1. Introduction

The SM provides a very good description of all the phenomena related to hadron and lepton colliders. This includes the Higgs boson which appears as elementary scalar and which arises through the breaking of electroweak symmetry. So, on 4 July 2012, the discovery of a new particle with a mass measured by CMS Collaboration to be 125.35 GeV was announced; physicists suspected that it was the Higgs boson. Since then, the particle has been shown to behave, interact, and decay in many of the ways predicted for Higgs particles by the Standard Model, as well as having even parity and zero spin, two fundamental attributes of a Higgs boson. This also means it is the first elementary scalar particle discovered in nature, ^[1]. Any discovery of a charged Higgs boson would be confirmation of new physics and the CLIC can admit or exclude such a probability.

Different types of Higgs bosons, if they exist, may lead us into new realms of physics beyond the SM. Since the SM leaves many questions open, there are various extensions. For instance, if the SM at high energies is restrained in the Grand Unified Theory (GUT), subsequently the Higgs bosons related with GUT symmetry breaking required masses of order MX∼O(1015) GeV. Supersymmetry ^[2] supplies a solution to hierarchy problem through the cancellation of the quadratic divergences via fermionic and bosonic loops contributions ^[3,4,5,6]. Furthermore, the Minimal Supersymmetric extension of the SM can be obtained as an effective theory of supersymmetric GUT ^[7,8,9].

There are also other class of models based on SU(3)C⊗SU(3)L⊗U(1)N gauge symmetry (3-3-1 model) ^[10,11,12], where the anomaly cancellation mechanisms occur when the three basic fermion families are considered and not family by family as in the SM. This mechanism is peculiar because it requires that the number of families is an integer multiple of the number of colors. This feature combined together with the asymptotic freedom, which is a property of quantum chromodynamics, requires that the number of families is three. Moreover, according to these models, the Weinberg angle is restricted to the value sW2=sin2θW<1/4 in the version of heavy-leptons ^[10]. Thus, when it evolves to higher values, it shows that the model loses its perturbative character when it reaches to mass scale of about 4 TeV ^[13]. Hence, the 3-3-1 model is one of the most interesting extensions of the SM and is phenomenologically well motivated to be probed at the CLIC and other accelerators.

Future TeV-scale linear colliders will cause a significant degradation of the energy of the center of mass during the collision of electrons and positrons, so corrective effects such as Beamstrahlung (BS) ^{[14,15,16,17]}, that is, the Radiation from a beam of charged particles in a linear collider, due to its interaction with the electromagnetic field of the other beam, causes a bending in the path of the particles under the influence of such electromagnetic fields. During this bending, the particles radiate photons, causing a loss of beam energy.

Another correction that must be made in the production of new particles is due to initial and final state radiation (ISR) ^[18], which is perturbative, it occurs in any process that contains electrically charged or colored particles, either in the initial state, where one of the incoming particles emits radiation before the interaction with the others, so reducing the beam energy previously to the momentum transfer, as in the final state and manifests with gluon or photon radiation.

So, these corrective effects such as BS and ISR will make it possible to have more precise measurements of the cross section, near the point of resonance.

This paper is organized as follows. In Sec. 2 we discuss the relevant features of the model. In Sec. 3 we compute the total cross sections and introduce the ISR and BS effects. Finally in Sec. 4 we present our concluding remarks.

2 Relevant features of the model

If there exist any model beyond the standard model (BSM), then it is no strange that it is still hidden in the scalar sector. As one of the simplest BSM models, is the 3—3—1 model, which has been extensively investigated in the literature.

These is a promissory model which is based on the SU(3)C⊗SU(3)L⊗U(1)N (3-3-1 for short) semi simple symmetry group ^[10], which contains both charged Higgs bosons. In this model the three Higgs scalar triplets

generate the fermion and gauge boson masses in the model. The neutral scalar fields develop the vacuum expectation values (VEVs) 〈η0〉=vη, 〈ρ0〉=vρ and 〈χ0〉=vχ, with vη2+vρ2=vW2=(246   GeV )2. The pattern of symmetry breaking is

and so, we can expect vχ≫vη,vρ. The η and ρ scalar triplets give masses to the ordinary fermions and gauge bosons, while the χ scalar triplet gives masses to the new fermions and new gauge bosons. The most general, gauge invariant and renormalizable Higgs potential is

Here f is a constant with dimensions of mass and the λ_i, (i = 1,…,10) are dimensionless constants with λ₃ < 0 from the positivity of the scalar masses. The term proportional to λ₁₀ violates lepto-barionic number so that, it was not considered in the analysis of the Ref. ^[19] (another analysis of the 3-3-1 scalar sector are given in Ref. ^[20] and references cited therein). We can notice that this term contributes to the mass matrices of the charged scalar fields, but not to the neutral ones. However, it can be checked that in the approximation vχ≫vη,vρ we can still work with the masses and eigenstates given in Ref. ^[19]. Here this term is important to the decay of the lightest exotic fermion. Therefore, we are keeping it in the Higgs potential.

We present the Yukawa Lagrangians that respects the gauge symmetry and that is given in several papers (see, for example, ^[21]).

