One-loop contributions for and in the extension of the standard model

We present one-loop contributions for with and in the extension of the standard model. In the phenomenological results, the signal strengths for at the Large Hadron Collider and for at future lepton colliders are analyzed in the physical parameter space for both the vector and chiral models. We found that the contributions from the neutral gauge boson to the signal strengths are rather small. Consequently, the effects will be difficult to probe at future colliders. However, the impacts of charged Higgs and CP-odd Higgs in the chiral model on the signal strengths are significant and can be measured with the help of the initial polarization beams at future lepton colliders.


I. INTRODUCTION
The precise measurements of the decay rates and production cross-sections of the standard-model-like Higgs boson (SM-like Higgs boson) h are primary targets at the High-Luminosity Large Hadron Collider (HL-LHC) [1,2] as well as future lepton colliders (LCs) [3].From the measured data, we can probe accurately the properties of the SM-like Higgs boson.In this perspective, the nature of the scalar Higgs potential can be discovered.In other words, we can deeply understand the electroweak spontaneous symmetry breaking (EWSB).The updated measurements for the Higgs boson production cross-sections and decay rates at ATLAS and CMS can be found in Refs.[4,5] and the references therein.Moreover, the loop-induced processes and have been measured at the LHC [6][7][8][9][10][11][12].More recently, the decay processes with have also received significant attention at the LHC [13][14][15][16].Together with , the decay processes provide important information for testing the standard model (SM) [17][18][19] and the constraining parameters in many of beyond the standard models (BSMs) [19].
One-loop contributions for the decay processes have been performed in Refs.[20][21][22][23][24][25][26][27] in the SM.In the framework of the two Higgs doublet models (THDM), one-loop decay processes have been evaluated in Refs.[28,29].Recently, one-loop expressions for in general BSM frameworks have been presented in Refs.[30,31].We have derived alternative presentations for one-loop contributions of in the Higgs extension of the standard model (HESM) [32].In [32], the computations have addressed the 't Hooft-Feynman (HF) gauge and confirm the results in our previous studies [30,31] in the unitary gauge.We note that all contributions of singly and doubly charged Higgses propagating in the loop diagrams have been taken into account in Ref. [32].It is well-known that there also exist many new gauge bosons exchanging in the loop diagrams of the decay processes in other BSMs.For example, in the simplest extension for the SM [33][34][35][36][37][38][39][40][41][42], a neutral gauge boson exists.In the left-right models (LR) constructed from the [43][44][45], 3-3-1 models ( ) [46][47][48][49][50][51][52], 3-4-1 models ( ) [52][53][54][55][56][57], etc., new charged gauge bosons as well as new neutral gauge bosons are included.Therefore, one-loop formulas for the decay rates of in the above-mentioned models are of great interest.The calculations for in the HF gauge have several advantages as discussed in [32].It is difficult to derive general one-loop formulas for in the HF gauge for the arbitrary BSM as we have performed in [31].Because of the couplings of new particles to Goldstone bosons, ghost particles depend on the models under consideration.Owing to gauge invariance, the contribution from Goldstone bosons, ghost particles propagating in the loop will be cancelled out at the final results.The cancellations within the vector boson loop with their Goldstone bosons.The ghost particles exchanging in the loop may not be the same in the different BSMs.For this reason, in this work, we compute the oneloop contribution for with and in the extension of the SM in the HF gauge.In comparison with the work in [31], we present alternative expressions for the one-loop contribution for in the HF gauge within the models.This work also extends the results in [32] by adding the new contributions of the neutral gauge boson as well as heavy neutrinos in the models.In another aspect, the couplings of the Higgs boson with can be probed by measuring the cross-sections of the Higgs boson production associated with a photon at future LCs.It is important to note that the treelevel cross-section for this process is proportional to the electron mass, and the process follows the electromagnetic gauge symmetry.Therefore, it is a significant contribution from the one-loop level.As a result, the cross-section is rather small because, compared with the LHC, the LC has a cleaner background.The new physic signals are thus easily extracted from the background.With the highluminosity designed at future LCs [3], the signal of Higgs boson production associated with a photon can be probed.Many computations for one-loop corrections to at future LCs in the SM are available [58][59][60], such as those in many frameworks of the HESM [61][62][63] and in the minimal supersymmetric standard model [64].In this work, we show that cross-sections of can be derived by using one-loop form factors in .The results are also valid in the SM, within the extension of the standard model, and can be extended to other BSMs.
In the phenomenological studies, we present the first results of analyzing the signal strengths for at the LHC and for at future LCs in the physical parameter space for both vector and chiral models.For the decay processes , we focus on the possibility to probe the heavy neutral gauge boson as well as the charged Higgs and CP-odd Higgs at the LHC.For this reason, we are interested in the case of heavy neutrinos scenario ( ).For the production process at future LCs, the signal strengths are examined by including the initial beam polarization effects.The contributions of the neutral gauge boson , charged Higgs, CP-odd Higgs, as well as additional neutrinos are investigated in further detail at future LCs.

