Differential Higgs Boson Pair Production at Next-to-Next-to-Leading Order in QCD
Daniel de Florian, Massimiliano Grazzini, Catalin Hanga, Stefan Kallweit, Jonas M. Lindert, Philipp Maierhöfer, Javier Mazzitelli, Dirk Rathlev
DDESY 16-107FR-PHENO-2016-007ICAS 08/16MITP/16-061ZU-TH 20/16
Differential Higgs Boson Pair Production atNext-to-Next-to-Leading Order in QCD
Daniel de Florian ( a ) , Massimiliano Grazzini ( b ) , Catalin Hanga ( b ) , Stefan Kallweit ( c ) , Jonas M. Lindert ( b ) , Philipp Maierhöfer ( d ) , Javier Mazzitelli ( a ) , Dirk Rathlev ( e )( a ) International Center for Advanced Studies (ICAS), UNSAM, Campus Miguelete25 de Mayo y Francia, (1650) Buenos Aires, Argentina ( b ) Physik-Institut, Universität Zürich,Winterthurerstrasse 190, CH-8057 Zürich, Switzerland ( c ) PRISMA Cluster of Excellence, Institute of Physics,Johannes Gutenberg University, D-55099 Mainz, Germany ( d ) Physikalisches Institut, Albert-Ludwigs-Universität Freiburg,79104 Freiburg, Germany ( e ) Theory Group, Deutsches Elektronen-Synchrotron, D-22607 Hamburg, Germany
Abstract
We report on the first fully differential calculation for double Higgs boson productionthrough gluon fusion in hadron collisions up to next-to-next-to-leading order (NNLO)in QCD perturbation theory. The calculation is performed in the heavy-top limit ofthe Standard Model, and in the phenomenological results we focus on pp collisions at √ s = 14 TeV. We present differential distributions through NNLO for various observablesincluding the transverse-momentum and rapidity distributions of the two Higgs bosons.NNLO corrections are at the level of − with respect to the next-to-leading order(NLO) prediction with a residual scale uncertainty of − and an overall mild phase-space dependence. Only at NNLO the perturbative expansion starts to converge yieldingoverlapping scale uncertainty bands between NNLO and NLO in most of the phase-space.The calculation includes NLO predictions for pp → HH + jet + X . Corrections to thecorresponding distributions exceed with a residual scale dependence of − . a r X i v : . [ h e p - ph ] J un Introduction
The discovery of the Higgs boson [1, 2] during Run I of the Large Hadron Collider (LHC) opened thedoor towards direct tests of electroweak symmetry breaking. To this end the search for the productionof Higgs boson pairs is one of the main goals of ongoing and future runs of the LHC. Only thisproduction mode allows for direct tests of Higgs trilinear self-couplings, whose knowledge in turn isnecessary to reconstruct the scalar potential responsible for electroweak symmetry breaking.As it is the case for single Higgs boson production, the dominant production mode for Higgs bosonpairs in the Standard Model (SM) proceeds at hadron colliders via gluon fusion, mediated by heavy-quark loops. At the leading order (LO) in QCD [3–5] there are two interfering production mechanisms:either the two Higgs bosons couple directly to a heavy-quark loop via a box diagram, gg → HH , orthey couple via the trilinear Higgs coupling λ to an off-shell Higgs boson, which in turn is producedvia a triangular loop, similarly to single Higgs boson production, gg → H ∗ → HH .Due to the loop suppression of the LO process and additional large accidental cancellations be-tween “triangle” and “box” contributions in the scattering amplitude [3], signal rates for Higgs bosonpair production are only at the level of few fb at TeV and of tens of fb at − TeV. These smallrates, together with large irreducible backgrounds in the relevant b ¯ bγγ [6–8], b ¯ bτ ¯ τ [7–9], b ¯ bW + W − [10]and b ¯ bb ¯ b [11, 12] final states, pose a true challenge to experimental searches for Higgs boson pair pro-duction. Consequently, currently only upper limits exist, constraining the cross section for Higgs bosonpair production to the level of about times the SM prediction [13–18]. However, the productioncross section can significantly be altered by new-physics effects, for example due to new loop contri-butions [19], altered t ¯ th or novel t ¯ thh couplings [20], or due to new resonances [21]. Thus, futuremeasurements (together with precision predictions) on Higgs boson pair production, and in particularratios of cross-section measurements [22], do not just serve as stringent tests of electroweak symmetrybreaking in the SM, but might also open the door towards physics beyond the SM.Given the loop-induced nature of the scattering process