# Top-pair production via gluon fusion in the Standard Model Effective Field Theory

LLMU-ASC 02/21February 2021

Top-pair production via gluon fusion in theStandard Model Eﬀective Field Theory

Christoph M¨uller Ludwig-Maximilians-Universit¨at M¨unchen, Fakult¨at f¨ur Physik,Arnold Sommerfeld Center for Theoretical Physics, D-80333 M¨unchen, Germany

Abstract

We compute the leading corrections to the diﬀerential cross section for top-pairproduction via gluon fusion due to dimension-six operators at leading order in QCD.The Standard Model ﬁelds are assumed to couple only weakly to the hypotheticalnew sector. A systematic approach then suggests treating single insertions of theoperator class containing gluon ﬁeld strength tensors on the same footing as expli-citly loop suppressed contributions from four-fermion operators. This is in particularthe case for the chromomagnetic operator Q ( uG ) and the purely bosonic operators Q ( G ) and Q ( ϕG ) . All leading order dimension-six contributions are consequentlysuppressed with a loop factor 1 / π . E-mail: [email protected] a r X i v : . [ h e p - ph ] F e b Introduction

After the discovery of the Higgs-boson in 2012, a lot of attention has been paid to theunknown physics beyond the Standard Model (SM). While at proton colliders like theLarge Hadron Collider (LHC) focus has mostly been put on the direct detection of newresonances by virtue of high center-of-mass energies, searching for indirect evidence ofnew particles through their virtual eﬀects on the interactions between SM particles mightrepresent a fruitful alternative. The latter approach has the advantage of being able toprobe regimes that are beyond the kinematic energy bounds of the LHC. As the physicsbeyond the SM is a priori unknown, one has to employ a bottom-up eﬀective ﬁeld theoryin order to systematically parameterize the new physics. A model independent approachwith as few assumptions about the new physics sector as possible is provided by theStandard Model Eﬀective Field Theory (SMEFT), which essentially enlarges the at mostfour-dimensional operators of the SM with non-renormalizable higher dimensional ones.A complete basis of up to operator dimension six has been given in [1, 2] and is commonlyreferred to as the ”Warsaw basis”. Recent developments have included complete sets ofdimension eight and nine [3, 4]. However, within the framework of a bottom-up eﬀectiveﬁeld theory further assumptions about the underlying nature of the new and modiﬁedinteractions still have to be made. In this context several questions should be addressed,in particular the question whether higher loop orders have to be included for a givenﬁxed canonical order calculation. This crucially depends on the coupling strengths of theSM ﬁelds to the new sector. For top-quark decay it has already been noted that higherloop order corrections might be important [5], in which case the respective running ofthe SMEFT operators should be taken into account. For instance, the renormalizationgroup equations for magnetic-moment-type operators [6] suggest that a cancellation ofthe unphysical renormalization scale can only be achieved upon adding the correct fourfermion-operators to the analysis.This work about gluon fusion top-pair production serves as an example of how to treat theoperator class containing gluon ﬁeld strength tensors in general with the only assumptionabout the new physics sector being its weak coupling to the SM. Higher loop order cor-rections associated with four-fermion operators are in fact necessary to obtain consistentresults.As we intend to focus on the main calculational aspects, we have made some phenomeno-logical simpliﬁcations for the sake of clarity. First, since cross sections in QCD factorizeinto soft and hard parts of the interaction, we may focus on the pure parton level reaction.In reality, parton distribution functions for the initial hadrons as well as jets associatedwith the ﬁnal quark pair play crucial roles for experimental constraints that should betaken into account for a full phenomenological analysis. Second, we concentrate on thegluon fusion channels, since for instance at the LHC quark fusion becomes less importantthan gluon fusion with increasing energy. An in-depth review for top-pair production isgiven in [7]. All contributions of the tree-level operators for gluon fusion top-pair pro-duction in SMEFT together with a comment on the assumptions about the underlyingtheory can be found in [8]. 1his paper is organized as follows: The general notational setup together with a summaryof the SM calculations for gluon fusion top-pair production is done in section 2. In section3 we identify the relevant SMEFT operators and their respective contributions. We alsodiscuss how to treat the chiral projectors and comment on the renormalization program.A brief phenomenological study of some of the new contributions is given in section 4.All our analytical results for the cross sections are listed in an appendix.

