t\bar{t}H production at NNLO: the flavour off-diagonal channels
Stefano Catani, Ignacio Fabre, Massimiliano Grazzini, Stefan Kallweit
ZZU-TH 3/21ICAS 061/21 t ¯ tH production at NNLO:the flavour off-diagonal channels Stefano Catani ( a ) , Ignacio Fabre ( b,c ) , Massimiliano Grazzini ( b ) , and Stefan Kallweit ( d )( a ) INFN, Sezione di Firenze and Dipartimento di Fisica e Astronomia,Universit`a di Firenze, I-50019 Sesto Fiorentino, Florence, Italy ( b ) Physik Institut, Universit¨at Z¨urich, CH-8057 Z¨urich, Switzerland ( c ) International Center for Advanced Studies (ICAS), ICIFI and ECyT-UNSAM, 25 de Mayoy Francia, (1650) Buenos Aires, Argentina ( d ) Dipartimento di Fisica, Universit`a degli Studi di Milano-Bicocca andINFN, Sezione di Milano-Bicocca, I-20126, Milan, Italy
Abstract
We consider QCD radiative corrections to the associated production of a heavy-quark pair ( Q ¯ Q ) with a generic colourless system F at hadron colliders. We discussthe resummation formalism for the production of the Q ¯ QF system at small values ofits total transverse momentum q T . The perturbative expansion of the resummationformula leads to the explicit ingredients that can be used to apply the q T subtractionformalism to fixed-order calculations for this class of processes. We use the q T subtraction formalism to perform a fully differential perturbative computation forthe production of a top-antitop quark pair and a Higgs boson. At next-to-leadingorder we compare our results with those obtained with established subtractionmethods and we find complete agreement. We present, for the first time, the resultsfor the flavour off-diagonal partonic channels at the next-to-next-to-leading order.February 2021 a r X i v : . [ h e p - ph ] F e b Introduction
The observation of Higgs boson production in association with a top quark–antiquark ( t ¯ t ) pairwas reported by the ATLAS and CMS collaborations in 2018 [1, 2]. This production modeallows for a direct measurement of the top-quark Yukawa coupling.The first theoretical studies of the production of a top–antitop quark pair and a Higgsboson ( t ¯ tH ) were carried out in Refs. [3, 4] at leading order (LO) in QCD perturbation theory,and in Refs. [5–10] at next-to-leading order (NLO). NLO EW corrections were reported inRefs. [11–13]. The resummation of soft-gluon contributions close to the partonic kinematicalthreshold was considered in Refs. [14–16]. The uncertainty of current theoretical predictions for t ¯ tH cross sections is at the O (10%) level [17]. To match the experimental precision expectedat the end of the high-luminosity phase of the LHC, next-to-next-to-leading order (NNLO)predictions in QCD perturbation theory are required.This Letter is devoted to the NNLO (and NLO) QCD calculation of t ¯ tH production. Atthe partonic level, the NNLO calculation of t ¯ tH production requires the evaluation of tree-levelcontributions with two additional unresolved partons in the final state, of one-loop contributionswith one unresolved parton and of purely virtual contributions. The required tree-level andone-loop scattering amplitudes can nowadays be evaluated with automated tools. The two-loopamplitude for t ¯ tH production is not known. Being a five-leg amplitude involving particles withdifferent masses, its computation is at the frontier of current possibilities [18].Even having all the required amplitudes, their implementation in a complete NNLO calcula-tion at the fully differential (exclusive) level is a highly non-trivial task because of the presenceof infrared (IR) divergences at intermediate stages of the calculation. In particular, these di-vergences do not permit a straightforward implementation of numerical techniques. Variousmethods have been proposed and used to overcome these difficulties at the NNLO level (seeRefs. [18–21] and references therein).In this work we will use the transverse-momentum ( q T ) subtraction method [22]. The q T subtraction formalism is a method to handle and cancel the IR divergences in QCD compu-tations at NLO, NNLO and beyond. The method uses IR subtraction counterterms that areconstructed by considering and explicitly computing the q T distribution of the produced final-state system in the limit q T →
0. If the produced final-state system is composed of non-QCD(colourless) partons (such as vector bosons, Higgs bosons, and so forth), the behaviour of the q T distribution in the limit q T → q T subtraction formalism up to NNLO for this entire class of processes . The re-summation formalism can, however, be extended to the production of final states containinga heavy-quark pair [31–33]. Exploiting such extension, the NNLO computations of top-quarkand bottom-quark production were recently completed [34–37]. For this class of processes transverse-momentum resummation has recently been extended [24–28] to next-to-next-to-next-to leading order (N LO), thus allowing first N LO applications [29, 30] of the q T subtractionmethod.
1n this Letter we consider the associated production of a top-quark pair with a Higgs boson.Since the Higgs boson is colourless, the structure of transverse-momentum resummation for thisprocess is closely analogous to that for heavy-quark production. The only important differenceis that the emission of a Higgs boson off the top-quark pair changes the kinematics, and the topand antitop quarks are not back-to-back anymore at Born level. We show that this feature canbe controlled through the knowledge of appropriate resummation coefficients. This allows usto present first results on the application of the q T subtraction method to the NLO and NNLOcomputations of t ¯ tH production in hadron collisions. We exploit the formulation of transverse-momentum resummation in Ref. [33] that includes the complete dependence on the kinematicsof the heavy-quark pair. This dependence and, in particular, the complete control on the heavy-quark azimuthal correlations are essential (see Sec. 2) to extract all the NNLO counterterms ofthe q T subtraction method. Although the structure of transverse-momentum resummation for t ¯ tH production is fully worked out up to NNLO, the explicit NNLO results for the hard-virtualfactors [33] in the flavour diagonal partonic channels q ¯ q → t ¯ tH + X and gg → t ¯ tH + X ( X denotes the unobserved inclusive final state) are not yet known. Their evaluation requires thetwo-loop amplitudes for the partonic processes q ¯ q → t ¯ tH and gg → t ¯ tH , as well as related softcontributions whose computation is available only for heavy-quark pair production. Therefore,in the NNLO calculation of this paper we present numerical results for all the flavour off-diagonal channels ab → t ¯ tH + X , with ab = qg (¯ qg ) , qq (¯ q ¯ q ) , qq (cid:48) (¯ q ¯ q (cid:48) ) , q ¯ q (cid:48) (¯ qq (cid:48) ) ( q and q (cid:48) denotequarks with different flavours).The paper is organised as follows. In Sect. 2 we recall the transverse-momentum resum-mation formalism for the production of a high-mass system containing a heavy-quark pair anddiscuss the perturbative ingredients needed for the NNLO