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-fusion Higgs production in the Effective Theory Nicolas Deutschmann, Claude Duhr, Fabio Maltoni, Eleni Vryonidou

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Nicolas Deutschmann, Claude Duhr, Fabio Maltoni, Eleni Vryonidou. Gluon-fusion Higgs production in the Standard Model . JHEP, 2017, 12, pp.063. ￿10.1007/JHEP12(2017)063￿. ￿hal-01704888￿

HAL Id: hal-01704888 https://hal.archives-ouvertes.fr/hal-01704888 Submitted on 20 Nov 2018

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JHEP12(2017)063 R q Φ L frame- ¯ Q d Φ Springer † August 28, 2017 : November 6, 2017 December 13, 2017 November 23, 2017 : : : aMC@NLO , Received Revised Accepted Published and Eleni Vryonidou b MadGraph5 Published for SISSA by https://doi.org/10.1007/JHEP12(2017)063 claude.duhr@.ch Fabio Maltoni , b,c . Focusing on the top--Higgs couplings, we GG [email protected] Φ † , Claude Duhr, a,b . 3 at NLO in QCD, which entails two-loop virtual and one-loop real con- G 1708.00460 R The Authors. σq c NLO Computations, Phenomenological Models

We provide the complete set of needed to achieve NLO accuracy in Φ , L [email protected] ¯ Q Nikhef, Park 105, 1098 XG,E-mail: Amsterdam, The Netherlands [email protected] Centre for Cosmology, ParticleUniversit´ecatholique de and Louvain, (CP3), Chemin du Cyclotron 2, 1348Theoretical Physics Louvain-La-Neuve, Belgium Department, CERN, CH-1211 Geneva 23, Switzerland Univ. Lyon, Universit´eLyon 1, CNRS/IN2P3,F-69622, IPNL, Villeurbanne, France b c d a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: consider the phenomenological impactrelevant operators of by the implementing the NLO resultswork. corrections into This the allows in us constraining toat compute the NLO total three that cross can sections as be well directly as employed to in perform experimental event analyses. generation the Standard Model Effective Fieldfusion. Theory In at particular, dimension we six computeoperator for for Higgs the production first in the gluon tributions, contribution of as the well chromomagnetic asand renormalisation the and gluon-fusion mixing operator with the Φ Yukawa operator Φ Abstract: Nicolas Deutschmann, Gluon-fusion Higgs production inEffective the Field Standard Theory Model JHEP12(2017)063 14 3 ], all of which so far support the 5 – 1 10 6 ]. In particular, higher-order corrections in the – 1 – 9 18 18 7 20 14 6 14 1 ], i.e., the SM augmented by higher-dimensional operators. Among the most 8 – 6 The steady improvement in the precision of the current and forthcoming Higgs mea- 4.1 Cross-section results 4.2 Differential distributions 4.3 Renormalisation effects 3.1 Computation of3.2 the two-loop amplitudes UV &3.3 IR pole structure Analytic results3.4 for the two-loop Renormalisation amplitudes group running of the effective couplings strong typically entail large effects at the LHC both in the accuracy in the absence of(SMEFT) resonant [ BSM production isinteresting provided features by of the this SM framework Effectivethe is Field gauge the Theory possibility couplings, to thusreduction compute allowing of radiative for corrections the systematic in theoretical uncertainties improvements [ of the predictions and a nances, as widely pursued atHiggs the LHC, to but the alsois known via that, SM indirect despite effects . being on much Thetheoretically, the more it most couplings challenging has than of appealing the direct the potential aspect searchesical to of both probe reach such experimentally of new and an the physics scales LHC. approach that A are powerful beyond and the predictive kinemat- framework to analyse possible deviations vector and to the electricallyand charged the third first generation evidence hasor of been ) the confirmed, could coupling arrive to in the second coming generation years, fermions ifsurements (either SM-like. invites charm to quark explore physics beyond the SM not only via the search of new reso- 1 Introduction Five years into itsof discovery the at the high-energy LHC,its physics the properties community. Higgs by boson the Apredictions ATLAS is of wealth and the still CMS of Standard the Model experiments information (SM). centre [ has of In particular, attention been the size collected of the on couplings to the weak 5 Conclusion and outlook 4 Phenomenology 3 Virtual corrections Contents 1 Introduction 2 Gluon fusion in the SM Effective Field Theory JHEP12(2017)063 R q Φ L ]. , while ¯ ]. The Q t 26 49 Φ) , – or † b 48 24 have also ap- ¯ tγ t , ¯ tZ t ]. An important conclusion 12 ]. At the differential level, phe- has appeared in the erratum of 3, are known only at NLO in the ], while at NNLO only subleading 44 , H , 30 43 – → = 2 production. 27 n gg LO [ 3 ¯ tH t ] and leading contributions to the top/bottom -jets, ]. The LO contribution of the chromomagnetic n 34 – 2 – – + 49 . The former can be accounted for by a straightfor- , 31 ]. H ] and at N LO in the limit of a heavy [ 12 ], this operator sizably affects Higgs production, both 3 23 GG 42 – 14 ) and Higgs- operator has appeared [ t ]. Φ) 19 † + was computed in ref. [ -even interactions) needed to achieve NLO accuracy in the 41 , H and 40 CP b , and for top-quark associated production ¯ t ]. Recently, the calculation of the Higgs spectrum at NLO+NNLL t 47 – ]. For top-quark production, NLO results for EW and QCD inclusive and ] are known. Beyond inclusive production, the only available NNLO 12 45 – , tj 36 ] was that even when the most stringent (and still approximate) constraints , 10 12 [ ]. The effect of dimension-six operators has also become available recently 12 35 ], while cross sections for 18 to inclusive Higgs production at NLO in QCD, thereby completing the set of ¯ tH – t 39 G – 13 production are considered [ R ] and later confirmed in refs. [ ¯ 37 t t σq 50 The purpose of this work is to provide the contribution of the chromomagnetic operator In the SMEFT, most studies have been performed at LO, typically using approxi- The situation is somewhat less satisfactory for gluon fusion, which, despite being a Φ L ¯ ref. [ operator of the top-quarkdrawn to in ref. [ from in gluon fusion (single and double Higgs) and level for the Yukawa (both Q predictions (involving only SMEFT for this process.of chromomagnetic The operator of first the correct top computation quark at to one-loop of the contribution including modified top and bottom Yukawainteraction couplings and are an available additional at direct NNLO Higgs-gluons nomenological [ studies at LOregion have of shown the the