We consider the parton level process gg −→ t ¯ t where two initial gluons merge to producea top- and an antitop quark (see Figure 1). The unpolarized diﬀerential cross section inthe center-of-mass frame in terms of the amplitude M is then given by the formula dσd Ω = |M| π s | p || k | (1)where in |M| an average over initial and a sum over ﬁnal spins and colors is understoodand | p | and | k | denote the spatial momenta of the outgoing top-quark and either in-coming gluon. It can generally be written as a function of the relevant coupling constants,the Mandelstam variables s , t and u and the masses of the involved particles. The SMresult is straightforwardly obtained by evaluating three diagrams and is given by [9] (cid:18) dσd Ω (cid:19) SM = α s s (cid:114) − m t s m t − m t ( t + u ) + 4 t − tu + 4 u s ( m t − t ) ( m t − u ) (cid:16) tu ( t + u )+ − m t + m t (3 t + 14 tu + 3 u ) − m t ( t + u )( t + 6 tu + u ) (cid:17) (2)where α s = g s / π is the strong ﬁne structure constant and m t is the top-mass. The ﬁrst contributions to gg −→ ¯ tt from operators of mass (canonical) dimension greaterthan four appear at dimension six and are accordingly suppressed by 1 / Λ , where Λdenotes a potentially large scale of new physics. In our calculation we consider the inter-ference terms of these new operators with the SM amplitude |M| = |M | + 1Λ ( M ∗ M + M ∗ M ) + O (cid:0) / Λ (cid:1) (3)where the indices 0 and 1 refer to the SM and SMEFT amplitudes, respectively. Withinthis notation the canonical dimension of the various terms is manifest. However, it isimportant to notice that for eﬀective theories a more general power counting prescriptionneeds to be speciﬁed, even if it is linearly realized [10–12]. Before we dive into the actualcalculation, we review some aspects of power counting in SMEFT.2igure 1: Kinematic setup for gluon fusion top-pair production with incoming momenta k and k and outgoing momenta p and p . The circle in the middle represents all possibleinteractions for the theory under consideration. There are three Feynman diagrams withinthe SM at leading order. The SMEFT diagrams will be discussed below. Higher dimensional operators are naturally organized in terms of their canonical dimen-sion that indicates their relative suppression with respect to the new physics scale Λ (seeEquation (3) above). This corresponds to an expansion in the dimensionless parameter E/ Λ, where E is a typical energy scale of the process, and from the point of view ofa renormalizable ﬁeld theory, the canonical dimension remains the strongest tool for asystematic approach. However, canonical dimensions alone do not provide enough infor-mation for a consistent treatment of the new operators. In addition, one needs to specifythe loop order of a speciﬁc operator within the relevant process by including the expan-sion parameter 1 / π to the analysis. The necessity of keeping track of loop orders hasbeen put on solid grounds in [13] and can be discussed quantitatively in terms of chiraldimensions in a more general manner, see [11, 14]. For instance, in [13] it has been arguedthat on the one hand, operators containing ﬁeld strength tensors cannot be generated attree-level when the underlying theory is assumed to be a weakly coupled gauge theory.In fact, the matching procedure has to occur at the one-loop-level at least. This has alsobeen noted in [2]. On the other hand, four-fermion operators can in fact be generated attree-level.The arguments presented in [13] remain true for any underlying theory as long as it cou-ples only weakly to the SM ﬁelds, which is the case for a wide class of possible high energyscenarios. The respective Wilson coeﬃcients of operators containing ﬁeld strength tensorsshould consequently be suppressed by the loop factor 1 / π when compared to the onesof the four-fermion operators in this case. Likewise it is easy to construct model theo-ries that contradict the na¨ıve assignment of an O (1)-number to every Wilson coeﬃcient.When a model independent parameterization like SMEFT is chosen, it therefore seems3igure 2: Feynman diagrams for the tree contributions in SMEFT, where dimension-sixinsertions are denoted by black squares. Crossings are not displayed. a - c)

