Perturbative Unitarity Bounds in Composite 2-Higgs Doublet Models

Perturbative Unitarity Bounds in Composite 2-Higgs Doublet Models

Stefania De Curtis INFN, Sezione di Firenze, and Department of Physics and Astronomy, University of Florence, Via G. Sansone 1, 50019 Sesto Fiorentino, Italy    Stefano Moretti School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, United Kingdom    Kei Yagyu School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, United Kingdom    Emine Yildirim School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, United Kingdom
Abstract

We study bounds from perturbative unitarity in a Composite 2-Higgs Doublet Model (C2HDM) based on the spontaneous breakdown of a global symmetry at the compositeness scale . The eight pseudo Nambu-Goldstone Bosons (pNGBs) emerging from such a dynamics are identified as two isospin doublet Higgs fields. We calculate the -wave amplitude for all possible 2-to-2-body elastic (pseudo)scalar boson scatterings at energy scales reachable at the Large Hadron Collider (LHC) and beyond it, including the longitudinal components of weak gauge boson states as the corresponding pNGB states. In our calculation, the Higgs potential is assumed to have the same form as that in the Elementary 2-Higgs Doublet Model (E2HDM) with a discrete symmetry, which is expected to be generated at the one-loop level via the Coleman-Weinberg (CW) mechanism. We find that the -wave amplitude matrix can be block-diagonalized with maximally submatrices in a way similar to the E2HDM case as long as we only keep the contributions from and in the amplitudes, where and GeV, which is an appropriate approximation for our analysis. By requiring the C2HDM to satisfy perturbative unitarity at energies reachable by the LHC, we derive bounds on its parameters such as and the masses of extra Higgs bosons present in the scenario alongside the Standard Model (SM)-like Higgs state discovered in 2012.

I Introduction

The search for additional Higgses, after the one discovered so far Aad:2012tfa (); Chatrchyan:2012ufa () and the possible evidence of a new (pseudo)scalar state with mass around 750 GeV Diphoton (), is one of the most important tasks of Run 2 of the Large Hadron Collider (LHC). It is widely known that extra spinless states with or without Standard Model (SM) quantum numbers can induce sizeable tree-level effects in the couplings of the discovered state, which have been under close scrutiny for three years now. It is also true that direct searches for new Higgs states, as shown by the aforementioned recent preliminary results, could have a dramatic impact on LHC activities. These two facts seem already remarkable motivations to study the phenomenology of extra Higgses at the present CERN machine.

Despite the obvious far-reaching consequences of a discovery of even a single additional scalar, the presence of another Higgs would not be, by itself, an evidence for the naturalness of the weak scale: such a defining situation would still be pending upon the whole subject. Just like for the case of a single Higgs doublet, for which the hierarchy problem can be explained by its pseudo Nambu-Goldstone Boson (pNGB) nature, we would like to link the presence of extra Higgs particles to natural theories of the Fermi scale. In particular, we have in mind composite Higgs models, where the mass of the lightest Higgs state is kept naturally lighter than a new strong scale around TeV by an approximate global symmetry Dugan:1984hq (), broken by SM interactions in the partial compositeness paradigm Kaplan:1991dc (); Contino:2006nn ().

In the minimal composite Higgs model MCHM (); Contino:2006qr (), the only light scalar in the spectrum is a pNGB, surrounded by spanned composite resonances roughly heavier by a loop factor. The underlying symmetries protect the Higgs mass from quantum corrections thus giving a simple solution to the hierarchy problem. The only robust way to expect new light (pseudo)scalars in the spectrum is to make them also pNGBs. Even in the case they are not expected to be as light as the SM Higgs, it is interesting to find a mechanism for describing all the Higgses as pNGBs and to explain their mass differences. Last, but not least for importance, in the case of extra Higgs doublets with no Vacuum Expectation Value (VEV) nor couplings to quark and leptons, one could also have the possibility to describe neutral light states as possible composite dark matter candidates IDM ().

