Coloured spin-1 states in composite Higgs models

Strong dynamics for composite Higgs models predict spin-1 resonances which are expected to be in the same mass range as the usually considered top-partners. We study here QCD-coloured vector and axial-vector states stemming from composite Higgs dynamics in several relevant models based on an underlying gauge-fermion description. These states can come as triplet, sextet and octet representation. All models considered have a colour octet vector state in common which can be singly produced at hadron colliders as it mixes with the gluon. We explore the rich and testable phenomenology of these coloured spin-1 states at the LHC and future colliders.


Introduction
The exploration of composite Higgs models has garnered significant attention in the realm of theoretical particle physics, as these models offer a possible explanation for the nature of the Higgs boson discovered at CERN and a dynamical origin for the breaking of the electroweak symmetry in the Standard Model (SM) [1][2][3].By positing the Higgs boson as a composite state that originates from a new strongly interacting sector, composite Higgs models provide a potential solution to the problem of hierarchy between the electroweak scale and the Planck scale: like in quantum chromodynamics (QCD), the breaking scale is dynamically generated via confinement and condensation of a new interaction.This idea is as old as the SM itself [4,5], starting from the first Higgsless (Technicolor) theories [6] and their effective Lagrangian counterparts [7], to models where the Higgs emerges as a meson [8,9].Composite model building has resumed in the early 2000's thanks to the idea of holography [10][11][12], freely adapted from supersymmetric string theory inspired duality conjectures [13].
The varied and rich phenomenology of composite Higgs models has been extensively studied, both from the point of view of holography-inspired effective models [14,15] and from models based on underlying gauge-fermion theories [16], the latter close in spirit to QCD.While we do not attempt to summarise the main features, which have been described several times, we want to recall the essential ingredients of such theories, which are directly related to the electroweak symmetry breaking.The Higgs typically emerges as a pseudo-Nambu Goldstone boson (pNGB) [10] from the spontaneous breaking of the global symmetry in the strong sector.Its potential and mass are generated by explicit breaking terms: the gauging of the electroweak symmetry, the couplings of the top quark [11] and (eventually) a mass term for the underlying fermions [17,18].In this framework, composite spin-1 states matching the electroweak gauge bosons and top partners have been widely considered.From the Higgs sector point of view, they are the minimal components required from the strong sector.
Nevertheless, the strong dynamics of composite Higgs models is much richer than this.Whether it consists of an unspecified conformal field theory in the holographic approach, or of a well-defined gauge-fermion theory, a more extended spectrum is a generic prediction.In particular, the fact that top partners [19] need to be charged under QCD interactions implies that other coloured resonances beyond the top partners must exist.This implies the presence of coloured spin-0 and spin-1 mesons, as well as fermions carrying unusual colour charges.In this work we will focus on coloured spin-1 resonances, which are expected to exist in all types of composite Higgs models.In holographic models, they emerge as Kaluza-Klein resonances of the gluon field [20].As we will show, however, a richer set of coloured spin-1 states is to be expected.
For definiteness, we will focus on theories based on an underlying gauge-fermion description, where the properties and quantum numbers of the resonances can be classified.A systematic list of models describing the minimal resonances needed by the Higgs sector has been presented by Ferretti and Karateev [21].Consistent models with a single species of fermions can only be based on SU (3) [22] -like in QCD -or G 2 [21] with fermions in the fundamental.However, models with two separate species in different irreducible representations (irreps) of the gauge group offer the intriguing possibility of sequestering QCD interactions from the sector responsible for the electroweak symmetry breaking [21,23].Theoretical and phenomenological considerations lead to the definition of 12 minimal models, whose characteristics are fully specified [24,25] in terms of the confining gauge group and the irreps and multiplicities of the two species of fermions.Upon confinement, both fermion species condense, as confirmed by Lattice results for SU(4) and Sp(4) gauge symmetries [26,27], hence generating two sets of pNGBs [24] (plus one coming from a global anomaly-free U(1) [25]).The symmetry breaking patterns are uniquely determined by the type of irrep the two species belong to [18], leading to the classification in Table 1.The top partners emerge as so called "chimera" baryons formed of the two species, where two different patterns can be realised: ψψχ and ψχχ, where ψ only carry electroweak charges while χ carry QCD colour and hypercharge.In the former case, the χ's QCD triplet carries hypercharge 2/3, in the latter case −1/3.The phenomenology of the resonances from these 12 models have been studied in the literature, covering some of the resonance types.So far, studies have focused on the pNGBs charged under electroweak quantum numbers [24,28,29], the singlets stemming from the global U(1)'s [24,25,30,31], QCD coloured pNGBs [25,32,33], top partners with non-standard decays [34][35][36] or colour assignment [37], and spin-1 resonances carrying electroweak charges [38].We also note that the spectra and couplings of such resonances can be computed on the Lattice, and some results are available for models based on Sp(4) [39][40][41][42][43][44][45][46], like models M5 and M8, and based on SU(4) [47][48][49][50][51][52][53], like models M6 and M11.Computations based on holography are also available [54][55][56][57][58][59].
In this work, we will focus on the phenomenology of spin-1 resonances that carry QCD charges and emerge as bound states of the χ species.Their properties emerge from three types of cosets, SU(6)/SO (6), SU(6)/Sp (6) and SU(3)×SU(3)/SU(3), and the hypercharge assignment for the colour triplet χ, which stems from the types of chimera baryons.The spectrum contains both a set of vectors V µ and of axial-vectors A µ , which decay respectively into two or three pNGBs.The latter property originates from the symmetric nature of the cosets.Mixing of the ubiquitous octet with the QCD gluons will also generate direct couplings to quarks, while the colour triplets and sextets may or may not couple to a pair of quarks depending on their baryon number.To properly characterise the phenomenology of these states, we will employ the hidden symmetry approach [60] to write an effective Lagrangian, and use the results to study their collider phenomenology.
The paper is structured as follows: in Section 2 we briefly review the hidden symmetry approach and present results in the allowed cases.In Section 3 we analyse the phenomenology at the LHC and future high energy proton colliders.Finally, we offer our conclusions in Section 4.

