Higgs boson decays h → MZ in the TNMSSM

We study the SM-like Higgs boson decays h → MZ in the Triplet extended NMSSM (TNMSSM), where M is a vector meson ( ρ, ω, ϕ, J/ Ψ , Υ). Compared to the minimal supersymmetric standard model (MSSM), the TNMSSM includes two new SU(2) triplets with hypercharge ± 1 and a SM gauge singlet which are coupled to each other. The indirect contributions to the decays h → MZ are produced from the effective hγZ vertex, and they are more important than the direct contributions. The results of this work would encourage a detection on h → Zγ at the future high energy colliders for exploring new physics beyond the SM.


I. INTRODUCTION
Since the Higgs was discovered by the ATLAS and CMS collaborations in 2012[1, 2], many questions regarding its properties remain unanswered.According to the latest experimental data the measured mass of the Higgs boson is [3].m h = 125.25 ± 0.17 GeV As a new elementary particle, h is largely consistent with the neutral Higgs boson predicted by the SM.However many questions have been raised that challenge the SM framework.
Weak scale supersymmetry (SUSY) is a promising extension of the Standard Model (SM): it naturally explains why the electroweak (EW) symmetry breaking scale is much smaller than the Planck scale, and solve the gauge hierarchy problem [4,5].But, minimum supersymmetric SM (MSSM) cannot fully solve the hierarchy problem.And it has another problem named "the µ problem".Hence, extensions of the MSSM have been proposed to solve these problems.For example the extension of the MSSM by adding a (SM) gauge singlet which is coupled to Higgs doublets (the NMSSM) has been proposed to solve the µ problem.
But unfortunately the NMSSM with all the couplings being perturbative up to the GUT scale also does not really solve the little hierarchy problem [6][7][8][9].If taking little hierarchy problem seriously, then one should consider another source of Higgs quadruple coupling, which will not decouple in the large tan β limit [10].The model which extended of the MSSM by adding SU(2) triplets(TMSSM) [11][12][13] possess such a Higgs quartic coupling naturally.
Combined the advantages of the NMSSM and the TNMSSM, they can solve each other's problems [10].In the triplet extended NMSSM (TNMSSM), the singlet interactions do not play any important role in raising the physical Higgs mass: one rely on triplets instead in achieving this goal.So, one also do not face the usual little hierarchy problems of the NMSSM.
There is no hZγ coupling at tree level, but it can be contributed by loop diagram [14].The first evidence for the process h → γZ is presented by the ATLAS and CMS.The obseved signal strength at the 68% confidence level is µ = 2.2 +1.0 −0.9 for ATLAS analysis, µ = 2.4 +1.0 −0.9 for the CMS analysis, and µ = 2.2 ± 0.7 for their combination [21].Due to the process h → γZ had observed and the results are shifted from the SM.This presupposes the existence of a new physics, whose contribution to the process may be able to explain the deviations between observed decays and SM predictions, and thus the associated decay process deserves to be investigated.This coupling is important for probing new physics.In the TNMSSM, there are additional coupling of the Higgs boson to additional charged scalars and charged fermions.
They contribute to the hZγ coupling by loop diagrams.
The paper is organized as follows.we briefly present the main ingredients of the TNMSSM in Sec.II.We give the Higgs boson decays h → Zγ and h → M Z formulas in Sec.III.We show the input parameters and numerical results in Sec.IV.In the last section we give the discussion and conclusion.Finally, some relate formulas are shown in Appendix.

II. THE TNMSSM
Compared to the MSSM, the TNMSSM includes two new SU(2) L triplet superfields T , T with hypercharge ±1 and a SM gauge singlet superfield ŝ which are coupled to each other.
The superpotential of the TNMSSM can be written as: Here the triplet superfields with hypercharge Y = ±1 are defined as: The soft SUSY breaking terms are shown as the follow where the respective definitions of the products between two SU (2) L doublets and between a SU (2) L doublets and a SU (2) L triplet are given as follows: Once the electroweak symmetry is spontaneously broken, the neutral scalar fields can be define as and we define the ratio Since we introduce a single state and two triplet states, we have five minimization equations, including the usual upper and lower Higgs.In general, the vacuum expectation value of the triplet states must be small to avoid large ρ-parameter corrections [10].