calculation. In Sect. 3 we present ournumerical results for ttH production at NLO and NNLO. In Sect. 4 we summarise our findings.In the Appendix we report the explicit expressions of the resummation coefficients required forthe calculation and highlight ensuing dynamical features related to azimuthal correlations andasymmetries. q T subtrac-tion formalism for Q ¯ QF production We consider the associated production of a heavy-quark pair ( Q ¯ Q ) and an arbitrary colourlesssystem F in hadron collisions. The system F consists of one or more colourless particles, suchas vector bosons, Higgs bosons, and so forth. We denote by M and q T the invariant mass andtransverse momentum of the Q ¯ QF system, respectively.At small values of q T (i.e., q T (cid:28) M ) the perturbative QCD computation for this class ofproduction processes is affected by large logarithmic terms of the type ln n ( M/q T ). These termscan formally be resummed to all perturbative orders.In the case of Q ¯ Q production (i.e., no accompanying system F ) the transverse-momentumresummation formalism was developed in Ref. [33] at arbitrary logarithmic accuracy, and byincluding the azimuthal-correlation contributions (see also Refs. [31, 32] for the azimuthally2veraged case up to next-to-leading logarithmic accuracy). The resummation formalism ofRef. [33] can be extended to Q ¯ QF associated production since the system F is formed bycolourless particles. The extension is straightforward at the formal level and, at the practicallevel, it requires the explicit computation of process-dependent resummation factors at thenecessary perturbative order. In the following we briefly summarize the key steps and theingredients that are involved in this extension.Transverse-momentum resummation is performed through Fourier transformation from im-pact parameter space, where the impact parameter vector b is the Fourier conjugated variableto the transverse-momentum vector q T . The general resummation formula for both Q ¯ Q and Q ¯ QF production is given in Eq. (5) of Ref. [33]. The process-dependent contributions to thisresummation formula are the LO partonic cross section (cid:104) dσ (0) c ¯ c (cid:105) ( c = q, ¯ q, g ) and the resumma-tion factor (cid:0) H∆ (cid:1) , while all the other contributions are process independent. These process-independent contributions and the corresponding resummation coefficients are the same termsthat control the production of a colourless high-mass system [23] (see below). The factor (cid:0) H∆ (cid:1) has an all-order process-independent structure (see Eqs. (10)–(16) and (26) in Ref. [33]) thatis controlled by resummation coefficients that can be explicitly computed at the required per-turbative order. These resummation coefficients are ( i ) the soft anomalous dimension matrix Γ t , ( ii ) the radiative factor D and ( iii ) the subtraction operator (cid:101) I c ¯ c → (cid:101) F .( i ) The resummation factor ∆ (see Eqs. (15)–(18) in Ref. [33]) depends on the soft anomalousdimension matrix Γ t , whose perturbative expansion in the QCD coupling α S reads Γ t ( α S , { p i } ) = α S π Γ (1) t ( { p i } ) + (cid:16) α S π (cid:17) Γ (2) t ( { p i } ) + O ( α ) . (1)( ii ) The term ∆ also depends on the radiative factor D ( ˆb , α S , { p i } ), which embodies az-imuthal correlations of soft origin and, therefore, it depends on the direction ˆb of the impactparameter vector b . Its perturbative expansion reads D ( ˆb , α S , { p i } )) = 1 + α S π D (1) ( ˆb , { p i } ) + O ( α ) , (2)with the constraint (cid:104) D ( ˆb , α S , { p i } )) (cid:105) av . = 1 , (3)where (cid:104) ... (cid:105) av . denotes the azimuthal average over ˆb (i.e., the average over the azimuthal angle φ ( b ) of the transverse vector b ).( iii ) The subtraction operator (cid:101) I c ¯ c → (cid:101) F has the following perturbative expansion, (cid:101) I c ¯ c → (cid:101) F ( α S ( M ) , (cid:15) ; { p i } ) = ∞ (cid:88) n =1 (cid:18) α S ( µ R )2 π (cid:19) n (cid:101) I ( n ) c ¯ c → (cid:101) F ( (cid:15), M /µ R ; { p i } ) , (4)where µ R is the renormalisation scale of the QCD coupling α S ( µ R ). Here (cid:101) F generically denotesthe observed final-state system (i.e., (cid:101) F = Q ¯ Q for heavy-quark pair production, or (cid:101) F = Q ¯ QF for the associated production process) with total invariant mass M . The operator (cid:101) I c ¯ c → (cid:101) F em-3odies IR-divergent contributions that are regularized by the customary procedure of analyticcontinuation in d = 4 − (cid:15) space-time dimensions. This subtraction operator contributes tothe resummation factor H (see Eqs. (12) and (13) in Ref. [33], and Eqs. (8) and (13) in thefollowing) through the definition of the (IR-finite) hard-virtual amplitude (cid:102) M c ¯ c → (cid:101) F (see Eq. (26)in Ref. [33] and Eq. (15) in the following) of the partonic production process c ¯ c → (cid:101) F .The general transverse-momentum resummation formula for (cid:101) F = Q ¯ Q, Q ¯ QF productioninvolves a sole additional ingredient that is process dependent, namely the scattering amplitude M c ¯ c → (cid:101) F of the partonic production process c ¯ c → (cid:101) F .The resummation quantities Γ t , D and (cid:101) I c ¯ c → (cid:101) F have a ‘minimal’ process dependence, whichhas a soft origin: they depend on the momenta p i and colour charges T i of the colour-chargedpartons of the process c ¯ c → (cid:101) F (namely, the colliding partons c and ¯ c and the produced heavyquarks and antiquarks). Such dependence is simply denoted by the argument { p i } in Eqs. (1),(2) and (4). We also recall that Γ t , D and (cid:101) I c ¯ c → (cid:101) F are actually colour-space operators that acton the colour indices of the corresponding partons. The explicit expressions of the perturbativeterms Γ (1) t , Γ (2) t , D (1) and (cid:101) I (1) c ¯ c → (cid:101) F for Q ¯ Q production were presented in Ref. [33], and we reportthe corresponding expressions for Q ¯ QF production in the Appendix of this Letter.According to the q T subtraction method [22], the formulation of transverse-momentum re-summation for Q ¯ QF production allows us to write the (N)NLO partonic cross section dσ Q ¯ QF ( N ) NLO as d ˆ σ Q ¯ QF ( N ) NLO = H Q ¯ QF ( N ) NLO ⊗ d ˆ σ Q ¯ QFLO + (cid:104) d ˆ σ Q ¯ QF +jet( N ) LO − d ˆ σ Q ¯ QF, CT ( N ) NLO (cid:105) , (5)where dσ Q ¯ QF +jet( N ) LO is the Q ¯ QF +jet cross section at (N)LO accuracy. To apply Eq. (5) at NLO,the LO cross section dσ Q ¯ QF +jet LO can be directly obtained by integrating the corresponding tree-level scattering amplitudes. To apply Eq. (5) at NNLO, dσ Q ¯ QF +jet NLO can be evaluated by usingany available NLO method to handle and cancel the corresponding IR divergences, if therelevant tree-level and one-loop QCD amplitudes are available. Therefore, dσ Q ¯ QF +jet( N ) LO is IRfinite, provided q T (cid:54) = 0. The square bracket term of Eq. (5) is IR finite in the limit q T → dσ Q ¯ QF +jet( N ) LO and dσ Q ¯ QF, CT ( N ) NLO , are separately divergent. The IR-subtraction counterterm dσ Q ¯ QF , CT ( N ) NLO is obtained from the (N)NLO