Revisiting the $t\bar{t}hh$ channel at the FCC-hh

The exploration of the scalar sector of the Standard Model is at the core of current and future science programs at collider experiments, with increasing focus on the self-interaction of the Higgs boson. This important parameter of the Higgs sector can be measured in various channels, among the production of a Higgs boson associated with a top-quark pair, $\bar{t}thh$. In this paper we study this channel and its potential to measure or constrain the self-coupling and possible new physics contributions at a future 100 TeV proton-proton collider. Analysing this highly complex final state adds to the sensitivity for enhanced self-coupling interactions, and we argue that a measurement of this process is a necessity to constrain blind directions in the multi-dimensional parameter space of well-motivated new physics scenarios.


Introduction
After the discovery of the Higgs boson [1,2] with a mass of m h 125 GeV, the focus of contemporary and future particle phenomenology shifted to the determination of its properties. Early on, spin and CP properties have already been found to be in extremely good agreement with Standard Model expectations [3][4][5][6], and the couplings of the Higgs boson to other particles, in particular the heavy gauge bosons and the third-generation fermions, are increasingly precisely measured [7][8][9], thereby reducing the parameter space for extensions of the Standard Model [10][11][12][13]. This leaves the form and parameters of the Higgs potential, and in particular the Higgs self-coupling, as the experimentally least constrained sector of the Standard Model. It is therefore not surprising that measurements of or constraints to the triple-Higgs coupling are one of the center pieces of ongoing efforts for the high-luminosity run of the LHC and an important part of particle phenomenology at possible future collider experiments. If the Higgs self-coupling is the only modification to the Standard Model, various ways have been proposed at existing and future colliders to search for this interaction. These approaches can be classified into three categories: in processes without Higgs bosons in the final state, electroweak precision observables can set a limit to λ hhh [14][15][16]; higher-order corrections in single-Higgs production processes [17][18][19] constrain the Higgs self-coupling; and double-Higgs production processes will provide direct sensitivity on this coupling in upcoming LHC and possible future high-energy collider runs [20][21][22][23][24][25], while the latter are expected to provide the best sensitivity on λ hhh during the LHC's high-luminosity runs.
Within the class of multi-Higgs production processes, the overwhelming focus to date was directed towards the channel with the largest cross section, i.e. Higgs-boson pair production in gluon fusion, while other channels, such as Higgs-pair production in association with other particles, e.g. pp → hhjj [26][27][28] or pp → tthh [29,30], have been somewhat neglected. In Refs. [29,30] it has been found that the tthh channel at the 14 TeV high-luminosity run of the LHC may provide welcome additional statistical power for a determination of the trilinear Higgs coupling. The feature that sets this channel apart from the gluon-induced Higgs-pair production process or the weak-boson induced production of hhjj arises due to the absence of a reduced cross section for large values of λ hhh [31]. Thus, tthh could be particularly useful in setting a stringent limit to enhanced self-interactions of the Higgs boson.
A further motivation to measure tthh final states arises when modifications of Higgs interactions originate in models where the Higgs field is realised in a nonlinear way, e.g. composite Higgs models [32][33][34]. There, the tth and tthh couplings are decorrelated [35,36], leading to a blind direction in the parameter space of effective operators when only probing them through the top-associated single Higgs production process, pp → tth. Thus, to rule out such a scenario conclusively, measuring the tthh process during future LHC runs or at future colliders is not optional but a necessity.
In the present work we revisit the proposal of [29] by extending it to the potential future FCC-hh 100 TeV pp collider and including the study of contributions from effective tthh interactions. We will focus on the scenario where both Higgs bosons decay into bottom quarks while one of the top quarks decays leptonically and the other hadronically. Owing to the increase in energy, we will see that this channel is competitive with various other di-Higgs channels [25,[37][38][39][40][41][42] in constraining the trilinear Higgs self-coupling at the 100 TeV collider. This increase in cross section due to energy is of course also a feature of the backgrounds, and we therefore substantially increase their discussion.
Using the formalism of effective field theories, with a strongly-coupled UV completion in mind, in Sec. 2 we describe why obtaining a direct measurement of the tthh final state at current or future colliders is of importance to obtain meaningful constraints in the top-Higgs sector. In Sec. 3 we describe the technical framework used. The analysis steps, reconstruction efficiencies and kinematic features of the signal and the background are detailed in Sec. 4. Finally, we offer our conclusions in Sec. 5.

