NLO QCD corrections to off-shell top-antitop production with leptonic decays in association with a Higgs boson at the LHC
aa r X i v : . [ h e p - ph ] N ov Prepared for submission to JHEP S EPTEMBER
24, 2018
NLO QCD corrections to off-shell top–antitopproduction with leptonic decays in association witha Higgs boson at the LHC
Ansgar Denner, Robert Feger
Universität Würzburg, Institut für Theoretische Physik und Astrophysik, Emil-Hilb-Weg 22,97074 Würzburg, Germany
E-mail: [email protected] , [email protected] Abstract:
We compute the hadronic production of top–antitop pairs in association witha Higgs boson at next-to-leading-order QCD, including the decay of the top and antitopquark into bottom quarks and leptons. Our computation is based on full leading andnext-to-leading-order matrix elements for e + ν e µ − ¯ ν µ b ¯ bH ( j ) and includes all non-resonantcontributions, off-shell effects and interferences. Numerical results for the integrated crosssection and several differential distributions are given for the LHC operating at 13 TeV usinga fixed and a dynamical factorization and renormalization scale. The use of the dynamicalinstead of the fixed scale improves the perturbative stability in high-energy tails of mostdistributions, while the integrated cross section is hardly affected differing by only aboutone per cent and leading to almost the same K factor of about 1.17. ontents After the discovery of a new boson of a mass around 125 GeV by the CMS and ATLASCollaborations [1, 2] at the LHC, the determination of its quantum numbers and couplingsto other particles has become a high priority in particle physics. Results from the first runof the LHC strongly support the hypothesis that this particle is the Higgs boson predictedby the Standard Model (SM) of particle physics. In the Brout–Englert–Higgs symmetrybreaking mechanism of the SM the Higgs boson is the key for understanding the originof mass. Since in this framework the Higgs boson couples to fermions with a strengthproportional to their mass via Yukawa interactions, its coupling to the heaviest quark, thetop quark, is of particular interest. The main production mechanism of the Higgs boson inthe SM is gluon fusion, gg → H. This process is sensitive to the top-quark Yukawa coupling,but possible heavy particles beyond the SM running in the loop bias its determination. Onthe other hand, the production of a SM Higgs boson in association with a top-quark pairallows a direct access of the top-quark Yukawa coupling already at tree level, disentanglingit from possible beyond-SM contributions.Leading-order (LO) predictions for t ¯ tH production for stable Higgs boson and topquarks have been presented in Refs. [3–7]. Since LO predictions of QCD processes suf-fer from large perturbative uncertainties higher-order corrections have to be taken intoaccount for adequate theoretical predictions. The cross section for t ¯ tH production at next-to-leading-order (NLO) QCD is known for more than 10 years [8–12]. Meanwhile, NLOQCD corrections have been matched to parton showers [13–15] and recently electroweak– 1 –orrections to t ¯ tH production have been computed [16–18]. NLO QCD corrections forthe important background processes t ¯ tb ¯ b and t ¯ tjj production have been worked out inRefs. [19–22] and Refs. [23–25], respectively, and matched to parton showers in Refs. [26–28]and Ref. [29].Several searches for the associated production of the Higgs boson with a top-quark pairfor a variety of decay channels have been published by ATLAS [30–34] and CMS [35–40].These searches are challenging due to a large background from t ¯ tb ¯ b and t ¯ tjj production, andso far no evidence of a t ¯ tH signal over background has been found. The current ratio of themeasured t ¯ tH signal cross section to the SM expectation quoted by ATLAS is µ = 1 . ± . [34] and by CMS µ = 1 . +1 . − . [40] for a Higgs-boson mass of 125 GeV.In this article we present the calculation of the NLO QCD corrections to the hadronicproduction of a positron, a muon, missing energy, two b jets and a SM Higgs boson, which weassume to be stable (pp → e + ν e µ − ¯ ν µ b ¯ bH) at the 13 TeV LHC, which includes the resonantproduction of a top–antitop-quark pair in association with a Higgs boson with a