Vector and Axial-vector resonances in composite models of the Higgs boson

We provide a non-linear realisation of composite Higgs models in the context of the SU(4)/Sp(4) symmetry breaking pattern, where the effective Lagrangian of the spin-0 and spin-1 resonances is constructed via the CCWZ prescription using the Hidden Symmetry formalism. We investigate the EWPT constraints by accounting the effects from reduced Higgs couplings and integrating out heavy spin-1 resonances. This theory emerges from an underlying theory of gauge interactions with fermions, thus first principle lattice results predict the massive spectrum in composite Higgs models. This model can be used as a template for the phenomenology of composite Higgs models at the LHC and at future 100 TeV colliders, as well as for other application. In this work, we focus on the formalism for spin-1 resonances and their bounds from di-lepton and di-boson searches at the LHC.


I. INTRODUCTION
Effective Lagrangian approaches have played a major role in various physics applications to model unknown sectors or situations that are simply difficult to treat. In more mature fields, they also provide a simple and more easily calculable tool describing a detailed and complex underlying theory. This is, for example, the case for the strong interactions of Quantum ChromoDynamics (QCD) that are described, at low energy, by a chiral Lagrangian incorporating the light composite degrees of freedom of the theory. The power of the chiral Lagrangian stands on a well defined expansion scheme that allows for very accurate calculations. A similar pattern can be followed in the electroweak sector, and attempts to describe the Standard Model (SM) in this way have been numerous since the the beginning [1, 2].
In particular, shortly after the theoretical establishment of the SM, the idea that QCD itself may play the role of a template for a composite origin of the electroweak symmetry breaking has been gaining popularity, leading to rescaled-QCD Technicolour models [3][4][5].
In this set up, the longitudinal degrees of freedom of the massive W and Z are accounted for as Goldstone bosons, i.e. pions, of the strong sector. While nowadays it is accepted that rescaled-QCD does not describe the physical reality, especially due to its Higgs-less nature, issues with generating quark masses [2, 6] and the correct flavour structure [7] and precision tests [8], new versions of such theories are gaining momentum.
The main break-through can be traced back to the idea that the Higgs too can be described as a pseudo Nambu-Goldstone boson (pNGB) of an enlarged flavour symmetry [9,10], and the coset SU(5)/SO(5), containing a singlet and a 9-plet of the custodial SO(4)∼SU(2) L ×SU(2) R together with the Higgs doublet, has been one of the first candidates [11]. For recent reviews on the developments occurred in the last decade, we refer the reader to Ref.s [12,13]. The realisation that the minimal symmetry breaking, SO(5)/SO(4), embedding custodial symmetry, contains only a Higgs boson at low energy [14] has inspired the construction of effective descriptions of the electroweak (EW) sector of the SM which do not contain more states [15][16][17]. Other composite states which are not pNGBs, like spin-1 resonances, or the so-called top partners [12] (needed in the partial compositeness scenario [18]), are typically heavier than a few TeV and thus their effect at low energy can be embedded in higher order operators. Yet, at the energy reached by the LHC, their direct production is a crucial test of the theories. A very rich literature is already available, 2 and we list here a forcibly incomplete list of papers addressing various issues on the experimental tests of composite top partners [19][20][21][22][23][24][25], aka vector-like fermions [26,27], spin-1 resonances [28][29][30], or additional scalars [31,32].
Models beyond the minimal case are interesting as they contain additional light scalars [33][34][35], among which a Dark Matter candidate may arise [36,37]. One possible way to discriminate among them is the requirement that they arise from an underlying theory consisting on a confining fermionic gauge theory. In this sense, the coset SU(4)/Sp (4) (equivalent to SO(6)/SO(5)) can be considered the minimal one, also arising from a very simple underlying theory [38,39] based on a confining SU(2) Yang-Mills theory with 2 Dirac fermions in the fundamental representation. Besides being a template for a Composite Higgs model [40], this theory has also been used as a simple realisation of the SIMP Dark Matter candidate [41] (for a critical assessment, see [42]). In this paper we want to extend the effective field theory studies of this template by adding the lowest-lying spin-1 resonances, i.e. vector and axial-vector states. We follow the CCWZ [43,44] prescription by employing the hidden symmetry technique [45]. While we focus on the Composite Higgs scenario with the scope of studying the phenomenology of such states at the LHC and at future higher energy colliders, our construction can be also applied to other phenomenological uses of this simple theory [41,[46][47][48].
After reviewing the basic properties of the SU(4)/Sp(4) coset, in Section II we construct an effective Lagrangian for the spin-1 states. In Section III we provide details of the properties of the physical states and connect them with the simplest underlying theory in Section IV. Finally, we briefly study the collider phenomenology in Section V, focusing both on di-lepton constraints at the LHC and on prospects for the future 100 TeV proton collider.