In this paper we aim at identifying among the lightest scalars at least two Higgs doublets as this would lead to a Composite 2-Higgs Doublet Model (C2HDM) so6 (). The latter represents the simplest natural two Higgs doublet alternative to supersymmetry. The composite Higgses arising from a new dynamics at the TeV scale ultimately drive the Electro-Weak (EW) symmetry breaking. To include them as pNGBs, one has basically two different and complementary approaches: (i) to write down an effective Lagrangian (e.g., a la Strongly Interacting Light Higgs (SILH) SILH ()) invariant under SM symmetries for light composite Higgses; (ii) to explicitly impose a specific symmetry breaking structure containing multiple pNGBs. We take here the second approach. In particular, we will study in detail models based on the spontaneous global symmetry breaking of  so6 (). We will focus on their predictions for the structure of the (pseudo)scalar spectrum and the deviations of their couplings from those of a generic renormalizable Elementary 2-Higgs Doublet Model (E2HDM). In the limit the pNGB states are in fact identified with the physical Higgs states of doublet scalar fields of the E2HDM111For an updated review of the theory and phenomenology of E2HDMs see branco (). . Deviations from the E2HDM are parametrized by , with the SM Higgs VEV.

Once the strong sector is integrated out, the pNGB Higgses, independently of their microscopic origin, are described by a non-linear -model associated to the coset. We construct their effective low-energy Lagrangian according to the prescription developed by Callan, Coleman, Wess and Zumino (CCWZ) ccwz (), 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 (we assume that they come from the couplings of the 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 so6 ().

Here we will focus on the unitarity properties of a C2HDM222For the discussion of unitarity in minimal composite Higgs models see Kanemura:2014kga (). , namely, we will derive the bounds on the parameters of the model by requiring perturbative unitarity to hold at the energies reachable by the LHC. In fact, contrarily to the E2HDM, which is renormalizable, the C2HDM is an effective theory. The pNGB nature of the Higgses leads to a modification of their couplings to matter with respect to the E2HDM case and, as a consequence, forces to a non-vanishing -dependence of the scattering amplitudes. This means that the C2HDM is not unitary for energies above a critical value or, alternatively said, one needs to consider other new physics contributions (e.g., new composite fermions and gauge bosons) to make the model unitary above that energy scale. Since the fermion content of the model is not playing a role in the present investigation, we will not specify the fermion representation and we will not calculate the Higgs potential generated by the radiative corrections. Instead, we will assume the same general form of the Higgs potential as in the E2HDM with a symmetry, the latter imposed in order to avoid Flavor Changing Neutral Currents (FCNCs) at the tree level GW (). Therefore, in the energy region where the E2HDM and C2HDM are both unitary, it is interesting to compare the bounds on the additional Higgs masses. In fact, due to a compensation amongst mass- and energy-dependent contributions, we find that regions not allowed in the E2HDM are instead permitted in the C2HDM for the most general configuration of their parameter spaces.

The paper is organized as follows. In Section II we describe the C2HDM based on . We separately discuss two scenarios: the active one in which both Higgs doublet fields acquire a VEV and the inert one in which only one does. In Section III, the unitarity properties of the C2HDM are discussed by calculating all the 2-to-2-body (pseudo)scalar boson scattering amplitudes and by deriving constraints through all these channels. Conclusions are drawn in Section IV. Some technical details of the derivation of the pNGB kinetic terms are given in Appendix A.

Ii The Model

ii.1 Higgs Doublets as pNGBs

We first discuss how we obtain two isospin scalar doublets from the spontaneous breakdown of the global symmetry, i.e., . In order to clarify this, we introduce the following fifteen generators:

 TaL,R=−i2[12ϵabc(δbiδcj−δbjδci)∓(δaiδ4j−δajδ4i)], TS=−i√2(δ5iδ6j−δ5jδ6i), T^a1=−i√2(δ^aiδ5j−δ^ajδ5i),T^a2=−i√2(δ^aiδ6j−δ^ajδ6i), with  (a,b,c=1-3),  (i,j=1-6),  (^a=1-4). (1)

The above generators are classified into the seven unbroken generators and and the eight broken generators and . We can confirm that and are the subalgebras which generate the subgroup by looking at the following commutation relations:

 [TaL,TbL]=iϵabcTcL,[TaR,TbR]=iϵabcTcR,[TaL,TbR]=[TaL,TS]=[TaR,TS]=0, (2) [TaL,TΦα]=−12σaTΦα,[T3R,TΦα]=−12TΦα, (3)

where

 TΦα=(T2α+iT1αT4α−iT3α),α=1,2. (4)

Eq. (2) tells us that the commutation relations among and are closed plus that () generates the () subgroup of which is identified as the custodial symmetry of the SM Higgs sector. Furthermore, Eq. (3) shows that transforms as the doublet with charge 333The overall minus sign is a conventional as the charge should be +1/2 to get and for the upper component of the Higgs doublet.. Therefore the broken generators are associated with the pNGBs, transforming as a (4,2) of .

