$HZ$ associated production with decay in the Alternative Left-Right Model at CEPC and future linear colliders

In this study, Higgs and Z boson associated production with subsequent decay is attempted in the framework of alternative left-right model, which is motivated by superstring-inspired $E_6$ model at CEPC and future linear colliders. We systematically analyze each decay channel of Higgs with theoretical constraints and latest experimental methods. Due to the mixing of scalars in the Higgs sector, charged Higgs bosons can play an essential role in the phenomenological analysis of this process. Even though the predictions of this model for the signal strengths of this process are close to the standard model expectations, it can be distinct under high luminosity.


Introduction
With the discovery of Higgs boson by ATLAS [1] and CMS [2] at Large Hadron Collider(LHC) in 2012, the standard model(SM) has achieved great accomplishment. Higgs production with subsequent decay plays an essential role not only in the precision test of the Higgs property but provides a window to new physics beyond the SM(BSM). The study of Higgs and Z boson associated production and decay at Higgs factory such as Circular Electron Positron Collider(CEPC) and future linear colliders is significantly important in measuring gauge and Yukawa interactions, so more and more theorists and experimenters are motivated to investigate this process in new physics sceneries [3][4][5][6]. CEPC was proposed by Chinese scientists at about 240 GeV center-of-mass energy mainly for Higgs studies with two detectors situated in a very long tunnel more than twice the size of the LHC at CERN. A future linear collider, such as International Linear Collider(ILC) or Compact Linear Collider(CLIC), at center-of-mass energy √ s = 500 GeV or even higher to TeV energy scale, will allow the Higgs sector to be probed with high precision significantly beyond that at High-Luminosity LHC [7][8][9]. CEPC and future linear colliders are e + e − colliders, and will be very important facilities for precision Higgs physics research, which may be an order of magnitude more precise than that achievable at LHC. Such measurements may be necessary to reveal BSM effects in Higgs sector. Moreover, e + e − colliders provide the opportunity to measure Higgs couplings, rather than ratios with a cleaner background. In addition, an e + e − collider operating at 1TeV or above, for example CLIC or an upgraded ILC, will have the sensitivity to top quark Yukawa coupling and Higgs selfcoupling parameters, thus will provide a direct probe of Higgs potential.
As for the discovery of neutrino masses and neutrino oscillations, it confirmed that SM is still incomplete. To provide a proper explanation for the measured neutrino masses, theorists have made many attempts to expand SM, such as supersymmetry, extra dimensions, Two Higgs Doublet Model(2HDM), Left-Right Model (LRM) and so on. The Alternative Left-Right Model(ALRM) [10][11][12][13], motivated by the superstringinspired E 6 model, is one type of left-right model [14][15][16][17]. ALRM is based on can break at TeV scale, so that allowing several interesting signatures at Large Hadron Collider(LHC). In ALRM all non-SM particles can couple with SM fermions and Higgs which will lead to low energy consequences. Due to the rich Higgs sector, there are four neutral CP-even and two CP-odd Higgs bosons, in addition to two charged Higgs bosons, which come from one bi-doublet and two left-handed and right-handed doublets and most of these Higgs bosons can be light, of order electroweak scale. As for the couplings of the SM-like Higgs with the fermions and gauge bosons, there are small changes compared to the corresponding ones in SM. In the literature, many studies have been done which primarily focus on dark matter or hadron production of ALRM [13,[18][19][20].
In this paper we will mainly pay attention to the weak production of Z boson and Higgs. Firstly, each channel of Higgs decay modes has been analyzed in ALRM and compare them with the recent results reported by ATLAS and CMS experiments [21][22][23][24][25]. There are possible discrepancies between their results of signal decay strengths in each channel. We analyze through five main Higgs decay channels: H → bb, cc, ττ , ZZ and W W in both ALRM and SM correspondingly. We find that the signal strength of Higgs decay channel H → ZZ is consistent with SM expectation. However the decay channel H → bb is more sensitive to the mass of charged Higgs, where may exists discrepancy with SM. Secondly, HZ associated production has been systematically explored in this model at CEPC and future e + e − colliders. The couplings of new heavy bosons to the known fundamental particles will be a crucial test of SM and may support a opportunity to establish physics BSM. We find that the discrepancies of cross sections between ALRM and SM are of a few percent at √ s = 240GeV , 500GeV and 1T eV . Finally we study the subsequent decay of the final state Higgs into a pair of bottom quarks and Z boson into a leptonic pair, where l = e, µ and τ , associated with ALRM and the required integrated luminosity when the discovery significance is 5σ.
