Signal of doubly charged Higgs at $e^+e^-$ colliders

Masses and signals of the production of Doubly charged Higgses(DCH) in the framework of the supersymmetric reduced minimal 3-3-1 model(SUSYRM331) are investigated. In the DCH sector, weprove that there always exists a region of the parameter space where the mass of the lightest DCH is in order of $\mathcal{O}(100)$ GeV even when all other new particles are very heavy.The lightest DCH mainly decays to two same-sign leptons while the dominated decay channels of the heavy DCHs are those decay to heavy particles. We analyze each production cross section for $e^+e^- \rightarrow H^{++} H^{--}$ as a function of a few kinematic variables, which are useful to discuss the creation of the DCHs in the $e^+e^-$ colliders as a signal of new physics beyond the Standard Model. The numerical study shows that the cross sections for creating the lightest DCH can reach values of few pbs. The two other DCHs are too heavy, beyond the observable range of experiments. The lightest DCH may be detected by the International Linear Collider (ILC) or the Compact LInear Collider (CLIC) by searching its decay to a same-sign charged lepton pair.


I. INTRODUCTION
The detection of the Higgs boson with mass around 126 GeV by experiments at LHC [1,2] has again confirmed the success of the Standard Model (SM). On the other hand, this model needs to be extended to cover other problems which cannot be explained in this framework. Many well-known models beyond the SM have Higgs spectrum containing Doubly Charged Higgses (DCHs), for example the left-right model [3], the Zee-Babu model [4], the 3-3-1 models [5,6,11],... and their supersymmetric versions [15][16][17][18]. The appearance of the doubly charged Higgses will really be one of the signals of new physics. Hence, there have been numbers of publications predicting this signal in colliders such as the LHC, International Linear Collider (ILC) [33] and Compact LInear Collider (CLIC) [34]. Recent experimental searches for these DCHs have been doing at the LHC [13], through the decays of them to four leptons. The signals of the DCHs have been investigated in many above models: left-right symmetric model [19,20] and the supersymmetric version [21], 3-3-1 models [22]. It is noted that the Higgs sectors in supersymmetric (SUSY) models seem to be very interesting because they do not contain unknown self-couplings of four Higgses in the superpotential, unlike the case of non-SUSY models where this kind of couplings directly contribute to Higgs masses. As a consequence, some Higgses will get masses mainly from the D-term, namely from the electroweak breaking scale, leading to values of Higgs masses being in order of O(100) GeV at the tree level. This happens in SUSY models such as the MSSM, supersymmetric versions of the economical 3-3-1 (SUSYE331) and reduced minimal 3-3-1 (SUSYRM331) models [17,18,25]. It was shown that there is at least one neutral CP-even Higgs inheriting a tree-level mass below mass of Z boson, m Z = 92 GeV.  [25] for the SUSYE331 that parameters characterized for two scales can cancel each other to create light mass of the lightest singly charged Higgs. In this work, we will investigate the DCHs in the SUSYRM331 and prove that there may exist a light DCH in the limit of large values of both soft and SU(3) L parameters. This light DCH will be lighter than almost of new particles in the model and will be decay mainly to same-sign lepton pairs. So this will increase the possibility to detect the lightest DCH in the LHC, ILC and CLIC. In the left-right symmetric model, the cross sections for the DCHs creation at the LHC are predicted below 5 fb for mass values being more than 200 GeV. In the SUSY left-right model they are estimated below 10 fb with collision energy is 14 TeV [19] and the mass of DCH is smaller than 450 GeV. The cross sections for the DCH creation will decrease if their masses increase. In the framework of the 3-3-1 model, the cross section for creating DCHs can reach the value that is smaller than 10 2 fb in e + e − colliders [22]. Our work will concentrate on the signals of detection DCHs in the ILC and CLIC because of their very high precision. In addition the collision energy of the ILC and CLIC are smaller than that of the LHC but the total cross sections for creating the DCHs at the ILC and CLIC are larger than those at the LHC.
Let us remind the reason for studying the models based on gauge group SU(3) C ⊗SU(3) L ⊗ U(1) X which are called for short by 3-3-1 models. Despite the great success, the SM is unlikely to be a truly fundamental theory due to some well-known problems which cannot be solved in its framework. Some of these problems are the non-zero neutrino mass, the existence of three generations of fermions, electric charge quantization, dark matter,... Among the extended SMs, the 3-3-1 models [5][6][7] have many intriguing properties. In order to make the models anomaly free, one of the quark families must transform under SU(3) L differently from other two. This leads to a consequence that the number of fermion generations has to be multiple of the color number which is three. In combination with the QCD asymptotic freedom requiring the number of quark generations must be less than five, the solution is exactly three for the number of fermion generations as required. Furthermore, the 3-3-1 models give a good explanation of the electric charge quantization [8].
It is to be noted that the unique disadvantage of the 3-3-1 models is the complication in the Higgs sector, which reduces their predictability. Recently, there have been some efforts to reduce the Higgs contents of the models. The first successful attempt was to the 3-3-1 model with right-handed neutrinos [7] giving the model with just two Higgs triplets. The model is called by the economical 3-3-1 model [10]. The similar version to the minimal 3-3-1 model with Higgs sector containing three triplets and one sextet is the reduced minimal 3-3-1 model with again just two Higgs triplets [11,27]. However, to give masses to all fermions in the 3-3-1 models with the minimal Higgs sector, ones have to introduce the effective couplings which are non-renormalizable. This problem can be avoided within supersymmetric versions which have been built and studied such as SUSYE331 [12], SUSYRM331 [17],... In addition, one of the intriguing features of supersymmetric theories is that the Higgs spectrum is quite constrained.
Our paper is organized as follows. In section II, we will briefly review the SUSYRM331 model and concentrate on the Higgs sector. In this section, it will be shown that there are some very interesting properties of the SUSYRM331, for example: (i) the soft and the SU(3) L parameters should be in the same order; (ii) the model contains a light neutral CPeven Higgs with the values of the squared tree-level mass of m 2 Here γ is defined as ratio of two vacuum expectation of two Higgses ρ and ρ ′ and ǫ is defined as a quantity characterized for the ratio of the electroweak and SU(3) L scales. Section III is devoted for investigation in details the masses and other properties of DCHs. From this we prove that there exists a region of parameter space containing light DCHs. In Section IV we discuss on the creation of DCHs in e + e − colliders such as the ILC and CLIC. Specifically, we establish formulas of the cross sections for reactions e + e − → H ++ H −− in collision energies from 1 to 3 TeV and calculate the events of DCHs creations. These cross sections and the Higgs masses are presented as functions of very convenient parameters such as masses of neutral CP-odd Higgses , mass of the heavy singly charged gauge boson, tan γ and tan β, which will be defined in the work. This will helps one more easily predict many properties relating to the DCHs as well as relations among the mass of particles in the model. With each collision energy level of 1.5, 2 and 3 TeV, we discuss on the parameter space where the masses of three DCHs can satisfy the allowed kinetic condition, namely the mass of each DCH must be smaller than half of the collision energy. Then we estimate the amplitudes of the cross sections in these regions of parameter space. Finally, the branching ratios of the DCHs decay to pairs of the same-sign leptons are briefly discussed.