Hidden gauge symmetry approach
The hidden local symmetry method is based on the idea that the nonlinear σ model on the manifold G/H is gauge equivalent to the σ model based on G × H local .The gauge bosons corresponding to the local symmetry can be identified with composite spin-1 mesons.The general procedure for building an effective Lagrangian including these new spin-1 resonances [7,61,62] consists, therefore, in a generalised group structure that splits the unitary matrix U (x) describing the Goldstone bosons into factors transforming under an extended symmetry G ′ .
The generators of the group G can be indicated with T A where A = 1, . . ., d G and d G is the dimension of the group G.These generators can be separated into two classes, T A = {S a , X I }: the unbroken generators S a with a = 1, . . ., d H belonging to the unbroken subgroup H ⊂ G, and the broken generators X I with I = 1, . . ., d G − d H belonging to the coset G/H.The elements of G are of the form g = e iα A T A and those of H of the form h = e iβ a S a .The elements of G can be parameterised by g = U h with U in the coset G/H U = e iπ I X I . (2.1) For cosets of the type SU(N )/SO(N ), SU(2N )/Sp(2N ) and SU(N ) L × SU(N ) R /SU(N ) V , the two classes of generators are determined by the following constraints: see [63][64][65] for details.The Lagrangian in the condensate ("chiral") phase is built using the standard chiral Lagrangian elements: with Ω µ the Maurer-Cartan form and j µ the current j µ = v a µ S a + a I µ X I .The form Ω µ can be further decomposed into projections along the unbroken and broken parts: ) which will be explicitly used in writing the Lagrangian.The notation for the current j µ indicates that vector resonances v a µ are associated to the unbroken generators of H, while axial-vectors a I µ to the broken ones.This is a formal definition, while a direct correspondence to vector and axial currents of fermions is only recovered in QCD-like cases based on SU(N ) L × SU(N ) R group symmetries.
For concreteness, in the rest of the section we will provide some details on the effective construction for one of the cosets, based on SU(6)/SO (6).We will show how to extend the results to the other two cosets (c.f.Tab. 1) at the end of the section.