A. Vacuum alignment structure
The vacuum structure of the SU(4)/Sp(4) model has already been extensively studied [33,39,49], so here we will briefly recap the main features. In this work, we will follow the prescription that the pNGB fields are defined around a true vacuum which includes the source of electroweak symmetry breaking, as in Ref.s [39,40]. It can be thus shown that the vacuum alignment can be described in terms of a single parameter, θ, and in the SU (4) 3 space it looks like where u H is an SU(4) rotation along a direction in the space defined by the generators transforming like a Brout-Englert-Higgs doublet. For θ = 0, i.e. u H (0) = 1, the electroweak symmetry is unbroken once the SU(2) L and U(1) Y generators are embedded in SU(4) as In the phase where θ is non-zero, the gauged generators of SU (4) are no more aligned with the 10 unbroken generators V a defined as where Y a are the 5 broken ones (explicit matrices can be found in [40]). As mentioned above, the pNGBs are defined around the θ-dependent vacuum Σ 0 as 1 where the would-be Higgs boson is identified with h = π 4 , the singlet η = π 5 , and the remaining 3 are exact Goldstones eaten by the massive W and Z. Also, f π corresponds to the decay constant 2 of the pNGBs, and it is related to the electroweak scale via θ: The alignment along θ, together with the masses of the two physical pNGB, is then fixed by a potential generated by explicit breaking terms of the SU(4) flavour symmetry: the gauging of the electroweak symmetry, Yukawa couplings for the top (above all) and a mass for the underlying fermions (which is allowed by the symmetries as the underlying theory is vectorial). The details of this potential are not essential for this paper: details can be found in [40] for the case where the fermion mass is used as a stabiliser, and in [49] for the case where top partners are present and used to fine tune the top loops.
1 Other parameterisation have been used in the literature where the pNGBs are defined around the θ = 0 vacuum, and the "Higgs" one is then assigned a vacuum expectation value [49]. The main differences lie in higher order interactions, see [37]. We prefer this approach because it sequesters the explicit breaking of the Goldstone shift symmetry to the potential terms that generate the pNGB masses. 2 Here, we adopt the standard normalisation used in Technicolour literature, and other composite Higgs literature: the difference with the f used in [40,50] is f π = 2 √ 2f .

II. EFFECTIVE LAGRANGIAN
To describe the new strong sector and remain as general as possible, a chiral-type theory can be constructed on the basis of custodial symmetry and gauge invariance. The simplest construction one can imagine uses a local copy of the global SU(2) "chiral" symmetry and builds the relevant invariants [51]. The same results follow from the hidden gauge symmetry approach [45]. Furthermore the global flavour symmetry can be enlarged in different ways, depending on the required model-building features (see for example the early attempts in [52,53]). To this basic idea one can add the Higgs boson as a pseudo-Nambu-Goldstone boson [9,10] or as a massive composite state, or as a superposition of both [40]. In the following we shall consider a model with vector and axial-vector particles: for a template description of these resonances based on the SU(2) group see [54]. In order to describe this kind of spectrum, we introduce a local copy of the global symmetry. When the new vector and axial-vector particles decouple, one obtains the non-linear sigma-model Lagrangian, describing the Goldstone bosons associated to the breaking of the starting symmetry to a smaller one. The approach we use is the standard one of the hidden gauge symmetry [45] (for an alternative, equivalent, way, see [55]) 3 .
In our specific case, i.e. the minimal model with a fermionic gauge theory as underlying description, the global symmetry SU(4)/Sp(4) is extended in order to contain, initially, two SU(4) i , i = 0, 1. The SU(4) 0 corresponds to the usual global symmetry leading to the Higgs as a composite pNGB, and the electroweak gauge bosons are introduced via its partial gauging. The new symmetry SU(4) 1 allows us to introduce a new set of massive "gauge" bosons, transforming as a complete adjoint of SU (4), which correspond to the spin-1 resonances in this model.

