Effective field theory and scalar extensions of the top quark sector
QQMUL-PH-19-19
Effective field theory and scalar extensions of the top quark sector
Christoph Englert, ∗ Peter Galler, † and Chris D. White ‡ SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK Centre for Research in String Theory, School of Physics and Astronomy,Queen Mary University of London, 327 Mile End Road, London E1 4NS, UK
Effective field theory (EFT) approaches are widely used at the LHC, such that it is importantto study their validity, and ease of matching to specific new physics models. In this paper, weconsider an extension of the SM in which a top quark couples to a new heavy scalar. We findthe dimension six operators generated by this theory at low energy, and match the EFT to thefull theory up to NLO precision in the simplified model coupling. We then examine the rangeof validity of the EFT description in top pair production, finding excellent validity even if thescalar mass is only slightly above LHC energies, provided NLO corrections are included. In theabsence of the latter, the LO EFT overestimates kinematic distributions, such that over-optimisticconstraints on BSM contributions are obtained. We next examine the constraints on the EFT andfull models that are expected to be obtained from both top pair and four top production at theLHC, finding for low scalar masses that both processes show similar exclusion power. However,for larger masses, estimated LHC uncertainties push constraints into the non-perturbative regime,where the full model is difficult to analyse, and thus not perturbatively matchable to the EFT. Thishighlights the necessity to improve uncertainties of SM hypotheses in top final states.
I. INTRODUCTION
The potential discovery of new physics beyond theStandard Model (BSM) remains one of the principal mo-tivations of contemporary high energy physics research,in both theory and experiment. Much attention focuseson the top quark and its antiparticle, given that these arethe heaviest particles in the SM, whose behaviour is thuslikely to be particularly sensitive to BSM effects. Fur-thermore, they are of fundamental importance when dis-cussing the naturalness (or otherwise) of the electroweaksymmetry breaking scale, such that typical BSM scenar-ios necessarily involve modifications of the top sector.Their effects may then be easier to investigate experimen-tally than purely electroweak processes (e.g. Higgs pro-duction), owing to the relatively large production crosssections of top particles at the Large Hadron Collider(LHC) (see e.g. Ref. [1] for a recent review).As is well-known there are, broadly speaking, two mainways to investigate possible new physics. The first is toassume a particular BSM theory, and to look for associ-ated signatures, such as the resonant production of newparticles e.g. decaying to a top pair. This is necessar-ily highly model-dependent, and lack of convincing newphysics signatures at the LHC to date instead motivatesthe use of model-independent approaches. Chief amongstthese is perhaps effective field theory (EFT), in which oneconsiders the SM Lagrangian to be the leading term inan expansion in gauge-invariant higher-dimensional op-erators. One may then extend the SM Lagrangian bythese higher dimensional terms, where dimension six is ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] the state of the art [2–8] (for a review see [9]). Each cor-rection to the SM is suppressed by one or more inversepowers of the new physics scale, and thus such a frame-work is only applicable if this scale (e.g. a new particlemass) exceeds the typical energy scales that are probedin a particular collider of interest.An intermediate approach between EFT and concreteUV scenarios is represented by so-called simplified mod-els (for reviews see [10–12]), which aim to reproduce abroad class of kinematic properties of the full UV the-ories parametrised by only a few additional propagat-ing degrees of freedom and their couplings. The relationto the EFT approach is that certain classes of opera-tors have then been resummed to arbitrarily high massdimension, and certain extensions of the SM might beparticularly generic or well-motivated, e.g. Higgs mixingmodels. Furthermore, experimental collaborations willin any case investigate large classes of simplified models.It is then instructive, for particular examples, to com-pare the two techniques, particularly with regards to howconstraints from the approaches to