U(1) B
Our work is organized as follows.In Sec.II, the extension of the SM is reviewed in more detail.In Sec.III, we present the concrete evaluations for oneloop form factors for the decay channels and for the process .The phenomenological results for the processes are discussed in Sec.IV.The conclusions and outlook are included in Sec.V.In the appendices, we calculate all the couplings related to the processes under consideration.

II. REVIEW OF EXTENSION OF THE STANDARD MODEL
U(1) In this section, following [33], we review in detail the model, one of the simplest extensions of the SM.The model enlarges the SM gauge group with a new gauge symmetry .The extension model then follows the gauge group .As a result, the Yang-Mills Lagrangian including the mixing of two abelian gauge groups is modified as follows: where correspond to the field strengths of the gauge groups , whereas and are the field strengths of the gauge groups and , respectively.Considering the mixing term (kinematic mixing case), we have to perform the following rotation for two gauge bosons in the groups of , : Here, κ is the mixing parameter between the gauge groups and .We note that ( ) are the couplings of the , ( ) gauge groups, respectively.In this paper, we note the hypercharge Y for the gauge group and for the gauge group.The model is classified into two types of models, the vector model and the chiral model [33].These models are studied in further detail in the following subsections.
In the vector model, we have three additional right handed neutrinos (RHNs, for ) and a new complex scalar singlet field χ, which is required for Here, the kinematic part reads as and the scalar potential is given by The fields Φ and χ are parameterized for spontaneous symmetry breaking as follows: Here, the Goldstone bosons are giving the masses for the bosons, whereas the fields are mixed together, being neutral Goldstone bosons .They play a role for giving masses to the gauge bosons Z and , respectively.The minimization for the scalar potential leads to a system of equations as follows: From the minimization conditions for , one can present in terms of the remaining parameters in the potential.As a result, the mass matrix for the neutral scalar components is collected on the basis of as follows:

(h, H)
To diagonalize the matrix in Eq. ( 9) for obtaining the Higgs physical masses of neutral Higgses, one first performs the rotation that shows the relation of and with mixing angle The masses of the CP-even Higgs scalars take then the form of In this work, we assume that , h being the SMlike Higgs boson with GeV.From the kinematic term of the scalar sector, we collect the mass matrix of the neutral gauge bosons.This term is given by where  .In this paper, we note as three RHNs for later uses.
, −1 ( ) , , 0 ( ) When we consider the mixing of two gauges, one first uses the rotation matrix in Eq. ( 2) for obtaining the basis .To find the physical masses of gauge bosons, we have to diagonalize the mass matrix in the basis .First, the mass matrix of the neutral gauge bosons in the kinetic term diagonalized basis can be collected as follows: where with the new coupling .By changing the basis from , to the one in , , we have the mass eigenstates for physical gauge bosons.The relation is shown in the following rotation matrix as Here, and are mixing angles of gauge bosons.In the above formulas, we have the following parameters: The masses of the physical gauge bosons A, Z, and are expressed as where

