LHC Phenomenology of Composite 2-Higgs Doublet Models

We investigate the phenomenology of Composite 2-Higgs Doublet Models (C2HDMs) of various Yukawa types based on the global symmetry breaking $SO(6)\to SO(4)\times SO(2)$. The kinetic term and the Yukawa Lagrangian are constructed in terms of the pseudo Nambu-Goldstone Boson (pNGB) matrix and a 6-plet of fermions under $SO(6)$. The scalar potential is assumed to be the same as that of the Elementary 2-Higgs Doublet Model (E2HDM) with a softly-broken discrete $Z_2$ symmetry. We then discuss the phenomenological differences between the E2HDM and C2HDM by focusing on the deviations from Standard Model (SM) couplings of the discovered Higgs state ($h$) as well as on the production cross sections and Branching Ratios (BRs) at the Large Hadron Collider (LHC) of extra Higgs bosons. We find that, even if the same deviation in the $hVV$ ($V=W,Z$) coupling is assumed in both scenarios, there appear significant differences between E2HDM and C2HDM from the structure of the Yukawa couplings, so that production and decay features of extra Higgs bosons can be used to distinguish between the two scenarios.


I. INTRODUCTION
After the discovery of a Higgs boson in July 2012 [1,2], an intense period of analysis of its properties has begun and is bearing fruits. We now know that this object is very consistent with the spinless scalar state embedded in the SM. Following the precision measurement of its mass, around 125 GeV, its couplings to all other states of the SM can be derived and compared with experimental data. Agreement between the SM and experimental results is presently within a few tens of percent at worse, thus leaving some scope for a Beyond the SM (BSM) Higgs sector.
By bearing in mind that the discovered Higgs state has a doublet nature, in the class of the many new physics scenarios available embedding such a structure those among the easiest to deal with are clearly the 2-Higgs Doublet Models (2HDMs). Furthermore, these scenarios always include a neutral scalar Higgs state that can play the role of the discovered one, which -as intimated -is very SM-like. Furthermore, they are also easily compliant with past collider data (from LEP/SLC and Tevatron) as well as present ones (from the LHC) while still offering a wealth of new Higgs states and corresponding signals that can be searched for by the ATLAS and CMS collaborations. In fact, a significant amount of experimental effort at the LHC is presently being spared on direct searches for new Higgs bosons, in parallel with the one of extracting their possible presence indirectly from the aforementioned precision measurements.
However, 2HDMs per se do not have the ability to solve the so-called hierarchy problem of the SM. An elegant way to do so though, is to presume that the Higgs boson discovered in 2012 and its possible 2HDM companions are not fundamental particles. This approach is not unreasonable as any other (pseudo)scalar state found in Nature eventually revealed itself to be a (fermion) composite state, i.e., a mesonic state of the now standard theory of strong interactions (QCD). Specifically, one can construct 2HDMs in which all Higgs bosons, both neutral and charged, both scalar or pseudoscalar, are not fundamental, rather composite. A phenomenologically viable possibility, wherein the mass of the lightest Higgs state is kept naturally lighter than a new strong scale (of compositeness, f , in the ∼ TeV region) is, in particular, the one of assigning to them a pNGB nature. In essence, we have in mind those Composite Higgs Models (CHMs) arising from the spontaneous symmetry breaking around the TeV scale, of the global symmetry of the strong sector [3]. The resisual symmetry is explicitly broken by the SM interactions through the partial compositeness paradigm [4,5].
In the minimal CHM [6,7], the composite version of the SM Higgs doublet, the only light scalar in the spectrum is indeed a pNGB (surrounded by various composite resonances, both spin-1/2 and spin-1, generally heavier). Hence, it is natural to assume that the new (pseudo)scalar Higgs states of a C2HDM are also pNGBs. In fact, even in the case in which they are eventually found to be heavier than the SM-like Higgs state, compositeness could provide a mechanism to explain their mass differences with respect to the latter. Finally, in the case of extra Higgs doublets with no Vacuum Expectation Value (VEV) nor couplings to quark and leptons, one could also have neutral light states as possible composite dark matter candidates [8]. Another example for a composite scalar dark matter candidate emerging as a pNGB is given in [9].
C2HDMs embedding pNGBs arising from a new strong dynamics at the TeV scale, ultimately driving Electro-Weak Symmetry Breaking (EWSB), can be constructed by either adopting an effective Lagrangian description (see example [10]) invariant under the SM symmetries for light composite SU(2) Higgses or explicitly imposing a specific symmetry breaking structure containing multiple pNGBs. We take here the second approach. In detail, we will analyse 2HDMs based on the spontaneous global symmetry breaking of an SO(6) → SO(4) × SO(2) symmetry [11]. Within this construct, which we have tackled in a previous paper [12], one can then study both the deviations of C2HDM couplings from those of a generic renormalisable E2HDM [13] as well as pursue searches for new non-SM-like Higgs signals different from the elementary case. In the f → ∞ limit the pNGB states are in fact identified with the physical Higgs states of doublet scalar fields of the E2HDM and deviations from the E2HDM are parametrised by ξ = v 2 SM /f 2 , with v SM the SM Higgs VEV. Once the new strong sector is integrated out, the pNGB Higgses, independently of their microscopic origin, are described by a non-linear σ-model associated to the coset. In Ref.
[12], we have constructed their effective low-energy Lagrangian according to the prescription developed by Callan, Coleman, Wess and Zumino (CCWZ) [14,15], which makes only few specific assumptions about the strong sector, namely, the global symmetries, their pattern of spontaneous breaking and the sources of explicit breaking (in our case they come from the couplings of the new strong sector with the SM fields). The scalar potential is in the end generated by loop effects and, at the lowest order, is mainly determined by the free parameters associated to the top sector [11].
However, both in Ref. [12] and here, we will not calculate the ensuing Higgs potential a la Coleman-Weinberg (CW) [16] generated by such radiative corrections, instead, we will assume the same general form as in the E2HDM with a Z 2 symmetry, the latter imposed in order to avoid Flavor Changing Neutral Currents (FCNCs) at the tree level [17]. We do so in order to study the phenomenology of C2HDMs in a rather model independent way, as this approach in fact allows for the most general 2HDM Higgs potential 1 It is our intention to eventually construct the true version of the latter through the proper CW mechanism [18]. However, first we intend to infer guidance in approaching this task from the study of theoretical (i.e., perturbativity, unitarity, vacuum stability, etc. -the subject of Ref. [12]) and experimental (one of the subjects of the present paper) constraints, specifically, by highlighting the parameter space regions where differences can be found between the E2HDM and C2HDM. This will inform the choice of how to construct a phenomenologically viable and different (from the E2HDM) realisation of a C2HDM in terms of underlying gauge symmetries, their breaking patterns and the ensuing new bosonic and fermionic spectrum, that is, indeed, to settle on a specific model dependence.
The paper is organised as follows. In Section II we describe the C2HDM based on SO(6)/SO(4) × SO (2). In Section III, the LHC phenomenology is discussed in presence of both theoretical and experimental constraints. Conclusions are drawn in Section IV. In Appendix A, relevant Feynman rules for the phenomenological study are presented.