A. Lagrangian
Following the prescription of the hidden gauge symmetry formalism, we enhance the symmetry group SU(4) to SU(4) 0 × SU(4) 1 , and embed the SM gauge bosons in SU(4) 0 and the heavy resonances in SU(4) 1 . The low energy Lagrangian is then characterised in 3 In that approach only one set of pNGBs is used for the CCWZ prescription, and the mass term for the axial-vectors f 2 a 2∆ 2 (g a a µ − ∆d µ ) 2 will give rise to a bilinear mixing of a 4 µ ∂ µ h.

5
terms of the breaking of the extended symmetry down to a single Sp(4): the SU(4) i are spontaneously broken to Sp(4) i via the introduction of 2 matrices U i containing 5 pNGBs each. The remaining Sp(4) 0 × Sp(4) 1 is then spontaneously broken to Sp(4) by a sigma field K, containing 10 pNGBs corresponding to the generators of Sp(4).
The 5+5 pNGBs associated to the generators in SU(4)/Sp(4) are parameterised by the following matrices: that transform nonlinearly as where g i is an element of SU(4) i and h the corresponding transformation in the subgroup Sp(4) i . It is convenient to define the gauged Maurer-Cartan one-forms as where D µ are the appropriate covariant derivatives The spin-1 fields are embedded in SU(4) matrices as where W k µ (k = 1, 2, 3) and B µ are the elementary electroweak gauge bosons associated with the SU (2) L and U (1) hypercharge groups. The vector V j µ (j=1 to 10) and axial-vector A l µ (l=1 to 5) are the composite resonances generated by the strong dynamics and associated to the unbroken V a and broken Y a generators as defined in eq. (3). The projections to the broken and unbroken generators are defined respectively by while p µ i transforms homogeneously and can be used to construct invariants for the effective Lagrangian.
The K field is introduced to break the two remaining copies of Sp(4), Sp(4) 0 × Sp(4) 1 to the diagonal final Sp(4): and it transforms like thus its covariant derivative takes the form The 10 pions contained in K are needed to provide the longitudinal degrees of freedom for the 10 vectors V j µ , while a combination of the other pions π i act as longitudinal degrees of freedom for the A l µ . It should be reminded that out of the 5 remaining scalars, 3 are exact Goldstones eaten by the massive W and Z bosons, while 2 remain as physical scalars in the spectrum: one Higgs-like state plus a singlet η.
To lowest order in momentum expansion, and including the scalar singlet σ, the effective Lagrangian is given by We have introduced the singlet field σ for generality, as it may be light in some theories, via generic functions in front of the operators in the strong sector: in the following, however, we will be interested to the case where it's heavy and thus we will replace the functions by the first term in the expansion, i.e. κ X (σ) = 1 and r(σ) = r. The field strength tensors are defined by for V µ = B µ , W µ , and F µ = V µ + A µ . The canonically normalised fields are g B µ , g W k µ , g V k µ and g A k µ . Due to the presence of the r term in the Lagrangian, the pions π 0,1 do not have proper kinetic terms. Calling the normalised fields π A and π B , they are given by As already mentioned, a linear combination of the two sets of 5 pions is eaten by the vector states A µ once they pick up their mass. The eaten Goldstones π a U , and the 5 physical ones π a P before the EW gauging, are given by π a B = sin α π a P + cos α π a U , where the mixing angle α is Combining the above redefinitions, we get 4 Note that only the pions associated with Y 4 and Y 5 are physical, as the remaining 3 are exact Goldstones eaten by the W and Z. In the following, we will associate one with the Higgs boson, π 4 P = h, and the other with the additional singlet π 5 P = η of the SU(4)/Sp(4) coset.

