Prospects of searching for composite resonances at the LHC and beyond

Composite Higgs models predict the existence of resonances. We study in detail the collider phenomenology of both the vector and fermionic resonances, including the possibility of both of them being light and within the reach of the LHC. We present current constraints from di-boson, di-lepton resonance searches and top partner pair searches on a set of simplified benchmark models based on the minimal coset $SO(5)/SO(4)$, and make projections for the reach of the HL-LHC. We find that the cascade decay channels for the vector resonances into top partners, or vice versa, can play an important role in the phenomenology of the models. We present a conservative estimate for their reach by using the same-sign di-lepton final states. As a simple extrapolation of our work, we also present the projected reach at the 27 TeV HE-LHC and a 100 TeV $pp$ collider.

CMS with integrated luminosity L = 77.3 fb −1 for the electron channel and L = 36.3 fb −1 for the muon channel [12], and the search for the pair production of top quark partners with charge-5/3 at CMS with integrated luminosity L = 35.9 fb −1 [13]. In addition, we paid close attention to scenarios in which the spin-1 resonances and top partners can be comparable in mass. In this case, cascade decays in which one composite resonance decays into another, can play an important role [14][15][16]. In particular, the channels ρ + L → tB/X 5/3t or ρ + L → X 5/3X2/3 and ρ 0 L → X 5/3X5/3 can have significant branching ratios for models with quartet top partner, if ρ L is in the intermediate mass region M Ψ < M ρ < 2M Ψ or the high mass region M ρ > 2M Ψ , respectively. Such cascade decays can lead to the same-sign di-lepton (SSDL) signals. Since these are relative clean signals, which have already been used for LHC searches, we use them in our recast and estimate the prospective reach on the M ρ −M Ψ plane. They are comparable in some regions of the parameter space to the di-boson searches for the spin-1 resonances and the pair-produced top partner searches at the LHC.
For the models with a singlet top partner, the cascade decay channel T → tρ X → ttt in the single production channel can play an important role in the mass region M T > M ρ X .
The reach at the LHC is also estimated in the SSDL channels. The projections made based on only the SSDL channel are of course conservative. Other decay modes of the cascade decay channels mentioned above can further enhance the reach, such as the ones including more complicated final states like 1 + jets channels. We leave a detailed exploration of such additional channels for a future work.
The paper is organized as follows. In Section II, we summarize the main phenomenological features of the models, including the couplings of the particles in the mass eigenstates, and the production and the decay of the resonances. The details of the models are presented in Appendix A and Appendix B. In Section III, we show the present bounds from the LHC searches and extrapolate the results to the HL-LHC with an integrated luminosity of L = 3 ab −1 . An estimate of the reach at the 27 TeV HE-LHC and 100 TeV pp collider is also included. We conclude in Section IV.

II. PHENOMENOLOGY OF THE MODELS
We begin with a brief review of the composite Higgs models under consideration. We will describe the particle content, and give a qualitative discussion of the sizes of various couplings. The details of the models are presented in Appendix A and B.
We will consider models similar to those presented in Ref. [14]. The strong dynamics is assumed to have a global symmetry SO (5), which is broken spontaneously to SO(4) SU (2) L × SU (2) R . The resulting Goldstone bosons, parameterizing the coset SO(5)/SO(4), contain the Higgs doublet. This is the minimal setup with a custodial SU (2) symmetry.
The composite resonances furnish complete representations of SO(4).
Particle content R , and it is embedded in an an incomplete representation, 5, of SO (5). It can also be a fully composite resonance, denoted as t (F) R , and it is assumed to be an SO(4) singlet massless bound state. Their representations under the unbroken SO(4) are also presented in the table. Lower table: the models with different combinations of the composite spin-1 resonances ρ and the fermionic resonances Ψ considered in our paper. P (F) denotes the partially (fully) composite right-handed top quark.
There are two well-studied ways of dealing with the right handed top quark. First, it can be treated as an elementary field, and embedded into a 5 representation of SO(5) (see Eq. (A22)) [5]. We call this the partially composite right-handed top quark scenario, and denote right-handed top as t (P) R . It is also possible that it is a massless bound state of the strong sector and a SO(4) singlet, denoted as t (F) R [17]. We call this the fully composite right-handed top quark scenario. We will consider both of these cases. In principle, many of the composite resonances can be comparable in their masses in a given model. Rather than getting in the numerous combinations, we consider a set of simplified models in which only one kind of spin-1 resonance(s) and one kind of top partner(s) are light and relevant for collider searches. For example, model LP 4 involves the strong interactions between the ρ L and the quartet top partner Ψ 4 and the partially composite right-handed top quark. In comparison, model LF 4 is different only in the treatment of the right handed top quark which is assumed to be fully composite.
In the following, we will first discuss all the most relevant interactions and their coupling strengths in Section II A. The production and decay of the resonances at the LHC are presented in Section II B. The mass matrices of different models and their diagonalizations are discussed in Appendix C, where we also list the expressions all the mass eigenvalues.

A. The couplings
Scale f , similar to the pion decay constant in QCD, parameterizes the size of global symmetry breaking. The parameter ξ = v 2 /f 2 measures the hierarchy between the weak scale and the global symmetry breaking scale in the strong sector. It has been well constrained from LEP electroweak precision test (EWPT) and the LHC Higgs coupling measurements to be ξ 0.13 [18,19]. In the expressions for the couplings, we will keep only terms to the leading order in ξ.
The interactions of the spin-1 resonances in the strong sector are characterized by several couplings, (g ρ L , g ρ R , g ρ X ), sometimes collectively denoted as g ρ . Typically, they are assumed to be much larger than the SM gauge couplings, i.e. g ρ g , g. We will keep only terms to the leading order in g/g ρ in the expressions of the couplings 1 . Similar to Ref. [20], we will also introduce an O(1) parameter for each representation of the spin-1 resonances, defined as a ρ L,R,X = m ρ L,R,X g ρ L,R,X f . (1) In most of the cases, we will fix a ρ .
The sector of fermionic composite resonances involve another strong coupling, g Ψ , defined as: and the same definition applies to c θ L , c θ R , t θ L , t θ R . The interactions of the spin-1 resonances and the fermions are summarized in Table II (for the charged sector) and Table III, Table IV (for the neutral sector).