Where G_R = 1 + γ₅, the Cabbibo-Kobayashi-Maskawa mixing matrix, Uνl,UPl,VνP,VlP are arbitrary mixing matrices, Ml=diagme  mμ  mτ, MP=diagmE  mM  mT.

Symmetry breaking is initiated when the scalar neutral fields are shifted as φ=vφ+ξφ+iζφ, with φ=η0,ρ0,χ0. Thus, the physical neutral scalar eigenstates H10,H20,H30 and h0 are related to the shifted fields as

and in the charge scalar sector we have

with the condition that vχ≫vη,vρ^[19].

In this work we introduce the ISR and BS as corrections effects to the cross section around the resonances points: mZ'=5976.43 GeV, mZ'=6830.21 GeV and mZ'=8539.47 GeV for vχ=3.5(4.0;5.0) TeV, at the CLIC with 3 and 5 TeV ^[22].

3 Cross section production

The process e-e+→H2-H2+, take place through the exchange of the bosons γ, Z and Z’ in the s channel. Then using the Yukawa Lagrangians written above we evaluate the differential cross section for the H2±

Writing separately the cross-sections, we will have

where we have defined the Łstu parameter

and the βH2± is the Higgs velocity in the c.m. of the process which is equal to

and t and u are

where θ is the angle between the Higgs and the incident electron in the CM frame.

The primes (′) are for the case when we take a Z’ boson, ΓZ'^[25,23], are the total width of the Z’ boson, g_V,A are the standard lepton coupling constants, g_V’,A’ are the 3-3-1 lepton coupling constants, s is the center of mass energy of the e^—e⁺ system, g=4  π  α/sinθW and α is the fine structure constant, which we take equal to α= 1/128.. For the Z’ boson we take mZ'=5.9-8.6 TeV ^[26,27], since mZ' is proportional to the VEV v_χ ^[19,11,12]. For the standard model parameters, we assume Particle Data Group values, i.e., m_z = 91.19 GeV, sin2θW=0.2315, and MW=80.33 GeV ^[28], the velocity of the Higgs in the c.m of the process we denote through βH2±, t and u are the kinematic invariants. We have also defined the ΛγH2-H2+ as the coupling constants of the γ to Higgs H2±, the ΛZH2-H2+ as the coupling constants of the Z to Higgs H2± and the ΛZ'H2-H2+ as being the coupling constants of the Z’ to Higgs H2±.

Where

And

We will now consider the effects of ISR and BS. The ISR corrections to the cross section for e⁺e^— annihilations at high energies are presented. These corrections are not at tree-level but with accuracy up to the α² terms. The cross section will be folded with the distributions given by ^[14,18]. Then we have

where σZ' represents the resonance cross section, χ=1-4me/s and H(χ,s) is a radiator function that summarize the results of the ISR corrections.

where we have for

On the other side, BS is an effect that occurs in high-energy e⁺e^— collisions, when each beam of e⁺(e^—), due to its high density, generates a strong electromagnetic field. So when the other beam passes through this electromagnetic field, its path is curved and the curved particles will radiate photons. This energy loss is called BS.

The effective differential cross section for a final state with the effect of the BS is related to the effective differential cross section without effect, by means of the following formula:

where

is the function that describes the probability density of the BS and the variable y is the fraction of energy lost through emitted photons from the BS, ϒ is the effective beamstrahlung parameter, which is given by

where r_e is the radius of the electron, N the number of particles per beam, γ the Lorentz factor, α the fine structure constant and (σ_x, σ_y, σ_z) are the beam parameters.

Therefore, the effective cross section, considering the BS effect is:

4 Results and conclusions

Precision tests of the 3-3-1 theory around the Z’ are in this context a relevant question. In particular a significant role is played by the ISR and BS corrections to the initial electron-positron state. The physics involved in the problem of ISR and BS corrections is related to the emission of photons in the process of annihilation of electrons and positrons and its effect on the resonance peak. That is, in the production of a resonance of mass mz' with colliding beams of total energy of the center of mass s=2E, the emission of hard and soft photons occurs.

The impact of the initial state radiaton (ISR) and beamstrahlung (BS) on precision measurements strongly affects the behaviour of the production cross section around the resonance peaks, modifying as the shape as the size, so Figs. 1, 2 and 3 shows the cross section with and without ISR + BS around the resonances points: mZ'=5976.43 GeV, mZ'=6830.21 GeV and mZ'=8539.47 GeV for vχ=3.5(4.0;5.0) TeV, at the CLIC with 3 and 5 TeV. As can be seen the peak of the resonance shifts to the right and is lowered as a result of the ISR + BS effects.

Figure 1 Effective cross section for resonance production mZ'=5976 GeV with and without effect of BS and ISR for vχ=3.5 TeV. 

Figure 2 Effective cross section for resonance production mZ'=6830 GeV with and without effect of BS and ISR for vχ=4.0 TeV. 

Figure 3 Effective cross section for resonance production mZ'=8539 GeV with and without effect of BS and ISR for vχ=5.0 TeV.