Setup for SU(6)/SO(6)
Following the hidden symmetry prescriptions, we consider a model based on the symmetry G ′ = SU(6) 0 × SU(6) 1 , where SU(6) 0 is partly gauged by the SM gauge bosons (gluons and hypercharge) and SU(6) 1 is fully gauged by the heavy resonances.The enlarged symmetry is broken to SO(6) 0 ×SO(6) 1 by two sets of pNGBs, π 0 and π 1 , so that a linear combination of them gives mass to the axial resonances.Furthermore, SO(6) 0 × SO(6) 1 is broken to the diagonal subgroup SO(6) by a second set of pNGBS, k, which gives mass to the vector resonances.
We parameterise the two sets of pNGBs as transforming under SU(6) i as We also define a Maurer-Cartan form for each sector: with covariant derivatives where the gauge fields act via the commutator, [G µ , U 0 ] etc, and where T X and T a G are the generators of SO(6) 0 corresponding to hypercharge and QCD colour, respectively.The colour multiplets are embedded in the SO(6) matrices as ) where ϕ 8 = 1 2 ϕ a 8 λ a with the Gell-Mann matrices λ a , ϕ 6 = ϕ T 6 , and ϕ 3 = −ϕ T 3 .To employ the CCWZ construction [66,67], we define the components of the Maurer-Cartan forms d i,µ and e i,µ parallel and orthogonal to SO (6) i as in Eqs.(2.4) and (2.5).They transform under SU(6) i as We refer the reader to Appendix A.2 for the explicit calculation of the CCWZ symbols.

The Lagrangian
From the previous considerations, and in a similar way to what was obtained in the corresponding SU(4) case in [38], the most general, leading-order Lagrangian reads: where contains all the massive resonances.We recall that for a generic gauge field V µ , where g is the appropriate coupling.In the unitary gauge, where K = 1, the kinetic term for K simplifies to Expanding the above Lagrangian will allow us to compute the mass eigenstates (elementary vectors and resonances do mix) and their couplings.

Vector boson masses and mixing
The masses and mixing of the vector resonances stem from the pNGB matrix K.The three terms in Eq. (2.22) read Tr(e 1,µ e µ 1 ) Tr(e 0,µ e µ 1 ) where, from the last line, we see that the colour octet and singlet components mix with gluons and the hypercharge gauge boson, respectively.The Lagrangian contains a simple mass term for the colour-triplet state: For the other states, a mixed mass term emerges.Starting with the colour octets: where

28)
Diagonalising the mass matrix, we find a massless eigenstate, which corresponds to the physical gluon octet, and a massive state, corresponding to the octet vector resonance.The latter have mass With some abuse of notation, we can switch to the physical mass eigenstates by replacing (2.30) Finally, the gauge coupling associated to the massless gluons reads and this corresponds to the physical coupling of QCD interactions.
A similar mixing pattern emerges in the singlet, leading to with the caveat that the hypercharge will also mix with a spin-1 resonance stemming from the electroweak sector of the composite theory.Such a mixing has been studied for the SU(4)/Sp(4) coset in [38].Combining the two sectors will, therefore, lead to a more complicated mixing pattern.We will not further pursue the analysis of the electroweak sector in this work, as we are interested in the phenomenology of the coloured resonances, which are more abundantly produced at hadron colliders.

Axial masses and scalar mixing
From the d 2 1 term, we obtain a mass for the axial vectors: while a mixing with π 0 is generated by the d 0,µ d µ 1 term.The mixing terms can be removed with an appropriate choice of gauge fixing, leaving a common mass term for all the axial vectors: (2.34) The mesons π 0 and π 1 undergo a non-trivial mixing, analogous to the case studied in [38], hence we will simply recall the basics here.As the d i forms give at leading order in the expansion, the Lagrangian contains a kinetic mixing of the form: Hence, one can define decoupled and canonically normalised fields π A and π B as ) A linear combination of these states is eaten by the A µ .The physical π P and unphysical π U states are given by where Combining the above redefinitions yields In the unitary gauge, only the π P remain in the spectrum, and they correspond to the pNGBs from the coset SU(6)/SO(6).