for Higgs boson pair production, higherorders in perturbation theory are extremely difficult to calculate. Only very recently a complete next-to-leading order (NLO) calculation became available [23], where the required multi-scale two-loopscattering amplitudes have been evaluated via numerical integration. Prior to the pioneering work ofRef. [23] the possible impact of the NLO corrections has been studied in Refs. [24–28] via asymptoticexpansions of the two-loop virtual diagrams in the inverse top-quark mass.Assuming the top quark to be heavy and all other quarks to be massless, an effective theory canbe formulated, introducing a tree-level coupling of gluons and Higgs bosons. In this heavy-top limitNLO corrections to Higgs boson pair production have been presented in Ref. [29], where a rescalingwith the exact Born cross section was performed. The obtained NLO corrections in the so-called Born-improved heavy-top limit increase the total cross section by about with sizable remaining scaleuncertainties. In contrast, the exact NLO computation presented in Ref. [23] yields a result that issmaller by .In order to further improve on these predictions, and in particular to reduce the remaining scaleuncertainties, the effective theory allows for the computation of perturbative corrections beyond NLO.To this end, employing the amplitudes derived in Ref. [30] (supplemented by Ref. [24]) a calculationof Higgs boson pair production at next-to-next-to-leading order (NNLO) accuracy in the heavy-topapproximation was presented in Ref. [31]. At the inclusive level the NNLO corrections increase thecross section by about with respect to the NLO prediction, leaving scale uncertainties at thelevel of − . Besides inclusive NNLO cross sections the calculation of Ref. [31] offers differential1redictions in the invariant mass of the produced Higgs boson pair indicating a rather mild phase-spacedependence of the NNLO corrections. Still working in the heavy-top limit, soft-gluon resummationup to next-to-next-to-leading logarithmic (NNLL) accuracy has been carried out, and the results werematched to the NLO [32] and the NNLO [33] fixed-order computations. At NNLL+NNLO accuracy thetheoretical uncertainties on the inclusive cross section due to QCD effects are reduced to about [33].Furthermore, extending the SM with additional dimension-6 operators [34], NLO corrections in theheavy-top limit have been presented in Ref. [35]. Notably, in Refs. [36, 37] a reweighting technique hasbeen presented, allowing to combine exact one-loop real corrections of Higgs boson pair productionwith the corresponding virtual contributions, the latter computed in the effective theory. On the otherhand, the real corrections, which imply the evaluation of one-loop amplitudes with an extra partonin the final state, have been computed in an exact way and used to merge LO samples for HH + 0 , jets [38, 39] in order to obtain more reliable exclusive distributions.In this paper we extend the calculation of Ref. [31] providing fully differential NNLO predictions forHiggs boson pair production in the heavy-top approximation of the SM via a flexible Monte Carlo im-plementation. The calculation is based on the combination of the q T subtraction formalism [40] withthe Monte Carlo framework Munich † , supplemented by tree and one-loop amplitudes from Open-Loops [41]. Employing these tools we provide NNLO predictions for various kinematic distributionsthat are relevant for searches and precision measurements of Higgs boson pair production at the LHC.The calculation includes NLO predictions for pp → HH + jet + X . Corresponding differential dis-tributions are studied in detail. In our study we refrain from a reweighting using exact LO or NLOmatrix elements or cross sections. Such a reweighting should eventually be performed employing theresults of Ref. [23]. For the time being we focus on the differential NNLO/NLO correction factors ob-tained in the effective theory, which are the main result of our paper. Such correction factors providevaluable information that can directly be applied to any pp → HH + X NLO prediction in differentapproximations.The paper is organized as follows. In Sect. 2 we introduce the heavy-top limit for multi-Higgsproduction at higher orders in perturbation theory together with the technical ingredients of ourcalculation. Numerical results are presented in Sect. 3, and in Sect. 4 we summarize our results.