B X
The covariant derivative with the kinetic mixing can be expressed in terms of the orthogonal fields and as Here, .In the case of the non-kinematic mixing gauge, taking , one has the covariant derivative as The neutrino masses in this model are generated by a seesaw mechanism that will be shown in the Yukawa Lagrangian, namely, generates the Dirac masses for the RHNs.The fermion Lagrangian is given by All couplings involving the processes under consideration are presented in the following Tables 2, 3, 4, 5.In Table 2, the third column is presented for the couplings in the second column by changing the bare parameters to physical parameters accordingly.The last column presents the results of the corresponding couplings in the limit of non-kinematic mixing.
Z ′ Table 3 presents the couplings of the SM-like Higgs to Z and related to the computed processes.The last column lists the results of the corresponding couplings in the case of non-kinematic mixing.
In Table 4, three gauge boson vertices related to the processes are presented.The corresponding couplings in non-kinematic mixing are listed in the last column results.
In Table 5, all the couplings of and are presented.Hypercharge takes the corresponding values for f presented in Table 1.Here, and .Again, the corresponding couplings in the case of non-kinematic mixing are presented in the last Table 2. Couplings related to the processes under consideration.In some cases, we have used the following relations: and .

Vertices Mixing
Mixing (physical parameters) Non-mixing 3. Couplings of the SM-like Higgs to Z and related to the processes under consideration.In some cases, we have used the following relations: .Here, . Vertices . Couplings of three gauge boson vertices related to the processes under consideration.This model also adds an additional scalar Dark matter , which plays a role of dark matter.In general, the scalar sector takes the form of

Vertices Mixing Non-mixing
Here, the covariant derivative is defined in (23).The scalar potential is expressed as follows: The fields , and σ can be written for the SSB as follows: In this role, the fields Φ and φ break the electroweak symmetry, whereas the scalar fields σ and φ break the symmetry.The minimization for the scalar potential leads to the following system of equations: The mass matrix for charged Higgs is then collected in Table 5. Couplings related to the processes under consideration.In some cases, we have used the following relations: . Hypercharge takes the corresponding values for f presented in Table 1.Here, and .
Vertices Mixing Non-mixing Table 6.Matter fields and their quantum numbers of the chiral model.We omit the index of generations of matter particles in this Table .In this work, we note right handed neutrinos as for for latter uses.
Tran Dzung Tri, Thanh Huy Nguyen, Khiem Hong Phan Chin.Phys.C 48, 053101 (2024) 053101-6 term of the basics as After symmetry breaking, this model contains Goldstone bosons and charged Higgs , which are mixed by .The Goldstone bosons give the masses to the bosons.The mixing matrix is given by The mass matrix in the basis can be written as For the neutral components of Higgs fields, we have the following relation: give the masses for Z and , respectively.The remaining physical field becomes a CP-odd Higgs boson.The masses of the charged and CPodd Higgses are determined as follows: The mass matrix for neutral scalars is collected in the basis .It can be expressed in the form of M 2