subsequentleptonic decay of the top and the antitop quark. Our calculation includes all NLO QCDcorrection effects in t ¯ tH production and top decays and also takes into account all off-shell,non-resonant and interference effects of the top quarks. We consider the fixed renormal-ization and factorization scale used in Ref. [9] and alternatively the dynamical scale choicefrom Ref. [13] and investigate their quality in reducing the dependence of the integratedcross section as well as differential distributions on the factorization and renormalizationscales. The phase-space integration is performed with a newly implemented in-house multi-channel Monte Carlo program, using phase-space mappings similar to Ref. [41]. The MonteCarlo implements the dipole subtraction method [42–45] for the computation of the realcorrections and is linked to the matrix-element generator Recola [46] for the computationof the LO and NLO matrix elements as well as colour- and spin-correlated squared matrixelements needed for the evaluation of subtraction terms.The calculation follows in many respects the one of pp → e + ν e µ − ¯ ν µ b ¯ b in Ref. [47].In particular, since the additional Higgs boson in the final state has no colour charge,the contributing Catani–Seymour dipoles [42, 44] are the same. For the hadronic processpp → e + ν e µ − ¯ ν µ b ¯ bH we find a qualitative similar behaviour of NLO corrections to theintegrated cross section as well as for differential distributions of the same observables.Where appropriate we compare results and point out differences. Moreover, we compareour results for the total cross sections to those of existing calculations for on-shell Higgsboson and top quarks [9, 13].The paper is organised as follows. In Section 2 we discuss details of the calculationsuch as contributing subprocesses and Feynman diagrams and explain some technical as-pects of the real (Section 2.1) and virtual (Section 2.2) corrections. We present numericalresults for the LHC operating at √ s = 13 TeV in Section 3. In Section 3.1 we list and ex-plain the input parameters, jet definition, cuts and scale choices for our calculation, whileresults for integrated cross sections and a discussion of the scale dependence are providedin Section 3.2. In Section 3.4 we display and discuss several differential distributions for afixed and a dynamical scale choice. We have performed several checks of our calculationwhich we present in Section 4. Our conclusions are given in Section 5.– 2 – a) gg t Ht t bW + e + ν e ¯t ¯bW − ¯ ν µ µ − (b) gg g t Ht bW + e + ν e ¯b¯t W − ¯ ν µ µ − (c) u¯u g t Ht bW + e + ν e ¯b¯t W − ¯ ν µ µ − (d) u¯u g b H¯b W − W − ¯ ν µ µ − t bW + e + ν e (e) gg g b H¯b t t W − b¯ ν µ µ − W + e + ν e (f) gg bt Ht t bW + e + ν e ¯bW − ¯ ν µ µ − (g) gg bb b¯bW − ¯ ν µ µ − HZ Z W + ν e e + (h) gg bttb b¯bW − ¯ ν µ µ − HW + ν e e + (i) u¯u dd HW − W − µ − ¯ ν µ g ¯bbW + ν e e + Figure 1 : Representative tree-level Feynman diagrams with (a)–(c) two (top), (d)–(f) one(middle) and (g)–(i) no top-quark resonances (bottom).
We compute the QCD corrections to the full hadronic processpp → e + ν e µ − ¯ ν µ b ¯ bH . (2.1)We consider the tree-level amplitude at O (cid:0) α s α / (cid:1) including all resonant, non-resonant,and off-shell effects of the top quarks and all interferences. Neglecting flavour mixing aswell as contributions from the suppressed bottom-quark parton densities and counting u, d,c and s quarks separately, we distinguish 5 partonic channels for the LO hadronic process:the gluon-induced process gg → e + ν e µ − ¯ ν µ b ¯ bH and four processes from q ¯ q → e + ν e µ − ¯ ν µ b ¯ bHby substituting different quark flavours ( q = u , d , c , s). Throughout this paper we considerthe bottom quark massless, implying no contribution from tree diagrams involving theHiggs–bottom-quark coupling. The gg process involves 236 and the q ¯ q processes 98 treediagrams each under these prerequisites. In Figure 1 we show sample diagrams grouped bythe number of top-quark resonances. – 3 – a) gg tt H bt W + e + ν e ¯t ¯bW − ¯ ν µ µ − g (b) gg g t Ht bW + e + ν e ¯b¯t W − ¯ ν µ µ − g (c) u u¯u g Ht bW + e + ν e ¯b¯b¯t¯t W − ¯ ν µ µ − g Figure 2 : Representative hexagon and heptagon one-loop Feynman diagrams with twotop-quark resonances.