Higgs relevance bosoninclusive of level in the [ order high to transverse resolve momentum degeneracies among operators present at the bution of two specific dimension-sixand operators, the gluon-fusion namely operator the (Φ Yukawaward operator modification (Φ of the Yukawathe coupling latter of the involves corresponding thelimit heavy computation of quark, of an contributions infinitely-heavy top identical quark. to SM Results calculations for in the the inclusive production cross section heavy top- expansion [ mate rescaling factors obtained fromconsidered SM when calculations. existing Higher-order SM resultssimplest calculations have examples only could been are be the readily inclusion used of within higher the orders SMEFT. in The the strong coupling to the contri- full quark-mass dependence is knownterms in up the to heavy top-mass NLO expansion [ interference [ [ result is the productionlimit of a [ in association with a jet in the infinite top-mass peared [ for top-quark and Higgs decays [ loop-induced process in the SM,provides is the highly most enhanced important by Higgs-production thecorrections gluon channel are density at now in the the known LHC. up and In to the N SM, the QCD processes involving operators oftop dimension six featuring and the theated Higgs , production bosons. the channels bottom Currently, forVBF and predictions the and for Higgs the boson mostproduction, are important i.e., available associ- in this framework, e.g., VH, and the precision. They are therefore being calculated for a continuously growing set of JHEP12(2017)063 4 (2.3) (2.4) (2.1) (2.2) spectrum T p ]. This allows ]. In this paper, 54 55 , 7 . framework [ ,  ], a new two-loop counterterm µν a 53 , , – G R t + h.c. 51 a µν a µν i of dimension six, Λ is the scale of new ˜ Φ G G i O L R O b i 2  we establish our notations and set up the Q t aMC@NLO 2 Λ C 2 2 µν v   σ 2 2 − a v i T – 3 – Φ − † ˜ Φ + Φ Φ L †  Q Φ SM 2 s s L  g g we describe in detail the computation of the two-loop MadGraph5 3 = = = = 1 2 3 O O O EFT L ) runs over a basis of operators are the (bare) Wilson coefficients, multiplying the effective operators. A 2.1 b i C The paper is organised as follows. In section At LO the chromomagnetic operator enters Higgs production in gluon fusion at one physics and complete and independent set ofwe operators are of only dimension interested six in is thosebottom known operators and [ that top modify quarks, thethere contribution to are of Higgs the three production heavy operators quarks, in of gluon dimension fusion. six that Focusing contribute on to the the top gluon-fusion quark, process, the SMEFT, i.e., the SM supplemented by a complete set of operators of dimension six, The sum in eq. ( and EFT expansion at theof total inclusive the level Higgs and via provide a predictions for NLO+PS the approach. 2 Gluon fusion inThe the goal SM of this Effective paper Field is Theory to study the production of a Higgs boson in collisions in need to be calculated.virtual In contributions section and the renormalisationexpressions procedure for and the we provide finiteleading compact logarithmic parts analytic renormalisation of group thewe running perform of two-loop a the amplitudes. phenomenological Wilson study coefficients. We at In also NLO, section in briefly particular discuss of the the behaviour of the QCD two-loop matrix elements into the us to compute totalparton shower cross (NLO+PS) sections that can as be well directly as employed in tocalculation experimental perform by analyses. identifying event the generation terms at in NLO the perturbative plus expansion that are unknown and mixing pattern of theenters SMEFT our at one computation, loop andpresent is very we known compact provide [ analytic its resultsFocusing for value on all for the possibly the relevant amplitudes anomalousconsider first up the contributions time to phenomenological in here. two impact loop top-quark-Higgsoperator of order. Moreover, the interactions, and we NLO we the corrections, then gluon-fusion including also operator the at Yukawa NLO by implementing the respective virtual loop. Therefore NLO correctionsbutions. in QCD The entail latter two-loop canformer, virtual nowadays and easily however, involve one-loop be a real computed non-trivial contri- loop using two-loop techniques an and computation automated a that approach. careful requiresboth The treatment analytic of of multi- the which renormalisation are and presented mixing in in the this SMEFT, work for the first time. In particular, while the full JHEP12(2017)063 , 2 . ) 2 4 MS Λ Λ / is the quark / 2 b O (1 2 a µν O C G 3 are real. , + are the bare 2 G  b t , ) . R for the is the usual m H 1 . Second, while σq ) (2.5) 3  } = 1 ) H Φ O ,m R -even contributions π b t L b i and is actually 2 ¯ ,m Q (4 m b t H (  → and , a diagram featuring the CP 3 E m EFT m G 1 b, γ ( R with L R 0 is the scale introduced by t − O A i b, e the Dirac . in 2 σq µ C A Φ, 2 b Φ 3 µ = 1 v O L γ 2Λ  C denotes the unrenormalised SM ¯  → Q S √ denotes the vacuum expectation )] b,i ˜ Φ 1 v )+ {  A · H 2 ], with p ν is the left-handed quark SU(2)-doublet ,m )( b t , γ L 2 0 it denotes the form factor with a single µ  , provide the leading contributions to the first m Q · GG t γ ( [ 1 2 i 2 p i > or b, Φ) ( † – 4 – Φ). b A = − 2 (Φ ) v is the generator of the fundamental representation 2 2 iσ b 2 µν  a Λ In this work, we focus on the · C σ 1 T 1  ˜ )+ Φ = )( = 0, the form factor 2 H i ], while for p ) and · is the right-handed SU(2)-singlet top quark, and 56 1 ,m ab b t p δ are not. R [( 1 2 t m H 3 ( b s 1 O α b, ) in the EFT, and so they are neglected. ] =  (1) the Euler-Mascheroni constant and 4 b 0 2 A ) we adopt the convention to include the hermitian conjugate for all operators, be Λ R − Γ 2 π and / 2 q ,T µ ) denotes the bare QCD coupling constant and − v 2.1 1 (1 a  ) contributions later, let us focus on the leading contributions coming Φ π b 1 2Λ c L O T O = C iS (4 ¯ √ Q . For this reason, all the Wilson coefficients ). The (unrenormalised) amplitude can be cast in the form / E 3 2 s + γ Φ) g ) = O † 2.4 is the (bare) strong coupling constant and = H (Φ . Representative diagrams contributing to gluon-fusion amplitudes with one insertion of )–( b s s g → and α 2.2 1 In the SM and at leading order (LO) in the strong coupling the gluon-fusion process g g is hermitian Note that in eq. ( According to our power counting rules, multiple insertions of an operator of dimension six correspond ( O 1 2 b (and possibly 2 g g A dimensional