Probing top quark neutral couplings in the Standard Model Effective Field Theory at NLO in QCD

Journal of High Energy Physics, May 2016

Top quark pair production in association with a Z-boson or a photon at the LHC directly probes neutral top-quark couplings. We present predictions for these two processes in the Standard Model (SM) Effective Field Theory (EFT) at next-to-leading order (NLO) in QCD. We include the full set of CP-even dimension-six operators that enter the top-quark interactions with the SM gauge bosons. For comparison, we also present predictions in the SMEFT for top loop-induced HZ production at the LHC and for \( t\overline{t} \) production at the ILC at NLO in QCD. Results for total cross sections and differential distributions are obtained and uncertainties coming from missing higher orders in the strong coupling and in the EFT expansions are discussed. NLO results matched to the parton shower are available, allowing for event generation to be directly employed in an experimental analyses. Our framework provides a solid basis for the interpretation of current and future measurements in the SMEFT, with improved accuracy and precision.

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Probing top quark neutral couplings in the Standard Model Effective Field Theory at NLO in QCD

JHE Probing top quark neutral couplings in the Standard Model E ective Field Theory at NLO in QCD Olga Bessidskaia Bylund 0 1 3 Fabio Maltoni 0 1 Ioannis Tsinikos 0 1 Eleni Vryonidou 0 1 Cen Zhang 0 1 2 Louvain-la-Neuve 0 1 Belgium 0 1 0 Upton , NY 11973 , U.S.A 1 SE-10691 Stockholm , Sweden 2 Department of Physics, Brookhaven National Laboratory 3 Oskar Klein Centre and Department of Physics, Stockholm University Top quark pair production in association with a Z-boson or a photon at the LHC directly probes neutral top-quark couplings. We present predictions for these two processes in the Standard Model (SM) E ective Field Theory (EFT) at next-to-leading order (NLO) in QCD. We include the full set of CP-even dimension-six operators that enter the top-quark interactions with the SM gauge bosons. For comparison, we also present predictions in the SMEFT for top loop-induced HZ production at the LHC and for tt production at the ILC at NLO in QCD. Results for total cross sections and di erential distributions are obtained and uncertainties coming from missing higher orders in the strong coupling and in the EFT expansions are discussed. NLO results matched to the parton shower are available, allowing for event generation to be directly employed in an experimental analyses. Beyond Standard Model; E ective eld theories - HJEP05(216) 1 Introduction 2 E ective operators 3 Calculation setup 4 Results for ttZ, tt and tt + 4.1 4.2 5 Results for gg ! HZ 6 Results for the ILC 7 Theoretical uncertainties 8 Discussion 9 Summary and conclusions A Connection with \anomalous coupling" approach B Ratios for comparing with measurements B.1 ATLAS | ttZ B.2 CMS | tt 1 Introduction Top quark measurements are an important priority in Run II at the LHC. Results from the Tevatron and the rst run of the LHC at 7 and 8 TeV have yielded precise measurements of the main top quark production channels, i.e. top-anti-top production and single top production. At the LHC, the high energy and luminosity open up new possibilities to access rarer production processes, such as the associated production of top pairs with a vector boson. These processes are particularly interesting, as they provide the rst probe of the neutral couplings of the top quark to the electroweak gauge bosons, which were not accessible at the Tevatron due to their high production thresholds. Therefore these channels could give important information about the top quark, which are complementary to top-pair and single-top production measurements as well as the top decay measurements. Measurements of tt have been performed at the Tevatron by CDF [1], and at the LHC by CMS [2] and by ATLAS [3]. Results for ttZ and ttW by CMS appear in [4, 5] and by ATLAS in [6]. { 1 { Measurements of these processes allow us to search for deviations from the Standard Model (SM) predictions. While these deviations are often interpreted in terms of anomalous top couplings, the SM E ective Field Theory (SMEFT) provides a much more powerful framework [7{9]. In this approach possible deviations can be consistently and systematically described by the e ects of higher-dimensional operators of the SM elds. By employing global analyses [10{12], experimental results can be used to determine the size of the deviations due to each e ective operator. The established deviations can then be consistently evolved up to high scales, and matched to possible new physics scenarios. In the absence of convincing evidence for new resonance states, the EFT provides the most model-independent approach to a global interpretation of measurements. With Run-II of the LHC, more and more precise measurements in the top-quark sector can be expected. In this respect, theoretical predictions matching the foreseeable precision of the experimental determinations are required to extract correct and useful information about deviations in the top-quark sector. For this reason, recently fully di erential NLO QCD corrections to top-quark processes within the top quark EFT have started to become available, for example for the top-decay processes including the main decay channel and the avor-changing channels [13, 14], and for single-top production triggered by avor-changing neutral interactions of the top [15]. More recently, the two main production channels in the SM, top-quark pair production and single top production, have also become available at dimension-six at NLO in QCD [16, 17]. QCD corrections are found to have nontrivial impact on SMEFT analyses [17]. In this work, we pursue this line of research further. We provide NLO QCD predictions for the ttZ and tt channels at the LHC and tt production at the ILC, including the full set of dimension-six operators that parametrise the interactions between the top-quark and the SM gauge bosons. Note that results for pp ! tt at NLO appear here for the rst time, while pp ! ttZ and e+e ! tt have been calculated at NLO in QCD in refs. [18, 19] in the anomalous coupling approach, albeit with the omission of the chromomagnetic dipole operator. As we will see, this operator gives a very important contribution to both the ttZ and tt processes. In addition, we also present results for the top-loop induced HZ production, which involves the same operators. An important feature of our approach is that NLO predictions matched to the parton shower (PS) are provided in an automatic way. Our results are important not only because predictions are improved in accuracy and in precision, but also because NLO results can be used directly in an experimental simulation, allowing for a more dedicated investigation of all the features of any potential deviations, with possibly optimised selections and improved sensitivities to probe EFT signals. Our approach is based on the MadGraph5 aMC@NLO (MG5 aMC) [20] framework, and is part of the ongoing e orts of automating NLO EFT simulations for colliders [21]. The paper is organised as follows. In section 2, we present the relevant dimensionsix operators. In section 3, we present our calculation setup. Results for the ttZ, tt , gg ! HZ processes at the LHC and tt production at the ILC are given in sections 4{6, the sensitivity of the various processes on the operators in light of the corresponding LHC measurements. We draw our conclusions and discuss the outlook in section 9. { 2 { E ective operators In an EFT approach, SM deviations are described by higher-dimensional operators. Up to dimension-six, we consider the following operators [22, 23]: O'(3Q) = i y O'(1Q) = i y 21 t2 'y D!I ' (Q 21 t2 'y D! ' (Q O't = i y 21 t2 'y D! ' (t t) I Q) Q) OtW = ytgw(Q OtB = ytgY (Q OtG = ytgs(Q I t)'~W I t)'~B T At)'~GA ; L = X Ci i 2 Oi + h:c: yt = p where Q is the third generation left-handed quark doublet, ' is the Higgs eld, gW , gY and gs are the SM gauge coupling constants, yt is the top-Yukawa coupling, de ned by 2mt=v where v is the Higgs vacuum expectation value and mt is the pole mass (and so yt does not run). At lowest order in perturbation expansion, the Lagrangian is modi ed by these operators as follows: Note that the Hermitian conjugate of each operator is added. The above operators form a complete set that parameterises the top-quark couplings to the gluon and the electroweak gauge bosons of the SM, which could contribute at O( 2). In this work, we focus on their contributions to top production processes at colliders calculated at NLO in QCD. The rst three operators are tree-level generated current-current operators. They modify the vector and axial-vector coupling of the top quark to the electroweak gauge bosons. The other three are dipole operators, that are more likely to be loop induced. OtW and OtB give rise to electroweak dipole moments, and OtG is the chromomagnetic dipole operator, relevant for the interaction of the top quark with gluons. Up to order 2, the cross sections and di erential observables considered in this work do not receive CP-odd contributions, so in the following we assume the coe cients of OtW;tB;tG to be real. The three current operators are Hermitian so their coe cients are always real. The operators enter the vertices are marked out on the example Feynman diagram for the ttZ, tt processes in gure 1. A complete study of the processes considered here involve more operators at dimensionsix. For example, four-fermion operators featuring top-quark pairs will also contribute to these processes. They are the same set of seven operators that contribute to top pair production as discussed in [24, 25]. Additional four-fermion operators could enter and modify poning this to future studies. Operators involving the gauge bosons and light quarks could in principle contribute to these processes, but as they receive stringent constraints from { 3 { (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) g ¯ t Z/γ enter the gtt vertex (OtG), the tt vertex (OtW ; OtB) or the ttZ vertex (O t; O(3Q) ; O(1Q) ; OtW ; OtB). HJEP05(216) precision observables, we consider their e ect to these processes to be negligible compared to the top operators. Another operator that contributes to the ttZ=tt processes is OG, which would enter by modifying the gluon self-interactions. As this is not a top-quark operator, we will not consider it further here, assuming also that its contribution is su ciently suppressed due to constraints from the accurately measured tt and dijet cross sections. In our approach, we also take into account an additional operator, O'b (identical to O't with b replacing t), which does not involve a top quark, but does contribute to, for example, NLO ttZ production through a bottom loop or b quarks in the initial state as well as HZ production in gluon fusion through the bottom loops. We include it in this study mainly as an option to cancel the ggZ chiral anomaly induced by modi cations to the ttZ interaction. Various constraints can be placed on the Wilson coe cients of the top quark operators of eqs. (2.1){(2.6) both from direct measurements and from electroweak precision measurements. For = 1 TeV, at 95% con dence level, CtG is constrained from top pair production to be within the range [-0.77,0.4] in ref. [26], and in ref. [16] [-0.56,0.41] at leading order (LO) and [-0.42,0.30] at NLO. CtW is constrained from W helicity fractions in top-decay measurements and single top production, to be in the interval [-0.15,1.9] [27]. The Z ! bb decay constrains the sum of C(3Q) + C(1Q) to be [-0.026, 0.059] [28]. The other three operator coe cients, C(3Q) C(1Q), C t and CtB receive indirect constraints from precision electroweak data, which lead to the following limits [28, 29]: C(3Q) C(1Q) : [ 3:4; 7:5] C t : [ 2:5; 7] CtB : [ 16; 43] : Note that indirect bounds should be interpreted carefully. The presented bounds here are marginalised over the S and T parameters, with all other operator coe cients assumed to vanish. We note here that comparable limits have been set on these operators by the recent collider based global analyses of [11, 12]. Furthermore, RG-induced limits can also be found in [30]. Finally, let us stress that even though we work in the context of the SMEFT, the NLO calculations presented in this work can be directly used in analyses employing an { 4 { anomalous couplings parametrisation, under the condition that CtG = 0 is assumed at all scales. In this case, operators do not mix under RG ow, and they only contribute via anomalous couplings in ttV , bbV and tbW vertices, and our NLO results can be translated into the anomalous coupling approach. The relations between the anomalous couplings and the e ective operator coe cients are given in appendix A. 3 Calculation setup Our computation is performed within the MG5 aMC framework [20], where all the elements entering the NLO computations are available automatically starting from the SMEFT Lagrangian [31{36]. NLO results can be matched to parton shower programs, such as PYTHIA8 [ 37 ] and HERWIG++ [38], through the MC@NLO [39] formalism. Special care needs to be taken for the UV and R2 counterterms, which are required for the virtual corrections. The R2 terms are obtained automatically through the NLOCT package [33], and have been checked against analytical calculations. The UV counterterms depend on the renormalisation scheme. For the SM part, we use M S with ve- avor running of s with the top-quark subtracted at zero momentum transfer. The bottom quark mass is neglected throughout. Masses and wave-functions are renormalised on shell. The operator OtG gives additional contributions to the top-quark and gluon elds, as well as s renormalisation [16]. The operator coe cients are subtracted with the M S scheme. They are renormalised by C0 i ! Zij Cj = 1 + (1 + ")(4 ) 1 2 " 1 "UV ij Cj ; where the anomalous dimension matrix has non-zero components for the dipole operators OtG, OtW , and OtB. The anomalous dimensions for these three operators are [14, 40{42] (3.1) (3.2) = 0 16 0 0 1 2 s BBBB 13 13 0 CC : C C The other operators do not have an anomalous dimension at order O( s) due to current conservation. Results in this work are presented in terms of operators de ned at the renormalisation scale, which we take as mt for pp ! ttV and e+e pp ! HZ. If the operator coe cients are known at the new physics scale ! tt, and mH for , the above anomalous dimension matrix can be used to evolve them down to the renormalisation scale, to resum the large log =mt terms. Hence results presented in this work are free of such large log terms. In general, we nd that NLO results cannot be approximated using the renormalisation group equations of the operators. Operators that modify the ttZ axial coupling may induce a chiral anomaly in the ggZ three point function, which has an e ect in ttZ and gg ! HZ production. The cancellation of the anomaly depends on the details of the underlying model. To cancel this anomaly { 5 { within the EFT framework, one option is to include the operator O b, which modi es the bbZ coupling, and require so that the change in ttZ and bbZ vertices cancel each other in the ggZ function. In this work, we keep this anomaly in the calculation, and take the point of view of [43], i.e. the chiral anomaly in an e ective theory is allowed, provided the corresponding gauge boson is massive. We have checked that, in either case, the numerical e ect is negligible. Note that the SU(3)C gauge is not a ected, and related Ward Identities have been veri ed. As a cross-check of our implementation we have compared our (LO) results with those presented in ref. [19], and have found agreement. In this section, we consider the inclusive ttZ, tt and ttl+l cross sections including the dimension-six operators. The ttl+l cross section includes the contribution of o -shell photons and the interference of ttZ and tt . In fact, this is the process that is experimentally accessible at the LHC, though the di erence between ttl+l and ttZ with leptonic Z decay is small for a lepton pair invariant mass close to the Z boson mass. We work up to O( 2), generating Feynman diagrams with at most one e ective vertex. The cross section can then be expressed in the form: with the sum running over all operators in eqs. (2.1){(2.6). Here i(1) is the cross section of the interference of diagrams with one EFT vertex with diagrams from the SM. The cross section i(j2), corresponds to the interference of two diagrams with one EFT vertex each or the squares of the amplitudes with one e ective vertex for i = j. the O( O( Our implementation allows the extraction of the O( 4) contribution i(j2). While the latter is formally higher-order with respect to the 2) accuracy of our computation in the SMEFT, it is important for several reasons. 