The real correction process pp → e + ν e µ − ¯ ν µ b ¯ bHj receives contributions from the 13 partonicsubprocesses gg → e + ν e µ − ¯ ν µ b ¯ bHg ,q ¯ q → e + ν e µ − ¯ ν µ b ¯ bHg , g q → e + ν e µ − ¯ ν µ b ¯ bH q, g ¯ q → e + ν e µ − ¯ ν µ b ¯ bH ¯ q, (2.2)where the gg process involves 1578 tree diagrams and the q ¯ q , g q and g ¯ q processes, all relatedby crossing symmetry, 614 tree diagrams each.Gluon Bremsstrahlung in the real corrections gives rise to IR divergences by soft orcollinear configurations, which cancel for the final state for infrared-safe observables uponcombination with the virtual corrections. Singularities from collinear initial-state split-ting factorize and can be removed by MS redefinition of the parton distribution functions.We employ the Catani–Seymour subtraction formalism [42, 44] for the regularization andanalytical cancellation of IR singularities. Both the amplitudes for the real-correction sub-processes as well as the colour and spin-correlated amplitudes of the subtraction terms havebeen calculated with Recola . The partonic subprocesses for the virtual QCD corrections can be identified with thoseat LO. We compute the virtual corrections in the ’t Hooft–Feynman gauge, where thegg process involves 9074 loop diagrams and the q ¯ q processes 2404 loop diagrams each.The most complicated one-loop diagrams are heptagons (Sample diagrams are displayed inFigure 2).The resonant top quarks, Z bosons and W bosons are treated in the complex-massscheme [48–50], where the masses of unstable particles are consistently treated as complexquantities leading in particular to a complex weak mixing angle, µ W = M W − i M W Γ W , µ Z = M Z − i M Z Γ Z , cos θ w = µ W µ Z . (2.3)For the renormalization we use the on-shell renormalization scheme as described in Ref. [49]for the complex-mass scheme. – 4 –or the computation of the matrix elements for the virtual corrections we employ Recola [46] in dimensional regularisation, which integrates the
Collier [51, 52] libraryfor the numerical evaluation of one-loop scalar [53–56] and tensor integrals [57–59]. We com-pared our results for the virtual NLO contribution to the squared amplitude, M ∗ M ,for many phase-space points with MadGraph5_aMC@NLO [60] (see Section 4 for details).
We present results for integrated cross sections and differential distributions for the LHCoperating at √ s = 13 TeV. For the computation of the hadronic cross section we employLHAPDF 6.05 with CT10NLO parton distributions at LO and NLO QCD. We use the valueof the strong coupling constant α s as provided by LHAPDF based on a one-loop (two-loop)accuracy at LO (NLO) with N F = 5 active flavours. In the renormalization of α s the top-quark loop in the gluon self-energy is subtracted at zero momentum. The running of α s inthis scheme is generated by contributions from light-quark and gluon loops only. For thefixed renormalization and factorization scale µ fix = 236 GeV, defined below in (3.10), wefind α s ( µ fix ) = 0 . . . . . (3.1)We neglect contributions from the suppressed bottom-quark parton density.The electromagnetic coupling α is derived from the Fermi constant in the G µ scheme[61], α = √ π G µ M W (cid:18) − M W M Z (cid:19) , G µ = 1 . × − GeV . (3.2)We compute the width of the top quark Γ t for unstable W bosons and massless bottomquarks according to Ref. [62]. At NLO QCD it reads Γ NLOt = G µ m t √ π M W Z d y γ W (1 − y/ ¯ y ) + γ W (cid:18) F ( y ) − α s π F ( y ) (cid:19) (3.3)with γ W = Γ W /M W , ¯ y = ( M W /m t ) , α s = α s ( m t ) | NLO and F ( y ) = 2(1 − y ) (1 + 2 y ) (3.4) F ( y ) = 2(1 − y ) (1 + 2 y ) (cid:2) π + 2 Li ( y ) − Li (1 − y ) (cid:3) + 4 y (1 − y − y ) ln( y ) + 2(1 − y ) (5 + 4 y ) ln(1 − y ) − (1 − y )(5 + 9 y − y ) . (3.5)For the top-quark width at LO we neglect the correction term − α s F ( y ) / (3 π ) .As input, we employ the following numerical values for the masses and widths: m t = 173 GeV , Γ LOt = 1 . . . . GeV , Γ NLOt = 1 . . . . GeV ,M OSZ = 91 . GeV , Γ OSZ = 2 . GeV ,M OSW = 80 . GeV , Γ OSW = 2 . GeV ,M H = 126 GeV , (3.6)– 5 –hich includes the measured value of the Higgs-boson mass with zero width since we assumeit to be stable. We neglect the masses and widths of all other quarks and leptons.We convert the measured on-shell values (OS) for the masses and