Gluon-fusion Higgs production in the Standard Model Effective Field Theory

Journal of High Energy Physics, Dec 2017

We provide the complete set of predictions needed to achieve NLO accuracy in the Standard Model Effective Field Theory at dimension six for Higgs production in gluon fusion. In particular, we compute for the first time the contribution of the chromomagnetic operator \( {\overline{Q}}_L\varPhi \sigma {q}_RG \) at NLO in QCD, which entails two-loop virtual and one-loop real contributions, as well as renormalisation and mixing with the Yukawa operator \( {\varPhi}^{\dagger}\varPhi\ {\overline{Q}}_L\varPhi {q}_R \) and the gluon-fusion operator Φ†Φ GG. Focusing on the top-quark-Higgs couplings, we consider the phenomenological impact of the NLO corrections in constraining the three relevant operators by implementing the results into the MadGraph5_aMC@NLO frame-work. This allows us to compute total cross sections as well as to perform event generation at NLO that can be directly employed in experimental analyses.

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Gluon-fusion Higgs production in the Standard Model Effective Field Theory

HJE Gluon-fusion Higgs production in the Standard Model Nicolas Deutschmann 0 1 2 4 5 Claude Duhr 0 1 2 3 5 Fabio Maltoni 0 1 2 5 Eleni Vryonidou 0 1 2 5 0 CH-1211 Geneva 23 , Switzerland 1 Chemin du Cyclotron 2 , 1348 Louvain-La-Neuve , Belgium 2 F-69622 , Villeurbanne , France 3 Theoretical Physics Department , CERN 4 Univ. Lyon , Universite Lyon 1, CNRS/IN2P3, IPNL 5 Science Park 105 , 1098 XG, Amsterdam , The Netherlands We provide the complete set of predictions needed to achieve NLO accuracy in the Standard Model E ective Field Theory at dimension six for Higgs production in gluon fusion. In particular, we compute for the rst time the contribution of the chromomagnetic relevant operators by implementing the results into the MadGraph5 aMC@NLO framework. This allows us to compute total cross sections as well as to perform event generation at NLO that can be directly employed in experimental analyses. Universite catholique de Louvain - E operator QL qRG at NLO in QCD, which entails two-loop virtual and one-loop real contributions, as well as renormalisation and mixing with the Yukawa operator y QL qR and the gluon-fusion operator y GG. Focusing on the top-quark-Higgs couplings, we consider the phenomenological impact of the NLO corrections in constraining the three 1 Introduction 2 3 3.1 3.2 3.3 3.4 4.1 4.2 4.3 4 Phenomenology Cross-section results Di erential distributions Renormalisation group e ects 5 Conclusion and outlook 1 Introduction Gluon fusion in the SM E ective Field Theory Virtual corrections Computation of the two-loop amplitudes UV & IR pole structure Analytic results for the two-loop amplitudes Renormalisation group running of the e ective couplings Five years into its discovery at the LHC, the Higgs boson is still the centre of attention of the high-energy physics community. A wealth of information has been collected on its properties by the ATLAS and CMS experiments [1{5], all of which so far support the predictions of the Standard Model (SM). In particular, the size of the couplings to the weak vector bosons and to the electrically charged third generation fermions has been con rmed, and the rst evidence of the coupling to second generation fermions (either charm quark or muon) could arrive in the coming years, if SM-like. The steady improvement in the precision of the current and forthcoming Higgs measurements invites to explore physics beyond the SM not only via the search of new resonances, as widely pursued at the LHC, but also via indirect e ects on the couplings of the Higgs boson to the known SM particles. The most appealing aspect of such an approach is that, despite being much more challenging than direct searches both experimentally and theoretically, it has the potential to probe new physics scales that are beyond the kinematical reach of the LHC. A powerful and predictive framework to analyse possible deviations in the absence of resonant BSM production is provided by the SM E ective Field Theory (SMEFT) [6{8], i.e., the SM augmented by higher-dimensional operators. Among the most interesting features of this framework is the possibility to compute radiative corrections in the gauge couplings, thus allowing for systematic improvements of the predictions and a strong coupling constant typically entail large e ects at the LHC both in the accuracy { 1 { and the precision. They are therefore being calculated for a continuously growing set of processes involving operators of dimension six featuring the Higgs boson, the bottom and top quarks and the vector bosons. Currently, predictions for the most important associated production channels for the Higgs boson are available in this framework, e.g., VH, VBF and ttH [10{12]. For top-quark production, NLO results for EW and QCD inclusive production, i.e., tj and tt, and for top-quark associated production ttZ, tt have also appeared [13{18]. The e ect of dimension-six operators has also become available recently for top-quark and Higgs decays [19{23]. The situation is somewhat less satisfactory for gluon fusion, which, despite being a loop-induced process in the SM, is highly enhanced by the gluon density in the proton and corrections are now known up to N3LO in the limit of a heavy top quark [24{26]. The full quark-mass dependence is known up to NLO [27{30], while at NNLO only subleading terms in the heavy top-mass expansion [31{34] and leading contributions to the top/bottom interference [35, 36] are known. Beyond inclusive production, the only available NNLO result is the production of a Higgs boson in association with a jet in the in nite top-mass limit [37{39], while cross sections for H + n-jets, n = 2; 3, are known only at NLO in the heavy top-mass expansion [40, 41]. In the SMEFT, most studies have been performed at