perturbative expansion (see,e.g., Ref. [38]) of the resummation formula of the logarithmically enhanced contributions tothe corresponding q T distribution [31–33]. The explicit form of dσ Q ¯ QF, CT ( N ) NLO can be completelyworked out up to NNLO accuracy. It depends on the resummation coefficients that controltransverse-momentum resummation for the production of a colourless final-state system and,additionally, on the first two coefficients Γ (1) t and Γ (2) t of the soft anomalous dimension matrixin Eq. (1).The explicit expression of the coefficient Γ (1) t for Q ¯ QF production is given in the Appendix.The expression of Γ (2) t can be determined (see the Appendix) by exploiting the relation [33]between Γ t and the IR singularities of the virtual scattering amplitude M c ¯ c → Q ¯ QF [39–43].The IR-finite function H Q ¯ QF in Eq. (5) corresponds to the coefficient of the δ (2) ( q T ) contri-bution in the expansion of the resummation formula. It reads [33] H Q ¯ QFc ¯ c ; a a = (cid:104) [( H D ) C C ] c ¯ c ; a a (cid:105) av . , (6)4here the perturbative functions C and C are process independent and describe the emissionof collinear radiation off the incoming partons. In Eq. (6) we have explicitly denoted the partonindices { c ¯ c, a , a } that are implicit in Eq. (5). The indices c and ¯ c correspond to the incomingpartons of the LO partonic cross section d ˆ σ Q ¯ QFLO . The indices a and a are those of the partondensities f a and f a of the colliding hadrons. The partonic cross section in Eq. (5) depends onthe renormalisation scale µ R of α S and on the factorisation scale µ F of the parton densities. InEq. (6) and in the following (see Eqs. (9) and (12)) the explicit structure of the function H Q ¯ QF ispresented by setting µ R = µ F = M . The exact dependence on µ R and µ F can straightforwardlybe recovered by using renormalisation group invariance and evolution of the parton densities.In the quark annihilation channel ( c = q, ¯ q ) the functions C and C do not depend on b ,and the symbolic factor [( H D ) C C ] c ¯ c ; a a takes the form[( H D ) C C ] c ¯ c ; a a = ( H D ) c ¯ c C ca C ¯ ca ( c = q, ¯ q ) (7)with ( HD ) c ¯ c = (cid:104) (cid:102) M c ¯ c → Q ¯ QF | D | (cid:102) M c ¯ c → Q ¯ QF (cid:105) α p S ( M ) |M (0) c ¯ c → Q ¯ QF ( { p i } ) | ( c = q, ¯ q ) . (8)The factor α p S |M (0) c ¯ c → Q ¯ QF | in the denominator is the LO contribution to the squared amplitude |M c ¯ c → Q ¯ QF | for the process c ¯ c → Q ¯ QF (note that the power p depends on the process, seeEq. (16)). The IR-finite hard-virtual amplitude (cid:102) M c ¯ c → Q ¯ QF in Eq. (8) is defined in terms of theall-order renormalised virtual amplitude M c ¯ c → Q ¯ QF through an appropriate subtraction of IRsingularities (see Eq. (15)). By using Eq. (3) the contribution of the azimuthal factor D toEq. (6) becomes trivial, and we obtain H Q ¯ QFc ¯ c ; a a = (cid:104) (cid:102) M c ¯ c → Q ¯ QF | (cid:102) M c ¯ c → Q ¯ QF (cid:105) α p S ( M ) |M (0) c ¯ c → Q ¯ QF ( { p i } ) | C ca C ¯ ca ( c = q, ¯ q ) . (9)In the gluon fusion channel ( c = g ) the collinear functions C and C can be decomposedas [44] C µνga ( z, p , p , ˆb ; α S ) = d µν ( p , p ) C ga ( z ; α S ) + D µν ( p , p , ˆb ) G ga ( z ; α S ) , (10)where the tensors d µν and D µν , which multiply the helicity-conserving and helicity-flip compo-nents C ga and G ga , read ( b µ = (0 , b ,
0) with b µ b µ = − b ) d µν ( p , p ) = − g µν + p µ p ν + p ν p µ p · p , D µν ( p , p , ˆb ) = d µν ( p , p ) − b µ b ν b . (11)Therefore, the function H Q ¯ QFgg ; a a reads H Q ¯ QFgg ; a a = (cid:104) ( H D ) gg ; µ ν ,µ ν C µ ν ga ( ˆb .... ) C µ ν ga ( ˆb .... ) (cid:105) av . , (12)5here ( H D ) gg ; µ ν ,µ ν = (cid:104) (cid:102) M ν (cid:48) ν (cid:48) gg → Q ¯ QF | D | (cid:102) M µ (cid:48) µ (cid:48) gg → Q ¯ QF (cid:105) d µ (cid:48) µ d ν (cid:48) ν d µ (cid:48) µ d ν (cid:48) ν α p S ( M ) |M (0) gg → Q ¯ QF ( { p i } ) | . (13)The functions C ca ( z ; α S ) ( c = q, ¯ q, g ) and G ga ( z ; α S ) have perturbative expansions C ca ( z ; α S ) = δ ca δ (1 − z ) + ∞ (cid:88) n =1 (cid:16) α S π (cid:17) n C ( n ) ca ( z ) , G ga ( z ; α S ) = ∞ (cid:88) n =1 (cid:16) α S π (cid:17) n G ( n ) ga ( z ) . (14)The helicity-conserving coefficients C ( n ) ca ( z ) are known up to n = 2 [45–48], and they are thesame that contribute to Higgs boson [22] and vector-boson [49] production. Recently, theircomputation has been extended to the third order ( n = 3) [26–28]. The helicity-flip coefficients G ( n ) ga ( z ) are known up to n = 2 [50, 51].The hard-virtual amplitude (cid:102) M c ¯ c → Q ¯ QF in Eq. (9) and (13) is expressed in terms of the all-order renormalised virtual amplitude M c ¯ c → Q ¯ QF as | (cid:102) M c ¯ c → Q ¯ QF ( { p i } ) (cid:105) = (cid:104) − (cid:101) I c ¯ c → Q ¯ QF ( α S ( M ) , (cid:15) ; { p i } ) (cid:105) |M c ¯ c → Q ¯ QF ( { p i } ) (cid:105) , (15)where (cid:101) I c ¯ c → Q ¯ QF is the subtraction operator whose perturbative expansion is given in Eq. (4). Thegeneral expression of the first order coefficient (cid:101) I (1) c ¯ c → Q ¯ QF in Eq. (4) is known (see Appendix),while the result for the second-order coefficient is available only in the case of heavy-quarkproduction [35, 52, 53].The quantity M c ¯ c → Q ¯ QF ( { p i } ) on the right-hand side of Eq. (15) is the renormalised on-shellscattering amplitude and has the perturbative expansion M c ¯ c → Q ¯ QF ( { p i } ) = α p/ ( µ R ) µ p(cid:15)R (cid:34) M (0) c ¯ c → Q ¯ QF ( { p i } )+ ∞ (cid:88) n =1 (cid:18) α S ( µ R )2 π (cid:19) n M ( n ) c ¯ c → Q ¯ QF ( { p i } ; µ R ) (cid:35) . (16)The perturbative expansion of (cid:102) M c ¯ c → Q ¯ QF is completely analogous to that in Eq. (16), with (cid:102) M (0) c ¯ c → Q ¯ QF = M (0) c ¯ c → Q ¯ QF and the replacement M ( n ) c ¯ c → Q ¯ QF → (cid:102) M ( n ) c ¯ c → Q ¯ QF ( n ≥ (cid:102) M (1) c ¯ c → Q ¯ QF = M (1) c ¯ c → Q ¯ QF − (cid:101) I (1) c ¯ c → Q ¯ QF M (0) c ¯ c → Q ¯ QF . (17)Having discussed the perturbative ingredients entering the function H Q ¯ QFc ¯ c ; a a , we can nowexamine its NLO and NNLO expansions. By inspecting Eqs. (9) and (12) we see that the NLOtruncation of H Q ¯ QFc ¯ c ; a a receives contributions only from the tree-level and one-loop hard-virtualamplitudes M (0) c ¯ c → Q ¯ QF and (cid:102) M (1) c ¯ c → Q ¯ QF , and from the first-order helicity-conserving coefficients C (1) ca ( z ). Indeed, at this perturbative order the azimuthally dependent terms in D (1) and C µνga do not contribute to Eq. (12) because of the azimuthal average. The hard-virtual one-loopamplitude can be computed with available one-loop generators such as OpenLoops [54–56]or