We then introduce the following two doublet scalar fields associated with as pNGBs:

 Φα≡1√2(h2α+ih1αh4α−ih3α)≡(ω+αvα+hα+izα√2), (5)

where the ’s are the VEVs of . The relation among , and the Fermi constant will be discussed in Section II-B. Notice that, in order to assign the right hypercharge to fermions, one has to introduce also an extra . The electric charge will be then defined as usual by with the hypercharge given by where is the charge. In this paper, we do not discuss the fermion sector which is not relevant to the following analysis, so we can omit to include this extra group.

ii.2 Kinetic Lagrangian

In general, once the coset space has been chosen, the low-energy Lagrangian is fixed at the two-derivative level, the basic ingredient being the pNGB matrix which transforms non-linearly under the global group.

The kinetic Lagrangian invariant under the symmetry can be constructed by the analogue of the construction in non-linear sigma models developed in ccwz (), as

 Lkin=f24(d^aα)μ(d^aα)μ, (6)

where

 (d^aα)μ=itr(U†DμUT^aα). (7)

Here is the pNGB matrix:

 U=exp(iΠf),  with  Π≡√2h^aαT^aα=−i⎛⎜ ⎜⎝ccc04×4h^a1h^a2−h^a100−h^a200⎞⎟ ⎟⎠. (8)

In Eq. (7), the covariant derivative is given by

 Dμ =∂μ−igTaLWaμ−ig′YBμ. (9)

The expressions for up to are given in Appendix A.

In order to see how the gauge boson masses are generated, let us consider the fourth components of the Higgs fields:

 h^a1=(0,0,0,~h1),h^a2=(0,0,0,~h2), (10)

with

 ~h1=h1+v1,~h2=h2+v2. (11)

In this case, the matrix defined in Eq. (8) takes a simple form,

 U =⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝14×4−(1−cos~hf)44~h^a1~hsin~hf~h^a2~hsin~hf−~h^a1~hsin~hf1−~h21~h2(1−cos~hf)−~h1~h2~h2(1−cos~hf)−~h^a2~hsin~hf−~h1~h2~h2(1−cos~hf)1−~h22~h2(1−cos~hf)⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (12)

where

 ~h≡√~h21+~h22. (13)

The 2-gauge boson terms are extracted from Eqs. (6), (7) and (12) as

 Lmasskin =f28(2g2W+μW−μ+g2ZZμZμ)sin2~hf, (14)

and thus the gauge boson masses are given by

 m2W =g24f2sin2vf,m2Z=g2Z4f2sin2vf, (15)

where and with being the weak mixing angle. Notice here that the VEV is different from the one in the SM as long as we take a finite value of . The relationship among , and is expressed as follows

 v2SM≡1√2GF=f2sin2vf≃(246 GeV)2. (16)

The ratio of the two VEVs is defined by .

Similarly to the E2HDM, we can define the so-called Higgs basis HB () in which only one of the two doublet fields contains the VEV and the Nambu-Goldstone states and absorbed into the longitudinal components of and bosons, respectively:

 (Φ1Φ2)=R(β)(ΦΨ),R(x)=(cosx−sinxsinxcosx), (17)

where

 (18)

In the above expressions, and are the physical charged and CP-odd neutral Higgs boson, respectively, while and are the CP-even Higgs bosons which in general can be mixed with each other. In this basis, the two-derivative terms for scalar bosons are extracted up to :

 L2-derkin =(1−ξ3)[|∂μG+|2+12(∂μG0)2+12(∂μh′2)2]+|∂μH+|2+12(∂μA)2+12(∂μh′1)2, (19)

where

 ξ=v2SMf2. (20)

We see that the kinetic terms for , and are not in canonical form and we need to rescale the fields:

 G+→(1−ξ3)−1/2G+,G0→(1−ξ3)−1/2G0,h′2→(1−ξ3)−1/2h′2. (21)

After this shift, we can define the mass eigenstates for the CP-even scalar bosons by introducing the mixing angle as:

 (h′1h′2)=R(θ)(hH), (22)

where is assumed to be the observed Higgs boson with a mass of 125 GeV. The mixing angle is determined by the mass matrix for the CP-even states calculated from the Higgs potential, which will be discussed in the next subsection.

ii.3 Higgs Potential

The Higgs potential is generated through the Coleman-Weinberg (CW) mechanism cw () at loop levels. There are two types of contributions to the potential, coming from gauge boson loops and fermion loops. The former contribution can be calculated without any ambiguities and it generates a positive squared mass term in the potential MCHM (). Thus, EW symmetry breaking does not occur by the gauge loops alone. Fermion loops can provide a negative contribution to the squared mass term, so their effect is essentially important to trigger EW symmetry breaking. However, the contribution from fermion loops depends on the choice of the representation of fermions.

The structure of the Higgs potential in the model has been studied in Ref. so6 () assuming several representations of fermion fields. They also assume that the explicit breaking of the global symmetry is associated with the couplings of the strong sector to the SM fields, that is, gauge and Yukawa interactions. This assumption, dictated by minimality, allows one to parameterize the Higgs potential, at each given order in the fermion and gauge couplings, in terms of a limited number of coefficients. If this assumption is relaxed, the parameter space of the C2HDM could be significantly enlarged. The form of the potential they obtain is given by the general E2HDM one, but each of the parameters is expressed in terms of those in the strong sector (mainly associated to the top dynamics). In our paper, however, we do not explicitly calculate the CW potential and we do not specify the fermion representations, making our analysis applicable to different choices of them. In fact, while the coupling of the vector bosons is fixed by gauge invariance, more freedom exists in the fermion sector and, to specify the model, one must fix the quantum numbers of the strong sector operators which mix with the SM fermions, in particular with the top quark. The CW potential clearly depends on these choices. Instead of performing the explicit calculation, we assume here the same form of the Higgs potential as that in the E2HDM. Our results on the unitarity properties of the C2HDM will be expressed as bounds on the masses of the Higgses which are free parameters in the E2HDM. Once explicitly specified the model, we will have the possibility to check, by calculating the CW potential, if the composite Higgs spectrum of that particular configuration satisfies the unitarity bounds.

In order to avoid FCNCs at the tree level, a discrete symmetry GW () is often imposed onto the potential, which is what we also do here444In Ref. so6 (), the symmetry ( and ) is referred as the symmetry whose transformation can be expressed by a diagonal matrix form acting on the pNGB matrix given in Eq. (8).. Under the symmetry, the two doublet fields are transformed as . This symmetry can also avoid a large contribution to the EW -parameter which could emerge in C2HDMs from the dimension 6 operator in the kinetic Lagrangian555The issue of anomalous contribution to the -parameter and to FCNCs in C2HDMs is faced also in enrico () where they discuss -safe models based on different cosets, in particular SO(9)/SO(8). . Depending on the nature of the symmetry, i.e., softly-broken or unbroken, the properties of the Higgs bosons can drastically change. In the following, we first discuss the softly-broken case and then we consider the unbroken case. For the latter case, the VEV of must be taken to be zero to avoid the spontaneously breakdown of the symmetry. In analogy with the E2HDM we will refer to the former scenario as the active C2HDM, while the latter describes the inert C2HDM.

ii.3.1 Active C2HDM

The Higgs potential under the gauge symmetry with the softly-broken symmetry is given by

 V(Φ1,Φ2) =m21Φ†1Φ1+m22Φ†2Φ2−m23(Φ†1Φ2+h.c.)+12λ1(Φ†1Φ1)2+12λ2(Φ†2Φ2)2 +λ3(Φ†1Φ1)(Φ†2Φ2)+λ4|Φ†1Φ2|2+12λ5[(Φ†1Φ2)2+h.c.], (23)

where and are generally complex, but we assume them to be real for simplicity. It is useful to rewrite the soft-breaking parameter through  KOSY () as follows:

 M2=m23sβcβ, (24)

where and . In the following, we use the shorthand notations and for an arbitrary angle .