II. THE COMPOSITE TWO HIGGS DOUBLET MODEL
We construct the Lagrangian of the C2HDM based on the spontaneous breaking of the global symmetry SO(6) → SO(4) × SO(2) at a scale f . In this model, eight (pseudo)scalar fields emerge as pNGBs from such a breaking pattern, which constructs two isospin doublet fields. In our approach, we do not specify the physics at any scale above a (large) cutoff Λ which is expected to be ∼ 4πf from a naïve dimensional analysis [19], i.e., we do not fix the concrete structure of the gauge and matter contents. Even in this setup, the kinetic term of 1 This choice is also motivated by the fact that, in the case in which the SM fermions are embedded in a 6-plet representation, the leading order terms in the perturbative and loop expansion of the potential do not provide EWSB in the composite scenario. As shown in [11], one ought to also include next order terms thus generating unrelated contributions to different operators, leading to the most general potential of the elementary version.
the pNGBs is uniquely determined by the structure of the global symmetry breaking. For the Yukawa sector though, we need to assume an embedding scheme for the SM fermions into SO(6) multiplets to build the Lagrangian at low energy. Although in this framework the scalar potential is generated via the CW mechanism at loop level [16], as intimated, we assume here its renormalisable form of the E2HDM. This gives a sort of more general approach to the potential, namely, once the CW potential is calculated in a fixed configuration, all the potential terms can be translated into the strong sector parameters. We therefore adopt the same setup of [12], to which we refer the reader for further details of the model construction.