LO, typically using approximate rescaling factors obtained from SM calculations. Higher-order results have only been considered when existing SM calculations could be readily used within the SMEFT. The simplest examples are the inclusion of higher orders in the strong coupling to the contribution of two speci c dimension-six operators, namely the Yukawa operator ( y )QL qR and the gluon-fusion operator ( y )GG. The former can be accounted for by a straightforward modi cation of the Yukawa coupling of the corresponding heavy quark, b or t, while the latter involves the computation of contributions identical to SM calculations in the limit of an in nitely-heavy top quark. Results for the inclusive production cross section including modi ed top and bottom Yukawa couplings and an additional direct Higgs-gluons interaction are available at NNLO [42] and at N3LO [43, 44]. At the di erential level, phenomenological studies at LO have shown the relevance of the high transverse momentum region of the Higgs boson in order to resolve degeneracies among operators present at the inclusive level [12, 45{47]. Recently, the calculation of the Higgs spectrum at NLO+NNLL level for the Yukawa (both b and t) and Higgs-gluons operator has appeared [48, 49]. The purpose of this work is to provide the contribution of the chromomagnetic operator QL qRG to inclusive Higgs production at NLO in QCD, thereby completing the set of predictions (involving only CP -even interactions) needed to achieve NLO accuracy in the SMEFT for this process. The rst correct computation at one-loop of the contribution of chromomagnetic operator of the top quark to gg ! H has appeared in the erratum of ref. [50] and later con rmed in refs. [12, 49]. The LO contribution of the chromomagnetic operator of the top-quark to H+jet was computed in ref. [12]. An important conclusion drawn in ref. [12] was that even when the most stringent (and still approximate) constraints from tt production are considered [14], this operator sizably a ects Higgs production, both in gluon fusion (single and double Higgs) and ttH production. { 2 { At LO the chromomagnetic operator enters Higgs production in gluon fusion at one loop. Therefore NLO corrections in QCD entail two-loop virtual and one-loop real contributions. The latter can nowadays easily be computed using an automated approach. The former, however, involve a non-trivial two-loop computation that requires analytic multiloop techniques and a careful treatment of the renormalisation and mixing in the SMEFT, both of which are presented in this work for the rst time. In particular, while the full mixing pattern of the SMEFT at one loop is known [51{53], a new two-loop counterterm enters our computation, and we provide its value for the rst time here. Moreover, we present very compact analytic results for all the relevant amplitudes up to two loop order. Focusing on possibly anomalous contributions in top-quark-Higgs interactions, we then leading logarithmic renormalisation group running of the Wilson coe cients. In section 4 we perform a phenomenological study at NLO, in particular of the behaviour of the QCD and EFT expansion at the total inclusive level and provide predictions for the pT spectrum of the Higgs via a NLO+PS approach. 2 Gluon fusion in the SM E ective Field Theory The goal of this paper is to study the production of a Higgs boson in hadron collisions in the SMEFT, i.e., the SM supplemented by a complete set of operators of dimension six, O1 = O2 = gs2 y y v 2 2 v 2 2 QL ~ tR ; G a Ga ; O3 = gs QL ~ T a tR G a ; { 3 { LEFT = LSM + X Cb i 2 Oi + h.c. : i The sum in eq. (2.1) runs over a basis of operators Oi of dimension six, is the scale of new physics and Cib are the (bare) Wilson coe cients, multiplying the e ective operators. A complete and independent set of operators of dimension six is known [7, 55]. In this paper, we are only interested in those operators that modify the contribution of the heavy quarks, bottom and top quarks, to Higgs production in gluon fusion. Focusing on the top quark, there are three operators of dimension six that contribute to the gluon-fusion process, (2.1) (2.2) (2.3) (2.4) g H (Φ† Φ) Q¯LΦ qR (Φ† Φ) GG Q¯LΦ σqRG where gs is the (bare) strong coupling constant and v denotes the vacuum expectation value (vev) of the Higgs eld ( ~ = i 2 ). QL is the left-handed quark SU(2)-doublet containing the top quark, tR is the right-handed SU(2)-singlet top quark, and Ga is the gluon eld strength tensor. Finally, T a is the generator of the fundamental representation of SU(3) (with [T a; T b] = 12 ab) and = 2i can be obtained by simply making the substitutions f , tR ! bRg. Second, while O2 is hermitian O1 and O3 are not.1 In this work, we focus on the CP -even contributions of O1 and O3. For this reason, all the Wilson coe cients Ci with i = 1; 2; 3 are real. Representative Feynman diagrams contributing at LO are shown in gure 1. ~ ! In the SM and at leading order (LO) in the strong coupling the gluon-fusion process is mediated only by quark loops. This contribution is proportional to the mass of the corresponding quark and therefore heavy quarks dominate. While we comment on the b (and possibly c) contributions later, let us focus on the leading contributions coming from the top quark, i.e., the contributions from the operators of dimension six shown in eqs. (2.2){(2.4). The (unrenormalised) amplitude can be cast in the form i S C1b v2 2 b Ab(g g ! H) = s [(p1 p2) ( 1 2) (p1 2) (p2 1)] 1 v Ab;0(mtb; mH ) (2.5) + p 2 2 Ab;1(mtb; mH )+ C2b v 2 Ab;2(mtb; mH )+ p Cb 3 2 2 Ab;3(mtb; mH ) +O(1= 4) ; where sb = gs2=(4 ) denotes the bare QCD coupling constant and mH and mtb are the bare masses of the Higgs boson and the top quark. The factor S = e E (4 ) is the usual MS factor, with E = 0(1) the Euler-Mascheroni constant and is the scale introduced by dimensional regularisation. For i = 0, the form factor Ab;i denotes the unrenormalised SM