Decay channels
We are now ready to determine the main decay modes for the heavy spin-1 resonances.They are generated by three types of interactions: • Couplings to pNGBs from the chiral Lagrangian in the strong sector, Eq. (2.19); • Couplings to quarks via the mixing of the colour octet to gluons; • Partial compositeness couplings to top and bottom quarks.
The first type stems directly from the pNGB embedding in the effective Lagrangian.We recall that, in the unitary gauge, the relevant terms simplify to (2.42) We are interested in terms linear in the vector fields V µ /A µ and with the smallest number of pNGBs.It turns out that these only come as two independent traces: where V = V, G, B is a generic vector.We recover explicitly that vectors couple to two pNGBs, while axial resonances can only couple to three pNGBs.Both O V and O A are hermitian.After transforming the pions and vectors to the physical fields, we find that these operators come with coefficients where R = rf 1 /f 0 .The details of this calculation are presented in Appendix A.4.The colour structure of the couplings among the various components are determined uniquely by the above traces in the SU(6) space.
The second type of couplings originates from the mixing of the gluon with V 8 , hence yielding a universal coupling of the massive resonance to quarks: where t a 3 = λ a /2 are the colour generators for the fundamental irrep.Note that the massless octet inherits a coupling ĝs cos β 8 ≡ g s , hence consistent with QCD gauge invariance.A coupling to two gluons, instead, is not generated, as shown in Appendix A. 3.
Finally, the third type is generated by the coupling of the spin-1 resonances to the baryons [68,69] that mix to top quarks via the partial compositeness mechanism.While the couplings generated by the strong dynamics are inherently vector-like, the chiral mixing of the physical states generates chiral couplings to the mass eigenstates.Details of the origin of these couplings are presented in Appendix A.5.Such couplings always exist for the colour octet states, and they can be parameterised as where P L,R are chiral projectors and we only consider the electric part of the coupling.
In the models under consideration, we have that g ρ/abb,LL ≃ g ρ/att,LL while the bottom coupling is only left-handed at leading order.Note that all the above couplings are of order g, while the chiralities are distinguished by the different mixing angles from partial compositeness.The non-octet resonances, V 3 and A 6 , couple to a pair of quarks via partial compositeness only in models where the two resonances have baryon number 2/3 and charge ±4/3, hence leading to two top decay channels.The effective couplings can be parameterised as where the superscript c indicates charge conjugation.As the currents contain effectively one left-handed and one right-handed top, the couplings must be suppressed by the EW scale v divided by the Higgs decay constant as compared to the octet couplings.Note finally that, while the first two types of couplings are completely determined by the chiral Lagrangian in Eq. (2. 19), the third one is more model dependent.In fact, the value of the couplings depend on the quantum numbers of baryons that mix with the elementary top fields, and on the value of the mixing angles. 1 Hence, they cannot be predicted in a model-independent way and we will leave them as free parameters.

Independent parameters
The effective Lagrangian for the coloured spin-1 resonances contains six free parameters: f 0 , f 1 , f K , r, ĝs , and g (the mixing angles depend only on g).As we have seen, ĝs can be fixed by the physical coupling of the massless gluons, as in Eq. (2.31).We can trade f 1 and f K for masses: hence we can choose as input parameters M V 8 and the ratio Note that the relation between the two vector masses only depends on the octet mixing angle, i.e. on g, as M V 3 = M V 8 cos β 8 .We can further use as an input the physical decay constant of the pNGBs f χ , which enters the couplings of the physical π P states and reads: (2.53) Finally, another input parameter can be the coupling of the vectors to the pion, g ρππ , which can be measured on the lattice, for instance.It relates to the Lagrangian parameters as follows: .
(2.54) Solving Eqs.(2.53) and (2.54) for f 0 and r yields In summary, this leaves us with five independent input parameters: As already mentioned, in addition we have the couplings to top and bottom quarks generated by top partial compositeness.For the SU(6)/Sp(6) case, the computation of the effective Lagrangian follows the same patterns as described above, with the only difference in the broken and unbroken generators.
Effectively, this implies that colour charges of the non-octet states are interchanged: V 6 and A 3 (as well as π 3 ).The coefficients of the various couplings and mass values, however, follow the same results as above.
For the case SU(3) × SU(3)/SU(3), the action of the symmetries are slightly different in form.However, for this coset all vector and axial resonances, as well as the pNGBs, transform as octets.Hence, the effective interactions are the same as above, once the non-octet states are removed.