The couplings can be organized into four classes by their typical sizes. The first class includes the interactions generated directly from the strong dynamics and preserve the non- ρ + LT B, ρ + LX 5/3 X 2/3 , ρ + RX 2/3 B, ρ + RX 5/3 T, ρ 0 L,R,XT T , ρ 0 where T = T , B, X 5/3 , X 2/3 denotes the fermionic resonances in the quartet. The last term is for the case of a fully composite right-handed top quark. As will be discussed in the Between heavy resonances: Between heavy resonances and SM fermions: Between SM particles: where we have defined the SU (2) R current The Higgs doublet can be parameterized as Vertices Between heavy resonances: Between heavy resonances and SM fermions: Between SM particles: with φ ± , χ eaten by the SM W ± , Z bosons after EWSB. By the Goldstone equivalence theorem, the interactions involve φ ± , χ will determine the couplings of longitudinal modes of W ± and Z gauge bosons at high energy, leading to the following interactions with O(g ρ ): Hence, ρ L,R will primarily decay into the longitudinal gauge bosons and the Higgs bosons if the other strongly interacting decay channels (ΨΨ or Ψq) are not kinematically open. The other type in the first class is the interactions between the resonances Ψ 4 and t where we have integrated by parts before turning on the Higgs vacuum expectation value (VEV) and focused only on the trilinear couplings (see Ref. [17] for detail). M Q X , M Q are defined in Eq. (B12). In the limit M 4 /f y L , y 2L , these are the dominant interactions between the top partners and the SM fields. By using Goldstone equivalence theorem, we can easily derive the well-known approximate decay branching ratios for the top parnters: Br(T → th) Br(T → tZ) 50%, Br(X 2/3 → th) Br(X 2/3 → tZ) 50%, Taking into account the mixing effects, shown in Eq. (3), will not modify the conclusions significantly.
The second class of interactions are suppressed either by the left-handed top quark mixing s θ L or the right-handed top quark mixing s θ R defined in Eq. (3). These are the couplings of ρ to one top partner and one SM quark. These interactions preserve SM SU (2) L × U (1) Y gauge symmetries. Symmetry considerations select the following interactions: where the last term is only present for the partially composite right-handed top quark scenario. The interactions will play an important role in the kinematical region which leads to the same decay branching ratios as Eq. (10) for the quartet. For the singlet top partner, this gives The fourth class contains the interactions with coupling strengths suppressed by g/g ρ , g /g ρ .
These are the universal couplings between the ρ and the SM fermions, due to the mixings of ρ and SM gauge bosons which are present before the EWSB. These interactions include where f el denotes all the SM elementary fermions including the the first two generation quarks, b R , and all of the leptons. For the ρ L , the couplings are of O(g 2 /g ρ L ), while for ρ 0 R,X , they are of O(g 2 /g ρ R,X ). For the couplings between ρ and the third generation quarks, there are additional contributions of O(g ρ s 2 θ L,R ):  obtain the next-to-next-to-leading-order (NNLO) cross sections. See Appendix D for the cross sections at different proton-proton center-of-mass energies. For the decay widths, we have used the analytical formulae calculated by the FeynRules.

Production at the LHC
We start from the production of the vector resonances at the LHC. The vector resonances ρ will be dominantly produced via the Drell-Yan processes inspite of their suppressed couplings ∼ g 2 SM /g ρ to the valence quarks [14,29]. Although the ρ resonances are strongly interacting with the longitudinal SM gauge bosons, as shown in Eq. (8), the electroweak Vector-Boson-Fusion (VBF) production can barely play an useful role in the phenomenology of the ρ at the LHC [14,29]. For example, for g ρ L = 3 and M ρ L = 3 TeV, the W + W − → ρ 0 L fusion cross section is two orders of magnitude smaller than that of the Drell Yan process. In  sections are decreasing functions of the strong coupling g ρ , as expected from the coupling scaling in Tables II and III. The only exception is the production rare of the charged ρ ± R , whose couplings to the valence quarks arise after EWSB and are of order g ρ R a 2 ρ R M 2 W /M 2 ρ R . As we are fixing a ρ in the plot, the cross section is larger for larger g ρ R , as shown Fig. 1.

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We also notice that generally, ρ 0 L has one order of magnitude larger production rate than the ρ 0 R,X case because of the smallness of U (1) Y hyper-gauge coupling g in comparison with SU (2) L gauge coupling g.
In Fig 1, we have calculated the cross sections using the 4-flavor scheme. The inclusion of bottom parton distribution function (PDF) will increase the cross sections of ρ 0 L,R,X . As shown in Table III, the ρ 0 L,R,X b LbL couplings in models with quartet top partners have contributions of O(g ρ s 2 θ L ) due to the mixing of b L and B L , which can considerably enhance the cross section in some parameter space. For example, in LP 4 , for y L = 1 and M 4 = 1 TeV, g ρ L = 3, and M ρ L = 3 TeV, the bb fusion can increase σ(pp → ρ 0 L ) by 34%. In the following section, when we will study the bounds from the searches at the LHC, we also include the bb fusion production.
The production of fermion resonances can be categorized into QCD pair production and electroweak single production processes (see Ref. [17] for detail). The QCD production rate depends only on the mass of top partners. Since two heavy fermions are produced, the rate drops rapidly when the resonance's mass increases because of the PDF suppression. In contrast, the single production channels typically have larger rates in the high mass region, thus it can play an important role in the search for heavier resonance [17,[30][31][32][33][34][35][36][37][38]. This effect can be clearly seen from the For these plots, we have chosen the following parameters: where the parameter y R or y 2L is determined by the top mass requirement for the partially composite t (P) R in Eq. (B18) (the "P 4 scenario") or for the fully composite t (F) R in Eq. (B21) (the "F 4 scenario"), respectively. For the single production, we have combined the contribution of the top parters and their anti-particles. For example, for the charge-5/3 resonance X 5/3 in the quartet case, the tW fusion process is defined as The tW → B and tZ → T, X 2/3 processes are defined in a similar way. Figure 2 shows that, for both P 4 and F 4 scenarios, tW → X 5/3 has the largest production rate among the 4 single production channels of the quartet fermionic resonances, and it dominates over the QCD pair production channel for M 4 1 TeV. Although the tW → X 5/3 rates of those two scenarios are similar under our parameter choice, the rate of tW → B channel in P 4 scenario is less than that in F 4 scenario. This is because the former is from the composite-elementary R (see Eq. (B17)) and proportional to c 2 θ L , while the latter is mainly controlled by the strong dynamics term −( √ (B20)) without such suppression. As c θ L will increase with M 4 , we see the values of the two green lines in Fig. 2(a) and Fig. 2(b) become similar at large M 4 . By naively using the Goldstone equivalence theorem, we expect if y L f /M 4 1 and the mass splittings of the top partners become negligible. From the figures we find that in the F 4 scenario it is indeed the case, but in the P 4 scenario it is not. The reasons is that in the P 4 scenario, large M 4 requires large y R to correctly reproduce the mass of the top quark (see Eq. (B18)), which results in a large mixing between the T and We emphasize that the single production rates are more model-dependent. For example, the tZ/W fusion rates in P 4 scenario increase when y L decreases. This is because the constraint from observed top quark mass requires a larger y R as y L decreases, while the fusion rates are proportional to (y R ) 2 . But in the F 4 scenario, the cross sections are rather insensitive to y L , since they are mainly determined by the c 2 term.
Similar to the quartet case, the single production mechanism of the singlet top partner T dominates over the QCD pair production if it is heavier than O(1) TeV, as shown in Fig. 3.
Besides the tZ → T fusion, the singlet can also be produced by bW fusion: In fact, the cross section of this channel is about an order of magnitude larger than the tZ fusion due to the large bottom PDF, as can be seen from the red solid lines in Fig. 3.
Note that for the partially composite t Let's start from the ρ L (3, 1) resonances in models LP 4 and LF 4 . In Fig. 4, we have plotted the decay branching ratios of ρ ±,0 L as functions of M ρ ±,0 L , choosing the following parameters: The parameter f is determined by Eq. (B14) and the parameters y R (LP 4 ), y 2L (LF 4 ) are fixed by reproducing the observed top quark running mass M t = 150 GeV at the TeV scale.
