𝑍 󸀠 and Higgs Boson Production Associated with a Top Quark Pair as a Probe of the 𝑈(1) 𝐵−𝐿 Model at 𝑒 + 𝑒 − Colliders

,


Introduction
The Compact Linear Collider (CLIC) [1][2][3] is  +  − machine that operates at a center-of-mass energy of 0.5 − 3  and luminosities within L = 500 − 2000  −1 .This makes it a powerful tool to perform high precision studies of the properties of the Higgs boson in the Standard Model (SM) [4][5][6][7][8] and of new particles predicted by other extended models, such as the case of the (3) 퐶 × (2) 퐿 × (1) 푌 × (1) 퐵−퐿 (B-L) theory given in [9][10][11][12][13][14], which is the benchmark model and the starting point for our research.In the extended gauge sector, the B-L model contains an extra gauge boson  耠 and an additional heavy neutral Higgs boson , and the theory predicts the existence of heavy neutrinos ] 푅 .These three elements altogether make the model phenomenologically interesting.With its great capability to cover ample energy scales, one primary goal of the CLIC is to directly search for new particles, especially those coupled to SM particles.In addition, the CLIC can potentially produce directly new particles that could modify the Higgs properties.The CLIC can significantly improve the Large Hadron Collider (LHC) measurements due to its clean environment.
High-energy colliders benefit the high rate production of top-Higgs via the processes  +  − → ℎ and  +  − → .This is crucial to directly constrain the top Yukawa coupling, and it helps us to decipher the Higgs boson properties and to give light to new physics beyond the SM.The production of a Higgs boson in association with top quarks requires a large center-of-mass energy which is achieved at a linear collider as is the case of the CLIC.The Higgs boson in association with a top quark pair can be produced at other colliders, such as the LHC and the International Linear Collider (ILC) at √ = 500 , albeit in the former the main process occurs through gluon splitting.
In this paper, we study the simple production of Higgs bosons ℎ,  associated with the underlying  耠 resonance, and a top quark pair via the processes  +  − → (, ,  耠 ) → ℎ and  +  − → (, ,  耠 ) →  in the context of the B-L model [9][10][11][12][13][14][15][16].Hereafter, we denote the Higgs bosons ℎ and  in the SM and in the B-L model, respectively.We derive  complete analytical expressions for (i) the decays of the new heavy gauge boson  耠 to three bodies Γ( 耠 → ℎ, ) and (ii) analytical formulas for the total cross section of the processes  +  − → (, ,  耠 ) → ℎ, .Thus, our analytical results of the Higgs bosons production and decay can easily be implemented in searching for signatures of new physics and could be of scientific significance.The Feynman diagrams contributing to the processes  +  − → (, ,  耠 ) → ℎ,  are shown in Figure 1.
The structure of our paper is as follows.In Section 2, we give a brief review of the B-L model.In Section 3, we calculate the decay widths of the heavy gauge boson  耠 for the three body processes.Then, we present the calculation of the cross section for the processes  +  − → (, ,  耠 ) → ℎ,  in Section 4 and our results and conclusions are presented in Section 5.
After spontaneous symmetry breaking, the two scalar fields can be written as with V ≈ 246  and V 耠 being the electroweak and B-L symmetry breaking scales, respectively.These are real and positive valuated scales.Getting the minima of the Higgs potential, (4) gives the scalar mass eigenvalues and the mass eigenstates are linear combinations of  0 and  耠0 and written as where the scalar mixing angle  (−/2 ≤  ≤ /2) can be expressed as while coupling constants  1 ,  2 , and  3 are determined using ( 7)- (9).Interpreting the LHC data [26,27] by identifying ℎ with the recently observed Higgs boson, the scalar field mixing angle satisfies the constraint sin 2  ≲ 0.33(0.36)for  퐻 = 200(300)  as discussed in [28,29].