new physics comparewith each other in different kinematic regions.In this paper, we perform a case study of this idea inthe top quark sector, which has been the subject of anumber of recent EFT studies [13–42]. We will considera particular model, in which the SM is supplemented byan additional scalar, whose behaviour is parametrised en-tirely by its mass and couplings. We will calculate thetop pair production cross-section including the effects ofthis new particle up to one-loop order, showing explicitlywhich dimension six SM EFT operators are generatedupon taking the mass to be asymptotically large. Match-ing of the EFT (see also [43–49] for generic approaches)to the full theory can be performed at (next-to) leadingorder ((N)LO) in the coupling space of the latter, so thatwe have potentially three different descriptions of the newphysics: (i) the LO EFT description; (ii) the NLO EFT a r X i v : . [ h e p - ph ] F e b description, in which more effective operators are gener-ated; (iii) the full simplified model. We can then examinethe validity of each approach, and the ease of matchingEFT constraints to the full theory.We will first focus on top quark pair production,demonstrating explicitly that an EFT description canprovide an excellent approximation to the full model, asexpected. However, we will see that NLO corrections inthe EFT approach are particularly important, and that ana¨ıve LO approach tends to overestimate kinematic dis-tributions, such that its (invalid) application would leadto over-optimistic constraints on new physics.The operators examined in this paper also affect fourtop production [26, 50, 51], which is actively beingsearched for by both the ATLAS [52] and CMS [53, 54]collaborations. We examine the projected constraints onthis process (and top pair production) that are expectedto be obtained after the high luminosity LHC (HL-LHC)upgrade [55–57], and convert these into constraints onthe parameter space of the new physics model. We willsee that constraints from four top production are compet-itive with top pair production, suggesting that the twoprocesses would have roughly comparable weights in aglobal EFT fit. However, the extrapolated uncertaintiesfrom both top pair and four top production lead to con-straints that probe parameter space regions in which thefull theory is non-perturbative. For large scalar masses,the width of the scalar resonance increases, such that nomeaningful constraint on the coupling is obtained in thefull theory. Thus, whilst constraints in the EFT descrip-tion remain in principle valid and are possible, it becomesimpossible to match the EFT description to the full the-ory of new physics, given that perturbative computationsin the latter are not obtainable.The model considered here has been widely-studied ina number of different new physics scenarios. Thus, wehope that our results provide a useful case study for theapplication of EFT at the LHC, which will inform prag-matic discussions about how to apply this technique go-ing forwards, and what can be learned (or otherwise)about specific UV completions. The structure of ourpaper is as follows. In Sec. II we introduce the sim-plified model (of an additional scalar particle) that weare considering, and calculate the corrections to top pairproduction up to NLO. We furthermore explain how theEFT description is obtained at low energy (relative to thescalar mass). In section III, we present numerical resultsfor the top invariant mass distribution, and demonstratethe validity of the EFT description, even at LO, when thescalar mass is asymptotically large. We then quantify themass regime in which the NLO-matched EFT descriptionis a good approximation of the full theory. In section IV,we examine the projected uncertainties on top pair andfour top production at the LHC, and examine the con-straints obtained in the EFT at (N)LO, as well as thefull theory. Finally, in section V, we discuss our resultsand conclude. II. A SIMPLIFIED MODEL AND ITS EFTLIMIT
