced prediction for Higgs boson pair production via gluon fusion. By contrast, the NNLOB-proj is expected to be the less accurate, since it is based on a simple Born level reweighting procedure. Nevertheless, and for comparison purposes, we always present results for the three approximations described above. (2.4) Numerical stability Before presenting our quantitative predictions, we brie y discuss the numerical stability of our results. From the computational point of view, the most challenging of the three approaches to incorporate mass e ects at NNLO is the NNLOFTapprox procedure, as it involves loop-induced double-real contributions in the full theory. In particular the dominant gg ! hhgg amplitude comprises computationally very challenging six-point loop integrals with internal masses. In fact, these contributions have to be evaluated in the numerically intricate NNLO unresolved limits and to the best of our knowledge, the present calculation is the rst application of a six-point one-loop amplitude integrated over its IR divergent unresolved limits in an NNLO calculation. Thanks to the numerical stability of the applied algorithms in OpenLoops together with Collier, the bulk of the phase-space points remains stable in double precision when approaching qT ! 0, even close to the dipole singularity, i.e. in the NNLO doubleunresolved limits. On average the runtime per phase space point for the gg ! hhgg 2Strictly speaking, the reweighting is applied to the nite part of the loop amplitudes. However, at one-loop level this procedure reproduces the loop structure of the full theory. { 5 { 1.0 0.5 % [ 1 -LO 0.0 N N σ/ σ -0.5 cut, rcut, normalized with respect to the extrapolated rcut ! 0 result. The dotted lines indicate the symmetrized uncertainty coming from the extrapolation. subtraction cut, rcut, for p amplitude is 1 sec. In principle OpenLoops provides a rescue system, such that remaining numerically unstable phase-space points can be reevaluated in higher numerical precision based on reduction with CutTools [50]. However, the runtime of the loop-induced gg ! hhgg amplitude in OpenLoops is signi cantly increased when CutTools is used in quadruple precision (to the level of 10 minutes per phase-space point), rendering the quadruple precision stability system prohibitive for this amplitude for practical purposes.3 Therefore, we restrict the evaluation to double precision and replace potentially unstable phase-space points close to the dipole singularities, quanti ed by L-i = (pi pj =s^)min, where the minimum among all potential emitter parton combinations i and j is taken, with an approximation: below a technical cut L-i, cut we switch from the (loop-induced) double-real amplitude in the FT to the (tree-level) double-real amplitudes in the HEFT, reweighted at LO. This approach could in principle introduce a bias in the NLO hh+jet cross section, thereby hampering the low-qT cancellation of the NNLO computation. We have checked that this is not the case, as detailed in the following. For the predictions presented in section 3 we use L-i, cut = 10 4 and we varied this parameter in the range 10 3 to 10 5 , nding results that only di er at the per mille level or below and which are always compatible within the numerical uncertainties. In gure 1 we illustrate the resulting dependence of the NNLOFTapprox total cross section on the qT s = 14 TeV. Due to the previously discussed stability challenges, we considered values of rcut between 1% and 3:5%, which are larger than the ones typically used in previous qT -subtraction calculations (compared for instance with the default values in the public Matrix release [32]). Nevertheless our results present a good stability, with e ects that are below 0:2% in the whole qT =Q range under study, validating this choice. The rcut ! 0 extrapolation is performed using a linear least varying the upper bound of the interval (in this case starting from a minimum of 25 points, which corresponds to an upper bound of rcut = 1:6%, and up to rcut = 3:5%). Then, the result with the lowest 2=degrees-of-freedom value is taken as the best t, and the 2 t. The t is repeated 3Here we want to note that these stability issues will be strongly mitigated in the future based on the new OpenLoops on-the- y reduction method introduced in ref. [47]. { 6 { rest is used to estimate the extrapolation uncertainty.4 In the case shown in gure 1 the extrapolation uncertainty for rcut ! 