Probing top quark FCNC couplings in the triple-top signal at the high energy LHC and future circular collider

Our main aim in this paper is to present detailed studies to probe the top quark flavor changing neutral current (FCNC) interactions at $tqg$, $tq\gamma$, $tqH$ and $tqZ (\sigma^{\mu \nu}, \gamma_{\mu})$ vertices in the triple-top signal $p p \to t t\bar t \, (\bar t t \bar t)$ at the high energy proposal of Large Hadron Collider (HE-LHC) and future circular hadron-hadron collider (FCC-hh). To this end, we investigate the production of three top quarks which arises from the FCNC couplings taken into account the fast simulation at $\sqrt{s} = 27$ TeV of HE-LHC and 100 TeV of FCC-hh considering the integrated luminosities of 10, 15 and 20 ab$^{-1}$. All the relevant backgrounds are considered in a cut based analysis to obtain the limits on the anomalous couplings and the corresponding branching ratios. The obtained exclusion limits on the coupling strengths and the branching ratios are summarized and compared in details with the results in the literature, namely the most recent direct LHC experimental limits and HL-LHC projections as well. We show that, for higher energy phase of LHC, a dedicated search for the top quark FCNC couplings can achieve much better sensitivities to the triple-top signal than other top quark production scenarios. We found that the limits for the branching ratios of $tqg$ and $tqH$ transitions could reach an impressive sensitivity and the obtained 95\% CL limits are at least three orders of magnitude better than the current LHC experimental results as well as the existing projections of HL-LHC.

The top quark with the mass of m top = 173.0 ± 0.4 GeV [1] which is close to the electroweak symmetry breaking scale, is the most sensitive probe to search * Hamzeh.Khanpour@cern.ch for a new physics evidence beyond the Standard Model (SM) at hadron and lepton colliders [2,3]. Due to the large mass of the top quark, the productions and related theoretical and experimental studies are golden places to look for possible signatures of new physics at TeV scales. Whilst improving the precision of SM predictions is highly important in its own right, any studies on the top quark to probe signatures of new physics are also most welcome. In this respect, over the past few years, several dedicated studies have shown that the non-SM couplings of the top quark should be one of the key analysis program pursued at the Large Hadron Collider (LHC) [2,[4][5][6][7]. These dedicated studies have been done in the top quark related processes, most notably in the single top quark production [8] or in the double top quarks production [9,10] scenarios. Among them, the flavor changing neutral current (FCNC) interactions involving a top quark, other quark flavors and neutral gauge boson are much of interest.
The FCNC interactions of top quark are forbidden at the tree level and due to the Glashow-Iliopoulos-Maiani mechanism (GIM) [11], are highly suppressed in a loop level. The SM predictions of the top quark FCNC decays to the gluon, photon, Z and Higgs boson, and an up or charm quark are expected to be at the order of O(10 −12 − 10 −17 ) which are currently out of range of present and even future experimental sensitivity [12]. However, in the beyond SM scenarios such suppression can be relaxed which yield to couplings of the orders of magnitude larger than those of the SM [13][14][15][16][17][18][19][20]. Hence, the possible deviation from the SM predictions of FCNC couplings would imply the existence of new physics beyond the SM [21]. In recent years, there has been a growing number of analyses focusing on this topic, and to arXiv:1909.03998v1 [hep-ph] 9 Sep 2019 date, there are many phenomenological studies in literature that have extensively investigated the association production of top quark with a gluon, photon, Higgs and Z boson mainly through single or double top production at hadron and lepton colliders, see e.g. Refs. [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] for the most recent reviews.
At the LHC, the top pair pp → tt and single-top quark productions are the dominated processes due to the strong coupling of gg → tt subprocess [8,36]. The production of an odd number of top quarks, i.e. tripletop quarks pp → ttt (ttt), requires a tbW vertex in every diagram. Since it also often involves a b-quark in the initial state of the hard process, therefore, they lead to a significant suppression in comparison to the strong processes. At the 14 TeV energy of LHC, triple-top quarks production cross section, with σ 1.9 fb, is almost five orders of magnitude less than the top pair production which is the dominant mechanism of top productions at the LHC. This relatively small SM production cross section of three top quarks makes it an interesting channel for investigating any signal of new physics.