B. Other terms
The previous Lagrangian contains the low energy composite sector in terms of effective fields using the CCWZ formalism and the hidden symmetry one, allowing for a description of composite spin-0 pNGB and spin-1 vector and axial-vector resonances. The interactions among these states are, to a large extent, described by this formalism, however some extra terms can potentially be added. While we leave a detailed study to a future work, it is worth mentioning how they can affect the phenomenology when added.
A first set of contributions are those induced by the Wess-Zumino-Witten (WZW) anomaly [56,57]. In our case, these can be added in a similar way to what is done for chiral Lagrangians to describe, for example, the decay of a neutral pion into two photons.
These terms are relevant for di-boson final states, allowing a scalar pNGB to couple to the SM gauge bosons. Furthermore, anomalous couplings of the vectors will also be generated, thus potentially providing new decay channels. study. This part of the scalar sector is not the main focus here and a detailed description can be found in [50]. Other possible items in this list are symmetry breaking terms and kinetic mixing terms. All these points will not be discussed further here, and may deserve a separate study.

III. PROPERTIES OF VECTOR STATES
The intrinsic properties of the 15 spin-1 states introduced in eq. (11) determine the structure of masses, mixing, couplings and their contributions to electroweak precision tests. 9 The vector fields can be organised as a matrix in SU(4) space, defined by where the generators in the general vacuum, V a and Y a , are defined in eq. (3). Under the unbroken Sp(4), the two multiplets transform as a 10 and a 5 respectively. It is however more convenient to classify the states in terms of their transformation properties under a subgroup SO(4)⊂Sp (4), which corresponds to the custodial symmetry SU(2) L × SU(2) R of the SM Higgs sector in the limit θ → 0: Physically, however, the SM custodial symmetry is broken to the diagonal SU (2)  Technicolor limit, θ = π/2, we find that a 0,± µ , which has a component of axial-vector SU (2) A proportional to sin θ, can be associated to − → a µ ; on the other hand, in the pNGB Higgs limit, where the mass parameters M A and M V are defined in terms of Lagrangian parameters as The matrices M 2 C and M 2 N are given in eqs. (A4)-(A5). Upon diagonalisation, the interaction eigenstates are rotated to the physical vector bosons Approximate expressions for C and N are given in Appendix A 1 in an expansion for large g.
The eigenstate in the neutral sector which is exactly massless is identified to be the photon, and it is related to the interactions eigenstates (exactly in g) as 11 with 1/e 2 = 1/g 2 + 1/g 2 + 2/ g 2 .
Besides the photon, all the massive states mix with each other with mixing angles typically of order 1/ g, with the exception of v µ and s µ whose mixing is controlled by the angle θ. For instance, in the charged sector, see eq. (A4), it is clear that the combination cos θ v ± µ − s ± µ decouples from the other states and has a mass equal to M V . A similar situation is realised in the neutral sector where, however, residual mixings suppressed by 1/ g are present.
Approximate expressions for the masses of the charged states are given below, including leading corrections in 1/ g 2 : Similarly, in the neutral sector, we find: In all above expressions, s θ = sin θ and c θ = cos θ. Furthermore, v = 246 GeV is the value of the effective EW scale, obtained from the definition of the Fermi decay constant as: Replacing the masses with the Lagrangian parameters, we also obtain the relation where f π = f 2 0 − r 2 f 2 1 is the decay constant of the SU(4)/Sp(4) pions, as in eq. (5). As a consistency check, note that f π = f 0 for r = 0 and, as mentioned in the previous section,