In this work, we consider a simplified model (similarto Ref. [58]) with dominant couplings to the top quark L BSM = 12 ∂ µ S∂ µ S − m S S − ( c S ¯ t L t R S + h.c.) (1)where S is a scalar field of mass m S . Provided thelatter is greater than 2 m t , where m t is the top mass, thescalar S may directly decay into (anti)-top pairs, withcorresponding widthΓ( S → t ¯ t ) = 3 c S m S π (cid:115) − m t m S ≡ c S ˜Γ . (2)Further contributions to the width arise from the factthat S can couple to gluons and photons via a top quarkloop, analogously to the SM Higgs boson. Although weinclude the loop-induced decays for completeness, theprompt decay S → t ¯ t dominates over the entire consid-ered mass range.Our aim in this paper is to compare an EFT descrip-tion of the theory of eq. (1) at low energy, with the fulltheory, in order to assess the validity and interpretationof the former. To this end, let us consider how this theoryleads to corrections to top pair production up to NLO inthe coupling of the scalar i.e. up to and including O ( c S ).Comparison with the EFT will then allow us to matchthe two descriptions. Representative diagrams contribut-ing to the gluon-induced process gg → t ¯ t are shown infig. 1, where we do not consider SM electroweak contribu-tions [60] (see also [61, 62]). In the SM, for heavy Higgsbosons, it is known that the Higgs signal (with a largeQCD K factor [63, 64]) has sizeable interference effectswith the QCD continuum in gg → t ¯ t [65–67]. This in-fluences exclusion constraints, but is also a viable sourcefor new physics beyond the SM [20, 68–77]. The pre-dominant focus of previous work was therefore devotedto isolating the resonance shape and cross section, whichis not our focus here. Note, however, that loop effectsand their relation to (Higgs) effective field theory werefirst discussed in [78–81].For our analysis, we implement the leading or-der, virtual and counter term (fig. 2) contributionsfor q ¯ q, gg → t ¯ t production at O ( c S ) in a modi-fied version of Vbfnlo [82–85] which links
Form-Calc/LoopTools [86, 87]. Various analytical compar-isons against alternative calculations as well as numer-ical cross checks of leading order amplitudes have beenperformed using
MadGraph [88]. We use the on-shellrenormalisation scheme, and have verified UV finiteness Similar frameworks have been considered in FCNC studies,e.g. [59]. g g ttg Stt gg t tttt S g g ttttt Sgg ttt tt S gg ttttt S gg t tSt ttq q ttg Stt
FIG. 1: Representative one-loop Feynman diagram contributions to gg → t ¯ t arising in the simplified model of eq. (1). g g t tg gg ttt gg tttg g t tg g g t tg g gg tttt FIG. 2: Representative counter term contributions to gg → t ¯ t . both analytically and numerically for the gg and q ¯ q chan-nels independently. We use real masses throughout thiswork, but note that the discrimination of signal and back-ground can have shortfalls when the scalar width be-comes comparable to the resonance mass [89–93], whichis indicative of a loss of perturbative control [94].We now turn to the effective theory description of themodel of eq. (1) at low energies or, equivalently, whenthe scalar mass m S is taken to be large. Integrating outthe heavy scalar generates two dimension six operatorsthat enter the processes considered in this paper. Thefirst of these is a modified gluon- t ¯ t interaction, describedby the effective operator O tG = v ¯ t L T a σ µν t R G aµν (3)(and its Hermitian conjugate). Here t L and t R de-note left-handed and right-handed top quarks, T a arethe SU(3) generators in the fundamental representa-tion, σ µν = i [ γ µ , γ ν ] / G aµν is the QCD gauge fieldstrength tensor. Note, that we have scaled the operatorby an additional factor of the vacuum expectation value v of the SM Higgs boson. The second operator is a four-fermion operator involving four top quarks, and given by expanding the scalar propagator for large m S in relationto its four momentum q ,(¯ tt ) c S q − m S (¯ tt ) q (cid:28) m S −→ − c S m S (¯ tt ) = c tt Λ O tt (4)see e.g. fig. 3. Note that this operator is not part of theWarsaw (SM EFT) basis [7], but it is more convenientfor our purposes. For instance in four top production,the operator of eq. (4) enters at tree-level in the EFT,as illustrated in fig. 3. The contribution from the oper-ator O tG is suppressed with respect to the contributionfrom O tt because it is loop-induced and four-top contri-butions with one O tG insertion are of higher order in α s than four-top contributions with one insertion of O tt (seeappendix A). The situation is different in top pair pro-duction. Since O tt enters only through loops (see, e.g.fig. 4) there is no relative suppression with respect to O tG which is also loop-induced. Furthermore, tree-leveldiagrams with one O tG insertion (whose topology is thesame as the upper three diagrams in fig. 2) and one-loopdiagrams with one O tt insertion (as in fig. 4) contributeto the same perturbative order in α s . Hence, in top quark (a) gg ttS tt (b) gg ttS tt (c) gg tt tt FIG. 3: (a), (b) tree-level graph in the theory of eq.