The present and future of four top operators

We study the phenomenology of a strongly-interacting top quark at future hadron and lepton colliders, showing that the characteristic four-top contact operators give rise to the most significant effects. We demonstrate the extraordinary potential of a 100 TeV proton-proton collider to directly test such non-standard interactions in four-top production, a process that we thoroughly analyze in the same-sign dilepton and trilepton channels, and explore in the fully hadronic channel. Furthermore, high-energy electron-positron colliders, such as CLIC or the ILC, are shown to exhibit an indirect yet remarkable sensitivity to four-top operators, since these constitute, via renormalization group evolution, the leading new-physics deformations in top-quark pair production. We investigate the impact of our results on the parameter space of composite Higgs models with a strongly-coupled (right-handed) top quark, finding that four-top probes provide the best sensitivity on the compositeness scale at the future energy frontier. In addition, we investigate mild yet persisting LHC excesses in multilepton plus jets final states, showing that they can be consistently described in the effective field theory of such a new-physics scenario.


Introduction
The LHC has lived up to the expectations as a top quark factory. Not only have we learnt about the properties of the top thanks to the variety of processes studied, but also about the extent to which dynamics beyond the SM couples to the top.
The electroweak hierarchy problem is arguably the strongest motivation for top-philic new physics at the TeV scale. The top quark is singled out in view of its large coupling to the Higgs, which in turn suggests that the new physics has strong couplings to the top sector. This is particularly the case in composite Higgs models where the Higgs arises as a Nambu-Goldstone boson [1][2][3]. The top quark can in fact be regarded as the raison d'être of such a scenario, since its large couplings to the composite dynamics are what allows the Higgs field to eventually break the electroweak symmetry. A resulting prime phenomenological implication is that the top exhibits the characteristics of a partially composite state [4], to the point that in well-motivated realizations of this idea one of its chiral components couples to the new dynamics as strongly as the Higgs does [5,6]. This is the case for instance in composite twin Higgs models and generalizations thereof [7][8][9], where consistency with precision data and minimal fine-tuning favor a strongly-coupled t R .
It is of major phenomenological relevance that in these scenarios the anomalous properties of the top sector remain experimentally accessible even if the new resonances are too heavy to be directly produced at colliders. Such anomalous effects are best described by an effective low-energy theory in terms of higher-dimensional operators deforming the SM Lagrangian. In this work we mainly focus on the four-top operator c tt

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since it captures the genuine physics of a strongly-interacting right-handed top quark. Indeed, in the broad class of models we are interested in, this operator is always generated with the largest size and independently of the details of how t R couples to the new dynamics. In addition to the everlasting search for inner structure of the particles that are presently viewed as fundamental, a strong motivation for studying the fate of four-top contact interactions at future colliders is the realization that already at the LHC they give rise to a competitive constraint on the parameter space of composite Higgs (CH) models. Denoting with m * the mass of the new heavy degrees of freedom and with g * their coupling to the Higgs and the (right-handed) top, one expects on dimensional grounds that This is of the same size as the operator encoding the inherent deformations associated with a strongly-interacting Higgs field, O H = 1 2 (∂ µ |H| 2 ) 2 with coefficient c H /Λ 2 ∼ 1/f 2 . As we review in section 2, the current 95% CL constraint on c H from LEP and LHC Run 2 data translates into f | H 820 GeV [10], only slightly stronger than the bound from c tt : the operator eq. (1.1) modifies the rate of four-top production at the LHC with respect to the SM, and the absence of significant deviations yields f | tt 730 GeV [11]. Therefore, probes of O tt constitute, already today, a competitive test of CH models.
An added motivation for considering these scenarios is that they generically predict important deviations in processes involving tops and the Higgs or electroweak vector bosons, see e.g. [12]. Intriguingly, present LHC measurements of top-quark pair production in association with a W or Z boson, which are main backgrounds for four-top production, show a consistent pattern of slight excesses with respect to the SM predictions. In section 3, we show their compatibility with precisely the type of heavy new physics we are interested in. Far from a definitive conclusion, we nevertheless find that the excesses are roughly but compellingly consistent with the as yet absence of deviations in other top or Higgs analyses.
We emphasize at this point that while pp → tttt is a rare process in the SM, as already noted in [6] the contribution from four-top operators is mainly associated with a strong tt → tt scattering amplitude that grows with energy, i.e. M tt→tt ∼ c tt (E/Λ) 2 , resulting in an enhanced sensitivity at high-energy colliders. This high-energy behavior has been exploited in a variety of works exploring new physics in the top sector (e.g. [12][13][14]), and it is behind the spectacular reach of a potential proton-proton collider at a center of mass energy of 100 TeV. We present our analysis of four-top production at the proposed Future Circular Collider (FCC-hh) [15] in section 4. We concentrate mainly on multilepton final states, although we also explore the fully hadronic signature. We find a sensitivity on the compositeness scale f | FCC tt 6.5 TeV at 95% CL after 30 ab −1 of integrated luminosity, an order of magnitude above the LHC. The impact on the parameter space (m * , g * ) of CH models, with a comparison to other relevant probes, is shown in figure 10.
At future electron-positron colliders, a direct test of the four-top scattering amplitude is not feasible. Therefore, the naive expectation would be that the largest sensitivity on top quark compositeness does not arise from eq. (1.1), but from other operators directly modifying e + e − → tt. In section 5 we show that this is, in fact, not the case: renor-JHEP02(2021)043 malization group evolution makes the four-top operator dominate the expected size of the effective (ē L,R γ µ e L,R )(t R γ µ t R ) operators. The excellent experimental precision achievable at lepton colliders, combined with the possibility of large scattering energies, like the 3 TeV attainable at the Compact Linear Collider (CLIC) [16], results in a truly remarkable sensitivity on c tt and thus on the compositeness scale, e.g. f | CLIC tt 7.7 TeV for m * = 4πf at 95% CL and after 3 ab −1 of integrated luminosity. As we show in figure 11, at CLIC these constraints are superior to the corresponding ones from Higgs measurements.
The impact of our results is summarized in section 6. For CH models with a stronglyinteracting top, both hadronic and leptonic future high-energy colliders will be able to test, through the top quark, fine-tunings of the electroweak scale at the ξ = v 2 /f 2 ≈ 10 −3 level, a hundred-fold that of the LHC and certainly unprecedented in the realm of particle physics. This is a truly exceptional motivation for future discoveries that could await us at the high energy frontier.