widths of the W andZ boson into pole values for the gauge bosons ( V = W , Z) according to Ref. [63], M V = M OS V / q OS V /M OS V ) , Γ V = Γ OS V / q OS V /M OS V ) , (3.7)that enter the calculation.We use the anti- k T algorithm [64] for the jet reconstruction with a jet-resolution pa-rameter R = 0 . . The distance between two jets i and j in the rapidity–azimuthal plane isdefined as R ij = q ( φ i − φ j ) + ( y i − y j ) , (3.8)with the azimuthal angle φ i and the rapidity y i = ln E + p z E − p z of jet i , where E is the energyand p z the component of momentum along the beam axis. Only final-state quarks andgluons with rapidity | y | < are clustered into infrared-safe jets.After recombination we impose standard selection cuts on transverse momenta andrapidities of charged leptons and b jets, missing transverse momentum and distance betweenb jets according to (3.8). We require two b jets and two charged leptons in the final state,with bottom quarks in jets leading to b jets, andb jets: p T , b > GeV , | y b | < . , charged lepton: p T ,ℓ > GeV , | y ℓ | < . , missing transverse momentum: p T , miss > GeV , b-jet–b-jet distance: ∆ R bb > . . (3.9)We have identified the renormalization scale with the factorization scale µ = µ R = µ F and have considered a fixed reference scale set to half the partonic threshold energy for t ¯ tHproduction according to Ref. [9]: µ fix = µ R = µ F = 12 (2 m t + m H ) = 236 GeV . (3.10)Alternatively, we use a dynamical scale following Ref. [13] µ dyn = µ R = µ F = (cid:0) m T , t m T , ¯ t m T , H (cid:1) with m T = q m + p T , (3.11)which corresponds to the geometric mean of the top-quark, antitop-quark and Higgs-bosontransverse masses. For the comparison of the scale choices we compute the logarithmicscale average ¯ µ dyn of the dynamical scale, defined as ln ¯ µ dyn = R ln( µ dyn ) d σ R d σ . (3.12)The scale uncertainty of the LO and NLO cross section is determined by variation ofthe renormalization and factorization scales µ R and µ F around the central value µ = µ fix – 6 – ch. σ LO [fb] σ NLO [fb] K µ dyn gg . +33 . − . . +8 . − . q ¯ q . +24 . − . . +17 . − . q ( ) . +295% − pp . +30 . − . . +0 . − . µ fix gg . +33 . − . . +8 . − . q ¯ q . +24 . − . . +17 . − . q ( ) . +310% − pp . +31 . − . . +0 . − . Table 1 : Composition of the integrated cross section for pp → e + ν e µ − ¯ ν µ b ¯ bH ( j ) at theLHC at √ s = 13 TeV with both the dynamical ( µ dyn ) and the fixed scale ( µ fix ) as denotedin column one. In column two we list the partonic initial states, where q = u , d , c , s and q ( ) = q, ¯ q . The third and fourth column give the integrated cross sections in fb for LOand NLO, resp. The upper and lower variations correspond to the envelope of seven scalepairs ( µ R /µ , µ F /µ ) = (0 . , . , (0 . , , (1 , . , (1 , , (1 , , (2 , , (2 , . The last columnprovides the K factor with K = σ NLO /σ LO .and µ = µ dyn for the fixed and dynamical scale choice, respectively.. While varying therenormalization scale in PDFs and matrix elements, the top-quark width remains fixedas computed at the top-quark mass according to (3.3). For the investigation of the scaledependence of the integrated cross section in Figure 3 we vary the scale µ up and downby a factor of eight for the LO and NLO integrated cross section for the three cases: 1) µ R = µ F = µ , 2) µ R = µ , µ F = µ , 3) µ F = µ , µ R = µ . While we show all three cases forthe dynamical scale, we show only the first case for the fixed scale choice in Figure 3. For allother results, i.e. those in Table 1 and Figures 5–7, the scale uncertainties are determinedfrom factor-two variations as follows. We compute integrated and differential cross sectionsat seven scale pairs, ( µ R /µ , µ F /µ ) = (0 . , . , (0 . , , (1 , . , (1 , , (1 , , (2 , , (2 , .The central value corresponds to ( µ R /µ , µ F /µ ) = (1 , and the error band is constructedfrom the envelope of these seven calculations. In Table 1 we present the integrated cross sections with fixed (3.10) and dynamical scale(3.11) at the LHC at √ s = 13 TeV corresponding to the input parameters (3.1), (3.2) and(3.6) and the cuts as defined in (3.9). The results include only contributions of O (cid:0) α s α / (cid:1) for LO amplitudes and the corresponding O ( α s ) QCD corrections. We neglect possiblecontributions to q ¯ q processes of O (cid:0) α / (cid:1) for LO amplitudes, which we determined to beabout 2 per mille of the integrated cross section at LO for the setup described above. Wealso do not include partonic channels with incoming bottom quarks. At LO and using the– 7 –