Several comments are in order. In the low mass region M ρ L < M 4 , ρ L can only decay into SM final states. Since we are interested in the mass region M ρ M W,Z,h , we can neglect all the SM masses. Hence, the decaying branching ratios are completely determined by the couplings among ρ L and SM particles. As discussed above, only couplings belong to the first class and are enhanced by the strong coupling g ρ L . Besides this, there are ρ LqL q L couplings, where q L are third generation left-handed quarks. They are of O(g ρ L s 2 θ L ) and can be relevant for the moderate size of s θ L . Therefore, the dominant decay channels for this mass region are as shown in Fig. 4. There are no significant differences between the two models in this kinematical region. From the Goldstone equivalence theorem, the decay branching ratio of (8)). We only plot the sum of the two channels in Fig. 4. The same argument applied to the W + W − , Zh decay channels of ρ 0 L . We also notice that for the SM light fermion channels, we have the accidental relations Br(ρ + L → jj) = 2 × Br(ρ + L → + ν ) and Br(ρ 0 L → jj) = 2 × Br(ρ 0 L → + − + ν ν ) as illustrated by Ref. [14,39].
For the intermediate mass region, i.e. M 4 < M ρ L < 2M 4 , the decay channels with one third generation quark and one top partner (the "heavy-light" channels) are open kinematically. For the charged resonance ρ + L , we have plotted the sum of branching ratios of the decay channels tB and Tb and the sum of the decay channels X 5/3t and X 2/3b . For the neutral resonances ρ 0 L , we have combined the channels tT and bB and their charge conjugate processes. Let's start the discussion from the model LP 4 . The branching ratios of such channels grow quickly once they are kinematically open. This rapid increase is due to the strong coupling enhancement. At the same time, there is also a difference between the tB+Tb channels and the X 5/3t +X 2/3b channels. The branching ratio for the former increases as M ρ + L becomes larger, while the branching ratio of the latter increases at the beginning then decreases as the mass of ρ + L increase. We first note that the couplings ρ + L X 5/3t , ρ + L X 2/3b are suppressed by the fine-tuning parameter ξ = v 2 /f 2 (see Table II). Since g ρ L and a ρ L are fixed, increasing mass M ρ + L will result in an increasing of the decay constant f and a smaller ξ parameter. The same behavior is also observed in the neutral resonance ρ 0 L decay channels of tT +bB and X 2/3t and their charge conjugates due to similar reasons. There is a difference here between the two models LP(F) 4 . For the partially composite t (P) R scenario, the decay channels ρ + L → X 5/3t + X 2/3b and ρ 0 L → X 2/3t +X 2/3 t can become sizable ∼ 10%. However, for the fully composite t (F) R , their branching ratios are below 1%. This is due to the fact that the couplings ρ + as can be seen clearly from Table II and Table III. We also notice that ρ + L → tb, ρ 0 L → tt + bb decay channels are always sizable even in the intermediate mass region and the high mass region M ρ L > 2M 4 . This is due to the fact that we are fixing y L and M 4 . Hence, increasing M ρ L will also increase f . As a result, the left-handed mixing angle s θ L becomes larger. The branching ratio ranges from 20% to 40% in the intermediate mass region and above 10% in the high mass region.
For the mass region of M ρ L > 2M 4 , the pure strong dynamics channels are kinematically allowed. Since their couplings are of O(g ρ ) and we expect that they will dominate. Among those channels, the ρ + L → X 5/3X2/3 channel has the largest branching ratios (above 60%), because they are the first and second lightest top partners. Note that the decaying channel into TB opens very slowly. In the parameter space under consideration, its branching ratio is always below 10% and smaller than those of the decay channels tb and tB + Tb. This behavior is due to the particular choice of our parameters in Eq. (20). In particular, the masses of T, B are roughly given by Even for large M ρ L , the masses of T, B are ∼ 0.47 × M ρ L and the decay into TB suffers from phase space suppression. We also expect that other choices of the parameters (for example smaller value of y L ) will make this channel more relevant. Things are similar in the case of ρ 0 L , where the decay channels intoX 5/3 X 5/3 ,X 2/3 X 2/3 are dominant (> 60%) andT T +BB decaying channels are below 10%.
Next we turn to the (1, 3) resonances ρ ±,0 R . The benchmark point is the same as that in the ρ ±,0 L case, with the replacement ρ L → ρ R . Unlike ρ + L , the ρ + R does not mix with SM gauge bosons before EWSB because of its quantum number. Consequently, its decay branching ratios to SM light fermions are tiny. For example, it is less than 10 −3 for the parameter space shown in Fig. 5(a) and 5(c). The decaying branching ratio into tb is also suppressed because the corresponding coupling arises after EWSB and is of order O(g ρ R ξ). As a consequence, the ρ + R mainly decays into di-boson channels W + h + W + Z. In the intermediate mass region, the decaying into X 5/3t + X 2/3b channels dominate over all the other channels with branching ratio larger than 90% in both model LP 4 and LF 4 , as their left-handed couplings arise before EWSB. The decay channels into tB + Tb are very small (2% − 4%) for model LP 4 and below 10 −3 for model LF 4 . In the high-mass region, the dominant decaying channels are X 5/3T + X 2/3B and X 5/3t + X 2/3b with similar branching ratios. It is interesting to see that the heavy-light decay channel is still sizable in the high-mass region, as the mixing angle s θ L becomes larger for larger ρ + R mass and the mass of T , B increase with M ρ R as discussed before. The neutral resonance ρ 0 R mixes with the SM Hypercharge gauge boson before EWSB, resulting in the relation Br(jj) = 22/27 × Br( + − + ν ν ) [39], as shown in Fig. 5(b) and 5(d). The branching ratios of the other decay channels of ρ 0 R are very similar to those of ρ 0 L , and we will not discuss them further. Finally, we study the (1, 1) resonance ρ 0 X . As an SO(4) singlet, the ρ 0 X can couple either to quartet Ψ 4 or to the singlet Ψ 1 , and the corresponding models are XP(F) 4 and XP(F) 1 , respectively. In our plots, the parameters chosen are very similar to the benchmark point of ρ ±,0 L , except for XP 4 where we choose y L = 1.5. For the XF 4,1 model, there is another parameter c 1 describing the direct interaction between the fully composite t (F) R and the ρ 0 X resonance, and it is set to be 1. For the XF 1 model, we further set c 1 (the parameter describing the interaction between t (F) R and the ρ 0 X , Ψ 1 resonances) to be 1. Since the U (1) X has no direct connection to the dynamical symmetry breaking SO(5) → SO(4), its corresponding spin-1 resonance ρ 0 X does not couple to the Goldstone boson H before EWSB. Consequently, the decaying branching ratios into SM di-bosons W + W − + Zh are very small (< 10 −4 ). The di-fermion decay channels of XP(F) 4 are very analogous to those of ρ 0 R in RP(F) 4 . The most relevant channels are ρ 0 X → tt + bb in the low-mass region, ρ 0 X → tT +tT + bB +bB in the intermediate mass region, and ρ 0 in the high-mass region . In models with singlet top partner XP(F) 1 , since the b quark does not mix with the resonance, we classify as one of the "SM light fermions". Therefore, we have Br(jj) = Br( + − + ν ν ), as shown in the bottom panel of Fig. 6. In model XP 1 , the dominant decaying channels are ρ 0 X → tt in the low-mass region, ρ 0 X →t T + T t (∼ 70%) in the intermediate mass region and ρ 0 X → T T (∼ 70%) in the high mass region. The situation is similar in the model XF 1 except that in the high-mass region, the ρ 0 X → tt and ρ 0 X →t T + T t decaying channels are also relevant. Their branching ratios are around 20% and 40%, respectively.