2) contribution i (1) as well as First, as this term is of higher-order one can decide to include it without changing the accuracy of the prediction of the central value. Arguments in favour of this approach in the SMEFT have been put forward, see e.g. [44, 45]. Finally, the O( to associate an uncertainty to missing higher-orders in the EFT expansion. For these reasons, we quote results for i(i2) (i.e. the squared contribution from Oi), to either improve the central value predictions or to (partly) assess the size of the theoretical uncertainties 4) terms are useful associated to the contribution of O( 4) and higher terms. In this context, we point out that the relative size of i(i2) with respect to i (1) cannot be used to infer the breaking down of the EFT expansion which even in the case where (2) ii i (1) could still be valid. One reason is that i (1) is an interference term and various cancellations could occur accidentally. We will see this is indeed the case for several { 6 { (3.3) (4.1) operators in ttV production. On the other hand, the EFT expansion in E2= 2 could still be well-behaved, or at least can be controlled by applying kinematic cuts on the total energy E of the process. In this respect, as we were mentioning above, a legitimate and motivated way to proceed is to always include both interference and squared contributions, and separately estimate the theoretical error due to missing dimension-eight operators. Another interesting possibility is in the presence of \strong interactions", i.e. when Ci2 E44 > Ci E22 > 1 > E22 . In this case, the squared contribution dominates over the interference one, without invalidating the E2= 2 expansion, which is parametrically independent of the size of the coe cients. In a phenomenological analysis and in a global t, all such cases should be always kept in mind and carefully analysed on the basis of the resulting bounds on the HJEP05(216) well as i(i2), the contribution of the O( Ci's. As the main goal of this paper is to present a framework to perform calculations in the SMEFT at NLO accuracy and study the results for the neutral top interactions, we do not discuss any further the issue related to the size of the coe cients and the validity conditions of the EFT itself. On the other hand, we stress that our implementation/framework can provide the elements necessary to make a detailed study. For example, we present the full results at O( due to neglecting all i(j2) terms. Note that if necessary, any i(j2) term can be also computed. In practice, to extract the values of i(1), we set one of the Ci coe cients to 1 and all the others to zero. Using the two values and the SM cross-section, we can extract i(1), as 2), characterised by i(1), together with i(i2) as an estimation of uncertainties 2) amplitudes squared. In order to improve the statistical signi cance of the interference for the operators where the interference is small, we nd the value of Ci which maximises it compared to the total cross-section and use that value for the runs instead of Ci = 1. The results are obtained using the 5- avour scheme, with the MSTW2008 [46] parton distribution functions. The input parameters are: mt = 173:3 GeV ; discussion of scale choices for the ttV processes see [47]. Scale variations are obtained by independently setting R and F to =2, and 2 , obtaining nine ( R; F ) combinations. For the ttZ process no cuts are applied on the nal state particles and no Z or top decays are considered, while for tt , pT ( ) > 20 GeV is required. We employ the photon isolation criterium of ref. [48] with a radius of 0.4. Finally, for the tt + process a cut of 10 GeV is set on the minimum invariant mass of the lepton pair. The SM predictions at LO and NLO in QCD1 for the processes considered here are summarised as a reference in table 1, where uncertainties from scale variation, PDF uncertainties, and the K-factors are shown for the LHC at 8 and 13 TeV. The scale uncertainties are signi cantly reduced at NLO. The PDF uncertainties are small compared to the scale uncertainties even at NLO and therefore we will not consider them any further. 1Note that the SM results for the ttZ process have been presented at NLO in the QCD and electroweak coupling expansion in [49]. { 7 { 8TeV 13TeV [fb] SM;LO SM;NLO K-factor SM;LO SM;NLO K-factor ttZ 1.12 1.18 HJEP05(216) and p uncertainties. Inclusive cross section results for ttZ production at the LHC at 8 and 13 TeV for the di erent operators are shown in tables 2 and 3. We include the LO and NLO results for i(1) and i(i2), the corresponding K-factors, the ratio of the dimension-six contribution over the SM and the ratio of the squared O( 4) contributions over the O( 2) one. Statistical uncertainties are not shown unless they are comparable to the scale uncertainties. The scale uncertainties are signi cantly reduced at NLO similarly to the SM predictions. We note that the ratios over the SM are signi cantly less sensitive to scale variations compared to the cross-section numbers. (and tan4 w for the squared contributions). In the tables, we include the O(3Q) operator but not O Q (1) . Results for these two operators di er by a sign at O( 2) and are identical at O( 4).2 Similarly at O( 4) the contributions of O(3Q) and O t are identical. This can be traced back to the way the operators modify the ttZ vertex as shown in eq. (A.1). Similarly we do not include the results for OtB, as they can be obtained from those of OtW by multiplying by a factor of tan2 w with The largest contribution is given by the chromomagnetic operator both at 8 and 13 TeV, (1) reaching almost 40% of the SM. We nd that while O(3Q) and O t give contrii butions of 6-10% of the SM for Ci = 1, OtW and consequently OtB give extremely small contributions reaching at most the per mille level. While the NLO predictions have significantly reduced theoretical uncertainties, we nd that the various ratios of cross-sections considered are generally stable with respect to QCD corrections (apart from OtW ), and also su er from much smaller scale uncertainties compared to the cross-sections. This fact can be exploited to extract information on the Wilson coe cients. The theoretical errors due to neglecting squared operator contributions ii (2) are characterised by the last two rows in the table. The results indicate that for coe cients of order one, neglecting squared 2This is only approximately true at the cross-section level. There is a small contribution from the bbZ vertex which spoils the minus sign relation between the two operators. The bbZ vertex contributes as we are working in the 5- avour scheme. Nevertheless this contribution is in practice numerically negligible and therefore the two operators give opposite contributions at O( 2). { 8 { 8TeV (1) 76:1+41:9% 27:1% 78:1+4:1% 10:0% shown in brackets if these are comparable in size to the scale uncertainties. shown in brackets if these are comparable in size to the scale uncertainties. contributions is safe for all operators except for OtB and OtW . When placing limits, this assessment should be done for the interval of where the limits are placed. We note here the extremely small contribution of the OtW operator, which also leads ence independently of the other two contributions. In