II. REVIEW OF THE SUSYRM331 MODEL
This work bases on the model presented in [17,18]. For convenience, we summarize important results which will be used in our calculation. Through the work we will use the notation of two-component spinor for fermions, where ψ denotes for a particle and ψ c denotes the corresponding anti-particle. Both ψ and ψ c are left-handed spinors. In case of the Majorana fields, where ψ = ψ c , we will use ψ notation.

A. Lepton and quark sectors
The lepton sector is arranged based on the original non-supersymmetric version [9], In parentheses it appears the transformation properties under the respective factors (SU(3) C , In the quark sector, the first quark family is put in a superfield which transforms as a triplet of the SU(3) L group,Q Three respective anti-quark superfields are singlets of the SU(3) L group, The two remaining quark families are included in two corresponding superfields transformed as antitriplets of SU(3) L , and the respective anti-quark superfields are transformed as followŝ B. Higgs sector The scalar superfields which are necessary to generate the fermion masses arê To remove chiral anomalies generated by the superpartners of the scalars, two new scalar superfields are introduced to transform as anti-triplets under the SU(3) L , namelŷ The pattern of the symmetry breaking of the model is given by the following scheme (using the notation given at [23]) For the sake of simplicity, all vacuum expectation values (VEVs) are supposed to be real.
When the 3-3-1 symmetry is broken, i.e, SU(3) C ⊗ U(1) Q , VEVs of the scalar fields are defined as follows Because of the symmetry breaking happen through steps given in (8), the VEVs have to satisfy the condition w, w ′ ≫ v, v ′ . The constraint on the W bosons mass leads to the consequence that