regularisation. For contribution to gluon fusion [ they hermitian or not. This means that theto overall contributions contribution of from where of the Higgs bosonfactor, and with the top quark. The factor from the top quark,eqs. i.e., ( the contributions from the operators of dimension six shown in Representative Feynman diagrams contributing at LO are shown inis figure mediated only bycorresponding quark quark loops. andb therefore This heavy contribution quarks is dominate. proportional to While the we comment mass on of the the of SU(3) (with [ Two comments are in order.can First, be the obtained corresponding operators byO simply making theof substitutions four point gluon-quark-quark-Higgs interaction is also present (not shown). where value (vev) of thecontaining Higgs the field top Φ quark, ( gluon field strength tensor. Finally, Figure 1 the three relevant operators.and third Heavy amplitudes. quarks, Note that for chromomagnetic operator, JHEP12(2017)063 ]. is 1 69 b, (2.6) – A and at cannot 60 T are loop- H p 3 contributes → 2 O are related to O gg 2 ). We emphasise and O ], where effects on 1 2.4 74 . O ) )–( ]. At NLO, we need to H 2.2 53 as well as real corrections , m – b t are also known, because they 51 H m ( 2 ) k O → ( b,i , are explicitly factored out. Each A v g g k  b s α  2 . π − contributes at two loops in 3 µ ], we have confirmed that the Higgs  µ cλ – 5 – S 75 G  λ bν G =0 ∞ k X ν aµ G ] (without the higher-order corrections to the matching ) = ]. The normalisation of the amplitudes is chosen such that . In the case of EW interactions, by just looking at the SM , which only enters at tree level at LO. The renormalisation abc 59 H ]. The UV divergence is absorbed into the effective coupling 2 is ultraviolet (UV) divergent, and thus requires renormalisa- is inserted. The computation of this ingredient, which is one f H s 57 O 3 3 , g , m 50 b t only amounts to a rescaling of the Yukawa coupling, i.e., , → O O ]. The NLO contributions from ], it is easy to see that many operators featuring the Higgs field 1 50 m , ( 49 gg O 58 , , 71 48 b,i , . The latter process has been considered in ref. [ 12 ]. In the case of QCD interactions, operators not featuring the Higgs A 70 30 , Hg 73 , ), because they mix under renormalisation [ 28 , → 72 2.4 27 inserted [ gg i )–( O is proportional to the SM amplitude, the corresponding NLO corrections can be 2.2 1 O As a final comment, we note that starting at two loops other operators of EW and The goal of this paper is to compute the NLO corrections to the gluon-fusion process For example, the operator one loop in the transverse momentum of the Higgsreproduced were these studied. results For the in sakethis of our operator completeness, framework, from we have multi-jet and observables by [ considering the recent constraints on QCD nature will affect EW contributions [ will enter, which in aself few coupling cases [ could alsofield lead will to enter, constraints, which, see, in e.g., general, the can trilinear be Higgs more efficiently bounded from other observables. coefficient). In particular,the QCD the form virtual factor, correctionsHence, which is to the known the only throughchromomagnetic three insertion missing operator loops of ingredient in is theof strong the the coupling NLO main [ results contributions of to this the paper, will process be where presented in the detail in the next section. from obtained from the NLOdependence corrections [ to gluon-fusion inare the proportional SM to including the thetop NLO full corrections quark top-mass to is gluon-fusion infinitely in heavy the [ SM in the limit where the that a complete NLOin computation eq. requires ( one toconsider consider both the virtual set correctionsdue of to all to the three the operators LO emissionpartonic process of channels an with additional a parton in quark the in final the state. initial Starting state from contribute. NLO, also Since the contribution tion, already at LO [ that multiplies the operator at NLO will be discussed in detail in section with an insertion of one of the dimension six operators in eqs. ( Some comments about thesebreaking, amplitudes the are operator in order.simply proportional to First, the after bareat electroweak SM amplitude. tree Second, level, at whileinduced. LO the the operator Finally, SM this amplitudechromomagnetic process operator and has the the contributions unusual from feature that the amplitude involving the all coupling constants, as wellform as factor all admits powers a of perturbative the expansion vev in the strong coupling, operator JHEP12(2017)063 , ) 58 2.4 (3.5) (3.1) (3.2) (3.3) (3.4) ). For )–( 2.5 2.2 ] we expect 77 , 76 . ]. After reduction, t , 85 1 ) – z 1 + ; 81 [ w ) = , t 1 , . . . , a − 2 ( z . a , ], LiteRed ( w 2 . 89 ) H 1 x ) log of the HPL. The only non-trivial func- and x − + 1 ! , t − , f 1 1 w t 1 a /τ /τ (1 ( 4 4 FIRE − ) = ] and Mathematica using a custom-made code. weight − − ) = dt f z = – 6 – 1 1 ; , t z 80 } 0 2 t 0 , 2 H p p (0 , all processes are loop-induced, and so the virtual Z m 2 m 79 [ = {z O ,..., = ) = x w | z (0 τ ; , f FORM w H t is called the 1 − w ’s are zero, we define, 1 i a , . . . , a 1 a ) = ( , t H (1 f ], and we use the latter to generate all the relevant Feynman diagrams. The 78 ), and we can simply project each onto the transverse polarisation ] in terms of harmonic polylogarithms (HPLs) [ 2.5 The complete set of one- and two-loop master integrals is available in the literature [ 88 – or equivalently The number of integrations tional dependence of themasses, and master it integrals is is useful through to the introduce the ratio following of variable, the Higgs and the top with In the case where all the master integrals. 