Operators, expectations and current constraints
In this section we define the dimension-six operators we will be considering throughout this work (see table 1) and discuss their expected size in theories with a strongly-interacting top quark. We review our current knowledge on the corresponding experimental constraints, paying special attention to those operators leading to the largest sensitivity on the parameter space of CH models. The current status is summarized in figure 1.
Searches for the production of four top quarks at the 13 TeV LHC have provided important constraints on the idea of top quark compositeness. From the absence of significant deviations in the total cross section, which have been searched for using ≈ 36 fb −1 of data in the single-lepton [11,17,18], opposite-sign dilepton [11,18], and same-sign dilepton and multilepton final states [19][20][21], the combined ATLAS observed (expected) bound on the four-fermion operator eq. (1.1) is |c tt |/Λ 2 < 1.9 (1.6) TeV −2 at 95% CL [11]. A similar bound is obtained by CMS [18]. Besides, very recently both experiments have updated their multilepton searches to ≈ 140 fb −1 [22,23], observing mild but intriguing excesses with respect to the SM predictions; we will discuss these separately in section 3. Constraints comparable to the one on c tt are obtained for the full set of four-top operators [18], which also involve the third generation left-handed quark doublet, where t A = λ A /2. In this work our focus is on strongly-interacting right-handed top quarks, for which eq. (1.2) sets the size of the associated four-top operator. Then, the generic expectation in scenarios dealing with the generation of the top Yukawa coupling, such as CH models [24], is that operators involving q L are generated as well, yet with coefficients proportional to y t , i.e. c tq , c (8) tq /Λ 2 ∼ y 2 t /m 2 * , and c qq , c (8) qq /Λ 2 ∼ y 2 t (y t /g * ) 2 /m 2 * , thus not as enhanced as c tt /Λ 2 for large new-physics couplings g * y t . We note that the H parameter [25] effectively contributes to four-top production just as the O tq operator does.

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Since our main interest is in new-physics scenarios with a strongly-interacting Higgs, top-Higgs operators should also be present and in principle with large coefficients. This is the case of c Ht which leads to a zero-momentum deformation of the Zt R t R coupling. However, we point out that, other than the fact that the experimental sensitivity on such anomalous couplings has been typically weak (see however section 3), this operator turns out to be suppressed by an accidental discrete symmetry [26] in models where the right-handed top does not induce radiative contributions to the Higgs potential, as preferred by fine-tuning considerations. Although such a symmetry is eventually broken, the coefficient of O Ht would be expected to be small in these cases, c Ht /Λ 2 ∼ N c (y t /4πf ) 2 . Similar statements can be made for the analogous operators with q L , namely O Hq and O (3) Hq (see table 1). The combination c Hq + c (3) Hq induces a correction to the Zb L b L coupling which, although constrained at the per-mille level at LEP, is also typically protected by P LR symmetry [26,27]. Measurements of deviations in the Zt L t L and W t L b L couplings from the SM, associated with c Hq − c (3) Hq and c (3) Hq respectively, do not reach the level of precision to be competitive with the four-top operator eq. (1.1), in particular at large m * , because c Hq /Λ 2 ∼ y 2 t /m 2 * . Similarly for the still poor measurements of the Higgs coupling to the top, which probe the Yukawa-like dimension-six operator O yt , even if c yt /Λ 2 ∼ g 2 * /m 2 * .

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Despite the fact that operators with SM gauge field strengths and top quarks are generated with coefficients that are not enhanced, or are even suppressed at strong coupling, 1 they could be relevant in situations where direct probes of the four-top operators are not feasible, as it is the case of future lepton colliders (see section 5). Such operators are with c tD /Λ 2 ∼ g /m 2 * and c (8) tD /Λ 2 ∼ g s /m 2 * , as well as * for a strongly-coupled right-handed top. These operators are equivalent to a particular combination of four-fermion operators, since by the equations of motion, For an example of the potential relevance of this class of operators in deciphering the composite nature of the top quark, let us consider probes of O (8) tD at the LHC. This operator affects top-pair production through a qq → tt amplitude that grows with energy [13]. Given the expectation c (8) tD /Λ 2 ∼ g s /m 2 * , one could naively conclude that the new-physics effects do not depend on g * for fixed m * . However, renormalization group evolution implies that at relevant scale, µ, the coefficient of O (8) tD is [29] c (8) tD (µ) = c (8) (2.8) Therefore, one-loop diagrams with one insertion of the four-top contact interaction eq. (1.1) dominate the amplitude at large g * , M qq→tt ∼ g 2 s (g * /4π) 2 (E/m * ) 2 log(m 2 * /E 2 ). Although current LHC searches in top-pair production yield c (8) tD /Λ 2 < 0.7 TeV −2 at 95% CL [30] and are therefore not sensitive enough to yield a relevant constraint on the (m * , g * ) plane, we show in section 5 that this changes at high-energy lepton colliders, due to the superior precision in top-pair production.
The main conclusion of the previous discussion is that probes of the four-top operator eq. (1.1) are the most relevant ones concerning a strongly-interacting (right-handed) top quark. 2 The impact of current LHC bounds from four-top production on the (m * , g * ) parameter space is shown in figure 1, where we also present a comparison with the main universal tests of CH models. The latter comprise searches for anomalous Higgs couplings, primarily controlled by O H = 1 2 ∂ µ |H| 2 2 and constrained by Higgs and electroweak precision data. The current exclusive (one operator at a time) 95% CL bound