Higgs Boson Production Associated with a Top Quark Pair
We now proceed to calculate the total cross section of the processes  +  − → ℎ, .The Feynman diagrams contributing to the processes  +  − → (, ,  耠 ) → ℎ,  are shown in Figure 1.
We express the total cross section of the processes mentioned before in compact form as a sum of the different contributions; that is, where The explicit form of the functions  푖 ( 1 ,  2 ),  = 1, 2, . . ., 10, and of their corresponding limits of integration  ± 1 ( ± 2 ) are given in Appendices A and B.
Eqs. ( 16) and (17) determine the cross section with the exchange of the  photon and the  boson, while the expressions in ( 18)-( 21) quantify the cross section contributions of the B-L model and of the interference, respectively.The expression for the cross section of the reaction  +  − → ℎ in the SM can easily be recovered in the decoupling limits  耠 = 0,  耠 1 = 0 and  = 0. Thus (15) reduces to the SM [41][42][43][44][45].

Results and Conclusions
We have addressed a study of the total cross section  푇표푡 =  푇표푡 (√,  푍  ,  耠 1 ,  耠 , ) for the processes of simple Higgs bosons production  +  − → (, ,  耠 ) → ℎ and  +  − → (, ,  耠 ) →  in association with the underlying  耠 boson and a top quark pair within the context of the (3) 퐶 × (2) 퐿 × (1) 푌 × (1) 퐵−퐿 model [9][10][11][12][13][14].This has been done as a function of the parameters of the model, which will be used by the CLIC; that is, √ = 0.5 − 3  and  = 500 − 2000  −1 .In addition, in our study we adopt the data reported in Tables 1 and 2, and the relationship between  푍  and  耠 1 is given by [33,46,47] The constraint given by ( 22) means that  푍  and  耠 1 can no longer be considered as independent parameters of the B-L model.Therefore, for our analysis we can only fix one of them.Such a relationship may restrict the search range of the mass of the new boson  耠 in the colliders.
Figures 3-5 illustrate our results regarding the sensitivity of the  耠 heavy gauge boson of the B-L model as a Higgs boson source through the process  +  − → (, ,  耠 ) → ℎ, including both the resonant and nonresonant effects.In Figure 3, we show the different contributions to the total cross section ( +  − → ℎ) as a function of the center-of-mass energy √, considering  耠 = 10 −3 and  耠 1 = 0.290.We see that the cross section corresponding to  푍 ( +  − → ℎ)  decreases for large √.In the case of the cross section of the B-L model ( 18) and the total cross section (15), respectively, there is an increment for large values of the center-of-mass energy, reaching its maximum value at the resonance  耠 heavy gauge boson, which is √ = 2000 .
For the reaction  +  − → ℎ, we plot in Figure 4 its associated cross section as a function of the center-ofmass energy √, for heavy gauge boson masses of  푍  = 2000, 3000, 4000  and  耠 1 = 0.290, 0.435, 0.580, respectively.The choice of the values for  푍  and  耠 1 is done by keeping the relationship between  푍  and  耠 1 of ( 22). Figure 4 shows that the cross section is sensitive to not only the free parameters, but also the height of the resonance peaks for the boson  耠 , corresponding to the value of √ =  푍  .The resonances are broader for larger values of  耠 1 , since the total width of the  耠 boson increases with  耠 1 , as shown in Fig. 3 of [13].
