Higgs boson production at the LHC: fast and precise predictions in QCD at higher orders

Eur. Phys. J. C (2022) 82:492 https://doi.org/10.1140/epjc/s10052-022-10436-4 Special Article - Tools for Experiment and Theory Higgs boson production at the LHC: fast and precise predictions in QCD at higher orders Stefano Camarda1, Leandro Cieri2, Giancarlo Ferrera3,a, Jesús Urtasun-Elizari3 1CERN, 1211 Geneva, Switzerland 2Instituto de Física Corpuscular, Universitat de València-CSIC, Parc Científic, 46980 Paterna, Valencia, Spain 3Dipartimento di Fisica, Università di Milano and INFN, Sezione di Milano, 20133 Milan, Italy Received: 2 March 2022 / Accepted: 13 May 2022 / Published online: 27 May 2022 © The Author(s) 2022 Abstract We present a new numerical program, HTurbo, which provides fast and numerically precise predictions for Higgs boson production cross sections. The present version of the code implements the perturbative QCD expansion up to the next-to-next-to-leading order also combined with the resummation of the large logarithmic corrections at small transverse momenta up to next-to-next-to-leading logarithmic accuracy and it includes the Higgs boson production through gluon fusion and decay in two photons with the full dependence on the final-state kinematics. Arbitrary kinematical cuts can be applied to the final states in order to obtain fiducial cross sections and associated kinematical distributions. We present a benchmark comparison with the predictions obtained with the numerical programs HRes and HNNLO programs for which HTurbo represents an improved reimplementation. After the discovery of the Higgs boson [1,2], a foremost goal of the physics program at the Large Hadron Collider (LHC) has become the direct investigation of the electroweak symmetry breaking mechanism. In particular, precision studies are a key tool for searches of possible deviations from Standard Model (SM) predictions in the Higgs sector. In this paper we consider the production of the SM Higgs boson through gluon fusion and its decay to γγ. The gluon fusion subprocess gg →H[3], through a heavy-quark (mainly, top-quark) loop, is the main production mechanism of the SM Higgs boson at hadron colliders and its dynamics is driven by strong interactions. Therefore, it is essential to study the effects of higher-order QCD radiative corrections and to provide accurate theoretical predictions of Higgs boson cross sections and associated distributions. The QCD radiative corrections to the total cross sections and differential distributions have been calculated up to next-to-next-to-next-to-leading order (N3LO) within the ae-mail: [email protected] (corresponding author) framework of the large-Mtop approximation [4–17]. The NNLO parton-level calculation of H+jet production has been performed in Refs. [18–20]. The combination of the QCD transverse-momentum (qT) resummation formalism of logarithmically enhanced contribution with fixed-order perturbative results at different levels of theoretical accuracy, have been obtained in [16,21–26] (see also references therein). Theoretical predictions depend on several different parameters and on various inputs such as parton density functions (PDFs), renormalization and factorization scales and SM parameters. In order to obtain precise theoretical predictions with a reliable estimate of the associated uncertainties it is therefore crucial to develop computing codes which allow for fast calculations with small numerical uncertainties. The HTurbo program, which is presented in this paper, provides fast and numerically precise predictions of the Higgs boson production and decay cross sections. It follows the structure of the DYTurbo code [27] developed for Drell– Yan lepton pair production. Performance enhancements are primarily achieved through the use of numerical integrations with quadrature rules based on interpolating functions, software profiling optimization and multi-threading implementation. HTurbo calculates higher-order QCD corrections of Higgs boson cross sections at fully differential level in the four momenta of the final states by implementing the qTresummation formalism developed in Refs. [28–31] and the qTsubtraction method of Ref. [11] in a completely independent way from the original numerical programs HqT [29,32], HNNLO [11] and HRes [23]. Besides an improvement in performances and numerical precision, this novel implementation has the aim of facilitating the inclusion of N3LO corrections along the lines of Ref. [33] and the fiducial perturbative power corrections within the qTsubtraction method 123 492 Page 2 of 8 Eur. Phys. J. C (2022) 82 :492 exploiting the recoil procedure of Ref.