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is c H /Λ 2 < 1.5 TeV −2 [10], which constitutes, given that c H /Λ 2 ∼ g 2 * /m 2 * , the leading constraint at strong coupling. Note however that such a bound is largely correlated with other contributions to the electroweak parameters, in particular S and T, controlled by the operators O W , O B , and O T respectively (see table 1). This is why the marginalized bound on O H (i.e. letting S and T float), determined mainly from LHC data, is significantly weaker, f | LHC H 550 GeV. These last operators are in fact very important in CH models, giving rise to constraints that are independent of the new-physics coupling, since c W,B /Λ 2 ∼ 1/m 2 * and c T ∼ N c y 2 t (y t /4π) 2 /m 2 * , the latter of one top-loop size because of custodial symmetry [33]. The region of parameter space covered by the bound (c W + c B )/Λ 2 < 0.07 TeV −2 [10,34], corresponding to m * > 3.7 TeV, is also shown in figure 1. The other set of relevant bounds are associated with non-standard effects that are largest at weak g * . They are described in terms of the parameters Z, W, Y [35], or equivalently by the operators O 2G , 3 We find that the current main sensitivity arises from LHC dijet searches, which lead to the bound c 2G /Λ 2 < 0.01 TeV −2 [36][37][38]. As shown in figure 1, this is superior to the LEP and LHC limits derived from c 2W and c 2B [10]. This remains the case at the high-luminosity phase of the LHC, even if with more statistics the constraint on W from pp → ν is expected to reach a comparable level to that on Z [39]. Finally, one of the most relevant constraints on CH models, connected as well with the top sector, comes from the CP-violating operator O γ = H † HB µν B µν , with coefficient of one top-loop sizec γ /Λ 2 ∼ g 2 N c (y t /4π) 2 /m 2 * and which itself contributes at one loop to the EDM of the electron. The current constraint on m * , taking the power counting estimate at face value, is m * > 20 TeV at the 95% CL [40,41]. While this is the strongest bound independent of g * , let us note that being CP-violating it is of a qualitatively different nature with respect to the previous ones; for this reason, we do not show it in figure 1.
To conclude this section we would like to stress that our approach is markedly driven by a well-motivated yet specific framework. This is why in figure 1, as well as in the rest of the paper, the constraints on the (m * , g * ) parameter space follow from experimental limits on a single operator at a time, each corresponding to the leading new-physics contribution to the relevant observable(s). This viewpoint is therefore orthogonal to that of global top-sector fits, for which the reader is directed to [42][43][44][45][46]. 2. The ATLAS four-top measurement [23] finds σ tttt = 24 +7 −6 fb, and also observes an excess of ttW events relative to the SM reference, with best fit µ ttW = 1.6 ± 0.3 based on σ SM, R ttW = 601 fb. The normalization of ttZ is not left to vary in the fit.

A dedicated measurement of ttZ
Most measurements [22,23,47,50] are based on ≈ 140 fb −1 , whereas [48,49] use ≈ 80 fb −1 . In addition, we mention but do not discuss further the combined analysis of EFT operators in top associated production modes by CMS [51], as well as previous measurements of ttW , ttZ by CMS [52] and ATLAS [53], which are all based on a smaller data set, ≈ 40 fb The above overview shows that an extensive pattern of 2σ-level excesses with respect to the SM predictions is observed by both ATLAS and CMS. We summarize the status in figure 2, where the above-quoted experimental results are compared with the following SM theoretical predictions in femtobarns The discrepancies are mild, yet their somewhat coherent structure hints that they may not be due to mere statistical fluctuations. 5 Recent theoretical studies have focused on pushing the SM predictions to higher accuracy, especially for ttW [57][58][59][60][61][62][63], which however still exhibits the strongest disagreement between theory and experiment. For the time being, a complete NNLO QCD calculation remains out of reach.
Here we take a different point of view and entertain the possibility that the excesses are due to heavy new physics, described by the two effective operators O tt and O Ht . The former mediates tttt production, whereas the latter modifies the Zt R t R coupling, thereby leading to three distinct effects: it contributes to ttZ production at the leading O(α 2 s α), to ttW + jets at the formally subleading, but tW scattering-enhanced, O(α s α 3 ) [12], and to tttt production at O(α 2 s α 2 ). We consider these two operators as a motivated first approximation, but note that others should be added in a more general analysis that includes e.g. tth production, notably O yt given that this modifies the htt coupling (the sensitivity of tttt to O yt was studied in [64]). We concentrate on the CMS four-top analysis [22] because it provides a cut-and-count version and sufficient pre-fit information for us to perform a detailed, if simplified, reinterpretation.
The analysis selects events containing at least two leptons of the same sign, N j ≥ 2 and N b ≥ 2, H T > 300 GeV and p miss T > 50 GeV, with complete definitions and list of requirements reported in [22]. The cut-based analysis defines two control regions, CRW (where the JHEP02(2021)043 Once normalized to 544 fb [54], the ttW (QCD) sample requires a rescaling factor r ttW (QCD) = 0.92 to match the total CMS MC yield in CRW. After using 839 fb as normalization [55], the ttZ sample is rescaled by r ttZ = 0.75 to match the total CMS MC yield in CRZ.
contribution of ttW is enhanced) and CRZ (where ttZ is enhanced), and 14 signal regions. In our reinterpretation, for simplicity we combine all signal regions into a single one (SR). Signal and background events are generated using MadGraph5_aMC@NLO [65], implementing higher-dimensional operators via FeynRules [66]. The factorization and renormalization scales are set to the default dynamical value for all processes, the top mass is set to 172.5 GeV, and NNPDF23_lo_as_0130_qed parton distribution functions [67] are used. Parton showering and hadronization are performed by Pythia8 [68] and detector effects are parametrized using the CMS card in Delphes3 [69], but setting R = 0.4 for the anti-k t jet clustering algorithm [70], implemented via the FastJet package [71]. As a preliminary check of our simulation tool chain we reproduce the SM ttW (ttZ) event yields in the CRW (CRZ), obtained by CMS with full detector simulation. The results, reported in figure 3, show that after application of mild scaling factors to match the overall normalizations, our simulations reproduce reasonably well the results reported by CMS (where it should be kept in mind that we simulate at LO in QCD, whereas the CMS treatment is at NLO). Having thus gained confidence in our setup, we proceed to include the new physics effects. The impact of O Ht on ttZ production is captured by rescaling the CMS yields using the overall factor (note that we set Λ = v throughout this section) which we have checked to be a good approximation by simulating a set of samples with different values of c Ht (see [72,73] for NLO QCD analyses of the ttZ sensitivity to top electroweak couplings). In addition, we take into account the impact of c Ht on ttW j(EW); this piece was altogether neglected in the CMS simulation of ttW + jets [22]. For the tttt process, our simulation is simplified in two ways: we neglect interference of the O tt -mediated amplitude with the SM, 6 and neglect the contribution of O Ht . We do so because reliably JHEP02(2021)043  assessing these effects at the hadronic differential level goes beyond our computational resources, and besides it would be best performed by the LHC experiments directly. At the qualitative level, we note that the c tt -SM interference is suppressed at high energies, whereas the impact of c Ht on four-top production is generally expected to be moderate, as the tt → tt amplitude does not grow with energy when c Ht = 0, in contrast with the aforementioned case of tW scattering. We provide an estimate of the expected size and pattern of these effects after presenting the results of our fit.
four-top production using the NLO QCD-only cross section of 11.1 fb [54]. However, very recently the SMEFT@NLO framework [74] has enabled the calculation at full NLO in QCD of the contributions of fourtop operators to tttt production (including interference with the SM). In particular, K < 1 was obtained for the O(c 2 tt ) piece. Due to the different scale choices, our approximate-NLO cross section turns out to be numerically very close to the exact-NLO result quoted in [74].