0, indicated with the dotted lines, is 0:14%. A further uncertainty arises due to the numerical evaluation of the two-loop integrals with full top-quark mass dependence in the virtual corrections of the NLO contribution. The error of the numerical integration of the amplitudes is propagated to the total cross section using Monte Carlo methods, varying the amplitude level results according to the corresponding error estimates. This leads to changes of the NLO cross section at the per mille level. Furthermore, we have checked that, within this uncertainty, results based on the grid for the virtual amplitude are consistent with the ones directly obtained from the amplitude results calculated in refs. [16, 17]. We want to point out that the uncertainties can be somewhat larger in di erential results, in particular in the tails of pT and invariantmass distributions. This discussion shows that the uncertainties due to the qT -subtraction method and the numerical evaluation of the NLO virtual contribution and grid interpolation are clearly under su cient control. 3 Results In this section we present our numerical predictions for inclusive and di erential cross sections for Higgs boson pair production in pp collisions. We consider centre-of-mass energies of 13, 14, 27 and 100 TeV. For the sake of brevity, di erential distributions are presented only for 14 TeV and 100 TeV. We use the values Mh = 125 GeV for the Higgs boson mass and Mt = 173 GeV for the pole mass of the top quark.5 We do not consider bottom quark loops, whose contribution at LO is below 1%. We also neglect top quark width e ects, which at LO are at the level of 2% for the total cross section [22]. We use the PDF4LHC15 sets [53{58] of parton distribution functions (PDFs), with parton densities and uated at each corresponding perturbative order (i.e., we use the (k + 1)-loop running at NkLO, with k = 1; 2). As renormalization and factorization scales, we use the central value 0 = Mhh=2, and we obtain scale uncertainties via the usual 7-point scale variation. S eval S 3.1 Inclusive cross sections In table 1 we present results for the total cross sections at NLO and NNLO in the various approximations. At NLO we report the exact result, including the full Mt dependence, and also the FTapprox result. By comparing the two NLO predictions, we see that the FT approximation overestimates the exact NLO result by 4% (6%) at 14 (100) TeV. At NNLO the largest prediction is obtained in the NNLOB-proj approximation, resulting in an 4We note that, in the current Matrix release, the rcut ! 0 extrapolation and the ensuing uncertainty estimation is only performed for inclusive ( ducial) cross sections. However, no signi cant e ects have been observed for kinematic distributions in various dedicated studies (see for instance ref. [51]). 5We note that it is not yet possible to fully assess the e ect of the top mass scheme in the calculation as the top quark mass is xed in the existing NLO two-loop virtual amplitudes. In the case of single Higgs boson production, using an MS mass instead of the pole mass for the top quark changes the NLO cross section by less than one per mille (see e.g. [52]). In the case of Higgs boson pair production, where the sensitivity to the top quark mass is much stronger, we expect the e ect to be larger but not to exceed the other uncertainties a ecting our calculation. { 7 { NLO [fb] NLOFTapprox [fb] NNLONLO i [fb] NNLOB proj [fb] NNLOFTapprox [fb] Mt unc. NNLOFTapprox NNLOFTapprox/NLO 13 TeV 1147 +190:9:7%% 1220 +1110::96%% 1337 +45::14%% 1406 +02::58%% 1224 +03::92%% 4:6% 1:067 predictions is also presented. The uncertainties due to the qT -subtraction and the numerical evaluation of the virtual NLO contribution are both at the per mille level. increase with respect to the exact NLO result of about 20% at 14 TeV. For this collider energy, the increase within the NNLONLO-i approach (which is computed based on the Mhh distribution) is smaller, being about 18%. Finally, the NNLOFTapprox prediction is the lowest one, with a 12% increase with respect to the NLO cross section at 14 TeV. For all the considered approximations and collider energies the scale uncertainties are signi cantly reduced when including the O( S4 ) NNLO corrections. This reduction is largest for the NNLOB proj and NNLOFTapprox approximations.6 For instance at 14 TeV, the total scale uncertainty is reduced from about 13% at NLO to +2% 5% at NNLOFTapprox, i.e. by about a factor of three. This reduction of the scale uncertainties is stronger as we increase the collider energy, being close to a factor of ve at 100 TeV. As is well known, scale uncertainties can only provide a lower limit on the true perturbative uncertainties. In particular, from table 1 we see that the di erence between the NNLO and NLO central predictions is always larger than the NNLO scale uncertainties (although within the NLO uncertainty bands). In any case, the strong reduction of scale uncertainties, together with the moderate impact of NNLO corrections, suggests a signi cant improvement