III. THE PRESENT LIMITS AND PROSPECTIVE REACHES AT THE LHC
In this section, we present the current limits and prospective reaches for the simplified models at the LHC.

A. Making projections
For the projections at the high luminosity or high energy LHC, we extrapolate from the current LHC searches using a similar method as in Ref. [40]. We described the method in detail in Appendix E.
There have been a number of searches for beyond the SM (BSM) resonances at the LHC, providing constraints to the composite Higgs models. To use a more generic and uniform notation in describing the searches, we denote the spin-1 resonances as ρ and the spin-1/2 resonances as F Q , where Q is the electric charge. The results at the 13 TeV LHC can be classified into two main groups. The first group is the Drell-Yan production and two-body decay of ρ, its various final states can be summarized as follows, 1. SM di-fermion final states, including di-lepton, di-jet, and the third generation quarkinvolved channels. We list the relevant measurements in Table V.  [67,68]. Hereafter, for simplicity the charge conjugate of the particle decay final state is always implied; for example, tF 2/3 denotes both tF 2/3 andtF 2/3 .
For the new channels we propose in this paper, especially the cascade decays of the ρ resonances to the heavy fermionic resonances, there have been no dedicated searches. We estimate their exclusion by recasting existing searches using the SSDL final states ± ± +jets. In The results of LP 4 and LF 4 are shown in Fig. 7(a) and 7(b), respectively. Since g ρ L is fixed, f is determined by M ρ L , and we use its value to label the top horizontal axis.
We plot the existing bounds from LHC searches and their extrapolations at 300 (3000) fb −1 in colored shaded regions. Besides the direct searches for resonances, the measurement of and ξ parameter can provide an indirect constraint. Currently LHC results imply ξ 0.13 [18,19], while the further constraints are expected to be as good as 0.066 (0.04) with 300 (3000) fb −1 of data [86][87][88]. We also plot the constraints on ξ in the figures as vertical black thin lines.
Putting all the constraints and projections together, we see that the future data at the LHC will explore the parameter space of LP(F) 4 extensively 4 . The constraints are similar in the two models LP(F) 4 . For a relatively large value of g ρ L (for example, g ρ L = 3 in our benchmark point), the most sensitive channel in the M ρ L < 2M X 5/3 region is the W ± Z/W + W − search with boosted di-jet channel performed by the ATLAS Collaboration with integrated luminosity L = 79.8 fb −1 in Ref. [11]. If g ρ L 2, the di-lepton + − channel by CMS with L = 77.3 fb −1 (e + e − ) + 36.3 fb −1 (µ + µ − ) [12] gives the strongest limit. Because of the large experimental uncertainty, the ρ → tt, bb and tb channels are not able to give competitive limits, although they have significant branching ratios. In Fig. 7, we only show the present limits and prospective reaches from ATLAS di-boson boosted jet channels in Ref. [11]. It is clear from the figure that the interactions with light top-partner has affected 3 Smaller value of g ρ L will make Drell-Yan di-lepton resonance search more relevant. Large g ρ L will make the production cross section too small to have relevant effects at the LHC. 4 The left hand side of the figures start from M ρ L = 2 TeV. TheŜ-parameter constraint roughly gives us M ρ L > 1.9 TeV in our parameter choice and we didn't show it [89,90].   QCD pair production of X 5/3X5/3 [13]. The event number contours for N ( ± ± + jets) = 20 are plotted in solid (dashed) lines for 300 (3000) fb −1 , to set a prospective limit for the proposed channels, including ρ ± L → tB/X 5/3t (denoted as ρ L → tF ), ρ ±,0 L → X 5/3X2/3 /X 5/3X5/3 (denoted as ρ L → F F ) and tW → X 5/3 . See the main text for more details.
the phenomenology of ρ L significantly. In particular, the present bound is relaxed from 4.2 TeV to 2.6 TeV for our benchmark parameters in Eq. (23) as the mass of top partner In the mass region of M X 5/3 < M ρ L 2M X 5/3 , the decays of ρ L into one top partner and one SM particles are kinematically allowed. The ttZ final state from the decay channel ρ 0 L → tT has been studied both experimentally [67,91] and theoretically [92], but current experimental results are still too weak to be visible in Fig. 7. The ρ ± L → Tb → tbZ channel is studied phenomenologically in Ref. [93]. In this work, we propose that the ttW ± → ± ± + jets final state from ρ ± L → tB/tX 5/3 can also be a good channel to probe such a heavy-light decay. In Fig. 7, we have plotted the contours for the constant number (= 20) of SSDL events summing all these decay channels at 300 fb −1 and 3 ab −1 LHC. These channels have sensitivity to the parameter space up to M ρ L = 3.8 TeV at 3 ab −1 LHC, but it still can't compete with the di-boson jet searches. This is due to the fact that the branching ratios into the heavy-light channels are not significantly larger than the di-boson channel and the decaying branching ratios to the SSDL are very small. It is interesting to explore other more complicated final states like 1 + jets and we leave this for future possible work.
In the mass region of M ρ L > 2M X 5/3 , the spin-1 resonances will decay dominantly into pairs of top partners, as discussed in detail in Sec. II B 2. We focus here on the decay channels resulting in the SSDL final states: ρ ±,0 L → X 5/3X2/3 /X 5/3X5/3 (see also Refs. [93,94] for the study of these channels). We plot the contours with 20 SSDL events, summing over all the above decay channels for 300 fb −1 and 3 ab −1 LHC. The prospective for the cascade decay channels are very promising and comparable with direct searches for the pair produced X 5/3 .
If the top partner is around 1 TeV, these channels can be promising to discover the heavy spin-1 resonance 5 . Note that in such region the Γ ρ L /M ρ L can be as large as 30%. It is interesting to study the effects of large decay width on the resonance searches and we leave this for a future work.
We have shown the present bounds and the prospectives of the searches for QCD pair produced X 5/3X5/3 in the 1 + jets final state by CMS [13] 6 . The single top partner production may play an important role in the relatively high top parter mass region as discussed in Sec. II B 1. Currently, the tZ → T /X 2/3 channel has been searched by CMS at 35.9 fb −1 [67], and the tW → X 5/3 channel has been searched by CMS at 35.9 fb −1 [83] in 1 + jets final state and by ATLAS at 36.1 fb −1 [78] in SSDL final state. However, the mass reaches of all those searches are still too low to be visible in our figures. Instead, in Fig. 7 we present the contours with constant number of events (= 20) in the tW → X 5/3 → ± ± +jets final states as a projection for the future run of the LHC. The reach in model LP 4 range 5 If the first generation light quarks have some degrees of compositeness as studied in Ref. [95], the cascade decay channels are more important as the Drell-Yan cross sections of ρ L are enhanced by the extra piece of coupling of O(g ρ s θ1q ). Here θ 1q is the mixing angle between the first generation quark and the corresponding partners. 6 See also Refs. [96,97] for the phenomenological study of these channels.