C. Higgs spectrum
As usual, the scalar Higgs potential is written as in [17], except V soft which is added b-type terms [18] to guarantee the vacuum stability of the model and to avoid the appearance of many tachyon scalars. Therefore we have where m ρ , m χ , m ρ ′ and m χ ′ have the mass dimension; b ρ as well as b χ are assumed to be real and positive to make sure the non-zero and real values of VEVs. The expansions of the neutral scalars around their VEVs are The minimum of the Higgs potential corresponds to the vanishing of all linear Higgs terms in this potential. As the results, it leads to four independent equations shown in [18] which reduce four independent parameters in the original Higgs potential. We will use notations chosen in [17] for this work. Especially, two independent parameters are chosen as Two scale electroweak and SU(3) L relate with masses of W and V boson [17,18], respectively We can choose m V as an independent parameter. On the other hand, there is a heavy doubly charged boson in the model, denoted as m 2 U = m 2 V + m 2 W . If m V ≫ m W , there will appear a degeneration in two heavy boson masses, The constraint of m V has been investigated recently by studying the flavor changing current [29], the muon Anomalous Magnetic Moment [30] in the RM331 model, where m W /m V ≤ 0.1. This is a rather good limit for our approximation used in this work. The minimum conditions of the superpotential are series of four equations below with t 2 ≡ 6s 2 W /(1 − 4s 2 W ). To estimate the scale of parameters in Eqs. (17), based on the calculation in [25], it is useful to write two equations (15) and (16) in the new forms as follows It was shown in [25] that because of the hierarchy between two breaking scales SU ( This conclusion explains why the values of parameters b ρ and b χ are chosen in [18] in order to get consistent values of the lightest CP-even neutral Higgs mass. For the review, we will list the Higgs spectrum as follows 1. CP-odd neutral Higgs. Two massless Higgses eaten by two neutral gauge bosons are Two massive Higgses are presented according to the original Higgses as follows and their masses are 2. Singly charged Higgs. Two massless eigenstates of these Higgses are They are eaten by the singly charged bosons. Two other massive states are 3. CP-even Neutral Higgses. In the basis of (H ρ , H ρ ′ , H χ , H χ ′ ) mass term of the neutral scalar Higgses has form of where Analytic formulas of elements of the matrix are listed in [17,18]. We remind that the eigenvalues of this matrix, λ = m 2 H 0 , must satisfy the equation Defining ǫ = (m 2 W /m 2 V ) ≤ (80.4/650) 2 = 0.015, we can estimate masses of these neutral Higgs as the following It is necessary to noted that the lightest mass with tree-level value of m Z | cos 2γ| ≤ m Z = 92 GeV. This is consistent with numerical result shown in [18]. Thus, the mass including loop corrections of this Higgs will increase to the current value of 125-126 GeV.
At the tree level, the above analysis indicates that the Higgs spectrum can be determined by unknown following independent parameters: α, β, , m V , m A 1 and m A 2 . Furthermore, the squared mass matrices of both CP-even neutral and DCHs explicitly depend on c 2β , s 2β , c 2γ and s 2γ but not explicitly depend on t 2γ , t 2β . Hence it can be guessed that Higgs masses will not very large when t α and t β are very large. And in some case we can take the limits c 2β,2γ → −1 and s 2β,2γ → 0 without any inconsistent calculation. In addition, we will limit m V ≥ 650 GeV based on [29,30]. The m A 1 and m A 2 are the same order of m V so we set This can be understood from the reason that all quantities we consider below depend on α, 2α, β and 2β only by factors of sine or cosine but not tan functions.

III. DOUBLY CHARGED HIGGS BOSONS AND COUPLINGS
A.

Mass spectrum
Mass term of the doubly charged boson: where the elements of the mass squared matrix was shown precisely in [18]. Taking a rotation characterized by a matrix we get new mass matrix From the mass matrix (25), the Goldstone Higgs boson eaten by the doubly charged gauge boson can be presented exactly in term of original Higgs basis, namely Because m W ≪ m U , m V ≃ m U and s γ ≤ 1, the doubly charged gauge boson couple weakly to light Higgses but strongly with heavy Higgses.
From the square mass matrix of the DCHs shown in (25) it can be realized that if there ) then the contributions of the off-diagonal elements in the matrix (25) are large. So it is difficult to find an analytic formula for both mass eigenstates and eigenvalues of these Higgses in this case. Note that apart from Goldstone boson (26), there are three other states corresponding to square matrix (25): We assume that three physical massive DCHs can be related to (H ′±± The relation between original Higgs basis and the mass basis now is and the Goldstone boson was indicated in (26). To estimate the values of elements of the matrix Λ, we firstly find out some properties of mass eigenvalues of the DCHs. Three remain where This equation gives three solutions corresponding to three values of physical DCHs masses at tree level. To avoid the appearance of doubly charged Higgs tachyons we deduce that Furthermore, from (25) if m 2 A 1 is enough close to m 2 V then there may appear one light DCH, while two others values always in the SU(3) L scale. To find the analytic formula of vertex factors V 0 H ++ H −− , we investigate the mass of the DCHs using techniques presented in [17], where masses can be expanded as The main contributions to three DCH masses are Comparing the element M 2 in the first line of (36), it can be . As a result, the main contributions to the mass eigenstate of m 2 . This is very useful to find the formulas of coupling between these DCHs with neutral gauge bosons.
From the first line of (33), the Viet theorem gives a relation Combining with m 2 . This is the consequence deduced from the formula (34): the condition of existing light Higgs It leads to m 2  Finally, the following of this section will calculate the coupling of H ++ H −− V 0 based on the above results.