this case, the impact of the EFT amplitude squared is much larger than its interference with the SM. The small size of the { 9 { OtW 0:077(8)+4463::62%% 0:32(2)+3697::13%% -4.2 4:14+50:1% 30:7% 4:81+6:2% 12:5% 0:00037(4)+3432::65%% 0:0014(1)+3516::48%% 54(6)+8249::71%% 15(1)+3463::95%% OtW 0:20(3)+82830:0:0%% 1:7(2)+3419::31%% 171:5+3285::66%% 218:9+1133::36%% 564:6+3254::41%% 765+1143::04%% errors are shown in brackets if these are comparable in size to the scale uncertainties. A pT ( ) > 20 GeV cut is imposed. interference is a result of various e ects. The most important reason is that the dipole interaction, q , involves the momentum of the Z boson, and leads to a suppression because the Z tends to be soft in ttZ production at the LHC. The same is true also for the tt production, as we will see. By crossing and g, we have explicitly checked that in the process g ! ttg this suppression e ect becomes an enhancement, as a large momentum for is guaranteed in the initial state. Apart from this, an additional suppression occurs due to an accidental cancellation between the contributions of the gg and qq channels, as they are similar in size but come with an opposite sign. This cancellation leads to a nal result that is an order of magnitude smaller than the individual contributions. Finally, an additional reason could be that the OtW vertex does not produce the Z boson in its longitudinal state, which is expected to dominate if it has large momentum. Finally, comparing 8 and 13 TeV we notice a small increase in the K-factors. The a signi cant increase of the ratio O( 4) over O( 2) as the O( rapidly with energy, as will be evident also in the di erential distributions. ratios of the O( 2) terms over the SM do not change signi cantly. For OtG we notice 4) contribution grows The corresponding tt results are shown in table 4. In this case, a minimum cut of 20 GeV is set on the transverse momentum of the photon. We note that here only three operators contribute: OtG, OtW and OtB. For this process, OtW and OtB are indistinguishable and therefore only OtB is included in the table. The K-factors in this process are larger than those of ttZ, reaching 1.3 for the SM and OtG operator but lower for OtB. This is due to the soft and collinear con gurations between the photon and the additional jet at NLO, which however cannot happen if the photon is emitted from an OtB vertex. γ g ¯ t with a blob, has generally a higher momentum transfer here than in tt production. ! ttg. The tt vertex, marked Similar conclusions to the ttZ can be drawn for tt regarding the operator contributions. The chromomagnetic operator contributes the most. Neglecting squared contributions is safe for Ci . 1, at both 8 and 13 TeV, but starts to become questionable (and therefore the corresponding uncertainty is increased) as the coe cients reach order of a few, with the relative contribution of i(i2) increasing from 8 to 13 TeV. The contribution of the OtW and OtB operators are 1% of the SM and signi cantly smaller than the OtG one. While the OtW and OtG operators lead to the same structure in the tt and ttg vertices respectively, similar to ttZ production, the e ect of OtW on the gg ! tt amplitude at typical LHC energies is suppressed compared with that of OtG. By examining the crossed amplitude, g ! ttg, illustrated in gure 2, we see that the two operators give contributions of the same order, as they both enter in the production side of the process and more momentum passes through the EFT vertices. We also note here that the K-factors for the operators are not the same as those as for the SM contribution which implies that combining the SM K-factor and LO EFT predictions does not provide an accurate prediction for the EFT contribution at NLO in QCD. We next examine ttl+l . For an invariant mass of the lepton pair around the Z mass, this process is dominated by ttZ with leptonically decaying Z, the mode that the ATLAS and CMS experiments at the LHC are most sensitive to. Generally it also includes the contribution of tt . As the EFT operators we study do not enter the vertices connected to leptons, we restrict our attention to tt + . 3 We collect the results for tt + at LO and NLO at 8 and 13 TeV in tables 5 and 6. In this case, the photon and Z amplitudes and their interference is included. For the tt + results, the scale and PDF choices are identical to those for the inclusive ttZ=tt processes. A lower cut of 10 GeV is imposed on the invariant mass of the lepton pair. No other cuts are imposed on the leptons. All six operators contribute to this process. Results for O(1Q) di er from those of O Q (3) by a sign at O( 2) and are identical at O( 4), therefore we show only one of the two. The cross-section is dominated by the region close to the Z mass peak and therefore the K-factors and relative contributions of the operators are similar to those of the ttZ process. The chromomagnetic operator contributes at the 35% level, while the other three current operators give a contribution at the 4-7% level. 3We note here that a contribution from 4-fermion operators describing the tt + interaction enter in study of these operators to future work. (3) O Q 0:613+4258::26%% 0:683+51:14:%3% dimension-six operators. An m(``) > 10 GeV cut is applied to the lepton pair. Percentages correspond to scale uncertainties. Integration errors are shown in brackets if these are comparable in size to the scale uncertainties. OtB 0:0101+4237::26%% 0:012(1)+81:29:%2% size to the scale uncertainties. nant compared to the O( 4 The contributions of OtW and OtB at O( of extracting the interference contribution we are always very limited statistically. Even maximising the interference contribution by choosing the appropriate value of the coe cient is not enough to give us good statistics, in particular at NLO which is evident in the quoted statistical and scale uncertainties. ) contributions. E ectively this means that with our method 2 ) are at the per mille level and subdomiNLO, μ=mt, CtG=1, Λ=1 TeV σSM+Cσ(1) σSM+Cσ(1)+C2σ(2) n i b / 1 ( σSM 1.8 1.4 1.8 n i b / T 0.001 1M 2 G G d d a a M M _ _ 5 5 h h G C C + + 1 1 for the chromomagnetic operator for CtG = 1 and = 1 TeV. Scale uncertainty bands are shown. 4.2 Di erential distributions tt + Di erential distributions are obtained at NLO for the pp ! ttZ, pp ! tt and pp ! processes. This can be done also at NLO with matching to the PS, and with top quarks decayed while keeping spin correlations [50], all implemented in the MG5 aMC framework. Hence our approach can be used directly in realistic experimental simulation, with NLO+PS event generation, which allows for more detailed studies of possible EFT signals. In this work, for illustration purpose, we keep results simple by only presenting xed order NLO distributions. No kinematical cuts are applied except for the m(``) > 10 GeV and pT ( ) > 20 GeV generation cuts. We show results obtained with one non-zero operator coe cient at a time, with = 1 TeV, and SM results are given for comparison. We start by showing the distributions obtained for the OtG operator at 8 and 13 TeV in gure 3. We show as a reference the invariant mass distribution of the top quark pair and the transverse momentum of the Z. In the plots we show the SM prediction SM, n i n i b / ) / 1 ( NLO, μ=mt, Cφt=2, Λ=1 TeV σSM+Cσ(1) σSM+Cσ(1)+C2σ(2) NLO, μ=mt, Cφt=2, Λ=1 TeV σSM+Cσ(1) σSM+Cσ(1)+C2σ(2) n i d / σ d n i b / b d / T dσ0.01 C C M M a a _ _ 5 5 h h p p a a r r G S S S 13 TeV for the O t operator for C t = 2 and = 1 TeV. Scale uncertainty bands are shown. the result for CtG = 1; = 1 TeV i) adding only the interference i the interference and the squared terms i(i2). We also include the corresponding ratios over the SM prediction and the scale uncertainty bands. It is clear that while the interference (1) and ii) adding both contribution is not changing the distribution shape, the O( fast at high energies with the e ect being more evident at 13 TeV in both distributions shown here. Similar observations can be made for other observables, such as the transverse 