The LHC at CERN and its luminosity upgrade, (HL-LHC) [37][38][39][40][41][42], have been actively carried out and will still continue the journey on searching for any signal of new physics in the next two decades. In addition to the HL-LHC, there are other proposals for the future higher energy hadron colliders to perform the direct searches at the energy frontier. These include the energy upgrade for the LHC to 27 TeV center-of-mass energy (HE-LHC) [37][38][39] and the future circular collider of about 100 TeV centerof-mass energy FCC-hh [43]. They will collect datasets corresponding to integrated luminosities of 10-20 ab −1 and 10 ab −1 , respectively. The high energy and high luminosity reach of these colliders strongly motivated to search for the FCNC couplings of top quark. Considering the need of these proposed colliders and our discussion on three top productions, one can conclude that the triple-top signal at HE-LHC or FCC-hh may potentially provide clear evidence for the top quark FCNC couplings. In this paper, we set out an initial study for the triple-top signal and present a detailed study to probe top quark FCNC interactions at tqg, tqγ, tqH and tqZ(σ µν , γ µ ) transitions at HE-LHC and FCC-hh. It should be mentioned here that, the triple-top quark signal that we are interested to investigate makes it possible to study all the top quark FCNC interactions tqX.
On the experimental side, lots of efforts performed earlier at the Tevatron at Fermilab and now at the 13 TeV LHC have failed to reveal any interesting observation of FCNC transitions. However, the obtained bounds on such couplings from the mentioned experiments are very strong. Most recently, considering the 13 TeV data from CMS and ATLAS, the exclusion limits on the top quark FCNC transitions have significantly improved by the LHC experiments. CMS and ATLAS Collaborations at CERN reported the most stringent constrains through the direct measurements [44][45][46][47][48][49][50][51][52][53][54][55][56][57].
These collaborations have set upper limits on the tqH FCNC couplings in the top sector at √ s = 13 TeV considering an integrated luminosity of 36.1 fb −1 (AT-LAS) and 35.9 fb −1 (CMS). Considering the analyses of the different top FCNC decay channels, the 95% confidence level (CL) upper limits have been found to be Br(t → uH) < 0.19% and Br(t → cH) < 0.16% from the ATLAS [45], and Br(t → uH) < 0.34% and Br(t → cH) < 0.44% from the CMS [56] Collaborations. In addition to this direct collider measurement for tqH couplings, single top quark production in the t channel is used to set limits for the top quark FCNC interactions with gluon tqg considering the data taken with the CMS detector at 7 and 8 TeV correspond to the integrated luminosities of 5.0 and 19.7 fb −1 . The upper limits on the branching fractions of Br(t → ug) < 0.002% and Br(t → cg) < 0.041% have been measured [50]. A search for FCNC through single top quark production in association with a photon also have been performed by CMS at √ s = 8 TeV corresponding to an integrated luminosity of 19.8 fb −1 . Upper limits at the 95% CL on tqγ anomalous couplings are measured to be Br(t → uγ) < 0.013% and Br(t → cγ) < 0.17% [47]. Finally, search for the FCNC top quark decays of t → qZ in proton-proton collisions at √ s = 13 TeV have been done both by CMS and AT-LAS Collaborations through different channels. Upper limits at 95% CL level on the branching fractions of top quark decays can be found to be Br(t → uZ) < 0.015% and Br(t → cZ) < 0.037% from the CMS [46], and Br(t → uH) < 0.024% and Br(t → cH) < 0.032% from the ATLAS [52] Collaborations for the integrated luminosities of 35.9 and 36.1 fb −1 , respectively.
In this paper, we shall try to investigate in details the projected sensitivity and discovery prospects of the HE-LHC and FCC-hh to the top quark FCNC transitions within the model independent way using an effective Lagrangian framework. To this end, we follow the strategy presented in [12,58] and quantify the expected sensitivity of the HE-LHC and FCC-hh to the top quark FCNC couplings tqX. The realistic detector effects are included in the production of the signal and background processes with the most up-to-date experimental studies carefully considering the upgraded CMS detector performance [59] for the HE-LHC and the FCC-hh baseline detector configuration embedded into Delphes. As we will demonstrate, the expected constraints for tqg and tqH from the HE-LHC and FCC-hh are significant and fully complementary with those from the LHC and HL-LHC.
This article is arranged as follows. In Sec. II, we present the theoretical framework and the effective Lagrangian approach for the top quark FCNC couplings. The details of the analysis strategy applied in this investigation are clearly discussed and presented in Sec. III. This section also includes the signal and background estimations, the simulations and detector effects for HE-LHC and FCC-hh. We detail in Sec.IV the statistical method we assume, together with the numerical calculations and distributions for the HE-LHC and FCC-hh. Sec. V includes the numerical results and findings in details. The 95% 95% confidence level (CL) limits of HE-LHC and FCC-hh are compared with the LHC measured limits and the other studies in literature. Finally, in Sec. VI, we conclude and summarize our main results and findings.