in the perturbative convergence as we move from NLO to NNLO. It is also worth mentioning that the three approximations have a di erent behaviour with ps. For instance at 100 TeV, the increase with respect to the NLO prediction for the NNLOB-proj and NNLONLO-i approaches is 23% and 17%, respectively, values that are close to the ones for 14 TeV (20% and 18%, respectively). By contrast, the NNLOFTapprox result increases the NLO prediction by 7% at 100 TeV, i.e. the correction is smaller by almost a factor of two than at 14 TeV (12%), which also means a larger separation with respect to the other two NNLO approximations. The smaller size of the NNLO corrections in the FTapprox at higher energies is also consistent with the observed reduction of scale uncertainties. 6The scale uncertainty of the NNLONLO-i prediction is de ned as the relative uncertainty of the HEFT result. { 8 { As was mentioned already in section 2.2, the NNLOFTapprox result is expected to be the most accurate one among the approximations studied in this work, and therefore it is considered to be our best prediction. In order to estimate the remaining uncertainty associated with nite top quark mass e ects at NNLO, we start by considering the accuracy of the FTapprox approximation at NLO. At 14 TeV the NLO FTapprox result (see table 1) overestimates the full NLO total cross section by only about 4%, or equivalently by about 11% of the pure O( S3) contribution. If we assume that FTapprox performs analogously at one order higher, we obtain a 11% uncertainty on the O( S4 ) contribution.7 Given that the relative weight of the O( S4) contributions to the total NNLO cross section is de nitely smaller than the weight of the O( S3) contributions to the NLO cross section, we obtain a signi cantly smaller overall uncertainty, in this case of 1:2%. In order to be conservative, we can increase this estimate by a factor of two. The relative di erence between the FTapprox and the full NLO result slightly increases with the collider energy. However, at the same time the relative size of the O( S4) correction decreases. The NNLO uncertainty obtained with this procedure ranges from 2:3% at 13 TeV to 3:1% at 100 TeV. We can repeat the above procedure to estimate the uncertainty of the NNLOB proj approximation, which displays the largest di erences with respect to the NNLOFTapprox result. Similarly to what we do for FTapprox, we can assign an uncertainty to the NNLOB proj result by relying on the accuracy of the same approximation at NLO, and conservatively to 36% at p multiplying by a factor of two. The ensuing uncertainties range from s = 100 TeV. We nd that the NNLOFTapprox prediction (always evaluated at R = F = 0) is fully contained in the NNLOB proj uncertainty band. Actually, there is a large overlap between the two approximations, which includes in all the cases the central value of the NNLOFTapprox, even when the conservative factor of two is not included. This can be regarded as a non-trivial consistency check for our procedure. We may be tempted to conclude our discussion by adopting the above procedure for the uncertainty estimate 14% at p s = 13 TeV of our NNLOFTapprox result. However, we have already pointed out that, as ps increases, the di erence between the 5:2% at p s = 13 TeV, and it becomes 9:2% at p NNLOFTapprox and the other approximations increases. In particular, the di erence between the NNLOFTapprox result and our \next-to-best" NNLO prediction, NNLONLO i, is s = 100 TeV. The signi cant increase of this di erence with the collider energy suggests us a more conservative approach. Our nal estimate for the nite top quark mass uncertainty of our NNLOFTapprox result is de ned as half the di erence between the NNLOFTapprox and the NNLONLO i approximations, and is reported in table 1 for the di erent values of ps. At p s = 13 and 14 TeV these uncertainties are 2:6% and 2:7%, and thus very similar to the ones obtained with the method discussed above. At p s = 100 TeV, however, the uncertainty increases to 4:6%, which appears to be more conservative than the 3:1% obtained with the previous procedure. 7We point out that in order to obtain the pure O( S4) corrections, we have subtracted the lower order contributions computed with NNLO parton distributions and strong coupling. The corresponding numbers are a few percent lower than the ones given in table 1 for the NLO results. { 9 { 6 5 )4 V e G / d / σd2 1 0 1.4 1.3 LO1.2 d / together with the NLO prediction, at 14 TeV (left) and 100 TeV (right). The lower panels show the ratio with respect to the NLO prediction, and the lled areas indicate the NLO and NNLOFTapprox scale uncertainties. 