86 The tensor structure ofeq. the ( amplitude istensor. fixed by The resulting gauge-invariance to amplitudeswhich all are are loop then reduced classified orders, to into cf. we distinct master can integral integrals topologies, express using all LO and NLO amplitudes as a linear combination of one and two-loop exception of the contributionscorrections from require the computation ofquark two-loop loop form and two factor external integrals gluons.into with QGraf We a have [ implemented closed the heavy- operatorsQGraf in output eqs. is ( translated into 3 Virtual corrections 3.1 Computation ofIn the this two-loop section amplitudes wethe describe sake the of virtual the correctionsdiscuss to presentation later the we on LO focus how amplitudes to here in obtain on eq. the the ( corresponding calculation results involving for a the top quark. and With the paper. Four- operators alsothese contribute starting modify at observables two related loopsthem to to to top gluon quark be fusion physics but independentlysignificantly as at constrained affect leading and gluon order fusion. work [ under the assumption that they cannot be significantly affected. For this reason we do not discuss further this operator in this JHEP12(2017)063 (3.9) (3.6) (3.7) (3.8) ) and which (3.10) 2  µ ( i . C MS scheme, ) i ), 2 m , µ ) ( s = 1, and so it can , µ | α sign( , x , H | , ≡ θ i , m ) even odd 2 µ w w ( t + + , σ k k ) , m z ) , if if ; 2 ) k µ , 2 , , , ( ) of the HPL. i µ t , σ   2 ( whose coefficients are linear combina- } j 0 iθ iθ µ ,C  ( e e δm C ) s 2 1) times k k α {z < θ < π . µ depth throughout this section. We can decompose |− s ( ) + ,..., C,ij 0 k s α 2 2 0 | Z α m ,...,m ,...,m µ . In the kinematic range that we are interested Z µ |  ( 1 1 ( ( i  x – 7 – t 2 a m m A ] and references therein), , µ µ m 1 H ,..., H 93 1 − g iθ = = = – e Z Re Im b b t b , σ s i ) has the advantage that the master integrals can be } = 0 α C 90 m ( is a unimodular complex number, ,  x ) = 3.4 S x 1) times 58 {z H ) = |− ,..., θ 1 | ( , m m ( 0 b t k | ( H 1). Unless stated otherwise, all renormalised quantities are as- , m , b i ,...,m 0 denotes the renormalisation scale. The renormalised parameters are 1 , ) = m ,C µ z b s ( Cl , the variable k α ( 2 t b m of non-zero indices is called the ) = (3 A ,...,m 4 3 1 k m < , a 2 is the field renormalisation constant of the gluon field and H 2 H , a g 1 Z a ) are the renormalised strong coupling constant, Wilson coefficients and top mass in < m 2 Inserting the analytic expressions for the master integrals into the amplitudes, we can MS scheme, and µ ( t with ( sumed to be evaluated at thethe arbitrary renormalised scale amplitude into the contributions from the SM and the effective operators, m the related to their bare analogues through and we write the bare amplitudes as a function of the renormalised amplitudes as, where next section. 3.2 UV & IRIn pole this section structure we discussNLO the amplitudes. UV We renormalisation start and by the discussing IR the pole UV structure singularities. of We the work in LO the and The number express each amplitude as ations Laurent of the expansion special in functionsare we of have just both described. ultraviolet The (UV) amplitudes and have infrared poles (IR) in nature, whose structure is discussed in the where we used the notation In terms of thisof) angle, Clausen the functions master (cf. integrals refs. can [ be expressed in terms of (generalisations written as a linear combination ofin, HPLs 0 in be conveniently parametrised in this kinematics range by an angle The change of variables in eq. ( JHEP12(2017)063 , ) 4 (3.14) (3.15) (3.16) (3.17) (3.12) (3.13) Λ / (1 O +  ) H , , ) , 2 s ,m ) t 2 s ) α into the counterterm. 2 s ( α m ,  ( ], where the renormali- α ( ) O  ( 3 ) (3.11) 2 s O 2 t . 12 H O + A α ) ( + t 2 ) H /m + t ,m O 2 3 t 2 t µ 2Λ 2.5 C v m , m + m . t v m 2 √ (  ) , v m 0 2 m 2 s  √ ) ( 2 A √ 2 s 2 t  ) α )+ , 2 ( k  √ α 1 v µ ( i  f H m ( 2  O t 2  2 A  N  2 t O 2 = 5 is the number of massless flavours. µ ,m m )] 2 3 +  k t µ  1 m + 2 t 1 t f  2   6 m − ·   s  µ ( N m m 2 s π  1 c 2 α  π p  1 α s  2 t 3 2  N 2 A 3 π – 8 – α )( 3 2 µ  Λ C 1 m − v 2 11 2 Λ C =0  ∞ − 2 s 3 2 0 k · X  Λ ]. The renormalisation of the strong coupling and  π s C 1 α β = Λ C = π  α p 1 59 0 s 6 ( s π − ) = β π )+ α 4 α + s SM t − H π H ) α − SM 2 − ,  SM δm s function, , m · ,m = = g, α t t 1 SM t β  m m δZ δZ denote the one-loop UV counterterms in the SM, ( SM SM ( )( , i 1 δm s g, 2 ]. Similarly, the renormalisation of the top mass is modified by A α p A · SM = , 50 δZ 2 1 s 2 t δZ p α v = 1 + = 1 + [( 1 2Λ s g δm δZ α s C Z √ Z π + iα and ) = SM = 3 is the number of colours and H g, c is the one-loop QCD → N δZ 0 β g g The presence of the effective operators alters the renormalisation of the SM parameters. ( A where the SM contribution is As a result onlythe massless top flavours contribute quark to effectivelythe the decouples gluon running [ field of are the modifiedcancel strong by each coupling, the other presence while out ofthe the [ presence dimension of six the operators, effective but operators, the effects where We work in a decoupling scheme and we include a factor and where Throughout this section wesation closely of follow the the operators approachstrong at of coupling one ref. constant loop and [ was the described. gluon The field one-loop are given UV by counterterms for the and each renormalisedstrong amplitude coupling admits constant, a perturbative expansion in the renormalised similar to the decomposition of the bare amplitude in eq. ( JHEP12(2017)063 (3.22) (3.18) (3.19) (3.20) (3.21) , denotes a renormalised of counterterms can be  0  C (1) 2 β Z . A + ) , which is proportional to the 2 s 2 3 is a process-dependent remain- at the end of this section. 3 corresponds to the counterterm α  , ( O . requires renormalisation at LO ). We have R  23 23 , ) is universal (in the sense that it O z z        3     ( − 2 R + is more involved, because the opera- b, ]. 3.18 2 t into the counterterm in order to de- t b  i A (1)   v + 23 m  53 (1) 1 I 0 C m  v C 6 2 z i  8 – ] 2 2 t π (0) δZ − − 51 2 √ 94 A 16 s µ 0 to eq. ( /m π ) 12 α 2  s  ( 1 µ 0 0 0 0 (1) C + − – 9 – 0 00 0 0 0 0 0 is non-trivial at LO in the strong coupling, − (1)  I     δZ (0) C      (0) C ) =  =  = δZ δZ E − γ (1) + (0) C 0. The quantity (1) − C A 1 e δZ Γ(1 → δZ =  − C Z ) = in our case. This counterterm is not available in the literature, yet  ( 2 (0) b, ) mix under renormalisation. The matrix (1) , all the entries are known [ I A 23 denotes the center-of-mass energy squared of the incoming gluons. 