II. THEORETICAL FRAMEWORK AND ASSUMPTIONS
This section presents the theoretical framework and assumptions applied in this analysis to study the top quark FCNC transitions at HE-LHC and FCC-hh. The possibility of the top quark anomalous FCNC couplings with light quarks (q = u, c) and a gauge bosons (g, H, Z, γ) is explored in a model-independent way considering the most general effective Lagrangian approach [58,60]. In the search of anomalous FCNC interactions at high energy colliders, this approach has been extensively studied in literature for lepton and hadron colliders [21][22][23][24][25][26][27][28][29][30][31][32][61][62][63][64][65][66][67][68][69][70][71]. In this framework, these FCNC vertices are described by higher-dimensional effective operators L tqX FCNC independently from the underlying theory. Up to dimension-six operators, the FCNC Lagrangian of the tqg, tqH, tqZ(σ µν ), tqZ(γ µ ) and tqγ interactions can be written as [12,58,60]: In Eq. (1), the real parameters ζ qt , η qt , κ qt , X qt and λ qt represent the strength of FCNC interactions of a top quark with gluon, Higgs, Z and γ, respectively. q indicates an up or charm quark as well. At the tree-level, in the SM, all the above coefficients are zero and in the presence of the anomalous FCNC vertices the straightforward way is to set limits on these couplings strength and the corresponding branching fractions. In the above equation, g s is the strong coupling constant and P L(R) denotes the left (right) handed projection operators. In this study, we assume no specific chirality for the anomalous FCNC interactions, and hence, we set ζ L qt = ζ R qt = ζ qt , η L qt = η R qt = η qt , κ L qt = κ R qt = κ qt , X L qt = X R qt = X qt and λ L qt = λ R qt = λ qt . As we mentioned earlier in the introduction, the triple-top quark signal includes all the top quark FCNC interactions, and hence, make it possible to study all these coefficients in this signal topology.
In this study, we consider pp → ttt (ttt) signal process to search for anomalous FCNC tqX(X = g, H, Z, γ) interactions in the presence of effective Lagrangian of Eq. (1). In order to provide more details on the FCNC vertices, we present in Fig. 1, representative Feynman diagrams contributing to this signal process at tree level. As one can see from Fig. 1, this Feynman diagram contains tqg vertices (red circle) in which make it possible to study the top quark FCNC coupling in this signal process. We consider the leptonic decays of the W boson originating from the same-sign top quarks which lead to same-sign dilepton final states. Another top quark decays hadronically. In order to provide more insight on the available FCNC transitions in triple-top productions, we present in Fig. 2, a set of Feynman diagrams contributing to FCNC vertices q → tH, q → tγ and q → tZ. The FCNC vertices are shown as a red circle.
In Fig. 3, we show the total cross sections σ(t → qX) in the unit of fb in the presence of anomalous tqX couplings versus the top quark FCNC branching ratios Br(t → qX) for five different signal scenarios.
The following conclusions can be drawn from the results presented in Fig. 3. As it can be seen, in term of individual tqX coupling, the largest contributions for the triple-top signal mainly come from the tqg coupling, and then the tqH. This indicates the large parton distribution functions (PDFs) of u-quark and gluon in the calculation of cross section at high center-of-mass energy. In our study, we show that the sensitivity to the branching ratio of tqg and tqH channels are much better than the current LHC experimental limits, and even they are much better than the projected limits on top FCNC couplings at HL-LHC with an integrated luminosity of L int = 3000 fb −1 . These findings suggest that the measurement of tqg and tqH FCNC couplings through triple-top productions at future high energy collider would carry a significant amount of information, and hence, are the most sensitive probe to search for a new physics beyond the SM.

III. ANALYSIS STRATEGY AND NUMERICAL CALCULATIONS
As we mentioned, in this study, we plan to investigate the discovery potential of future HE-LHC with 27 TeV C.M. and FCC-hh with 100 TeV C.M. to the top quark FCNC transitions. To this end, we follow an strategy based on an effective Lagrangian approach to describe the top quark FCNC in a model independent way. There are a lot of studies in literature that have been done in new physics searches to enrich the physics motivations of such proposed colliders [37][38][39]59]. One of our main goals in this paper is to study the impact of HE-LHC and FCC-hh to the top quark FCNC coupling determinations. After introducing our theoretical framework and assumptions in previous section, we now present the analysis strategy and numerical calculations related to our study. We first discuss the tqX signal and background analysis. Then, we present the simulation and realistic detector effects for both HE-LHC and FCC-hh. Figure 1: The Feynman diagram for the triple-top quark production containing tqg anomalous FCNC vertex. As we described in the text, we consider the leptonic decays of the W boson originating from the same-sign top quarks which lead to same-sign dilepton final states.