Phenomenology
Table 2: Properties of the spin-0 (π), spin-1 (V µ , A µ ) and spin-1/2 (Ψ) lightest resonances in the 12 models, grouped in 5 classes.Each class is determined by the properties of the χ species, listed in the second column by irrep type (R for real, Pr for pseudo-real and C for complex).For the resonances, the colours indicate the baryon numbers, with black for B = 0, red for B = ±1/3 and blue for B = ±2/3.In the last column we indicate the bosons that can decay into a di-quark state (tt).
The twelve models under consideration allow us to predict the quantum numbers of the lightest coloured resonances.Following the properties of the fermion species χ, they can be grouped into five classes, as shown in Tab. 2. For the fermionic states, the electroweak charges depend on the configuration of the ψ fermions inside the chimera baryons, and a full classification is possible, but beyond our purposes.In fact, we will assume here a lattice and QCD inspired mass hierarchy, where the baryon-like states are heavier than the spin-1 states, which are heavier than the pNGBs.Henceforth, the heavy baryons do not have a direct relevance for the phenomenology of the spin-1 states, except for the fact that their couplings can generate a direct coupling of the spin-1 resonances to a pair of tops via the top partial compositeness mixing, as discussed in the previous section.
The coloured spin-1 resonances, therefore, can be produced via their QCD interactions at hadron colliders.This leads to pair production for all types of states.The only one that also features single-production is the vector colour octet, as it inherits a universal coupling to all quarks via its mixing to the gluons.As the masses of the spin-1 resonances are expected to be of the same order, we will first study the LHC limits on the vector colour octet to determine the smallest allowed mass.Before doing that, however, it is important to recall the properties of the coloured pNGBs, which appear in the decays of all spin-1 resonances.Finally we will present first results for future high energy hadron colliders, which could access pair production of all the resonances.

Coloured pNGB decays
The phenomenology of the coloured pNGBs have been studied in several works and contexts [25,32,33], hence we will here only remind their main features.
A colour octet pNGB is ubiquitous to all models.It always features two types of couplings: a coupling to gauge bosons generated by a topological anomaly and one to tops generated by partial compositeness [25,33].Which one dominates, however, depends crucially on the details of the model, as their origin is rather different in nature.Note that the anomaly dominantly consists of couplings to two gluons, however it also generates suppressed couplings to gγ and gZ, which provide interesting and clean final states [33,70].Nevertheless, to simplify the analysis and focus on existing searches, we will neglect the single-gluon decay channels in the following.
The decays of the non-octets depend crucially on the scenario at hand.Following the classification in Table 2, we distinguish four cases: C2 : C3 : C4-5 : In C2, the sextet has baryon number 2/3 and charge 4/3, hence partial compositeness will generate an unsuppressed coupling to two right-handed tops [32].In C1 and C3, the sextet and triplet have baryon number ∓1/3 and charges ∓2/3, respectively, hence they are not allowed to decay into standard model fermions by partial compositeness alone.Their decays must, therefore, be generated by specific operators that need to violate either baryon or lepton number.Considering the standard model gauge quantum numbers [71], the allowed final states are listed above.The di-quark final state violates baryon number by one unit, ∆B = 1, and we consider preferential couplings to heavier flavours (while this is not strictly required).For the triplet, decays to a quark and a lepton can be envisioned, violating lepton number by one unit, ∆L = 1.They can be generated in some models by partial compositeness extended to leptons [37], hence naturally involving the third (heavier) family.In such case, the triplet effectively behaves like a composite leptoquark [72].We remark that the B or L violating couplings can be rather small, however they provide the only decay channel for the sextet and triplet states.Depending on the value of these couplings, therefore, they decay promptly, as we consider in the following, or lead to displaced vertices and anomalously massive hadronic tracks [73].
The ubiquitous colour octet can be searched at the LHC via QCD pair production: in the following, we will assume dominant top couplings, hence leading to a four tops final state.A recent reinterpretation [74] of a CMS search [75] leads to a conservative lower bound of 1.25 TeV for the colour octet mass.Note that in C2 models, the contribution of the sextet can further push up this limit.Dedicated searches also exist for the leptoquark decays of the triplet in C3 models, where both bτ and tν final states have been searched for by ATLAS and CMS [76][77][78][79][80] yielding bounds between 1.25 and 1.46 TeV, depending on the branching ratios in the two channels.The bounds on this mass are significantly lower of about 770 GeV if π 3 decays dominantly into light quarks [81,82].