The tadpole conditions for and fields, assuming and , are given by:

 m21+12v2(λ1c2β+λ345s2β)−M2s2β=0, (25) m22+12v2(λ2s2β+λ345c2β)−M2c2β=0, (26)

where . The mass matrices for the charged states in the basis of (,) and the CP-odd scalar states in the basis of (,) are diagonalized as

 RT(β)M2±R(β)=diag(0,m2H±),RT(β)M2oddR(β)=diag(0,m2A), (27)

where and are the squared masses of and :

 m2H± =M2−v22(λ4+λ5),m2A=M2−v2λ5. (28)

The massless states correspond to the modes and . The mass matrix for the CP-even scalar states is also calculated in the basis of () as

 M2even=((Meven)211(Meven)212(Meven)212(Meven)222), (29)

where each of matrix elements is expressed by

 (Meven)211 =v2(λ1c4β+λ2s4β+2λ345c2βs2β), (30) (Meven)222 =(1+ξ3)[M2+v2(λ1+λ2−2λ345)s2βc2β], (31) (Meven)212 =v2(1+ξ6)[−λ1c2β+λ2s2β+c2βλ345]sβcβ. (32)

This matrix can be diagonalized by the rotation introduced in Eq. (22) as

 m2h=c2θ(Meven)211+s2θ(Meven)222+2sθcθ(Meven)212, (33) m2H=s2θ(Meven)211+c2θ(Meven)222−2sθcθ(Meven)212, (34) tan2θ=2(Meven)212(Meven)211−(Meven)222. (35)

Now, we can rewrite all the parameters of the potential (23) in terms of the masses of the physical Higgs bosons and the mixing angle as follows:

 λ1 =1v2c2β[m2hc2β+θ+m2Hs2β+θ−M2s2β+ξ3sβ(m2hcβ+θsθ−m2Hsβ+θcθ)], (36) λ2 =1v2s2β[m2hs2β+θ+m2Hc2β+θ−M2c2β−ξ3cβ(m2hsβ+θsθ+m2Hcβ+θcθ)], (37) λ3 =1v2[2sβ+θcβ+θs2β(m2h−m2H)+2m2H±−M2−ξ3s2β(m2hsθc2β+θ−m2Hcθs2β+θ)], (38) λ4 =1v2(M2+m2A−2m2H±), (39) λ5 =1v2(M2−m2A). (40)

There are in total 9 independent parameters which can be expressed as , , , , , , (or ), and . The latter two parameters will be fixed in our analysis by requiring GeV and GeV.

ii.3.2 Inert C2HDM

The Higgs potential is given as in Eq. (23) without the term. Because of the absence of the VEV of , we have only one tadpole condition for , and the parameter will set the scale for the mass of the inert Higgs. Thus, the mass relations are the following:

 m2H± =m22+υ22λ3, (41) m2A =m22+v22(λ3+λ4−λ5), (42) m2H =(1+ξ3)(m22+v22λ345), (43) m2h =λ1v2. (44)

From teh above four relations, the , , and parameters can be rewritten in terms of the four mass parameters and :

 λ1 =m2hv2, (45) λ3 =2v2(m2H±−m22), (46) λ4 =1v2[m2A−2m2H±+m2H(1−ξ3)], (47) λ5 =1v2[m2H(1−ξ3)−m2A]. (48)

We note that the parameter is not determined in terms of the Higgs masses, just like the parameter.

The 8 independent parameters in the potential can be expressed as , , , , and (or ), and . Similar to the active case, and will be fixed by 125 GeV and by requiring GeV, respectively.

Iii Unitarity Bounds

In this section, we discuss the bound from perturbative unitarity in our C2HDM. We consider all possible 2-to-2-body bosonic elastic scatterings. The procedure to obtain the unitarity bound is similar to that in elementary models such as the SM LQT () and E2HDM KKT (); Akeroyd (); Ginzburg_CPV (); Ginzburg (); KY (). Namely, we compute the -wave amplitude matrix, derive its eigenvalues and then impose the following criterion HHG () for each of these:

 |Re(xi)|≤1/2. (49)

The most important difference between the unitarity bound in elementary models and that in composite models is that there is a squared energy dependence in the -wave amplitude for the latter. This is exactly canceled in elementary models among the diagrams with the gauge boson mediation, the Higgs boson mediation and the contact interactions. In composite models, however, this cancellation does not work, because the sum rule of the Higgs-Gauge-Gauge type couplings is modified from that in the elementary ones. For example, in the E2HDM, the squared sum of the and () couplings is the same as the squared coupling in the SM, while in the C2HDM, that is modified by the factor . The energy dependence of the -wave amplitudes leads to unitarity violation and asks for an Ultra-Violet (UV) completion of the C2HDM. The study of the unitarity bounds in this effective theory therefore gives an indication of the scale at which the onset of other effects of the strong sector become relevant.

iii.1 The W+LW−L→W+LW−L Reference Process

In order to clearly show the difference of perturbative unitarity properties in the E2HDM and those in the C2HDM, let us calculate the elastic scattering of the longitudinal component of the boson scattering, i.e., , in the active case.