(1) contributing to four top production; (c) tree-level contribution in theEFT description upon integrating out the heavy scalar, where the grey blob represents the operator of eq. (4). g g ttg tt gg t tttt g g tttttgg ttt tt gg ttttt ! q q ttg tt FIG. 4: Representative one-loop Feynman diagram contributions to gg → t ¯ t arising in the effective theory formulation of eq. (1),the shaded region represents a four top insertion. pair production both operators are relevant and have tobe included in a consistent EFT calculation.In particular, including the four top contribution re-quires the calculation of the loop diagrams of fig. 4. Thesecontributions require additional UV counterterms whichare not present in the full theory but which renormalisethe EFT operators. For illustrative purposes, we showthe counterterm graphs in the gluon channel in figure 2.Renormalisation of the UV divergences related to EFToperators is performed in the MS scheme, and we havechecked UV finiteness of all of our expressions for the fi-nal amplitude. The sum of tree-level contributions from O tG and the one-loop contributions from O tt yields theresults for observables of top quark pair production atNLO EFT in terms of the Wilson coefficients c tG and c tt .Furthermore, our use of a specific model of new physicsat high energy means that we can fix the values of theWilson coefficients by matching the full theory and NLOEFT calculations at a suitable matching scale µ M takingthe operator mixing between O tG and O tt into account.We choose µ M = m S / c tG as the finite remainderafter subtracting the MS-renormalised four fermion one-loop insertion from the EFT operator that is induced bythe propagating S . Note that it does not require UV . . . . m S [TeV] r a t i o . . . . − − − µ M = 1 TeV µ M = m S / µ M = m S m S [TeV] | c t G | / Λ [ G e V − ] FIG. 5: Matched value of c tG for different matching scalechoices and scalar masses ( c S = 1) as detailed in the text. renormalisation as opposed to the four fermion insertion. . . . . . . . − − m ( t ¯ t ) [TeV] | r a t i o w r s t f u ll | . . . . . . . − − − − − − − effectivefullnaive effective m ( t ¯ t ) [TeV] | d σ / d m ( t ¯ t ) | [ f b / G e V ] p ( g ) p ( g ) −→ t ¯ t , interference only ✞✝ ☎✆ m S = 2 TeV 2 . . . . . . . − − m ( t ¯ t ) [TeV] | r a t i o w r s t f u ll | . . . . . . . − − − − − − − effectivefullnaive effective m ( t ¯ t ) [TeV] | d σ / d m ( t ¯ t ) | [ f b / G e V ] p ( q ) p (¯ q ) −→ t ¯ t , interference only ✞✝ ☎✆ m S = 2 TeV FIG. 6: BSM interference contribution as a function of the invariant t ¯ t mass for gluon fusion (left) and q ¯ q annihilation, neglectingthe Z contribution. As the interference changes sign we choose to plot the absolute value of the interference for clarity. Wechoose m S = 2 TeV and c S = 0 . The dependence of c tG on the matching scale is shown infigure 5. As the matching scale is related to a renormali-sation scale choice (appendix A), the cross section has alogarithmic dependence on the µ M .As for the full simplified model calculation describedabove, we have implemented our matched NLO calcula-tion in a modified version of Vbfnlo [82–85].
III. VALIDITY OF EFT AT (NEXT-TO)LEADING ORDER
In the previous section, we outlined a particular sim-plified model for new physics in the top quark sector, andexplained how this can be matched to an EFT descrip-tion at low energies. In this section, we analyse the rangeof validity of the latter, as the mass of the scalar particleis lowered towards LHC energies. We will illustrate ourresults using the invariant mass distribution of the finalstate tops in top pair production, although similar resultswould be obtained for other kinematic distributions.In fig. 6, we show the contribution to the invariantmass m t ¯ t stemming from the interference between thenew physics process, and the SM contribution,d σ ( t ¯ t ) ∼ (cid:16) M SM t ¯ t M ∗ virt/d6 t ¯ t (cid:17) (5)where virt/d6 represents the propagating- S contributionsor their dimension six EFT counterparts, for a scalarmass of m S = 2 TeV. Three different curves are shown.The blue curve shows the result obtained from the fulltheory of eq. (1), with all dynamics correctly included.The red curve shows the results of our NLO-matched EFT calculation. We see that the EFT and the full com-putation agree well, as long as we are away from the