[34] as performed in Ref. [35].1 The present version of the program includes the Higgs boson production through gluon fusion and its decay in a photon pair, implementing the resummation of the QCD contributions logarithmically-enhanced in the small-qTregion at leading-logarithmic (LL), next-to-leading-logarithmic (NLL), and NNLL accuracy as well as the corresponding finiteorder contributions at next-to-leading order (NLO) and NNLO both in the smalland large-qTregions.2The fullydifferential fixed-order QCD calculation has been implemented up to next-to-next-to-leading order (NNLO). The H+jet predictions have been reimplemented from the MCFM program [36,37], as encoded in HRes and HNNLO.The HTurbo program is based on a modular C++ structure (with few Fortran functions interfaced) with multi-threading implemented with OpenMP and through the Cuba library [38]. The parameters of the calculation can be set in a flexible way via input file and/or command line options. The HTurbo program will be made publicly available. We briefly summarize the relevant formulae which have been implemented in the HTurbo program.3The fullydifferential Higgs boson cross section, completely inclusive over the final-state QCD radiation, is described by six kinematic variables corresponding to the momenta of the two photons. Therefore we can express the cross-section as a function of the transverse momentum qT, the rapidity yand the invariant mass mof the Higgs boson (or photon pair), and three angular variables corresponding to the polar θand the azimuthal φangles of the photon decay in a given Higgs boson rest frame and to the azimuth φHof the Higgs boson in the laboratory frame. Given the spin-0 nature of the SM Higgs boson the cross section factorizes in two independent factors for the Higgs boson production and decay subprocesses. We treat the Higgs boson within the narrow-width approximation, H/mH→0(His the Higgs boson total decay width), and thus we have m=mH. Moreover in (unpolarised) hadron collisions the initial-state hadrons, i.e. the incoming beams, are to very good approximation azimuthally symmetric and therefore the cross section does not depend on the absolute value of φH. Therefore in the following we will consider the cross section averaged over φHat fixed values of the additional kinematical variables of the final-state system. 1We note however that in order to reach this goal a necessary ingredient is the NNLO QCD prediction for H+jet production [18–20]which,at present, is not available in a public form. 2Sometimes in the literature this is referred respectively as NLLand NNLLaccuracy. 3The reader interested on the details of the qTresummation and qT subtraction formalisms is referred to the original literature [11,28–31]. The qTresummed cross section can be written as dσH NnLL+NnLO =dσres NnLL −dσasy Nn−1LO +dσf.o. Nn−1LO ,(1) with n=1,2,3,... (in the following we do not explicitly consider the lowest order predictions at LL accuracy: dσH LL =dσres LL ). The term dσres in Eq.(1)istheresummed component, dσasy is the asymptotic contribution (that is the fixed-order expansion of dσres), and dσf.o. is the finiteorder cross section integrated over final-state QCD radiation (which can be obtained from the H+jet cross section). The resummed term dσres dominates at small qT(qTm) while the finite-order component dσf.o. describes the largeqTregion (qT∼m). An accurate description of the region of intermediate qTrequires a consistent match between the resummed and finite components. The resummation of the logarithmic contributions has been carried out in the impact-parameter space b(which is the Fourier-conjugate variable to qT)[39] in order to fulfill the constraint of transverse-momentum conservation for multi-parton radiation. Moreover convolution with PDFs is more conveniently expressed by considering double Mellin moments of the corresponding partonic functions[30]. The resummed and asymptotic terms in Eq.