Contributionsfrom Q ( uG ) . The local interaction c) does not appear in the SM. d) Contribution from Q ( G ) which modiﬁes the s-channel process of the SM. e) New contribution from Q ( ϕG ) . natural to stick to a weakly coupling new sector and to adopt the corresponding rules for asuperﬁcial estimation of the Wilson coeﬃcients. Applying them to gluon fusion top-pairproduction, we ﬁnd that there does not exist a pure tree-level contribution within theSMEFT at leading order in QCD at all. On the contrary, the ﬁrst non-vanishing cor-rections are suppressed with respect to both 1 / Λ and / π , i.e. they are all one-loopsuppressed. We will provide a complete list of the relevant operators and their particularcontributions in the next section. As the new physics is expected to mainly aﬀect the third particle generation, we restrictour calculation to this sector and neglect eﬀects of the CKM-matrix. We also ignore CP-odd operators and impose baryon number conservation. In the Warsaw basis the followingoperators are relevant for gluon fusion top-pair production at tree-level Q ( G ) = f ABC G Aνµ G Bρν G Cµρ Q ( ϕG ) = ϕ † ϕG Aµν G Aµν Q ( uG ) = (¯ qσ µν T A t ) ˜ ϕG Aµν (4)4here q is the left-handed third generation quark-doublet, t is the right-handed top-quark, ϕ is the Higgs-doublet ( ˜ ϕ i = (cid:15) ij ϕ ∗ j ) and G Aµν is the A th component of the gluonﬁeld strength tensor with respect to the SU (3) generators T A = λ A /