turn-on of the scalar Breit-Wigner distribution. The greencurve in fig. 6 shows the results of a bottom-up approachto EFT where we assume no knowledge about the fulltheory. Specifically, we perform a LO EFT calculationof t ¯ t production taking only tree-level diagrams with oneinsertion of O tG into account. We treat the Wilson co-efficient c tG as a free parameter in the EFT and fit c tG to Monte Carlo data that was generated using the fulltheory. This approach simulates an EFT fit where theEFT prediction is calculated at LO and applied to datawhich contains the signatures of the simplified model ofeq. (1). This na¨ıve approach based on fitting c tG alonenever reproduces the correct shape. This becomes evenmore transparent when we push the scalar mass to largervalues, e.g. m S = 5 TeV in fig. 7. The full theory and theNLO EFT calculation agree very well, with the turn-onof the scalar exchange only leading to mild correctionsfor large values of m ( t ¯ t ) in the (dominant) gluon fusioncomponent. Again as expected, the LO EFT approachnow deviates significantly. In particular, fixing the coef-ficient of c tG at low energies where the m t ¯ t distribution ismeasured more precisely leads to a drastic mismodellingof the shape of the invariant mass distribution, with asignificant overestimate of the high mass tail. As we willsee in the following section, this can lead to an overlyoptimistic constraint on possible new physics effects, forthe model that we consider here.In fig. 8, we indicate the validity range when com-paring full theory and NLO EFT computation (for ageneral discussion see [95]). The parameter m max ( t ¯ t )denotes the energy scale at which the NLO EFT and . . . . . . . − m ( t ¯ t ) [TeV] | r a t i o w r s t f u ll | . . . . . . . − − − − − − − effectivefullnaive effective m ( t ¯ t ) [TeV] | d σ / d m ( t ¯ t ) | [ f b / G e V ] p ( g ) p ( g ) −→ t ¯ t , interference only ✞✝ ☎✆ m S = 5 TeV 2 . . . . . . . − − m ( t ¯ t ) [TeV] | r a t i o w r s t f u ll | . . . . . . . − − − − − − − effectivefullnaive effective m ( t ¯ t ) [TeV] | d σ / d m ( t ¯ t ) | [ f b / G e V ] p ( q ) p (¯ q ) −→ t ¯ t , interference only ✞✝ ☎✆ m S = 5 TeV FIG. 7: Like fig. 6 but choosing m S = 5 TeV and again c S = 0 . full computations deviate beyond the indicated percent-ages for c S = 1. In this comparison we also include thesquared resonance contribution. Note that for this cou-pling choice, the width remains at (cid:39) . m S leading tothe turn-on of the Breit-Wigner distribution becomingresolvable in the direct comparison. This turn-on cannotbe resolved when background uncertainties are included(see below).Our results in this section confirm the possibility ofobtaining an accurate EFT description of the simplifiedmodel of eq. (1), which is generic enough to apply in mul-tiple contexts, including singlet Higgs mixing scenariosand multi-Higgs doublet extensions. A key issue facingcontemporary global EFT fits is whether or not to pursuethe effort of carrying out a full NLO calculation for allprocesses and observables considered. The latter requiresa considerable effort (see e.g. [37, 96] for recent exam-ples), although the intermediate possibility also exists ofincluding renormalisation group mixing effects betweendimension six operators, but neglecting additional con-tributions that are non-logarithmic in the matching scale.The importance of NLO effects in the present case is ulti-mately due to the fact that of the two operators that aresourced in the low energy description, one is tree-inducedbut the other is loop-induced. Our example thus clearlyshows the need to bear such considerations in mind whentrying to match EFT constraints to specific new physicsmodels. (cid:7)(cid:6) (cid:4)(cid:5) c S = 1 m S [TeV] m m a x ( t ¯ t ) [ T e V ] FIG. 8: Validity within percentage of the EFT computa-tion when compared to the full model including squaredHiggs resonance contribution. m max ( t ¯ t ) indicates the invari-ant (anti-)top mass, at which the relative difference becomeslarger than the indicated percentage. IV. LHC COMPARISONS, RESULTS ANDEXTRAPOLATIONS
Both top pair and four top production are being ac-tively measured at the LHC, and will play a crucial rolein searching for new physics in the top quark sector inthe coming years. To this end, it is instructive to ex-amine the sensitivity that the LHC is likely to achievefollowing its high luminosity upgrade, with an expected3 ab − of data. We will do this here for two scenar-ios. Firstly, we will constrain the full simplified modelof eq. (1) directly. Assuming a given uncertainty for theabove-mentioned processes leads to exclusion contours inthe ( m S , c S ) parameter space, shown in fig. 9, where any-thing above a given curve (i.e. for stronger couplings c S )is excluded. Secondly, we will assume that an NLO EFTanalysis has been applied, leading to constraints on thecoefficients of the new physics operators O tG and O tt .By matching with the full theory as described previously,constraints on the operator coefficients can also be con-verted to curves in the ( m S , c S ) plane.The top pair production cross section is currentlyknown at NNLO precision [97, 98] (see also [99]). Giventhe large cross section, the theoretical uncertainty willbe the limiting factor of physics in the top sector (seealso [24]). In fig. 9, we show the sensitivity of the LHCunder the assumption that the unfolded m t ¯ t distributioncan be described at an optimistic 3% level using a binned χ test as detailed in Ref. [15]. For this particular er-ror choice the EFT and full theory agreement happensto be slightly above the perturbative unitarity limit of c S (cid:39) π that can be derived from t ¯ t → t ¯ t scattering inthe full model (i.e. with propagating S ). A larger er-ror budget quickly pushes the constraints deeply into thenon-perturbative regime. On the other hand sensitivityto c S (cid:39) s -channel scalar contribution with anapproximate K factor (cid:39) . S . Notwithstand-ing the accuracy at which the EFT manages to approx-imate the full computation, we see that hadron collidersystematics do seriously curtail precision physics in thetop sector when contrasted with certain classes of top-philic BSM models. The simplified model highlights thisthrough Fig. 9. Gaining sensitivity in such an instancecrucially rests on more precise SM predictions that allowconstraints to be pushed into the perturbative limit ofthe model.One might argue that discovering a contrived top-philicnew physics scenario is difficult to achieve in the firstplace. However, for the scenario that we have stud-ied there is the possibility to investigate four top finalstates similar to existing analyses [26, 50, 51]. The ex-periments have also performed extrapolations to the HL-LHC, e.g. [55–57]. As the cross sections for this pro-cess are relatively small, O (10 fb) [100, 101], statisticaland experimental uncertainties will be important. Thereis reason to believe that the latter can be brought un-der sufficient control and e.g. ATLAS have shown that a sensitivity of 11% around the SM expectation can beachieved [56] which is smaller than the current theoreti-cal precision. It is not unreasonable to expect that the-oretical predictions can be improved and we assume a18% accuracy in the extraction of the unfolded t ¯ tt ¯ t crosssection, which is slightly worse than the ATLAS extrap-olation and the lowest bound provided by CMS [55].We simulate four top events using MadEvent [88] keep-ing track of destructive interference effects that arise be-tween the QCD and new scalar contributions. In the fourtop case, these are much smaller than for gg → t ¯ t , wefind a typical mild correction of O ( − m S [ GeV ] c S FIG. 9: 95% confidence level exclusion contours for the sim-plified model of eq. (1) as a function of its mass m S and topcoupling c S . The blue solid contour shows the full result (i.e.propagating S at NLO) while the blue dashed line correspondsto the EFT calculation. For pp → t ¯ t we assume a flat uncer-tainty of 3%. The solid red line represents a pp → t ¯ tt ¯ t analysisof the simplified scenario using the extrapolation of Ref. [56]while the red dashed line represents the (LO) EFT four top re-sults. The shaded band shows the region where perturbativeunitarity is lost, c S > ∼ √ π which we obtain from an explicitpartial wave projection calculation of t ¯ t → t ¯ t in the full model,i.e. with propagating S . Note that this is precisely the regionwhere Γ( S → t ¯ t ) (cid:39) m S according to Eq. (2). Finally, theblack dashed line is the unitarity constraint on the effectivefour top interaction, below which unitarity is preserved (fordetails see text). around m S (cid:39) . t ¯ t system for large enough c S behavesas ∼ ˜Γ − (see eq. (2)), i.e. flattens out as a function of c S such that the cross section constraint for large enough c S is determined by the mass m S . This means that thesensitivity translated to our simplified