C. The results of RP 4 and RF 4
We now turn to discuss the models RP(F) 4 . Similar to the cases of LP(F) 4 , we have set the following parameters as and scanned over (M ρ R , M X 5/3 ). The results are plotted in Fig. 8. The meanings of the shaded regions and contour lines are similar to those in Fig. 7. Note that we have started from M ρ R from 1 TeV. Because the production cross sections of charged ρ ± R resonances are very small, we only use the searches for the Drell-Yan production of ρ 0 R at the LHC. Similar to the search for the ρ L resonances, the di-boson channel provides the strongest constraints in the region of M ρ R < 2M 4 . Among the existing limits, we found that the diboson resonance searches by ATLAS in the semi-leptonic channel [54] and in the fully hadronic channel in [11] give the strongest constraints, and their results are similar. Here we show the limits from results of Ref. [54]. As expected, due to the smallness of hypercharge gauge coupling, the bound is weaker than the ρ L resonances. The present bound is around 1.6 TeV and will reach 3.8 TeV at the HL-LHC. In the mass region of M X 5/3 < M ρ R 2M X 5/3 , the ρ 0 R → tX 2/3 → ttZ may be relevant, but the current search in Ref. [67] is still not possible to put any relevant constraint in our parameter space. Thus, it is not shown in the figure.
In the mass region of M ρ R > 2M X 5/3 , the cascade decay channel ρ 0 R → X 5/3X5/3 in the SSDL final state is not comparable with the searches for the QCD pair X 5/3 production, due to the smallness of the production cross section. We can also read from the figure that the electroweak precisionŜ-parameter measured by LEP sets a strong constraint on the models with ρ R , requiring M ρ R 1.95 TeV, which is heavier than current experimental reach. However, the reach of LHC with an integrated luminosity of 300 fb −1 could surpass this constraint. The bounds for the top parters are the same as models LP(F) 4 and not discussed here anymore.

D. The results of XP 4 and XF 4
We now turn to the models with a singlet vector resonance ρ 0 X . In this subsection we will discuss its interactions with the quartet top partner in models XP(F) 4 , while in the next subsection we will investigate its interactions with the singlet top partner XP(F) 1 . As discussed in Ref. [14], ρ X only contributes to the Y -parameter of the electroweak precision test (see also Eq. (B33)). Due to the (g /g ρ X ) 2 suppression, the indirect constraint on the ρ X is weak. As a result, ρ X could be very light especially in the case of large g ρ X . We choose the benchmark values for the parameters as g ρ X = 3, a 2 ρ X = 1 4 , y L = 1, c 1 = 1, c 1 = c 2 = 1 (for XF 4 only), and scan over (M ρ X , M X 5/3 ) in Fig. 9. Note that we have chosen a slightly smaller value of a ρ X in order to relax the bound from ξ measurement. Here we can see a difference [QCD] [13] are plotted as shaded regions. The green regions come from ttρ 0 X associated production, by the phenomenological study of Ref. [98]. The purple regions represent the limit from the + − search [12] and its extrapolations. The contours for N ( ± ± + jets) = 20 are drawn with solid (dashed) lines for 300 (3000) fb −1 , as a prospective reach for the ρ 0 X → X 5/3X5/3 (denoted as ρ X → F F ) and the tW → X 5/3 channels. See the text for more details.
between the partially composite t (P) R and the fully composite t (F) R scenario. While the dilepton channel [12] can play an important role in model XP 4 in the large M X 5/3 region (i.e. M X 5/3 > M ρ X ), it won't put any significant constraint on the model XF 4 . This is due to the fact that the branching ratio of di-lepton in the model XP 4 scales like [g /(g ρ X s θ L )] 4 , while in model XF 4 , it scales like (g /g ρ X ) 4 . As we fix y L , larger value of M X 5/3 will induce smaller value of s θ L and an enhancement of the di-lepton branching ratio in model XP 4 .
Note that in the region M ρ X M X 5/3 where ρ 0 X only decays to SM particles, the tt and bb channels dominate. The sensitivity in these channels at the 13 TeV LHC is roughly three order of magnitude worse than the di-lepton channel, assuming the same branching ratios.
Thus they can only play a role in the large g ρ X region. However, large g ρ X will lead to small Drell-Yan production cross section and make tt, bb channels not relevant in our parameter space. In contrast, the authors of Ref. [98] have pointed out that the pp → ttρ 0 X → tttt channel with the SSDL final states can probe the fully composite t (F) R scenario very well, as the production cross section scales like g 2 ρ X . In Fig. 9, we have reinterpreted the results of Ref. [98] in our parameter space in model XF 4 . We see that ρ 0 X with mass below 2 (2.4) TeV can be probed at 300 (3000) fb −1 LHC with our choice of g ρ X = 3 in model XF 4 . While for model XP 4 , the bound (not shown in the figure) is weaker (∼ 1.0 TeV at 3 ab −1 ) due to the suppression of ρ X tt couplings either by the t L − T L mixing or the B µ − ρ Xµ mixing. We can also see that the limits from ttρ X channel become stronger in the low M X 5/3 region in model and the choice for y L in Eq. (26) has fixed s θ R ∼ 0.6. This means that the couplings of the interactions ρ XtR t R , ρ XtR T R are roughly constants with varying mass of the top partner (see Table IV). In both models, the Drell-Yan production of the ρ X can't play an important role in our interested parameter space, because of the lack of the sensitivity to the dominant decay channel tt and the suppression of the decay branching ratio into the di-lepton final state. In Fig. 10, we have shown the reach from the ttρ X production with the SSDL channel, including the analysis of Ref. [98]   the strong interaction in the fully composite t (F) R scenario (c 1 term in Eq. (B36)). For the top partner, we present the current limits and prospective reaches coming from the ATLAS searches for the QCD pair production of the top partner with the bW +b W − (1 + jets) final states [71]. Note that the single top partner searche performed by ATLAS in Ref. [81] with integrated luminosity L = 3.2 fb −1 using the bW (→ ν) decay channel is not sensitive to our parameter space yet 7 . Instead, we find that the cascade decay of the top partner T into ρ X t with ρ X decaying into top pair in the single production channel can become relevant in the mass region of M T > M ρ X . For example, for M T = 2 TeV and M ρ X = 1 TeV, the branching ratio can reach 65.8% (93.8%) for XP(F) 1 in our parameter choice, due to the large coupling of ρ X t R T R in both models. Moreover, it will lead to the SSDL signature. In Fig. 10, we have estimated the reach of this channel with SSDL searches at the LHC with integrated luminosities 300 fb −1 and 3 ab −1 . This channel is very promising, and can become comparable with the four top final states in both models, especially in XP 1 . This is due to the fact that in model XP 1 , the branching ratio of this cascade decay channel is further enhanced by the s 2 θ R suppression oft R ρ 0 X coupling, as can be seen from Table IV. is useful for XP 4 , while the ttρ 0 X associated production is useful for XF 4 and XF(P) 1 , as the cross section scales like g 2 ρ X (g 2 ρ X s 4 θ R ) and it can lead to four top final states with SSDL signature. We have recasted the analysis of Ref. [98] in this SSDL channels in our parameter space. The cascade decaying channels (heavy-light and heavy-heavy) in models XP(F) 4 can rarely play an important role because the cross section is small in the high mass region, and the very light top partners have already been excluded by the present experiments. In models XP(F) 1 , we find that the SSDL final states from the bW → T → tρ 0 X process can be very important in the M ρ X < M T region, while the SSDL channel of ttρ X → ttt T can be relevant in intermediate mass region. Finally, the QCD pair production of top partners offers a robust probe for the models. At the same time, the singly produced channels have a much higher mass reach. For example, for the models with quartet top partners, the QCD pair channel and tW → X 5/3 channel could probe the parameter M X 5/3 up to ∼ 2 TeV and (a) The results of LF 4 . reaches of the mass scale from present and future searches at he LHC are summarized in Fig. 12 (for models LP(F) 4 and RP(F) 4 ) and in Fig. 13 (for models XP(F) 4 and XP(F) 1 ).