S
where The matrix can be diagonalized by an orthogonal matrix as Here, the relation between the two basics is given by For CP-even scalar Higgses, the mass eigenstates are ordered by their masses .In this notation, is identified as the SM Higgs of 125 GeV.The rotation matrix is defined as , where each matrix is expressed as follows: where with .
The rotation matrix for neutral gauge bosons is A, Z, Z ′ and the masses of photon bosons are obtained as follows: Here, It is noted that we have not considered the kinematic mixing of the two gauges and .However, in this report, we assume the mixing case of two neutral gauge bosons Z, .
The Yukawa Lagrangian of the model can be written as where the φ field generates a Dirac neutrino mass for the RHNs.This model does not have a Majorana mass term ν R for .
Finally, the mass of dark matter from the Higgs potential is given as In this version, we have two CP-even Higges, one CP-odd Higgs, and two charged Higgses.Furthermore, we have three more right handed neutrinos and a neutral gauge boson .All these new particles are exchanged in the loop Feynman diagrams of the processes under consideration.
All couplings relating to the the computed processes are presented in the following Tables 7, 8, 9, 10, and 11.
In several cases, we have used .In Table 7, the third column results correspond to the couplings in the second column by replacing the bare parameters in the Lagrangian with the physical parameters.The last column shows for the results of the non-mixing two gauge bosons.
The couplings of and are given by Couplings related to the processes under consideration.In some cases, we have used .The vector can be obtained by taking the limits of and .
Vertices Mixing Mixing (physical parameters) Non-mixing Tran Dzung Tri, Thanh Huy Nguyen, Khiem Hong Phan Chin.Phys.C 48, 053101 (2024) 053101-8 We stress that the corresponding couplings in the vector can be derived by taking the limits of , , , and .In these limits, for example, the results of the third column in Table 7 are returned to the third column couplings in Table 2.

AND IN THE MODELS
One-loop form factors for and in the models are presented in detail in this section.First, we presents the calculations for one-loop form factors for .We then show that the cross-sections for can be derived by using one-loop form factors in the decay process .
h → ℓ lγ A. Form factors for the decay process In this section, we present the calculations for oneloop form factors for the decay channels in the extension of the SM.In this computation, all the new couplings that appear in the models are denoted as . For the cases of the couplings of the SMlike Higgs to the vector boson pair and fermion pair, we parameterize as B − L All the couplings from the models involving the Couplings in the chiral model.We have already taken into account .The vector can be obtained by taking the limits of and .
Within the HF gauge, all one-loop Feynman diagrams contributing to the decay process are presented in the following paragraphs.These one-loop diagrams can be classified into several groups.In group (as shown in Fig. 1), we include all one-loop Feynman diagrams having -poles contributing to the decay process.This means that we have involved in these diagrams.In the current work, can be , and .In the groups (as presented in Figs. 2, 3), other one- loop Feynman diagrams with -poles are also plotted.In the HF gauge, we also take into account all fermions, charged Higgs, vector bosons, Goldstone bosons, and ghost particles propagating in the loop.As proved in Ref. [30], we only collect one-loop form factors, which are proportional to the (for ) appearing in Eq. (50).Hence, all the diagrams in Fig. 2 and Fig. 3 can be ignored in the present work.
In addition, three types of one-loop Feynman diagrams without -poles (called non -pole diagrams hereafter) contribute to the processes under consideration.The first classification includes non--pole diagrams with two neutral gauge bosons exchanging in the loop.In the calculation, can be and in the loop (as depicted in Fig. 4).The second group includes non--pole with W bosons propagating in the loop (as presented in Fig. 5).The last category includes non-pole diagrams with charged Higgs propagating in the loop (as plotted in Fig. 6).
In general, the one-loop amplitude for the decay process of SM-like Higgs can be expressed in terms of the Lorentz structure as follows: where .Owing to the on-shell photon at Fig. 1.Group One-loop Feynman diagram -poles contributing to the processes.All the fermions, W bosons, charged Higgs, Goldstone bosons, and ghost particles exchanging in the loop are considered.can be , and in this calculation.
the final state, the amplitude needs to follow ward identiy, and we can derive the relation .As a result, the decay rates for the processes can be computed via one of the above form factors.It is advantageous to select one-loop form factors for the later analysis because these form factors do not have the ultraviolet divergent ( -divergent).Therefore, all diagrams in Fig. 2 and Fig. 3 can be ignored in this work, as we stated previously.Hereafter, .The one-loop form factors are separated into two parts as follows: The related kinematic variables for this process calcula-tions are included as follows: The kinematic variables also follow the relation

B − L FeynArt
We first implement the models into the package [65].One-loop amplitude can be generated automatically by this package.The amplitude is then decomposed into one-loop tensor integrals.The tensor integrals are then reduced to scalar one-loop PV-functions Fig. 3. Group One-loop Feynman diagram -poles contributing to the processes.All the fermions, W bosons, charged Higgs, Goldstone bosons, and ghost particles exchanging in the loop are considered.can be , and in this calculation.

