We describe how a simulation of Higgs boson production accurate at
next-to-next-to-leading order and matched to a parton shower can be built by
combining the POWHEG and MiNLO methods and using HNNLO results as input.Comment: 6 pages, 2 figures. Submitted to the proceedings of the LC13
  Workshop, September 16-20, 2013, ECT*, Trento, Ital

Re, Emanuele

English

arXiv

We describe how a simulation of Higgs boson production accurate at next-to-next-to-leading order and matched to a parton shower can be built by combining the POWHEG and MiNLO methods and using Hnnlo as input

Re, E.

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2014-11747-9Colloquia: LC13IL NUOVO CIMENTO Vol. 37 C, N. 2 Marzo-Aprile 2014Reaching NNLOPS accuracy with POWHEG and MiNLOE. ReRudolf Peierls Centre for Theoretical Physics, Department of PhysicsUniversity of Oxford - Oxford, OX1 3NP, UKricevuto il 22 Gennaio 2014Summary. — We describe how a simulation of Higgs boson production accurateat next-to-next-to-leading order and matched to a parton shower can be built bycombining the POWHEG and MiNLO methods and using Hnnlo as input.PACS 12.38.Bx – Perturbative calculations.PACS 14.80.Bn – Standard-model Higgs bosons.1. – IntroductionDuring the last decade a major research effort in the Monte Carlo community hasbeen devoted to the development of NLOPS tools, i.e. tools that allow a matching ofnext-to-leading order (NLO) computations with parton showers (PS), thereby bringingNLO accuracy into standard Monte Carlo event generators [1]. Among many proposals,there are currently two well-established NLOPS approaches, namely POWHEG [2, 3] andMC@NLO [4], which have now become the methods of choice used by experimental collabo-rations in many searches being carried out at the LHC. Part of this success was possibledue to the progress in the automation of NLO computations, in the development of semi-automated or fully-automated NLOPS frameworks [5-8], as well as in the standardizationof well-defined interfaces [9, 10] between programs that operate different tasks.A topic of research that has received much attention during the last 2 years isthe merging of multiple NLOPS simulations for different jet multiplicities. These ad-vances represent the NLO generalization of well-established tree-level multileg mergingapproaches [11,12], and their relevance for future LHC phenomenology is clear, since theywill allow a significant improvement in the simulation of processes where a heavy systemis produced in association with multiple jets, which is the generic background for manynew-Physics searches. There have been several proposals aiming at this goal [13-20],among which the MiNLO approach [21,19].After a short review of the POWHEG and MiNLO approaches, I will describe how theircombination can be used to match NNLO computations with PS, and show recent resultsobtained for inclusive Higgs production [22].c© Societa` Italiana di Fisica 137138 E. RE1.1. POWHEG. – The POWHEG method is a prescription to interface NLO calculationswith parton shower generators avoiding double counting of real emissions and virtualcorrections. In the POWHEG formalism, the generation of the hardest emission is performedfirst, according to the distribution given by(1) dσ = B¯ (ΦB) dΦB[ΔR(pminT)+R (ΦR)B (ΦB)ΔR (kT (ΦR)) dΦrad],where B (ΦB) is the leading order contribution,(2) B¯ (ΦB) = B (ΦB) +[V (ΦB) +∫dΦrad R (ΦR)]is the NLO differential cross section integrated on the radiation variables while keepingthe Born kinematics fixed (V (ΦB) and R (ΦR) stand, respectively, for the virtual and thereal corrections), and ΔR (pT ) = exp[− ∫ dΦrad R(ΦR)B(ΦB) θ (kT (ΦR)− pT )]is the POWHEGSudakov. With kT (ΦR) we denote the transverse momentum of the emitted particle off aBorn-like kinematics ΦB , and, as usual, the cancellation of soft and collinear singularitiesis understood in the expression within the square bracket in eq. (2). Partonic eventswith hardest emission generated according to eq. (1) are then showered with a kT -vetoon following emissions. Subject to these conditions, it can be shown that such eventsexhibit the features typical of PS when the chosen observable probes the soft-collinearregions (Sudakov suppression), reproduce the exact fixed-order results in the regionswhere emissions are widely separated, and, crucially, they preserve NLO accuracy forinclusive observables. From the NLOPS-matching point of view, the more challengingprocesses currently described with this approach are 2→ 3 and 2→ 4 processes, with atmost 2 light jets at LO [23-26].For the benefit of the following discussion, the (unregulated) B¯ function of the stan-dard POWHEG simulation of H + 1 jet can be written schematically as(3) B¯ HJ = α3S(μR)[B + αSV (μR) + αS∫dΦradR],where we have made explicit the dependence of all terms upon αS and the renormalizationscale μR. It is also worth recalling that when one or more jets are present at LO (as inthe H +1 jet case) the associated B¯ function needs to be regulated from the divergencesarising when jets in the LO kinematics become unresolved [27]: as a consequence, astandard POWHEG simulation of H + 1 jet cannot be used to describe inclusive Higgsproduction.1.2. MiNLO. – It is known that a common issue present in multileg NLO computationsis the choice of the factorization (μF ) and renormalization scale: ultimately the problemis due to the fact that these computations are characterized by kinematical regimesinvolving several different scales, and, although some choices are clearly pathologic (asthey can lead for instance to negative cross sections), in general there is no procedure toa priori choose μR and μF , being the scale dependence of the result just an artefact oftruncating the perturbative expansion.The MiNLO procedure [21] was originally defined as a prescription to address thisissue, and it works by consistently including CKKW-like corrections into a standardREACHING NNLOPS ACCURACY WITH POWHEG AND MiNLO 139NLO computation. By clustering with a kT -measure the momenta of each phase-spacepoint occurring in the computation, one can define the “most-probable” branching historythat would have produced such a kinematics: the argument of each power of αS is thenfound from the transverse momentum of the splitting occurring at each nodal point ofthe skeleton built from clustering, and a prescription for μF is given as well. The resultis also corrected by means of Sudakov form factors (called MiNLO-Sudakov FFs in thefollowing) associated to internal lines, accounting for the large logarithms that arise whenthe clustered event contains well separated scales.Because of the presence of MiNLO-Sudakov FFs associated to the Born-like kinematics,the integration over the full phase space ΦB can be performed without generation cuts:a MiNLO-improved computation yields finite results also when jets in the LO kinematicsbecome unresolved. As a consequence, the MiNLO procedure can be used within thePOWHEG formalism to regulate the B¯ function for processes involving jets at LO, withoutusing external cuts or variants thereof.MiNLO-enhanced POWHEG simulations have been presented in refs. [21,19,28,29] and, inparticular, in the H +1 jet case, the master formula for generating the hardest emissioncontains the following B¯ functionB¯ HJ−MiNLO = α2S(MH)αS(qT )Δ2g(qT ,MH)(4)×[B(1− 2Δ(1)g (qT ,MH)) + αSV (μ¯R) + αS∫dΦradR],that should be contrasted with eq. (3). In eq. (4) qT is the Higgs transverse momentum(in the underlying-Born kinematics), MH is its virtuality, μ¯R is set to (M2HqT )1/3 in ac-cordance with the MiNLO prescription and Δg(qT , Q) = exp{−∫ Q2q2Tdq2q2αS(q2)2π[Ag log Q2q2 +Bg]}is the MiNLO-Sudakov FF associated to the jet present at LO. At NLL, the A1,g,A2,g and B1,g terms in the expansion of Ag and Bg need to be included [21]. Theterm in brackets multiplying B is needed to avoid double-counting of NLO factors:Δ(1)g (qT , Q) = −αS2π[12A1,g log2 Q2q2T+ B1,g log Q2q2T]corresponds to the O(αS) expansionof Δg.The B¯ function in eq. (4) can be integrated over the full phase space associated withthe “LO” jet, yielding a finite cross-section for inclusive Higgs production. The formalaccuracy of the result so obtained was carefully addressed in ref. [19], by means of acomparison with the NNLL qT -resummation of the Higgs transverse momentum. It wasfound that, in order to reach NLO accuracy for the total inclusive Higgs production, theNNLL B2,g term should be included in the MiNLO-Sudakov FF, and qT should be used asfactorization scale and as the argument of the power of αS associated to R, V and Δ(1)g(i.e. the power of αS where no argument was specified in eq. (4)). If such terms are notincluded properly, spurious terms of order α3.5S are generated upon integration over theentire Higgs pT spectrum, violating the requirement [dσHJ−MiNLO]integrated − σNLO(gg →H) = O(α4S), which is needed to claim NLO accuracy for fully-inclusive Higgs production.2. – Higgs production with NNLOPS accuracyThe H + 1 jet POWHEG implementation enhanced with the improved MiNLO procedurepreviously outlined can be used to reach NNLOPS accuracy. In fact, since such a simula-tion gives a NLO-accurate prediction of the Higgs rapidity (y), then the function W (y),140 E. REdefined as(5) W (y) =(dσ/dy)NNLO(dσ/dy)HJ−MiNLO,can be used to reweight each HJ-MiNLO-generated event, thereby obtaining a NNLOPSsimulation of inclusive Higgs production. By NNLOPS we mean