Effective Field Theory formalism
Extensions of the Standard Model can lead to various modifications of Higgs interactions. Some of the most popular are composite Higgs models, which assume that the Higgs boson is a pseudo-Nambu-Goldstone boson of a strongly coupled UV completion. The most general effective field theory that describes the low-energy effects of a strongly-coupled embedding of the Standard Model is the electroweak chiral Lagrangian (ewχL) [43][44][45][46][47]. Here, the SU (2) × U (1) symmetry is non-linearly realised, with the Goldstone bosons φ a (a=1,2,3) and the Pauli matrices σ a . After introducing a scalar field that transforms linearly under the custodial symmetry, the Lagrangian contains 1 Focusing on contributions of effective operators to the top-Higgs sector we find 5 operators to be of imminent importance, i.e., the ones associated with the coefficients k g , k 2g , c, c 2 and d 3 . While k g and c can be constrained in various single-Higgs production processes, e.g. gluon-fusion, vector-boson fusion or top-associated single-Higgs production, the coefficients k 2g , c 2 and d 3 rely at leading-order predominantly on double-Higgs production processes to be tensioned with data. Thus, to overconstrain the parameter space of L ewχ it is necessary to access as many double-Higgs channels as possible, i.e. pp → hh, pp → hhjj and pp → tthh. The process pp → tthh is of particular relevance to constrain c 2 , as it is the only process of appreciable cross section where this coefficient can be constrained at tree-level. The Feynman diagrams showing the modified vertices are shown in Fig. 1.  Figure 1. Feynman diagrams [49] showing the impact of the three effective vertices, viz., hhh, tthh and gghh.
In this paper, we work with the simplified Lagrangian Table 1, we show the relations between the various bases. For reference, we also include the relationship with the SILH basis [36,48,50,51], which corresponds to a linearised sigma model.
We show the ratio of signal cross-sections with respect to the SM expectation as a function of κ λ , κ tthh and κ gghh in Fig. 2. One can see that the cross-section increases with λ > λ SM . We validated our setup by checking this ratio (σ/σ SM ) at 14 TeV with Ref. [31]. It is interesting to note that the nature of the growth of the cross-section for negative values of λ changes significantly upon going from a 14 TeV machine to a 100 TeV machine. In the SILH Lagrangian, the coupling modifying the gghh coupling also contributes to the ggh coupling. Allowing for a 10% modification in the 14 TeV gg → h cross-section, we find that κ gghh is very strongly constrained. Thus, in the analysis, we only consider the couplings modifying the trilinear Higgs coupling and the tthh vertex.

The Monte Carlo Setup
The final state in our analysis results from the decays of the two top quarks and the two Higgs bosons into six b-tagged jets, one isolated lepton, missing transverse momentum, and at least two extra jets which are not b-tagged. This leaves a wide range of backgrounds to be considered, see below. In all channels, potential additional jets may give rise to required final state particles, either by jet radiation mimicking the light jets stemming from the hadronically decaying top quark, or by gluons 2 The Higgs potential in the SM can be written as splitting into b-quark pairs, yielding b-tagged jets. In addition, light and charm jets may produce fake b tags. Apart from our signal ttHH we therefore include as backgrounds processes where • Z and Higgs bosons decay into b-quark pairs, such as ttZZ and tthZ; • one or both pairs of b-quarks are produced through gluon splittings in QCD, like tthbb, ttZbb, or ttbbbb; • the leptonically decaying top quark is mimicked by a W plus four b-jets, such as W ± bbbb+jets; and • sub-dominant or fake backgrounds which can contribute to the total background yield, for example ttcccc and W ± cccc, misidentifying charm as b-quarks, or ttbb, tth, ttZ, and W ± bb, all associated by light jets.
Due to their complexity and their large final-state multiplicity we chose to simulate signal and backgrounds with leading order matrix elements and consistently combine Process category Table 2. Renormalisation and factorisation scales used for the various processes them with subsequent parton showers, to capture the effect of QCD radiation and, in particular, of gluon splittings into heavy quarks. We use SHERPA v2.2.5 [52,53] with the COMIX matrix element generator [54] and the parton shower based on Catani-Seymour splitting kernels [55,56]. The central CT14NLO PDF set [57] is used throughout. All jets contributing to the process classification including b and c-jets, are defined with the anti-k T algorithm [58] with We also require a minimal invariant mass m bb/bc/cc ≥ 50 GeV for all possible b and c pairs. Where necessary, we add matrix elements for final state with more jets through multijet merging according to [59][60][61], and a merging cut of Q cut = 40 GeV.