contribution to gluon fusion [56], while for i > 0 it denotes the form factor with a single2 1Note that in eq. (2.1) we adopt the convention to include the hermitian conjugate for all operators, be they hermitian or not. This means that the overall contribution from O2 in LEFT is actually 2C2O2= 2 2According to our power counting rules, multiple insertions of an operator of dimension six correspond . to contributions of O(1= 4) in the EFT, and so they are neglected. { 4 { operator Oi inserted [48, 50, 57]. The normalisation of the amplitudes is chosen such that all coupling constants, as well as all powers of the vev v, are explicitly factored out. Each form factor admits a perturbative expansion in the strong coupling, Ab;i(mtb; mH ) = 1 X k=0 S 2 b k s A(bk;i)(mtb; mH ) : (2.6) Some comments about these amplitudes are in order. First, after electroweak symmetry breaking, the operator O1 only amounts to a rescaling of the Yukawa coupling, i.e., Ab;1 is simply proportional to the bare SM amplitude. Second, at LO the operator O2 contributes at tree level, while the SM amplitude and the contributions from O1 and O3 are loopinduced. Finally, this process has the unusual feature that the amplitude involving the chromomagnetic operator O3 is ultraviolet (UV) divergent, and thus requires renormalisation, already at LO [12, 49, 50]. The UV divergence is absorbed into the e ective coupling that multiplies the operator O2, which only enters at tree level at LO. The renormalisation at NLO will be discussed in detail in section 3. The goal of this paper is to compute the NLO corrections to the gluon-fusion process with an insertion of one of the dimension six operators in eqs. (2.2){(2.4). We emphasise that a complete NLO computation requires one to consider the set of all three operators in eq. (2.2){(2.4), because they mix under renormalisation [51{53]. At NLO, we need to consider both virtual corrections to the LO process g g ! H as well as real corrections due to the emission of an additional parton in the nal state. Starting from NLO, also partonic channels with a quark in the initial state contribute. Since the contribution from O1 is proportional to the SM amplitude, the corresponding NLO corrections can be obtained from the NLO corrections to gluon-fusion in the SM including the full top-mass dependence [27, 28, 30, 58]. The NLO contributions from O2 are also known, because they are proportional to the NLO corrections to gluon-fusion in the SM in the limit where the top quark is in nitely heavy [59] (without the higher-order corrections to the matching coe cient). In particular, the virtual corrections to the insertion of O2 are related to the QCD form factor, which is known through three loops in the strong coupling [60{69]. Hence, the only missing ingredient is the NLO contributions to the process where the chromomagnetic operator O3 is inserted. The computation of this ingredient, which is one of the main results of this paper, will be presented in detail in the next section. As a nal comment, we note that starting at two loops other operators of EW and QCD nature will a ect gg ! H. In the case of EW interactions, by just looking at the SM EW contributions [70, 71], it is easy to see that many operators featuring the Higgs eld will enter, which in a few cases could also lead to constraints, see, e.g., the trilinear Higgs self coupling [72, 73]. In the case of QCD interactions, operators not featuring the Higgs eld will enter, which, in general, can be more e ciently bounded from other observables. For example, the operator gsf abcGa G b G c contributes at two loops in gg ! H and at one loop in gg ! Hg. The latter process has been considered in ref. [74], where e ects on the transverse momentum of the Higgs were studied. For the sake of completeness, we have reproduced these results in our framework, and by considering the recent constraints on this operator from multi-jet observables [75], we have con rmed that the Higgs pT cannot { 5 { be signi cantly a ected. For this reason we do not discuss further this operator in this paper. Four-fermion operators also contribute starting at two loops to gluon fusion but as these modify observables related to top quark physics at leading order [76, 77] we expect them to be independently constrained and work under the assumption that they cannot signi cantly a ect gluon fusion. Virtual corrections Computation of the two-loop amplitudes In this section we describe the virtual corrections to the LO amplitudes in eq. (2.5). For HJEP12(07)63 the sake of the presentation we focus here on the calculation involving a top quark and discuss later on how to obtain the corresponding results for the bottom quark. With the exception of the contributions from O2, all processes are loop-induced, and so the virtual corrections require the computation of two-loop form factor integrals with a closed heavyquark loop and two external gluons. We have implemented the operators in eqs. (2.2){(2.4) into QGraf [78], and we use the latter to generate all the relevant Feynman diagrams. The QGraf output is translated into FORM [79, 80] and Mathematica using a custom-made code. The tensor structure of the amplitude is xed by gauge-invariance to all loop orders, cf. eq. (2.5), and we can simply project each Feynman diagram onto the transverse polarisation tensor. The resulting scalar amplitudes are then classi ed into distinct integral topologies, which are reduced to master integrals using FIRE and LiteRed [81{85]. After reduction, we can express all LO and NLO amplitudes as a linear combination of one and two-loop master integrals. The complete set of one- and two-loop master integrals is available in the literature [58, 86{88] in terms of harmonic polylogarithms (HPLs) [89], Z z 0 H(a1; : : : ; aw; z) = dt f (a1; t) H(a2; : : : ; aw; z) ; mm2Ht2 = p1 p1 (1 x x)2 ; 4= 4= + 1 1 : { 6 { 1 t ; 1 w! H(0; : : : ; 0; z) = |w t{izmes} logw z : f (1; t) = 1 1 t ; In the case where all the ai's are zero, we de ne, f (0; t) = f ( 1; t) = 1 1 + t : The number of integrations w is called the weight of the HPL. The only non-trivial functional dependence of the master integrals is through the ratio of the Higgs and the top masses, and it is useful to introduce the following variable, (3.1) (3.2) (3.3) (3.4) (3.5) The change of variables in eq. (3.4) has the advantage that the master integrals can be written as a linear combination of HPLs in x. In the kinematic range that we are interested in, 0 < m2H < 4mt2, the variable x is a unimodular complex number, jxj = 1, and so it can be conveniently parametrised in this kinematics range by an angle , x = ei ; In terms of this angle, the master integrals can be expressed in terms of (generalisations of) Clausen functions (cf. refs. [58, 90{93] and references therein), Clm1;:::;mk ( ) = ( Re Hm1;:::;mk e Im Hm1;:::;mk e i ; if k + w even ; i ; if k + w odd ; (3.6) (3.7) HJEP12(07)63 where we used the notation (jm|1j {1z) ti}mes (jm|kj {1z) ti}mes Hm1;:::;mk (z) = H( 0; : : : ; 0 ; 1; : : : ; 0; : : : ; 0 ; k; z) ; i sign(mi) : (3.8) The number k of non-zero indices is called the depth of the HPL. Inserting the analytic expressions for the master integrals into the amplitudes, we can express each amplitude as a Laurent expansion in whose coe cients are linear combinations of the special functions we have just described. The amplitudes have poles in which are of both ultraviolet (UV) and infrared (IR) nature, whose structure is discussed in the next section. 3.2 UV & IR pole structure In this section we discuss the UV renormalisation and the IR pole structure of the LO and NLO amplitudes. We start by discussing the UV singularities. We work in the MS scheme, and we write the bare amplitudes as a function of the renormalised amplitudes as, Ab( sb; Cib; mtb; mH ) = Zg 1 A( s( 2); Ci( 2); mt( 2); mH ; ) ; (3.9) where Zg is the eld renormalisation constant of the gluon eld and mt( 2) are the renormalised strong coupling constant, Wilson coe cients and top mass in the MS scheme, and denotes the renormalisation scale. The renormalised parameters are s( 2), Ci( 2) and related to their bare analogues through (3.10) S sb = Cib = 2 Z s s( 2) ; ai ZC;ij Cj ( 2) ; mtb = mt( 2) + mt ; { 7 { with (a1; a2; a3) = (3; 0; 1). Unless stated otherwise, all renormalised quantities are assumed to be evaluated at the arbitrary scale 2 throughout this section. We can decompose the renormalised amplitude into the contributions from the SM and the e ective operators, similar to the decomposition of the bare amplitude in eq. (2.5) A(g g ! H) = and each renormalised amplitude admits a perturbative expansion in the renormalised strong coupling constant, Ai(mt; mH ) = s k Ai(k)(mt; mH ) : The presence of the e ective operators alters the renormalisation of the SM parameters. Throughout this section we closely follow the approach of ref. [12], where the renormalisation of the operators at one loop was described. The one-loop UV counterterms for the strong coupling constant and the gluon eld are given by 1 X k=0 s C3 1 2 s C3 1 2 2 mt2 Zg = 1 + Zg;SM + Z s = 1 + Z s;SM 2 mt2 2 mt2 p p 2 v mt + O( s2) ; 2 v mt + O( s2) ; Zg;SM = Z s;SM = s 1 6 4 s 0 + O( s2) ; s 1 6 2 mt2 + O( s2) ; where Zg;SM and Z s;SM denote the one-loop UV counterterms in the SM, (3.11) (3.12) where Nc = 3 is the number of colours and Nf = 5 is the number of massless avours. We work in a decoupling scheme and we include a factor 2=mt2 into the counterterm. As a result only massless avours contribute to the running of the strong coupling, while the top quark e ectively decouples [59]. The renormalisation of the strong coupling and the gluon eld are modi ed by the presence of the dimension six operators, but the e ects cancel each other out [50]. Similarly, the renormalisation of the top mass is modi ed by the presence of the e ective operators, mt = 2 2 v mt2 + O( s2) ; where the SM contribution is mtSM = s mt + O( s2) : { 8 { In eq. (3.16) we again include the factor into the counterterm in order to decouple the e ects from operators of dimension six from the running of the top mass in the 2=mt2 MS scheme. written in the form The renormalisation of the e ective couplings Cib is more involved, because the operators in eqs. (2.2){(2.4) mix under renormalisation. The matrix ZC of counterterms can be ZC = 1 + Z(0) + C s Z(1) + O( s2) : C We have already mentioned that the amplitude Ab;3 requires renormalisation at LO in the strong coupling, and the UV divergence is proportional to the LO amplitude A(b0;2) [12, 49, 50]. As a consequence, ZC(0) is non-trivial at LO in the strong coupling, At NLO, we also need the contribution ZC(1) to eq. (3.18). We have Z(0) = BB 0 0 C 8mt2 1 v2 C z23 1 6 C ; C A where, apart from z23, all the entries are known [51{53]. z23 corresponds to the counterterm that absorbs the two-loop UV divergence of the operator O3, which is proportional to the tree-level amplitude A(b0;2) in our case. This counterterm is not available in the literature, yet we can extract it from our computation. NLO amplitudes have both UV and IR poles, and so we need to disentangle the two types of divergences if we want to isolate the counterterm z23. We therefore rst review the structure of the IR divergences of NLO amplitudes, and we will return to the determination of the counterterm z23 at the end of this section. A one-loop amplitude with massless gauge bosons has IR divergences, arising from regions in the loop integration where the loop momentum is soft or collinear to an external massless leg. The structure of the IR divergences is universal in the sense that it factorises from the underlying hard scattering process. More precisely, if A one-loop amplitude describing the production of a colourless state from the scattering of (1) denotes a renormalised two massless gauge bosons, then we can write [94] A (1) = I(1)( ) A(0) + R ; where A(0) is the tree-level amplitude for the process and R is a process-dependent remainder that is nite in the limit ! 