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To shed light on the compatibility of the data with our BSM hypothesis, we form a χ 2 from 18 non-overlapping bins. 7 We use the uncertainties on event counts as read from figures 2 and 3 in [22], averaging over positive and negative directions, and neglect theoretical uncertainties. The results of a two-parameter fit to (c Ht , c tt ) are shown in the upper panels of figure 4, while in the lower panel we show for comparison a fit where no higher-dimensional operators are introduced but the signal strengths (µ ttW , µ ttZ ) are left floating, which is similar to the treatment performed by CMS. The best-fit point of the latter fit is (µ ttW , µ ttZ ) ≈ (1.3, 1.2). We note that the two EFT coefficients parametrizing the effects of heavy new physics provide a reasonable fit to the data, with comparable goodness of fit to the ad-hoc signal strengths. The best fit is given by (c Ht , c tt ) ≈ (0.21, ±0.054), corresponding to scales f | Ht ≈ 540 GeV and f | tt ≈ 1.1 TeV if the respective coefficients are set to unity. The impact of the BSM contributions on the CRW, CRZ and SR are shown in figure 5 taking the best-fit values of the coefficients.
Next, to gain some insight on the effects of the approximations we made in our description of the four-top process, we consider parton level tttt production (with undecayed tops) including the full LO amplitude for the SM plus O tt and O Ht . We split the cross section into a low-energy and a high-energy region according to The boundary value M T = 1.15 TeV is chosen to roughly match H T = 800 GeV at hadronic level, which we have verified splits the SR into two sub-regions of comparable sensitivity in our fit to CMS data (see the bottom panel of figure 5). Equation (3.4) confirms the expectation that at high energies, it is reasonable to neglect all BSM terms except for the O(c 2 tt ) one: for example, plugging in the best fit point, we find σ > /σ SM > = 1.79 whereas our approximation gives 1.86. For the low-energy region, using eq. (3.3) we find σ < /σ SM < = 1.15 versus the approximate value 1.20. This apparently reasonable agreement is, however, actually the result of a compensation between different corrections arising from c Ht and c tt , suggesting that the shapes of our fit contours could be somewhat affected by a fully accurate description of BSM effects in the low-H T bins of SR.
Finally, we remark that O Ht mediates BSM contributions to additional processes, including for instance pp → tthj at O(α s α 3 ) and tZW at O(α s α 2 ). The analysis of such subleading effects was initiated in [12] and later expanded in [75]. Based on their findings we do not expect the O Ht dependence of these and other analogous processes, which is neglected here, to have a significant impact on our results. Nonetheless, a detailed assessment would be of interest to obtain a complete picture of heavy new physics effects in LHC multilepton plus jets final states. In summary, the main messages we derive from the fit are: • The O(α s α 3 ) ttW j(EW) contribution to ttW + jets is important and should be consistently included at the differential level, as originally pointed out in [12] and later analyzed in depth in [54,59].
• An interpretation of the CMS data [22] in terms of the O Ht and O tt operators gives a goodness of fit comparable to the application of constant rescaling factors to the ttW and ttZ cross sections, while having a stronger physical motivation.
• While it is too early to draw any conclusions, it is intriguing that a scale f ∼ 750 GeV improves the fit to multilepton + jets data, while being roughly consistent both with four-top constraints from the single lepton and opposite-sign dilepton final states [11], and with measurements of the Higgs couplings [10].
A more comprehensive study, including a wider set of signal regions, would be strongly desirable to obtain further insight. Nevertheless, we regard the coincidence of scales suggested by our analysis as an additional motivation to further investigate heavy top-philic new physics.

JHEP02(2021)043 4 Future proton-proton colliders
Let us remark that under well-motivated assumptions, current searches for strong tttt production enjoy a higher reach on the compositeness scale 4πf than probes of the Higgs sector at the LHC. This fact motivates our sensitivity studies at future colliders. We begin in this section with hadron colliders, first discussing shortly the high-luminosity phase of the LHC and then analyzing in detail the 100 TeV FCC-hh [15].
To estimate the HL-LHC sensitivity to c tt /Λ 2 we perform a simple extrapolation of the CMS four-top search in multileptons [22]. We focus on the signal region (bottom panel of figure 5), adopting the H T -binning chosen by CMS and rescaling their MC predictions for all SM processes to a luminosity of 3 ab −1 . As in the previous section, we include the missing SM ttW j(EW) contribution among the backgrounds and simulate the signal neglecting interference with SM four-top production. Assuming as systematics on the two main backgrounds (δ tttt , δ ttW ) = (8.5%, 7.5%), which correspond to half the current theoretical uncertainties [54], and applying a mild PDF rescaling factor [76] to account for the increase of collider energy to 14 TeV, we obtain at 95% CL Λ/ |c tt | > 1.3 TeV (no syst.: 1.4 TeV) .
(HL -LHC) (4.1) We view this as a conservative estimate, as the actual HL-LHC analysis will capitalize on the ≈ 20 times larger statistics by refining the binning at larger H T , thus increasing slightly the sensitivity. 8 Furthermore, a caveat is that we have assumed agreement of data with the SM predictions, although as discussed in section 3 this is somewhat unclear for current multilepton measurements. We now turn to the analysis of the four-top final state at the FCC-hh. The decays of the four tops give rise to a complex set of possible final states. The same-sign dileptons (SSL) and trileptons (3L) signatures both benefit from suppressed SM backgrounds, while retaining not-too-small branching ratios of 4.1% and 2.6%, respectively. These numbers do not include contributions from leptonic τ decays, which are systematically neglected in our FCC analysis (whereas they are always included when we quote LHC results). 9 Conversely, the fully hadronic signature has a large branching ratio of 20%, but suffers from challenging backgrounds. In this work we thoroughly analyze the SSL and 3L signatures, and perform an exploratory study of the fully hadronic final state.
For the SSL and 3L final states we partly build on the latest LHC searches for four-top production in multilepton + jets [22,23], and on the LHC study [77] which focused on SM tttt production and BSM effects mediated by relatively light new physics (see also [78] for a thorough analysis of resonant signals in the four-top final state at the LHC). Signal and background events are generated using MadGraph5_aMC@NLO [65], using a Feyn-Rules [66] model where O tt is added to the SM. The factorization and renormalization scales are set to M T /2 for all processes, where M T is the sum of transverse masses. The signal 8 In addition, rescaling the current statistical-only 95% CL bound Λ/ |ctt| > 0.93 TeV using Collider Reach [76] would give an estimate of 1.7 TeV at the HL-LHC. 9 As taus dominantly originate from W and Z decays, they give approximate equal contributions to both signal and backgrounds, hence neglecting them makes our FCC results slightly conservative.