4) contribution is growing momentum of the top. Results for the O t and O(1Q) are shown in gure 4 and 5 respectively. In this case, we set the Wilson coe cients to 2, in order to obtain visible deviations from the SM. These values are allowed by the current constraints. For these operators the O( is signi cantly smaller than the O( 2) and does not signi cantly alter the shape of the 4) contribution di erential distributions as seen in the at ratios for both the tt invariant mass and Z pT distributions. Results for O(3Q) are identical to those of O(1Q) (with a relative sign for i(1)), so we do not include them for brevity. n i 0.001 n i b / dσ0.01 σSM+Cσ(1) σSM+Cσ(1)+C2σ(2) σSM+Cσ(1) σSM+Cσ(1)+C2σ(2) n i d / σ d n i b / b /dσ0.01 d h p p a a r r G G d d a a M M O N N C C M M a a _ _ 5 5 h h p p a a r r G d a a M M M M S S HJEP05(216) M S S 13 TeV for the O(1Q) operator for C(1Q) = 2 and = 1 TeV. Scale uncertainty bands are shown. For the OtW and OtB operators the EFT contributions are very small compared to the SM. In this case, we resort to CtB = 4 to demonstrate the e ect of the OtB operator in gure 6. For this operator the interference with the SM is much smaller than the O( 4 ) terms which are rising with the energy, as evident in the ratio plots. For tt , the results for OtG operators are shown in gure 7 for 8 and 13 TeV. We notice that, in contrast with ttZ, where the squared terms grow rapidly with the energy, that contribution is smaller for tt and does not lead to signi cant changes in the shapes of the two observables shown here. A comparison of the two processes can be made at the partonic cross-section level as shown in gure 8. In this plot the total cross-section is shown, i.e., schematically SM + C (1) + C2 (2) for the chromomagnetic operator. The ttZ cross-section is decomposed into the transverse and longitudinal Z contributions. The only component that is rising with the energy is the longitudinal one, which explains why the photon distributions do not show any increase with the energy, while those for the Z rise fast. In fact in ttZ, the Higgs eld in OtG always takes its vacuum expectation value, and so by power counting the squared amplitude scales at most as sv2= 4 for ttZT and n i ) / 1 n i b / ) / 1 σSM 1.2 1 1 σSM+Cσ(1) σSM+Cσ(1)+C2σ(2) NLO, μ=mt, CtB=4, Λ=1 TeV σSM+Cσ(1) σSM+Cσ(1)+C2σ(2) n i d / σ d n i b / b d / T dσ0.01 p a a r r G G d d a a M C C M M a a _ _ 5 5 h h p p a a r r G d a a M S S S 13 TeV for the OtB operator for CtB = 4 and = 1 TeV. Scale uncertainty bands are shown. tt , which is not enough for the cross section to rise at high energy. On the other hand, in ttZL the longitudinal polarisation vector contributes an additional factor of (E=mZ )2, leading to a nal s2= 4 scaling of the squared amplitude. According to the Goldstoneboson equivalence theorem, the process pp ! ttG0, where G0 is the neutral Goldstone boson, has the same energy dependence. This dependence comes solely from the diagram with a ve-point contact interaction, ggttG0, from OtG, and because here the Higgs eld is dynamical by simple power counting the square amplitude indeed scales as s2= 4. To rises as verify this reasoning, we have checked that in the process gt ! tZL, the squared amplitude s2= 4, and the leading term in the energy expansion can be fully reproduced by a single diagram with a contact gttG0 interaction. An analogous example of a high-energy growth is discussed in [51] where tW ! tW scattering in ttW j electroweak production is employed to provide information on the top-Z couplings. The corresponding distributions for OtB are shown in gure 9 for 8 and 13 TeV. As setting CtB = 1 does not give any visible deviations from the SM, we resort to CtB = 4 for these plots. While the squared term does not rise with m(tt), it increases fast with m(t-t) [GeV] n i b / p [ /bb 0.1 0.01 0.1 NLO, μ=mt, CtG=1, Λ=1 TeV σSM σSM+Cσ(1) σSM+Cσ(1) σSM+Cσ(1)+C2σ(2) O L N N 5 a a r r G 5 a M M 1 1 n i b / b p d / σ 1 13 TeV for the chromomagnetic operator for CtG = 1 and = 1 TeV. Scale uncertainty bands are shown. the photon transverse momentum. This is again related to the amount of momentum passing through the EFT vertex. High top pair invariant mass does not correspond to high momentum through the EFT vertex for the OtB operator, in contrast with the situation for OtG. For OtG there is a strong correlation between the m(tt) and the energy in the EFT vertex leading to a rising distribution. For the tt + process, we examine the angular separation between the leptons and the invariant mass distribution of the two leptons m(``) for the OtG operator in gure 10 for 8 and 13 TeV. The angular separation between the two leptons is highly correlated with the transverse momentum of the vector boson. This implies that at low , the behaviour matches that of the high vector transverse momentum region, since for a boosted vector boson, the leptons are collimated. As expected, the behaviour close to the Z mass peak resembles that of the ttZ process, while at low invariant mass of the lepton pair it approaches that of tt . [p0.7 g σg0.6 _ 5 h p a r G d a M HJEP05(216) 1000 1500 2000 2500 3000 s [GeV] chromomagnetic operator. The ttZ cross section is decomposed to transverse and longitudinal Z contributions. current operators, as they are similar to O Q The corresponding results for O(1Q) are shown in gure 11. Again the behaviour of the ratios follows that of the ttZ close to the Z mass peak, while at low masses the dimensionsix contribution approaches zero as this operator has no e ect on the tt process which dominates at low m(``). The distributions are rather at similarly to those of the pT (Z) for the same operator. For brevity we do not show the results for the rest of the We conclude our tt + discussion by showing the results for the OtB operator operator in gure 12. The size of the interference with the SM increases at high lepton pair invariant masses while it is constant as a function of the angular separation between the leptons. The squared terms rise at high invariant mass and low angular separation in agreement with the observations made for the tt and ttZ distributions. We conclude this section by commenting on the di erential K-factors of the EFT contributions. As already mentioned in section 4.1, by comparing table 1 with tables 2{6, one can see that the SM global K-factors are in general di erent from the K-factors of the EFT operators. This shows that already at the cross-section level, using the SM K-factor to estimate the NLO QCD corrections of the EFT contribution is not a reliable approximation. The same applies at the di erential level. To demonstrate this observation, we present in gure 13 a comparison of the di erential K-factors for the SM and EFT contributions. We focus on the tt and ttZ processes at 13 TeV and show four representative observables. In /dσ0.01 ) / 1 ( d / 0.01 0.1 NLO, μ=mt, CtB=4, Λ=1 TeV σSM σSM+Cσ(1) σSM+Cσ(1) σSM+Cσ(1)+C2σ(2) σ σ p [ T σ d C + 1M 1.6 /σS 1.2 σ d / σ ) / 1 M L N N _ 5 a a r r G M _ 5 5 h h p p a a r r G d a a M M 13 TeV for the OtB operator for CtB = 4 and = 1 TeV. Scale uncertainty bands are shown. the main panels we show the NLO and LO results of the SM and SM + C (1). In the rst two insets we present the di erential K-factors of i) the SM (K( SM )) and ii) the operator contribution (K( (1))), while in the third inset we show the ratio (R) between the two. The comparison between table 1 and table 4 shows that the OtB global K-factor is lower than the SM one for the tt process. On top of that, in the top left plot of gure 13 we see that at di erential level the ratio of K-factors is not at. For the same process at the cross-section level the OtG contribution and the SM have similar K-factors. However, the top right plot reveals that the ratio R is again not at. Therefore even a bin-by-bin rescaling of the LO OtG distribution with the SM di erential K-factor would lead to the mismatch depicted in the third inset. In the two lower plots we show results for the ttZ process for the OtG operator where similar observations can be made, highlighting the need for NLO QCD predictions for the EFT contributions. 