LoopTools
LoopTools using the package [66].The one-loop form factors and are collected and presented as functions of the kinematic variables and the squared internal masses.The analytical expressions for the form factors are written in terms of scalar PVfunctions in the notations of [67].As a result, one-loop decay rates can be evaluated numerically by using .In the next paragraphs, we show the ana-lytical results for the form factors group by group of the Feynman diagrams.
V * 0 -pole contributions : We first consider all -pole diagrams as shown in Fig. 1 including the decay of in these diagrams.The form factors can be expressed in the form of for .From the fermions f-loop contributions, the form factor is given as follows: Here, denotes the couplings of with the fermionpair in the loop.We next regard the W-boson loop contributions.This is different from our previous work in [32], where we included new parameters to unify the one-loop form factors for both the photon and Z-poles in one analytic expression.In this work, to avoid definning new parameters, we present one-loop form factors for -pole and -or -pole separately.In detail, the form factors are shown for -pole as follows: Here, is noted for -pole.The one form factors for -pole and -pole take the form of Here, we have used .this formulas, shows the couplings of Z or with the W-pair, which are presented in Tables 4 and 9.
In addition, the one-loop form factor with the charged Higgs bosons in the loop reads Here, it is noted that stands for the couplings of with and .
We emphasize the following inportant points.First, for the case of the -pole being -and -poles, we reduce the PV functions in Eqs. ( 55)−( 58) to scalars oneloop functions, and then we reprodure the corresponding results in [32].Obviously, we have to replace all the couplings factored out in all the above formulas from this model to the HESM respectively.Second, the contributions from the -pole are new results from this work.Third, we derive alternative presentations for one-loop form factors for in comparison with our previous work in [31], where the results are shown in the unitary gauge.Last but not least, taking the limits of to on-shell γ or on-shell Z, we have one-loop form factors for the loop-induced and in the extension of the SM.
Non V 0 -polecontributions : We turn to one-loop diagrams without -pole contributions.The form factors can be separated into the form of We arrive at one-loop diagrams with contributions of two neutral gauge bosons, and .The form factors are also expressed in terms of PV-functions as follows: We note that, in this case, in the decay channels and in the decay channels .Finally, we consider one-loop diagrams with non-pole , where the charged Higgs bosons and right handed neutrino are exchanging in the loop.
It is important to note the following arguments.When we take only in the non--pole diagrams and replace the overall couplings respectively to the HESM, we then reprodure the corresponding results in [32].Furthermore, including all the contributions from and charged scalar Higgs in the non--pole diagrams, we derive alternative results from our previous reference [31].

UV IR UV IR IR UV
Having all the one-loop form factors, we can check the calculations by verifying the finiteness and finiteness of the form factors.We verify that all the form factors presented in the calculations are -finite and -finite.As there is no virtual photon exchanging in the loop, we do not have the -divergence in the processes.We have confirmed the finiteness for the form factors analytically in [30,32].
h → ℓ lγ confirming the correctness of the results, the differential decay rate for including one-loop contributions can be computed as follows After integrating over and , one then obtains the total decay rates.At the LHC, the signal strengths for can be derived as follows: Here, .In the scenario of a heavy neutral gauge boson and heavy neutrinos ( TeV), we then confirm that .Furthermore, we verify that Here, is defined as in Eq. ( 49).Therefore, the signal strengths can be evaluated as In the scope of this work, we are interested in this limit and take the signal strengths in Eq. ( 66) for the phenomenological analysis at the LHC.
e − e +