2.4 z 2 p )–( ]. As a consequence, 1 p 2.2 50 ) we again include the factor , is the tree-level amplitude for the process and = 2 49 (0) , 3.16 12 s A 12 [ A one-loop amplitude with massless gauge bosons has IR divergences, arising from The renormalisation of the effective couplings 2 . We therefore first review the structure of the IR divergences of NLO amplitudes, and (0) b, 23 der that is finitedoes in not the depend limit on the details of the hard scattering) and is given by where one-loop amplitude describing thetwo production massless of gauge bosons, a then colourless we state can from write the [ scattering of where we will return to the determination of the counterterm regions in the loop integrationmassless where leg. the loop The momentum structure is offrom soft the the or underlying IR collinear hard divergences to scattering is an process. universal external in More the precisely, sense if that it factorises that absorbs the two-loop UVtree-level amplitude divergence of the operator we can extract it fromso our we computation. need NLO to amplitudes disentangle the havez both two types UV of and divergences IR if poles, we and want to isolate the counterterm where, apart from At NLO, we also need the contribution We have alreadyin mentioned the that strong theA coupling, amplitude and the UV divergence is proportional to the LO amplitude couple the effects from operatorsMS of scheme. dimension six from the running of thetors top in mass eqs. in ( the written in the form In eq. ( JHEP12(2017)063 3 )– O MS 2.2 ) as a (3.25) (3.23) (3.24) (3.26) ). The ) satisfies 3.23 . There is a 3.21 b t 3.24 /m b 1 ) in dimensional 2 C  ) remains valid for ( ]. The amplitude is O 59 3.21 , . 0 3 defined in eq. ( β . i . 3 ≤ . 2 H i R  ) iπ  2 s m ≤ 23 4 α ( 0 = , instead, we can use eq. ( + O = 16 2 2 µ 2 (1) 3  + 5 R A 6 , 11 i are renormalised using a different scheme −  b R 1  (1) C and C + 2 Z – 10 – 2 √ (0) i s π t v π α A 2 2 m ) π √  = ( MS-scheme that we use, the renormalised amplitudes 16 32 (1) SM t I − = = = δm 23 z (1) i (0) 2 A A will in general not hold after renormalisation. only contributes at one loop at NLO, and agrees (up to normalisation) (1) 1 2 A O . We find 23 and z (1) 0 A The operator Let us conclude our discussion of the renormalisation with a comment on the rela- Since in our case most amplitudes are at one loop already at LO, we have to deal ) included. We show explicitly the one-loop amplitudes up to . We know that the corresponding unrenormalised amplitudes are related by a simple 1 2.4 all scales are fixed to the mass of thewith Higgs the boson, one-loop correctionsindependent to of Higgs the production top via mass gluon-fusion through [ one loop, and so it evaluates to a pure number, In this section wethe present computation the of analytic the results gluon-fusion( cross for section the at renormalised NLO amplitudesregularisation, with that the as operators enter in well eqs.amplitudes as ( have the been renormalised finite using two-loop the scheme remainders described in the previous section and If the top masswhich and breaks this the relation Wilson betweenplitudes coefficient the counterterms, the simple relation between the3.3 am- Analytic results for the two-loop amplitudes are still related bycounterterms this are simple related scaling. by This can be traced back to the fact that the tionship between the renormalised amplitudesO in the SM andrescaling, the and insertion the of constant of the proportionalitypriori operator is no proportional reason to why the such ratio procedure. a simple relationship In should (the be variant preserved by of) the the renormalisation Note that the coefficientdimension of of the the effective double couplingsthis pole criterion, to which be is is finite. in a We strong fact have consistency fixed checked check that by on eq. our requiring ( computation. the anomalous We have checked thathave our the results correct for IRconstraint pole amplitudes on structure which the at singularities do of NLO.counterterm not the amplitude. For involve This the allows operator us to extract the two-loop UV with two-loop amplitudes atindependent NLO. of However, the since details thetwo-loop of structure amplitudes the of describing the underlying loop-induced IR processes, hard singularities and scattering, is we eq. can ( write JHEP12(2017)063 . ). ] ) θ θ 3.6 . We 3 ( 3 (3.31) (3.27) (3.28) (3.29) (3.30) ) 2 , − θ O − , ( )  2  θ Cl 4 τ − (  , θ ) and ( 16 2 2 4 − − , − # 252 3.4 ) ) 1 τ − 3 θ 4 −   ( 21 3 2 Cl ( 3 τ ζ τ )+3Cl + O − ) θ Cl τ τ ) ( θ 2 + + 6 3 τ θ θ +24log − − 2 , 2 48 log  2 2 − θ 3 1 0 ζ [ θ +log (4 128 − 2 2 3 8 2 2 13 τ τ a θ p θ − 2 ) + ) log(4 defined in eq. ( + + τ 25 48 θ θ )+6Cl ) ( θ (  θ log 6 θ τ [ +28 3 τ + . 3 2 ) ( 1 2 τ − 3 τ θ 2 τ is genuinely new and is presented  − − Cl 2 θ 24 − 4 τ , − Cl + and π (4 2 1 log 2 log 3 . θ (1) 3 θ 1 55 12 13 (4 τ τ − 2 2 p 2 − Cl A θ θ θ MS-scheme the renormalised amplitudes 1 τ (1) 0 − 3 θ τ a 4 )+ 6 128  θ 3 θ θ A ] + 59 48 12 +5 ( " 3 3 t τ 3 3 − − ) ζ ζ + 1 ζ −  m ) 3 τ ) )+ 4 + log θ 3 θ 4 θ θ τ 32 Cl ( 2 ( ( − − – 11 – 2 = 2 1 a − 5 8 3 ) R 64 − − 1 )+28 θ (4  , Cl (1) 1 ( ) − 2 θ  − 2  2 2 ) ( θ p  − 2 4 A ( θ 3 θ 8 − ζ 4 τ τ 4 ( Cl − 2 3 Cl − 28 + ) + θ 51 16  θ 0  Cl τ 4 28Cl ). The remaining amplitudes have a non-trivial functional 4 τ 2 + θ 3 72 Cl 6 3 − π ) + τ a − ζ ) 4 − θ − +4 17 2 ( θ 3.15 1 ) + 2 2 ) ) ( 2 θ 31 12 θ θ θ 2  − ( ( ( )+ are given by 4 14 + 3 2 + log − 4 θ τ Cl i Cl − ( − )  − τ − 1 ]. The two-loop amplitude a θ 2 2 τ τ a 1 τ 2 − θ ( − , 28 θ log 58  2 Cl , [4 Cl 4 − Cl 3 3 − 16 (4 θ ζ  + 0 )]  τ τ   − + 4 30 2 + 96 Cl 0 4 3 9 4 p θ , 2 4 τ are related by a simple rescaling, ( 2 4 τ a τ 1 2 16 3 − + 3 τ iπ β θ − − = ) = − 28 (1) 1 − , − + θ − 1 1 is defined in eq. ( ( A = (0) 0 2  27  8 Cl 2 0 R θ 0 A τ β 1 1 τ 2 τ − R and The one-loop amplitude in the SM can be cast in the form = = = 0 1 2 (1) 0 a a a where we have defined the function The finite remainder of the two-loop SM amplitude is where the coefficients We therefore only present results for thehave SM checked contribution that and the our contribution result from of for refs. the [ two-loop amplitudehere in for the the SM time. agrees with the results dependence on the topWe mass have argued through in the the previous variables A section that in the where JHEP12(2017)063 τ ) θ ) ( 2 τ log 2 (3.32) (3.33) θ  − 3 ) 62 . θ (4 (  # 2 + 2 + 12 θ  ) + 3Cl 3 τ 2 71 θ , log θ ( 3 2 2 2 2 ) 100  − θ 74 Cl θ ) π + 12 − log ( 4 3 − θ 2  ζ − − Cl ( 5 3 2 2 ) 4 ζ ) Cl 3 π θ O ) + 6 3 32 θ θ ( + θ 5 3 151 ( 4 2 ( + + 3 − 1 − R ζ 3 − + − −  ) + 5 " θ 8 , ) 3 Cl τ  τ τ 2 t τ θ ζ ) 2 3 τ − ( −  3 m − 3 τ 64 τ ) , ) − ) + 2 − θ 2 log τ ] ) log ( − τ − + 80 1 θ 3 θ 2 3 48 Cl ( τ −  τ −  (4 3 log(4 − 96Cl +  (4 ) τ p 2 Cl 0 3 ) θ log − is + 35) (3.34) 16 16 Cl 3 4 τ + 2 log π ( τ 4 3 ) log ) t − 3 is τ 2 τ b 0 θ − θ + ζ − (1) 3 − (  − ( 2 m # 3 − ) ) log(4 2  τ 2 1 τ b θ A θ 2 ) O 3 − ( (4 ( 3  θ log . Cl 104 3 2 2 Cl ( 238 + 2 + log τ Cl p 1 # 1 2 − − 1 3 , (16 log  + 62 ) θ − − 3 3 + t τ  + log 2 3 − ζ ) + 92 τ – 12 – , 48 θ Cl 2 1  τ m 2 1 6 ) + − τ 71 b τ 24 Cl τ 1 4 θ 2 − ) 8 3  − 63  − ( − θ t τ τ b (4 4  ) τ 1 − τ − , + 2 − θ 2 2 m ) τ − + θ ( p + 4 16 log τ ) τ )) + 32 Cl − θ 3 1 6 θ θ ( Cl τ θ  1 ( − log (4 ( log(4 − 2 − , log + log 2 τ 8 3 2 ) + log 2 −  ) − θ 1 4 − p 64 2 θ [ θ − , 3 1 6 ( τ b Cl 4 3 ( − 2 2 ζ  ) 3 3 τ Cl  )  τ − 2 + 62 − 4 3 − 2 θ θ 4 θ τ + 4 log )  2 ( ) ) + + − τ θ 2 + 2 log τ θ − 0 θ 96 48 Cl τ are given by + 0 ( − b + − 2 64 Cl 2 2 , 2 β i ) + 24 Cl (4  3 − τ 6 − 1 1 13 − − θ b π − t θ τ t −  2 )  p ) ) + 8 Cl ) θ − m π m − τ 9 θ θ τ θ ( τ 2 ) ) − ( ( 1 ) ( ) 3 48 Cl log(4 = 2 2 τ (2 Cl τ 3 τ log(4 τ −   , − − − − − t + 2 + 2 1 8 3 τ θ − − 0 3 − 4 − (0) 3 θ 16   log(4 3 2 Cl θ Cl β  3 θ A (4 32 (4 t  θ m ζ θ (4 θ − (4 τ 2 48Cl " 3 " 61 48 64 4  θ 96 Cl θ t τ p t p p  12 τ τ 1 6 iπ θ m + − + − m − iπ m 2 6 2 τ − + + 72 − + − + + = = = = 2 0 1 b b b 3 R The finite remainder of the two-loop amplitude where the coefficients The one-loop amplitude involving the operator JHEP12(2017)063 , ]. x x . x 0 n and 96 , ], any x (3.35) (3.36) (3.37) log x > ! 