G. Future colliders
Before we conclude our study, we make some estimates of the prospective reaches on the mass scales in our models at the 27 TeV HE-LHC and 100 TeV pp collider. In Fig. 11, we have used the method described in Appendix E to extrapolate, based one the di-boson boosted-jet resonance searches at ATLAS [11] and the pair top partner searches in the 1 + jets channel at CMS [13] in model LF 4 . We present the results with the integrated   1, 1). In addition, we have also studied the two scenarios depending on whether the right-handed top quark is elementary or fully composite.
We have categorized the couplings of the composite resonances into four classes according  Table II, Table III and   Table IV. Based on the discussion of the couplings, we have studied different production and decay channels for the composite resonances, paying special attention to the relevance of the cascade decay channels between the composite resonances. We have shown the present and future prospective bounds on our parameter space in the M ρ − M Ψ plane in different models, focusing on the moderate large coupling g ρ = 3. We found that the cascade decay channels into one top partner and one top quark tΨ or two top partners ΨΨ strongly affect the phenomenology of the ρ if they are kinematically open. Their presence significantly weakens the reach of the channels with only SM particles, such as the di-boson channel. In addition, the decay channels ρ + L → tB/X 5/3t and ρ + L → X 5/3X2/3 , ρ 0 L,R,X → X 5/3X5/3 can lead to the SSDL final states, which are used as an estimate of the reach on the M ρ − M Ψ plane.
We found that they are comparable in some regions of the parameter space to the di-boson searches or the top partner searches at the HL-LHC, especially for the ρ L models LP(F) 4 .
For the ρ R,X models RP(F) 4 , XP(F) 4 , because the Drell-Yan production is suppressed by the smallness of the hypercharge gauge coupling, the cascade decay channels play less important roles. We also find that the SSDL channels in the single production of the charge-5/3 top partner X 5/3 can always play an important role in our parameter spaces. In the models involving the singlet spin-1 resonance XP(F) 4 and XP(F) 1 , the associated production of top pair and the ρ X with the four top final states can play an important role, as the coupling between ρ X andtt is of O(g ρ ) for the fully composite t (F) R models and O(g ρ s 2 θ L ) or O(g ρ s 2 θ R ) for the partially composite t (P) R models. We have recast the analysis in the SSDL channel by Ref. [98] in our parameter space. In models XP(F) 1 , the single production of the top partner T , followed by cascade decaying into tρ X (tt) can be important in the region M T > M ρ X , and we have explored its sensitivity in the SSDL channel. It can be better than the ttρ X (tt) SSDL channel in model XP 1 . In the mass region M T < M ρ X < 2M T , the tt fusion production of ρ X , which decays into t T , can lead to the tttbW + final state with SSDL signature. We have used this to explore its sensitivity. In Fig. 12 and Fig. 13, we have summarized the prospective reach on the mass scale M ρ and M Ψ by the different existing searches at the LHC and by various SSDL channels from the cascade decays.
Several directions should be explored further. Among the various cascade decay channels, we have only considered the SSDL final state. The reach obtained this way is conservative.
Other decay final states, such as 1 +jets, should also be studied in detail. The final kinematical variables are usually very complicated, and new techniques such as machine learning may be useful to enhance the sensitiy. We hope to address the issues in a future work.

V. ACKNOWLEDGEMENT
We would like to thank Andrea Tesi for the collaboration in the early stage of this work. The last gauge field X µ , corresponding to the U (1) X group, is introduced to give correct hypercharge for the fermions, and the Goldstone bosons are neutral under this symmetry.
The full formulae of d µ and e µ symbols can be obtained as follows [104] where the covariant derivative is given by: and the matrices t a L/R are defined in Eq. (A2). Because of Eq. (A5), the leading Lagrangian of the Goldstone fields is simply For the fermionic heavy resonances, they fall into the irreducible representations of the unbroken group SO(4)×U (1) X SU (2) L ×SU (2) R ×U (1) X . We will consider two irreducible representations: the quartet 4 2/3 and the singlet 1 2/3 as the lightest top partners. They are parametrized as follows: and transform as Ψ → H r Ψ ⊗ G X Ψ, where r Ψ is the SO(4) representation of Ψ, and G X denotes the group element of U (1) X . From the transformation rules in Eq. (A5), we can construct a covariant derivative acting on the composite fermionic fields Ψ: Taking into account of the U (1) X group, the covariant derivative becomes (∇ µ − ig 1 XB µ ).

The matching to the Higgs doublet notation
The CCWZ operators and the effective Lagrangians for the composite resonances can be written in terms of the fields that have the definite quantum number under the SM gauge group SU (2) L × U (1) Y . To see this, we first notice that the SM Higgs doublet with hypercharge Y = 1/2 can be written as follows: It is related with the quartet notation h by an unitary matrix P with determinant -1: The SO(4) generators can be converted to the doublet notation by using P : Consequently, the h covariant derivative term can be rewritten as: where the D µ in the right-hand side of the equation is the normal SM covariant derivative: where hypercharge Y is given by Y = T 3 R + X. Using above results, we can easily rewrite the leading Lagrangian in Eq. (A9) in the doublet notation: For further convenience, we list the following useful identities: where the ↔ D µ is defined as: The quartet top partner fields, Ψ 4 can be decomposed as two SU (2) L doublets with hypercharge Y = 1/6, 7/6 as follows: with the same P matrix as defined in Eq. (A13). The SM fermions are assumed to be embedded in the 5 X representation of SO(5) × U (1) X with hypercharge given by Y = T 3 R + X. We only consider the top sector in our paper. For the SM SU (2) doublet q L = (t L , b L ) T , we have the embedding: The q 5 L formally transforms under the G ∈ SO(5) and G X ∈ U (1) X as q 5 L → G ⊗ G X q 5 L . For the right-handed top quark, we will consider two possibilities: t R as an elementary filed or as a massless bound state of the strong sector. In the first case, we also embed it in the representation of 5 2/3 : For the fully composite right-handed top quark, we assume that it is a singlet of SO(4), denoted as t (F) R and its interactions preserve the non-linearized SO(5). We denote those two treatments as partially and fully composite t R scenario, respectively.