Figure 2:
The Feynman diagrams for the triple-top quark production in the presence of FCNC q → tH, q → tγ and q → tZ vertices.

A. The tqX signal and SM backgrounds
In the following section, our study on the pp →ttt(ttt) signal process including the FCNC tqX(X = g, H, Z, γ) couplings as well as the relevant SM backgrounds at the HE-LHC and FCC-hh are given. This signal provides searching for all FCNC couplings of tqg, tqH, tqZ and tqγ independently. We also do this study separately considering q = u and q = c. For the tripletop quark, we consider both hadronic (jj) and leptonic ( ν) decays of W boson by analyzing a very clean signature with two same-sign leptons ( ±± ), where the lepton could be an electron or a muon. Then, the signal analysis is performed with the following final states: . Therefore, these unique signal events are characterized by the presence of exactly two isolated same-sign charged leptons, (2 + or 2 − ). In addition, there should be a large missing transverse energy (MET) from the undetected neutrino. This signal also characterized by several jets in which three of them should come from b-quarks. As one can see from the Feynman diagrams (see Figs. 1 and 2), the top quark FCNC couplings can be understood by considering the appearance of subprocess diagrams like tqX →ttt(ttt) with q = u, c and X = g, H, Z, γ.
Considering these signal scenarios, the following relevant background processes which have similar final state topology need to be taken into account: ttZ in which Z decays to a pair of opposite-sign isolated leptons (Z → + − ) with semi-leptonic decay of one top quark and fully hadronic decay of another top quark. We also consider the ttW with leptonic decay of W boson (W → ν), semileptonic of one top and fully hadronic decay of another quark. In addition, the W W Z background also included in which Z decays to a pair of leptons (Z → + − ) and the W decays to quark-antiquark pair (hadronic) or to a charged lepton ( ± ) and a neutrino (leptonic). The mentioned backgrounds are the most important sources of backgrounds analyzed in this study. In addition to these relevant backgrounds, we also consider other source of backgrounds such as SM four top productions, ttH, ttW W and ttW W in our study of MET + jets + lep- tons final states. In order to minimize the contributions of these backgrounds, different selection cuts are applied which will be discussed in details in Sec. IV. We show that by applying a same-sign isolated dilepton and 3 b-jets selections, some of these SM backgrounds can be strongly reduced and safely ignored, and hence, they are not considered. Some selected examples of partonic Feynman diagrams for thettW , ttZ and W W Z backgrounds analyzed in this study are shown in Fig. 4.
In the next section, we present the simulation of signal ans backgrounds, and the realistic detector effects for the HE-LHC and FCC-hh.

B. The simulation and detector effects
In this section, we present the analysis of pp → ttt (ttt) signal process including the FCNC tqg, tqγ, tqH and tqZ(σ µν , γ µ ) vertices as well as all the relevant SM backgrounds with experimental conditions of the HE-LHC and FCC-hh. For the simulations of the HE-LHC and FCC-hh collider phenomenology, we use the FeynRules [72] to extract the Feynman rules from the effective Lagrangian of Eq. (1). The Universal FeynRules Output (UFO) files have been generated [73] and then UFO files fed to the Monte Carlo event generator MadGraph5_aMC@NLO [74,75] to generate the event samples for signal processes. Mad-Graph5_aMC@NLO [74,75] is also used to generate background processes. The sample are generated using the leading order (LO) NNPDF23L01 parton distribution functions (PDFs) [76][77][78] considering the renormalization and factorization scales are set be the threshold value of the top quark mass, µ = µ F = µ R = m top . For the parton showering, fragmentation and hadronization of generated signal and backgrounds events we utilized the Pythia 8.20 [79]. During the production of signal and backgrounds samples, all produced jets inside the events forced to be clustered using the FastJet 3.2 [80] considering the anti-k t jet clustering algorithm with a cone radius of R = 0.4 [81]. Finally, we pass all generated events through the Delphes 3.4.2 [82], which handles the detector effect.
We should emphasize here that, for the FCC-hh analysis, we use the default FCC-hh detector card configuration implemented into the Delphes 3.4.2 in order to consider the realistic detector effects of the FCC-hh baseline detector. Considering this configuration, the efficiency of b-tagging b (p T ), efficiency of c-jets c (p T ) and misidentifications rates for the light-jets are assumed to be jet transverse momentum dependent. They are given by [21,31] For the case of HE-LHC projections and in order to produce the Monte Carlo events, we also employed the DELPHES framework for performing a comprehensive high luminosity (HL) CMS detector response simulation. To this end, we have used the HL-LHC detector card configuration implemented into the Delphes 3.4.2 which includes high configuration of the CMS detector [36,39,42,59]. The b-tagging efficiency b (p T ) and misidentification rates for light-flavor quarks are assumed to be (3)

IV. STATISTICAL METHOD FOR THE tqX FCNC ANALYSIS
We detail below the statistical method we assume, together with the numerical calculations and distributions for the HE-LHC and FCC-hh. More details are provided in the next section. As we discussed in details in Sec. III A, the studied topology gives rise to the MET + jets + leptons signature characterized by five or more than five jets, and a missing transverse momentum from the undetected neutrino and exactly 2 same-sign isolated charged leptons. Among these jets, three of the them should be tagged as b-jets. Based on this signal topology, we follow a standard methodology to distinguish the signal signature from the corresponding SM backgrounds and considered some different preselection cuts as we describe here.
As we applied the leptonic channels of W boson for the top (antitop) quark pairs in the signal process, exactly two same-sign isolated charged leptons (electron or muon) are required, n = 2 ±± , with |η | < 2.5 and p T > 10 GeV. As we highlighted before, one of the key ingredients in the strategy pursued in the present study is the triple-top signal with the topology of two samesign isolated charged leptons. We will discuss in the next section that an additional cut of the same-sign dileptons invariant mass distributions (M ±± > 10 GeV) need to be taken into account to suppress events with pairs of same-sign energetic leptons from the heavy hadrons de-cays of backgrounds. Since we consider the doubly leptonic decay of W boson in the final state, triple-top signals include a substantial amount of missing transverse energy. For the case of missing transverse energy, we apply E miss T > 30. Our signal scenario also includes at least five jets n j ≥ 5 jets with |η jets | < 2.5 and p jets T > 20 GeV. We also considered the distance between leading leptons and jets ∆R( , j i ) = (∆φ ,ji ) 2 + (∆η ,ji ) 2 > 0.4 in which are azimuthal angle and the pseudorapidity difference between these two objects. The same selection also need to be taken into account between two jets, ∆R(j i , j j ) > 0.4. Among the selected jets, at least three of them need to be tagged as b jets, i.e. n b−jets ≥ 3.
After adopting our basic cuts and selection of signal and background events, in next section we study the signal and the backgrounds at the level of distributions and the numerical calculations for the HE-LHC and FCC-hh separately. We should notice here that, in our study that will be discussed in the next section, we only concentrate on the t → qg and t → qH modes as a reference throughout this work for presenting some selected distributions. We also choose those distributions which show a good potential to separate the signal for the SM backgrounds.

A. The signal and background analysis and distributions at HE-LHC
After introducing the simulation, detector effects and the event selection in previous sections, in the this section, we present the numerical calculations and distributions for the HE-LHC scenario.
Let us know present and discuss the cross sections of the triple-top signal and all SM backgrounds in order to provide a basic idea of their production rate. As we discussed before, ttZ, ttW and W W Z SM backgrounds are the main backgrounds considered in this study. We also include other source of SM backgrounds such as SM four top productions, ttW W , ttH and ttW W . However, we found that these backgrounds have small contributions in the total background composition. The cross-sections in the unit of fb for the ttZ, ttW and W W Z SM backgrounds passing sequential selection cuts are presented in Table I for the HE-LHC at √ s = 27 TeV. As one can see from Table I, these selection criterion could significantly suppress the large contributions of background events originating from the ttZ and ttW , and specially from the W W Z. Among the selection strategy, two same-sign isolated leptons selection n = 2 ±± reduces these backgrounds and lead to the selection efficiencies of 12%, 37% and 12% for ttZ, ttW and W W Z backgrounds, respectively. Selecting jets and b-jets, considerably affect all backgrounds as well. For example, the cut efficiency of b-jets selection is about 0.3% for ttZ, 0.6% for ttW , and 0.001% for W W Z backgrounds which all have the same final state with the signal. These small efficiencies indicate that the three tagged b-jets selection can reduce the SM backgrounds strongly. The sum of the cross section for all SM backgrounds after all cuts is found to be 0.249 fb.