89 95 1 n , 89 ) = ) has a branch x . For example, x 0; ; x k . 0. As a consequence, )] = 1 and Im ,..., | /x x (0 | , . . . , a 1 H 1; 1 a < x < , ( 1 (0 H , 0, and so the correct analytic − 2 iπ . H i 3 π − + ) + ) 4. Above threshold, the variable x x x ) + x − 1; → , 0; τ > , (0; (0 x (0 H H [ H i → 1 2 ). − – 13 – x ) x ) = 3.37 = 0, and those of the form x (0; x 4. It is, however, not difficult to analytically continue 1; ) = log , ) can be expressed as a linear combination of products of x x iπ H (0 τ < 0; (0; H , k ) + H ]). ) and express all Clausen functions in terms of HPLs in /x = 0, and, using the shuffle algebra properties of HPLs [ 58 can always be expressed in terms of HPLs in , k 3.7 ) = Im a 1; 1 , . . . , a θ , /x 30 1 ( , a 2 (0 ( 28 Cl H ) is no longer a phase, but instead we have , H may develop an imaginary part. Indeed, when the threshold − 27 3.5 x ) = = 0 only if /x x ) = log 1; 1 x , Using the procedure outlined above, it is possible to easily perform the analytic con- We start from eq. ( Although the main focus of this paper is to include effects from dimension six operators (0 (0; H tinuation of our amplitudesgluon-fusion above process threshold. when The lighttaken resulting into quarks, amplitudes account. e.g., contribute Hence, to massivein although the this bottom we paper, focus and/or our primarily charm resultsquarks on can quarks, as the be well. effects are easily from extended the to top include quark effects from bottom and charm HPLs such that if their lasttherefore entry be is expressed zero, in then terms all of of two categoriesand its of so entries HPLs: do are not zero. those have whose a The last branch amplitudeswhich entry point can is at are non-zero continued according to eq. ( The previous rule isin sufficient to our perform results. the Indeed,point analytic it continuation at is of known all that HPLsHPL an of appearing HPL the of form the form The previous equation, however,H is not yet validapproaches above threshold, the because negative the realcontinuation logarithms of axis the from logarithms above, is Similar relations can be derived for all other HPLs in an algorithmic way [ HPLs evaluated at 1 one finds defined in eq. ( the Clausen functions mayone develop can an extract imaginary the correct part.(see imaginary also part In refs. of the [ the following amplitudes in we the describe region how above threshold its inverse, e.g., that affect the gluon-fusion cross sectionby through making the top a quark, comment letquark. about us The conclude effects amplitudes this from section presented inthan the this the bottom, section quark-pair and are threshold, only toour valid a if results the lesser to Higgs extent, the boson the is region charm lighter above threshold where JHEP12(2017)063 (3.40) (3.38) (3.39) in the t , ) m . 2 , ) ) ) 2 2 2 = 5 massless ) ) Q 2 and 2 ( f µ s s Q ( N ( α s α s ( , which in the fol- α α ( µ O ( O + O +  + )         2  ) ) to one loop, and we find Q 2 2 69 ( v 2 µ 2 t ) v ( . − t 2 m 3.38 ) 2 µ 2 2 m 2 ) v ( ) t µ 6 1 2 Q 2 2 Q m √ Q ( 8 2 ( 3 23 s π − C 5 log . It can, however, be desirable to choose α ( 32 γ C , F  O ) µ ) 2 = 2 ) + 8 + 1 0 Q 2 2 is non-zero. This latter example serves as a re- v Q 0 0 0 0 ( v 2 2 µ ( − t Q 3 – 14 – t µ ( Q         C m 1 dC m log ) C 2 2 2 2 2 d log µ  Q µ µ Q π ( ) 2 s 2 2 2 in two scenarios: one where all Wilson coefficients are α µ log Q π Q / ( 6 ) H s ) log + 2 2 2 log α m π Q         Q ) ) ( ( ) 2 2 3 3 2 16 Q C Q µ , and the anomalous dimension matrix is given by C π ( v ( ( T 3 2 t ) s 3 ) 2 C α 3 √ π m Q 2 ( the quantitative impact of running and mixing by varying the renormal- − ,C s ) + 2 ) + ) √ 2 192 α 2 1 2 2 2 ) cancels. We can solve the RGEs in eq. ( 2 π Q ,C Q Q ( 1 8 ( ( √ 3 2 1 C 3.24 C C C − 0 00 0 0 0 0 0 = (         ) = ) = ) = C 2 2 2 = µ µ µ Since we are working in a decoupling scheme for the top mass, the RGEs for the ( ( ( γ 3 1 2 C C C 4 Phenomenology 4.1 Cross-section results In this section we performfocusing a on phenomenological anomalous study contributions of Higgs coming production from in the the top SMEFT, quark. Results are obtained isation scale from 10 TeVequal to at 10 TeV and anotherminder where of only the need tological always consider analyses the as effect choosing of all a the single relevant operator operators to in phenomeno- be non-zero is a scale-dependent choice. We show in figure erm in eq. ( As already mentioned in the previous section, the double pole from the two-loop countert- where strong coupling constant andflavours. the top We have mass checked are thatMS identical we scheme, to correctly and the reproduce we SMeffective do the with couplings, not evolution we discuss of find them here any further. For the RGEs satisfied by the After renormalisation, our amplitudeslowing depend we explicitly identify with on the thedifferent factorisation scales scale scale for the strongIn coupling this constant, section the we topparameters. derive mass and and solve the the effective renormalisation couplings. group equations (RGEs) for these 3.4 Renormalisation group running of the effective couplings JHEP12(2017)063 ], ]. ij ) σ = 1 12 108 (4.1) (4.2) (4.3) 3 1 2 3 Mad- 10 TeV C ( ] when C C C and i 12 σ ]. ] formalism. . However, in 109 ], but we have [ = 0 and 107 2, and we work [ 2 12 , / C H = ]. Within the m ]. Starting from the 1 . μ C ij 12 104 σ , . aMCSusHi j 1876 GeV RGE Evolution 2 aMCSusHi . C MC@NLO i − 103 C . We note here that results for 4 = 91 ij GeV 4 σ Z 5 Λ are chosen as − ]. The computation builds on the 1TeV 10 54 j 2 / , m ≤ , and EFT H i × i X = 10 TeV. Right:

m µ

1.0 0.8 0.6 0.4 0.2 0.0 σ

i ], through the µ C + , i σ and 106 SM i [ 16637 σ . ) C F – 15 – 2 = 1 at framework [ , µ 3 = 125 GeV = 1 2 1 2 3 R C 10 TeV Λ ( F C C C H µ 1TeV = 2 ]. The two-loop amplitudes for the virtual corrections i C , m X 102 = + – ,G 1 HERWIG++ 9 C 97 . SM aMC@NLO σ ] and = framework NLO results can be matched to parton shower programs, μ production in the SMEFT were presented at LO in QCD in ref. [ = 127 = 173 GeV j σ 105 t [ + 1 RGE Evolution m − EW . We include effects from bottom-quark loops (top-bottom interference H α 1 MadGraph5 aMC@NLO . group evolution of the three Wilson coefficients between 10 TeV and PYTHIA8 2 / H 2 in two scenarios. Left: = 10 TeV.

Our results for the total cross section at the LHC at 13 TeV at LO and NLO are We parametrise the contribution to the cross section from dimension-six operators as Results are obtained for the LHC at 13 TeV with MMHT2014 LO/NLO PDFs [

m

0.9 1.5 1.4 1.3 1.2 1.1 1.0 1.6

/ i C µ H have been cross-checked with the NLO+PS implementation of shown in table and pure bottom contributions) intothis the first SM study, we neglect by bottom-quark using effects from dimension-six operators in Within our setup we cansingle obtain Higgs and results for The normalisation of the operatorsfound used full here agreement differs between fromthis the the difference one is LO taken in results into ref. account. presented [ Furthermore, here the and SM top-quark those results of obtained ref. here [ with the top mass in the on-shell scheme. for LO and NLO results respectively. The values of the input parameters are The values for the central scales for using a series ofare packages implemented [ in theGraph5 code through a reweightingsuch method as [ at within the implementation of the dimension-sixSMEFT Lagrangian, operators all presented tree-level in and one-loop ref. amplitudes [ can be obtained automatically Figure 2 m JHEP12(2017)063 i O (4.4) (4.5) ). can be , which i κ 4.3 κ K . By sub- = 10TeV), 2 t K m NLO SM 138 1.64 78 1.53 127 1.61 . . . 0 6 0 122 1.59 − − 1.13 1.56 3000 1.47 0.0038 1.59 2.28 1.65 114 1.55 − 1.0 1.71 σ/σ 9% 6% . . 9% 6% 9% 6% 1 . . . . 9% 6% 9% 6% . . . . 9% 6% 9% 6% 1 1 − . . . . 0% 7% 9% 6% 9% 6% 1 1 ...... 6+1 9 1 1 − − . The Wilson coefficient . . 1 1 1 − − 4+1 3 8+1 0 − − . . . . 6+1 5 8+1 0 − − − . . . . 18 6+1 9 9+1 6 H . . . . 3+2 4 0+1 6 4+1 0 , 20 20 ...... , +22 − 19 20 i µν a 19 19 +25 − +24 − 20 20 20 +23 − +24 − (1) i i (0) ). In the infinite top mass EFT, each κ +24 − +23 − G +25 − +26 − +26 − i 2 s 04 70 κ κ . 6 5 . 6 C α . . . s NLO at 13 TeV, as parametrised in eq. ( a µν those corresponding to each operator 5 248 141 4 ( . α i − − − σ O H κG κ X + + → U + -factor, which is exactly equal to – 16 – 0 can be expressed in terms of SMEFT parameters K LO pp (1) i κ K i κ = κ s = i SM 145 61 138 α is generated by the same Feynman diagrams both at . . . κ K 0 7 0 i + σ 131 4460 − − 1.25 41 3480 109100 0.0042 0 2.38 83 125 4130 − 1.0 36 σ/σ (0) i -factors that are slightly smaller then their SM counterpart, κ K 5% 5% . . 5% 5% 5% 5% 5% 5% = ...... 5% 5% 1 . . 5% 5% 5% 5% 1 1 1 . . . . 5% 5% 5% 5% 5% 5% i 1 − ...... 1 1 − − − 0+1 0 1 1 1 κ − in the infinite top mass limit numerically (setting . . 0+1 0 0+1 0 0+1 0 − − . . . . . Indeed, each . . 0+1 0 − − − . . 0+1 0 0+1 0 i 25 . . . . 0+1 0 0+1 0 0+1 0 25 25 ...... 25 3 25 +34 − K 25 25 +34 − +34 − +34 − 25 25 25 +34 − +34 − +34 − +34 − +34 − +34 − 08 93 . 6 5 . 3 . . . LO 3 162 0890 2 . 2800 − − 26 74100 0 50 2660 − 21 σ . Total cross section in pb for can be decomposed as to each i is the universal part of the 2 denotes the SM contribution and K K U 0 κ K 23 13 12 33 22 11 3 2 1 SM σ σ σ σ σ σ σ σ σ σ 13 TeV Table 1 -factor K not entirely universal asboth NLO from corrections diagrammatic to corrections thecan and infinite be top-mass corrections obtained EFT to byas amplitudes the matching illustrated come Wilson the in coefficients figure SMEFT amplitudeas to a the perturbative infinite series top mass amplitude, written as where in the SMEFT. AsLO a and result NLO each in the infinite top-mass EFT. The effect of radiative corrections is, however, where tracting present an argument whichsection for explains Higgs this boson observation.heavy production top to quark, a We because good can mosttheory accuracy of describe where by the the taking the production top the quark happens totalbe limit is near described cross integrated of threshold. out, by an In all the infinitely this contributions same from effective contact SMEFT interaction operators can as we assume themimplementation to can be be extended subleading. to alsofrom include As effective these operators mentioned effects. have above, We see ourwith that analytic a the contributions results residual and scale MC dependence that is almost identical to the SM. In the following we JHEP12(2017)063 (4.6) (4.7) (4.8) (4.9) (4.10) -factors K . . 21 48 . . 2 0 2 0 C ] < < 3 3 . C 110 terms on the limits is at 48 2 . < C 0 C + 3000 ii , σ 2 1  < 12 . C 3 3 0 σ ) contributions the linear combi- C + 122 3 − 4 3 C , 28 0038 Λ . . C / , + 1 (1 2 C 19 + 2 + 0 0043 ) terms. We therefore find that we can . σ . O 2 SM 2 2 3 0 0 σ Λ 138 C C C . /  < 0 + 28 1 (1 2 . . 1 − O σ 2 + 114 – 17 – 1 + 2 at the LHC. Whilst here we do not attempt to = 1 < C C 1 C 2 1 C C C  ggF ggF 78 µ 0025 µ 128 . . . 6 and check explicitly that these non-universal corrections 0 0 + 114 − − − = 1 + i 1 (1) i , κ 2 3 κ 2 < C . s C 2 ggF α 28 µ 13 . 128 . < . 0 0 1 − − + 1 < < C 8 . 28 . 3 0 − − . Diagrammatic description of the matching between the SMEFT and the infinite top For reference we note that if one includes the Our results can be used to put bounds on the Wilson coefficients from measurements most 10%. nation in the bound becomes a quadratic one: the size of each individualas Wilson actual coefficient. bounds Of onvalue. course the such We Wilson obtain estimates coefficients must not and be should taken only be considered ofFor illustrative these individual operator constraints, the impact of the While the correct methodto for consider the putting combined boundspresence contribution on of of the all unconstrained parameter relevant linear operatorsoperator space combinations to would of makes a be the it given bounded SMEFT interesting observable, the is to if consider the how others each were absent in order to obtain an estimate of where we set Λ =put 1TeV the and following kept constraint only on the the Wilson coefficients with 95% confidence level: simple fit. Forsignal illustration strength purposes, in we the use diphoton the channel by recent the measurement CMS of experiment the [ gluon-fusion which we compare tothe our experimental predictions selection for efficiency is this not signal changed strength by under BSM effects the assumption that are subdominant compared to thesimilarity of universal the diagrammatic effects corrections, of which radiative explains corrections the for eachof contribution. the gluon-fusion signalperform strength a rigorous fit of the Wilson coefficients, useful information can be extracted by a mass EFT at LOcontains (left) two and elements: atand diagrammatic NLO Wilson corrections, coefficient (right). corrections, which which The contribute are universally NLO non-universal. to amplitude in the thewe infinite could extract top-mass the EFT ratios Figure 3

JHEP12(2017)063 MadGraph5_aMC@NLO ] , ) 4 4 7 the Λ and 105 1 2 3 = 1, / O O O 1 SM 1 3 3 (1 O O C O 2 . Left: Higgs i σ are harder than 1 2 O H 0 y 2 the only contribution and / , assuming that -1 3 H O m spectrum. It is known that EFT -2 ], and we use PYTHIA8 [ µ = T =62.5 GeV p . The operator coefficients are µ 107 EFT µ -3 = 6 H in the EFT LHC13 F µ → = R Higgs y p p NLO+PS, MMHT2014NLO SM Interference with the µ spectrum, we match our NLO predic- -4 T 2 1 0 1.5 0.5

p

0.10 0.08 0.06 0.04 0.02 0.00 0.14 0.12

Ratio over the SM the over Ratio

σ σ /bin d 1/ MadGraph5_aMC@NLO – 18 – 450 )) contributions from the three operators in gluon- 1 2 3 O O O 2 ) contributions involving SM 4 400 Λ ]. Λ / / (1 for the transverse momentum distribution. We find that 109 350 (1 O 5 spectrum can be used to lift the degeneracy between O 2 and Λ = 1 TeV. While at 300 T / for the interference contributions. The impact of the p ) is satisfied. We find that large deviations can be seen in the H 4 give rise to harder transverse momentum tails, while for m 250 2 4.10 [GeV] O = H T p 200 . and EFT 1 µ 3 O 150 ]. For a realistic description of the O 100 =62.5 GeV ], as discussed in ref. [ 111 , EFT = 0 at T . Higgs distributions, normalised for the interference contributions from µ = H in the EFT LHC13 F 50 2 112 45 µ → = , R C p p Higgs p NLO+PS, MMHT2014NLO Interference with the SM µ Finally we show the transverse momentum distributions for several benchmark points The normalised distributions for the transverse momentum and rapidity of the Higgs 12 [ 0 = 0 -1 -2 -3 -4 2 1 0