A. Couplings
We assume here that the SM fermions only couple to the SM weak bosons, W µ and B µ : this is a reasonable assumption, as direct couplings to the composite resonances can only be induced by interactions external to the dynamics. The interaction with the heavy vectors, therefore, are generated via mixing terms. For the charged currents, we have where and f labels all the SM fermions. For the neutral currents with T 3 being the weak isospin and Y L,R the hypercharge of the left-handed doublet and the right handed singlet respectively. All the neutral vector couplings can be expressed like this, but for the photon gauge invariance requires that Note that eq. (45) and eq. (46) encode corrections to the couplings of SM fermions with respect to the SM predictions, that are strongly constrained by EW precision observables and can be encoded in the oblique parameters, as discussed in the following Section.
The Higgs couplings to weak bosons are phenomenologically important because they can constrain the model parameters, both from direct measurements and from its contribution to the electroweak parameters. Schematically, the couplings can be written as where R ± i and R 0 i encode all the charged and neutral vectors. In the gauge interaction basis, the couplings are provided in Appendix A, while in the mass eigenbasis we calculated expressions at leading order in 1/ g. We find that the Higgs couplings to at least one photon are automatically zero at the tree level, with the other couplings given by in agreement with previous studies [40], while the couplings to the resonances are (we list only the diagonal ones and the ones with one SM gauge boson) The charged heavy vector states contribute to the decay of the Higgs boson into two pho- where we approximate the amplitude of the heavy states to the asymptotic value A V (∞) = −7 (A f (τ f ) and A V (τ W ) being the standard amplitudes [58]), and κ t,W are the modification of the fermion and vector couplings of the Higgs normalised by the SM expectation (in our case, κ W ∼ cos θ, while κ t depends on the mechanism providing a mass for the top and is equal to κ t = cos θ in the simplest case): where both couplings and masses are defined in the mass eigenstate basis. By analysing the mass and coupling matrices, we found that the following sum rule holds, at all orders in 1/ g: where c hV + V − are the couplings of the Higgs in the interaction basis (see eq. (A6)) and we used exact matrices. At leading order in 1/ g, the sum rule is saturated by the W coupling κ W = cos θ, as shown in eq. (50). However, corrections arise at order 1/ g 2 : using eq. (51), we find We expect, therefore, the contribution of the mixing with the heavy resonances to be very small, as it is suppressed by sin 2 θ, 1/ g 2 , and it also vanishes for r = 1: the latter is a reminder of the fact that the two SU(4)'s decouple in this limit. This analysis shows that the effect of the heavy resonances on the Higgs properties can be neglected, thus the bounds from the measured Higgs couplings are the same as in [50], and they are typically less constraining than electroweak precision tests.
For completeness, a similar analysis can be done in the neutral sector, where we define and the exact mass and coupling matrices entail the following sum rule: where the reduced coupling,c hV 0 V 0 , and mass matrices,M 2 N , are 4×4 matrices obtained from the complete ones in eq. (A7) and eq. (A5) by removing the photon, i.e. the zero-mass eigenstate. Like for the charged case, κ Z is the deviation of the Higgs couplings to the Z boson normalised by the SM value, while κ 0,res encodes the contribution of the heavy resonances: We can see that the custodial violation due to the gauging of the hypercharge emerges here.
The off-diagonal Higgs and gauge couplings will contribute to the H-Z-γ vertice [59], but no bound is availabe at current LHC precision [60]. Other relevant interactions involve the η state, which couples to the "tilded" vectors. The production of heavy vector states can go through a cascade of decays with rich phenomenology. The interaction Lagrangian involving η and vector fields is given, at leading order in g/ g and sin θ, by An interesting collider signature would be the production of A µ with subsequent decay into S + η, then S → η + Z and the two η resonances decay for instance into top pairs.