All the effective Lagrangian in MCHMs can be rewritten in terms of the doublet notation easily using Eq. (A7), Eq. (A18), Eq. (A20), Eq. (A21) and Eq. (A22). The full results are tedious, thus we will not list them here; however, their LO expansions in H † H/f 2 order will be listed and discussed in Appendix B.
In this section, we briefly describe the models considered in our paper (see Refs. [14,17,20]). We focus on the minimal coset SO(5) × U (1) X /SO(4) × U (1) X of the strong sector, where the Higgs bosons are the pseudo-Nambu-Goldstone bosons associated with this global symmetry breaking.
1. The models involving ρ L (3, 1) and quartet top partners Ψ 4 (2, 2): LP(F) 4 We start from the models involving the ρ L and the quartet top partners Ψ 4 . The Lagrangian of the strong sector reads: where the field strength of the spin-1 resonance is defined as The Yukawa interactions between strong and elementary sector are: The fully Lagrangian is then written as [14,17,20] where we omitted the SM Lagrangians for the quark fields q L and t R . Note that the CCWZ covariant objects e a µ include the SM gauge fields: and we have written the formulae in terms of SM Higgs doublet H (see Appendix A for the definition and derivation). Note that the SM gauge interactions don't preserve the nonlinearly realized SO(5) symmetry and provide the explicit breaking, thus will contribute to the Higgs potential at one-loop level. The term with coefficient c 1 involves the direct coupling between the ρ L and the quartet top partners at the order of g ρ L . As discussed in Ref. [14], this interaction will have an important impact on the phenomenology of ρ L especially when m ρ L > 2M 4 and decaying into two top partners are allowed. In most of the case, we will choose c 1 = 1 as our benchmark point.
Note that the mass term for the ρ L in Eq. (B1) will induce a linear mixing between them and the SM W µ gauge bosons before EWSB. Diagonalizing the mass matrix will lead to the partial compositeness of O(g 2 /g ρ L ) for the W bosons. As a result, the SM SU (2) L gauge coupling will be redefined as follows: and the W -mass at the LO is given by (see Appendix C for detail): Due to the linear mixing, the mass of the ρ L will also be modified as follows: Note that this direct mixing mass term will also lead to contribution toŜ-parameter in the low energy observable. Actually, integrating out the ρ L at the LO, we will obtain the O W operator (see Ref. [14]), which leads to the contribution to theŜ parameter [89]: The ρ L resonance will be coupled to SM fermions universally with strength of O(g 2 /g ρ L ) due to the linear mixing. The non-universality comes from the linear mixing between the SM fermions and corresponding composite partners. Since the mixing is the source of the SM fermion masses after EWSB, it is roughly the order of the fermion Yukawa couplings.
Thus we expect that only the third generation mixings (especially the top quark) have the important impact on phenomenology of the ρ L , which is the reason we only focus on the top sector.
For the partially composite right-handed top quark scenario, we have two parameters y L , y R controlling the mixing between q L , t R and the top partner Ψ 4 . Similar to the SM gauge bosons, there will be direct mixing between q L and the composite SU (2) L doublet Q before EWSB proportional to y L : where the doublet Q = (T, B) T is defined in Eq. (A20). This motives us to define a lefthanded mixing angle θ L as follows: which measures the partial compositeness of the SM fermions q L . Due to the linear mixing, the mass formulae for the fermionic resonances before EWSB are given by: Note that y L breaks the SO(4) explicitly and will contribute to theT parameter at the loop level, thus can't be too large. In contrast, t (P) R is an SO(4) singlet so that y R term preserves the custodial symmetry can in principle can be large [107]. For the fully composite t (F) R , besides the mixing between q L and Ψ 4 (denoted also as y L ), we can write a direct coupling y 2L between q L and t (F) R . This term provides the main source of top quark mass. Since t (F) R belongs to the strong sector, there are also direct interactions between it and the composite resonances, which are written as the c 2 term in the L F 4 . As discussed in Ref. [17], this strong interaction term provides the dominant contribution to decay of the top partners, especially when the mixing parameters are small.
Note that it will be very useful to rewrite the Lagrangian in terms of SM SU (2) L × U (1) Y notation, where the SM gauge symmetries are manifest. By using the formulae of the Goldstone matrix U and the d µ , e µ in the Appendix A, we can write the Lagrangian L L 4 using the doublet notation as follows: where the · · · denotes the higher order terms in H † H/f 2 and we have defined the O(1) parameter a ρ L as in Ref. [20]: From the dimension-six operators involving the top partners and the Higgs fields, we can see that generally the gauge couplings of the top partners are modified at the O(ξ) after EWSB. Note that there is an accidental parity symmetry P LR in the kinetic Lagrangian for the quartet top partner defined as [108]: and the couplings between eigenstates of this parity (X 5/3 , B) and the SM Z gauge bosons will not obtain any modification after EWSB. This can be easily seen by using the formulae for the currents in the vacuum: remembering that T 3 L (X 5/3 ) = T 3 R (X 5/3 ) = 1/2 and T 3 L (B) = T 3 R (B) = −1/2. This is important because ZB LBL are not modified by the Higgs VEV means that after the mixing between b L and B L , the Zb LbL remains the same as the SM canonical couplings 8 . But the presence of c 1 term will break this parity, as a result, Zb L b L coupling obtains O(c 1 gs 2 θ L ξ) correction after EWSB (see Table III). Although this provides another strong bound on the left-handed top quark mixing, it can be relaxed by including the ρ R and impose the P LR parity.
Similarly, we can write the elementary-composite mixing Lagrangian L P 4 in the doublet notation: where we only keep the leading terms in the expansion of H † H/f 2 . We can see clearly that after EWSB only the mass matrix in the top sector obtains corrections of O(y L f ξ, y R v), while for the charge −1/3 and charge-5/3 resonances, their mass formulae are not modified 9 . After EWSB, the top mass is given by: where s θ L denotes sin θ L defined in Eq. (B11). The EWPT at the LEP prefers y L y R , thus y R mixing term is dominant. In the unitary gauge, this term becomes: So in the large y R limit, there will be a top partner (the heavier one) in the mass eigenstate, which will primarily decay into th and the other one will primarily decay into tZ. See Appendix C for detail, where we summarize the mass matrices and mass formulae. As we will discuss below, in our consideration, we will focus on the region y R 1, this effect will not be manifest. For the fully composite t (F) R case, we have: .