10 1.5 0.5 2.5

10 10 10 10 2

1 σ σ per bin [pb] bin per 1/ Ratio over the SM the over Ratio 4.3 Renormalisation group effects The impact ofwhere running we and show mixing the individualfusion between ( Higgs the production operators atC is LO demonstrated and NLO, in as figure a function of which respect thechosen total such cross-section that bounds eq.tails ( in of the figure distributions for coefficient values which respect the total cross-section bounds. boson are shown in figures terms is demonstrated in figure the operators shape is identical to thethe SM. rapidity distribution. The The dimension-six operatorsthose have involving no impact on the shape of tions to the partonfor shower with the the parton MC@NLO shower. methodMC@NLO, which [ gives Note results that thatof we are ref. in have [ good kept agreement the with shower the optimised scale scale at choice its default value in In the light ofthe differential impact Higgs of measurements themeasurements at of dimension-six the the operators Higgs LHC, onO it the is Higgs important to examine Figure 4 transverse momentum. Right: Higgstions rapidity. are SM displayed. contributions Lower and panels individual give operator the contribu- ratio over the SM. 4.2 Differential distributions

JHEP12(2017)063 MadGraph5_aMC@NLO 450 2 3 3 . We O O O SM 1 1 2 2 O O O 400 O 350

are induced

has a strong MadGraph5_aMC@NLO 2 3 300 C σ 450 250 and [GeV] H T p SM 1 400 200 C 150 350 B5 (5,0.004,0.2) B4 (2,-0.006,0.3) B1 (4,0.004,0.21) B2 (-4,0.005,-0.5) B3 (-1,0.004,-0.25) 100 =62.5 GeV EFT T µ 300 = H in the EFT LHC13 F 50 µ → = R p p Higgs p NLO+PS, MMHT2014NLO Interference between operators µ terms 4 0 0 -3 -4 -1 -2 2 1 0 Λ

10 2.5 1.5 0.5

250

10 10 10 10

σ σ per bin [pb] bin per 1/ Ratio over the SM the over Ratio [GeV] is only at the percent level, H T p

3

. SM contributions and operator contributions are MadGraph5_aMC@NLO 200 C – 19 – ij 450 σ 1 2 3 O O O SM 400 150 terms, solid: including 1/ 350 2 Λ 100 =62.5 GeV 300 EFT T µ = H in the EFT LHC13 F 50 250 µ → = [GeV] R H T p p Higgs p NLO+PS, MMHT2014NLO dashed: only 1/ µ p 200 0 2 1 0 -3 -1 -2 2 1 0 10 10 10

1.5 0.5

10 10 10

150 σ per bin [pb] bin per Ratio over the SM the over Ratio 100 =62.5 GeV EFT . Transverse momentum distributions of the Higgs for different values of the Wilson . Higgs transverse momentum distributions, normalised. Left: squared contributions T µ = H in the EFT LHC13 F 50 µ → = R p p Higgs p NLO+PS, MMHT2014NLO Squared contributions µ 0 0 -1 -2 -3 -4 2 1 0 . Right: interference between operators,

10 1.5 0.5 2.5

10 10 10 10

σ σ /bin d

1/ ii Ratio over the SM the over Ratio is coming from thescale. chromomagnetic operator, While this the contributiondependence effect changes of on rapidly the the with runningthrough scale. the of renormalisation group running, At which gives thefind rise that to same the large dependence contributions time from on non-zero the EFT values scale is of tamed when the sum of the three contributions Figure 6 coefficients. The lowerand panel the shows SM the scale ratio variation over band. the SM prediction for the various benchmarks Figure 5 σ displayed. Lower panels give the ratio over the SM. JHEP12(2017)063 -even . CP 2) = 1 which has a / 1000 H m ( 3 C [GeV] -even) operators entering the same process EFT µ CP – 20 – 100 LHC13 1 2 3 H σ σ σ 1 2 3 operator to the inclusive Higgs production at NLO in QCD. Since c c c → (total) G σ R pp solid: NLO, dashed: LO σq - Φ 2 we compute the EFT contributions at different scales, taking into account the 0 L / 50 50 ¯ H Q 100 150 200 150 100

−

m − −

[pb] σ . Contributions of the three operators to the inclusive Higgs production cross section = In this work we have computed for the first time the contribution of the ( EFT µ the SMEFT. part of the) the NLO correctionsare for available the in other the literature, two this ( calculation completes the SMEFT predictions for this couplings to the other SMcoming particles years. is The one interpretation ofcontext of the of main such an goals measurements, EFT, of and allowsnew the of one LHC interactions, possible to programme and deviations put of therefore in constraints the The on on the success the the of type scale and this of strength endeavourat new will of least physics, critically hypothetical match in depend a on the having model-independent precision theoretical way. of predictions the that experimental measurements, both in the SM and in LO and NLO we find that the relative uncertainty5 is reduced at NLO. Conclusion and outlook A precise determination of the properties of the Higgs boson and, in particular, of its is considered. This isweaker dependence the on physical the EFT cross scale.estimation section The of coming dependence the from of higher this order quantityas corrections on to an the the scale additional gives effective an uncertainty operators of and should the be predictions. reported By comparing the total contributions at Figure 7 at the LHCat at 13 TeV asrunning a and mixing function of the ofrespectively. operators. the LO EFT and NLO scale. predictions are shown Starting in from dashed and one solid non-zero lines coefficient JHEP12(2017)063 (and Mad- are very ¯ tH t where the GG ¯ tH t Φ † +jet, and and Φ G H,H R σt ˜ Φ L are similar and produce a shape ¯ Q cannot really be distinguished in GG Φ GG † Φ † and Φ G and Φ – 21 – R G σt R ˜ Φ L σt -even operators. As mentioned above and explained ¯ ˜ Φ Q L CP ¯ Q -factor is of the same order as that of the SM and of the other K has also allowed us to study the process at a fully differential level, is extremely weak. Therefore, we expect that GG , extending it to include anomalous couplings for lighter quarks, the bottom Φ † ) can effectively constrain the set of the three operators. 3 ¯ t aMC@NLO t production, the impact of the chromomagnetic operator cannot be neglected in ¯ t t In this work we have mostly focused our attention on the top-quark-Higgs boson in- The implementation of the finite part of the two-loop virtual corrections into -odd operators requires a new independent calculation. We reckon both developments vention “Fundamental Interactions” FNRS-IISNMarie 4.4517.08, Curie and Innovative Training by Networkresearch the programme MCnetITN3 of European 722104. the Union Foundation for E.V.is Fundamental Research is part on supported of (FOM), by the which the the Netherlands hospitality Organisation of for the Scientific CERN Research TH (NWO). department ND while acknowledges this work was carried out. worth pursuing. Acknowledgments This work was supported in part by the ERC grant “MathAm”, by the FNRS-IISN con- teractions and only considered in section and possibly the charm,CP is straightforward. On the other hand, extending it to include with a harder tail(which substantially are different the from same).gluon-fusion that Higgs of production, While they the do SMeffect contribute and in of a Φ the very Yukawa differentpossibly operator way to Graph5 including the effects oftransverse the momentum parton distributions shower of resummation theparameter SM and space and in the where particular three the tofound operators total compare that in the the the cross contributions region section from of bound the is respected. Once again, we have from global fits of thea Higgs three-operator couplings. fit As using aobservables the result, or total a processes Higgs two-fold to cross degeneracy constrain section is all and left three one unresolved of is by the forced operators. to look for other scale dependence also matchthat extremely the well. chromomagnetic operator Theis cannot result be of of neglected the purely for NLO theoreticalmuch at calculation dependent nature: least on confirms the two the EFT reasons. individual scale,scale The while effects choice. their first of sum The is second stable drawsuncertainties and from in only the inclusive mildly present Higgs affected status production by of the cross the section constraints. measurements and Considering the the constraints anomalous couplings of the bottomnew quark, physics we mostly have affects consideredbased in the on the top-quark previous detail calculations couplings. and the on Our caseat the where results general the features confirm inclusive of the level gluon-fusion Higgs the expectations two production: operators. 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