0.1 t­tμ+μ−, LHC8 C L N N C M M a a _ _ 5 5 h h p a r r G G d d a a M L N N C C M M a a _ _ 5 5 h h p p a a r r G G d d a a M d / ) / 1 ( σ 1 ( σ 1.4 1 C bution at 8 and 13 TeV for the chromomagnetic operator for the OtG operator for CtG = 1 and = 1 TeV. 5 Results for gg ! H Z A subset of the operators a ecting ttZ=tt enter also in the associated production of a HZ pair in gluon fusion, shown in the Feynman diagrams of gure 14. This process is formally part of the NNLO cross section for HZ production and contributes at the 10% level. It is nevertheless particularly important in the high Higgs pT regions where the experimental searches are most sensitive. This process has been studied within the SM, also including the contribution of additional jet radiation, which turns out to be important in the high pT regions [52]. In this work, we consider this process as it can provide additional information on the Wilson coe cients once combined with the corresponding HZ measurements at the LHC. In this section, we investigate the e ect of the operators presented above on this process. We note that we consider only the operators involving the top quark and ignore all other dimension-six operators, such as those a ecting the interaction of the Higgs with n i n i n i d / b / 1 ( σSM σSM+Cσ(1) σSM σSM+Cσ(1) L N N a _ _ 5 5 h p a a r r G G d d a M L N N a _ _ 5 5 h h p p a a r r G G d d a a M S S 0.1 tN­tμLO+μ,μ−=,mLtH,CC(φ1Q1)=32, Λ=1 TeV σSM σSM+Cσ(1) σSM+Cσ(1)+C2σ(2) 0.0028 t­tμ+μ−, LHC13 NLO, μ=mt, C(φ1Q)=2, Λ=1 TeV σSM σSM+Cσ(1) σSM+Cσ(1)+C2σ(2) S S 0 50 100 150 bution at 8 and 13 TeV for the O(1Q) operator for C(1Q) = 2 and the vector bosons. In addition to modifying the interactions in the SM-like diagrams of gure 14, the dimension-six operators introduce additional vertices and hence Feynman diagrams as shown in gure 15. For this process, the factorisation and renormalisation scale is set to mH = 125 GeV. Only LO results can be obtained as the NLO computation requires 2-loop multi-scale Feynman integrals which are currently not available. The results are shown in table 7 for both the SM and the dimension-six operators cross sections, the corresponding scale uncertainties and the corresponding cross-section ratios for 8 and 13 TeV. The OtW and O Q OtB operators do not contribute to this process, due to charge conjugation invariance. The (3) , O(1Q) and O t give the same contributions (with a relative minus sign as determined by eq. (A.3)) in the massless b-quark limit, as they a ect in the same way the axial vector coupling of the top to the Z, which is the only component whose contribution is allowed by charge conjugation symmetry. If one wants to cancel the chiral anomaly in the triangle n i d / b / σSM σSM+Cσ(1) 0.0006 NLO, μ=mt, CtB=4, Λ=1 TeV σSM σSM+Cσ(1) 0.1 tN­tμLO+μ,μ−=,mLtH,CCtB1=34, Λ=1 TeV σSM σSM+Cσ(1) σSM+Cσ(1)+C2σ(2) 0.0028 t­tμ+μ−, LHC13 NLO, μ=mt, CtB=4, Λ=1 TeV σSM σSM+Cσ(1) σSM+Cσ(1)+C2σ(2) n i n i L N N a _ _ 5 5 h p a a r r G G d d a M HJEP05(216) O O L L N N a _ _ 5 5 h h p p a a r r G G d d a a M bution at 8 and 13 TeV for the OtB operator for CtB = 4 and its Wilson coe cient set to C b = 2C(1Q) loop diagrams with the Z-boson in the s-channel, the O b operator can be included with C t. By appropriately xing the coe cient of O b, the axial-vector coupling of the bottom remains opposite to that of the top and the anomaly cancels. In practice this has a negligible numerical e ect on the results. The chromomagnetic operator gives a signi cant contribution reaching 35% of the SM cross section for CtG = 1 and = 1 TeV. The three current operators give contributions at the 6% level. In both cases, the contribution of the squared amplitudes are subdominant at the total cross section level. These results su er from large scale uncertainties as it is often the case with gluon fusion processes at LO. The invariant mass distribution for the HZ pair is shown in gure 16 for the SM and the dimension-six operators. For this process, we nd that both the interference with the SM amplitude and the squared contribution are growing with energy. μ=mt, CtB=4, Λ=1 TeV Κ(σ(1))/Κ(σSM) σSM,LO σSM,NLO+Cσ(N1L)O σSM,LO+Cσ(L1O) O O L L N N 5 h p a r r G G d d a M a _ _ 5 5 h h p p a a r r d a a M m(t-t) [GeV] 700 t-tZ, LHC13 μ=mt, CtG=1, Λ=1 TeV at 13 TeV for CtB = 4, CtG = 1 and = 1 TeV. Comparison between the SM and the interference term di erential K-factors. Scale uncertainty bands are shown. n i 0.01 )M 1.5 σ ( n i 0.01 )M 1.5 σ ( S t S S n i b / b [ / d m0.1 d )M 1.5 σ ( Κ 0.5 / σ d g presence of dimension-six operators. The new vertices originating from the dimension-six operators are denoted with a blob. [fb] SM 8TeV 29:15+4206::06%% 13TeV 93:6+3243::38%% (1) i (2) ii (1) i (2) ii i(1)= SM i(i2)= i(1) i(1)= SM i(i2)= i(1) OtG 10:37+4217::32%% 1:621+4258::17%% 0:356+00::98%% 0:156+22::60%% 34:6+3254::25%% 6:09+3296::21%% 0:370+00::79%% 0:176+22::91%% (1) O Q 1:719+4227::56%% 0:0469+4269::52%% 0:0590+11::84%% 0:0273+22::83%% 5:91+3264::49%% 0:182+4206::26%% 0:0631+11::65%% 0:0309+22::82%% 13 TeV for the SM and the dimension-six operators. Scale uncertainties are shown in percentages. 6 Results for the ILC The top-quark electroweak couplings can be accurately determined by future e+e colliders, using top-pair production, thanks to the clean background. Our approach can be applied to e+e p colliders as well, providing more accurate predictions for deviations that will be measured in this process. In this section we present results obtained for the ILC at s = 500 GeV for top pair production. For this process, the OtG operator contributes only at NLO, while the other operators contribute starting at LO. The results are presented in be computed only at NLO and are at the 1-2% level. Unlike the ttV processes, here we nd signi cant contributions from the dipole operators OtB and OtW , while the other operators are suppressed, with OtG, O(1Q) and O(3Q) at the percent level, and O t at the per mille level. This is mainly because the momenta of Z and are at least at the tt threshold, and so the same dipole structure, which suppresses ttV production at the LHC, enhances the tt production at the ILC. It follows that the ILC could provide useful information complementary to the LHC as discussed also in [18, 19]. We note here that the analysis of [18, 19] does not include the contribution from OtG, although (following an anomalous coupling approach) it does include the contribution of HJEP05(216) 0.1 1 5 /σS 3 M σ 1 gg HZ, LHC8 NLO, μ=mH, CtG=1, Λ=1 TeV σSM 5 5 h h p p a a r r G 5 5 h h p p a a r r G 0.1 operators. Scale uncertainty bands are shown. the squares of the amplitudes with the top anomalous couplings and therefore also the CP-odd contributions. 7 Theoretical uncertainties In this section we brie y discuss various theoretical uncertainties relevant to our results. In the SMEFT calculation there are two main types of theoretical uncertainties, those related to missing higher orders in the strong coupling and those from higher terms in the 1= expansion. In the former class, we can list Uncertainties due to parton-distribution functions. This type of uncertainty is also present in the SM calculations and can be treated in the same way, i.e. by following the procedures associated with the corresponding 500GeV (1) i;LO (1) i;NLO (2) i;LO (2) i;NLO 1.14 OtG 0 -6.22 N/A (3) 15.3 -18.0 -1.3 -1.0 scale uncertainties are not shown. They are only present at NLO and remain at the 1% level. PDF sets, as long as the scale of new physics is high enough and the EFT operators do not modify the DGLAP equations. Uncertainties due to missing higher orders in the s expansion as in the SM. This kind of uncertainty is typically estimated by varying the renormalisation and factorisation scales as done in SM calculations. All results presented in this work are provided along with uncertainties that are estimated by varying these two scales independently. Uncertainties due to missing higher orders in the