Colour octet single production at the LHC
Via mixing to the gluons, the vector octet V 8 inherits a coupling to all quarks, see Eq. (2.48), allowing it to be singly produced.This coupling is suppressed by a mixing angle that depends only on g, hence it is reduced for large g, see also Eq. (2.31).Nevertheless, even for moderate values, the single production cross section dominates over pair production.Henceforth, the colour octet is the main resonance to be hunted at hadron colliders, and it has been considered in the literature in various composite contexts [83].Typically, decays into two quarks are considered, while we will also include decays into two coloured pNGBs as shown in Fig. 1.
In the models under consideration, the possible decay modes of the vector octet can be classified as follows: C1-2 : C3 : C4-5 : where C1 and C2 are distinguished by the decays of the sextet pNGB.The decays into light quarks q = u, d, c, s feature flavour-independent branching ratios, while bottom and top quark channels receive additional contributions from partial compositeness, see Eq. (2.49), leading to different branching ratios.Finally, the relative strength of the pNGB channels is determined purely by colour factors, assuming their masses are equal, and we find The importance of each channel depends on the parameter space, and we provide some benchmarks in Fig. 2. The relevance of the decays into light quarks and pNGBs depends mainly on the g and g ρππ couplings.On the one hand, the partial width to light quarks is controlled by the mixing angle to gluons and it decreases for increasing g.On the other hand, the partial width to pNGBs receives a dominant contribution proportional to g ρππ : the dependence on g is such that this partial width also decreases for increasing g.For very small g, instead, the second term in Eq. (2.46) starts becoming relevant, thus explaining the drop in the qq branching ratio observed in Fig. 2. The scaling in g also explains why the total width of V 8 increases for small values and for large octet masses.Finally, the branching ratio to top (and bottom) receives a dominant contribution from partial compositeness, which do not scale with g and hence dominates for large values.One caveat is that the coupling g ρππ and the coupling to baryons are expected to be large in the strong theory, hence the colour octet will tend to have large width as compared to its mass.To quantify this important effect, we show in Fig. 3 curves of fixed width over mass ratios for different values of g as a function of g ρππ and the octet mass.The plots highlight the fact that the width can be larger than 50% of the mass, especially for small values of g, hence invalidating the treatment of the octet in a standard narrow width approximation.
Finally, the current bounds on the colour octet mass crucially depend on the branching ratios in the three channels: light quarks, pNGBs and tops, as they are controlled by different couplings.As a model-independent estimate of the bounds we, therefore, decided to show limits assuming 100% branching ratios in the three channels, as shown in Fig. 4 for the four nonequivalent classes C1, C2, C3 and C4-5.The lines are extracted from the following searches: • di-jet: Search for high mass di-quark resonances [84]; • di-top: Search for t t resonances [85]; • pNGBs: Recasts of SUSY searches [86][87][88] implemented in CheckMATE 2 .
The coloured heatmap indicates the Drell-Yan cross section, which only depends on g and the octet mass, while the region below and to the left of the lines is excluded.The results show that the mass limits are roughly the same for all cases, and comparable for the three decay modes.Hence, the mass limits are in the range of 4 to 5 TeV.Note that the region for small g ≲ 3 cannot be trusted as it corresponds to widths above 50% of the mass, c.f. Fig. 3.
2 For the simulation of signal events, we implemented the relevant interactions as a FeynRules [89] model at leading order.For each mass point, we generate 10 4 events using MadGraph5 aMC@NLO [90] version 3.5.3, in association with the parton densities in the NNPDF 2.3 set [91,92].We then interfaced the events with Pythia8 [93] for showering and hadronisation.The resulting showered signal events are analysed with CheckMATE [94,95] (commit number 1cb3f7).To this end, events are reconstructed using Delphes 3 [96] and the anti-kT algorithm [97] implemented in FastJet [98].We also ran the events against the searches and SM measurements implemented in MadAnalysis5 [99][100][101][102]    The High-Luminosity run at the LHC will certainly allow to further improve the mass limits on the vector octet, with the caveat that dedicated searches or reinterpretations will be needed to take into account the large width.However, even with the current bounds in Fig. 4, we can infer that pair production of all spin-1 resonances will be very small at the LHC, hence making their detection unlikely.In the next section, therefore, we will discuss pair production at a future high energy hadron collider.