18 14 12 σ [f b ] µ / µ LO µ = µ R = µ F LO µ = µ R , µ F = µ dyn LO µ = µ F , µ R = µ dyn NLO µ = µ R = µ F NLO µ = µ R , µ F = µ dyn NLO µ = µ F , µ R = µ dyn LO fix µ = µ R = µ F NLO fix µ = µ R = µ F Figure 3 : Scale dependence of the LO andNLO integrated cross section at the TeVLHC. The renormalization and factorizationscales are varied around the central valuesof the fixed ( µ = µ fix , dash-dotted lines)and dynamical scale ( µ = µ dyn , solid lines).For the dynamical scale the variation with µ R while keeping µ F = µ dyn fixed and viceversa is shown with dashed lines. ¯ σ [f b ] Γ t [GeV] LONLO2.22.32.42.52.62.7 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Figure 4 : Zero-top-width extrapolation ofthe LO and NLO cross section at the LHCat √ s = 13 TeV for fixed scale µ = µ fix .dynamical scale (3.11), these contribute . × − fb , i.e. . , to the integrated crosssection. Moreover, we calculated the integrated cross section at LO with finite bottom-quarkmasses, leading to a reduction of the cross section for our set of cuts by 0.03%.The use of the dynamical scale instead of the fixed scale increases the LO and NLOcross sections by only about 1 %, and the K factor is 1.172 for the dynamical scale and1.176 for the fixed scale. The similar quality of both scale choices is also supported by thelogarithmic scale average of the dynamical scale as defined in (3.12). With ¯ µ dyn = 222 . GeVit corresponds to a slight effective decrease of the fixed scale by only about 6 %.Since integrated cross sections and NLO effects are very similar, the following con-siderations hold true for both, the dynamical and fixed scale: The major contributionsto the cross section originate from the gluon-fusion process, with about 70 % at LO whileincreasing at NLO to 76 %. The contribution of the quark–antiquark annihilation dropsfrom about 30 % at LO to 19 %. At NLO the gluon–(anti)quark induced real-radiationsubprocesses contribute about 5 % to the integrated cross section. The inclusion of NLOQCD corrections reduces the scale dependence from 31 % to 5 %.We display the dependence of the integrated LO (blue) and NLO (red) cross sectionson the values of the fixed and dynamical scale in Figure 3. Solid lines for the dynamicalscale and dash-dotted lines for the fixed scale show the scale dependence for a simultaneousvariation of the renormalization and factorization scales and dashed lines the individual– 8 –ariation, where one of the scales is kept fix at the central value, for the dynamical scale only.While the largest scale variation is obtained when both scales are changed simultaneously,the smallest effect results if only the factorization scale is varied. The cross sections for thefixed and dynamical scale choices are uniformly shifted relative to each other by about 1 %as for the central scale µ both for LO and NLO except for µ < µ / , where the fixed scaleleads to a faster decrease of the cross section with µ as the dynamical scale. For the fixedand dynamical scale the maximum of the NLO cross section is near µ ≃ µ , justifying theuse of both scale choices to be stable against scale variations. The K factor equals one atthe slightly lower scale of about µ ≃ . µ To determine the effects of non-resonant and off-shell top-quark contributions on the in-tegrated cross section we perform a numerical extrapolation to the zero-top-width limit, Γ t → . To this end we plot ¯ σ LO/NLO (Γ t ) = σ LO/NLO (Γ t ) Γ t Γ LO/NLOt ! (3.13)in the range ≤ Γ t ≤ Γ LO/NLOt , where Γ LO/NLOt is the top-quark width at LO and NLO,resp., and extrapolate linearly to Γ t → , using a linear regression based on the computedLO and NLO integrated cross sections, as shown in Figure 4. The factor (Γ t / Γ LO/NLOt ) re-stores the physical top-decay branching fraction. Finite-top-width effects can be extractedby comparing the results for ¯ σ LO/NLO (Γ t → to σ LO/NLO (Γ LO/NLOt ) . At the LHC at √ s = 13 TeV for fixed scale µ = µ fix finite-top-width effects shift the LO and NLO crosssection by − . ± .
01 % and − . ± .