model calculationis no longer under perturbative control. This effect ismainly driven by the dynamics in the full theory of thechosen model, and is not a problem of setting constraintsusing EFT approaches. Hence, applying a cut on thetypical energy scale of the process as proposed for ex-ample in [95] would not resolve this problem. While thehigh mass region is plagued by large width effects in thefull theory the lower masses ( < c S < ∼ π m S E − m t , (6)where, for illustrative purposes, we have matched theconstraint on c tt to c S using Eq. (4). E max denotes theenergy where unitarity is broken in the leading order ap-proximation. This happens at the latest at E max (cid:39) m S when considering our scenario, which is depicted by theblack dashed line in Fig. 9. Comparing this bound withthe red dashed curve shows that the EFT constraints on c S (or rather c tt ) from four-top final states remain withinthe perturbative unitarity bounds of the EFT. Hence,analyses of the four-top final states do give rise to con-straints which are perturbatively meaningful in differentscenarios where matching is possible (e.g. [102–104]). V. SUMMARY AND CONCLUSIONS
Effective field theory approaches are becoming a newstandard for the dissemination of LHC physics results.In contrast to flavour physics where EFT methods havebeen successfully employed over decades (see e.g. [105]), The width being related to the resummation of the imaginary partof the top 2 point function signifies the relevance of higher ordercorrections in this model for the four top final states as well. Theseeffects would be interesting to study but are beyond the scope ofthis work. the non-obvious scale separation of hadron collider mea-surements that probe a broad partonic centre of massenergy range makes their implementation less straight-forward. In particular, operator mixing effects that aresensitive to whether dimension six operators are tree- orloop-induced in particular UV scenarios will shape thephenomenology at intermediate scales and has to be re-flected consistently in any limit setting procedure. Inthis work we have examined a particular scalar simpli-fied model with top-philic couplings that approximates abroad range of UV scenarios, with the particular aim togauge the sensitivity reach of top quark final states at theLHC. Top pair production processes with large cross sec-tions are prime candidates to look for new physics effectswith statistical control. We demonstrate that the NLOmatching of the EFT and full model allows a broad rangeof agreement of the two approaches, up to ∼ pp → t ¯ t is considered in isolation (i.e. no other compet-ing BSM effects are present). Pivotal to changing thissituation is the continued precision calculation efforts forSM processes, and t ¯ t production in particular in our con-text. In the concrete case of top-philic interactions asexpressed by the scalar model, subsidiary measurementssuch as four top final states can provide additional sensi-tivity. While these processes are considerably more rarethan top pair production at the LHC, they have directsensitivity to four top contact interactions which are clearsigns of top-philic interactions below their characteristicscale. Including four top final states in leading order fitsis therefore crucial to achieve sensitivity to the scenariodiscussed in this work, as an example for new physicsthat predominantly talks to the top sector. Acknowledgments
This work was supported by the Munich Institutefor Astro- and Particle Physics (MIAPP) of the DFGExcellence Cluster Origins ( ).CE and PG are supported by the UK Science andTechnology Facilities Council (STFC), under grantST/P000746/1. CDW is supported by the STFC, un-der grant ST/P000754/1, and by the European UnionHorizon 2020 research and innovation programme underthe Marie Sk(cid:32)lodowska-Curie grant agreement No. 764850“SAGEX”.
Appendix A: Notes on renormalisation and matching