Recola [57–59]. As a consequence, H Q ¯ QFNLO is available for the processes of interest, and, inparticular, for t ¯ tH production. 6he second-order coefficients H Q ¯ QFNNLO are not available in general. Indeed, they depend onthe hard-virtual amplitude (cid:102) M (2) c ¯ c → Q ¯ QF , which in turn requires the knowledge of the renormalisedtwo-loop amplitude, and of the subtraction operator (cid:101) I (2) c ¯ c → Q ¯ QF . The computation of the two-loop amplitude for Q ¯ QF production is at the frontier of current techniques, and, moreover, (cid:101) I (2) c ¯ c → Q ¯ QF is also not known yet. However, the NNLO contribution to H Q ¯ QFc ¯ c ; a a can be completelydetermined for all the flavour off-diagonal partonic channels ( a , a ) (cid:54) = ( c, ¯ c ). The perturbativeingredients entering the calculation in the quark–antiquark annihilation channel (see Eq. (9))are the corresponding tree-level ( M (0) c ¯ c → Q ¯ QF ) and one-loop ( (cid:102) M (1) c ¯ c → Q ¯ QF ) hard-virtual amplitudesand the first- and second-order helicity-conserving coefficients C (1) ab ( z ) and C (2) ab ( z ). In the gluonfusion channel, the azimuthally dependent first-order coefficients of soft ( D (1) ) and collinear( G (1) ga ) origin are also required. Indeed at NNLO such coefficients produce [33] non-vanishingmixed collinear–collinear and soft–collinear contributions in the expansion of Eq. (12). Thegeneral expression of the first-order coefficient D (1) ( ˆb , { p i } ) is explicitly known (see Appendix).The evaluation of the ensuing NNLO contributions can be performed by computing the corre-sponding spin and colour-correlated squared tree-level amplitudes for the process gg → Q ¯ QF .In summary, the current knowledge of transverse-momentum resummation for high-masssystems containing a heavy-quark pair and a colour singlet system allows us to use Eq. (5) toobtain the complete NLO corrections for this class of processes plus the NNLO corrections inthe flavour off-diagonal partonic channels. t ¯ tH production Having discussed the content of Eq. (5), we are in a position to apply it to t ¯ tH productionand to obtain the complete NLO results plus the NNLO corrections in all the flavour off-diagonal partonic channels. Our NLO implementation of the calculation has the main purposeof illustrating the applicability of the q T subtraction method to t ¯ tH production and, in par-ticular, of cross-checking the q T subtraction methodology by numerical comparisons with NLOcalculations performed by using more established NLO methods. Our NNLO results on t ¯ tH production represent a first step (due to the missing flavour diagonal partonic channels) towardsthe complete NNLO calculation for this production process.Our results are obtained with two independent computations, which show complete agree-ment. In the first computation, up to NLO, we use the phase space generation routines fromthe MCFM program [60], suitably modified for q T subtraction along the lines of the cor-responding numerical programs for Higgs boson [22] and vector-boson [49] production. Thenumerical integration is carried out using the Cuba library [61]. At NNLO accuracy the t ¯ tH +jet cross section is evaluated by using the Munich code , which provides a fully automatedimplementation of the NLO dipole subtraction formalism [62–64] and an efficient phase spaceintegration. The remaining flavour off-diagonal contributions at NNLO are evaluated with adedicated fortran implementation. The second computation is directly implemented within the Munich , which is the abbreviation of “MUlti-chaNnel Integrator at Swiss (CH) precision”, is an automatedparton-level NLO generator by S. Kallweit. [fb] 13 TeV 100 TeVLO 394 . . Madgraph5 aMC@NLO ) 499 . Matrix ) 499 . q T ) 499 . O ( α ) qg − . . . O ( α ) q (¯ q ) q (cid:48) . . ttH total cross section at LO and NLO, and its NNLO corrections in theflavour off-diagonal partonic channels. The numerical uncertainties at LO and NLO ( Mad-graph5 aMC@NLO , Matrix ) are due to numerical integration, while at NLO ( q T subtrac-tion) and NNLO they also include the systematics uncertainty from the r cut → Matrix framework [65], suitably extended to t ¯ tH production. In both implementations all therequired tree-level and one-loop amplitudes are obtained with OpenLoops [54–56], includingthe tree-level spin- and colour-correlated amplitudes required to evaluate the contributions inEq. (12).In order to numerically evaluate the contribution in the square bracket of Eq. (5), a technicalcut-off r cut is introduced on the dimensionless variable q T /M , where M is the invariant massof the t ¯ tH system. The final result, which corresponds to the limit r cut →
0, is extracted bycomputing the cross section at fixed values of r cut in the range [0 . , r max ]. Quadratic least χ fits are performed for different values of r max ∈ [0 . , χ / degrees-of-freedom, and the uncertainty is estimated bycomparing the results obtained by the different fits. This procedure is the same as implementedin matrix [65] and it has been shown to provide a conservative estimate of the systematicuncertainty in the q T subtraction procedure for various processes (see Sec. 7 in Ref. [65]).We consider pp collisions at the centre-of-mass energies √ s = 13 TeV and √ s = 100 TeV.We use the NNPDF31 [66] parton distribution functions (PDFs) with the QCD running coupling α S evaluated at each corresponding order (i.e., we use ( n + 1)-loop α S at N n LO, with n = 1 , m t = 173 . m H = 125 GeV,and the Fermi constant G F = 1 . × − GeV − . The renormalisation and factorizationscales, µ R and µ F , are fixed at µ R = µ F = (2 m t + m H ) /
2. Our predictions for the LO andNLO cross sections and for the NNLO corrections in the flavour off-diagonal channels arepresented in Table 1 together with their uncertainties due to the numerical integration andthe extrapolation to r cut →
0, computed as explained above. The NLO cross section computedwith q T subtraction is compared with the result obtained with Madgraph5 aMC@NLO [67],which uses FKS subtraction [68, 69] and with the corresponding result obtained with
Matrix ,which implements dipole subtraction [62–64].We start our discussion from the NLO results. The NLO corrections increase the LO resultby 27% (31%) at √ s = 13 TeV ( √ s = 100 TeV). The flavour off-diagonal qg + ¯ qg channel con-tributes about 15% (23%) of the total NLO correction. As expected, from Table 1 we observe ex-8 . . . . . . r cut = cut q T /m t ¯ tH [%] − . − . − . − . . σ / σ N L O − [ % ] σ MG5 aMC@NLONLO
FKS σ MATRIXNLO CS σ MATRIXNLO qT ( r cut → σ MATRIXNLO qT ( r cut ) pp → t ¯ tH @ 13 TeV, µ F = m t + m H , µ R = m t + m H . . . . . . r cut = cut q T /m t ¯ tH [%] − . − . − . − . . σ / σ N L O − [ % ] σ MG5 aMC@NLONLO
FKS σ MATRIXNLO CS σ MATRIXNLO qT ( r cut → σ MATRIXNLO qT ( r cut ) pp → t ¯ tH @ 100 TeV, µ F = m t + m H , µ R = m t + m H Figure 1: The r cut dependence (data points) at √ s = 13 TeV (left) and 100 TeV (right) ofthe NLO total cross section computed by using q T subtraction. The bands show the extrap-olated value at r cut → Madgraph5 aMC@NLO (using FKSsubtraction) and
Matrix (using dipole subtraction).cellent agreement between the NLO cross section obtained with
Madgraph5 aMC@NLO and
Matrix . The result obtained with q T subtraction also agrees with Madgraph5 aMC@NLO and
Matrix results. The quality of the r cut → r cut . In Figure 1 we investigate thisbehavior and show also the ( r cut independent) NLO result obtained with Matrix , by usingdipole subtraction, and