(1) can thus be written as:4 dσres NnLL =dˆσH LO ×HH NnLO ×exp{GNnLL}(2) dσasy Nn−1LO =dˆσH LO ×H(qT/Q)Nn−1LO ,(3) where Q∼mdenotes the so-called resummation scale [29], an auxiliary scale that is introduced in dσres and, consistently, in dσasy whose variations can be used to estimate the uncertainty from not yet calculated higher-order logarithmic corrections. The factor d ˆσH LO is the Born level cross section, the coefficient HH[40] is the (process dependent) hard-collinear function and the term exp(G)is the gluon Sudakov form factor[41–43] which resums in an exponential form the large logarithmic corrections in the impact-parameter space b.The function H(qT/Q), which embodies the singular behaviour of dσf.o. in the limit qT→0, can be obtained from the fixedorder expansion of the term HH×exp{G}. The Mellin moments of the hard-collinear function HH have been computed with the FORM [44] packages summer [45] and harmpol [46], using the method of Ref.[47]. The Mellin space evolution of PDFs and the Mellin moments of the splitting functions have been calculated with the package QCD-PEGASUS[48]. The HTurbo program includes also fixed-order predictions (without the resummation of logarithmically-enhanced contributions). Beyond the LO, the fixed-order cross section 4For the sake of simplicity we use a symbolic notation where convolution with PDFs, the sum over different initial-state partonic channels and the inverse Mellin and Fourier transformations are understood. 123 Eur. Phys. J. C (2022) 82 :492 Page 3 of 8 492 is computed through the qTsubtraction formalism [11] and is expressed as the sum of three terms as follows: dσH NnLO =HH NnLO ×dσH LO +dσH+jet Nn−1LO −dσCT Nn−1LO,(4) where the term dσH+jet is the H+jet cross section, and the counter-term dσCT Nn−1LO is given by dσCT Nn−1LO =dσH LO ×∞ 0 d2q TH(q T/m)Nn−1LO .(5) The singular behaviour of dσH+jet in the limit qT→0, known from the qTresummation formalism, is the same of the subtraction counter-term dσCT. Being the terms dσH+jet and dσCT in Eq.(4) separately divergent at qT=0a technical parameter qcut T>0 has to be introduced. For qT≥qTcut the sum of the terms in the square bracket of Eq. (4) is infrared finite (or, more precisely, integrable over qT) and the “exact” value of the cross section can be obtained evaluating the square bracket term in Eq. (4) in the limit qTcut →0. However for finite value of qcut Tthe cross sectioninEq.(4) contains perturbative power corrections ambiguities O((qcut T/M)p)[49–52], with p>0 which are particularly severe in the case of fiducial selection cuts which yield an acceptance that has a residual linear dependence on qcut T[50,53,54]. A method to remove such linear fiducial power corrections (FPC) within the qTsubtraction formalism has been proposed in Refs.[35,55]. An important feature of the resummation formalism of Ref. [29] is the so called unitarity constraint which leads to the following relation: ∞ 0 dq2 Tdσres NnLL+NnLO =HH NnLO ×dˆσH LO ,(6) which ensures that fixed-order results are exactly recovered upon integration over qTof the matched cross section. A consequence of the unitarity constraint is the reduction of resummation effects in the region of small impact parameter where it is clear that resummation cannot give an improvement over the accuracy of the fixed-order calculation. The effect of unjustified resummed contributions in the large-qTregion can be further reduced or eliminated by introducing a switching function w(qT,m)which multiplies the terms dσres NnLL and dσasy Nn−1LO in Eq.(1) above a given qTvalue. However because such switching violates the unitarity constraint of Eq. (6)it has to be included with some care. Within HTurbo the effect of a Gaussian switching function w(qT,m)chosen following Ref. [34] can be included. The perturbative form factor exp(G)is formally singular when transverse-momenta of the order of the scale of the Landau pole of the QCD coupling (b−1∼QCD)are approached. This is the indication of the breakdown of perturbation theory and of the onset of truly non-perturbative (NP) effects. In this region a model for NP QCD effects, which has to include a regularization of the Landau singularity, is necessary. We have explicitly implemented in the HTurbo program the so-called Minimal Prescription [56–58] which regularizes the Landau singularity in resummed calculations without introducing higher-twist power-suppressed contributions of the type O(QCD/Q). Alternatively it can be chosen the freezing procedure[59,60] known as the ‘b∗prescription’, which is implemented in HRes, consisting in the replacement b2→b2 ∗=b2b2 lim/(b2+b2 lim)(7) in the form factor exp(G). The value of the parameter blim has to be set to be slightly smaller than the Landau singularity b−1∼QCD. Power-suppressed contributions are expected to dominate at very small transverse-momentum (qT∼QCD) and to correctly describe the experimental data in