After our discussion on the numerical calculations for the backgrounds, let us know present our signal calculations. We should notice here that, the fixed values of ζ qt = 0.1, η qt = 0.1, κ qt = 0.1, X qt = 0.1 and λ qt = 0.1 with q = u, c are chosen for all coupling strength as the benchmark point if not stated otherwise. Taking these typical benchmarks input for the triple-top signal, the expected cross sections before and after the selection cuts are presented in Table II.
The following conclusions can be drawn from the cut flow table present in Table II. For all triple-top signal topologies, around 35% efficiency obtained considering the same-sign dilepton selection and 12-15% efficiency achieved after the jet selection strategy. As one can see, after the sample selections, around 6-7% of signal events could pass the selection criteria. Now let us discuss the triple-top signal and the SM backgrounds at the distribution level. Characteristic signature of the triple-top signal process analyzed in this study suggests to work with the events having at least two isolated same-sign lepton in which can be an electron or a muon, large missing transverse energy (MET), and at least 5 jets which three of them are required to be identified as jets originating from the b-quark. Considering these signal scenarios and all relevant SM backgrounds, in Fig. 5, we show the jets and b-jets multiplicities for triple-top signal and the main SM backgrounds events before applying the jet and b-jet selection. All the plots are unit normalized. As we mentioned before, we only concentrate on the t → qg and t → qH modes as a reference throughout this work for presenting some selected distributions. Hence, for the signal in these figures, only one coupling (ζ tq or η tq with q = u, c) at a time is varied from its SM value. It can be concluded from these plots, the requirement of at least five jets (n jets ≥ 5) is useful to reduce the contributions of the SM backgrounds. In addition to this requirement, selecting at least three btagged jets n b−jets ≥ 3 among those jets is also useful to suppress the contribution of the SM backgrounds.

B. The signal and background analysis and distributions at FCC-hh
In this section, we focus on the numerical calculations of the FCC-hh collider for the triple-top signal analyzed in this study. As we mentioned before, in addition to the HE-LHC, we also plan to investigate the potential of future FCC-hh collider to the top quark FCNC couplings at a center of mass energy of 100 TeV and present our study to set an upper limits on the anomalous top FCNC tqX(X = g, H, Z, γ) vertices including the realistic detector effects. However, one can even expect the higher center-of-mass energies of FCC-hh collider could lead to the improvement of these limits. For the FCChh analysis, we follow the same strategy applied for the case of HE-LHC, namely the the semileptonic final state with two same-charged W decaying to either electron or muon, and the other decaying hadronically. We also use the same selection cuts for the signal and backgrounds. We expect, and do find, a similar peak on the jets and b-jets multiplicity distributions for the signal and for the backgrounds as the case of HE-LHC presented in Fig. 5. The difference between the HE-LHC and FCC-hh mainly comes from the different detector configurations. As we discussed in Section III B, for the FCC-hh analysis, we use the default FCC-hh detector card configuration implemented into the Delphes 3.4.2 in order to consider the realistic detector effects of the FCC-hh baseline detector. All the SM backgrounds listed above for the HE-LHC are also belonging to the backgrounds of FCC-hh.
Notice that while the colliding energy of FCC-hh is different, we consider the same selection cuts which are optimized for the HE-LHC. We find that these selections also reasonable for the case of FCC-hh. Considering this point, in Table III, we present the cross section measurements of the SM backgrounds at FCC-hh before and after applying the selection cuts. As one can see, selection of two same-sign isolated charged lepton could significantly affects the ttZ and W W Z backgrounds with 11% of efficiency. This selection also leads to 43% for the ttW backgrounds. This finding indicates that, among all the backgrounds, the ttW is indeed hard to suppress. One can see that, the jets selection also affects the mentioned backgrounds considerably. Finally, our selection strategy leads to 0.8% of ttZ, 2% of ttW and 0.004% of W W Z backgrounds.
Our numerical calculations of the triple-top signal cross sections at leading order in the presence of the top quark FCNC vertices are presented in details in Table IV. This table shows the cut flow dependence of various signal scenarios studied here. One clearly sees that we achieved to the selection efficiency of 6% for tug, 10% for tcg and   Table II: Cross-sections in the unit of fb for the triple-top production at HE-LHC pp → ttt (ttt) with = e, µ for five signal topologies of tqg, tqH, tqZ(σµν ), tqZ(γµ) and tqγ before and after passing sequential selection cuts.
around 15% for all other signal scenarios.
For completeness, we also depict in Fig. 6, some selected distributions, including the cosine between two same-sing leptons cos( ± , ± ) (left) and the invariant mass distributions of dilepton (right) for tqg and tqH signal scenarios and the corresponding ttZ and ttW SM backgrounds for the FCC-hh collider. As we explained before, an additional cut on the same-sign dileptons invariant mass distributions (M ±± > 10 GeV) has been applied to suppress events with pairs of same-sign energetic leptons from the heavy hadrons decays of backgrounds.