Madgraph5 aMC@NLO , by using FKS subtraction. As expected,the r cut dependence is linear [70, 71], contrary to what happens in the case of the production ofa colourless final-state system (see Sec. 7 of Ref. [65]), where the power-like dependence of thetotal cross section on r cut is known [72–74] to be quadratic (modulo logarithmic enhancements).In Figure 2 we present the NLO results for several differential distributions at √ s = 13 TeVand compare them with those obtained by using Madgraph5 aMC@NLO . In particular weconsider the transverse-momentum (top left) and rapidity (top right) distributions of the Higgsboson, the transverse-momentum (center left) and rapidity (center right) distributions of thetop quark, and the invariant-mass distributions of the top-quark pair (bottom left) and of the t ¯ tH system (bottom right). We find excellent agreement between the two calculations, withbin-wise uncertainties at the percent-level or below. Our results are obtained by using a fixedvalue of r cut , r cut = 0 . µ R and µ F .We now move to considering the NNLO contributions to the total cross section. In Table 1we report our results for the O ( α ) contributions to the NNLO cross section from the flavouroff-diagonal partonic channels a a → t ¯ tH + X . The contribution from all the channels with a a = qg, ¯ qg is labelled by the subscript qg , and the contribution from all the channels with a a = qq, ¯ q ¯ q, qq (cid:48) , ¯ q ¯ q (cid:48) , q ¯ q (cid:48) , ¯ qq (cid:48) ( q (cid:54) = q (cid:48) ) is labelled by the subscript q (¯ q ) q (cid:48) . We see that the NNLOcorrections from both contributions are very small, at the few per mille level of the NLO crosssection. At √ s = 13 TeV they contribute with similar size and opposite sign, and, therefore,their overall quantitative effect in this setup is completely negligible. We also see that the9
50 100 150 200 250 3001020304050 [f b ] q T -subtractionMG5_aMC@NLO p T , H [GeV] M G M G [ % ] [f b ] q T -subtractionMG5_aMC@NLO y H M G M G [ % ] [f b ] q T -subtractionMG5_aMC@NLO p T , t [GeV] M G M G [ % ] [f b ] q T -subtractionMG5_aMC@NLO y t M G M G [ % ]
400 500 600 700 800 900 10001020304050607080 [f b ] q T -subtractionMG5_aMC@NLO
400 500 600 700 800 900 1000 m tt [GeV] M G M G [ % ]
500 600 700 800 90015202530354045 [f b ] q T -subtractionMG5_aMC@NLO
500 600 700 800 900 m ttH [GeV] M G M G [ % ] Figure 2: The NLO results of
Madgraph5 aMC@NLO for the cross section dependence onseveral kinematic variables at √ s = 13 TeV. The lower panels show the relative comparisonwith the corresponding results obtained by using q T subtraction.10 . . . . . . r cut = cut q T /m t ¯ tH [%] − − − ∆ σ / ∆ σ NN L O − [ % ] σ MATRIXNNLO ( r cut → σ MATRIXNNLO ( r cut ) pp → t ¯ tH ( gq ) @ 13 TeV, µ F = m t + m H , µ R = m t + m H . . . . . . r cut = cut q T /m t ¯ tH [%] − − − ∆ σ / ∆ σ NN L O − [ % ] σ MATRIXNNLO ( r cut → σ MATRIXNNLO ( r cut ) pp → t ¯ tH ( gq ) @ 100 TeV, µ F = m t + m H , µ R = m t + m H . . . . . . r cut = cut q T /m t ¯ tH [%] − . − . − . − . . ∆ σ / ∆ σ NN L O − [ % ] σ MATRIXNNLO ( r cut → σ MATRIXNNLO ( r cut ) pp → t ¯ tH ( q (¯ q ) q ) @ 100 TeV, µ F = m t + m H , µ R = m t + m H . . . . . . r cut = cut q T /m t ¯ tH [%] − . − . − . − . − . . . ∆ σ / ∆ σ NN L O − [ % ] σ MATRIXNNLO ( r cut → σ MATRIXNNLO ( r cut ) pp → t ¯ tH ( q (¯ q ) q ) @ 100 TeV, µ F = m t + m H , µ R = m t + m H Figure 3: The NNLO contribution ∆ σ of the gq (top) and q (¯ q ) q (cid:48) (bottom) partonic channelsto the total cross section at √ s = 13 TeV (left) and 100 TeV (right). The r cut dependence of∆ σ is normalized to its extrapolation ∆ σ NNLO at r cut → qg channel is rather large. This is due toa cancellation between the two terms in Eq. (5): the term H ⊗ d ˆ σ , which is r cut independent,and the term in the square bracket, which depends on r cut . This cancellation is observed at both √ s = 13 TeV and √ s = 100 TeV, but it is particularly severe at √ s = 13 TeV, downgradingthe numerical precision that can be obtained for the relative correction. Similar effects wereobserved for t ¯ t production in Refs. [34, 35].In Figure 3 we show the r cut -dependence of the NNLO corrections ∆ σ of the flavour off-diagonal channels to the total cross section for t ¯ tH production. The result is normalised toour extrapolation ∆ σ NNLO at r cut →
0. The r cut dependence confirms that our calculation cancontrol the NNLO contributions in the off-diagonal partonic channels at the few percent level.Based on the experience with the NNLO calculations for heavy-quark production [35–37],this should be fully sufficient to obtain precise NNLO results by using the q T subtraction methodonce the presently unknown soft contributions and the two-loop amplitudes become available. In this Letter we have considered the associated production of the SM Higgs boson with atop-quark pair, and, more generally, processes in which heavy-quark pairs are produced in11ssociation with a colourless final-state system F . We have pointed out that the transverse-momentum resummation formalism developed for Q ¯ Q production in Ref. [33] can be extendedto associated Q ¯ QF production. This extension, which requires the evaluation of the appropriateresummation coefficients at the necessary perturbative accuracy, is also sufficient to apply the q T subtraction method to this class of processes.Using the resummation coefficients presented in this paper and the current knowledge ofscattering amplitudes, it is possible to apply the q T subtraction formalism to Q ¯ QF productionup to NLO and to obtain the NNLO corrections in all the flavour off-diagonal partonic channels.We have implemented for the first time the q T subtraction formalism for t ¯ tH production,and we have presented first quantitative results at NLO and NNLO. The calculation is accurateat NLO in QCD, and the NNLO corrections have been computed for the flavour off-diagonalpartonic channels. At NLO we have checked the correctness of our implementation by compar-ing with the results obtained by using tools that are based on established subtraction methods.We found complete agreement for the total cross section and for single-differential distributions.Within the setup that we have considered, we have found that the NNLO contribution of theoff-diagonal partonic channels to the total cross section has a very small quantitative effect.The extension of this calculation to the diagonal channels requires further theoretical work tocompute the two-loop virtual amplitudes, and the NNLO soft contributions to the resummationcoefficients. Acknowledgements
We are grateful to Federico Buccioni, Jean-Nicolas Lang, Jonas Lindert and Stefano Pozzorinifor their continuous assistance on issues related to
OpenLoops during the course of this project,and for providing us with the specific spin- and colour-correlated amplitudes which were neces-sary to complete the calculation. This work is supported in part by the Swiss National ScienceFoundation (SNF) under contracts IZSAZ2 173357 and 200020 188464. The work of SK issupported by the ERC Starting Grant 714788 REINVENT.