3.2 Di erential distributions In this section we present predictions for di erential Higgs boson pair production at 14 TeV and 100 TeV. We consider the following kinematical distributions: the invariant mass (Mhh, gure 2) and rapidity (yhh, gure 3) of the Higgs boson pair, the transverse momenta of the Higgs boson pair and the leading jet (pT;hh and pT;j1, gures 4 and 5), the transverse momenta of the harder and the softer Higgs boson (pT;h1 and pT;h2, gures 6 and 7), and the azimuthal separation between the two Higgs bosons ( hh, gure 8). For the sake of clarity, we only show the scale uncertainty bands corresponding to the NLO and NNLOFTapprox predictions. We start our discussion from the invariant-mass distribution of the Higgs boson pair, reported in gure 2. We observe that the NNLOB-proj and NNLONLO-i approximations predict a similar shape, with very small corrections at threshold, an approximately constant K-factor for larger invariant masses, and only a small di erence in the normalization between them, which increases in the 100 TeV case. The NNLOFTapprox, on the other hand, presents a di erent shape, in particular with larger corrections for lower invariant masses, a minimum in the size of the corrections close to the region where the maximum of the distribution is located, and a slow increase towards the tail. The di erent behavior of the NNLOFTapprox in the region close to threshold is more evident at 100 TeV, where the increase is about 30% in the rst bin. Naively we could expect that if this region is dominated by soft parton(s) recoiling against the Higgs bosons, the Born projection and FTapprox should provide similar results. We have investigated the origin of this di erence, and we nd that in the region Mhh 2Mh the cross section is actually dominated by events with relatively hard radiation recoiling against the Higgs boson pair (for example, NNLOB-proj NNLONLO-i NNLOFTapprox NLO 0 yhh 10 5 b (f hh gether with the NLO prediction, at 14 TeV (left) and 100 TeV (right). 1.4 1.3 LO1.2 s = 100 TeV, the average transverse momentum of the Higgs boson pair in the rst 100 GeV at NLO). In this region the exact loop amplitudes behave rather di erently as compared to the amplitudes evaluated in the HEFT: as the production threshold is approached, they go to zero faster than in the mass-dependent case, thus explaining the di erences we nd. Within the NNLOFTapprox, the corrections to the Mhh spectrum range between 10% and 20% at 14 TeV. The scale uncertainty is substantially reduced in the NNLOFTapprox, and this reduction is particularly strong for large invariant masses. As observed at the inclusive level, the NNLOFTapprox corrections are smaller at 100 TeV (except only for the rst bin) and the di erence with respect to the other approximations is larger. Next we move to the rapidity distribution of the Higgs boson pair, reported in gure 3. The NNLO results are similar for all three approximations. This is not unexpected as the shape of the rapidity distribution is mainly driven by the PDFs. Besides the obvious difference in the normalization, the largest e ect in the shape of the NNLONLO i distribution is observed in the central region, which is particularly evident in the 100 TeV case. Again we observe a clear reduction of scale uncertainties over the whole range under study. More signi cant di erences between the three approximations are obtained in the pT;hh distribution, reported in gure 4. The NNLOB-proj approximation predicts huge corrections for large transverse momentum, the result being almost an order of magnitude larger than the NLO prediction and the other approximations for pT;hh 500 GeV. This behavior is hardly surprising since already at NLO the Born-projected result deviates from the exact NLO prediction in this way [17]. In fact, given that the pT;hh distribution is not de ned at LO, the NNLOB-proj corrections cannot inherit any information about the (full) lowestorder prediction for this distribution. This is of course not the case for the other two approximations, which in fact make an almost identical prediction at large pT;hh, with ) NNLOB-proj NNLONLO-i NNLOFTapprox NLO d / 3.0 2.5 LO2.0 toN1.5 d / 3.0 2.5 LO2.0 toN1.5 d / σd0.10 d / σd0.10 0.05 0.01 3.0 2.5 LO2.0 toN1.5 Here jets are clustered with the anti-kT algorithm [59] with R = 0:4 and pT;j1 > 30 GeV and j jj 4:4. large corrections that can be well above 50%, and sizable uncertainties at the level of 30%{ 40%, re ecting the NLO-nature of this observable. At lower transverse momenta, however, the NNLONLO i and NNLOFTapprox deviate from each other, and the latter approaches the NNLOB proj prediction. Once again, the di erent behavior of these approximations is more pronounced in the 100 TeV distribution, for which the central NNLONLO i curve lies outside the NNLOFTapprox uncertainty band below