2, where λ A arethe Gell-Mann matrices. We adopt the conventions found in [15]. These operators leadto the diagrams displayed in Figure 2. Although their contributions are eventually loopsuppressed by virtue of their Wilson-coeﬃcients, we will refer to the operators in Equation(4) as the ”tree-level operators”. Note that the local interaction between gluons and theHiggs-particle introduces a new s-channel contribution associated with the Higgs-mass m h . Also, the chromomagnetic operator Q ( uG ) induces a new local interaction betweentwo gluons and two quarks. Its role for gluon fusion Higgs production to higher looporders has been investigated in [16, 17]. For all tree-level operators we ﬁnd full agreementwith the analytic expressions for the diﬀerential cross sections of gluon fusion top-pairproduction found in [8, 18].In view of the last section the tree-level operators have to be supplemented by the followingfour-fermion operators Q (1)( qd ) = (¯ qγ µ q )(¯ bγ µ b ) Q (8)( qd ) = (¯ qγ µ T A q )(¯ bγ µ T A b ) Q (1)( ud ) = (¯ tγ µ t )(¯ bγ µ b ) Q (8)( ud ) = (¯ tγ µ T A t )(¯ bγ µ T A b ) Q (1)( qu ) = (¯ qγ µ q )(¯ tγ µ t ) Q (8)( qu ) = (¯ qγ µ T A q )(¯ tγ µ T A t ) Q (1)( qq ) = (¯ qγ µ q )(¯ qγ µ q ) Q (3)( qq ) = (¯ qγ µ τ I q )(¯ qγ µ τ I q ) Q ( uu ) = (¯ tγ µ t )(¯ tγ µ t ) Q (1)( quqd ) = (¯ q j t ) (cid:15) jk (¯ q k b ) Q (8)( quqd ) = (¯ q j T A t ) (cid:15) jk (¯ q k T A b ) (5)where b denotes the right-handed bottom-quark and τ I are the Pauli-matrices. This listrepresents a complete set of four-fermion operators consistent with the assumptions madeabove and leads to the loop-diagrams shown in Figure 3. The operators Q ( uG ) , Q (1)( quqd ) and Q (8)( quqd ) are not hermitian. However, only the real parts of their Wilson coeﬃcients enterthe ﬁnal expression for the cross section at O (1 / Λ ). The subtleties regarding minus signsarising from ordering ambiguities in the four-fermion operators are discussed in [19]. γ The Dirac structure of the four-fermion operators include chiral projectors which are ac-companied by the strictly four-dimensional object γ . There have been many discussionsconcerning the question of how to regularize divergent amplitudes with a consistent treat-ment of γ , in particular for triangle diagrams like the ones appearing in this work, seee.g. [20] for a recent review. After all, a na¨ıve application of dimensional regularization(NDR) is at ﬁrst sight incompatible with a straightforward continuation of γ whose re-lation to the other Dirac matrices are unambiguously deﬁned in four dimensions only. In5igure 3: Feynman diagrams for the one-loop contributions in SMEFT. Again, crossingsare not displayed. These diagrams are needed to cancel the implicit dependence on therenormalization scale µ in the diagrams of Figure 2 a - c).fact, an anti-commuting γ cannot be extrapolated to arbitrary dimensions together withthe trace formula tr ( γ α γ β γ γ γ δ γ ) = − i(cid:15) αβγδ (6)where we have chosen the default sign convention implemented in FeynCalc [21]. How-ever, despite being inconsistent in the ﬁrst place, NDR is known to lead to correct ﬁnalresults when the potential hard anomalies are separately taken into account as has beenreviewed in [22].Irrespective of the particular regularization method, most approaches usually rely on shift-ing the loop momenta at some point of the calculation, so linearly divergent diagrams areexpected to produce non-vanishing surface terms. These in general spoil the underlyinggauge invariance and should consequently be adjusted by hand in order to keep gaugeinvariance intact as was demonstrated in [23]. For instance, in the textbook example ofa three-point-function of two vector- and one axial-vector current for massless fermions,one can shuﬄe the anomaly around by virtue of boundary terms in a way that the twovector (gauge) currents are conserved, but the axial current - which in our case corre-sponds to the four-fermion interaction - is not. Having this example in mind, we ﬁx thesuperﬁcially divergent boundary terms by requiring manifest gauge invariance at eachcalculational step. Since in the case at hand the SMEFT operators do not modify theSM gauge couplings within the triangle diagrams (as is the case for example in [24]), noinduced gauge anomalies are expected to play a role. Also, SMEFT alone does not makeany assumptions about the new physics sector, so anomalies in the latter are beyond thescope of this paper and are therefore neglected. More comments about the four-fermiontriangle diagrams can be found in [25]. Further discussions about calculational aspectsregarding γ can be found in [5, 26–28]. 6 .4 Renormalization The self-energy corrections of the quark lines induced by the four-fermion operators donot depend on the momenta and are therefore negligible when on-shell renormalizationis performed. In general, however, we also need to renormalize the Wilson coeﬃcients ofthe SMEFT operators, which is done in the

M S scheme.Most of the contributions are already ﬁnite by themselves and do not need any renormal-ization. Operator mixing only appears between the operators Q ( uG ) , Q (1)( quqd ) and Q (8)( quqd ) ,as the latter two produce UV-divergences in the form of the ﬁrst one. The tree-leveloperator Q ( uG ) is therefore the only one that needs special treatment. To the order underconsideration an implicit dependence of its Wilson coeﬃcient on the renormalization scale µ is introduced, which has to cancel the explicit logarithmic dependence associated with Q (1)( quqd ) and Q (8)( quqd ) .The relevant part of the renormalization group equation is given by [6, 29] dC ( uG ) d ln( µ ) = − g s m b π √ v (cid:18) C (1)( quqd ) − C (8)( quqd ) (cid:19) (7)where v is the vacuum expectation value of the Higgs-ﬁeld and C i denotes the respectiveWilson coeﬃcient of the operator Q i . When this equation is applied to our total diﬀeren-tial cross section, we ﬁnd that our expressions are indeed independent of the renormaliza-tion scale µ . Note that all mixing disappears when the bottom-mass m b is sent to zero.On the other hand, it provides a useful consistency check when the bottom-mass is fullytaken into account. m t m b m H v α s ( M Z ) Λ173 GeV 4 .