The organization of this paper is as follows: In section 2 we briefly describe the related theory of ALRM. In section 3 we perform the numerical analysis for Higgs and Z boson associated production with decay in ALRM. Finally a short summary is given in section 4.

Alternative Left-Right Symmetric Model
ALRM is an standard model extension based on SU (3) C ×SU (2) L ×SU (2) R ×U (1) (B−L)/2 ×S gauge symmetry, the discrete symmetry S is to distinguish the scalar bi-doublet from its dual scalars. The details of ALRM are presented in the literature [19]. Following, we only introduce the formulas used in our calculations.
Let's start from the most general left-right symmetric Yukawa Lagrangian: In Eq.(1), Φ and χ L,R are the duals of the bi-doublet Φ and the doublets χ L,R , which are defined as Φ = τ 2 Φ * τ 2 and χ L,R = iτ 2 χ * L,R . From this Lagrangian, We can get the masses of the fermions, which are quarks u, d, d ′ , the charged leptons ℓ, and the addition singlet fermion n called scotino: The mixing angle and the vacuum expectation are set as tan β = k/v L and v 2 L + k 2 = v ≡ 246 GeV. From Eq.(1), we also get the Yukawa couplings of the SM-like Higgs where the T Φ , T L and T R are the mixing parameters of the SM-like Higgs with the gauge eigenstates φ 0R 2 , χ 0R L and χ 0R R , respectively. Similarly to the Yukawa coupling, the specific couplings of the SM-like Higgs with the massive EW gauge bosons can also be derived from the Lagrangian of the scale sector [19].
In the gauge sector, W ± L and W ∓ R can not be mixing with each other due to < φ 0 1 >= 0. We get the mass eigenstates are W ± = W ± L which are the SM gauge bosons and the heavy charged bosons W ′± = W ± R . The masses of these bosons are given by: At present, a lot of measurements focused on the heavy bosons have been done. Up to date a search for highmass resonances using an integrated luminosity of 36.1 f b −1 by the ATLAS collaboration offer a lower mass limit of W ′ as M W ′ > ∼ 3.7 TeV [26][27][28]. This lower bounded on M W ′ is applicable in our following calculation in this paper.
For the neutral gauge bosons, the masses of two massive bosons can be calculated by where mixing angle ϑ is defined as xxxx-2 and From Eqs. (6-7), we know that mixing angle ϑ are strongly influenced masses of Z and Z ′ , and when ϑ → 0, Z ≃ Z L and Z ′ ≃ Z R . The latest LHC experiments using a data sample corresponding to an integrated luminosity of 36.1f b −1 from protonCproton collisions give a limitation for the Z ′ gauge boson as M Z ′ > ∼ 2.42 TeV [28][29][30], and we use the constraint in our numerical calculation.
Ref. [17] gives the most general Higgs potential with the symmetry invariance, We follow the theorems [31,32] The lightest neutral eigenstate H is the SM-like Higgs, whose mass is fixed to be 125.09 GeV. For charged Higgs H ± 1 , the diagonalizable matrix is related to angle β with tan β = k/v L . However for H ± 2 , the diagonalizable matrix is related to angle ζ with tan ζ = k/v R . The vevs of v R is much larger than v L . From this point, H ± 1 couples to the SM particles stronger than H ± 2 . The LEP experiments have searched for charged bosons via pair charged Higgs production. The data statistically combined by four experiments (ALEPH, DELPHI, L3 and OPAL) [33] showed that the mass of charged Higgs boson must be greater than 80 GeV. This lower limit will be used as a reference in our calculations. Recently, Atlas and CMS have also search the charged boson masses ranging from 200 to 2000 GeV [34,35], and constrains for some models such as hMSSM are given.

ALRM effects in Higgs decay
Each channel of discovered Higgs has already been detected by CMS and ATLAS experimental groups. The decay signal strengths of Higgs to ZZ bosons and bb pair are given in Table 1, independently by CMS [21,22], ATLAS [23,24].