V. ANALYSIS RESULTS AND 95% CONFIDENCE LEVEL LIMITS AT HE-LHC AND FCC-HH COLLIDERS
In this section, we present the main results of this study, namely the upper limits on the coupling strengths obtained from the fast simulation of the triple-top signal at HE-LHC and FCC-hh. First, we discuss the upper limits obtained in this study focusing on the 95% CL limits of the HE-LHC and FCC-hh, and then we compare the results with other studies in literature. Finally, we detail a number of updates and improvements that should be foreseen for the future work.
Considering the optimized selections of signal and backgrounds that we discussed in section IV and having at hand the signal efficiencies and the number of backgrounds, we set 95% CL upper limits on the anoma-lous FCNC couplings and determine the expected limits on the FCNC branching fractions Br(t → qX) using a Bayesian approach [83]. The 95% CL constraints on various FCNC branching fractions including the gluon, Higgs boson, Z boson and photon are detailed and summarized in Table V for HE-LHC working at three scenarios of integrated luminosities of L int = 10, 15 and 20 ab −1 of data. The most recent experimental constraints on the corresponding branching ratio of the top quark FCNC transitions obtained at the ATLAS and CMS with 95% CL are also presented as well [45][46][47]50].
A few remarks concerning the results presented in this table are in order. As one can see, with an integrated luminosity of 15 ab −1 , the sensitivity to the branching ratio of tug and tcg channels are three order of magnitude better than the available experimental limits form CMS Collaboration [50]. For the tuH and tcH, the limits obtained in this study are two and one orders to magnitude better than the most recent direct limits on the corresponding branching ratios reported by ATLAS Collaborations at CERN with an integrated luminosity of 36.1 fb −1 at √ s = 13 TeV. As a short summary and based on the HE-LHC projections, one can conclude that in the cases of tqg and tqH FCNC transitions, the most stringent constraints have been obtained, and hence, the limits through the tree-top signal can be much better than other channel studied in the literature. For other FCNC transitions, namely tqγ and tqZ, we determined comparable branching ratios with the recent experimental limits [46,47]. This finding indicates that the tree top signal analyzed in this study is not much sensitive to  these particular FCNC vertices. At this stage, we turn to present and discuss upper limits on the signal rates at 95% CL for the case of FCChh scenario. We exactly follow the statistical method used for the HE-LHC study to set upper limits on the coupling strengths and the resulting branching fractions. The branching fractions Br(t → qX) for the FCC-hh at the luminosity of 10 ab −1 are detailed in Table VI. Similar conclusions as those for the HE-LHC can be drawn for the FCC-hh. However, one can expect further improvements with a higher center-of-mass energy collider. Our study suggests that improvement on the upper limits of all analyzed FCNC couplings can be obtained at FCC-hh collider; which is found to be around 80% for the tug and one order of magnitude for the tcg coupling with respect to the HE-LHC. The improvements on these bounds are likely due to the higher center of mass energy of FCC-hh. In the light of the results on the limits on the FCNC coupling strengths in triple-top quark signal presented in Tables V and VI, we confirm the remarkable sensitivity that HE-LHC and FCC-hh would have on these couplings, especially on those of tqg and tqH, which are comparatively much more constrained than other results in literature.
As a final point, the present research explores for the first time, the sensitivity of triple-top signal at 27 TeV HE-LHC and 100 TeV FCC-hh to probe the top quark FCNC couplings with a gluon, photon, Higgs and Z boson. We have presented and discussed the sensitivity of such collider and shown that the triple-top signal signature improves the current limits of LHC for most of top quark FCNC interactions, especially for the tqg and tqH FCNC couplings. We have then examined and highlighted limits for HE-LHC working at different scenario of integrated luminosity. In this study, we have introduced some methodological improvements aimed to improve the current limits on the top quark FCNC branching fractions. Regarding the results presented here, a number of important updates and improvements are foreseen. The main scope of the present paper is the beyond SM scenarios study of triple-top quark productions leading to anomalous tqX vertices. Along with the phenomenological method presented here, additional observables can be use to suppress the background contributions and enhance the signal significance in extracting the reach of the couplings. A further improvement for future investigation is the use of multivariate technique [84]. Such technique is expected to improve the limits of coupling  Table IV: Cross-sections in the unit of fb for the triple-top production at FCC-hh pp → ttt (ttt) with = e, µ for five signal topologies of tqg, tqH, tqZ(σµν ), tqZ(γµ) and tqγ before and after passing sequential preselection cuts.  Figure 6: The cosine between two same-sing leptons cos( ± , ± ) (left) and the invariant mass distributions of dilepton (right) for tqg and tqH signal scenarios obtained from MadGraph5_aMC@NLO [74,75] at leading order for FCC-hh at 100 TeV. The main SM backgrounds ttZ and ttW also are presented as well.
strengths. Finally, one could also consider another channel of top quark production to study top quark FCNC transitions at the HE-LHC and FCC-hh.