s expansion of the EFT operators. In the SMEFT an additional uncertainty, related to the scale at which the operators are de ned, should be considered as well. It characterises the uncancelled logarithmic terms in the renormalisation group running and mixing of the operators. We did not evaluate these uncertainties explicitly even though it is possible in our framework. For the operators we have studied in this work, they are expected to be negligible compared to the rst two scale uncertainties [17]. This is because the anomalous dimensions of the relevant operators happen to be smaller by roughly an order of magnitude compared to the beta function of s (see ref. [17] for a discussion of the operator scale uncertainty in the single-top processes). We now consider uncertainties due to missing O( Up to this order, the cross section (or any other observable) can be written as: 4) contributions, also discussed in [53]. = SM + X Cdim6 i i Cdim8 i ( =1TeV)4 i (1;dim8) (7.1) The last two terms are formally O( 4) contributions, and could in principle be neglected as they are expected to be suppressed for O(1) coe cients. One should then consider Impact of the squared contributions i(j2;dim6) coming from dimension-six operators. These contributions can be explicitly calculated with our approach, even though obtaining the complete results can be time consuming. In this work, we have always provided the results for i(i2) for each operator Oi, for not only total cross sections but also for distributions. In fact, one could include these squared contributions in the central values as part of the theoretical predictions, if only one operator is taken to be non-zero at a time. As we have mentioned, this can be justi ed for cases where the expansion in E2= 2 is under control but the squared contribution may still be large, due to less constrained operator coe cients, i.e. if Ci2 E44 > Ci E22 > 1 > E22 is satis ed. In any case, our results for the i(i2) terms can provide useful information for the evaluation of the uncertainties, if the squared contributions are neglected or only partly included. case-by-case basis. As we have discussed already, the relative size of i(i2) compared to i(1) does not imply anything about the validity of the EFT and careful assessment should be done on a Validity of the EFT, i.e. contributions from missing higher-dimensional operators. The second contribution at O( 4), (1;dim8), comes from interference between SM and dimension-eight operators. These contributions cannot be computed in our approach, and will have to be neglected. A corresponding uncertainty should be taken into account. This can be easily done at the LO by calculating the interference contribution from typical dimension-eight operators. Alternatively, by simple power counting, these uncertainties may be estimated to be of order Cdim6=( =1TeV)2 i(1;dim6)s= 2. In this work, we do not assume a speci c value of , i and so evaluating such uncertainties is not possible without additional assumptions. However, in a real analysis, for any given , one can always apply a cut smax on the centre-of-mass energy of the process, so that this uncertainty remains under control. 8 Discussion In this section we explore the sensitivity of the top processes discussed above to the various operators. Experimental results from [2, 5, 6, 54] are used. For the ttZ measurement by ATLAS [6] and the tt measurement by CMS [2], a direct comparison is di cult, because of the way in which the measured cross sections are de ned. We thus de ne the \R" ratios in order to facilitate a direct comparison between the quoted experimental measurements and our theory predictions, as explained in appendix B. These ratios are always taken into account when experimental results on tt + and tt are used. On the other hand, the other measurements can be directly compared with our predictions. We rst examine the OtG operator, which a ects all production of top quark pairs with a vector boson, as well as tt production. The sensitivity of various processes to the HJEP05(216) Δ -50 OtG sensitivity EFT@NLO ΔSM ΔExp Δ OtG(CtG=1) HJEP05(216) ttW ttZ denotes the percentage di erence from the SM theoretical prediction for each process. Theory predictions for all ttV processes are at NLO in QCD while for tt the NNLO result of [55] is employed. Experimental measurements are also shown along with the corresponding experimental uncertainties taken from [54] for tt, [5] for ttW and ttZ, [6] for tt + and [2] for tt . OtG operator is demonstrated in gure 17. In the plot we include the percentage deviation from the SM predictions for top pair production, and top pair production in association with a W; Z boson or a photon, as well as the tt`+` process for CtG = 1 and All SM predictions and uncertainties are given at NLO, apart from the top pair production cross-section, which is given at NNLO+NNLL [55]. We also present the experimental measurements and the corresponding uncertainties (systematic and statistical uncertainties added in quadrature). Only the O(1= 2) contribution is included. The OtG operator a ects all processes considered here in a similar way, at the 30% level for = 1 TeV and CtG = 1. At present, the most stringent direct constraints on this operator are obtained from the top pair production measurement, which is by far the most accurate one. The relative sensitivity of the top processes to all operators can be summarised in gure 18, where the results for C = 1 are shown as a ratio over the SM NLO cross sections, for the 6 operators considered here both at LO and NLO, along with the corresponding K-factors in the lower panel. The reduction of the theoretical uncertainties at NLO is also evident in the plot. The corresponding sensitivity plot for 13 TeV is shown in gure 19, in which similar observations can be made. Using the experimental measurements, one can further explore the sensitivity of the tt and ttZ processes on the various operators as shown in gures 20, 21 and 22. In the contour OtG NLO at 8 TeV. K-factors are also shown for i the K-factors for the OtB and OtW operators in the ttZ and tt + accidental cancellations lead to large or even negative K-factors. (1) as well as the scale uncertainties. We do not show Oi sensitivity LHC 13 TeV Oi sensitivity 10-2 10-3 ()1 i,LNO )(1 i,LO1.2 σ σ 1 M S σ ) / (1 i σ10-1 10-2 10-3 ()1 i,LNO )(1 i,LO1.2 σ σ 1 O(φ3Q) φQ Oφt 10×|OtW| 10×OtB OtG NLO at 13 TeV. Details as in gure 18. σi(,1N)LO ttW ttZ ± ttμ+μ HJEP05(216) processes, as in this case σi(,1N)LO ttW ttZ ± ttμ+μttγ 1/Λ2,C(φ1Q),1/Λ4 -3.75 -3.01 -2.28 -1.54 -0.81 -0.07 -7.57 Z t (σ300 200 100 O(φ1Q), OtB SM Exp 500 1000 1500 the coe cient we show the cross-section including i) only the interference term ( lled triangles) and ii) both the interference and the squared contribution (un lled triangles). The range for the Wilson coe cients is determined by the current constraints as discussed in section 2. The experimental measurements used in this plot are taken from [2] and [5] for tt and ttZ respectively. The squared contribution of the OtB operator is very large, and therefore we employ a separate smaller interval to obtain cross sections within the boundaries of this plot. plots we include the experimental results of [2] for tt and [5] for ttZ and the corresponding one and two sigma contour plots. In this case, we assume there is no correlation between the two measurements. The SM NLO predictions and the corresponding scale uncertainties are also shown in the plots. We plot the cross section obtained by varying the Wilson coe cients of the various operators. For clarity and to avoid overcrowding the contour plots, we present the operators in pairs. For the coe cients, we employ the current constraints to de ne our interval. Vertical lines in the plots indicate that the tt process is not a ected by the speci c operator, i.e. O t; O(3Q) and O Q (1) . Cross sections with and without adding the O(1= 4) contributions from the squared EFT amplitudes are compared. The OtB operator is very loosely constrained, and therefore including the squared term for the large allowed values of the Wilson coe cient has an enormous e ect on the cross sections, as (1) the O(1= 4) contribution scales like Ct2B. For the more constrained current operators O Q and O(3Q) , the squared contribution becomes important only at the edges of the allowed intervals. We also notice that for the O t and OtG operators the O(1= 4) contribution is important for a sizeable part of allowed interval, in the rst case because the constraints are rather loose and in the second case because contour plots qualitatively demonstrate the size of the experimental uncertainties needed for these processes to have an impact on the allowed values of the coe cients. In that respect we observe for example that the OtW operator receives very stringent constraints t(G2) is large. Finally, we note that the Z t ( σ300 200 100 f t ( σ300 200 100 O φQ , O SM Exp tG 500 1000 LHC8, Λ=1 TeV O , O φt tW SM Exp 1/Λ2,C(3) φQ 1/Λ2,C ,1/Λ4 -1.70 -0.92 -0.14 -0.60 -0.44 -0.27 -0.10 1/Λ2,C ,1/Λ4 tW gure 20. from top decay, and it is not expected to be further constrained by ttV measurements even with a signi cant reduction of the experimental uncertainties. 