18 GeV 125 GeV 246 GeV 0 . Values of the input parameters used for the analysis taken from [9].This section contains an exploratory analysis of the SMEFT corrections to the SM crosssection. The input parameters are given in Table 1. In addition we assume a cut-oﬀ scaleof Λ = 1 TeV, which is well beyond the parameters of the SM and set µ = m t . As QCDcorrections at next-to-leading order within the SM (see [30, 31] for the computation) donot change the shape of the total cross section as a function of the center-of-mass energy[7], the overall normalization of our results can possibly be adjusted to ﬁt the more realisticcurves. Within the SM the K-factor from next-to-leading order QCD corrections is about1 . α s ( M Z ) = 0 . SM diﬀerential cross section ( dσ/d cos θ ) SM . b - d) Selected SMEFT correc-tions to the SM result b) ( dσ/d cos θ ) Q ( ϕG ) , c) ( dσ/d cos θ ) Q ( uG ) and d) ( dσ/d cos θ ) Q ( uu ) .The diﬀerential cross sections dσ/d cos θ are given in units of pb. The center-of-massenergy was chosen to be √ s = 1 TeV.we used Version 3 of RunDec [34] to determine the M S -value α s (1 TeV) = 0 . √ s is unavoidable in this context. However, as we intend to onlypresent indicative estimations of the new eﬀects when compared to the SM predictions,this should not impact the overall message.In accordance with our discussion about loop counting above we choose the values ofthe Wilson coeﬃcients of the operators displayed in Equation (4) to be 1 / π , whereasthe ones of the four-fermion operators in Equation (5) are set to 1. Although the positivesign for all coeﬃcients is a mere convention, it does not aﬀect the qualitative outcomeof our analysis as we focus only on the magnitudes of the new eﬀects. Of course, de-spite being well motivated these assumptions may only serve as preliminary estimationsfor a more complete analysis and should not be taken at face value. Note that withinthe chosen framework experiments are actually only sensitive to the ratio C i / Λ . Fixingthe cutoﬀ-scale Λ and the Wilson coeﬃcients C i separately therefore corresponds to anartiﬁcial split of the actual experimental coeﬃcients.8igure 5: Selected relative corrections to the SM diﬀerential cross section at √ s = 1 TeV with a) f = ( dσ/d cos θ ) Q ( ϕG ) / ( dσ/d cos θ ) SM , b) f = ( dσ/d cos θ ) Q ( uG ) / ( dσ/d cos θ ) SM and c) f = ( dσ/d cos θ ) Q ( uu ) / ( dσ/d cos θ ) SM . We plot the expected corrections to the SM diﬀerential cross section dσ/d cos θ of theoperators Q ( ϕG ) , Q ( uG ) and Q ( uu ) for √ s = 1 TeV in Figures 4 and 5. Integrating overthe residual angular dependence gives the total cross section σ that is plotted againstthe center-of-mass energy √ s in Figure 6. The plots for the remaining operators havesimilar shapes and are not displayed. In comparison to the SM all SMEFT correctionsare suppressed by both 1 / Λ and 1 / π , so the overall eﬀects are rather small. For therelevant energy regimes, i.e. √ s ≈ − to the SM diﬀerential cross section. However,there are some interesting qualitative observations: • Only the operators Q ( G ) , Q (1)( quqd ) and Q (1)( qu ) have their largest relative impact for highscattering angles. In contrast, all other operators, in particular