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samples only contain the O(c 2 tt ) contribution, as interference with the SM tttt amplitude is a small effect in our signal region; we provide a quantitative assessment of this at the end of the section. The SM four-top production is simulated at full LO, namely O(α i s α j ) with i, j ∈ {0, . . . , 4} and i+j = 4, while as normalization we use the complete NLO (QCD+EW) calculation of [54]. The normalization of the signal is rescaled by the ratio of the NLO (QCD+EW) and LO QCD cross sections as calculated for SM production, which equals 1.8.
Parton showering and hadronization are performed by Pythia8 [68] and detector effects are parametrized using Delphes3 [69] adopting the FCC card. Within Delphes, jets are clustered with the FastJet package [71] using the anti-k t algorithm [70] with R = 0.5. The b-tagging performance is described through the following efficiencies,  .2), χ denotes the characteristic function. As our signals feature highly boosted tops, as well as a generally large amount of hadronic activity, we apply lepton isolation using a variable cone, following the mini-isolation proposal [79]: an electron (muon) is said to be isolated if p cone and we employ NNPDF23_lo_as_0130_qed parton distribution functions [67].

Same-sign dileptons
In this channel, the main background beyond the irreducible SM tttt is the production of ttW + jets, which is in fact also primarily measured in the SSL final state. Secondary backgrounds with genuine SSL include ttZ and tth, as well as some other processes listed in table 2, together with the MC generation-level cross sections. In all cases we generate processes giving rise to at least a SSL pair and four jets at the matrix element level; for a few backgrounds, we are able to include additional jets within computing limitations. Some important processes, including SM tttt and ttW production, are normalized to the best available predictions that include both QCD and EW corrections [54,80,81].
In addition, there are important reducible backgrounds: either a jet is mis-identified as a "fake" lepton, or one lepton belonging to an opposite-sign pair has its charge mismeasured (Qflip); both of these originate mainly from tt + jets. The fake lepton component can be estimated by applying a probability for a given jet to be misidentified as a lepton (in general, the probability depends on the jet flavor and p T ), and a transfer function relating the properties of the daughter lepton to those of the parent jet [82]. The probability and transfer function parameters need to be tuned against data. Here we follow a simplified approach, assuming a constant probability for both heavy flavor and light jets, and that JHEP02(2021)043 the fake lepton inherits the full four-momentum of the jet it originates from, whereas the lepton charge and flavor are assigned randomly and independently. The probability is fixed to ε fake = 3.7 × 10 −5 by comparing a sample of 13 TeV semileptonic tt + jets, normalized to a cross section of 832 pb [83], to the "nonprompt" yields in the control region CRW of the CMS four-top search [22] (see left panel of figure 5). 10 The Qflip component is estimated from MC events containing an e + e − or e ± µ ∓ pair and applying a constant probability for the charge of each electron with p e T > 10 GeV to be mismeasured (the probability of flipping the charge of a muon is negligible). The probability ε flip = 2.2 × 10 −4 is taken from [77] and further validated by checking that a 13 TeV fully leptonic tt + jets sample reproduces the "charge misID" yields in the control region CRW of [22]. The processes we include in our estimates of the fake lepton and Qflip backgrounds at the FCC are listed in table 2.
We now turn to the event selection. First, we identify the lepton and jet candidates satisfying Next, to prevent assignment of a single detector response to both a lepton and a jet, we apply to the selected candidates an overlap removal procedure, following closely [11]. To avoid the double counting of energy deposits as electrons and jets, for each electron the closest jet within ∆R < 0.2 (if any) is removed; however, if the next-to-closest jet is within ∆R < 0.5 of the electron, then the electron is removed and the previously removed jet is reinstated. For muons we apply a different criterion, aimed at distinguishing muons arising from hadron decays within proper jets, from muons that undergo bremsstrahlung radiation inside the calorimeter and are accidentally reconstructed as jets, typically characterized by a very small number of matching tracks. If a jet satisfies ∆R(µ, j) < 0.04 + 10 GeV/p µ T and it has at least three tracks, the muon is rejected; otherwise, the jet is removed. The baseline selection is completed by the following requirements, exactly two SSL with p 1 , 2 T > 40, 25 GeV , where 1 ( 2 ) denotes the (sub)leading lepton. We expect that the above requirements on the lepton transverse momenta will allow for a high efficiency of an FCC-hh dilepton trigger. At this stage, for a reference BSM scale Λ/ |c tt | = 6 TeV, we have S/B ∼ 10 −3 , as shown in table 3. Therefore we search for additional cuts tailored to the signal, which is characterized by a hard tt → tt scattering. We find as optimal variables the p T of the leading lepton and S T , defined as the scalar sum of the transverse momenta of the SSL pair and of all jets. Normalized distributions of these variables after the baseline selection are shown for the signal and the main backgrounds in figure 6. We apply the cuts p 1 T > 275 GeV and S T > 3 TeV, and divide the remaining events into three S T bins, with S/B ranging from 8 × 10 −3 to 5 × 10 −2 . To derive a bound on Λ we construct a χ 2 , accounting for the systematic uncertainties on the two main SM backgrounds, namely tttt and ttW . For n JHEP02 (2021) Table 2. SSL signal and background processes at √ s = 100 TeV. Samples with different jet multiplicities have been merged using the MLM prescription with matching scale of 30 GeV. The cuts p j T > 50 GeV and |η j | < 5 are imposed on jets arising from QCD radiation, but no cuts are applied yet to decay products of heavy particles. The subsequent baseline selection, discussed in section 4.1, requires ≥ 5 jets among which ≥ 3 are b-tagged. The higher-order cross sections we use for normalization always assume µ = M T /2 (note that in [54] this is not the central choice for tttt). The † indicates that p j T > 100 GeV was exceptionally required, to match [81] (we have checked that this different initial cut has negligible impact on the event yield after the complete selection). Whenever they do not appear in tt or bb pairs, the symbols t and b refer to either particles or antiparticles. To the tt + jets samples used to estimate the fake lepton and Qflip backgrounds we apply a K tt = 1.4, calculated for inclusive tt production using the NNLO cross section of 34.7 nb [84]. bins, the χ 2 is defined as where the number of signal events N S i ∝ c 2 tt Λ −4 , and   with σ stat i = N SM i , σ sys,A i = δ A N A i and we assume that each systematic uncertainty δ A is fully correlated across bins, namely ρ ij = 1 for all i, j. We have also assumed that the observed number of events will match the SM expectation. We take (δ tttt , δ ttW ) = (8.5%, 7.5%) as reference values, obtained by halving the current theoretical uncertainties on the SM predictions [54]. The resulting 95% CL bound with L = 30 ab −1 is Λ/ |c tt | > 6.1 TeV (no syst.: 6.9 TeV) .