Appendix
In this Appendix we report the explicit expressions [75] of the resummation coefficients relevantfor the production of an arbitrary number of heavy quarks Q j (with colour charges T j , polemasses m j and transverse momenta p jT ) accompanied by a colourless system F with totalmomentum p F . More precisely, we consider the partonic process c ( p )¯ c ( p ) → Q ( p ) ¯ Q ( p ) . . . Q N − ( p N − ) ¯ Q N ( p N ) F ( p F ) . (18)where the final-state system is produced by either massless q ¯ q annihilation ( c = q ) or gg fusion( c = g ). All the momenta p i ( i ≥
1) are in the physical region, i.e. four-momentum conservationreads p + p = p + ...p N + p F , while colour is outgoing, such that (cid:80) Ni =1 T i = 0. In particularwe have T = T = C c , where C c = C F if c = q and C c = C A if c = g . This notationcorresponds to that used in Ref. [33] for the simpler case of heavy-quark pair production,12 ( p )¯ c ( p ) → Q ( p ) ¯ Q ( p ). We note that the massless and heavy quarks and antiquarks of theprocess in Eq. (18) can have different flavours (e.g., we can have u ¯ d → t ¯ bF ).The coefficients that are presented below refer to perturbative expansions (see Eqs. (1),(2) and (4)) in which α S ( µ ) denotes the renormalised QCD coupling in the MS scheme withdecoupling of the heavy quarks Q j [76]. In the case of t ¯ tH production α ( µ ) is the QCDcoupling in the five-flavour scheme.The first-order soft anomalous dimension Γ (1) t ( { p i } ) (see Eq. (1)) reads Γ (1) t ( { p i } ) = − (cid:88) j ≥ T j (1 − iπ ) + (cid:88) i =1 , j ≥ T i · T j ln (2 p i p j ) M m j + (cid:88) j,k ≥ j (cid:54) = k T j · T k (cid:20) v jk ln (cid:18) v jk − v jk (cid:19) − iπ (cid:18) v jk + 1 (cid:19)(cid:21) , (19)where v jk = (cid:115) − m j m k ( p j p k ) . (20)The first-order subtraction operator (cid:101) I (1) (see Eq. (4)) can be written as (cid:101) I (1) ( (cid:15), M /µ R ; { p i } )) = − (cid:18) M µ R (cid:19) − (cid:15) (cid:40) (cid:88) i =1 , (cid:20)(cid:18) (cid:15) + iπ (cid:15) − π (cid:19) T i + 1 (cid:15) γ i (cid:21) − (cid:15) Γ (1) t ( { p i } ) + F (1) t ( { p i } ) (cid:27) , (21)where the coefficients γ i ( i = q, ¯ q, g ) originate from collinear radiation and read γ q = γ ¯ q = 3 C F / γ g = (11 C A − N f ) / N f being the number of flavours of massless quarks. The function F (1) t ( { p i } ) in Eq. (21) is F (1) t ( { p i } ) = (cid:88) j ≥ T j ln (cid:18) m j + p jT m j (cid:19) − (cid:88) i =1 , j ≥ T i · T j Li (cid:18) − p jT m j (cid:19) + (cid:88) j,k ≥ j (cid:54) = k T j · T k v jk L jk , (22)where L jk = 12 ln (cid:18) v jk − v jk (cid:19) ln (cid:20) ( m j + p jT )( m k + p kT ) m j m k (cid:21) − (cid:18) v jk v jk (cid:19) −
14 ln (cid:18) v jk − v jk (cid:19) + (cid:88) i =1 , (cid:34) Li (cid:32) − (cid:115) − v jk v jk r jk,i (cid:33) + Li (cid:32) − (cid:115) − v jk v jk r jk,i (cid:33) + 12 ln r jk,i (cid:35) (23)13ith r jk,i ≡ m k m j p i · p j p i · p k . (24)The function Li ( z ) in Eq. (23) is the customary dilogarithm function, Li ( z ) = − (cid:82) dtt ln(1 − zt ).The coefficient D (1) ( ˆb , { p i } ) (see Eq. (2)) reads D (1) ( ˆb , { p i } ) = (cid:88) j ≥ T j ˆb · p jT (cid:114) m j + (cid:16) ˆb · p jT (cid:17) (cid:34) arcsinh (cid:32) ˆb · p jT m j (cid:33) + iπ (cid:35) −
12 ln (cid:18) m j + p jT m j (cid:19) + (cid:88) i =1 , j ≥ T i · T j (cid:32) arcsinh (cid:32) ˆb · p jT m j (cid:33)(cid:34) arcsinh (cid:32) ˆb · p jT m j (cid:33) + iπ (cid:35) + 12 Li (cid:18) − p jT m j (cid:19)(cid:33) + (cid:88) j,k ≥ j (cid:54) = k T j · T k D jk , (25)with the function D jk that is given in terms of the following one-fold integral representation: D jk = (cid:90) dx p j · p k w jk ( x ) ˆb · w jkT ( x ) (cid:114) w jk ( x ) + (cid:16) ˆb · w jkT ( x ) (cid:17) arcsinh ˆb · w jkT ( x ) (cid:113) w jk ( x ) + iπ − ln (cid:32) w jkT ( x ) w jk ( x ) (cid:33)(cid:33) , (26)where the four-vector w µjk ( x ) is defined as w µjk ( x ) = xp µj + (1 − x ) p µk . (27)We note that the expressions of the resummation coefficients in Eqs. (19), (21), (22) and(25) for the process in Eq. (18) have no explicit dependence on the flavour of the (masslessand heavy) quarks and on the colourless system F . In particular, there is no dependence onthe quantum numbers of F and on its momentum p F (though the dependence on p F entersimplicitly through momentum conservation, p + p = p + ...p N + p F ).Using Eqs. (19), (21), (22) and (25), we also note that we can recover the resummationcoefficients of Ref. [33] for the simpler process of heavy-quark pair production , c ( p )¯ c ( p ) → Q ( p ) ¯ Q ( p ). To this purpose we can simply use the corresponding constraints m = m , T + T = − ( T + T ) and, importantly, the constraint p T = − p T that follows from momentumconservation (i.e., p F = 0). Indeed, to a large extent, the main difference between heavy-quark pair production and the process in Eq. (18) is due to the fact that the N transversemomenta p jT (3 ≤ j ≤ N ) in Eq. (18) are independent kinematical variables (since p F T (cid:54) = 0). In this case, the one-fold integral in Eq. (26) is expressed through dilogarithms in Eqs. (36)–(38) of Ref. [33]. b -space resummationcoefficient D (1) (which is due [33] to soft-parton radiation from the final state and from initial-state/final-state interferences) depends on the relative azimuthal angles φ jb = φ ( p jT ) − φ ( b ).This dependence produces ensuing azimuthal correlations [33] with respect to the observableangles φ jq = φ ( p jT ) − φ ( q T ) of the q T -differential cross section at q T (cid:54) = 0. In the limit q T → D (1) depends on cos φ b = cos φ b [33], and thisimplies that it produces azimuthal-correlation divergences only in the case of even harmonics(i.e., harmonics with respect to cos k φ q with k = 2 , , , . . . ) [70]. This dependence of D (1) is due to the kinematical relation p T (cid:39) − p T at q T →
0. In the case of the process inEq. (18), the transverse momenta p jT are kinematically independent even if q T →
0, and itturns out that D (1) depends on both cos φ jb and cos φ jb (the dependence on cos φ jb is due to thecontributions that are proportional to ‘ iπ ’ in the right-hand side of Eqs. (25) and (26)). Thisdependence implies that the process in Eq. (18) is affected by lowest-order (and higher-order)divergent azimuthal correlations for both even and odd harmonics (i.e., harmonics with respectto cos k φ q with k = 1 , , , . . . ). A quite general discussion of azimuthal correlations at small q T is presented in Ref. [70], where is also pointed out that lowest-order divergent odd harmonicscan occur in other processes, such as V + jet and dijet production.We conclude this Appendix by recalling [33] that the second-order term Γ (2) t of the softanomalous dimension Γ t (see Eq. (1)) for the process in Eq. (18) is related to the correspond-ing term of the anomalous dimension matrix Γ ( µ ) that controls the QCD IR divergences ofscattering amplitudes with massive external particles [39–43]. We have Γ t ( α S ; { p i } ) = 12 Γ sub . ( α S , { p i } ) − (cid:16) α S π (cid:17) (cid:16) (cid:104) Γ (1) t ( { p i } ) , F (1) t ( { p i } ) (cid:105) + πβ F (1) t ( { p i } ) (cid:17) + O ( α ) , (28)where 12 πβ = 11 C A − N f , Γ (1) t ( { p i } ) and F (1) t ( { p i } are given in Eqs. (19) and (22), and Γ sub . is given below. The perturbative expansion of the right-hand side of Eq. (28) includesboth the first-order and second-order terms Γ (1) t and Γ (2) t (obviously, Γ sub . = 2( α S /π ) Γ (1) t + O ( α )), while terms at O ( α ) and beyond are neglected. The ‘subtracted’ anomalous dimension Γ sub . ( α S ; { p i } ) is Γ sub . ( α S , { p i } ) = Γ ( µ ) − (cid:20)