In the heavy-top approximation effective tree-level couplings between gluons and Higgs bosons areintroduced via the effective Lagrangian [29, 42, 43] L HEFT = − G µν G µν (cid:18) C H Hv − C HH H v (cid:19) , (1)where v (cid:39) GeV is the vacuum expectation value of the Higgs field. In this effective Lagrangianonly couplings relevant for our calculation are shown, while in general within this effective theory thereare also further couplings for any number of Higgs bosons to gluons. The matching coefficients C H † Munich is the abbreviation of “MUlti-chaNnel Integrator at Swiss (CH) precision”—an automated parton level NLOgenerator by S. Kallweit. In preparation. C HH can be expanded in powers of α S via the following parametrization, C X = − α S π (cid:88) n ≥ C ( n ) X (cid:16) α S π (cid:17) n , with X = H, HH . (2)The perturbative expansion for both coefficients is known up to O ( α ) and reads [24, 42, 44, 45] C (0) H = C (0) HH = 1 , (3) C (1) H = C (1) HH = 114 ,C (2) H = 2777288 + 1916 ln µ R m t + n F (cid:18) − µ R m t (cid:19) ,C (2) HH = C (2) H + 3524 + 2 n F , where n F is the number of light quarks, µ R the renormalisation scale and m t the pole mass of the(heavy) top quark. As can be seen from Eq. (3), up to O ( α ) we have C H = C HH .As already discussed, in the full theory Higgs boson pair production at LO is governed by “box”and “triangle” contributions. In the heavy-top limit the corresponding scattering amplitudes manifestas tree-level diagrams with one double-Higgs and one single-Higgs operator insertion, respectively. AtNLO, in the perturbative expansion we have the usual real and virtual contributions, where the formerincludes gluon and quark bremsstrahlung, and the latter are given by one-loop corrections to thediagrams mentioned before. However, at the same order of perturbation theory there is an additionalcontribution with Born-level kinematics, originating from amplitudes with two single-Higgs operatorinsertions in interference with the LO amplitude [29]. In the full theory such contributions correspondto reducible double-triangle two-loop diagrams.This pattern also appears at higher orders, and in particular the NNLO virtual contributions haveto include both two-loop corrections to amplitudes with one single- or double-Higgs operator insertion,and one-loop corrections to amplitudes with two single-Higgs operator insertions [30]. These NNLOvirtual contributions have to be combined via an appropriate subtraction scheme with double-real andreal–virtual contributions of the same perturbative order. Similarly to what was discussed before, thereal–virtual contributions, i.e. the virtual amplitudes for HH + jet production, have to be extended toinclude two single-Higgs operator insertions in interference with the corresponding tree-level amplitude.More details on the technical implementation of such double-operator insertions in our calculation aregiven in Sect. 2.3. q T subtraction In order to handle infrared singularities in the NNLO calculation, we apply the q T subtraction for-malism [40]. In this approach the separation between genuine NNLO singularities, located where thetransverse momentum of the Higgs pair, q T ,HH , is zero, from NLO-like singularities in the HH + jet contribution is explicit. As a consequence, the contribution d σ HH+jetNLO in the q T subtraction formula, d σ HH NNLO = H HH NNLO ⊗ d σ HH LO + (cid:104) d σ HH +jetNLO − d σ CTNNLO (cid:105) , (4)can be evaluated using any well-established subtraction procedure at NLO. The remaining divergencein the limit q T ,HH → is cancelled by the process-independent counterterm d σ CTNNLO . The implemen-tation is fully