Pair production at future high-energy hadron colliders
Future hadron collider projects are expected to reach energies well above the LHC, with expectations up to 100 TeV [106,107].At such energies, pair production of the vector and axial resonances will be accessible.In principle, single production of the vector octet remains the leading channel, with the caveat that at large masses the width will also increase and hence affect the search strategy.In this section, we will focus on pair production.The cross sections only depend on the QCD quantum numbers of the spin-1 states, and they are the same for vector and axial-vector states.In Fig. 5 we show them as a function of the masses for pp collisions at 100 TeV centre of mass, using the NNPDF 2.3 PDF set [91,92].Hence, the most abundantly produced states will be the sextets, followed by octets, while triplet production is about one order of magnitude below.
In the following, we describe the main features to be expected from pair production, leaving a detailed analysis for future work.Firstly, we should stress that single production of the vector octet remains the leading discovery channel, including non-resonant effects that are relevant in the case of very large width.Hence, at a future 100 TeV collider, one would expect to discover the octet before pair production becomes relevant.
The sextets, which feature the largest pair production cross sections, are present in the classes C1, C2 and C3.The decays can be classified as follows: A 6 → π 8 π 8 π 6 (t tt tbb) and π 6 π c 6 π 6 ( bb bbbb) in C1 , (3.9) A 6 → tt or π 8 π 8 π 6 (t tt ttt) and π 6 π c 6 π 6 ( tt tttt) in C2 , (3.10) The two-body decay of V 6 in C3 is driven by g ρππ , hence is will likely produce a large decay width.Instead, thanks to the three body final state for A 6 in pNGBs, we expect the axial widths to remain small compared to the mass.In the cases C2, a competitive decay into two tops is also present: the two-body top decay width is, in fact, suppressed by v 2 /f 2 ≤ 0.04 (for f ≥ 1 TeV), while the three-pNGB channel is suppressed by a phase space factor as compared to the two-body channel.Hence, we expect the two to lead to competitive branching ratios.In both cases C1 and C2, the pair production of the sextet will lead to final states with many top and bottom jets.The axial colour octet A 8 also has a sizeable pair cross section, and it also has a leading decay channel into two tops in all cases.Hence, it will generate 4-top final states with potentially large width effects.
Finally, the colour triplets, present in C1, C2 and C3, lead to the following decays: ) A 3 → π 8 π 8 π 3 (t tt t + bs or ql) or π 3 π c 3 π 3 ( bsbs bs or qlqlql) in C3 . (3.14) For the vectors, the decays will be largely dominated by the π 8 π c 6 channels, as the di-top coupling in C2 is suppressed by v/f .The caveat remains that the vector widths may be large in most of the allowed parameter space.Instead, the axial in C3 remains narrow, leading to interesting final states rich in tops and possibly leptons, if the π 3 decays violate lepton number.