0. The quantity I(1)( ) is universal (in the sense that it does not depend on the details of the hard scattering) and is given by I(1)( ) = e E (1 ) s12 i0 2 3 2 + 2 0 ; where s12 = 2p1p2 denotes the center-of-mass energy squared of the incoming gluons. { 9 { (3.18) (3.19) (3.20) (3.21) (3.22) Since in our case most amplitudes are at one loop already at LO, we have to deal with two-loop amplitudes at NLO. However, since the structure of the IR singularities is independent of the details of the underlying hard scattering, eq. (3.21) remains valid for two-loop amplitudes describing loop-induced processes, and we can write Ai (1) = I(1)( ) Ai (0) + Ri ; 0 i We have checked that our results for amplitudes which do not involve the operator O3 have the correct IR pole structure at NLO. For A(31), instead, we can use eq. (3.23) as a constraint on the singularities of the amplitude. This allows us to extract the two-loop UV Note that the coe cient of the double pole is in fact xed by requiring the anomalous dimension of the e ective couplings to be nite. We have checked that eq. (3.24) satis es this criterion, which is a strong consistency check on our computation. Let us conclude our discussion of the renormalisation with a comment on the relationship between the renormalised amplitudes in the SM and the insertion of the operator O1. We know that the corresponding unrenormalised amplitudes are related by a simple rescaling, and the constant of proportionality is proportional to the ratio Cb=mtb. There is a 1 priori no reason why such a simple relationship should be preserved by the renormalisation procedure. In (the variant of) the MS-scheme that we use, the renormalised amplitudes are still related by this simple scaling. This can be traced back to the fact that the MS counterterms are related by mtSM = s Z(1) C 11 + O( s2) : If the top mass and the Wilson coe cient C1b are renormalised using a di erent scheme which breaks this relation between the counterterms, the simple relation between the amplitudes A0 (1) and A1 (1) will in general not hold after renormalisation. 3.3 Analytic results for the two-loop amplitudes In this section we present the analytic results for the renormalised amplitudes that enter the computation of the gluon-fusion cross section at NLO with the operators in eqs. (2.2){ (2.4) included. We show explicitly the one-loop amplitudes up to O( 2) in dimensional regularisation, as well as the nite two-loop remainders Ri de ned in eq. (3.21). The amplitudes have been renormalised using the scheme described in the previous section and all scales are xed to the mass of the Higgs boson, 2 = m2H . The operator O2 only contributes at one loop at NLO, and agrees (up to normalisation) with the one-loop corrections to Higgs production via gluon-fusion [59]. The amplitude is independent of the top mass through one loop, and so it evaluates to a pure number, p A2 (0) = 32 2 2 and R2 = 16 i 3 0 ; (3.26) where 0 is de ned in eq. (3.15). The remaining amplitudes have a non-trivial functional dependence on the top mass through the variables and de ned in eq. (3.4) and (3.6). We have argued in the previous section that in the MS-scheme the renormalised amplitudes A(01) and A(11) are related by a simple rescaling, We therefore only present results for the SM contribution and the contribution from O3. We have checked that our result for the two-loop amplitude in the SM agrees with the results of refs. [27, 28, 30, 58]. The two-loop amplitude A(31) is genuinely new and is presented The one-loop amplitude in the SM can be cast in the form (3.27) (3.28) (3.29) (3.30) (3.31) A(00) = a0 + (a1 + log a0) + 2 a2 + log a1 + log2 a0 + O( 3) ; where we have de ned the function 1 2 where the coe cients ai are given by a0 = a1 = a2 = The nite remainder of the two-loop SM amplitude is R0 = 0 2 28 Cl2( ) 28 Cl3( )+28 3 +5 2 log +28 2 48 log 252 R( )+ (4 12 p(4 ) ) 13 2 +24 log 16 ; R( ) = 16 2 Cl 2( ) + 28 2 Cl2( ) + 96 Cl 4( ) 72 Cl4( ) 128 3 + p(4 2 ) where the coe cients bi are given by 2 2 p(4 1 4 ; 4 Cl 2( ) + 8 Cl 3( ) + 2 The nite remainder of the two-loop amplitude A(31) is R3 = i mt 0 1 + 2 log + 2 m2t (16 log + 35) The one-loop amplitude involving the operator O3 is [ log(4 ) 2 Cl 2( ) its inverse, e.g., Although the main focus of this paper is to include e ects from dimension six operators that a ect the gluon-fusion cross section through the top quark, let us conclude this section by making a comment about e ects from the bottom, and to a lesser extent, the charm quark. The amplitudes presented in this section are only valid if the Higgs boson is lighter than the quark-pair threshold, < 4. It is, however, not di cult to analytically continue our results to the region above threshold where > 4. Above threshold, the variable x de ned in eq. (3.5) is no longer a phase, but instead we have 1 < x < 0. As a consequence, the Clausen functions may develop an imaginary part. In the following we describe how one can extract the correct imaginary part of the amplitudes in the region above threshold We start from eq. (3.7) and express all Clausen functions in terms of HPLs in x and 1 2i Cl2( ) = Im H(0; 1; x) = HPLs evaluated at 1=x can always be expressed in terms of HPLs in x. For example, one nds 2 3 ; H(0; 1; 1=x) = H(0; 1; 1=x) + i H(0; x) H(0; 0; x) + jxj = 1 and Im x > 0 : Similar relations can be derived for all other HPLs in an algorithmic way [89, 95, 96]. The previous equation, however, is not yet valid above threshold, because the logarithms H(0; x) = log x may develop an imaginary part. Indeed, when crossing the threshold x approaches the negative real axis from above, x ! x + i0, and so the correct analytic continuation of the logarithms is H(0; x) = log x ! H(0; x) + i : The previous rule is su cient to perform the analytic continuation of all HPLs appearing in our results. Indeed, it is known that an HPL of the form H(a1; : : : ; ak; x) has a branch point at x = 0 only if ak = 0, and, using the shu e algebra properties of HPLs [89], any HPL of the form H(a1; : : : ; ak; 0; x) can be expressed as a linear combination of products of HPLs such that if their last entry is zero, then all of its entries are zero. The amplitudes can therefore be expressed in terms of two categories of HPLs: those whose last entry is non-zero and so do not have a branch point at x = 0, and those of the form H(0; : : : ; 0; x) = which are continued according to eq. (3.37). 