The top mass to the leading order is given by: where c θ L denotes cos θ L defined in Eq. (B11). So that the top Yukawa coupling is mainly determined by y 2L , which is different with partially composite t (P) R case. For the ρ R models, the effective Lagrangians read: (B22) 9 Since we don't include the right-handed bottom quark mixings with bottom partners, the bottom quark remains massless.
where the definition of ρ a R µν is the same as in Eq. (B2) with (L → R). The effective Lagrangians in models RP(F) 4 are given by: where the Lagrangians L P(F) 4 are the same as in Eq. (B1). In terms of doublet notation, we have: where we only show the terms involving the ρ R and defined: Note that similar with ρ L , there is a direct mixing between ρ 3R µ and the hypercharge field B µ . So the U (1) Y gauge coupling is redefined as follows: and the Z-mass to the LO is given by: Note that this direct mixing mass term will also lead to contribution toŜ-parameter in the low energy observable: integrating out the ρ R will result in the O B operator and As can been seen from Eq. (B24), for the neutral resonance ρ 3 R , it has the universal coupling of O(g 2 /g ρ R ) to the SM fermions, while for the charged ρ R , its coupling arise from O(ξ). This makes ρ 0 R more produced at the LHC than the charged one and thus the most stringent constraint on the ρ R models comes from the neutral spin-1 resonance searches. Because of the smallness of U (1) Y gauge coupling g compared with SU (2) L gauge coupling g, its constraints are weaker than ρ L . For the direct interactions with the fermionic resonances (the c 1 term), they are similar to the ρ L interactions except that the charged currents are between Q and Q X .
3. The models involving ρ X (1, 1) : XP(F) 4 and XP(F) 1 For the models involving the ρ X and the quartet Ψ 4 , the Lagrangian containing the ρ X are given by: where ρ Xµν = ∂ µ ρ Xν − ∂ ν ρ Xµ , and where the Lagrangians L P(F) 4 are the same as in Eq. (B1). Similar to ρ 3 R µ , ρ Xµ is mixing with the hypercharge gauge field B µ , thus will have a universal coupling of O(g 2 /g ρ X ) to the SM elementary fermions. The U (1) Y gauge coupling g is redefined as: (B31) Similar to the case of ρ L,R , we will also define the O(1) parameter a ρ X as follows: ρ X will not contribute toŜ-parameter because of its singlet nature, but will contribute to the Y -parameter (defined in Ref. [89]) as follows: The extra suppression factor (g /g ρ X ) 2 will make the constraint on the mass of the ρ X from EWPT much weaker than ρ L,R . For the case of fully composite right-handed top quark, a direct interaction term between ρ X and t (F) R can be written down. The coefficient is denoted as c 1 in Eq. (B30). This term is special in the sense that it can affect the decay of ρ X and also can lead to a new production mechanism of ρ X : tt fusion. The decay of ρ X into a pair of top quark will result in four top final states, which can be probed using the SSDL final state [98].
Finally, we consider the models involving ρ X and the singlet Ψ 1 . The Lagrangian involving the heavy resoances read: The mixing term is given by: and the effective Lagrangians in models XP(F) 1 are: Note that here besides the c 1 term, we also have the non-diagonalized interaction, i.e. the c 1 term. The mixing term between the elementary SM quarks and the composite fields can be rewritten in terms of doublet notation. The results read: For the model XP 1 , the linear mixing term between t (P) R and the singlet T will lead to the partial compositeness of the right-handed top quark with mixing angle θ R : The top partner mass and the top mass will become: For the fully composite t (F) R , the top mass is simply: In both XP 1 and XF 1 models, the y L mixing term controls the top partner T decay, as this is the leading term with trilinear interactions violating the top partner fermion number. By using the Goldstone equivalence theorem, we can easily see the following branching ratios for the decay of the singlet T : where the factor 2 in the branching ratios comes from the √ 2 suppression of the real scalar fields compared with complex scalar fields.
Appendix C: The mass matrices and the mass eigenstates Before EWSB, the mixing between the composite resonances and SM particles can be easily and exactly solved, as stated in Appendix B of this paper. However, after EWSB, i.e. h = (0, 0, 0, h ) T , all particles with the same electric charge and spin will be generally mixed, and it is impossible to analytically resolve the mixing matrices exactly. In this section, we list all mass matrices after EWSB, and use perturbation method to derive the mass eigenvalues up to ξ = v 2 /f 2 level.

The spin-1 resonances
Due to the SM gauge quantum number, ρ a L L mixes with W a L , while ρ 3 R R and ρ 0 X mix with B before EWSB, and the mixing angles are determined by tan θ ρ = g SM /g ρ . The VEV of Higgs will provide O(ξ) modifications to such pictures. Below, we will give the mass eigenvalues up to ξ level for the vector bosons.
a. The ρ L (3, 1) resonance After EWSB, the mass terms of vector bosons are where M 2 L± =   1 4 f 2 g 2 2 a 2 ρ L −ξ + 2 and By using ξ as the expanding parameter, we can diagonalize above matrices perturbatively.
Up to ξ order, the mass eigenvalues of the SM gauge bosons are and the photon is massless, due to the residual electromagnetic gauge invariance. Note that theT -parameter is 0, as expected. For the spin-1 resonances, the mass eigenvalues are We can obtain the mass terms from the Lagrangian as follows: between b L and B L is not affected by the EWSB and has been exactly solved in Appendix B; while in the singlet case, b L quark has no mixing in the unitary gauge (in our massless b approximation). Below we just discuss the mass matrices of charge-2/3 fermions.
a. The Ψ 4 (2, 2) resonance In the quartet case, the charge-2/3 mass term of top sector is where the mass matrices are Those M P(F) 4 2/3 's are not symmetric. Thus, instead of diagonalization, we should do the singular value decomposition, i.e. finding unitary matrices U t and V t such that U † t M P(F) 4 2/3 V t is diagonal. Up to ξ level, for partially composite t (P) R scenario we have while for fully composite t In this scenario, the lightest charge-2/3 top partner X 2/3 has degenerate mass with X 5/3 up to ξ order.
b. The Ψ 1 (1, 1) resonance The fermion mass term is where Singular value decomposition is used to find the mass eigenvalue, and up to ξ order for P 1 , and for F 1 , Appendix D: The NNLO cross sections for QCD pair production of the top partners In this appendix, we list the cross section for the QCD pair production of the top parters.
They are calculated using Top++2.0 package, at NNLO level with next-to-next-to-leading logarithmic soft-gluon resummation [23][24][25][26][27][28]. The results are shown in Table VIII.  In this appendix, we sketch the method we used to extrapolate the existing searches to the future high luminosity or high energy LHC. We refer the reader to Ref. [40] for the detailed description of the method. The basic assumption of the method is that the same number of background events in the signal region of two searches with different luminosity and collider energy will result in the same upper limit on the number of signal events. To be specific, from an existing resonance search at collider energy √ s 0 with integrated luminosity L 0 , we can obtain the 95% CL upper limit on the σ × Br for a given channel for the mass Note that Eq. (E1) can be further expressed as an identity involving the parton luminosities associated with the background [40]: where dL ij /dŝ is the parton luminosity defined as [40,109]: We have chosen the factorization scale to be the partonic center-of-mass energy √ŝ . Note that if the signal and the main background come from the same parton initial states, the method is the same as in Ref. [110].
For the QCD pair production of top partners, we have chosen an invariance mass square window around (2M F ) 2 , where M F is the mass of the top partner under consideration.