22 % , respectively, which are within the expectedorder of Γ t /m t . The strong suppression of finite-top-width effects is related to the require-ment of a final state with two hard b jets. Finite-width effects are much more sizable incalculations where phase-space regions allowing for associated single-top plus W-boson pro-duction are included [65]. Such calculations require massive bottom quarks to regularizecollinear singularities. In this section we present various differential distributions with two plots for each observ-able: The upper plot showing the LO (blue, dashed) and NLO (red, solid) predictions withuncertainty bands from the envelope of scale variations by seven pairs, ( µ R /µ , µ F /µ ) =(0 . , . , (0 . , , (1 , . , (1 , , (1 , , (2 , , (2 , . The lower plot displays the LO (blue)and NLO (red) predictions normalized to the LO results at the central scale, i.e. K LO = d σ LO ( µ ) / d σ LO ( µ ) and K NLO = d σ NLO ( µ ) / d σ LO ( µ ) . Thus, the central red curve corre-sponds to the usual NLO correction factor ( K factor), defined as K = σ NLO ( µ ) /σ LO ( µ ) .The blue band shows the relative scale uncertainty of the LO differential cross section.Most of the displayed differential distributions were obtained using the dynamical scale µ dyn , except for Figure 5 which illustrates the effect of the fixed-scale choice on transverse-momentum distributions. – 9 – a) d σ d p T , e + (cid:2) f b G e V (cid:3) K f ac t o r p T , e + [GeV] LONLO10 − − − − − (b) d σ d p T , b1 (cid:2) f b G e V (cid:3) K f ac t o r p T , b [GeV] LONLO10 − − − − Figure 5 : Transverse-momentum distributions at the LHC at √ s = 13 TeV for fixed scale µ = µ fix : (a) for the positron (left) and (b) for the harder b jet (right). The lower panelsshow the K factor.Where appropriate we compare NLO effects in differential distributions to those of therelated process pp → e + ν e µ − ¯ ν µ b ¯ b presented in Ref. [47]. In general we find similar NLOeffects for most of the distributions, but being often more distinct for pp → e + ν e µ − ¯ ν µ b ¯ b inRef. [47].Figures 5a and 5b display the transverse-momentum distributions of the positron andthe harder b jet, resp., for the fixed scale µ = µ fix . The K factor of the transverse-momentum of the positron drops by about 50 % within the plotted range. In the high- p T tail the NLO predictions move outside the LO band with a scale variation of almost thesame size as the LO one. The K factor of the transverse-momentum of the harder b jetexhibits the same tendency, but not as drastic as for the positron. It moves outside the LOband for p T < GeV with larger scale variation as at the average p T of around 90 GeV.In Figure 6 we collect several transverse momentum distributions obtained using thedynamical scale µ = µ dyn . Figures 6a and 6c show the transverse-momentum distributionsof the positron and the harder b jet, resp., to compare with the fixed-scale distributionsin Figures 5a and 5b described above: They show clearly that the dynamical-scale choiceimproves the perturbative stability. The K factor changes only slightly (within 20 %) overthe displayed range, and the NLO band lies within the LO band. The residual scale variationis at the level of
10 % at NLO.In Figure 6b the distribution of missing transverse momentum, defined as p T , miss = (cid:12)(cid:12) p T ,ν e + p T , ¯ ν µ (cid:12)(cid:12) , is shown. The K factor rises for p T , miss & GeV up to about 1.5.Figures 6c and 6d display the distribution of the transverse momentum of the harderand softer b jet, resp. While the K factor for the harder b jet exhibits a minimum at themaximum of the distribution and slightly rises towards its tail, the K factor of the softerb jet decreases by about 30 % in the plotted range.The distribution of the transverse momentum of the b-jet pair in Fig. 14 of Ref. [47]exhibits a strong suppression of the t ¯ t cross section at LO above p T , b ¯ b & GeV. This– 10 – a) d σ d p T , e + (cid:2) f b G e V (cid:3) K f ac t o r p T , e + [GeV] LONLO10 − − − − − (b) d σ d p T , m i ss (cid:2) f b G e V (cid:3) K f ac t o r p T , miss [GeV] LONLO10 − − − − − (c) d σ d p T , b1 (cid:2) f b G e V (cid:3) K f ac t o r p T , b [GeV] LONLO10 − − − − (d) d σ d p T , b2 (cid:2) f b G e V (cid:3) K f ac t o r p T , b [GeV] LONLO10 − − − − − (e) d σ d p T , b1b2 (cid:2) f b G e V (cid:3) K f ac t o r p T , b b [GeV] LONLO10 − − − − (f) d σ d p T , H (cid:2) f b G e V (cid:3) K f ac t o r p T , H [GeV] LONLO10 − − − − Figure 6 : Transverse-momentum distributions at the LHC at √ s = 13 TeV for dynamicalscale µ = µ dyn : (a) for the positron (upper left), (b) for missing energy (upper right),(c) for the harder b jet (middle left), (d) for the softer b jet (middle right), (e) for theb-jet pair (lower left) and (f) for the Higgs boson (lower right). The lower panels show the K factor. – 11 – a) d σ d M t ¯ t H (cid:2) f b G e V (cid:3) K f ac t o r M t¯tH [GeV] LONLO10 − − − (b) d σ d M b1b2 (cid:2) f b G e V (cid:3) K f ac t o r M b b [GeV] LONLO10 − − − − − (c) d σ d c o s θ e + µ − [ f b ] K f ac t o r cos θ e + µ − LONLO0.80.911.11.21.31.41.51.61.70.811.21.4 -1 -0.5 0 0.5 1 (d) d σ d φ e + µ − (cid:2) f b ◦ (cid:3) K f ac t o r φ e + µ − [ ◦ ]LONLO0.0050.010.0150.020.0250.030.811.21.4 0 20 40 60 80 100 120 140 160 180 (e) d σ d y t [ f b ] K f ac t o r y t LONLO00.20.40.60.811.20.811.21.4 -3 -2 -1 0 1 2 3 (f) d σ d y H [ f b ] K f ac t o r y H LONLO00.20.40.60.811.20.811.21.4 -3 -2 -1 0 1 2 3