B. Couplings between DCHs with neutral scalars and gauge bosons
It is noted that the process e + e − → H ++ H −− through virtue neutral Higgses involves with the coupling e + e − H 0 . In the SUSY version [11], this kind of the coupling is 2gm e /(m W c γ ).
While the SUSY version [17] has no this kind of coupling at the tree level. In this work we will use the case in [11]. Corresponding with this, we consider the coupling H ++ H −− ρ ′0 .
Couplings H ++ H −− H 0 comes from the D-term of the scalar potential (11) , namely Because the contribution from neutral Higgs mediation only relates to ρ ′0 , so the contribution to the e + e − → H ++ H −− amplitude is proportional to This contribution is smaller than one from neutral gauge boson mediation a factor of m e / √ s, so we can neglect it.
The Higgs-Higgs-gauge boson vertices come from the covariant kinetic terms of Higgs The interactions among neutral gauge bosons and doubly charged Higgses can be written as where D µ = ∂ µ − igV a µ T a − ig ′ XT 9 B µ , T a = 1 2 λ a or − 1 2 λ a * corresponding to triplet or anti-triplet representation of Higgses, T 9 = 1  Table I.  (CM) frame, the differential cross section for each DCH is given by where s = (p 1 + p 2 ) 2 = E 2 cm and M is the scattering amplitude, θ is the angle between k 1 and p 1 . The detail calculation is presented in the appendix B. The final result is where with total width of the Z ′ given in appendix A and with G a L , G a H are the couplings of the neutral gauge bosons with two leptons and two DCHs, respectively.
Similarly, in the case of H ±± 2,3 , we have The total cross section for this reaction is We remind that if the above process happen, then √ s > 2m H ±± . As mentioned above, the heaviest DCH, namely m H ±∓ it is hardly to create heavy DCHs. Noting that with the chosen values of m A 2 = 500 GeV, so this Higgs appears only when √ s ≥ 1.6 TeV. Therefore, we just consider the lightest DCH when √ s ≤ 1.5 TeV. is small enough, even very close to the below constraint of current investigation [29,30]. All above numerical investigation shows that the cross sections for creation of DCHs in e + e − colliders are in range 10 −1 − 1 pb. This will be a good signal for detection DCHs in near future colliders [33,34]. In particularly, the ILC with the collision energy of 1 TeV will correspond to the integrated luminosity of 1000 fm −1 [33], the events for creation the lightest DCH will be 10 5 − 10 6 . With the CLIC [34] where the collision energy will increase up to 3 TeV or more, the heavy DCHs may appear. Furthermore, the estimated integrated luminosity targets will be 500 fb −1 at 500 GeV, 1.5 ab −1 at 1.4 (1.5) TeV and 2 ab −1 at 3 TeV collision energy. The events corresponding with these are 50 − 500, 1.5 × 10 5 − 1.5 × 10 6 and 2 ×10 5 −2 ×10 6 . If we look for the events from the decay of DCHs to same-sign lepton pairs, the decay of the lightest DCH is the most promising signal. The reason can be explained as follows. In this model, all DCHs have the lepton number two so the total lepton number of all final states must be the same. Therefore, if a DCH decays to two SM particles, the final states are only two same-sign charged leptons. If the final states of the decay contains new particles, from the correlation between masses of the lightest DCH and all other particles mentioned in section III, one can see that the lightest DCH always couples with at least one heavier particle having non-zero lepton number. It can be seen precisely in tables VI and VII where vertex couplings relating with DCHs are listed. In addition, because the lightest DCH is contributed mainly from the Higgses ρ and ρ ′ and couplings between Higgses and charged leptons are shown in table VI, it is easy to prove that the partial decay of this DCH to a pair of same-sign leptons is Γ(H ±± → l ± i l ± i ) ∼ (m l i /m W ) 2 with l i = e, µ, τ . As a result, we obtain Br(H ±± → τ ± τ ± ≃ 1), i.e, the events of four-tauon observable signal are equal to the events of creation the lightest DCH at e + e − colliders. It is emphasized that the events of creation lightest DCH will increase when including the decays of other heavier doubly charged particles such as other DCHs or gauge bosons. For the case of DCHs with mass of m 2 A 2 + m 2 W , it gets contribution mainly from χ and χ ′ , which weakly couple to leptons so the decay of this Higgs to leptons is much small. Also the heaviest DCHs will decay mainly to other particles than leptons.