the ones in Figures 4 - 6appear to be more signiﬁcant for low scattering angles. • The relative correction to the total SM cross section (not plotted) increases most rapidlyfor the operator Q (1)( qu ) , reaching the percent level at around √ s = 4 TeV. Of course, pos-sible resonances above 1 TeV could spoil the validity of the eﬀective theory in this energyregime. • There is a change of sign in the correction of the total cross section just after thethreshold energy √ s = 2 m t for the operators Q (1)( quqd ) , Q ( uu ) , Q (1)( qq ) , Q (3)( qq ) , Q (1)( qu ) and Q (8)( qu ) .The last aspect might in particular be useful for new constraints of the purely right-handed operator Q ( uu ) (see Figure 6 d for the curve). This operator has recently beeninvestigated in [35] where an emphasis was put on four-top production at hadron colliders.The combined upper limit from the ATLAS experiment for four-top production is givenby | C ( uu ) | ≤ . ∼ π tothe new physics in the top-sector would lead us to superﬁcially expect a numerical value9round | C ( uu ) | ≈ π if - na¨ıvely trusting perturbation theory - the four-fermion operatorwere generated by new exchange processes. The smallness of the actual value, however,indicates signiﬁcant limitations for such strong coupling scenarios and thus also for thevalues of C ( G ) , C ( ϕG ) and C ( uG ) (see below).Apart from the top-sector, the Higgs-sector is predestined for strong couplings in the newphysics domain that could enhance certain Wilson coeﬃcients involving the Higgs-ﬁeld.Scenarios with strong dynamics of electroweak symmetry breaking can be described infull generality by the Higgs-Electroweak Chiral Lagrangian [10, 12]. In particular, it waspointed out that matching the latter to SMEFT generates the operator class of Q ( ϕG ) without the extra loop factor 1 / π , in which case its Wilson coeﬃcients should betreated as O (1)-numbers. Indeed, keeping track of all possible weak coupling constantsfor a given operator generated in a strongly coupled Higgs scenario reveals that the chiraldimensions, i.e. the loop order is eﬀectively reduced by the Higgs-ﬁeld. The operator Q ( ϕG ) then represents the dominant SMEFT correction to gluon fusion top-pair produc-tion with relative impact to the SM diﬀerential cross section in the low percent range.In contrast, the operator Q ( uG ) can still only be generated at next-to-leading order, sohere our discussion above remains valid even for strong couplings in the Higgs-sector. Seealso [37] and the comments in Chapter 2.1 in [38] for gluon fusion Higgs-pair produc-tion. Within the Higgs-Electroweak Chiral Lagrangian, the gluon-gluon-Higgs coupling isparameterized by a coeﬃcient c g , that can directly be translated to C ( ϕG ) via the formula C ( ϕG ) = Λ α s πv c g ∼ . c g (8)where the parameters are deﬁned in Table 1. Keep in mind, that in a strongly coupledscenario, there is no decoupling of the eﬀective theory from its cut-oﬀ scale Λ s.c. , as it isrelated to the low energy scale v by Λ s.c. = 4 πv . A loop factor 1 / π can therefore betraded against the expansion parameter v / Λ s.c. and vice versa. The experimental valuefor c g can be found in [39] and is approximately given by c g = − . ± .