a fully-exclusive MonteCarlo simulation of Higgs-production which is NNLO accurate when one is fully inclusiveon extra radiation, as well as LO (NLO) accurate for H + 2(1) jet observables [19, 22].Since we are reweighting with W , the Higgs rapidity is NNLO accurate by construction,whereas the NLO accuracy of the 1-jet region, inherited from the underlying HJ-MiNLOsimulation, is not spoiled, because the first non-controlled terms in the whole simulationare O(α5S): this follows from the fact that W (y) = 1 +O(α2S), as can be seen expandingnumerator and denominator in eq. (5).In ref. [22] the following generalization of eq. (5) was used:(6) W (y, pT ) = h(pT )∫dσNNLOδ(y − y(Φ))−∫dσBHJ−MiNLOδ(y − y(Φ))∫dσAHJ−MiNLOδ(y − y(Φ))+ (1− h(pT )) ,where we have split the HJ-MiNLO differential cross section among dσA = dσ h(pT ) anddσB = dσ (1−h(pT )), with h(pT ) = (βmH)2(βmH)2+p2T. The profiling function h controls wherethe NLO-to-NNLO correction is spread: as 2nd argument of W the transverse momentumof the leading jet was used, and we have chosen β = 1/2, which implies that the NNLOcorrecting factor W is effectively applied in the region pT  mH/2 (for pT  mH ,W (y, pT )→ 1). With the choice in eq. (6) one also has that (dσ/dy)NNLOPS reproduces(dσ/dy)NNLO exactly, without O(α5S) ambiguities.2.1. Results. – In our simulation, the central value for dσNNLO was obtained withHnnlo [30,31], setting μR = μF = mH/2. We refer to ref. [22] for details on how scaleswere varied to obtain uncertainty bands.In fig. 1 a comparison between our NNLOPS simulation and Hnnlo is shown: asexpected, the NNLOPS simulation reproduces extremely well the NNLO results for the10−210−1100101dσ/dy[pb]Ratioydσ/dy[pb]RatioNNLOPSHNNLO0.91.01.1-4 -3 -2 -1 0 1 2 3 410−210−1100101dσ/dy[pb]Ratioydσ/dy[pb]RatioHNNLONNLOPS0.91.01.1-4 -3 -2 -1 0 1 2 3 4Fig. 1. – Comparison of the NNLOPS (red) and Hnnlo (green) results for the Higgs fullyinclusive rapidity distribution. On the left (right) plot only the NNLOPS (Hnnlo) uncertaintyis displayed. The lower left (right) panel shows the ratio with respect to the NNLOPS (Hnnlo)prediction obtained with its central scale choice.REACHING NNLOPS ACCURACY WITH POWHEG AND MiNLO 14110−310−210−1100dσ/dpH T[pb/GeV]RatiopHT[GeV]dσ/dpH T[pb/GeV]RatioNNLOPSHQT0.61.01.40 50 100 150 200 250 30010−310−210−1100dσ/dpH T[pb/GeV]RatiopHT[GeV]dσ/dpH T[pb/GeV]RatioHQTNNLOPS0.61.01.40 50 100 150 200 250 300Fig. 2. – Comparison of the NNLOPS (red) with the NNLL+NNLO prediction of HqT (green)for the Higgs transverse momentum. In HqT we keep the resummation scale Qres always fixedto mH/2 and vary μR and μF . On the left (right), the NNLOPS (HqT) uncertainty band isshown. In the lower panel, the ratio to the NNLOPS (HqT) central prediction is displayed.Higgs rapidity both in the central value and in the uncertainty band obtained by scalevariation.Figure 2 shows the Higgs transverse momentum pHT . We compare our simulation withHqT [32,33], whose central value is obtained with Qres = mH/2 and μR = μF = mH/2.The HqT result corresponds to a NNLL prediction of pHT , matched to the fully inclusivecross section at NNLO. Here we notice that the two results are almost completelycontained within each other’s uncertainty band in the region of low-to-moderatetransverse momenta. The central values at small momenta also exhibit a very goodagreement, supporting our choice for β. The difference in the large-pT tail is not areason of concern, and it is expected since the two predictions use different scales atlarge pT , as explained in ref. [22].Finally, we also mention that a comparison among NNLOPS and NNLL+NNLOpredictions from JetVHeto [34] was successfully carried out for the jet veto efficiency,defined as the cross section for Higgs boson production events containing no jets withtransverse momentum greater than a given value (pT,veto), divided by the respectivetotal inclusive cross section. The central predictions of the two programs are never outof agreement by more than 5–6%, and the two sets of predictions lie within each other’serror bands essentially everywhere over all values of pT,veto, as shown in ref. [22].∗ ∗ ∗NNLOPS results presented here have been obtained in ref. [22], in collaboration withK. Hamilton, P. Nason and G. Zanderighi. The original proposal of reaching NNLOPSaccuracy from MiNLO-merged NLOPS simulations was outlined in ref. [19], which wasco-authored by C. Oleari. The author acknowledges G. Corcella and L. 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Reaching NNLOPS accuracy with POWHEG and MiNLO

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Reaching NNLOPS accuracy with POWHEG and MiNLO

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