For the various processes we use the renormalisation and factorisation scales listed in Table 2, where for merged samples we cluster back to the relevant core process before determining the scales. Details of the generation cross-sections for all signal and background samples are listed in Tab. 3.

Analysis
In Ref. [29], the tthh channel was studied in the context of the 14 TeV high-luminosity run of the LHC. In the present work we revisit this analysis by focussing on the prospects of observing this channel at a possible future FCC-hh 100 TeV pp collider. While our analysis strategy here is reminiscent of Ref. [29], we study the backgrounds in significantly greater detail.
At leading order, the pp → tthh cross-section increases by a factor ∼ 75 for the SM scenario upon going from 14 TeV to 100 TeV. However, from Fig. 3, we can see that the large increment in the total cross-section does not translate into a significantly enhanced distribution for large transverse momenta. Hence, any advantage in the analysis is due to the increased total rate and to an improved reconstruction efficiency for highly energetic final-state objects.
For the b-tagged jets we demand that the distance between B-hadron and jet center fulfils ∆R j,B < 0.2 (4.2) and that |y b | < 2.5 . For the 100 TeV collider study, we assume a b-tagging efficiency of 80%. To estimate the effect of fake b-tags, we assume a mistagging probability of 10% for c-jets and of  1% for light jets. We require exactly one lepton in each event with p T ( ) > 10GeV and |y | < 2.5 . To isolate the leptons we demand that the total hadronic activity around a cone of ∆R = 0.3 to be less than 10% of its p T . From Tab. 3, it is clear that some of the backgrounds are much larger in crosssections than others. We note that the tth+jets process already contains tthbb. The same is true for ttZ+jets and ttZbb. Thus, in order to avoid double counting, we focus on the tth+jets and tt+jets channels. For the tth/Z+jets, we consider the merged sample, where additional jets stem from QCD radiation, including the gluon splitting into b-quark pairs. In order to ensure that none of the additional jets contains more than one B-mesons, we implement a further criterion, namely that the B-hadron closest to the jet axis satisfies This condition reflects the b-quark fragmentation, which is characterised by relatively low energy or momentum losses due to QCD radiation and, correspondingly, the fact that B-hadrons in a jet stemming from a b-quark carry the dominant fraction of the jet momentum. Obviously, this is not true for those jets, where a gluon splits into two b-quarks, which typically have relatively symmetric momentum fractions. Consequently, this criterion effectively suppresses "doubly-tagged" b-jets, which contain two b-hadrons. b-tagged jets failing to fulfil this criterion are considered as light jets, and we call those jets that that fulfil it "good" b-jets. We therefore require events with exactly 6 good b-tagged jets and all pairs of b-tagged jets must have an invariant mass greater than 50 GeV. We convinced ourselves that these additional conditions on "good" b-jets render the effect of gluon jets tagged as b-jets due to gluon splitting negligibly small. In order to ascertain that the events are ensuing from a tthh topology, reconstructing most of the electroweak resonances is of essence. We follow Ref. [29] and define our two Higgs boson candidates by minimizing where i = j = k = l run over all the 6 b-tagged jets. As parameters for this minimisation we use m h = 120 GeV and ∆ h = 20 GeV. The strange choice for m h warrants an elucidation. Because the Higgs bosons decay to b-quarks which essentially hadronise to B-mesons, the invisible decays of the latter shifts the reconstruction of the Higgs peak to smaller values. A different value of m h can be chosen after correcting explicitly for jet energy effects in b-jets. After minimising χ 2 HH , we require The reconstructions of the hardest and the second hardest Higgs bosons are shown in Fig. 4 (left).   After finding the two Higgs bosons, we are left with two b-tagged jets. Because of the uncertainty in the longitudinal momentum p z of the neutrino, we only reconstruct the hadronic top, t h . We consider the two remaining b-tagged jets and all other light jets to minimise the following χ 2 .
where i runs over the remaining two b-tagged jets and k = l denote the indices of all the light jets. For this minimisation, we take m t = 170 GeV and ∆ t = 40 GeV. We allow for a larger uncertainty as we demand the hadronic top to be reconstructed from three jets. After minimising Finally, instead of fully reconstructing the leptonic top, we reconstruct the invariant mass of the last b-jet which is neither part of the two Higgs reconstructions nor of the hadronic top and the single isolated lepton. We further require the visible invariant mass to satisfy m t vis lep < m t . The reconstruction of the hadronic and the visible part of the leptonic top masses (before imposing the cuts) are shown in Fig. 4 (right). These two reconstructions ensure the complete obliteration of the W ± +jets backgrounds. In Tab. 4, we show the effects of the various cuts for three signal scenarios (κ λ = 1, 2 and κ tthh = −0.003) and the four dominant backgrounds, i.e. ttbbbb, tthbb, ttZh and ttZbb.