A. Two Higgs doublets as pseudo Nambu-Goldstone bosons
We construct the 6 × 6 pNGB matrix U using the eight broken generators 2 of SO(6) Tâ α (α = 1, 2 andâ = 1-4) as The eight real spinless fields πâ α associated with the broken generators can be expressed through two complex doublets as where the π 4 α 's acquire the non-zero VEVs: π 4 α = v α . Their ratio is expressed as tan β = v 2 /v 1 and we define v ≡ v 2 1 + v 2 2 . The EW scale, v SM , related to the Fermi constant G F , is expressed by f and v as follows: We here introduce the Higgs basis in which the physical Higgs states are separated from the NG boson states G ± and G 0 , which are absorbed into the longitudinal components of the W ± and Z bosons, as The doublet Ψ contains the physical CP-odd Higgs boson (A) and a pair of charged Higgs bosons (H ± ). As noted in Ref. [12], in the Higgs basis the G 0 , G ± and h ′ 2 fields do not yield the kinetic terms in canonical form, hence we shift these fields so as to render them to canonical up to O(1/f 2 ) by In general, the two CP-even scalar states h ′ 1 and h ′ 2 can mix with each other. Their mass eigenstates can be defined by where −π/2 < θ ≤ π/2. We identify the mass eigenstate h as the Higgs boson with a mass of 125 GeV discovered at the LHC.
The kinetic terms of the eight pNGB fields can then be written in terms of Σ as follows The covariant derivative D µ is given by where with R and θ W being the weak mixing angle. In Appendix A, we give all the Feynman rules relevant to the discussion on Higgs phenomenology, which are derived from the kinetic term given in Eq. (10)

B. Yukawa Lagrangian
In this subsection, we construct the low-energy (below the scale f ) Yukawa Lagrangian.
In order to do this, we need to determine the embedding scheme of the SM fermions into SO(6) multiplets. This embedding can be justified via the mechanism based on the partial compositeness assumption [4], where elementary SM fermions mix with composite fermions in the invariant form under the SM SU(2) L × U(1) Y gauge symmetry but not under the global SO(6) symmetry. Through the mixing, the SO(6) invariant Yukawa Lagrangian given in terms of Σ and composite fermions turns out to be the SM-like Yukawa Lagrangian after integrating out the (heavy) composite fermions.

Fermion embeddings
We discuss the embeddings of the SM quarks and leptons using 6-plet representations of SO (6). In order to reproduce the correct electric charge of the SM fermions, we introduce an additional U(1) X symmetry and assign its appropriate charge to 6-plets. The electric charge Q is thus given by 3 In the SO(6) basis, the 6-plet fermion Ψ X , with the U(1) X charge X expressed as a mixture of the states in the SU(2) L × SU(2) R basis, is obtained as follows: All the left-handed fermions Q u L , Q d L and L L are transformed as even under C 2 . In the third column, the symbol , while ψ 00 and ψ ′ 00 are singlets under SU(2) L and SU(2) R , respectively. From this relation, we can embed the SM quarks and leptons into the 6-plet representation Ψ X as follows:

Yukawa Lagrangian
The Yukawa Lagrangian at low energy is given in terms of the 15-plet of pNGB fields Σ and the 6-plet of fermions defined in the previous subsection: We note that the Σ 3 term is equivalent to the −Σ term, thus the terms with the cubic and more than cubic power of Σ do not give any additional independent contributions to the Yukawa Lagrangian. The parameters a f and b f should be understood as 3 × 3 complex matrices in flavour space. This Lagrangian is rewritten, up to the order 1/f 2 , using the complex doublet form of the Higgs fields defined in Eq. (2), as The fermion mass terms, at the same order, are then extracted to be: Clearly, the existence of two independent Yukawa matrices, a f and b f , for f = u, d, e, introduces FCNCs at the tree level. As it is well known, they are induced by the fact that both doublets Φ 1 and Φ 2 couple to each fermion types. This property is common to the E2HDM. In fact, the Yukawa Lagrangian in Eq. (21), in the limit f → ∞, reproduces to so-called Type-III E2HDM.
In order to avoid FCNCs at the tree level, we impose a discrete C 2 symmetry [11] as follows: where C 2 = diag(1, 1, 1, 1, 1, −1). By this definition, πâ 1 and πâ 2 have a C 2 -even and C 2 -odd charge, respectively. Depending on the C 2 charge assignment of the right-handed fermions, we can define four independent types of Yukawa interactions, just like the softly-broken Z 2 symmetric version of the E2HDM [20][21][22], as shown in Tab. I. For example, the Type-I Yukawa interaction is obtained by taking a f = 0.
In the C 2 symmetric case, we obtain the following interaction terms in the mass eigenbasis of the fermions: In the limit of ξ → 0, these coefficients get the same form as the corresponding ones in a softly-broken Z 2 symmetric version of the E2HDM [22].

C. Potential
We adopt the same form of the potential as in the E2HDM. We have in total eight parameters, which can be translated into eight physical inputs, as explicitly done in Ref. [12]: In Fig. 1, we show the allowed regions (green shaded) in the (sin θ, tan β) plane using HiggsBounds. We can see that larger values of ξ give more excluded regions in the Type-I C2HDM, but in the other three models the ξ dependence is not so significant. In particular, (corresponding to ξ = 0 and shown in the first column of Fig. 1). In the Type-X C2HDM, additional exclusion parameter regions appear for larger ξ, which is mostly due to the same reason as in the Type-I C2DHM. In the Type-II and -Y C2HDMs, the excluded regions almost do not depend upon the ξ value. In Tab. II we list the Higgs search channels most responsible for the exclusions.
In Fig. 1   to the LHC Higgs data. Overall, the ξ dependence is only marginally evident, being more pronounced for Type-I.

B. Deviation in the Higgs boson couplings
In both the E2HDM and the C2HDM, the Higgs boson couplings can deviate from the SM predictions. However, the pattern of deviations can be different between these two scenarios. In order to discuss these, it is convenient to define the scaling factor κ X for the hXX couplings by κ X = g NP hXX /g SM hXX and ∆κ X = κ X − 1. In the C2HDM, these deviations are given at the tree level by Those for the E2HDM can be easily obtained by taking ξ → 0 corresponding to f → ∞. We can see that there are two sources giving κ X = 1 in the C2HDM, i.e., non-zero values of ξ and θ. Conversely, only θ = 0 gives κ X = 1 in the E2HDM 4 . Therefore, for a given measured value of κ X , the value of θ is determined in the E2HDM while only the combination (θ, ξ) is determined in the C2HDM.
In Fig. 2, we plot the contour for ∆κ V as a function of ξ and sin θ. We note that there is As a result, we will find a significant difference in the two scenarios for the decay Branching Ratios (BRs) of the extra Higgs bosons for a given value of ∆κ V , which will be discussed in the succeeding subsections.
As it has been discussed in Ref. [38], the type of Yukawa interactions can be determined by looking at the correlation between ∆κ E and ∆κ D in the E2HDM, where E and D represent a charged lepton and a down-type quark, respectively. Now, let us discuss the correlation between ∆κ E and ∆κ D in the C2HDM.
BP1 corresponds to the E2HDM case, while BP2 and BP3 are two possible C2HDM cases, the latter corresponding to zero-mixing angle.
Before studying the BRs, we survey the allowed parameter regions by bounds from the perturbative unitarity and the vacuum stability. Details of these bounds have been discussed in Ref. [12]. Concerning to the unitarity bound, we take into account all the elastic scatterings of 2 body to 2 body scalar boson processes up to O(s 0 ) dependences, where √ s is the scattering energy. Differently from E2HDMs, the s-wave amplitude matrix has terms proportional to s ξ, thus indicating that an UV completion of the theory is needed at high energy. Here we fix √ s = 1 TeV.
In Fig. 4    Concerning the BRs of A (Fig. 6), it is seen that their behaviour is drastically changed The mass dependence on the BRs of H ± is shown in Fig. 7. We see that the H + → tb For the calculation of the gluon fusion cross section, we use the following equation: where h SM is the SM Higgs boson with the mass artificially set at m φ 0 . We adopt the value of the gluon fusion cross section σ(gg → h SM ) in the SM from Ref. [39]. For the other calculations of the production cross sections, we use CalcHEP [40] and adopt the CTEQ6L [41] for the parton distribution functions with factorisation/renormalisation scale set at Q = √ŝ .
We note that the lepton Yukawa coupling is not relevant for the calculation of the production cross sections, so that the result in the Type-I (Type-II) and Type-X (Type-Y) models are the same with each other. As in the previous subsection, we take BP1, BP2 and BP3 given in Eq. (30) and tan β = 2 for the numerical analysis. In Fig. 8, we show the gluon fusion production cross section as a function of the mass of the produced Higgs boson. In this process, the dependence of the type of Yukawa interactions is almost negligible, because only the top Yukawa coupling is important to determine the size of the cross section. The results for BP1, BP2 and BP3 are respectively shown as the solid, dashed, and dotted curves. We find differences in the cross section of gg → H among the three benchmark points, which comes from the s θ term in ζ H or ξ H given in Eq. (26).
In contrast, the cross section for A is essentially the same for the three benchmark points.
In Fig. 9, we show the cross section of the bottom quark associated production as a function of the mass of the produced Higgs boson. Typically, the cross section is more than one order of magnitude smaller than the gluon fusion production process because of the smallness of the bottom Yukawa coupling and the three body phase space. Differently from the gluon fusion, the dependence of the type is important, because the bottom Yukawa coupling determines the size of the cross section. In fact, the cross section in Type-II and Type-Y is almost one order of magnitude greater than that in Type-I and Type-X. Similar to the case for the gluon fusion, a larger discrepancy of the cross section among BP1, BP2 and BP3 is seen for the production of H state, as for the A one differences are marginal. In Fig. 10, we show the gluon-bottom fusion production cross section for the H ± state.
Similar to the gluon fusion process, the dependence of the type of Yukawa interactions is almost negligible, because of involving the top Yukawa coupling. The differences among the three benchmark points are negligibly small.
To summarise, in this section, we have discussed the differences between the E2HDM and C2HDM by focusing on the deviations in the SM-like Higgs boson couplings from the SM predictions as well as the decay BRs and production cross sections at the LHC. We have shown that, even if both the E2HDM and C2HDM give the same value of the deviation in the hV V coupling, we can find significant differences in the correlation of ∆κ E -∆κ D in the two scenarios (elementary and composite). In addition, through the combination of the differences in the decay BRs and production cross sections for the extra Higgs bosons, we may be able to distinguish these two hypothesis on the nature of the Higgs bosons responsible for EWSB.

IV. CONCLUSIONS
In this paper, we have continued our exploration of C2HDM scenarios, started with Ref. [12], assuming four different types of Yukawa interactions, wherein the nature of all Higgs states is such that they are composite objects. Specifically, they are the pNGBs from the global symmetry breaking SO(6) → SO(4) × SO(2), induced explicitly by interactions between a new strong sector and the SM fields at the compositeness scale f . Such pNGBs, for which we adopt the same scalar potential as in the E2HDM, then trigger EWSB governed by the SM gauge group. Under the assumption of partial compositeness, it is rather natural that one of the emerging physical Higgs fields, the lightest one, is the 125 GeV state, h, discovered at CERN.
Within this construct, we then proceed to carry out a phenomenological study aiming at establishing the potential of the LHC in disentangling the two hypotheses, E2HDM versus C2HDM, by exploiting the fact that drastically different production and decay patterns for the four heavy Higgs states (H, A and H ± ) may onset in the composite scenario with respect to the elementary one, even when the properties of SM-like Higgs state are the same (within experimental accuracy) in the two scenarios. This has been done after imposing both theoretical (already derived in Ref. [12]) and experimental (obtained here by suitably modifying numerical toolboxes used in E2HDM analysis to also embed the C2HDM option) constraints, the latter revealing a marked dependence upon ξ only for the case of Type-I Yukawa interactions. Specifically, the most dramatic situation could occur when, e.g., in the presence of an established deviation of a few percents from the SM prediction for the hV V of the H ± state, a similar role is played by the H ± → W ± h decay. Obviously, for both these states too, intermediate situations are also possible, so that a precise study of these two channels would be a further strong handle to use in order to disentangle the two hypotheses.
As far as A and H ± production modes which are accessible at the LHC, i.e., gluon-gluon fusion and associate production with bb pairs (for the A) and associated production with bt pairs (for the H + ), are concerned though, practically no difference appears. The actual size of all these differences between the E2HDM and C2HDM is governed by the value of the ξ = v 2 SM /f 2 parameter, the larger the latter the more significant the former. Finally, although there are quantitative differences between the usual four Yukawa types (I, II, X and Y, in our notation) when predicting the yield of both the E2HDM and C2HDM, the qualitative pattern we described would generally persist. In fact, a similar phenomenology would emerge if deviations were instead (or in addition) established in the Yukawa couplings of the h state to b-quarks and/or τ -lepton.
In short, if deviations will be established during Run 2 of the LHC in the couplings of the discovered Higgs state with either SM gauge bosons or matter fermions, then, not only a thorough investigation of the 2HDM hypothesis is called for (as one of the simplest nonminimal version of EWSB induced by the Higgs mechanism via doublet states, like the one already discovered) but a dedicated scrutiny of the decay patters of all potentially accessible heavy Higgs states could enable one to separate the E2HDM from the C2HDM.