B. Electroweak Precision Tests
The precise measurements near the Z-pole performed at several high energy experiments, especially at LEP [61], are crucial tests for any kind of model of New Physics. These effects can be parameterised via the so-called oblique parameters [62,63], expressed explicitly in terms of the weak boson self energies in eq. (A55).
At tree level, the vector contribution to the oblique parameters are given bŷ where the other EW observables vanish,T = 0,Û = 0. For θ = π/2 these expressions agree with [64] once one identifies 1 − χ = r and sets the hyper-charge y = 0. In our analysis, we are going to use the notation adopted by the Particle Data Group (PDG) and rescale S = 4s 2 WŜ /α EW , T =T /α EW and U = −4s 2 WÛ /α EW . Note that, assuming treelevel vector meson dominance, the above contributions can replace the contribution of the strong dynamics, estimated in [8,50] as a loop of the underlying fermions. For r ∼ 1 the S parameter vanishes and higher order parameters, W ,Y and X will play the dominant role.
This situation is similar to the Custodial Vector Model described in [52,65].
Additionally, deviations in the Higgs coupling w.r.t. the SM ones also bring additional contributions to the S and T parameters. The modification in the Higgs coupling, eq. (51), produce approximately the following deviations in the S and T parameters. In the above formulas, the couplings of the SM gauge bosons to the Higgs, κ V , include corrections up to order 1/ g 2 from eq. (56) in order to be consistent with the tree-level effects (for simplicity, we neglect the term in g 2 so that κ V ∼ κ W ∼ κ Z ). In principle, the heavy resonances also contribute at one-loop level: naively, the loops with a Higgs boson have an additional suppression 1/ g 2 · m 2 W /M 2 V ∼ 1/ g 4 . Pure loops of the heavy resonances may be unsuppressed, however their effect should be small as the dynamics is custodial invariant. Furthermore, due to the intrinsic strong interactions among resonances, such loop calculations are not reliable in general because perturbative expansions cannot be trusted.
Attempts to compute their effect exist in the literature [66,67], and they can be considered a modelling of the contribution of the strong dynamics (see also [68,69]).
The experimental values from PDG for S and T (leaving U to be free), with a strong correlation coefficient 0.90, at 1σ deviation are [70]: The corresponding limits on the model parameters are shown in fig. (1). For r 1 the vector partially cancels the Higgs contribution allowing a larger parameter space: in some areas of the parameter space, therefore, the most constraining bound comes form the measurements of the Higgs couplings, which give constraints on θ of the order of θ 0.6 [50]. Another effect that may significantly modify the EWPT is the presence of the σ state, that will in general mix with h, with an un-calculable mixing angle α, and can potentially alleviate the constraints from EWPT [50].

IV. A MINIMAL FUNDAMENTAL GAUGE THEORY
The effective model characterised in the previous sections, can originate from a very simple scalar-less underlying theory [38,39]: it consists of a gauged and confining G HC = SU (2) with two light Dirac flavours transforming as the fundamental representation. Following the notation of [38,40], the 2 Dirac fermions, U and D, can be arranged in a flavour SU (4) multiplet as where α is the spin Lorentz index, i is a flavour index and a is a hyper-colour index. The

A. Scalar sector
The scalar sector of the SU (4)/Sp(4) models was studied in [40]. In general we can write a scalar matrix

B. Vector sector
The composite spin-1 states can be defined in terms of the underlying fermions via the flavour adjoint left-current: where E ab is the antisymmetric tensor making a hyper-colour singlet. Note the first line is non-standard notation for the left bilinears, but the one that directly implements the the flavour transformation structure F → g · F · g † . The last line is the standard bilinear notation. After some current algebra, the components in the vector matrix from tab. (I) can be associated to currents in terms of the underlying quarks, as detailed in tab. (II).
We first notice that this decomposition matches with the interpretation we provided at the beginning of the previous section: the triplet − → v µ corresponds to the "vector" current Qγ µ Q, typically associated to the ρ µ meson in QCD, while − → s µ and − → a µ contain an "axial" current component, associated with a µ meson in QCD, proportional to the cos θ and sin θ respectively.

C. Discrete symmetries
The action of space-time discrete symmetries on the composite states can be derived from the transformation properties of the underlying quarks. However, the gauging of the EW interactions break P and C individually, but preserves CP in the strong confining sector.
Under CP , the bound state fields transform as where (−1) µ = 1 for µ = 0, and −1 on spacial directions. The parities associated with the spin-1 resonances are summarised in tab. (II): in our notation, a vector has CP = −, while Field Fermion currents P C G GP Once combined with P , the new symmetry defines a parity acting on the composite fields From tab. (II) and tab. (III) we see that all the tilded fields are odd under GP , as well as η, thus this is the symmetry preventing decays of such field directly into SM ones. Note, however, that GP is violated by the anomalous WZW term which generates decays for η.
Additional decay channels will also be generated for the vectors. The neutral resonances A, Z, A 0 , V 0 , S 0 ,S 0 andṼ 0 are introduced as one particle class of VN, with two additional neutral states X 0 ,X 0 into another particle class of VX.
The charged resonances W + , A + , V + , S + andS + are put into one particle class of VC.
The model file is loaded using FeynRules package which exports the Lagrangian into UFO format [73]. We implement one python code as the parameter card calculator, to conduct the numerical rotation and write all block information into a param card.dat.