VI. SUMMARY AND CONCLUSIONS
We now turn to present our summary and conclusions. We first compare our limits with the projections of HL-LHC. Then, we conclude this section with a plot showing a summary of branching fraction limits for top quark FCNC interactions in compassion to the SM predictions, as well as to the various beyond SM scenarios.
All the proposed future lepton [85] and hadron [43] colliders including the high energy large hadron collider (HE-LHC) [86] will likely face critical decision making in the coming years [87]. On the theory side, investigations of the top quark FCNC coupling to photons, gluon, Z and Higgs boson offer one of the important alternatives to explore new physics beyond the SM. Determination of new physics potentials of such proposed colliders at the energy frontier, in particular the reach on the top quark FCNC couplings measurements, is an important topic for the high energy physics community. Continued efforts are needed to be done to investigate the new physics accessible at these proposed colliders [88]. In recent years, there has been a considerable amount of literature that published to highlight the need for future high energy colliders.
Currently, the most stringent limits on top quark FCNC branching fractions Br(t → qX) have been measured with ATLAS and CMS Collaboration at the LHC. The results obtained at √ s = 13 TeV of LHC significantly improve the upper limits set with the 7 and 8 TeV data. It is worth mentioning here that, due to availability of large number of experimental results on the top quark FCNC transitions from the LHC through pp collision, it could leads to a good prospects for pushing the top quark   FCNC boundaries to even much higher constraints using future colliders. This paper focused to present phenomenological investigations to analyze the sensitivity of the triple-top quark signal at HE-LHC and FCC-hh to the top quark FCNC couplings, tqX with X = g, H, Z, γ. To this end, we have studied the triple-top production pp → ttt (ttt) at 27 TeV of HE-LHC and 100 TeV of FCC-hh taken into account the unique signal signature of two same-sign isolated charged lepton.
Numerical values are provided in Table V and VI. Considering the 27 TeV HE-LHC projections, our results clearly show that with an integrated luminosity of 15 ab −1 , it is likely that while the LHC can be obtained limits on the top quark FCNC couplings of tqg and tqH up to a sensitivity of the order 10 −5 and 10 −3 , the limits on these couplings can reach up to a sensitivity of the order 10 −8 and 10 −5 , respectively. Consistent with the literature, this study confirmed that most stringent limits can be obtained for the case of tqg.
The reach we obtain on the upper limits of top quark FCNC branching ratios Br(t → qg) and Br(t → qH) are highlighted in Fig. 7, for the two future hadron-hadron colliders considered in this analysis, HE-LHC and FCChh.
As one can see from Fig. 7, our limit on the tqg and tqH branching ratios are much better than the projected limits on top FCNC couplings at HL-LHC [40,41]. While some of other calculated limits sound approximately similar to that are projected in the case of HL-LHC, note that our analysis considered the triple-top signal which could let us to examine all the FCNC couplings. We believe that our study and the results presented here have clearly brought out the advantages of the HE-LHC and FCC-hh colliders in probing the top quark FCNC couplings with gluon, photon, Z and Higgs boson, which could complement the information that could be extracted from the LHC and HL-LHC.
In order to have a better estimate of the extracted limits, let us conclude our discussions by comparing the branching fractions obtained in this study with theoretical predictions in the SM and other new physics models beyond SM. In Fig. 8, our obtained results are compared with the theoretical predictions in the SM, as well as with various new physics models. In comparison with those of new physics models, our findings for future projections of HE-LHC and FCC-hh represent good prospects for pushing top FCNC boundaries to even higher constraints.

ACKNOWLEDGMENTS
We acknowledge fruitful discussions with Daniel Schulte, Mogens Dam, Marco Zaro, Giovanni Zevi and Frederic Deliot in customizing the detector card used in this analysis. Author are thankful to Mojtaba Mohammadi for reading this manuscript and many helpful discussions and comments. Author thanks School of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM) and University of Science and Technology of Mazandaran for financial support of this research. Author also is thankful the CERN theory department for their hospitality and support during the  preparation of this paper.