08, indicatingthat the overall eﬀects are still rather small.Experimental constraints for the remaining tree-level operators can be found in [18, 40–44] and are at best given by | C ( uG ) | ≤ .

78 and | C ( G ) | ≤ .

037 for Λ = 1 TeV, dependingon the ﬁtting procedure. While the latter value seems more plausible in light of ourassumptions, the experimental bounds of the former are not as constraining. As a matterof fact, both numbers are still well above their natural value of around 1 / π ≈ . SM total cross section σ SM . b - d) Selected SMEFT corrections to the SMresult b) σ Q ( ϕG ) , c) σ Q ( uG ) and d) σ Q ( uu ) . The center-of-mass energies √ s are given inunits of GeV, whereas the total cross sections σ are again given in units of pb. In this paper we have computed the diﬀerential cross section for gluon fusion top-pairproduction in SMEFT including single insertions of operators of canonical dimension six.Systematic power counting rules relying on realistic assumptions about the new physicssector lead us to consider the tree-level contributions on the same footing as one-loopcontributions arising from four-fermion operators. Our calculation serves as an exampleof how to treat operators featuring gluon ﬁeld strength tensors in a consistent mannerwithin the perturbative expansion. In particular, it illustrates how this class needs to beaccompanied by four-fermion operators with explicit loop suppression to ensure workingwith a complete set of operators for a given loop order. As a result, the overall SMEFTeﬀects are rather small. Meanwhile, since next-to-leading order QCD corrections areexpected to be of great importance to the process under concern, a broader analysis shouldinclude them as well. We postpone a general phenomenological discussion together witha more complete analysis concerning the parameter space of the Wilson coeﬃcients tofuture works. 11 cknowledgments

I would like to thank Gerhard Buchalla for useful discussions at the diﬀerent stages of thiswork and valuable comments on the manuscript. I am supported by the Studienstiftungdes Deutschen Volkes and by the Deutsche Forschungsgemeinschaft (DFG, German Re-search Foundation) under grant BU 1391/2-2 (project number 261324988) and by theDFG under Germanys Excellence Strategy EXC-2094 390783311 ORIGINS.

A Corrections to the SM diﬀerential cross section

This appendix features a complete list of the analytic expressions for the corrections to theSM diﬀerential cross section due to the SMEFT operators. The diﬀerential cross sectionincluding the SMEFT corrections can be written as (cid:18) dσd Ω (cid:19) SM + (cid:18) dσd Ω (cid:19) SMEF T (9)where ( dσ/d Ω) SMEF T denotes the sum of all contributions of the dimension-six operatorsunder concern.The relevant real parts of the Feynman parameter integrals are given by S ( a i ) = Re (cid:40)(cid:90) dz (cid:90) − z dy − yza i − iη (cid:41) = − ln (4 a i ) + π a i + O (cid:18) a i (cid:19) (10) S ( a i ) = Re (cid:40)(cid:90) dz (cid:90) − z dy y − yza i − iη (cid:41) = − ln(4 a i ) + 24 a i + O (cid:18) a i (cid:19) (11)where a i = s/ m i for the top- and the bottom-mass and η is a small positive number anda center-of-mass energy above the threshold for top-pair production is assumed.Deﬁning (cid:101) C i = C i / Λ and β = (cid:112) − m t /s , the single analytic expressions for the contri-butions of the SMEFT operators are given by the following list: (cid:18) dσd Ω (cid:19) Q ( G ) = − (cid:101) C ( G ) α s m t β g s s t − u ) ( m t − t )( m t − u ) (12) (cid:18) dσd Ω (cid:19) Q ( ϕG ) = (cid:101) C ( ϕG ) α s m t β π s ( m t − t )( m t − u )( m h − s ) (13) (cid:18) dσd Ω (cid:19) Q ( uG ) = − Re { (cid:101) C ( uG ) ( µ ) } α / s vm t β √ πs m t − m t ( t + u ) + 4 t − tu + 4 u ( m t − t )( m t − u ) (14)12 dσd Ω (cid:19) Q (1)( quqd ) = − Re { (cid:101) C (1)( quqd ) } π α s m t β m b s m t − t )( m t − u ) ·· (cid:18) s (cid:16) m t S ( a b ) + m b β (cid:0) S ( a b ) − (cid:1)(cid:17) ++ 8 m b (cid:0) m t − m t ( t + u ) + 4 t − tu + 4 u (cid:1) ln µ m b (cid:19) (15) (cid:18) dσd Ω (cid:19) Q (8)( quqd ) = − Re { (cid:101) C (8)( quqd ) } π α s m t β m b s m t − t )( m t − u ) ·· (cid:18) s (cid:16) m t S ( a b ) + m b β (cid:0) S ( a b ) − (cid:1)(cid:17) + − m b (cid:0) m t − m t ( t + u ) + 4 t − tu + 4 u (cid:1) ln µ m b (cid:19) (16) (cid:18) dσd Ω (cid:19) Q (1)( qd ) = − (cid:101) C (1)( qd ) π α s m t sβ