pT;hh 200 GeV. Of course, in order to obtain reliable results in the low-pT;hh region, the corresponding logarithmically enhanced contributions need to be properly resummed to all orders in the strong coupling constant. NNLOB-proj NNLONLO-i NNLOFTapprox NLO 6 5 ) V e /G4 b (f ph1T , 3 d / NNLOB-proj NNLONLO-i NNLOFTapprox NLO HJEP05(218)9 d / 0.05 0.00 The transverse momentum distribution of the leading jet pT;j1, reported in gure 5, has similar features as the pT;hh distribution. Again we observe the unphysical excess predicted by the NNLOB proj approximation, which can be understood using the same arguments as presented for the pT;hh distribution, and the agreement between NNLOB proj and NNLOFTapprox at low pT;j1. The di erence between the NNLONLO i and NNLOFTapprox results is more pronounced here, with the FTapprox predicting a softer spectrum for this observable, and small corrections that are almost always contained in the NLO scale uncertainty band. The transverse-momentum distributions of the harder and the softer Higgs boson are reported in gures 6 and 7, respectively. As can be expected from the pT;hh spectrum, the NNLOB-proj result for pT;h1 features very large corrections as pT;h1 increases. The e ect, however, is less severe than the one observed in pT;hh because the pT;h1 observable is already well de ned at LO. The NNLONLO-i curve is overall in good agreement with the NNLOFTapprox prediction: it shows moderate corrections with respect to the NLO result which increase as pT;h1 increases, while the scale uncertainties are about 15%. At very small pT;h1 the higher-order corrections become perturbatively unstable as the available phase space for the real radiation is severely restricted in this regime yielding large logarithms that should be resummed in order to get a reliable prediction, see also the discussion in section 3.4 of ref. [19]. For the transverse momentum of the softer Higgs boson, pT;h2, the NNLO e ect is rather uniform in all three approximations, especially at 14 TeV. The NNLOFTapprox predicts small corrections of order 10%, while the other two approximations show larger corrections with a similar shape. In the tail of the distribution the scale uncertainty at NNLO is larger than at NLO, most likely due to an accidentally small size of the NLO scale variation (in fact, in this region the NLO corrections almost vanish). Finally, the distribution in the azimuthal angle between the two Higgs bosons, hh, is shown in gure 8. At LO we have hh = , due to the back-to-back production of ) V V b b NNLOB-proj NNLONLO-i NNLOFTapprox NLO d / 1 0.1 3.0 2.5 LNO2.0 too1.5 i tra1.0 0.5 0.00.0 pT,h2 (GeV) d / 2 0 d / 100 10 3.0 2.5 LNO2.0 Azimuthal angular separation between the two Higgs bosons at 14 TeV (left) and the two Higgs bosons at Born level. Real contributions allow hh to be smaller than , and again we observe that the NNLOB-proj approximation predicts larger corrections in the region dominated by hard radiation compared to the other two results, which again are in good agreement with each other in that region, whereas they start to deviate for larger angles. For values of hh close to , this observable receives large corrections from soft-gluon emission, and the corresponding large logarithms should be resummed in order to get a reliable prediction. We conclude this section by adding a few comments on the nite-Mt uncertainties at NNLO for the various di erential distributions. The analysis that was performed for the total cross section cannot be easily extended to di erential distributions. On one hand, any accidental agreement between the FTapprox and the full result at NLO in a given phase-space region would likely lead to an underestimation of the top quark mass e ects; on the other hand, the regions in which the NLO corrections are very small due to cancellations between di erent contributions can present very large relative di erences in the O( S3) contribution of the NLOFTapprox and NLO results, thus leading to arti cially large uncertainties at NNLO. In addition, there are observables that are by de nition reproduced in an exact way by the FTapprox at NLO (in our case pT;hh, pT;j1 and hh), and the uncertainty estimate procedure that we de ned for the inclusive case is therefore not applicable. Despite these facts, and based on the performance of the FTapprox at NLO [17] as well as on the observed di erences between our NNLO approximations, we can try to assess the order of magnitude of the expected missing Mt e ects for the distributions presented above. In the Higgs boson pair invariant-mass distribution, for values of Mhh below 500 GeV the level of accuracy of the FTapprox at NLO is similar to the inclusive case, and therefore the Mt uncertainty at NNLO is