Figure 5 describes the correlation between the heavy gauge boson mass  푍  and the  耠 1 coupling of the (1) 퐵−퐿 model for the cross section of  푇표푡 = 7, 8, 9, 10  (top panel) with √ = 2000 ,  푇표푡 = 7, 8, 9, 10  (central panel) with √ = 3000 , and  푇표푡 = 7, 8, 9, 10  (bottom panel) with √ = 4000 .From there, we see that there is a strong correlation between the gauge boson mass  푍  and the new gauge coupling  耠 1 .The sensitivity of the total cross section is evident with respect to the value of the gauge boson mass  푍  , centerof-mass energy √, and  耠 1 , which is the new (1) 퐵−퐿 gauge coupling as shown in Figures 3-5.The total cross section increases with the collider energy, reaching a maximum at the resonance of the  耠 gauge boson.In Table 3 we present the ℎ number of expected events for the center-of-mass energies of √ = 1000, 2000, 3000 , integrated luminosities L = 500, 1000, 1500, 2000  −1 , and heavy gauge boson masses  푍  = 1000, 2000, 3000  with  耠 1 = 0.145, 0.290, 0.435, respectively.The possibility of observing the process  +  − → (, ,  耠 ) → ℎ is very promising as shown in Table 3, and it would be possible to perform precision measurements for both the  耠 and Higgs boson in the future high-energy and high-luminosity linear  +  − colliders experiments.Table 3 also indicates to us that the cross section rises with the energy once the threshold for ℎ production is reached, until  耠 is produced resonantly at √ = 1000, 2000 and 3000 , respectively.Afterwards it decreases with rising energy due f(  , g  1 )   to the  and  耠 propagators.Another promising production mode for studying the  耠 boson and Higgs boson properties of the B-L model is  +  − → (, ,  耠 ) → , which is studied next.We analyze the impact of the parameters of the B-L model on the process  +  − → (, ,  耠 ) → .In Figure 6 we present the  푍푍  퐻 ( 耠 ,  耠 1 ) (in  unit) coupling and ( 耠 ,  耠 1 ), ( 耠 ,  耠 1 ) functions as a function of , with  耠 = 10 −3 , 10 −4 .From this figure it is clear that ( 耠 ,  耠 1 ) and ( 耠 ,  耠 1 ) are almost  independent, while  푍푍  퐻 ( 耠 ,  耠 1 ) is dependent on the scalar mixing angle .
The total cross section for the process under study is presented in Figure 7, as a function of the collision energy for  ℎ = 125 ,  퐻 = 800 ,  ]  = 500 ,  푍  = 2000 , and  耠 1 = 0.290 .Each curve in figure holds for the different contributions, and they are plotted as a function of the center-of-mass energy √.We can see that the cross section corresponding to  푍 ( +  − → ) decreases for large √.We note again that, for the cross section of the B-L model Eq. ( 18) and the total cross section Eq. ( 15), there is an increment for large values of the center-of-mass energy, reaching its maximum value at the resonance  耠 heavy gauge boson, which is quite near √ = 2000 .
We now explore the effects of  耠 1 ,  푍  over the total cross section of the process  +  − →  as a function of the center-of-mass energy √.We explore the cases  푍  = 1000  with  耠 1 = 0.145,  푍  = 2000  with  耠 1 = 0.290, and  푍  = 3000  with  耠 1 = 0.435.Our results are plotted in Figure 8.For √ =  푍  the resonant effect dominates and the cross section is sensitive to the free parameters.
In Figure 9, we show the correlation between the boson mass  푍  and the  耠 1 coupling for the cross section values of  푇표푡 = 1, 2, 3, 4  (top panel),  푇표푡 = 1, 2, 3, 4  (central panel), and  푇표푝 = 1, 2, 3, 4  (bottom panel).The plots expose a strong correlation between  푍  and  耠 1 .Altogether, Figures 6-9 clearly show how sensitive the total cross section is, to the value of the boson mass  푍  , to center-of-mass energy √ and  耠 1 , and how it increments with the collider energy reaching a maximum at the resonance of the  耠 gauge boson.The  number of events which are expected to be observed are shown in Table 4; these were obtained for center-of-mass energies of √ = 1000, 2000, 3000 , integrated luminosity L = 500, 1000, 1500, 2000  −1 , and heavy gauge boson masses of  푍  = 1000, 2000, 3000  with  耠 1 = 0.145, 0.290, 0.435, respectively.These numbers encourage the possibility of observing the process  +  − → (, ,  耠 ) → .We observe from Table 4 that the cross section grows with the energy once the threshold for  production is reached, until the  耠 is produced resonantly at √ = 1000, 2000 and 3000 .Afterwards it decreases with rising energy due to the  and  耠 propagators.