1 n! logn x, Using the procedure outlined above, it is possible to easily perform the analytic continuation of our amplitudes above threshold. The resulting amplitudes contribute to the gluon-fusion process when light quarks, e.g., massive bottom and/or charm quarks, are taken into account. Hence, although we focus primarily on the e ects from the top quark in this paper, our results can be easily extended to include e ects from bottom and charm quarks as well. (3.36) (3.37) 3.4 After renormalisation, our amplitudes depend explicitly on the scale , which in the following we identify with the factorisation scale F . It can, however, be desirable to choose di erent scales for the strong coupling constant, the top mass and the e ective couplings. In this section we derive and solve the renormalisation group equations (RGEs) for these parameters. Since we are working in a decoupling scheme for the top mass, the RGEs for the strong coupling constant and the top mass are identical to the SM with Nf = 5 massless avours. We have checked that we correctly reproduce the evolution of s and mt in the MS scheme, and we do not discuss them here any further. For the RGEs satis ed by the e ective couplings, we nd dC d log 2 = C ; where C = (C1; C2; C3)T , and the anomalous dimension matrix is given by = BB0 0 0 B B B B 0 0 0 0 0 0 1 mt( 2) CCCC + v 1 C C A 0 s( 2) BBBB 0 0 C C A 23 As already mentioned in the previous section, the double pole from the two-loop counterterm in eq. (3.24) cancels. We can solve the RGEs in eq. (3.38) to one loop, and we nd C1( 2) = C1(Q2) C2( 2) = C2(Q2) + p2 C3(Q2) 16 2 log Q2 s(Q2) log 2 Q2 C1(Q2) + 8 C3(Q2) mt2(Q2) v2 + O( s(Q2)2) ; p 2 s(Q2) 192 3 C3(Q2) log Q2 5 log 2 Q2 69 + O( s(Q2)2) ; C3( 2) = C3(Q2) + C3(Q2) s(Q2) 6 log Q2 + O( s(Q2)2) : We show in gure 2 the quantitative impact of running and mixing by varying the renormalisation scale from 10 TeV to mH =2 in two scenarios: one where all Wilson coe cients are equal at 10 TeV and another where only C3 is non-zero. This latter example serves as a reminder of the need to always consider the e ect of all the relevant operators in phenomenological analyses as choosing a single operator to be non-zero is a scale-dependent choice. 4 4.1 Phenomenology Cross-section results In this section we perform a phenomenological study of Higgs production in the SMEFT, focusing on anomalous contributions coming from the top quark. Results are obtained 2 mt(Q2) 2 mt(Q2) v v 2 (3.38) (3.39) (3.40) C C C 1 2 3 within the MadGraph5 aMC@NLO framework [54]. The computation builds on the implementation of the dimension-six operators presented in ref. [12]. Starting from the SMEFT Lagrangian, all tree-level and one-loop amplitudes can be obtained automatically using a series of packages [97{102]. The two-loop amplitudes for the virtual corrections are implemented in the code through a reweighting method [103, 104]. Within the MadGraph5 aMC@NLO framework NLO results can be matched to parton shower programs, such as PYTHIA8 [105] and HERWIG++ [106], through the MC@NLO [107] formalism. Results are obtained for the LHC at 13 TeV with MMHT2014 LO/NLO PDFs [108], for LO and NLO results respectively. The values of the input parameters are mt = 173 GeV ; mH = 125 GeV ; mZ = 91:1876 GeV ; E1W = 127:9 ; GF = 1:16637 10 5 GeV 2 : The values for the central scales for R; F and EFT are chosen as mH =2, and we work with the top mass in the on-shell scheme. We parametrise the contribution to the cross section from dimension-six operators as = SM + X 1TeV2 Ci i + X 1TeV4 2 4 CiCj ij : i i j Within our setup we can obtain results for SM, i, and ij . We note here that results for single Higgs and H + j production in the SMEFT were presented at LO in QCD in ref. [12]. The normalisation of the operators used here di ers from the one in ref. [12], but we have found full agreement between the LO results presented here and those of ref. [12] when this di erence is taken into account. Furthermore, the SM top-quark results obtained here have been cross-checked with the NLO+PS implementation of aMCSusHi [109]. Our results for the total cross section at the LHC at 13 TeV at LO and NLO are shown in table 1. We include e ects from bottom-quark loops (top-bottom interference and pure bottom contributions) into the SM prediction by using aMCSusHi. However, in this rst study, we neglect bottom-quark e ects from dimension-six operators in i and ij C C 1 2 3 (10 TeV) HJEP12(07)63 (4.1) (4.2) (4.3) 13 TeV SM 1 2 3 11 22 33 12 13 23 Ki = KU + s i(0) ; (1) i as we assume them to be subleading. As mentioned above, our analytic results and MC implementation can be extended to also include these e ects. We see that the contributions from e ective operators have K-factors that are slightly smaller then their SM counterpart, with a residual scale dependence that is almost identical to the SM. In the following we present an argument which explains this observation. We can describe the total cross section for Higgs boson production to a good accuracy by taking the limit of an in nitely heavy top quark, because most of the production happens near threshold. In this e ective theory where the top quark is integrated out, all contributions from SMEFT operators can be described by the same contact interaction Ga Ga H. The