Finally after imposing all cuts, we are left with the cross-sections listed in Tab. 5. For the case of λ SM , we obtain a signal over background ratio of S/B ∼ 0.14 at leading order. For the design luminosity of 30 ab −1 , this translates into a statistical significance, S/ √ B ∼ 5.9 upon assuming no systematic uncertainties. Finally we feed these results into a log-likelihood CLs hypothesis test assuming the SM as the null hypothesis and also assuming no systematic uncertainties. At 68% confidence level, we find −4.12 < κ λ < 2.75 3/ab −3.01 < κ λ < 1. 65  Lastly, we also perform the same test on the tthh four point vertex. Upon resorting to a model independent bound on the coupling, we obtain the following bounds at 68% confidence level. −0.53 TeV −1 < κ tthh < 0.88 TeV −1 3/ab −0.24 TeV −1 < κ tthh < 0.60 TeV −1 30/ab (4.9) In order to see if we can gain additional sensitivity, we finally perform a multivariate analysis (MVA) with boosted decision trees (BDT) in the TMVA framework [63] with the following variables: reconstructed masses of both the Higgs bosons, h 1 , h 2 , reconstructed mass of the hadronic top-quark, visible part of the reconstructed leptonic top-quark, transverse momenta of these four objects, transverse momenta of the 6 b-tagged jets, hardest two light jets and the isolated lepton, the missing transverse energy and ∆R(h 1 h 2 /h i t h /h i t vis lep ), where i = 1, 2, and the total number of jets. We train the SM tthh sample with the dominant backgrounds, viz., ttbbbb, tthbb, tthZ, ttZbb and ttZZ. However, owing to a strong drop in efficiency due to the several reconstructions and requirements, we are left with an inadequate number of Monte Carlo events for a proper training [64]. However, the various vari-ables involved are mostly indiscernible and a BDT is not efficient in improving the sensitivity. For completeness, we find that our S/B improves to ∼ 0.17 with the statistical significance increasing to ∼ 6.4. Thus, we did not pursue the MVA analysis further.

Summary and Conclusions
One of the most important tasks after the discovery of the Higgs boson and after studying its couplings with gauge bosons and third-generation fermions is to understand the interactions of the scalar sector underlying electroweak symmetry breaking in more detail. One of its cornerstones is the trilinear interaction of the Higgs boson κ λ . Within most realistic extensions of the Standard Model one does not expect a modified Higgs self-interaction in isolation, but modifications of various couplings, i.e. the presence of many additional operators. Such new operators would simultaneously contribute to di-Higgs production processes, e.g. pp → hh, and would therefore result in blind directions in a global fit. To over-constrain the system of operators expected in extensions of the Standard Model it is consequently of crucial importance to measure as many multi-Higgs final states as possible.
We have revisited the sensitivity of the process pp → tthh at a future circular collider with √ s = 100 TeV. To take into account deformations of the Standard Model, we varied the two operators κ λ λ SM h 3 and κ tthh (t L t R h 2 + h.c.) independently. While the signal cross section increases by a factor of 75 between √ s = 14 TeV and √ s = 100 TeV, the total background before cuts increases by ∼ 40. Each background has a different enhancement factor and this total factor becomes ∼ 80 if we don't take into account the W ± +jets backgrounds which are completely negligible after the analysis. Hence, we surveyed a comprehensive list of backgrounds and found that the two operators can be constrained to −3.01 < κ λ < 1.65 and −0.24 TeV −1 < κ tthh < 0.60 TeV −1 for an integrated luminosity of 30/ab. While the limit on k λ is not competitive to the predicted limits from the processes pp → hh [39,41,65,66] or pp → hhj [67] for a 100 TeV collider, the fact that both coefficients κ tthh and κ tthh are contributing at tree-level to tthh production means that this process is of crucial importance to include in an agnostic global fit for dimension-6 effective operators.