that region they have to be (properly) included taking into account the delicate interplay with the leadingtwist term. We parameterize the NP QCD effects at low qT through a non-perturbative form factor with different functional forms (the simplest one is a Gaussian smearing factor, exp (−gNPb2), which depends on the non perturbative parameter gNP). In the following we show some benchmark numerical results obtained with HTurbo compared with corresponding results from HRes (up to NNLL+NNLO accuracy) and HNNLO (up to NNLO). In particular we consider the cross section differential in the Higgs boson qTin both the full final state diphoton phase space and in a given selected fiducial region. We also compare the time performance of the codes in order to assess the performance improvement of HTurbo. We consider Higgs boson cross sections in proton–proton collisions at √s=13 TeV using the NNPDF3.1 NNLO [61] set of parton density functions with αS(mZ)=0.118. The computation is performed by considering gg →Hproduction, through a top-quark loop, in the large-Mtop approximation. We use the same settings and input parameters in both the HTurbo and HRes codes. In particular the value of the renormalization (μR), factorization (μF) and resummation (Q) scales have been chosen to be equal to the Higgs boson mass mH. We start to present our benchmark results at inclusive level (i.e. integrating over the diphoton final state kinematics). In Fig.1we consider the resummed part of the qTdistribution (see Eq. (2)) at NLL accuracy (left panel) and at NNLL accuracy (right panel). The HTurbo results using quadrature integration (blue dots) have been compared with the HRes results (green histograms). The lower panels show the ratio between the results which are in agreement, within the numerical uncertainties of the codes, at better than 1% level. In Fig.2we consider the asymptotic term of the cross section (see Eq. (2)) at LO (left panel) and NLO (right panel). The asymptotic term diverges in the qT→0 limit and it becomes negative at large qT(we thus show the abso123 492 Page 4 of 8 Eur. Phys. J. C (2022) 82 :492 (a) (b) Fig. 1 Comparison of full-photon phase space differential cross sections computed with HRes and HTurbo at √s=13 TeV. Resummed component of the transverse momentum distribution at NLL (a)and NNLL (b) accuracy. The top panels show absolute cross sections, and the bottom panels show ratios of HTurbo to HRes results (a) (b) Fig. 2 Comparison of full-photon phase space differential cross sections computed with HRes and HTurbo at √s=13 TeV. Absolute value of the asymptotic component of the transverse momentum distribution at LO (a)andNLO(b) accuracy. The top panels show absolute cross sections, and the bottom panels show ratios of HTurbo to HRes results lute value of the results in logarithmic scale). The qTdistribution of the asymptotic term has been computed in the range 1 GeV <qT<mHand we obtained a sub-percent agreement between HTurbo (blue dots) using quadrature integration and HRes (green histograms) results. Finally, in Fig.3, we show the fixed-order term of the cross section at LO (left panel) and NLO (right panel) as obtained with HTurbo and HNNLO. As both HTurbo and HNNLO programs use the Vegas algorithm for numerical integration, we expect to observe similar results as is confirmed by the sub-percent agreement between HTurbo (blue dots) and HNNLO (green histograms) results. We then consider the case of fiducial cross sections. The fiducial phase space is defined by the photon transverse momenta pγ T>0.35 mHand the photon pseudorapidities |ηγ|<2.37. In Figs.4,5and 6we show the comparison between the resummed, asymptotic and fixed-order term as presented above for the inclusive phase space. Also in the case of fiducial phase space we observe a sub-percent agreement between HTurbo (blue dots), HRes and HNNLO (green histograms) results in the entire range of qTconsidered. We now briefly comment on various tests of time performance which have been performed on a machine with 3.50 GHz Intel Xeon CPUs. The computation time necessary to calculate cross-section predictions for HTurbo and HRes is compared and used to assess the performance improvement of HTurbo.TheHRes calculation for the resummed term of the inclusive cross-section at NLL accuracy with an uncertainty of 1% took around 0.5 h, while the analogous HTurbo calculation (without the multi-threading option) 123 Eur. Phys. J. C (2022) 82 :492 Page 5 of 8 492 (a) (b) Fig. 3 Comparison of full-photon