The number of events is given by  푆푀 = L 푖푛푡 ×  푆푀 , where L 푖푛푡 is the integrated CLIC luminosity.In Figures 10 and 11, we plot the  2 distribution as a function of  푍  ( 耠 1 ,  耠 , ).We plot the curves for each case, for which we have divided the  푍  ( 耠 1 ,  耠 , ) interval into several bins.
The most important systematic errors are due to the modelling of the signal and the background.For our analysis we choose a systematic error of  푠푦푠 = 10% [1,3,55,56], which is a reasonably moderate value.This value was chosen considering that the ℎ cross section can be measured with an accuracy of 12% in the semileptonic channel and 11% in the hadronic channel.The combined precision of the two channels is 8% [1,3,55,56].Furthermore, the machinerelated uncertainties, such as the knowledge of the centerof-mass energy of the collider and the luminosity, are also relevant for this study.We can assumed that the CLIC will be built in the coming years and the systematic uncertainties will be lower when considering the development of future detector technology.
It must be noted that the sensitivity for the parameters of the B-L model,  푍  ,  耠 1 ,  耠 , and , is good as is shown in Figures 10 and 11.However, it is worth mentioning that it is necessary to carry out a complete and detailed study on the sensitivity of the aforementioned parameters; for this, kinematic cuts must be applied on the particles of the final state to reduce the background and to optimize the signal sensitivity.
Future electron-positron linear colliders operating as Higgs factories, and having the advantages of a clean collider environment and large statistics, could greatly enhance the sensitivity to study in detail the production and decay processes of various particles.In addition, the sensitivity increases by increasing the center-of-mass energy and the integrated luminosity.Furthermore, if we consider the cleanest modes and new models beyond the SM, large cross sections and little background from the SM, combined with the high luminosity of the colliders, result in large data samples allowing precise measurements with high sensitivity.
The presented results definitely are inside the scope of detection in future experiment with improved sensitivity of the next generation of linear colliders.The processes  +  − → ℎ,  will be an important tool for the precision measurements of the top Yukawa coupling and to study some, or all, the implications of the CP violation Higgs-top coupling [57].As an application of this process, another point to study is the importance of the Higgs-top coupling in the hierarchy problem [58] and a deeper understanding of the vacuum stability of the SM [59,60].In this regard, we study the CP violation Higgs-top coupling in the context of the B-L model at future  +  − colliders energies [61].
Once we have extensively studied the production of light and heavy Higgs boson in the context of extension (1) 퐵−퐿 of the SM, with an additional  耠 boson, we can conclude that our results seem to be achievable with the capability of future linear colliders (ILC and CLIC) with center-of-mass energies of √ = 500 − 3000  and integrated luminosities of L = 500 − 2000  −1 .Our study covered the processes  +  − → (, ,  耠 ) → ℎ and  +  − → (, ,  耠 ) → , including resonant and nonresonant effects.We find that the total number of expected ℎ and  events can reach 69,067 and 52,578, respectively.Under this optimistic scenario it would be possible to perform precision measurements for both Higgs bosons ℎ and , for the  耠 heavy gauge boson, and for the full parameters of the models  耠 ,  耠 1 , and .The SM expression for the cross section of the reaction  +  − → ℎ can be recovered in the decoupling limit, when  耠 = 0,  耠 1 = 0, and  = 0.In this case, the terms depending on  耠 ,  耠 1 , and  in ( 15) are zero and therefore (15) reduces to the SM case [41][42][43][44][45].Our study complements other studies on the (1) 퐵−퐿 model and on the single Higgs bosons production processes  +  − → ℎ and  +  − → , which seems to be suitable for experiments at hadron and  +  − colliders.

Figure 1 :
Figure 1: Feynman diagrams for the Higgs bosons production in association with  pair  +  − → ℎ and  +  − →  in the B-L model.

Table 1 :
Current experimental data.