9 Summary and conclusions We have presented the NLO QCD predictions in the SMEFT framework for the associated production of a top-quark pair and a neutral gauge boson at the LHC. In addition, we have considered top-pair production in e+e colliders and the top loop-induced process gg ! HZ at the LHC. These processes are important because they directly probe the neutral gauge-boson couplings to the top quark, which are not well probed by other means. In our approach we have included the full set of dimension-six operators that parameterise these couplings. We have studied the contribution of each relevant dimension-six operator, in both total cross sections and di erential distributions. We have presented full results for O( contributions, along with the squared contribution of each operator at O( 4). The latter contribution can be used to estimate uncertainties coming from higher order O( contributions. Scale uncertainties are provided in all cases, and their reduction at NLO 2 ) ) re ects the increased precision of our predictions. In tt and ttZ, we nd that the operator that contributes the most, given our choice of operator normalisation, is the chromomagnetic one. This observation is particularly important in the context of a global EFT t, because it means that, when extracting information on operators modifying the top couplings with the weak gauge bosons, uncertainties due to a possible non-vanishing chromomagnetic operator should be carefully accounted for. We also nd that, the weak dipole operators give extremely suppressed contributions at O(1= 2), due to a momentum suppression from the operator structure, and in ttZ an additional accidental cancellation between gg and qq initial states. A subset of the operators a ects the associated production of the Higgs with a Z in gluon fusion, and we have considered their e ects on this process at the LHC. This might provide additional constraints on the operators once ZH production is measured accurately at the LHC. Again, we nd that the contribution of OtG is large, while all the current operators give the same contribution as they a ect the axial vector of the Z in the same way. The weak dipole operators do not contribute due to charge conjugation parity. We have also found that at the ILC, tt production is sensitive to weak dipole operators, and could provide information complementary to the LHC. We have studied the sensitivity of the processes to the various operators in light of the current experimental measurements, as well as the constraints currently placed on the operators from other top measurements and electroweak precision observables. A discussion of the relevant uncertainties coming from missing higher orders in QCD and in the EFT has also been presented. The NLO results provide a solid basis for current and future measurements to be analysed in an EFT approach. In summary, at NLO in QCD accuracy, deviations from the SM in the top sector can be extracted with improved accuracy and precision, keeping EFT uncertainties under control. As our calculation is based on the MG5 aMC framework, matching with the HJEP05(216) parton shower and top decays with spin correlations can be achieved in an automatic way. Therefore, the corresponding simulations can be directly used in experimental analyses in the future to provide reliable information on possible EFT signals. Furthermore, dedicated investigations of the features of deviations from the SM in these processes can be performed based on our results, with an expected improvement in sensitivity. Acknowledgments We acknowledge illuminating discussions with Christophe Grojean, Alex Pomarol, Francesco Riva on the SMEFT and its range of validity. We would like to thank Raoul Rontsch and Markus Schulze for discussions and helpful checks. C.Z. would like to thank Valentin Hirschi and Hua-Sheng Shao for valuable discussions about gauge anomaly. This work has been performed in the framework of the ERC Grant No. 291377 \LHCTheory" and has been supported in part by the European Union as part of the FP7 Marie Curie Initial Training Network MCnetITN (PITN-GA-2012-315877). C.Z. is supported by the United States Department of Energy under Grant Contracts DE-SC0012704. A Connection with \anomalous coupling" approach In order to compare with other work in the literature, we present here the connection of the Wilson coe cients with the top quark anomalous couplings. The anomalous coupling approach is followed in [18, 19] where the ttZ process is used to probe anomalous top couplings. Compared with the anomalous coupling parametrisation of the ttZ vertex, LttZ = eu(pt) C1Z;V + 5C1Z;A + C2Z;V + i 5C2Z;A v(pt)Z the relation between anomalous couplings and Wilson coe cients are: i q mZ C'(3Q) C'(1Q) C't C'(3Q) + C'(1Q) C't C1Z;V = C1Z;A = 2 2 C2Z;V = CtW c2W CtBs2W mt2 2sW cW mt2 2sW cW 2mtmZ 2sW cW Similar relations for the top photon interactions are: Ltt = eu(pt) Qt C2;V = (CtW + CtB) C2;A = 0 i 2mtmZ q mZ 2 C2;V + i 5C2;A v(pt)A The CP-odd operators are zero simply because we have assumed CtW and CtB are real. (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) Ratios for comparing with measurements of tt total prediction, here called processes as follows: The SM prediction employed by ATLAS for the ttZ process [6] also contains contribution in the channel where the vector boson decays into two charged leptons (ttl+l ). The AT LAS (ttZ), can be written in terms of the ttZ and ttl+l SM SM AT LAS (ttZ) = SM (tt`+` ; m(``) > 5 GeV) + SM (ttZ) [1 BR(Z ! `+` )] : (B.1) The BR(Z ! `+` ) is taken from MadSpin [50]. The branching ratio and the NLO cross sections including the absolute scale uncertainties, using our parameter settings, are SM (tt + SM (tt + ; m(``) > 5 GeV) = 11:63(1)+11::0308 fb ; m(``) > 10 GeV) = 9:83(1)+01::7153 fb SM (ttZ) = 226:5(6)+1255::13 fb BR(Z ! `+` ) = 0:1029 : Applying these results to eq. (B.1), the corresponding prediction when using the same scales, PDF sets and generation procedure as in this paper is: AT LAS (ttZ) = 238:1(6)+1266::68 fb : SM results with the ATLAS measurement we apply to the In order to compare our tt + experimental result the RAttZTLAS , de ned as RAttZTLAS = SM (tt + ; m(``) > 10 GeV) SM AT LAS (ttZ) = 0:0413(1)+00::00000031 : The corresponding value for 13 TeV is RAttZT;L1A3TSeV = 0:0408(1)+00::00000032 : B.2 CMS | tt The measurement of tt described in ref. [2] should be compared with the W +bW b SM cross section calculated with pT ( ) > 20 GeV and R( ; b=b) > 0:1. Our tt results are with pT ( ) > 20 GeV, but they do not include photon radiation from the t; t decay products (W ; b; b). 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Olga Bessidskaia Bylund, Fabio Maltoni, Ioannis Tsinikos. Probing top quark neutral couplings in the Standard Model Effective Field Theory at NLO in QCD, Journal of High Energy Physics, 2016, 52, DOI: 10.1007/JHEP05(2016)052