Figure 7 : Differential distributions at the LHC at √ s = 13 TeV for dynamical scale µ = µ dyn : invariant mass of (a) the t ¯ tH system (upper left) and (b) the b-jet pair (upperright), (c) the cosine of the angle between the positron and the muon (middle left), (d) theazimuthal angle between the positron and the muon in the transverse plane (middle right),the rapidity of (e) the top quark (lower left) and of (f) the Higgs boson (lower right). Thelower panels show the K factor. – 12 –s due to the fact that in narrow-top-width approximation the b quarks are boosted viatheir parent top and antitop quark, which have opposite transverse momenta resulting in asuppression of a b ¯ b system with high p T at LO. The lesser stringent kinematical constraintsat NLO result in an enhancement of the cross section and thus a large K factor for high p T , b ¯ b . For the t ¯ tH production at hand a Higgs boson acquiring transverse momentumsoftens the kinematical constraint already at LO leading to smaller NLO corrections forhigh p T , b ¯ b , which can be seen in Figure 6e. Furthermore, the K factor of the distributionresembles the K factor of the missing transverse momentum due to a kinematically similarconfiguration, but with a stronger increase to a value of 1.8 at p T ≃ GeV.Figure 6f displays the transverse-momentum distribution of the Higgs boson. The av-erage p T of the Higgs boson is around 70 GeV. The cross section decreases more moderatelywith p T in the plotted range than for other transverse momentum distributions presentedin Figure 6.In Figure 7 we present further differential distributions for other types of observables:the invariant mass of the t ¯ tH and b b system in Figure 7a and Figure 7b, resp., thecosine of the angle between the two charged leptons in Figure 7c, the azimuthal anglein the transverse plane between them in Figure 7d, and the rapidity of the top quarkand the Higgs boson in Figure 7e and Figure 7f, resp. Large NLO corrections appear inthe M t ¯ tH distribution below the t ¯ tH threshold. These arise dominantly from real gluonbremsstrahlung contributions, where the emitted gluon is not recombined with the bottomquarks and thus does not contribute to M t ¯ tH . The distribution of the azimuthal anglein the transverse plane between the two charged leptons exhibits sizeable NLO effects forsmall angles similarly as the distribution in the cosine of the angle between the two chargedleptons. The K factor varies by
40 % for these distributions. The rapidity distributionof the Higgs boson features about the typical NLO effects in the central detector region,which disappear going into forward or backward direction. The K factor of the top-quarkrapidity distribution on the other hand is almost flat at the level of the NLO effects of theintegrated cross section. We have performed several comparisons and consistency checks of our calculation. Wehave reproduced the LO hadronic cross sections with
MadGraph5_aMC@NLO [60] usingthe fixed scale choice. We also compared the virtual NLO contribution to the squaredamplitude, M ∗ M , for the gg and ¯ uu subprocesses computed by Recola [46] for manyphase-space points with
MadGraph5_aMC@NLO . In Figure 8a we plot the cumulativefraction of events with a relative difference larger than ∆ between M ∗ M obtained by Recola and
MadGraph5_aMC@NLO . The median of agreement is about − for the ggsubprocess and roughly × − for the ¯ uu subprocess. The agreement is worse than − for less than 0.3 % of gg- and less than 0.04 % of ¯ uu-subprocess events. For the gg process wechecked the accuracy of virtual matrix elements in satisfying the Ward identity, by replacingthe polarization vector of one of the initial-state gluons in the one-loop amplitude M byits momentum normalized to its energy: ǫ µ g → p µ g /p g . In Figure 8b we plot the cumulative– 13 – a) fr ac ti ono f e v e n t s maximum agreement ∆ ¯uugg10 − − − − − − − − − − − (b) f r a c t i o n o f e v e n t s maximum accuracy ∆10 − − − − − − − − − − Figure 8 : Checks of the virtual contributions for fixed scale µ = µ fix : (a) agree-ment between Recola and