32 2 S ( a b ) − m t − t )( m t − u ) (17) (cid:18) dσd Ω (cid:19) Q (1)( ud ) = (cid:101) C (1)( ud ) π α s m t sβ

32 2 S ( a b ) − m t − t )( m t − u ) (18) (cid:18) dσd Ω (cid:19) Q (8)( qd ) = − (cid:101) C (8)( qd ) π α s m t β s s (cid:0) S ( a b ) − (cid:1) + 3( t − u ) (cid:0) S ( a b ) − S ( a b ) − (cid:1) ( m t − t )( m t − u ) (19) (cid:18) dσd Ω (cid:19) Q (8)( ud ) = (cid:101) C (8)( ud ) π α s m t β s s (cid:0) S ( a b ) − (cid:1) − t − u ) (cid:0) S ( a b ) − S ( a b ) − (cid:1) ( m t − t )( m t − u ) (20) (cid:18) dσd Ω (cid:19) Q ( uu ) = (cid:101) C ( uu ) π α s m t β s s (cid:0) S ( a t ) − (cid:1) − t − u ) (cid:0) S ( a t ) − S ( a t ) − (cid:1) ( m t − t )( m t − u ) (21) (cid:18) dσd Ω (cid:19) Q (1)( qq ) = (cid:101) C (1)( qq ) π α s m t β s m t − t )( m t − u ) (cid:16) s (cid:0) S ( a b ) − (cid:1) ++ 13 s (cid:0) S ( a t ) − (cid:1) − t − u ) (cid:0) S ( a t ) − S ( a t ) − (cid:1)(cid:17) (22)13 dσd Ω (cid:19) (cid:101) C (3)( qq ) = (cid:101) C (3)( qq ) π α s m t β s m t − t )( m t − u ) (cid:16) s (cid:0) S ( a b ) − (cid:1) ++ 13 s (cid:0) S ( a t ) − (cid:1) − t − u ) (cid:0) S ( a b ) − S ( a b ) − (cid:1) + − t − u ) (cid:0) S ( a t ) − S ( a t ) − (cid:1)(cid:17) (23) (cid:18) dσd Ω (cid:19) Q (1)( qu ) = − (cid:101) C (1)( qu ) π α s β s m t − t )( m t − u ) (cid:18) m t s (cid:0) S ( a b ) − (cid:1) + 56 m t ++ 4 m t (cid:16) m t s (cid:0) S ( a t ) − (cid:1) − s S ( a t ) + 8 t − tu + 8 u (cid:17) + − m t s (cid:0) S ( a t ) − (cid:1) − m t ( t + u ) + 14 s S ( a t ) (cid:19) (24) (cid:18) dσd Ω (cid:19) Q (8)( qu ) = − (cid:101) C (8)( qu ) π α s β s m t − t )( m t − u ) (cid:18) m t s (cid:0) S ( a b ) − (cid:1) + − m t s (cid:0) S ( a t ) − (cid:1) − m t + 112 m t ( t + u )+ − m t (cid:16) − m t s (cid:0) S ( a t ) − (cid:1) + 11 s S ( a t ) + 8 t − tu + 8 u (cid:17) ++ 9 m t ( t − u ) (cid:0) S ( a b ) − S ( a b ) − (cid:1) + 44 s S ( a t ) (cid:19) (25)14 eferences [1] W. Buchmueller and D. Wyler, “Eﬀective Lagrangian Analysis of New Interactionsand Flavor Conservation,” Nucl. Phys. B (1986) 621–653.[2] B. Grzadkowski, M. Iskrzynski, M. Misiak, and J. Rosiek, “Dimension-Six Terms inthe Standard Model Lagrangian,”

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