where σ stands for the total Higgs production cross section at LHC and BR(H → XX) is the corresponding branching ratio. The total decay width of Higgs can be considered as the sum of some dominant Higgs partial decay widths. In Eq.(3), we can see the Yukawa couplings Y Htt and Y Hbb in ALRM may be changed by adding a factor of T Φ sin β and T L cos β respectively from the SM values. In Fig. 1, the effect of these two factors are plotted to show that both of them tend to be 1 while T L cos β of Y Hdd is bigger than 1 particularly in the region of small M H ± 1 with large tan β. Y Hbb strongly depends on the mass of H ± 1 . However, the total decay width of Higgs boson remains very close to the SM result, κ tot ≃ 1, when the mass of H ± 1 is big enough. As for κ gg , this channel is mainly propagated through the top quark triangle loop diagram and the extra quark d ′ can be neglected due to the suppression of its coupling with SM-like Higgs. From Fig.1, we can see that the adding factor to top Yukawa coupling can be almost 1 and make the top Yukawa coupling unchanged from the SM result. Therefore, the ratio κ gg = Γ(H → gg)/Γ(H → gg) SM can be considered as 1.
Now we turn to the SM-like Higgs decay into ZZ in ALRM. For the kinematics forbidden, we compute H → ZZ via H → ZZ * → Zf f , where f = e, µ, τ and u, d, c, s, b quark.
It is worth mentioning that the parameters λ 3 , α 12 and M H ± 2 is not sensitive in the numerical results, only 0 < λ 3 < √ 4π and α 12 > 0, to be consistent with the perturbative unitarity and the minimization and boundedness from below conditions Eqs. (21)(22)(23)(24) in ref. [19]. The relevant input parameters are chosen as [28]: We have used Feynrules [36,37] to generate the model files and MadGraph [38] to calculate the numerical values of cross sections. In Fig. 2 we display the results of κ ZZ = Γ(H → zz)/Γ(H → zz) SM as functions of tan β and M H ± 1 . This Figure confirms our theoretical expec-tation and shows that κ zz can slightly deviate from 1. In Fig. 2(a) κ ZZ is calculated by MadGraph and the relevant couplings κ ZZ = g(HZZ) 2 ALRM /g(HZZ) 2 SM respectively. Obviously the results are the same by that two ways. In Fig. 2(b) with the increasing of M H ± 1 , the trend is rapidly increasing and then stable while M H ± 1 200 GeV . So in the following calculation M H ± 1 is equal to 200 GeV, unless otherwise stated. In this case, it is clear that the signal strength µ zz is also close to the SM expectation and can be consistent with CMS experimental results. It is remarkable that all signal strengths of Higgs decay channels in ALRM are close to SM results with MadWidth [39] automatically computing decay widths. In Fig. 3(a), κ bb is plotted as a function of M H ± 1 . This figure shows that the decay channel H → bb is more sensitive to M H ± 1 than that in H → ZZ. In addition, it is remarkable that the decay width of this channel in ALRM is slightly larger than that in SM. In Fig. 3(b), κ bb is plotted as a function of tan β, it is clearly that tan β has little impact on κ bb . From Fig. 3, we find that the decay channel H → bb is more sensitive to M H ±

ALRM effects in e + e − → HZ
In this subsection the production of e + e − → HZ at CEPC and future e + e − colliders has been studied and Feynman diagrams of this process are displayed in Fig. 4. Due to the contributions from t channel are negligible small, we only give the Feynman diagrams in s channel. As can be seen from Fig. 4, a new  The cross section as a function of tan β with the mass of heavy boson W ′ varying from 4 TeV to 5 TeV and charged boson H ± 1 varying from 150 GeV to 300 GeV is shown in Fig. 5 at √ s = 240 GeV. Obviously the total cross sections are all less than SM from Fig. 5. From Ref. [19], the masses of A 2 become small when tan β and M H ± 1 are small, and M W ′ influence the heavy boson Z ′ . In Fig. 5, due to the large M W ′ , the contribution from Z ′ propagator is small and the contribution from A 2 is mainly in small tan β region. Fig. 5(b) shows the discrepancies from A 2 propagator become larger with smaller M H ± 1 . At the other two collision energies, the same tendency can be obtained which we didn't show. The total cross sections and corresponding relative ALRM discrepancies are given in Table 2 with tan β = 50, M H ± 1 = 200 GeV and M W ′ = 4 TeV at √ s = 240 GeV, 500 GeV, and 1 TeV respectively. The relative ALRM discrepancy is defined as δ=(σ ALRM −σ SM )/σ SM . In this table, we can see the relative ALRM discrepancies are increasing with √ s. It is worth mentioning that the Higgs sector in ALRM is very similar to that in 2HDM, where one Higgs doublet couples to up-quarks and the second couples to down-quarks. Therefore, it does not lead to any flavor changing neutral current