general, and it is based on the universality [46] of the hard-collinear coefficients H HH NNLO q T NLO ( r cut ) σ CSNLO σ / σ C S N L O − [ % ] pp → HH + X @ 14 TeV r cut = cut q T /q [%] 1.00.90.80.70.60.50.40.30.20.10+0 . . . . . − . − . − . − . − . σ q T NLO ( r cut ) σ CSNLO σ / σ C S N L O − [ % ] pp → HH + X @ 14 TeV r cut = cut q T /q [%] 1.00.90.80.70.60.50.40.30.20.10+0 . . . . . − . − . − . − . − . σ q T NNLO ( r cut ) σ NNLO σ dFMNNLO σ / σ d F M NN L O − [ % ] pp → HH + X @ 14 TeV r cut = cut q T /q [%] 1.00.90.80.70.60.50.40.30.20.10+0 . . . . . − . − . − . − . − . σ q T NNLO ( r cut ) σ NNLO σ NNLO σ / σ d F M NN L O − [ % ] pp → HH + X @ 14 TeV r cut = cut q T /q [%] 1.00.90.80.70.60.50.40.30.20.10+0 . . . . . − . − . − . − . − . Figure 1: Dependence of the pp → HH + X cross sections at 14 TeV on the q T -subtraction cut, r cut , for both NLO (left plot) and NNLO (right plot) results. NLO results are normalized to the r cut -independent NLO cross section computed with Catani–Seymour subtraction, and the NNLO resultsare normalized to the r cut -independent inclusive NNLO cross section calculated in the framework ofRef. [33]. The blue band indicates the NNLO result from q T subtraction in the limit r cut → , withan approriate extrapolation-error estimate.appearing in the first term on the right hand side of Eq. (4). The general structure of these coefficientsfor gluon-initiated processes has been presented in Refs. [47, 48]. Their process dependence is embodiedin a single perturbative hard factor which is obtained from the two-loop virtual correction, derived forthis process in Ref. [30], through an appropriate subtraction procedure [46].The difference in the square bracket in Eq. (4) is formally finite as q T → , but each term separatelyexhibits logarithmic divergences in this limit. In practice a small technical cut, r cut , needs to be appliedon r ≡ q T /Q , where Q is typically chosen as the invariant mass of the final-state system (so here Q = m HH ). After cancellation of these logarithms between the real contribution d σ HH+jetNLO and thecounterterm, the remainder shows a very slight r cut dependence below about r cut = 1% ; we thus usethe finite- r cut results to extrapolate to r cut = 0 , and conservatively assign an additional extrapolationerror to our results. We verified in detail that for Higgs boson pair production the NNLO result isindeed very stable when varying the cut parameter. More precisely, in the range below r cut = 1% , thevariation in the NNLO cross section is of O (0 . , and our extrapolated result is in good numericalagreement with the analytic result of Ref. [33] (see Fig. 1). OpenLoops
All tree and one-loop amplitudes, i.e. in particular the one-loop amplitude for the d σ HH +jetNLO contribu-tion, are provided by the publicly available ‡ OpenLoops amplitude generator [41], which is based ona fast numerical recursion for the generation of tree and one-loop scattering amplitudes [49].In order to extend
OpenLoops to one-loop corrections in the heavy-top limit of the SM, all rele-vant Feynman rules for single-Higgs [50] and double-Higgs [35, 51] production have been implemented ‡ The publicly available
OpenLoops process library includes all relevant matrix elements to compute NLO QCDcorrections, including colour- and helicity-correlations and real radiation as well as loop-squared amplitudes, for morethan a hundred LHC processes. Amplitudes for Higgs boson pair production (+1 jet) at NLO in the heavy-top limit havebeen made available together with this publication, while amplitudes for pp → H, pp → Hj, pp → Hjj and pp → Hjjj have been publicly available already for some time.