12 ( T + T ) γ cusp ( α S ) (cid:18) ln M µ − iπ (cid:19) + 2 γ c ( α S ) (cid:21) , (29)where the terms on the right-hand side are written by exactly using the notation of Eq. (5) ofRef. [42]. The term Γ ( µ ) is the anomalous-dimension matrix that controls the IR divergencesof the scattering amplitude M c ¯ c → Q ¯ Q ...F for the process in Eq. (18), while the square-bracketterm on the right-hand side of Eq. (29) is the corresponding expression of Γ ( µ ) for a genericprocess c ¯ c → F , where the system F is colourless. The expressions of Γ ( µ ) , γ cusp ( α S ) and γ c ( α S )up to O ( α ) are explicitly given in Ref. [42]. 15 eferences [1] ATLAS
Collaboration, M. Aaboud et al.,
Observation of Higgs boson production inassociation with a top quark pair at the LHC with the ATLAS detector , Phys. Lett. B (2018) 173–191, [ arXiv:1806.00425 ].[2]
CMS
Collaboration, A. M. Sirunyan et al.,
Observation of tt H production , Phys. Rev.Lett. (2018), no. 23 231801, [ arXiv:1804.02610 ].[3] J. N. Ng and P. Zakarauskas, A { QCD } Parton Calculation of Conjoined Production ofHiggs Bosons and Heavy Flavors in p ¯ p Collision , Phys. Rev. D (1984) 876.[4] Z. Kunszt, Associated Production of Heavy Higgs Boson with Top Quarks , Nucl. Phys. B (1984) 339–359.[5] W. Beenakker, S. Dittmaier, M. Kr¨amer, B. Plumper, M. Spira, and P. Zerwas,
Higgsradiation off top quarks at the Tevatron and the LHC , Phys. Rev. Lett. (2001) 201805,[ hep-ph/0107081 ].[6] W. Beenakker, S. Dittmaier, M. Kr¨amer, B. Plumper, M. Spira, and P. Zerwas, NLOQCD corrections to t anti-t H production in hadron collisions , Nucl. Phys. B (2003)151–203, [ hep-ph/0211352 ].[7] L. Reina and S. Dawson,
Next-to-leading order results for t anti-t h production at theTevatron , Phys. Rev. Lett. (2001) 201804, [ hep-ph/0107101 ].[8] L. Reina, S. Dawson, and D. Wackeroth, QCD corrections to associated t anti-t hproduction at the Tevatron , Phys. Rev. D (2002) 053017, [ hep-ph/0109066 ].[9] S. Dawson, L. Orr, L. Reina, and D. Wackeroth, Associated top quark Higgs bosonproduction at the LHC , Phys. Rev. D (2003) 071503, [ hep-ph/0211438 ].[10] S. Dawson, C. Jackson, L. Orr, L. Reina, and D. Wackeroth, Associated Higgs productionwith top quarks at the large hadron collider: NLO QCD corrections , Phys. Rev. D (2003) 034022, [ hep-ph/0305087 ].[11] S. Frixione, V. Hirschi, D. Pagani, H. Shao, and M. Zaro, Weak corrections to Higgshadroproduction in association with a top-quark pair , JHEP (2014) 065,[ arXiv:1407.0823 ].[12] Y. Zhang, W.-G. Ma, R.-Y. Zhang, C. Chen, and L. Guo, QCD NLO and EW NLOcorrections to t ¯ tH production with top quark decays at hadron collider , Phys. Lett. B (2014) 1–5, [ arXiv:1407.1110 ].[13] S. Frixione, V. Hirschi, D. Pagani, H. S. Shao, and M. Zaro,
Electroweak and QCDcorrections to top-pair hadroproduction in association with heavy bosons , JHEP (2015)184, [ arXiv:1504.03446 ]. 1614] A. Kulesza, L. Motyka, T. Stebel, and V. Theeuwes, Soft gluon resummation forassociated t ¯ tH production at the LHC , JHEP (2016) 065, [ arXiv:1509.02780 ].[15] A. Broggio, A. Ferroglia, B. D. Pecjak, A. Signer, and L. L. Yang, Associated productionof a top pair and a Higgs boson beyond NLO , JHEP (2016) 124, [ arXiv:1510.01914 ].[16] A. Kulesza, L. Motyka, T. Stebel, and V. Theeuwes, Associated t ¯ tH production at theLHC: Theoretical predictions at NLO+NNLL accuracy , Phys. Rev. D (2018), no. 11114007, [ arXiv:1704.03363 ].[17] LHC Higgs Cross Section Working Group
Collaboration, D. de Florian et al.,
Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector , arXiv:1610.07922 .[18] G. Heinrich, Collider Physics at the Precision Frontier , arXiv:2009.00516 .[19] J. Andersen et al., Les Houches 2017: Physics at TeV Colliders Standard Model WorkingGroup Report , arXiv:1803.07977 .[20] S. Amoroso et al., Les Houches 2019: Physics at TeV Colliders: Standard ModelWorking Group Report , arXiv:2003.01700 .[21] W. J. Torres Bobadilla et al., May the four be with you: Novel IR-subtraction methods totackle NNLO calculations , arXiv:2012.02567 .[22] S. Catani and M. Grazzini, An NNLO subtraction formalism in hadron collisions and itsapplication to Higgs boson production at the LHC , Phys. Rev. Lett. (2007) 222002,[ hep-ph/0703012 ].[23] S. Catani, L. Cieri, D. de Florian, G. Ferrera, and M. Grazzini, Universality oftransverse-momentum resummation and hard factors at the NNLO , Nucl. Phys. B (2014) 414–443, [ arXiv:1311.1654 ].[24] Y. Li and H. X. Zhu,
Bootstrapping Rapidity Anomalous Dimensions forTransverse-Momentum Resummation , Phys. Rev. Lett. (2017), no. 2 022004,[ arXiv:1604.01404 ].[25] A. A. Vladimirov,
Correspondence between Soft and Rapidity Anomalous Dimensions , Phys. Rev. Lett. (2017), no. 6 062001, [ arXiv:1610.05791 ].[26] M.-X. Luo, T.-Z. Yang, H. X. Zhu, and Y. J. Zhu,
Quark Transverse Parton Distributionat the Next-to-Next-to-Next-to-Leading Order , Phys. Rev. Lett. (2020), no. 9 092001,[ arXiv:1912.05778 ].[27] M. A. Ebert, B. Mistlberger, and G. Vita,
Transverse momentum dependent PDFs atN LO , JHEP (2020) 146, [ arXiv:2006.05329 ].[28] M.-X. Luo, T.-Z. Yang, H. X. Zhu, and Y. J. Zhu, Unpolarized Quark and Gluon TMDPDFs and FFs at N LO , arXiv:2012.03256 .1729] L. Cieri, X. Chen, T. Gehrmann, E. W. N. Glover, and A. Huss, Higgs boson productionat the LHC using the q T subtraction formalism at N LO QCD , JHEP (2019) 096,[ arXiv:1807.11501 ].[30] G. Billis, M. A. Ebert, J. K. L. Michel, and F. J. Tackmann, A Toolbox for q T and -Jettiness Subtractions at N LO , arXiv:1909.00811 .[31] H. X. Zhu, C. S. Li, H. T. Li, D. Y. Shao, and L. L. Yang, Transverse-momentumresummation for top-quark pairs at hadron colliders , Phys. Rev. Lett. (2013), no. 8082001, [ arXiv:1208.5774 ].[32] H. T. Li, C. S. Li, D. Y. Shao, L. L. Yang, and H. X. Zhu,
Top quark pair production atsmall transverse momentum in hadronic collisions , Phys. Rev.
D88 (2013) 074004,[ arXiv:1307.2464 ].[33] S. Catani, M. Grazzini, and A. Torre,
Transverse-momentum resummation forheavy-quark hadroproduction , Nucl. Phys.
B890 (2014) 518–538, [ arXiv:1408.4564 ].[34] R. Bonciani, S. Catani, M. Grazzini, H. Sargsyan, and A. Torre,
The q T subtractionmethod for top quark production at hadron colliders , Eur. Phys. J.
C75 (2015), no. 12581, [ arXiv:1508.03585 ].[35] S. Catani, S. Devoto, M. Grazzini, S. Kallweit, J. Mazzitelli, and H. Sargsyan,
Top-quarkpair hadroproduction at next-to-next-to-leading order in QCD , Phys. Rev.