→ hγ B. Form factors for
We turn our attention to evaluate cross-sections for at future LCs.Applying the same method for , the one-loop amplitude for the production process can be written in terms of the following Lorentz structure : Here, the one-loop form factors are collected as coefficient of .By comparing with the one-loop form factors , we find that the form factors can be derived from by applying the following relations: The above relations can be explained by applying crosssymmetry for converting the decay process to the production scattering ; see [63] for more detail.The differential cross section for the production process, including initial beam polarizations, can be expressed in terms of one-loop form factors as follows: where and are the polarized degrees of the initial and beams, respectively.In the above expression, are given by Here, is the angle between the initial electron and external photon.
The signal strengths for the Higgs production associated with a photon for the models at future LCs can be estimated as follows: √ s Here, is center-of-mass energy.In the next section, we present the phenomenological results for the calculations.

M
In the phenomenological results, we use the following input parameters: GeV, GeV, GeV, GeV, GeV, and GeV.The lepton masses are selected as GeV, GeV, and GeV.For quark masses, one takes GeV, GeV, GeV, GeV, GeV, and GeV.We work in the so-called -scheme in which the Fermi constant is taken as GeV and the electroweak coupling can be calculated appropriately as follows: As we show later in this paper, the mass of the boson is considered as a parameter.The total decay width of can be calculated as in [68].As indicated in the previous sections, the models contain an additional neutral gauge boson associated with the coupling in the gauge sector.The limit on and reported by Tevatron [69] is Moreover, from the electroweak precision test (EWPT) [70,71], one has the further constraint on the parameter space as follows: In the LHC era, we can probe via the processes .Recently, ATLAS [72] and CMS [73] reported a lower bound of , GeV.In the scalar sector, models include the neutral Higgs H and mixing angle of the SM-like Higgs with a scalar singlet χ.According to the results from LHC run II data in [74][75][76][77], we have the limits for 150 GeV GeV.We also have two heavy neutrinos in the neutrino sector.Direct searches for heavy neutrinos can be performed at LCs via channels , where , etc.At LEP and LEP II [78,79], the limits for are 10 GeV GeV.At the LHC [80-85], heavy neutrinos mixing to light neutrinos can be probed via processes such as .The data present the limits on as 1 GeV GeV.