4n the framework of the numerical open-loops recursion including UV renormalisation and the rationalcontributions of type R [52]. Combined with the OPP reduction method [53] implemented in Cut-Tools [54] and the scalar one-loop library
OneLOop [55] the employed recursion permits to achievevery high CPU performance and a high degree of numerical stability. The small fraction of numericallyunstable matrix elements is automatically detected and rescued through re-evaluation in quadrupleprecision.Technically, the effective field theory of Eq. (1) introduces various features which do not appearin the Standard Model. Most notably, the Feynman rules for the dimension-5 and -6 operators G µν G µν H ( H ) introduce, apart from 5- and 6-point vertices, the Lorentz structure p p g µν − p ν p µ (where p µ and p ν are the gluon momenta) which, if present in a loop, raises the tensor rank of theamplitude by . In the calculation of HH (+jet) production at one-loop level such operators enter onlyonce, leading to a tensor rank up to one higher than the number of loop propagators. The reduc-tion of such amplitudes with one “additional” tensor rank is supported by CutTools . Furthermore,the O ( α S ) contributions to the matching coefficients C H and C HH must be included. Consideringthe order of coupling powers it is natural to treat these contributions as counterterms. As discussedabove, at the same order of perturbation theory as the one-loop scattering amplitudes for HH (+jet)production, contributions from two single-Higgs operator insertions at tree-level in interference withthe LO tree-level amplitude with one double-Higgs operator insertion have to be considered. Similarlyto the O ( α S ) contributions to C H and C HH , these contributions are included via dedicated O ( α S ) pseudo-counterterms. One-loop amplitudes for single Higgs boson production plus up to two jets have been extensivelyvalidated against the results of Refs. [56–62] implemented in
Sherpa [63] and
MCFM [64] (via thecorresponding implementation in the
POWHEG-BOX [64, 65]). Due to the lack of publicly availablealternatives, the validation of the one-loop amplitudes for Higgs boson pair production plus jets hadto rely on various internal cross checks.We performed a calculation of pp → H + X up to NNLO in the heavy-top limit in the sameframework as employed for pp → HH + X , and compared against the results obtained with the inclusiveanalytical codes of Refs. [66–68], where agreement well beyond the per mill level was found. Due tothe similarity of the two processes this serves as a strong cross check for many technical ingredients ofthe calculation presented here.In order to validate all ingredients of the computation of Higgs boson pair production in theheavy-top limit presented in this paper, the LO, NLO and NNLO inclusive cross sections computedin Ref. [31] have been reproduced at the per mill level § (see Fig. 1). Additionally, mutual agreementhas been found for the invariant mass distribution m HH up to NNLO comparing against the results ofRef. [31]. Furthermore, the NLO results have been computed in the q T subtraction formalism and alsoemploying the dipole subtraction framework [69, 70] within Munich , where again we found mutualagreement far beyond the per mill level (see again Fig. 1). § In [31] the relation C (2) H = C (2) HH was assumed due to the lack of knowledge of C HH up to O ( α S ) at that time, whilefor this cross check the matching coefficients listed in Eq. (3) have been applied in both calculations. s [TeV] σ LO [fb] σ NLO [fb] σ NNLO [fb]
13 13 . +31 . − . . +17 . − . . +5 . − .
14 17 . +30 . − . . +17 . − . . +5 . − . Table 1: Inclusive cross sections for Higgs boson pair production for different centre-of-mass energiesat LO, NLO and NNLO. Numerical errors on the respective previous digits are stated in brackets,including the extrapolation error in the NNLO prediction. Scale uncertainties are obtained fromindependent variations of µ R and µ F around the central scale µ = m HH / . In the following we present predictions for Higgs boson pair production at the LHC including pertur-bative fixed-order corrections up to NNLO in the heavy-top limit. Inclusive results will be presentedfor centre-of-mass energies of √ s = 13 TeV and √ s = 14 TeV, while at the differential level we restrictourselves to √ s = 14 TeV. SM input parameters are chosen according to the recommendations of [71],which in particular implies v = 246 . GeV, m t = 173 . GeV and m H = 125 GeV . (5)Here, the top-quark mass does only enter via the NNLO contributions to the matching coefficients, asgiven in Eq. (3). For the calculation of hadron-level cross sections we employ the PDF4LHC15 [72]parton distribution functions (PDFs), and use the corresponding NLO PDFs for our LO and NLOpredictions and NNLO PDFs for