B. LHC Run-II
At the LHC Run-II, several resonances may be produced via the Drell-Yan production mechanism, with qq as initial states, therefore unfolding a delighting and rich phenomenol- where we find that the relevant parameter is r. We set the benchmark point to be M V = 2.5 TeV and M A = 3 TeV, and vary the other parameters ( g, θ, r) to inspect the region where the largest Γ/M among all resonances is less than 50%. Generally, in order to use NWA as an approximate analysis for the event line shapes (e.g. di-lepton invariant mass distribution) we require Γ/M < 10% so that interference effects with Z, γ can be safely neglected. Due to the small mass split between many resonances, off-diagonal width effects may also be important [75,76]. Furthermore, the small width region will be favoured in order to resolve the compressed multi-peaking structure in the spectrum. According to this criterion, for a small θ = 0.2, the NWA applies very well for 0 < r < 2.0, but with a larger value θ = 0.4, the resonance will become broad and we need at least to tune g > 6.0 for the NWA to be effective.
In fig. (3), we show the typical branching ratios for V 0 and where branching ratios into A 0 Z and X 0 h, mostly overlap with each other in the range of 0 < θ < 0.8 due to the global symmetry. We also explore the branching ratios as a function of r: the fermions spectrum goes to a maximum at r = 1, while the W W or hZ spectrum, instead, goes to a minimum since the coupling is ∝ (r 2 − 1) at 1/ g order. In either vector or axial resonance dominant case, the lower mass state displays a larger branching ratio into l + l − and W + W − rather than into final states containing a composite vector, therefore we can exploit the most recent LHC Run-II results to constrain the model parameters.
Since our model provides several candidates as a heavy Z or W , the LHC measurement for the Drell-Yan process and di-boson process would impose a stringent constraint on the parameter space. We calculate the theoretical cross section for pp → R 0 → l + l − and pp → R ± → W Z in this SU (4)/Sp(4) model and compare them with the 95% upper bound observed from the latest ATLAS measurement [77,78]. Similar results can be obtained by using the corresponding CMS searches [79,80]. The single lepton plus MET process is expected to require similar constraints to the di-lepton ones, thus we do not consider the lν channel in detail for simplicity. We derive the exclusion limits in the parameter space specified by ( g, θ, r ) after assuming M V = M A . Since we have not included the acceptance factor into this analysis, our result would be stronger than the exact 95% exclusion from the LHC Run-II searches. We show the exclusion contours from l + l − and W Z in fig. (4) prediction at 100 TeV [82]. Nonetheless, the cross sections present here can serve as a guideline. Similarly to the scenario described in the last section, the production rate for these states with r ∼ 1 is not large, around O(1) fb for θ ∼ 0.2, since the resonance coupling to SM quarks are generated via mixing. Departing from r = 1, weak boson fusion can be an important production mechanism, but we will not consider it in this work. It is also important to note that SM physics, jets, top production and other important background for the process will present quite peculiar aspects at a 100 TeV collider (see e.g. [83]) and must be taken into consideration for a more precise phenomenological analysis.
The value of r is constrained by perturbativity of the effective description. The consistent region is illustrated in fig. (6), with the largest ratio of width over mass extracted in the plane of ( g − r). We find that the region of r close to one is where all the resonances are narrow, thus it is valid to apply the NWA for event analysis. For r = 1 the coupling of heavy vector to longitudinal bosons rapidly grows as the width of the resonance approaches its mass, jeopardising perturbativity and the validity of the description [45].
We show the BR of V 0 state as a function of θ in the left panel of fig. (7). S 0 has similar decay structure as V 0 , while charged states present similar pattern, thus we do not show them here. At r = 1 they mainly decay into SM fermions, in particular into di-jets.
There will be small differences in the BR spectrum between V 0 and S 0 . We find that, in the channel of di-leptons, V 0 decays at ∼ 10% and S 0 at ∼ 40%. Moreover, for V 0 , the decay into W + W − is larger than hZ, while for S 0 the decay of hZ turns to dominate over W + W − . Varying r to be slightly larger than 1, notable changes happen as the di-boson and hZ channels rapidly overcome the fermion ones. For r = 1.1 the branching ratios are close to 45%, equally split between W + W − and hZ at small θ. Only small variation can be observed in θ 0, but the two channels will start to split exactly till θ 0.8.
The decay pattern of the A 0 resonance is shown in the right panel of fig. (7), with more channels opened. At r = 1, the fermion channel is subdominant, while the W W , hZ channels almost disappear. The decay into V ± /S ± W ∓ and V 0 /S 0 h become competitive, and we can observe dominant decays into ηS, with η further decaying into a pair of tops, or gauge bosons via the WZW anomaly term (as discussed in [50]). Since theS decay is nearly 100% to Zη for the lattice benchmark point, this will give rise to novel collider signature of 4t + Z final states. For r = 1.1 the di-boson and Higgs-strahlung channels enter into play, but this does not alter the picture dramatically.
It has been argued that the luminosity of this future machine should be at least a factor 50 larger than the LHC luminosity in order to profit from its full potential to find new physics [84,85]. An integrated luminosity of 3 − 30 ab −1 per year is therefore expected, leading to several heavy vector bosons produced and a promising phenomenology. We also stress that probing masses up to ∼ 50 TeV indirectly tests the models at small values of θ ∼ 0.05, where the high level of fine tuning renders the models unnatural and unappealing.
While an ultimate exclusion is not possible due to a decouplings limit θ → 0 (like in supersymmetry), in our opinion a 100 TeV collider can ultimately probe the "motivated" region of the parameter space in this class of composite Higgs models.