A. Numerical checks confirming gauge invariance
We first confirm the results of the decay rates of in vector in our previous work [31], in which the computations were performed in the unitary gauge.We use the same input parameters as in [31].The cross-check results are listed in Table 12.In the first column, we apply the cuts and , the same as in [31].As a result, the integration region is now and .The second column lists the results from this work in the HF gauge.The last column presents the results in [31].We find that the two results are good agreement.The checks confirm gauge invariance: the results remain unchanged in the different gauges.We then cross-check the cross sections for in this work with those of previous computations within the SM.The results for this test are presented in Table 13.In the first column, we select polarized degrees for initial beams of electron and positron.The second column lists the results from this work.The results in the last column are from Ref. [86].Both sets of results are in good agreement, up to the last digits.The effects of one-loop contributing to the differential decay rates of in the model were presented in [31].In this work, we focus on the signal strengths in both the vector and chiral models at the LHC.Moreover, we also examine the signal strengths for the Higgs production associated with a photon for the models at future LCs.
We arrive at the phenomenological results for the decay process at the LHC in the vector and chiral models.The signal strengths given in (66) are examined in the following subsections.
In the vector new physical parameters are included: mixing angle between two neutral Higgses ( ), coupling , a new neutral gauge boson , the kinematic mixing parameter κ, and the mixing angle ( ) of the two gauge bosons Z and .For the numerical results in this section, we only consider the non-mixing gauge bosons (non-mixing between two gauges) in which we apply the limits of .
In Fig. 7, the signal strengths are generated as functions of and .In this plot, we set .By considering experimental constraints such as TeV, the mass of the neutral gauge boson GeV.For this reason, the signal strengths are shown in the ranges of 490 GeV GeV.The mixing angle of two neutral Higgses is taken as .We find that the signal strengths are proportional to the mixing angle and depend slightly on .This shows that the contributions of the neutral gauge boson are rather small.The effects of the neutral gauge boson on the decay rates for the processes are difficult to probe at the LHC.
In this version of , together with the mixing angle ( ) of two gauge bosons of gauge groups, we include three more mixing angles of neutral Higgses as , mixing angle of charged Higgs with charged Goldstone boson , and mixing angle between neutral Goldstone bosons and CP-odd Higgs .The masses of the new particles in this version are .For the numerical results, considering the non-mixing gauge boson case (or ), we also take the simplest case as .The mixing angle between (treated as SM-like Higgs boson) and is .In this limit, the coupling of is then derived as follows: In this model, we have .For the phenomenological studies, we are interested in the the case (the SM limit) .
In Fig. 8, the signal strengths are presented as functions of and charged Higgs mass .The values of are generated in the ranges of and 600 GeV GeV.One selects , GeV, and masses of right handed neutrinos TeV.Furthermore, TeV and in combination with the values of GeV and , we then obtain and .Having the value of , and taking GeV, the value of μ can be then obtained appropriately (follows Eq. ( 37)).In principle, . As mentioned above, we are interested in the SM limit . For example, one takes for the following plot.In this plot, we find that the signal strengths are increased with a developing .At a fix value of , is also proportional to .The scatter plot shows that the contributions from charged Higgs to the processes are significant and the effects can be probed at the HL-LHC.
We are particularly interested in the contributions of the CP-odd Higgs to the signal strengths.In Fig. 9, the signal strengths are presented as functions of and CP-odd Higgs mass at the LHC.For the scatter plot, we take , GeV, and GeV.The values of are generated in the ranges of and 600 GeV GeV.The masses of the right handed neutrinos are selected as TeV and the mixing angle taken as for the following plot.We find that the signal strengths depend significantly on but change slightly with . This shows that the contribution from is rather small in comparison with those from chiral Higgs scalars.The indirect impacts of on the signal strengths can be explained as follows.By changing , the values of μ in Eq. ( 37) are varied appropriately.This leads to the change in the coupling in Eq. ( 76).This fact explains that the signal strengths change crucially with .

C. Process at future lepton colliders B − L
We turn to Higgs production associated with a photon at future LCs within vector and chiral versions.
As in previous cases, we also consider the non-mixing gauge bosons, , for the following numerical results.In Figs. 10, 11, the signal strengths as functions of and are presented at GeV and at GeV, respectively.In these plots, we take the same previous input parameters, that is, 0.07, 490 GeV GeV, and .
In ), whereas those in the polarization case ( ) are presented in the right panel.

At
GeV, we find that the signal strengths are proportional to the mixing angle and change slightly with . The plots indicate that the contributions from are rather small.As a result, the effects of to the decay processes are difficult to probe at future LCs.At GeV, around the -peak, 500 GeV, we also find the peak of the signal strengths.Beyond the -peak region, the same behavior of signal strengths as in a previous case observed: the signal strengths are proportional to the mixing angle and change slightly with .It is indicated that the contributions from are rather small beyond the -peak.Consequently, the impacts of on the decay rates for the processes are difficult to measure at future LCs.One may probe the contributions of to around the -peak at future LCs.
In Fig. 12 (and Fig. 13), the signal strengths are presented as functions of and charged Higgs mass at GeV ( GeV), respectively.In these plots, the values of are generated in the ranges of and 600 GeV GeV. Further, one selects , GeV, masses of right handed neutrinos GeV, and GeV.As mentioned above, we are also interested in the SM limit . For example, one takes for the following plots.
The left and right panels show the signal strengths for the and polarization cases, respectively.By implying the initial beam polarizations, there are no contributions of one-loop non -pole diagrams with exchanging W boson and charged Higgs in the loop for the case (see Figs. 5, 6).This explains the different behavior of in the LR and RL cases.This also explains why the values of in the case are bigger than those in the In Fig. 14 (and Fig. 15), the signal strengths are shown as functions of and right handed neutrino mass .In these plots, we present in the ranges of and 1 GeV GeV.We take 0.005, 500 GeV, and charged Higgs mass 1000 GeV.As in the previous case, having TeV and 800 GeV, the value of μ can be then derived appropriately.We are also very interested in the SM lim-