the NNLO predictions. ¶ Couplings are evaluated using the runningstrong coupling provided by the respective PDFs. All light quarks, including bottom quarks, are treatedas massless particles, i.e. n F = 5 , while the top quark does not contribute explicitly in the employedheavy-top limit. To define jets, we employ the anti- k T jet clustering algorithm [74] with R = 0 . andrequire p T j > GeV and | η j | < . . In all results the renormalisation scale µ R and factorisation scale µ F are set to µ R , F = ξ R , F µ , with µ = m HH / and ≤ ξ R , ξ F ≤ , (6)where m HH is the invariant mass of the produced Higgs boson pair. Our default scale choice cor-responds to ξ R = ξ F = 1 , and theoretical uncertainties are assessed by applying the 7-point scalevariations ( ξ R , ξ F ) = (2 , , (2 , , (1 , , (1 , , (1 , . , (0 . , , (0 . , . , i.e. omitting antipodal vari-ations. As shown in Ref. [33] the scale choice of Eq. (6) guarantees a good perturbative convergenceof the total cross section and of the m HH distribution in Higgs boson pair production.In Tab. 1 we report inclusive cross sections for √ s = 13 TeV and √ s = 14 TeV. No phase-space cutsare applied, and the quoted uncertainties are obtained from scale variations. Both at √ s = 13 TeV and TeV the NLO corrections increase the LO result by about , and the NNLO corrections have aneffect of about on top of the NLO result. Scale uncertainties are successively reduced from about − at LO (which largely underestimates the effect of higher-order corrections) to less than at NNLO.In Figs. 2–7 differential distributions for Higgs boson pair production at the LHC with √ s = 14 TeVare shown at LO, NLO and NNLO accuracy. In those distributions shown in Figs. 2–4, both NLO ¶ To be precise, we use the
PDF4LHC_nlo_30 and
PDF4LHC_nnlo_30 sets, interfaced through the
Lhapdf library [73]. pp → HH + X at √ s = 14 TeV. Shown are absolute LO (black), NLO (red) and NNLO (blue)predictions in the heavy-top approximation and corresponding relative corrections normalized to thecentral NLO prediction. Bands correspond to independent variations of µ R and µ F around the centralscale µ = m HH / as described in the text.and NNLO corrections are sizable, and only at NNLO the perturbative convergence becomes manifestwith overlapping scale uncertainty bands between the NLO and NNLO predictions in most of theconsidered phase-space regions. At the same time theoretical uncertainties estimated by the scalevariations described above are approximately halved when going from NLO to NNLO. The NNLOdistributions shown in Figs. 5–7 are effectively only of next-to-leading order as they are either trivialor not defined at LO. They can be considered as a computation of HH + jet at NLO. Nevertheless, inthe following discussion we always denote the highest considered perturbative order as NNLO (withrespect to pp → HH + X ).Differential distributions in the transverse momentum and the rapidity of the two Higgs bosons,ordered by their hardness in p T , are shown in Fig. 2 and Fig. 3, respectively. NNLO corrections areoverall at the level of − with a rather mild phase-space dependence. In particular, in thetransverse-momentum distribution of the harder Higgs boson, p T ,H , the NNLO corrections slowlyincrease as p T ,H increases, while for the softer Higgs boson the corrections are to a large extentindependent of p T ,H , except for the very low p T ,H region. As for the rapidity distributions, theNNLO effect is largely constant and at O (20%) for both y H and y H .In the left plot of Fig. 4 we show predictions for the invariant-mass distribution of the producedHiggs boson pair, m HH . NNLO corrections are at the level of with respect to NLO and hardly showany phase-space dependence. NNLO predictions in m HH have already been presented in Ref. [31], andthe corresponding results are overlaid in Fig. 4 (left). In the computation of Ref. [31] IR singularitiesare analytically cancelled, thereby leading to negligible numerical fluctuations in the shown distri-bution. Within statistical uncertainties the results obtained from the two completely independentimplementations agree perfectly. 