Wilson coe cient can be written as = 0 + X Ci i ; as a perturbative series i = K-factor Ki can be decomposed as where 0 denotes the SM contribution and i those corresponding to each operator Oi in the SMEFT. As a result each i is generated by the same Feynman diagrams both at LO and NLO in the in nite top-mass EFT. The e ect of radiative corrections is, however, not entirely universal as NLO corrections to the in nite top-mass EFT amplitudes come both from diagrammatic corrections and corrections to the Wilson coe cients i, which can be obtained by matching the SMEFT amplitude to the in nite top mass amplitude, as illustrated in gure 3. Indeed, each i can be expressed in terms of SMEFT parameters (0) + s i i (1) + O( s2). In the in nite top mass EFT, each where KU is the universal part of the K-factor, which is exactly equal to K2. By subtracting K2 to each Ki in the in nite top mass limit numerically (setting mt = 10TeV), (4.4) (4.5) mass EFT at LO (left) and at NLO (right). The NLO amplitude in the in nite top-mass EFT contains two elements: diagrammatic corrections, which contribute universally to the K-factors and Wilson coe cient corrections, which are non-universal. we could extract the ratios s ii and check explicitly that these non-universal corrections are subdominant compared to the universal diagrammatic corrections, which explains the similarity of the e ects of radiative corrections for each contribution. Our results can be used to put bounds on the Wilson coe cients from measurements of the gluon-fusion signal strength ggF at the LHC. Whilst here we do not attempt to perform a rigorous t of the Wilson coe cients, useful information can be extracted by a simple t. For illustration purposes, we use the recent measurement of the gluon-fusion signal strength in the diphoton channel by the CMS experiment [110] which we compare to our predictions for this signal strength under the assumption that the experimental selection e ciency is not changed by BSM e ects ggF = 1:1 0:19; ggF = 1 + C1 1 + C2 2 + C3 3 SM ; where we set = 1TeV and kept only the O(1= 2) terms. We therefore nd that we can put the following constraint on the Wilson coe cients with 95% con dence level: 0:28 < 0:128C1 + 114C2 + 2:28C3 < 0:48: While the correct method for putting bounds on the parameter space of the SMEFT is to consider the combined contribution of all relevant operators to a given observable, the presence of unconstrained linear combinations makes it interesting to consider how each operator would be bounded if the others were absent in order to obtain an estimate of the size of each individual Wilson coe cient. Of course such estimates must not be taken as actual bounds on the Wilson coe cients and should only be considered of illustrative (4.6) (4.7) (4.8) value. We obtain most 10%. 3:8 < C1 < 2:2; 0:0025 < C2 < 0:0043; 0:12 < C3 < 0:21 : (4.9) For these individual operator constraints, the impact of the ii terms on the limits is at For reference we note that if one includes the O(1= 4) contributions the linear combination in the bound becomes a quadratic one: 0:28 < 0:128C1 + 114C2 + 2:28C3 + 0:0038C12 + 3000C22 + 1:13C32 6:78C1C2 0:138C1C3 + 122C2C3 < 0:48 : (4.10) 10-1 e p 0 0 O2 O3 O L N M a _ 5 h p a r G d a M O L N p a r G d a M 0.12 pμRp=→μF=HμiEnFTth=e62E.F5TGLeHVC13 NLO+PS, MMHT2014NLO 0.10 Interference with the SM transverse momentum. Right: Higgs rapidity. SM contributions and individual operator contributions are displayed. Lower panels give the ratio over the SM. Di erential distributions In the light of di erential Higgs measurements at the LHC, it is important to examine the impact of the dimension-six operators on the Higgs pT spectrum. It is known that measurements of the Higgs pT spectrum can be used to lift the degeneracy between O1 and O2 [12, 45, 111]. For a realistic description of the pT spectrum, we match our NLO predictions to the parton shower with the MC@NLO method [107], and we use PYTHIA8 [105] for the parton shower. Note that we have kept the shower scale at its default value in MC@NLO, which gives results that are in good agreement with the optimised scale choice of ref. [112], as discussed in ref. [109]. The normalised distributions for the transverse momentum and rapidity of the Higgs boson are shown in gures 4 for the interference contributions. The impact of the O(1= 4 ) terms is demonstrated in gure 5 for the transverse momentum distribution. We nd that the operators O3 and O2 give rise to harder transverse momentum tails, while for O1 the shape is identical to the SM. The dimension-six operators have no impact on the shape of the rapidity distribution. The O(1= 4) contributions involving O3 and O2 are harder than those involving O1. Finally we show the transverse momentum distributions for several benchmark points which respect the total cross-section bounds in gure 6. The operator coe cients are chosen such that eq. (4.10) is satis ed. We nd that large deviations can be seen in the tails of the distributions for coe cient values which respect the total cross-section bounds. 