APPENDIX B: THE COUPLINGS
In this Appendix, we present all the related couplings to the processes under consideration in this paper.We begin with the vector model and show the couplings in the chiral model in the following subsections.
We first derive the coupling of Z to the fermion pair, .To illustrate this, we take with quark q in the first generation of matter contents.In detail, the couplings are derived as follows: We next calculate the couplings of as follows: The general couplings of and with arbitrary fermions f are presented in Table 5.

B − L
The components of Higgs kinetic for the vector model can be written as follows: We also have Here, we have already used the relation The Higgs potential is expanded and we collect the coupling of We have used the following relations: (1 + c 2θ ), (B12) As a result, we arrive at We are going to calculate all the couplings related to the processes in the chiral model.The fermion Lagrangian will be expanded as follows: The general formulas for the couplings of Z ( ) with the fermion-pair are given in Table 10.

B − L
The components of Higgs kinetic for the chiral model can be written as follows: The Yukawa Lagrangian is expanded as follows; In the limits of , the mixing angle between and is considered to be ; then, the coupling of takes the form of

Fig. 2 .
Fig. 2. GroupOne-loop Feynman diagram -poles contributing to the processes.All the fermions, W bosons, charged Higgs, Goldstone bosons, and ghost particles exchanging in the loop are considered.can be , and in this calculation.

)Fig. 7 .
Fig. 7. (color online) Signal strengths for the decay process in vector shown as functions of and at the LHC.

L
the left panel, we show the signal strengths in the polarization case (

Fig. 8 .Fig. 9 .
Fig. 8. (color online) Signal strengths for the decay process in chiral shown as functions of and charged Higgs mass at the LHC. t

Fig. 10 .Fig. 11 .
Fig. 10.(color online) Signal strengths for the process in the vector model at a center-of-mass energy GeV.The left panel shows the polarization case, and the right panel presents the polarization case.

Fig. 12 .
Fig. 12. (color online) Signal strengths for the process in the chiral model at GeV.The left panel presents the polarization case, and the right panel shows the polarization case. the

Fig. 13 .
Fig. 13.(color online) Signal strengths for the process in the chiral model at GeV.The left panel presents the polarization case, and the right panel shows the polarization case.

Fig. 14 .
Fig. 14. (color online) Signal strengths for the process in the chiral model at GeV.The left panel presents the polarization case, and the right panel shows the polarization case.The vertical grey lines appear in the Figure for distinguishing from the plotting boundary.

Fig. 15 .
Fig. 15.(color online) Signal strengths for the process in the chiral model at GeV.The left panel presents the polarization case, and the right panel shows the polarization case.The vertical and horizontal grey lines appear in the Figures for distinguishing from the plotting boundary.

)
Tran Dzung Tri, Thanh Huy Nguyen, Khiem Hong Phan Chin.Phys.C 48, 053101 (2024) 053101-22 Table of matter fields, scalar bosons with their charge quantum numbers of the vector model.We omit the index of generations of matter particles in this Table

Table 9 .
Couplings of three gauge bosons in the chiral model, which are related to the processes under consideration.The vector model can be derived by applying .

Table 10 .
Couplings of the Z and bosons to the fermion pair in the chiral model, which are related to the processes under consideration.The vector model can be derived by applying .Hypercharge takes the corresponding values for f presented in Table 1.Here, and .

Table 12 .
Decay rate (in KeV) for non-mixing in unitary and HF gauges.Here, , considering the case of , and GeV.The energy cut for the external photon is presented in GeV.

Table 13 .
Numerical results in femtobarn (fb) from crosschecking the cross section with various different beam polarizations at CMS energy GeV, compared with the numerical values in Table 1 in Ref. [86] within the framework of the Standard Model. +)