7igure 3: Distributions in the rapidity of the harder (left) and the softer (right) Higgs boson. Curvesand bands as in Fig. 2.Figure 4: Invariant-mass distribution m HH (left) and rapidity distribution y HH (right) of the producedHiggs boson pair. Curves and bands as in Fig. 2. Additionally, in the left plot we show the m HH distribution as obtained with the calculation of Ref. [31].In the right plot of Fig. 4 predictions for the rapidity of the Higgs boson pair, y HH , are presented.Again, we observe a mild phase-space dependence, with increasing NNLO corrections only for largerapidities. In all distributions in Figs. 2–4, NNLO scale uncertainties are reduced to the level of ± (5% − , compared to ± (15% − at NLO.8igure 5: Distributions in the transverse momentum of the Higgs boson pair (left) and of the hardestjet (right). Curves and bands as in Fig. 2.In Fig. 5 we show distributions in the transverse momentum of the Higgs boson pair, p T ,HH , andof the hardest jet, p T ,j . At NLO (which is effectively LO for non-vanishing transverse momenta)these two distributions are directly related, and in both distributions scale uncertainties reach almost . The NNLO effect is larger on the p T ,HH distribution than on the p T ,j distribution, reaching for p T ,HH ≈ GeV compared to for p T ,j ≈ GeV. The NLO nature of these NNLOcorrections is furthermore reflected by sizable scale uncertainties at the level of − . In thelimit p T ,HH → the perturbative expansion fails due to the appearance of large logarithmic terms ofthe form log n ( p T ,HH /m HH ) . Here, a proper resummation of such terms is required in order to achievea reliable theoretical prediction.At LO the two Higgs bosons are always produced back-to-back. However, at higher orders additionalQCD radiation allows for a non-trivial angular separation between the two Higgs bosons. In Fig. 6 weshow the corresponding distribution in the azimuthal angle between the two Higss bosons, ∆ φ HH . Inour fixed-order approach, NNLO corrections are large and positive in the back-to-back configuration,where they are driven by soft-gluon emission, and jump to negative values for ∆ φ HH (cid:46) π , due to themis-cancellation between real and virtual contributions. In this region of phase-space, large logarithmicterms should again be resummed for achieving a reliable theoretical prediction. Configurations at smallangles, i.e. ∆ φ HH → , are driven by hard gluon emission, and NNLO corrections are at the level of with respect to NLO.Finally, in Fig. 7 we investigate corrections to the ∆ R separation between the two Higgs bosonsand the hardest jet. Overall corrections to these observables are moderate at the level of − with largely overlapping uncertainty bands between NNLO and NLO. However, for small ∆ R H j separations, due to the ordering of the Higgs bosons according to their transverse momenta the entirephase-space opens up only at the NNLO level, inducing sizable correction factors at the NLO boundary ∆ R H j (cid:38) π/ . 9igure 6: Distribution in the angular separation between the two Higgs bosons ∆ φ HH . Curves andbands as in Fig. 2.Figure 7: Distributions in the ∆ R = (cid:112) (∆ φ ) + (∆ y ) separation between the harder Higgs boson andthe hardest jet (left), ∆ R H j , and the softer Higgs boson and the hardest jet (right), ∆ R H j . Curvesand bands as in Fig. 2. In this paper we have presented the first fully differential calculation for double Higgs boson productionthrough gluon fusion in hadron collisions. We worked in the heavy-top limit, and presented resultsfor differential distributions through NNLO QCD for various observables including the transverse-10omentum and rapidity distributions of the two Higgs bosons. NNLO corrections amount to about − with respect to the NLO prediction and are mildly dependent on the kinematics. Theresidual scale uncertainty at NNLO is about − . Only at NNLO the perturbative expansionstarts to converge, and the uncertainty bands obtained through scale variations at NLO and NNLOoverlap in most of the phase-space. The calculation includes NLO QCD predictions for pp → HH +jet + X . Corrections to the corresponding distributions exceed with a residual scale dependenceof about − .The calculation presented here is based on the combination of the q T subtraction formalism [40]with the Monte Carlo framework Munich , supplemented by tree and one-loop amplitudes from
Open-Loops [41]. This framework, to be integrated in the new numerical program
Matrix (cid:107) , which iscurrently under development, allows for an extremely flexible implementation.In the present paper we have limited ourselves to strictly work in the heavy-top limit of theSM. With the exact NLO virtual contributions available since recently, a combination of the exactresults at NLO accuracy with the NNLO calculation in the heavy-top limit should be performed in thefuture. This combination, together with the inclusion of the Higgs boson decays, will facilitate realisticphenomenological studies at an unprecedented level of precision, as required for future measurementsof the Higgs trilinear coupling.
Acknowledgements.
We thank Stefano Pozzorini and Marius Wiesemann for valuable discussions.This research was supported in part by the Swiss National Science Foundation (SNF) under contractsCRSII2-141847, 200021-156585, and by the Research Executive Agency (REA) of the European Unionunder the Grant Agreement number PITN–GA–2012–316704 (
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