D99 (2019),no. 5 051501, [ arXiv:1901.04005 ].[36] S. Catani, S. Devoto, M. Grazzini, S. Kallweit, and J. Mazzitelli,
Top-quark pairproduction at the LHC: Fully differential QCD predictions at NNLO , JHEP (2019)100, [ arXiv:1906.06535 ].[37] S. Catani, S. Devoto, M. Grazzini, S. Kallweit, and J. Mazzitelli, Bottom-quarkproduction at hadron colliders: fully differential predictions in NNLO QCD , arXiv:2010.11906 .[38] G. Bozzi, S. Catani, D. de Florian, and M. Grazzini, Transverse-momentum resummationand the spectrum of the Higgs boson at the LHC , Nucl. Phys. B (2006) 73–120,[ hep-ph/0508068 ].[39] S. Catani, S. Dittmaier, and Z. Trocsanyi,
One loop singular behavior of QCD and SUSYQCD amplitudes with massive partons , Phys. Lett. B (2001) 149–160,[ hep-ph/0011222 ].[40] A. Mitov, G. F. Sterman, and I. Sung,
The Massive Soft Anomalous Dimension Matrixat Two Loops , Phys. Rev. D (2009) 094015, [ arXiv:0903.3241 ].[41] A. Ferroglia, M. Neubert, B. D. Pecjak, and L. L. Yang, Two-loop divergences ofscattering amplitudes with massive partons , Phys. Rev. Lett. (2009) 201601,[ arXiv:0907.4791 ]. 1842] A. Ferroglia, M. Neubert, B. D. Pecjak, and L. L. Yang,
Two-loop divergences of massivescattering amplitudes in non-abelian gauge theories , JHEP (2009) 062,[ arXiv:0908.3676 ].[43] A. Mitov, G. F. Sterman, and I. Sung, Computation of the Soft Anomalous DimensionMatrix in Coordinate Space , Phys. Rev. D (2010) 034020, [ arXiv:1005.4646 ].[44] S. Catani and M. Grazzini, QCD transverse-momentum resummation in gluon fusionprocesses , Nucl. Phys. B (2011) 297–323, [ arXiv:1011.3918 ].[45] S. Catani and M. Grazzini,
Higgs Boson Production at Hadron Colliders: Hard-CollinearCoefficients at the NNLO , Eur. Phys. J. C (2012) 2013, [ arXiv:1106.4652 ].[Erratum: Eur.Phys.J.C 72, 2132 (2012)].[46] S. Catani, L. Cieri, D. de Florian, G. Ferrera, and M. Grazzini, Vector boson productionat hadron colliders: hard-collinear coefficients at the NNLO , Eur. Phys. J. C (2012)2195, [ arXiv:1209.0158 ].[47] T. Gehrmann, T. L¨ubbert, and L. L. Yang, Transverse parton distribution functions atnext-to-next-to-leading order: the quark-to-quark case , Phys. Rev. Lett. (2012)242003, [ arXiv:1209.0682 ].[48] T. Gehrmann, T. L¨ubbert, and L. L. Yang,
Calculation of the transverse partondistribution functions at next-to-next-to-leading order , JHEP (2014) 155,[ arXiv:1403.6451 ].[49] S. Catani, L. Cieri, G. Ferrera, D. de Florian, and M. Grazzini, Vector boson productionat hadron colliders: a fully exclusive QCD calculation at NNLO , Phys. Rev. Lett. (2009) 082001, [ arXiv:0903.2120 ].[50] D. Gutierrez-Reyes, S. Leal-Gomez, I. Scimemi, and A. Vladimirov,
Linearly polarizedgluons at next-to-next-to leading order and the Higgs transverse momentum distribution , JHEP (2019) 121, [ arXiv:1907.03780 ].[51] M.-X. Luo, T.-Z. Yang, H. X. Zhu, and Y. J. Zhu, Transverse Parton Distribution andFragmentation Functions at NNLO: the Gluon Case , JHEP (2020) 040,[ arXiv:1909.13820 ].[52] R. Angeles-Martinez, M. Czakon, and S. Sapeta, NNLO soft function for top quark pairproduction at small transverse momentum , JHEP (2018) 201, [ arXiv:1809.01459 ].[53] S. Catani, S. Devoto, M. Grazzini, J. Mazzitelli, in preparation.[54] F. Cascioli, P. Maierh¨ofer, and S. Pozzorini, Scattering Amplitudes with Open Loops , Phys. Rev. Lett. (2012) 111601, [ arXiv:1111.5206 ].[55] F. Buccioni, S. Pozzorini, and M. Zoller,
On-the-fly reduction of open loops , Eur. Phys. J.
C78 (2018), no. 1 70, [ arXiv:1710.11452 ].1956] F. Buccioni, J.-N. Lang, J. M. Lindert, P. Maierh¨ofer, S. Pozzorini, H. Zhang, and M. F.Zoller,
OpenLoops 2 , Eur. Phys. J.
C79 (2019), no. 10 866, [ arXiv:1907.13071 ].[57] S. Actis, A. Denner, L. Hofer, A. Scharf, and S. Uccirati,
Recursive generation ofone-loop amplitudes in the Standard Model , JHEP (2013) 037, [ arXiv:1211.6316 ].[58] S. Actis, A. Denner, L. Hofer, J.-N. Lang, A. Scharf, and S. Uccirati, RECOLA:REcursive Computation of One-Loop Amplitudes , Comput. Phys. Commun. (2017)140–173, [ arXiv:1605.01090 ].[59] A. Denner, J.-N. Lang, and S. Uccirati,
Recola2: REcursive Computation of One-LoopAmplitudes 2 , Comput. Phys. Commun. (2018) 346–361, [ arXiv:1711.07388 ].[60] J. M. Campbell, R. K. Ellis, and W. T. Giele,
A Multi-Threaded Version of MCFM , Eur.Phys. J. C (2015), no. 6 246, [ arXiv:1503.06182 ].[61] T. Hahn, CUBA: A Library for multidimensional numerical integration , Comput. Phys.Commun. (2005) 78–95, [ hep-ph/0404043 ].[62] S. Catani and M. H. Seymour,
The Dipole formalism for the calculation of QCD jetcross-sections at next-to-leading order , Phys. Lett.
B378 (1996) 287–301,[ hep-ph/9602277 ].[63] S. Catani and M. H. Seymour,
A General algorithm for calculating jet cross-sections inNLO QCD , Nucl. Phys.
B485 (1997) 291–419, [ hep-ph/9605323 ]. [Erratum: Nucl. Phys.B510, 503 (1998)].[64] S. Catani, S. Dittmaier, M. H. Seymour, and Z. Trocsanyi,
The Dipole formalism fornext-to-leading order QCD calculations with massive partons , Nucl. Phys. B (2002)189–265, [ hep-ph/0201036 ].[65] M. Grazzini, S. Kallweit, and M. Wiesemann,
Fully differential NNLO computations withMATRIX , Eur. Phys. J. C (2018), no. 7 537, [ arXiv:1711.06631 ].[66] NNPDF
Collaboration, R. D. Ball et al.,
Parton distributions from high-precisioncollider data , Eur. Phys. J. C (2017), no. 10 663, [ arXiv:1706.00428 ].[67] J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao,T. Stelzer, P. Torrielli, and M. Zaro, The automated computation of tree-level andnext-to-leading order differential cross sections, and their matching to parton showersimulations , JHEP (2014) 079, [ arXiv:1405.0301 ].[68] S. Frixione, Z. Kunszt, and A. Signer, Three jet cross-sections to next-to-leading order , Nucl. Phys. B (1996) 399–442, [ hep-ph/9512328 ].[69] S. Frixione,
A General approach to jet cross-sections in QCD , Nucl. Phys. B (1997)295–314, [ hep-ph/9706545 ]. 2070] S. Catani, M. Grazzini, and H. Sargsyan,
Azimuthal asymmetries in QCD hardscattering: infrared safe but divergent , JHEP (2017) 017, [ arXiv:1703.08468 ].[71] L. Buonocore, M. Grazzini, and F. Tramontano, The q T subtraction method: electroweakcorrections and power suppressed contributions , Eur. Phys. J. C (2020), no. 3 254,[ arXiv:1911.10166 ].[72] M. Grazzini, S. Kallweit, S. Pozzorini, D. Rathlev, and M. Wiesemann, W + W − production at the LHC: fiducial cross sections and distributions in NNLO QCD , JHEP (2016) 140, [ arXiv:1605.02716 ].[73] M. A. Ebert, I. Moult, I. W. Stewart, F. J. Tackmann, G. Vita, and H. X. Zhu, Subleading power rapidity divergences and power corrections for q T , JHEP (2019) 123,[ arXiv:1812.08189 ].[74] L. Cieri, C. Oleari, and M. Rocco, Higher-order power corrections in atransverse-momentum cut for colour-singlet production at NLO , Eur. Phys. J. C (2019), no. 10 852, [ arXiv:1906.09044 ].[75] S. Catani, M. Grazzini, A. Torre, unpublished.[76] W. Bernreuther and W. Wetzel, Decoupling of Heavy Quarks in the Minimal SubtractionScheme , Nucl. Phys. B197