V. CONCLUSIONS
We have investigated the Higgs sector of the SUSYRM331 model where the DCHs are specially concentrated on as one of signals to look for new physics at e + e − colliders. Here, masses of neutral CP-even, DCHs and the cross section of creating DCHs at e + e − colliders can be presented according to five unknown parameters: two masses m A 1 ,A 2 of neutral CPodd Higgses, mass of singly heavy gauge boson m V , t γ and t β . This will help us more easily to discuss on the relation between particle masses in the model. We have found the exact Lagrangian for the vertex of f f V was shown in [11,17,18,27]. Here we use the eigenvalues of gauge bosons given in [17,27], namely where relations between mass eigenstates and original eigenstates of neutral gauge bosons are given as follows Here c ζ ≡ cos ζ > 0, s ζ ≡ sin ζ > 0 where ζ is defined as a rotative angle satisfying .

(A3)
The parameter t is the ratio between g ′ and g, namely Masses of gauge bosons are follows Above analysis is enough to calculate vertex factors of charged leptons with neutral gauge bosons, as shown explicitly in table II. Here we only concentrate on the largest vertex couplings by assuming the case of flavor basis of leptons and quarks is the mass basis. then the total amplitude for each doubly charged Higgs is written as where m a V = 0, m Z , m Z ′ corresponding to photon, Z, Z ′ bosons. Squaring the amplitude and summing over the electron and positron spins, we have Now we use the fact that p 2 with E = E cm /2 = √ s/2. We define two pour-momenta of final particles as k 1µ = (E 1 , k 1 ) and k 2µ = (E 2 , k 2 ). Using the condition of four-momentum conversation, it is easy to prove some following results Inserting all results (B5) into (B6), we obtain The analytic forms of g φ Z ′ depend on the particular representation of φ. Specially, we have • SU(3) L adjoint representation relates with gauge bosons and their superpartners only.
The standard couplings of three gauge bosons can be written as shown in the table III. With gauginos, the vertices can also written in the form of Below we will calculate the partial decay width of Z ′ into three different class of particles.

Gauge boson
Analytic formulas can be found in [32]. For the purpose of estimation the total width decays of Z ′ as simple as possible, we only consider the largest contribution to each class of particles. In addition, all particles as gauginos, sleptons, squarks receiving masses from the soft terms are very heavy so that Z ′ cannot decay into. We assume the similar situations for cases of exotic quarks. The decay of the Z ′ relating with these particles deserve a further

Decay of Z ′ to fermions pairs
This kind of decay involves with the below Lagrangian where sum is over all fermions in the model that couple with Z ′ and satisfy the kinetic condition m Z ′ > 2m f . Formulas of g f Z ′ and g f c Z ′ were shown in the table II. The partial decay width corresponding to each fermion is [28], where N f c is the color factor being equal to 3 for quarks and 1 for all other fermions (leptons, quarks, Higgsinos and gauginos).

Decay of Z ′ to scalar pairs
Lagrangian relating with these decay is Vertex factor Vertex factor  If the momentum of the Z ′ boson is p µ then we have p 2 = m 2 Z ′ and p = k 1 + k 2 . The amplitude of the decay is with ε µ = ε µ (p, λ Z ′ ) being the polarization vector of the Z ′ .
Averaging over the Z ′ polarization using we obtain the squared amplitude Noting that k 2 1 = m 2 S i , k 2 2 = m 2 S j and (k 1 − k 2 ) 2 = 2 m 2 S i + m 2 S j − m 2 Z ′ we have formula of Γ(Z ′ → SS), namely for two distinguishable final states. For identical final states, there need an extra factor 1/2 to avoid counting each final state twice [31]. Therefore, if S i ≡ S j → S, then m S i = m S j = m S and denoting g S ij Z ′ = g S Z ′ we have more simple formula: Vertex factor Vertex factor and Z ′ ZH 0 . This part of Lagrangian has form g SV Z ′ Z ′ µ V µ S. Vertex factors are presented in Table V. The partial decay width for this case is