The contribution from the diagrams without the Higgs bosons is calculated as in the SM,

 M(W+LW−L→W+LW−L)% gauge =s2v2SM(1−cϕ)−g2Z4[(2cos2θW−1)(1+cϕ)−2tan2ϕ2]+O(s−1), (50)

where is the scattering angle and is the squared Center-of-Mass (CM) energy. The contribution from the Higgs boson mediation ( and ) is given by:

 M(W+LW−L→W+LW−L)% Higgs =−s2v2SM(1−cϕ)(1−ξ)−2v2SM(1−ξ)(m2hc2θ+m2Hs2θ)+O(s−1). (51)

Thus, in the total amplitude the dependence appears which vanishes in the limit of :

 M(W+LW−L→W+LW−L)tot =sξ2v2SM(1−cϕ)−2v2SM% (m2hc2θ+m2Hs2θ)(1−ξ)+O(g2,s−1). (52)

The -wave amplitude , defined by

 a0=132π∫1−1dcosϕM=−132π∫0πdϕsinϕM, (53)

is calculated for the process as

 a0(W+LW−L→W+LW−L) =s32πv2SMξ−18πv2SM(m2hc2θ+m2Hs2θ)(1−ξ)+O(g2,s−1). (54)

Therefore, -matrix unitarity is broken at a certain energy scale as long as we take .

We expect that exactly the same result as in Eq. (54), up to , is obtained by using the equivalence theorem et (), in which the mode is replaced by the Nambu-Goldstone mode . Let us check this. There are three relevant diagrams for the amplitude , i.e., the contact diagram (denoted by ), and the - and -channel diagrams (denoted by and , respectively) with the and exchanges. Each of these diagrams is calculated as:

 Mc(G+G−→G+G−) =s2(1−cϕ)(gG±G±,G∓G∓−gG+G−,G+G−)+λG+G−G+G−, (55) Ms(G+G−→G+G−) =−∑φ=h,H1s−m2φ[s2(2gG±φ,G∓−gG+G−,φ)+λG+G−φ]2, (56) Mt(G+G−→G+G−) =−∑φ=h,H1t−m2φ[t2(2gG±φ,G∓−gG+G−,φ)+λG+G−φ]2. (57)

In the above expressions, we introduced the scalar trilinear and quartic couplings from the potential as well as the scalar trilinear and quartic couplings with two derivatives coming from the kinetic Lagrangian. They are defined by

 λabcd≡−∂4V∂a∂b∂c∂d,λabc≡−∂3V∂a∂b∂c (58)

and

 gab,cd≡∂4Lkin∂(∂μa)∂(∂μb)∂c∂d,gab,c≡∂3Lkin∂(∂μa)∂(∂μb)∂c. (59)

While and are proportional to , the and couplings contain terms plus corrections proportional to . These scalar couplings appearing in Eqs. (55)–(57) are given by:

 λG+G−G+G− =−2v2SM(1+ξ3)(m2hc2θ+m2Hs2θ), (60) λG+G−h =−m2hvSM(1+ξ6)cθ,λG+G−H=m2HvSM(1+ξ6)sθ,, (61) gG+G−,G+G− =−ξ3v2SM,gG±G±,G∓G∓=2ξ3v2SM, (62) gG+G−,h =−2ξ3vSMcθ,gG±h,G∓=ξ3vSMcθ, (63) gG+G−,H =2ξ3vSMsθ,gG±H,G∓=−ξ3vSMsθ. (64)

By substituting in Eqs. (55)–(57), we find that the total amplitude of the process is exactly the same as that of the one given in Eq. (52). In the following, we calculate all the other 2-to-2-body scattering channels using the equivalence theorem.

iii.2 Generic Formulae for the 2-to-2-body (Pseudo)Scalar Boson Scatterings

We discuss here the general 2-to-2-body scattering process denoted by with and being (pseudo)scalar bosons. There are contributions from contact , -channel , -channel and -channel diagrams as shown in Fig. 1. Each of the amplitudes is calculated in the following way: