$Z'$ resonance and associated $Zh$ production at future Higgs boson factory: ILC and CLIC

We study the prospects of the B-L model with an additional $Z'$ boson to be a Higgs boson factory at high-energy and high-luminosity linear electron positron colliders, such as the ILC and CLIC, through the Higgs-strahlung process $e^{+}e^{-}\rightarrow (Z, Z') \to Zh$, including both the resonant and non-resonant effects. We evaluate the total cross section of $Zh$ and we calculate the total number of events for integrated luminosities of 500-2000\hspace{0.8mm}$fb^{-1}$ and center of mass energies between 500 and 3000\hspace{0.8mm}$GeV$. We find that the total number of expected $Zh$ events can reach $10^6$, which is a very optimistic scenario and it would be possible to perform precision measurements for both the $Z'$ and Higgs boson in future high-energy $e^+e^-$ colliders experiments.


I. INTRODUCTION
The discovery of a light scalar boson H of the ATLAS [1] and CMS [2] Collaborations at the Large Hadron Collider (LHC) compatible with a SM Higgs boson [3][4][5][6][7] and with mass around M h = 125 ± 0.4(stat.) ± 0.5(syst.) GeV has opened a window to new sectors in the search for physics beyond the Standard Model (SM). The Higgs boson might be a portal leading to more profound physics models and even physics principles. Therefore, another Higgs factory besides the LHC, such as the International Linear Collider (ILC) [8][9][10][11][12][13] and the Compact Linear Collider (CLIC) [14][15][16] that can study in detail and can precisely determine the properties of the Higgs boson is another important future step in high-energy and high-luminosity physics exploration.
The existence of a heavy neutral (Z ′ ) vector boson is a feature of many extensions of the Standard Model. In particular, one (or more) additional U(1) ′ gauge group provides one of the simplest extensions of the SM. Additional Z ′ gauge bosons appear in Grand Unified Theories (GUTs) [17], Superstring Theories [18], Left-Right Symmetric Models (LRSM) [19][20][21] and in other models such as models of composite gauge bosons [22]. In particular, it is possible to study some phenomenological features associated with this extra neutral gauge boson by considering a B − L (baryon number minus lepton number) model.
The B −L symmetry plays an important role in various physics scenarios beyond the SM: a) the gauge U(1) B−L symmetry group is contained in a GUT described by a SO(10) group [23]. b) the scale of the B − L symmetry breaking is related to the mass scale of the heavy right-handed Majorana neutrinos mass terms providing the well-known see-saw mechanism [24] to explain light left-handed neutrino mass. c) the B − L symmetry and the scale of its breaking are tightly connected to the baryogenesis mechanism through leptogenesis [25].
The B − L model [26,27] is attractive due to its relatively simple theoretical structure, and the crucial test of the model is the detection of the new heavy neutral (Z ′ ) gauge boson.
The analysis of precision electroweak measurements indicates that the new Z ′ gauge boson should be heavier than about 1.2 T eV [28]. On the other hand, recent bounds from the LHC indicate that the Z ′ gauge boson should be heavier than about 2 T eV [29, 30], while future LHC runs at 13-14 T eV could increase the Z ′ mass bounds to higher values, or may be lucky and find evidence for its presence. Further studies of the Z ′ properties will require a new linear collider [31], which will also allow us to perform precision studies of the Higgs sector.
The Higgs-stralung [40][41][42][43][44] process e + e − → Zh is one of the main production mechanisms of the Higgs boson in the future linear e + e − colliders experiments, such as the ILC and CLIC. Therefore, after the discovery of the Higgs boson, detailed experimental and theoretical studies are necessary for checking its properties and dynamics [45][46][47][48]. It is possible to search for the Higgs boson in the framework of the B-L model, however the existence of a new gauge boson could also provide new Higgs particle production mechanisms, which could prove its non-standard origin. In this work, we analyze how the Z ′ gauge boson of the U(1) B−L model could be used as a factory of Higgs bosons.
Our aim in the present paper is to study the sensitivity of the Z ′ boson of the B-L model as a Higgs boson factory through the Higgs-strahlung process e + e − → (Z, Z ′ ) → Zh, including both the resonant and non-resonant effects at future high-energy and high-luminosity linear e + e − colliders, such as the International Linear Collider (ILC) [8] and the Compact Linear Collider (CLIC) [14]. We evaluate the total cross section of Zh and we calculate the total number of events for integrated luminosities of 500-2000 f b −1 and center-of-mass energies between 500 and 3000 GeV . We find that the total number of expected Zh events for the e + e − colliders is very promising and that it would be possible to perform precision measurements for both the Z ′ and Higgs boson in the future high-energy e + e − colliders experiments. In addition, we also studied the dependence of the Higgs signal strengths (µ) on the parameters g ′ 1 and θ B−L of the U(1) B−L model for the Higgs-stralung process e + e − → Zh. This paper is organized as follows. In Section II, we present the theoretical framework.
In Section III, we present the decay widths of the Z ′ boson in the context of the B-L model.
In Section IV, we present the calculation of the process e + e − → (Z, Z ′ ) → Zh, and finally, we present our results and conclusions in Section V.

II. THEORETICAL FRAMEWORK
We consider an SU(2) L × U(1) Y × U(1) B−L model consisting of one doublet Φ and one singlet χ and briefly describe the lagrangian including the scalar, fermion and gauge sector.
The Lagrangian for the gauge sector is given by [36,37,49,50] where W a µν , B µν and Z ′ µν are the field strength tensors for SU(2) L , U(1) Y and U(1) B−L , respectively.
The Lagrangian for the scalar sector of the SU(2) where the potential term is [34], with Φ and χ as the complex scalar Higgs doublet and singlet fields, respectively. The covariant derivatives for the doublet and singlet are given by [32][33][34] where the doublet and singlet scalars are with G ± , G Z and z ′ the Goldstone bosons of W ± , Z and Z ′ , respectively.
After spontaneous symmetry breaking the two scalar fields can be written as, with v and v ′ real and positive. Minimization of Eq. (3) gives To compute the scalar masses, we must expand the potential in Eq. (3) around the minima in Eq. (6). Using the minimization conditions, we have the following scalar mass matrix: The expressions for the scalar mass eigenvalues (m H ′ > m h ) are and the mass eigenstates are linear combinations of φ 0 and φ ′ 0 , and written as, where h is the SM-like Higgs boson. The scalar mixing angle, α can be expressed as In the Lagrangian of the SU(2) L ×U(1) Y ×U(1) B−L model, the terms for the interactions between neutral gauge bosons Z, Z ′ and a pair of fermions of the SM can be written in the form [37,38] From this Lagrangian we determine the expressions for the new couplings of the Z, Z ′ bosons with the SM fermions, which are given by where g = e/ sin θ W and θ B−L is the Z − Z ′ mixing angle. The current bound on this parameter is |θ B−L | ≤ 10 −3 [51]. In the decoupling limit, that is to say, when g ′ 1 = 0 and θ B−L = 0 the couplings of the SM are recovered.

III. THE DECAY WIDTHS OF Z ′ IN THE B-L MODEL
In this section we present the new decay widths of the Z ′ boson [28,[52][53][54] in the context of the B-L model which we need in the calculation of the cross section for the process e + e − → Zh. The Z ′ partial decay widths involving vector bosons and the scalar boson are where The vacuum expectation value v ′ is taken as v ′ = 2 T eV , while α = π 9 for the Higgs mixing parameter in correspondence with Refs. [1,2,36,55]. In our analysis we take v = 246 GeV and constrain the other scale, v ′ , by the lower bounds imposed on the mass of the extra neutral gauge boson Z ′ . The mass of the Z ′ and of the heavy neutrinos depend on v ′ , and should be related to it, while the Higgs masses depend on the angle α, the value of which is completely arbitrary.
Finally, the decay width of the Z ′ boson to fermions is given by where N f is the color factor (N f = 1 for leptons, N f = 3 for quarks) and the couplings g ′f V and g ′f A of the Z ′ boson with the SM fermions are given in Eq. (14).
is the usual two-particle phase space function, while g e V , g e A , g ′e V , g ′e V , f (θ B−L ) and g(θ B−L ) are given in Eqs. (13), (14) and (17), respectively.
The expression given in the first term of Eq. (19) corresponds to the cross section with the exchange of the Z boson, while the second and third term come from the contributions of the B-L model and of the interference, respectively. The SM expression for the cross section of the reaction e + e − → Zh can be obtained in the decoupling limit, that is to say, when θ B−L = 0 and g ′ 1 = 0, in this case the terms that depend on θ B−L and g ′ 1 in (19) are zero and (19) is reduced to the expression given in Refs. [40,44] for the standard model.
in our numerical analysis, we obtain the total cross section σ tot = σ tot ( √ s, M Z ′ , g ′ 1 ). Thus, in our numerical computation, we will assume √ s, M Z ′ and g ′ 1 as free parameters. We do not consider the process e + e − → (Z, Z ′ ) → ZH ′ [35] in our study since in major parts of the U(1) B−L model parameter space the higgs boson H ′ is quite heavy, and it is difficult to detected the process e + e − → ZH ′ when the relevant mechanism is e + e − → Zh.
In Figs. 2-3 we present the total decay width of the Z ′ boson as a function of M Z ′ and the new U(1) B−L gauge coupling g ′ 1 , respectively, with the other parameters held fixed to three different values. From Fig. 2, we see that the total width of the Z ′ new gauge boson varies from a few to hundreds of GeV over a mass range of 500 GeV ≤ M Z ′ ≤ 3000 GeV , depending on the value of g ′ 1 . In the case of Fig. 3, a similar behavior is obtained in the range 0 ≤ g ′ 1 ≤ 1 and depends on the value of M Z ′ . The branching ratios versus Z ′ mass are given in Fig. 4 for different channels, that is to say, BR(Z ′ → ff ), BR(Z ′ → Zh) and BR(Z ′ → W + W − ),  Fig. 6. We can see that the relative correction reaches its maximum value between 0.1 ≤ g ′ 1 ≤ 2.5 and remains almost constant as g ′ 1 increases. The deviation of the cross section in our model from the SM one, δσ/σ SM is depicted in Fig. 7 as a function of M Z ′ for √ s = 1500 GeV and three values of the g ′ 1 , new gauge coupling. Figure 7 shows that the relative correction is very sensitive to the gauge boson mass M Z ′ and for the gauge parameter g ′ 1 = 0.2, 0.5, 0.8, the peak of the total cross section emerges when the heavy gauge boson mass approximately equals M Z ′ = 1500, 1450, 1300 GeV , respectively. Thus, in a sizeable parameter region of the B-L model, the new heavy gauge boson Z ′ can produce a significant signal, which can be detected in future ILC and CLIC experiments.
We plot the total cross section of the reaction e + e − → Zh in Figure 8  √ s = M Z ′ , the resonant effect dominates the Higgs particle production. A similar analysis was performed in Fig. 9, but in this case θ B−L = 10 −3 and g ′ 1 = 0.5. In both figures we show that the cross section is sensitive to the free parameters. Comparing Figs. 8 and 9, we observe that the height of the resonances is the same in both Figures, but the resonances are broader for larger g ′ 1 values, as the total width of the Z ′ boson increases with g ′ 1 , as it is shown in Fig.2.
Finally, in Fig. 10     From Figs. 5-10, it is clear that the total cross section is sensitive to the value of the gauge boson mass M Z ′ , center-of-mass energy √ s and g ′ 1 , the new U(1) B−L gauge coupling, increases with the collider energy and reaching a maximum at the resonance of the Z ′ gauge boson. As an indicator of the order of magnitude, we present the Zh number of events in Table I, for several gauge boson masses, center-of-mass energies and g ′ 1 values and for a luminosity of L = 500, 1000, 2000 f b −1 . We find that the possibility of observing the process e + e − → (Z, Z ′ ) → Zh is very promising as shown in Table I, and it would be possible to perform precision measurements for both the Z ′ and Higgs boson in the future high-energy linear e + e − colliders experiments. We observed in Table I that the cross section rises once the threshold for Zh production is reached, with the energy, until the Z ′ is produced resonantly at √ s = 1500, 2000 and 2500 GeV , respectively, for the three cases. Afterwards it decreases with rising energy due to the Z and Z ′ propagators. Another promising production mode for studying the Z ′ boson and Higgs boson properties is e + e − → (γ, Z, Z ′ ) → tth [57].

B. The Higgs signal strengths in the B-L model
Considering the Higgs boson decay channels, the Higgs signal strengths can be defined as The corresponding CMS collaboration result [59] is µ = 1.00 ± 0.13.
Good consistency is found, for both experiments, across different decay modes and analyses categories related to different production modes.
In the B-L model, the modifications of the hff (the SM fermions pair) and hV V (V = W, Z) couplings can give the extra contributions to the Higgs boson production processes.
On the other hand, the loop-induced couplings, such as hγγ and hgg, could also be affected.
Finally, beside the effects already seen in the Higgs-strahlung channel due to the couplings Eqs. (13), (14) and the functions given by Eq. (17), the exchange of s-channel heavy neutral gauge boson Z ′ also affected the production cross section. All effects can modify the signal strengths in a way that may be detectable at the future ILC/CLIC experiments.
In Fig. 11, we show the dependence of the Higgs signal strengths µ i (i = bb, γγ) on the parameter g ′ 1 and θ B−L for the Higgs-strahlung process e + e − → (Z, Z ′ ) → Zh, where (a) and (b) denote the Higgs signal strengths µ bb and µ γγ , respectively.
Using θ B−L = 10 3 for the mixing angle and M h = 125 GeV for the Higgs boson mass, the following bound on the signal strength is obtained: which is consistent with that obtained for the ATLAS [58] and CMS [59] collaborations, Eqs. (23) and (24), respectively.
In conclusion, we consider the Z  (19) is reduced to the expression given in Refs. [40,44] for the standard model. We also studied the dependence of the Higgs signal strengths (µ) on the parameters g ′ 1 and θ B−L of the U(1) B−L model for the Higgs-stralung process e + e − → Zh. We obtain a bound on (µ), which is consistent with that obtained for the ATLAS [58] and CMS [59] collaborations. In addition, the analytical and numerical results for the total cross section have never been reported in the literature before and could be of relevance for the scientific community.