problem and light charged Higgs is phenomenologically acceptable. In order to analyze the discovery significance which is calculated by the formula n S √ ntot , where n S = Ldt × (σ ALRM − σ SM ) is the number of discrepancy events and n tot = Ldt × σ SM is the number of total events. The discovery significance is plotted in Fig. 6 as a function of integrated luminosity for CEPC, ILC and CLIC respectively. From Fig. 6, one can see that the integrated luminosity of all three colliders can reach several hundred specifically at 619.72 f b −1 (CEPC), 553.80 f b −1 (ICL) and 443.93 f b −1 (CLIC) when the discovery significance is 5σ. By contrast CLIC seems have an advantage in detecting it. It's worth mentioning that the discovery significance shown in Fig. 6 is calculated with no kinetic cuts. It can not be treated as a seriously result. In dealing with the sequential Z-boson leptonic decay and Higgs decay, the naive narrow-width approximation(NWA) method is used to produce the total cross section. Hence cross section for this process can be approximately written as: In order to get the precise branch ratio of H → bb in ALRM, we set the K-factor of each channel is the same as which in SM. By adopting HDECAY program [40], dominant Higgs partial decay widths are computed and listed in Table 3. Here the scale µ of the Yukawa coupling y Q (µ) = m Q (µ)/v is used to be the mass of Higgs in the calculation, m Q (µ) is the running mass of heavy quark. The total cross section Γ tot = Γ bb + Γ cc + Γ τ + τ − + Γ ZZ + Γ W W + Γ gg = 3.93×10 −3 GeV . Obviously the branch ratio of H → bb is 58.27% in SM and experiment shows that SM prediction for the decay branching fraction of Higgs boson with mass around 125.09GeV to bb is 57.5% [25]. The leading order(LO) results in SM computed by MadWidth and the corresponding K-factors ( σ HDECAY σ MadWidth ) are also added in Table 3. The decay width of H → gg is directly used the SM result by HDECAY. To this way we can estimate the branch ratio of H → bb in ALRM as a function of M H ± 1 . As for the branch ratio of Z → l + l − , the result is 10.31% in both SM and ALRM which is independent of M H ± 1 . xxxx-6 The total cross sections and corresponding relative discrepancies are shown in Table 4 at √ s=240 GeV, 500 GeV and 1 TeV respectively, where the relative deviation is defined as δ=(σ ALRM −σ SM )/σ SM . From Table 4 one finds that with the increasing of √ s, the cross sections both in ALRM and SM are decreasing. While the corresponding relative discrepancies are increasing significantly, which will be phenomenologically accessible.  To analyze the feasibility of experiment, the discovery significance is chosen the same as last subsection: is the number of discrepancy events and n tot = Ldt × BR(Z → l + l − )×σ SM ×BR SM (H → bb) is the total events. In Fig. 8, we depict the integrated luminosity needed for 5σ discovery significance as a function of M H ± 1 for CEPC, ILC and CLIC determined by n S √ ntot . For e + e − → HZ → l + l − bb process, The discrepancies between SM and ALRM are mainly influenced by the cross section of e + e − → HZ and the branching ratio of Higgs to bb. The cross sections of e + e − → HZ in ALRM is a little smaller than that in SM but the branching ratio of H → bb is opposite. In the middle of M H ± 1 region, the two part of the contribution counteracts each other, the cross section of e + e − → HZ → l + l − bb in ALRM and SM are approaching. So in order to detect the discrepancies, the required integrated luminosity is very large, which means that the region is difficult to search new physics. This corresponds to the peak in Fig. 8.

Summary
In the present paper, we have analyzed Higgs decay in each channel, Higgs and Z boson associated production and decay at CEPC and future e + e − colliders in Alternative Left-Right model(ALRM), motivated by superstring inspired E 6 model. We found that the contribution of charged Higgs boson M H ± 2 to Higgs decay is quite negligible due to the large tan β, while M H ± 1 plays an essential role in the decay channel of H → bb due to the mixing of scalars. And the model predictions for the signal strengths of Higgs decay, in particularly of H → ZZ that are consistent with SM expectations. We also analyzed the discrepancies of cross sections about Higgs and Z boson production between ALRM and SM and found that it can enhance to 6.78% when √ s increase to 1 TeV. Finally we study the sequential decay of Higgs and Z boson respectively to bb and l + l − , where xxxx-7 l = e, µ and τ , with NWA method. We found that the cross sections of sequential decay are much more dependent on the branch ratio of H → bb. We also shown that the typical value of cross sections are of O(1) fb which can be measured by future colliders.