The UV divergent corrections of top pair productionin the simplified model are given by the vertex and prop-agator corrections depicted in Fig. 1. The on-shell renor-malisation of UV divergencies is determined only by topquark mass and wave function counterterms (these can befound in Ref. [106]). The cancellation of UV singularitiesalong these lines is expected by the gauge-singlet char-acter of S and the product-group gauge theory form ofthe SM. Hence, there is no renormalization of the gaugecouplings.The qualitative changes in the renormalisation proce-dure when comparing full and effective theory compu-tation is highlighted by considering the top quark two-point function. Approaching the limit m S → ∞ beforecarrying out the loop integration results in a schematicidentification B ( q , m t , m S ) ∼ g g t tg gg ttt gg tttg g t tg g g t tg g gg ttttg g t t m S →∞ −→ − m S g g t tg gg ttt gg tttg g t tg g g t tg g gg ttttg g t t ∼ − A ( m t ) m S , (A1)where A and B are the Passarino-Veltman one-pointand two-point scalar functions [106, 107]. Since the A function does not depend on the momentum of the two-point function there is no top quark wave function renor-malisation involved in the EFT calculation. Instead therenormalisation of the EFT calculation is performed inthe top quark mass and the Wilson coefficient c tG . TheEFT renormalisation of the top mass due to the fourfermion insertion is given by δm EFT t = c tt π Λ m t A ( m t ) . (A2) The one-loop EFT contributions (see fig. 4) give rise toUV singularities. After top mass renormalisation we areleft with the following UV divergence in the NLO EFTamplitude M ( gg → t ¯ t ) (cid:12)(cid:12) EFT ,m t -ren.NLO, div = − c tt g s y t π Λ ∆ UV (cid:104)O tG (cid:105) , (A3)where ∆ UV = (cid:15) − − γ E + log 4 π in dimensional regu-larisation with D = 4 − (cid:15) dimensions and y t denotesthe top Yukawa coupling (we have traded m t against thevacuum expectation value that apears in the normalisa-tion of eq. (3)). The amplitude (cid:104)O tG (cid:105) denotes all O tG operator insertions that contribute to gg → t ¯ t at tree-level including those with contact interactions ggt ¯ t . Thisshows that the one-loop insertion of the four-fermion op-erator O tt induces a renormalisation of the O tG operatorsince the LO EFT amplitude is given by M ( gg → t ¯ t ) (cid:12)(cid:12) EFTLO = (cid:104)O SM (cid:105) + c tG Λ (cid:104)O tG (cid:105) , (A4)where (cid:104)O SM (cid:105) represents the SM amplitude, which is in-dependent from (cid:104)O tG (cid:105) as a result of [7]. The divergencein eq. (A3) can be removed by including a c tG counterterm δc tG Λ = c tt g s y t π Λ (cid:0) ∆ UV + F ( µ ) (cid:1) , (A5)where F denotes renormalisation-scheme dependent fi-nite terms that will be fixed when we match the one-loop EFT amplitude with the on-shell renormalised one-loop result for propagating S at a matching scale µ M .The matching relation (which also addresses the quark-induced channels) is given by ttgQ tt + ttg Q = µ M ⟨O tG ⟩ , ren.= ttgQ Stt + t tg Q = µ M ⟨O tG ⟩ . (A6)Concretely this means that we first extract the Lorentz structure related to the operator insertion of O tg of therenormalised EFT as well as the full calculation. We then identify the coefficients of the O tg amplitudes (Lorentzstructures) at a matching scale µ M , which fixes the finite terms F ( µ M ) that correspond to a tree-level insertion of0 O tG after matching F ( µ M ) = − m S µ M − m t + m S µ M m t − m t ˜ A ( m S ) − m S m t ( µ M − m t ) ˜ A ( m t )+ m S (4 m t ( µ M − m t ) − m S ( µ M − m t )) m t ( µ M − m t ) ˜ B ( m t , m S , m t ) + (cid:18) − m S (2 m S + µ M − m t )( µ M − m t ) (cid:19) ˜ B ( µ M , m t , m t ) − m S ( m S + µ M − m t )( µ M − m t ) ˜ C ( m t , µ M , m t , m S , m t , m t ) , (A7)where ˜ A , ˜ B , ˜ C are the (cid:15) → c tG ( µ M )Λ = − c S g s y t π m S F ( µ M ) , (A8)which reflects the g s -loop induced nature of c tG in theconsidered simplified model. The inclusion of appropri-ately defined finite terms in the comparison of Sec. II iscrucial to obtain agreement between full and EFT-basedcomputation. This balances the c tG -related momentum transfer behaviour of (cid:104)O tG (cid:105) against the virtual contribu-tions of O tt . In a na¨ıve or bottom-up approach based on (cid:104)O tG (cid:105) alone without matching this balance is lost whichleads to overestimates in the tails of distributions longbefore √ s (cid:39) m S .Note that only the sum of loop-inserted c tt and tree-level c tG is defined as a consequence of eq. (A6) and wecan always move finite terms between the EFT coeffi-cients. The scheme that we adopt is fixing c tt throughleading order t ¯ t → t ¯ t scattering (accessible in four topfinal states), which leaves c tG determined as a functionof µ M . [1] K. Kr¨oninger, A. B. Meyer, and P. Uwer, in TheLarge Hadron Collider: Harvest of Run 1 , edited byT. Sch¨orner-Sadenius (2015), pp. 259–300, 1506.02800.[2] S. Weinberg, Physica
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