VI. CONCLUSIONS
In the present work we construct an effective Lagrangian that allows to describe vector spin-1 resonances in composite models of the Higgs boson. The framework adopted is the one of the hidden gauge symmetry approach, and we focus on a case with global symmetry structure based on the minimal case of an SU(4) symmetry broken to Sp(4). The chosen coset both satisfies the requirement of a custodial Higgs sector and allows for a fundamental composite description of the new resonances in terms of fermionic bound states. The SU (4) structure is promoted to SU(4) 0 ×SU(4) 1 in order to apply the hidden gauge symmetry idea and to obtain the vector and axial-vector states in the adjoint of the second SU(4).
The paper discusses in detail the effective Lagrangian for these states and their properties including mass matrices, mixing and couplings. The underlying fundamental realisation of the theory in terms of fermionic bound states is also discussed, together with the associated discrete symmetries, such as parity.
In the gauge eigenbasis, the vector mass matrices in the charged M C and neutral M N sectors are 32 where 2ω = f 2 0 /f 2 K − 1. In the same basis, we provide the couplings of one Higgs with charged vector bosons and for the neutral ones Similarly, the η-V -V interaction in gauge eigenstate are provided below: The above couplings are provided in the gauge eigenbasis, so one need to include the mixing matrices in order to extract couplings in the mass eigenstate basis. Approximate expressions for the mixing matrices are provided in the following section.

Perturbative diagonalisation of the mass matrices
The label of the physical states, W +µ , A +µ , V +µ and S +µ in the charged sector, and A µ , Z µ , A 0µ , V 0µ and S 0µ in the neutral sector (left hand side of eq. (32)), are defined as the ones with predominant component of the corresponding interaction eigenstates, W +µ , a +µ , v +µ and s +µ in the charged sector, and B µ , W 3µ , a 0µ , v 0µ and s 0µ in the neutral sector respectively. Therefore, in theory, the columns in C and N do not assume fixed expressions which can swap depending on the largest entry, i.e, the matrix is reorganised in such a way that the diagonal entry is the largest in each column. In practice, however, for the parameter values we consider, the columns 1 and 2 in C and 1,2 and 3 in N have fixed expressions, even though there are significant mixing between the photon, A µ and Z µ . On the other hand, the states V 0,± µ and S 0,± µ are highly mixed, and columns 3 and 4 in C and 4 and 5 in N can be swapped, depending on the parameters, to fulfil our definition of these states.
In the following we provide expressions for these mixing matrices, C and N , defined in eq. (32), keeping in mind that the last two columns may be swapped depending on the values of their entries.