phase space differential cross sections computed with HRes and HTurbo at √s=13 TeV. Fixed-order component of the transverse momentum distribution at LO (a)andNLO (b) accuracy. The top panels show absolute cross sections, and the bottompanelsshowratiosofHTurbo to HRes results (a) (b) Fig. 4 Comparison of full-photon phase space differential cross sections computed with HRes and HTurbo at √s=13 TeV. Resummed component of the transverse momentum distribution at NLL (a)and NNLL (b) accuracy. The top panels show absolute cross sections, and the bottom panels show ratios of HTurbo to HRes results with an uncertainty of 0.001% took around 20 s, yielding an improvement of around two orders of magnitude in the time performance and three orders of magnitude in numerical precision. Similar results were obtained including fiducial cuts or considering the resummed term at NNLL accuracy. The LO or NLO calculation of the asymptotic term took around 10 minutes with an uncertainty of 0.5-1% within HRes both at inclusive level and with fiducial cuts, while the analogous HTurbo single-thread calculation with a similar accuracy took from 3 s (inclusive case) up to 30 s (fiducial case), yielding an improvement of one or two orders of magnitudes in the time performance. Finally, the fixed-order term of the cross section represents the most time-consuming part of the calculation; the computation at NLO (LO) with 1% (0.05%) accuracy required 30 min (1 min) both within HNNLO and HTurbo single-thread. However we observe that the fixedorder part of the calculation could be computed by using fast interpolation techniques [62,63]. In conclusion, we have presented the HTurbo numerical program which provides fast and numerically precise predictions for Higgs boson production through a new implementation of the HqT,HRes and HNNLO codes, following the improvements of the DYTurbo program [27] for Drell– Yan lepton pair production. HTurbo implements the fullydifferential fixed-order QCD calculation for Higgs boson production (via gluon fusion) and decay as well as the resummation of the large logarithmic corrections at small transverse momenta. The present version of the code reaches 123 492 Page 6 of 8 Eur. Phys. J. C (2022) 82 :492 (a) (b) Fig. 5 Comparison of fiducial differential cross sections computed with HRes and HTurbo at √s=13 TeV. The fiducial phase space is defined in the text. Absolute value of the asymptotic component of the transverse momentum distribution at LO (a)andNLO(b) accuracy. The top panels show absolute cross sections, and the bottom panels show ratios of HTurbo to HRes results (a) (b) Fig. 6 Comparison of fiducial differential cross sections computed with HRes and HTurbo at √s=13 TeV. The fiducial phase space is defined in the text. Fixed-order component of the transverse momentum distribution at LO (a)andNLO(b) accuracy. The top panels show absolute cross sections, and the bottom panels show ratios of HTurbo to HRes results. the next-to-next-to-leading order and next-to-next-to-leading logarithmic accuracy, and it includes the decay of the Higgs boson into two photons. The enhancement in performance of HTurbo over the previous programs (which reaches two orders of magnitude for the resummed term) is achieved by optimizing the code, factorizing the cross section into production and decay variables, and using numerical integration quadrature rules based on interpolating functions. The resulting cross-section predictions are in agreement with the results of the original programs. The great reduction in computing time for performing cross-sections calculation opens new possibilities for Higgs boson physics and facilitates an efficient inclusion of N3LO corrections along the lines of Refs. [33,35]. Acknowledgements This project has been supported by the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Grant agreement number 740006). LC is supported by the Generalitat Valenciana (Spain) through the plan GenT program (CIDEGENT/2020/011) and his work is supported by the Spanish Government (Agencia Estatal de Investigación) and ERDF funds from European Commission (Grant no. PID2020-114473GB-I00 funded by MCIN/AEI/10.13039/501100011033). Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theoretical study and has no associated experimental data.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes 123 Eur. Phys. J. C (2022) 82 :492 Page 7 of 8 492 were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. 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