expected to be of a comparable size. In the tail of the distribution, however, the quality of the FTapprox decreases (see gure 5 of ref. [17]), and we thus expect the nite top quark mass e ects to be of O(10%) in this region. The shape of the rapidity distribution of the Higgs boson pair is correctly described by the FTapprox at NLO (see gure 8 of ref. [17]), and the di erence to the full result is only the overall normalization. Based on this, the estimated top quark mass uncertainty for the NNLOFTapprox result is constant in the whole yhh range and of the same size as for the inclusive cross section. The transverse momentum of the harder Higgs boson is very well described at NLO by the FTapprox (see gure 7 of ref. [17]), being always within the NLO scale uncertainty band. This fact, together with the close agreement between the NNLOFTapprox and NNLONLO i predictions, suggests that the missing top quark mass e ects at NNLO are probably of moderate size. The same holds true for the transverse-momentum distribution of the softer Higgs boson, except for the tail where at NLO the FTapprox overestimates the full NLO corrections, which in fact almost vanish in this region. The remaining distributions, which are either not de ned or trivial at LO, are by definition reproduced in an exact way by the FTapprox at NLO, and this makes the estimate of the missing top quark mass e ects at NNLO more di cult. In this case, a possible approach can be to use the di erence between the NNLOFTapprox and NNLONLO i prediction as an estimate of the uncertainty (as discussed before, the NNLOB proj prediction is not expected to be reliable in the regions dominated by hard real radiation, where it largely deviates from the other two approximations). This procedure would imply relatively low top quark mass uncertainties for the pT;hh and hh distributions, except for the low pT;hh and the hh regions, typically below the size of the scale uncertainties, and larger uncertainties for the leading-jet transverse momentum, for which the di erence between the two approximations is larger. In this work we considered Higgs boson pair production through gluon fusion in proton collisions. We presented new QCD predictions for inclusive and di erential cross sections, which include the full NLO contribution and also account for nite top quark mass e ects at NNLO. Our best prediction, denoted NNLOFTapprox, retains the full top quark mass dependence in the double-real emission amplitudes, while the remaining real-virtual and two-loop virtual HEFT amplitudes are treated via a suitable reweighting for the corresponding subprocesses with a given nal-state multiplicity. This approximation represents the most advanced prediction available to date for this process. The numerical results we obtained for the NNLOFTapprox are quantitatively di erent from the results obtained in previous combinations. In particular, as far as the total cross section is concerned, the corrections turn out to be smaller than previous estimates, increasing the NLO result by about 12% at 13 TeV and 7% at 100 TeV. The reduction of the scale uncertainties is signi cant, by about a factor of three for LHC energies. Given that our NNLOFTapprox prediction includes top quark mass e ects in an approximated way, it is important to assess the corresponding uncertainty. We carefully examined the performance of our approximations at both the inclusive and di erential levels. The uncertainty on our reference inclusive NNLOFTapprox prediction is estimated to be about 2:7% at 14 TeV, increasing with the collider energy to reach 4:6% at 100 TeV. Regarding di erential distributions, in most of the cases we can observe clear qualitative di erences with respect to the bin-by-bin reweighting procedure introduced in ref. [17], in the shape and/or the normalization. For some of the distributions, however, speci cally the tails of the pT;hh and pT;h1 spectra, both approximations are in very good agreement. We discussed an estimate of the uncertainty associated with top quark mass e ects at NNLO at the di erential level, and we found that in most of the cases its magnitude is comparable to the size of the scale uncertainties, except for the tails of some distributions where the uncertainty from missing Mt e ects can be dominant. Acknowledgments We thank Stefano Catani, Daniel de Florian, Ramona Grober, Andreas Maier, Stefano Pozzorini and Michael Spira for valuable discussions and comments on the manuscript. 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M. Grazzini, G. Heinrich, S. Jones, S. Kallweit, M. Kerner, J. M. Lindert, J. Mazzitelli. Higgs boson pair production at NNLO with top quark mass effects, Journal of High Energy Physics, 2018, 59, DOI: 10.1007/JHEP05(2018)059