4.3 Renormalisation group e ects The impact of running and mixing between the operators is demonstrated in gure 7, where we show the individual (O(1= 2)) contributions from the three operators in gluonfusion Higgs production at LO and NLO, as a function of EFT, assuming that C3 = 1, C1 = C2 = 0 at EFT = mH =2 and = 1 TeV. While at = mH =2 the only contribution 10-1 e p σ M 2 S O2O3 O L L N N _ 5 h h p p a a r r G G d d a a M M 10-1 ibn10-2 / σ d σ / 1 10-3 102 101 ] 100 b p [ n i b r e pσ10-1 10-2 10-3 2 M tehS 1.5 r veo 1 o i taR 0.5 0 0 50 ii. Right: interference between operators, ij. SM contributions and operator contributions are displayed. Lower panels give the ratio over the SM. Higgs pT SM B5 (5,0.004,0.2) B4 (2,-0.006,0.3) B3 (-1,0.004,-0.25) B2 (-4,0.005,-0.5) B1 (4,0.004,0.21) p p → H in the EFT LHC13 μR=μF=μEFT=62.5 GeV NLO+PS, MMHT2014NLO dashed: only 1/Λ2 terms, solid: including 1/Λ4 terms O L 5 h p a r G d a M 200pTH [GeV] 250 50 100 150 300 350 400 450 coe cients. The lower panel shows the ratio over the SM prediction for the various benchmarks and the SM scale variation band. is coming from the chromomagnetic operator, this contribution changes rapidly with the scale. While the e ect of the running of C3 is only at the percent level, 3 has a strong dependence on the scale. At the same time non-zero values of C1 and C2 are induced through renormalisation group running, which gives rise to large contributions from O2. We nd that the dependence on the EFT scale is tamed when the sum of the three contributions 200 150 100 50 0 −50 −100 −150 pp → H LHC13 solid: NLO, dashed: LO 100 μ EF T [GeV] 1000 . at the LHC at 13 TeV as a function of the EFT scale. Starting from one non-zero coe cient at EFT = mH =2 we compute the EFT contributions at di erent scales, taking into account the running and mixing of the operators. LO and NLO predictions are shown in dashed and solid lines respectively. is considered. This is the physical cross section coming from C3(mH =2) = 1 which has a weaker dependence on the EFT scale. The dependence of this quantity on the scale gives an estimation of the higher order corrections to the e ective operators and should be reported as an additional uncertainty of the predictions. By comparing the total contributions at LO and NLO we nd that the relative uncertainty is reduced at NLO. 5 Conclusion and outlook A precise determination of the properties of the Higgs boson and, in particular, of its couplings to the other SM particles is one of the main goals of the LHC programme of the coming years. The interpretation of such measurements, and of possible deviations in the context of an EFT, allows one to put constraints on the type and strength of hypothetical new interactions, and therefore on the scale of new physics, in a model-independent way. The success of this endeavour will critically depend on having theoretical predictions that at least match the precision of the experimental measurements, both in the SM and in the SMEFT. In this work we have computed for the rst time the contribution of the (CP -even part of the) QL qRG operator to the inclusive Higgs production at NLO in QCD. Since the NLO corrections for the other two (CP -even) operators entering the same process are available in the literature, this calculation completes the SMEFT predictions for this process at the NLO accuracy. Even though our results can be easily extended to include anomalous couplings of the bottom quark, we have considered in the detail the case where new physics mostly a ects the top-quark couplings. Our results con rm the expectations based on previous calculations and on the general features of gluon-fusion Higgs production: at the inclusive level the K-factor is of the same order as that of the SM and of the other two operators. The residual uncertainties estimated by renormalisation and factorisation scale dependence also match extremely well. The result of the NLO calculation con rms that the chromomagnetic operator cannot be neglected for at least two reasons. The rst is of purely theoretical nature: the individual e ects of QL ~ tRG and y GG are very much dependent on the EFT scale, while their sum is stable and only mildly a ected by the scale choice. The second draws from the present status of the constraints. Considering the uncertainties in inclusive Higgs production cross section measurements and the constraints from tt production, the impact of the chromomagnetic operator cannot be neglected in global ts of the Higgs couplings. As a result, a two-fold degeneracy is left unresolved by a three-operator t using the total Higgs cross section and one is forced to look for other observables or processes to constrain all three of the operators. The implementation of the nite part of the two-loop virtual corrections into MadGraph5 aMC@NLO has also allowed us to study the process at a fully di erential level, including the e ects of the parton shower resummation and in particular to compare the transverse momentum distributions of the SM and the three operators in the region of the parameter space where the total cross section bound is respected. Once again, we have found that the contributions from QL ~ tRG and y GG are similar and produce a shape with a harder tail substantially di erent from that of the SM and the Yukawa operator (which are the same). While QL ~ tRG and y GG cannot really be distinguished in gluon-fusion Higgs production, they do contribute in a very di erent way to ttH where the e ect of y GG is extremely weak. Therefore, we expect that H; H+jet, and ttH (and possibly tt) can e ectively constrain the set of the three operators. In this work we have mostly focused our attention on the top-quark-Higgs boson interactions and only considered CP -even operators. As mentioned above and explained in section 3, extending it to include anomalous couplings for lighter quarks, the bottom and possibly the charm, is straightforward. On the other hand, extending it to include CP -odd operators requires a new independent calculation. We reckon both developments worth pursuing. Acknowledgments This work was supported in part by the ERC grant \MathAm", by the FNRS-IISN convention \Fundamental Interactions" FNRS-IISN 4.4517.08, and by the European Union Marie Curie Innovative Training Network MCnetITN3 722104. E.V. is supported by the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scienti c Research (NWO). ND acknowledges the hospitality of the CERN TH department while this work was carried out. Open Access. 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Nicolas Deutschmann, Claude Duhr, Fabio Maltoni, Eleni Vryonidou. Gluon-fusion Higgs production in the Standard Model Effective Field Theory, Journal of High Energy Physics, 2017, 63, DOI: 10.1007/JHEP12(2017)063