MadGraph5_aMC@NLO for the partonic subprocesses ¯ uu → e + ν e µ − ¯ ν µ b ¯ bH and gg → e + ν e µ − ¯ ν µ b ¯ bH (left), (b) accuracy in satisfying the Ward identityfor gg → e + ν e µ − ¯ ν µ b ¯ bH (right), both showing the cumulative fraction of events with arelative difference/accuracy larger than ∆ .fraction of events with M ∗ ( ǫ g ) M ( ǫ g → p g /p g ) / M ∗ ( ǫ g ) M ( ǫ g ) > ∆ for virtualevents obtained by Recola with a median of about − .With the Monte Carlo code we developed for the calculation of the process pp → e + ν e µ − ¯ ν µ b ¯ bH, which employs Recola for the evaluation of matrix elements, we reproducedthe results of Ref. [47] for the closely related process pp → e + ν e µ − ¯ ν µ b ¯ b for the LHC at √ s =8 TeV using a dynamical scale. Since the Catani–Seymour dipoles [42, 44] are the same forthe process without the Higgs boson, this comparison allows us to test the implementationof the subtraction formalism. As we use the same renormalization procedure as in Ref. [47],this is also verified via this comparison. In Ref. [47] the one-loop matrix elements and the I -operator are computed in double-pole approximation for the two W-boson resonances usingphysical (i.e. real) W- and Z-boson masses and Γ W = Γ Z = 0 . Moreover, the M H → ∞ limit is adopted, i.e. closed fermion loops involving top quarks coupled to Higgs bosons areneglected. In contrast, the new code computes the full one-loop matrix elements and the I -operator without any approximation. The largest difference between both calculationsresults from the use of the double-pole approximation for the virtual corrections and is oforder α s Γ W / ( πM W ) ∼ . . We could reproduce the results of Ref. [47] for the integratedcross section at NLO within (2 − σ , which is at the level of 0.5 %, thus confirming bothcalculations basically within the accuracy of the numerical integration.We compared our predictions for the NLO total hadronic cross section with compu-tations of t ¯ tH production without decays of the top quarks at NLO for fixed [9] and dy-namical scale [13]. To this end we performed NLO computations using the references’parameters and PDF mappings and a minimal set of cuts on the decay products of thetop and antitop in our process. Since our calculation includes background processes tot ¯ tH production, a minimal set of cuts has to be maintained to ensure infrared safety: toavoid large contributions from possible collinear events with bottom quarks in forward di-rection (as induced by diagrams shown in Figure 1g or Figure 1h) we kept a small b-jet– 14 –ransverse-momentum cut ( p T , b > GeV) and a small b-jet distance cut ( ∆ R bb > . ) toavoid singular events from gluon splitting (g → b ¯ b) as in Figure 1i. We multiply appropri-ate branching ratios to the results of Refs. [9, 13] in the narrow-top-width approximation(NtWA), i.e. R d σ NtWA = σ t ¯ tH BR t → i BR ¯ t → j , and apply the NLO matching prescription ofSection 2.1.2 in Ref. [47] to our results. Thus, we find agreement of our NLO predictionswith Ref. [9] and Ref. [13] within 1 %, which is of the expected order of Γ t /m t , since ourcalculation includes off-shell and non-resonant top-quark effects. In this article we have presented the calculation of the next-to-leading-order QCD correc-tions to off-shell top–antitop production in association with a Higgs boson with leptonicdecay of the top quarks at the LHC, including all resonant, non-resonant, and off-shelleffects of top quarks as well as all interferences. For the computation of leading- and next-to-leading-order matrix elements we have utilised the recursive matrix-element generator
Recola linked to the one-loop integral library
Collier . The phase-space integrationhas been performed with an in-house multi-channel Monte-Carlo code that implements thedipole subtraction formalism.We provided integrated cross sections and several differential distributions for a 13 TeVLHC using a fixed and a dynamical scale that have both been used in the literature for thecomputation of NLO QCD corrections of t ¯ tH production with a stable Higgs boson andstable top quarks. We find almost the same integrated cross sections and scale dependencefor both scale choices at leading order as well as next-to-leading order QCD, with a similar K factor of 1.172 and 1.176 for the dynamical and fixed scale choice, resp. However, theuse of the dynamical scale instead of the fixed scale improves the perturbative stability inhigh-energy tails of most distributions, especially those of transverse momenta. Using thedynamical scale, we find K factors in the range . − . and residual scale uncertainties arethe level or
10 % for distributions. For the integrated cross section an extrapolation to thezero-top-width limit has been performed, indicating non-resonant and off-shell top-quarkeffects below one per cent. While this effect is small, our calculation is also the first oneto include NLO correction effects in the top–antitop–Higgs-boson production and the topdecay processes.Besides its phenomenological relevance, this calculation demonstrates the power of thetools
Recola and
Collier in performing complicated NLO calculations.
Acknowledgments
This work was supported by the Bundesministerium für Bildung und Forschung (BMBF)under contract no. 05H12WWE. – 15 – eferences [1]
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