Trileptons
In the trilepton channel the main backgrounds are the irreducible SM tttt production and the ttW , ttZ, tth+ jets processes. The full list of backgrounds we consider is given in table 4, together with the MC generation-level cross sections. We generate processes giving rise to three leptons and at least four jets at the matrix element level. The fake lepton background is generated using the same method as in the SSL analysis of section 4.  [80] other tZbjj W ± Z + − 145 Table 4. 3L signal and background processes at √ s = 100 TeV. Samples with different jet multiplicities were merged using the MLM prescription with matching scale of 30 GeV. The cuts p j T > 50 GeV and |η j | < 5 are imposed on jets arising from QCD radiation, but no cuts are applied to decay products of heavy particles. The subsequent baseline selection, discussed in section 4.2, requires ≥ 4 jets among which ≥ 3 are b-tagged. The higher-order cross sections we use for normalization always assume µ = M T /2 (note that in [54] this is not the central choice for tttt). The † indicates that p j T > 100 GeV was exceptionally required, to match [81] (we have checked that this different initial cut has negligible impact on the event yield after the complete selection). Whenever they do not appear in tt or bb pairs, the symbols t and b refer to either particles or antiparticles. To the tt + jets samples used to estimate the fake lepton background we apply a K tt = 1.4. a different set of processes. The Qflip background is negligible, since no requirement is imposed on the lepton charges. The event selection is analogous to the one for SSL: after lepton and jet candidates are identified as in eq. (4.4), we apply the same overlap removal procedure. In addition, the baseline selection requires exactly three leptons with p T > 25 GeV , and events where among the three leptons appears one opposite-sign, same-flavor lepton pair satisfying |m + GeV are vetoed, to suppress backgrounds containing a leptonic Z decay. The requirement of three leptons with p T > 25 GeV should allow for a straightforward triggering on these events. Notice that these selection requirements are orthogonal to those of the SSL analysis, which will ease the combination of the results.

Same-sign dileptons and trileptons combination and discussion
We now combine the results in the SSL and 3L final states, by considering a joint χ 2 with 6 orthogonal bins. We obtain Λ/ |c tt | > 6.5 TeV (no syst.: 7.3 TeV) , (FCC-hh, SSL + 3L) (4.11) from L = 30 ab −1 and with the reference systematic uncertainties (δ tttt , δ ttW ) = (8.5%, 7.5%). The impact of varying these uncertainties is shown in the left panel of figure 8; we stress that we assume full correlation of each uncertainty across bins. In the right panel of figure 8 we display the dependence of the combined bound on the fake lepton probability, whose value at FCC-hh is unknown and which we have fixed based on a fit to LHC data. The Qflip background affects only the SSL analysis and is 3 - than the fake lepton background in our benchmark scenario, so its impact remains small for any reasonable choice of the electron charge-flip probability ε flip .
In addition, we want to ensure that our bounds arise from regions of phase space where the EFT expansion is under control. For this purpose we show in figure 9 the combined SSL + 3L bound on Λ/ |c tt | obtained by discarding events where the largest parton-level invariant mass of a top quark pair is larger than m * , which represents the mass of new resonances. Since it is not possible to tell on an event-by-event basis whether the hard scattering involved a tt, tt, ortt pair, we make the conservative choice to discard events where the largest invariant mass of any such combination is larger than m * .
We now return to the role of the interference between the signal and SM tttt amplitudes. To quantify it, it is enough to work at the parton level, hence as rough proxy for our signal region we consider the process pp → tttt, followed by SSL decays and including the cuts where the coefficients are obtained by fitting to a set of cross sections calculated for varying c tt /Λ 2 , and the uncertainties on the SM and O(c 2 tt ) terms are negligible compared to the one on the linear term. This result confirms that interference can be safely neglected. The same conclusion applies to the 3L final state.
Finally, in figure 10 we show the impact of our combined SSL + 3L bound, eq. (4.11), on the (m * , g * ) parameter space of CH models, and compare it with other, complementary probes which will become available throughout the development of the FCC program.
Strikingly, four-top production at the FCC-hh provides the dominant sensitivity on the compositeness scale, f | FCC tt 6.5 TeV, outperforming tests of Higgs coupling deformations associated with O H , as combined in [85] which includes input from the HL-LHC and the FCC-ee, -eh, and -hh phases, resulting in f | FCC H 4.2 TeV at 95% CL. In addition, we show the projected constraint on O W , O B [85], namely m * > 17 TeV at 95% CL, as well as the expected FCC-hh bounds on O 2W and O 2G , derived from charged-and neutral-current dilepton production [85] (see also [39]), and high-energy dijet and inclusive jet production [38], respectively. These observables dominate the sensitivity for moderate strength of the new-physics coupling g * . Finally, we mention that strong constraints are also expected from CP-violating observables: the limit on O γ from the future measurement of the electron JHEP02(2021)043 EDM by the ACME III experiment [86] reaches m * > 115 TeV at 95% CL. However, this probe is left out of figure 10 due to its inherently different nature, as already done in figure 1.