Conclusions
We have investigated the phenomenology of spin-1 resonances in Composite Higgs Models carrying QCD charges, with particular attention on production and decay modes, LHC bounds, and future hadron collider prospects.We have in particular focused on models which allow for fermionic UV completions [24,25] as they provide detailed information on the quantum numbers and properties of the bound states.We have worked out their properties for three types of cosets, SU(6)/SO(6), SU(6)/Sp(6) and SU(3) × SU(3)/SU(3) and the most relevant production and decay channels at present and future pp-colliders.The considered cosets are symmetric and they, therefore, contain two sets of spin-1 resonances: vector states that couple to two pNGBs and axial-vector states that couple to three pNGBs.
In all scenarios, the vector V 8 in the adjoint representations of colour SU(3) is present, and it mixes with the QCD gluon octet.Thanks to the mixing, this state can be singly produced at hadron colliders via Drell-Yan, whereas all other states can only be pairproduced.The V 8 can either decay into a quark pair or into two pNGBs leading in all cases to the final states q q (q ̸ = t), t t and 4t.In a coset with triplet or sextet pNGBs, in addition one has a subset of the following final states: 4b, 2b2s or 2t2ν, 2b2τ and tνbτ .We have investigated in all cases bounds on the mass ranging from 3.5 TeV to 6 TeV from existing LHC data.We have focused on scenarios where V 8 has a sufficiently small decay width so that the narrow width approximation holds.Hence, pair production is only relevant for a future high energy hadron collider, where pairs of the sextets and octets will be abundantly produced.We classified all permitted final states, which are typically rich in top quarks and leptons.Further studies, including the large width case and specific prospects for future 100 TeV pp-collider, as well as more model dependent signatures will be analysed in a separate publication.DFG Cluster of Excellence PRISMA+ (Project ID 39083149) for its hospitality and support during the final stages of this work.This work has been supported by the "DAAD, Frankreich" and "Partenariat Hubert Curien (PHC)" PROCOPE 2021-2023, project number 57561441.M.K. and W.P. are supported by DFG, project no.PO-1337/12-1.M.K. is supported by the "Studienstiftung des deutschen Volkes".

A.1 Conventions
It is convenient to embed a field ϕ r in the r irrep of QCD within 3 × 3 matrices in the notation of [71]: where and The matrices are normalised as follows: For the SU(6)/SO(6) coset, the fields can be embedded within the symmetric and antisymmetric two-index irrep as follows: (A.7)

A.2 CCWZ symbols
In the first sector of the hidden-symmetry extended coset, the Maurer-Cartan form reads: where the dots indicate higher orders in the pNGB fields.Reading off the components for d 0,µ and e 0,µ , we find: For the second sector, containing the heavy spin-1 resonances, we find: Reading off the two components: The elements above are used as building blocks for the effective Lagrangian we used in the main text. 1 And finally e 0 e 1 : Finally we take into account the V µ 8 -G µ and the V µ 1 -B µ mixings: where V 3 (V 6 ) contains both 3 and 3 (6 and 6).All in all, the decays into pNGBs are described by with coefficients (gc 8 + g s s 8 ) + 2 1 + R 2 1 − R 2 g s s 8 , (A.51) (gc 8 + g s s 8 ) + 2 1 + R 2 1 − R 2 g s s 8 , (A.52) , (A.53) We recall that the singlet will have additional mixing in the electroweak sector of the theory, which we do not include here.Finally we have to calculate the operators O V/A .In the main text, we focus on the phenomenology of the V 8 , so we calculate O V 8 .In the SU(6)/SO (6)

Figure 1 :
Figure 1: Feynman diagrams of V 8 single production and decay into quarks or pNGBs.

6 V 8 3 c 3 Figure 2 :
Figure 2: Sample branching ratios for g ρππ = 1 and M V 8 = 4.5 TeV.We also fix m π = 1.4 TeV, f χ = 1 TeV and the coupling to top quarks to 1.Note that here q = u, d, s, c, b.

5 Figure 3 :
Figure 3: Isocurves of width/mass for ξ = 1.0 and f χ = 1 TeV for different values of g for classes C1-2.The coupling to top quarks is fixed to 1.

Figure 4 :
Figure 4: Bounds on vector octet single production for the model classes defined in Tab. 2. The heat map and the dotted contours indicate the single production cross section.The region to the left and below the coloured lines is excluded.The bounds are determined assuming 100% branching ratio into the indicated channel.For the decays in pNGBs, the branching ratios in Eq. (3.8) are taken into account.

pp 6 c 6 pp 3 c 3 Figure 5 :
Figure 5: Pair production cross section at √ s = 100 TeV for vectors in the sextet, octet and triplet representations.The same values apply for axial vectors.

Table 1 :
Classification of the 12 models based on the global symmetry breaking patterns in the electroweak and QCD sectors.In parenthesis, we indicate the template for the chimera baryons, representing the top partners.