Fully hadronic final state
Finally, we turn to the signature that arises when all four tops decay hadronically. This channel benefits from a large branching ratio of 20% and is intrinsically interesting because at the FCC-hh the hadronic tops will frequently possess multi-TeV transverse momenta, entering a kinematic regime that is only marginally accessible at the LHC (for which the fully hadronic signature was discussed in [78], albeit assuming a resonant signal). While this happens already in the SM, the relative importance of ultra-boosted tops increases further in the presence of heavy new physics that generates O tt . To obtain a first estimate of the reach, we perform a crudely simplified analysis that requires four top-tagged jets, relying on the performance of existing hadronic top tagging methods developed for the LHC, as studied by CMS [87]. As a first step, we generate the signal and the main backgrounds, which are tttt, ttjj and jjjj production in the SM, at parton level with a p T > 200 GeV cut on each undecayed top or jet. The interference between the BSM and SM four-top amplitudes is neglected, since we are interested in the high-energy regime. We then include the branching ratios for hadronic top decays and apply, on an event-by-event basis, the p T -dependent efficiencies and mistag rates extracted from [87]. 12 Finally, we select highly energetic events by requiring the total invariant mass of the four final-state objects to be larger than 5.5 TeV and the sum of the transverse momenta to be larger than The estimate eq. (4.13), although obtained by means of rough approximations, indicates a promising potential for the fully hadronic channel. However, requiring a large p T for all four tops, as necessary in order to apply the results of [87], severely suppresses the signal rate, ultimately limiting the sensitivity. This motivates pursuing a different strategy, where the two hardest tops are tagged using jet substructure algorithms whereas the two softest tops are identified from their resolved decay-product jets; this is in consonance with the topology of our signal, which is characterized by a high-energy tt → tt scattering mediated by O tt . The challenge of this approach is to retain a strong rejection capability against the ttjj background, in particular the configuration where the two tops have larger p T 's than the light jets', in which case the signal/background discriminating power must be obtained from the "soft" component of the event.
To study this problem, we generate SM four-top production and ttjj with hadronic top decays, using the MadGraph5_aMC@NLO -Pythia8 -Delphes3 chain. All final-state JHEP02(2021)043 partons are required to have p T > 100 GeV, whereas the leading (subleading) jet is required to have p T > 900 (800) GeV and the H T must exceed 2 TeV. The only notable setup differences compared to the multilepton analyses are that we use the default Delphes card and set R = 0.3 for the (anti-k t ) jet clustering, because using such narrower jets allows for a more efficient matching of the hadron-and parton-level objects, therefore easing the isolation of a ttjj sub-sample containing the configuration where the light jets are softer than the tops (which happens for O(10)% of the events). Two different strategies are investigated to separate this background from SM tttt production: one based on top invariant mass reconstruction, and one employing a neural network discriminant.
For the first strategy we implement an algorithm which first removes the two hard jets that are matched to partonic tops, and then identifies two sets of up to three jets each, whose invariant masses are closest to m t , with each set required to contain at least one b-tagged jet. This method results in an 8% efficiency per event on SM four-top production and 0.4% on the background. For the second strategy we use the same MC samples to train a three-layer neural network with 2910 neurons per layer, which takes as input features the p T , η, φ, mass, number of tracks, and b-tag flag of up to 26 jets with p T > 50 GeV in each event (including, in particular, the two hard jets which are matched to tops), as well as information on possible additional particles such as taus and photons, and on missing transverse energy. At the optimal threshold value the efficiency on SM four-top production is 16%, significantly higher than for the mass reconstruction procedure, but this comes at the price of a less effective background suppression of 4%. The above efficiencies are obtained neglecting systematic uncertainties.
Unfortunately, neither approach yields a satisfactory combination of signal efficiency and background rejection, resulting in weaker bounds on Λ/ |c tt | than the estimate in eq. (4.13). Nevertheless, we believe that our attempts have only scratched the surface of the fully hadronic four-top final state, while uncovering some of the main obstacles that need to be overcome. The sensitivity of this channel is thus still waiting to be untapped, for instance through the development and application of FCC-tailored and/or machine learning-based top tagging algorithms (see e.g. [88][89][90]) that encompass both the resolved and boosted top regimes. Judging from our preliminary estimates, this channel has the potential to give the strongest constraint on the new-physics scale at the FCC-hh, further improving on our multilepton results.

Future electron-positron colliders
In this section we show that future leptonic machines have much to inform on the fate of a strongly-interacting top quark. The colliders under consideration are CLIC [16], the International Linear Collider (ILC) [91], and the FCC-ee [15]. We will not be carrying out any new analysis towards the extraction of their sensitivity to the dimension-six effective operators of interest, since this has been the subject of a number of detailed and comprehensive studies. Instead, we merely yet crucially reinterpret the relevant results in terms of the expected effects associated with a strongly-interacting (right-handed) top quark, in JHEP02(2021)043 particular via the four-top operator eq. (1.1). 13 The different collider specifications can be found in the pertinent works: [93] for what regards the top sector, and [85] concerning universal effects, which we use to draw a comparison of both types of probes in the context of composite Higgs models. The runs from where most of the sensitivity to a composite top comes from are those at the highest energies: √ s = 3 TeV (L = 3 ab −1 ) at CLIC, √ s = 1 TeV (L = 1 ab −1 ) at ILC, and √ s = 365 GeV (L = 1.5 ab −1 ) at FCC-ee. 14 The reason for this is that at linear colliders the best process to probe such type of physics is top-pair production, e + e − → tt. In our new-physics oriented analysis we find that the largest effects are associated with the four-fermion operators c te where both e R and L correspond to first-generation leptons. Since we consider a negligible degree of lepton compositeness, as motivated by their small Yukawa couplings, the largest contribution to the coefficients in eq. (5.1) arises from operators of the form of eq. (2.3), in particular from O tD which, given the equation of motion eq. (2.5), yields c te = g c tD and c t = g c tD /2. What is important to notice is that at the relevant scale, µ = √ s, the coefficient of O tD is dominated by the RGE contribution from the four-top operator O tt , for a mildly strong coupling g * , since recall c tD /Λ 2 ∼ g /m 2 * and c tt /Λ 2 ∼ g 2 * /m 2 * at the scale m * , where the coefficients are generated. Therefore, a strongly-interacting (right-handed) top quark leads to a new-physics amplitude that scales like for m * = 4πf (to fix the size of the logarithm in eq. (5.3)). This is stronger than the expected sensitivity to be achieved in Higgs measurements via the operator O H , f | CLIC H > 4.3 TeV [85], also shown in figure 11 in the (m * , g * ) plane. At the ILC the sensitivity via the four-top operator is comparatively lower, f | ILC tt > 4.1 TeV (c te /Λ 2 < 7 × 10 −4 TeV −2 [93]), yet similar to that from the Higgs. Finally, the importance of high collision energies for this type of probes is reflected in FCC-ee bounds on c te , c t , which are approximately an order of magnitude weaker, yielding a significantly lower sensitivity f | FCC-ee tt > 1.6 TeV (c te < 4.3 × 10 −3 TeV −2 [93]), see figure 11. 13 To some extent, our analysis resembles that of [92]. However, as in previous sections, we focus on a single operator at a time, the one leading to the largest sensitivity in a given region of the (m * , g * ) parameter space, which is not always the same operator as claimed in that study. Besides, by considering exclusive constraints, we avoid issues associated with cancellations from different operators in a given observable.
14 Notice the mildly different assumptions made for the luminosities and energies of these machines in [93] and [85]. 15 The experimental sensitivity to O t is similar, but we neglect it in setting the limit because c t = cte/2.

JHEP02(2021)043
Let us note at this point that our analysis of one operator at a time must be interpreted with a certain care, particularly in the case where several operators enter a given process. For instance, while the one-loop contribution from O tt gives the leading non-standard effect in e + e − → tt at large g * , for small new-physics couplings other operators become comparable and eventually dominate, in particular the finite contribution to O tD generated at m * , see eq. (5.2) (loops from other four-top operators in eq. (2.1) are not enhanced by the strong coupling and thus always subleading). This implies that in the transition region cancellations could take place, reducing the sensitivity to new physics. Fortunately, this is not an issue that prevents us from probing those regions of parameter space, since they are tested in other processes via independent operators; specifically, tests of the operator O W +B in electroweak precision data are expected to provide at CLIC the bound m * > 19 TeV at 95% CL [85], independent of the new-physics coupling. The same holds at ILC and FCC-ee, even though, as shown in figure 11, the sensitivity to the resonance scale is somewhat lower.
The power of tests of the top sector in covering the parameter space of CH models goes beyond top-pair production. As already noted in [92], production of left-handed bottom pairs at lepton colliders is sensitive to effects that are enhanced at weak coupling, for instance via the operator O (3) qD in eq. (2.4) with c (3) qD /Λ 2 ∼ (y t /g * ) 2 g/m 2 * , which from the equations of motion contributes as a contact term to the amplitude for e + e − → bb. As we show in figure 11, this is superior to electroweak precision tests in the form of the W parameter, to be measured in e.g. e + e − → µ + µ − , because the coefficient of O (3) qD is enhanced by the top Yukawa coupling, c (3) qD /(g c 2W ) ∼ (y t /g) 2 , while the experimental precision in the two processes is expected to be comparable. In addition, it is worth noting that at CLIC and ILC, bottom-pair production could provide a non-negligible sensitivity to the masses of the composite resonances, independently of g * , because of RGE effects associated with the four-top operator O tq in eq. (2.1), c qD (µ) = c qD (m * ) + c tq (m * ) g 12π 2 log m 2 * µ 2 . (5.5) Given that c tq /Λ 2 ∼ y 2 t /m 2 * , we find m * | CLIC tq > 6.5 TeV, a significant constraint, yet weaker than the sensitivity to be achieved from the S parameter (O W +B ).
Let us finally comment on the potential sensitivity from measurements of anomalous top and bottom couplings to the Z boson. Under our assumptions, both the corrections to the Zt R t R and Zb L b L couplings, dominated by O Ht and O Hq + O (3) Hq respectively, do not receive large tree-level contributions, being protected by a P LR symmetry. This then implies that the dominant contributions arise from the RGE associated with where we set c Hq (m * ) + c (3) Hq (m * ) 0 and neglected gauge coupling terms, which are relatively suppressed by (g/y t ) 2 [94]. We find that the expected precision on these couplings [85,93] is not high enough to give rise to any constraint at the level of those already JHEP02(2021)043 at the experimental level, where a complete modeling of the impact of higher-dimensional operators can be achieved. A well-motivated set would include, beyond O tt and O Ht , the operator O yt , which controls non-standard contributions to the htt coupling; these three operators are weakly constrained by other measurements yet their coefficients are expected to be large. 16 In general, the current status of top data provides additional motivation to investigate the new-physics scenarios discussed in this work.
Looking ahead, our FCC-hh analysis of O tt in multilepton final states can be repurposed to derive the reach on other four-top operators, which may play a central role under different theoretical assumptions. On the other hand, exploiting the whole potential of the fully-hadronic signature requires a targeted study. Furthermore, the indirect sensitivity attainable at a multi-TeV muon collider remains to be explored [95].
Finally, we stress the importance of our results for composite Higgs models, where minimal fine-tuning and electroweak precision data point towards a fully-composite righthanded top quark. With the scales of compositeness that we have shown are to be reached, future high-energy colliders will push the concept of naturalness of the electroweak scale to a whole new level, perhaps one where the SM is no longer. 17