In the basis (H
), the definition of the mass squared matrix for charged Higgs is given by where This matrix is diagonalized by Z + : The mass of the SM-like Higgs boson in the TNMSSM can be written as: where m 0 h 1 is the lightest tree-level Higgs boson mass, and ∆m 2 h is the radiative correction.The two-loop leading-log radiative corrections can be given as: where α 3 is the running strong coupling constant, M S = √ m t1 m t2 with m t1,2 are the stop masses.Ãt = A t − µ cot β with A t = T u,33 /Y u,33 .There is no contribution to hZγ coupling at tree level in the TNMMSM, but it can be created by loop diagrams.The h → Zγ * process can be used to probe for New Physics.
So we will focus on discussing h → Zγ * .In the TNMSSM, the non-standard h 0 γZ vertex should be taken into account.The effective Lagrangian for hγZ is written as: with s W = sin θ W , c W = cos θ W .The decay width of h → Zγ deduced by the effective Lagrangian defined in Eq.( 6) is: The loop diagrams make additional contributions to h → M Z decays in new physics.And the decay width of h → M Z is given by: , m M is the mass of vector meson.
represent the CP-even longitudinal and transverse form factors, respectively F M Z ⊥ represent the CP-odd transverse form factors.For the vector mesons considered in this work, the mass ratio r M is very small, but it can make significant contributions to the transverse polarization states.In order to obtain better results we keep them in our study.
In Eq.( 8), where v q = T q 3 2 − Q q sin 2 θ W are the vector couplings of the Z boson to the quark q, κ Z is the ratio of the coupling of the SM-like Higgs boson to Z boson to the corresponding SM value.α s is the strong coupling constant.The flavor-specific decay constants f q M are defined by The calculations can be simplified by the following relationship The vector meson decay constants f M , Q M , v M are shown in Table I.
The concrete forms of C γZ and CγZ in Eqs.(9)(10)(11) can be written as [16,17,22] where    represent the SM contributions to h → Zγ. κq and κl represent the effective Higgs couplings to the quarks and the leptons.A f , B f , and A γZ W are loop function could find in Refs.[18,27].The numerical values of C SM γZ and CSM γZ are taken as : C SM γZ ∼ −2.395 + 0.001i, CSM γZ ∼ 0 in Ref. [16].
In the TNMSSM, the one loop diagrams contributing to h → γZ are shown in Fig. 2, where F represent the charged Fermions and S represent the charged scalars.The new contributions to C γZ originate from the exchanged particles:charginos, sleptons, squarks, and charged Higgs.
The CP-odd coupling C γZ is 0 in the SM.In the TNMSSM, the hγZ coupling can be written as F1 i(A + Bγ 5 )F 2 h, where A is the CP-even part and B is the CP-odd part [16,17].and P R = 1+γ 5  2 , the CP-even part is γZ in the TNMSSM.The expression of CP-even coupling C N P γZ in the TNMSSM is: where w )/c w , v f1 and v f2 represent up and down-quark sectors, T f 3 is the weak isospin of fermion f , θ f is the mixing angle of sfermins f1,2 .The function A 0 , A 1/2 can be found at [27,28].The concrete expressions of couplings As discussed in Ref. [23], the QCD corrections to the process h → Zγ are around 0.1% which medicates the QCD corrections can be neglected safely.In other words, we can safely neglect QCD corrections because they are very small.
Compared to the indirect contributions, the direct contributions are very different, and they can be calculated in a power series (m q /m h ) 2 or (Λ QCD /m h ) 2 .For the transversely polarized vector meson, leading-twist projections provide direct contributions.We can get the direct contributions by the asymptotic function In the calculations, it is found that the direct contribution is much smaller than the indirect contribution.Which indicates that the indirect contributions are more important than the direct contributions.The contributions for the decay width of h → M Z in SM are shown in Table II.Normalized to the SM expectation, the signal strengths for the Higgs decay channels can be quantified as where ggF stands for gluon-gluon fusion.And Higgs production cross sections can be written as Through Eqs.(18)(19)(20), the signal strengths for h → M Z and h → γγ can be quantified as where Γ h NP and Γ h SM denote total decay widths in the NP model and the SM respectively.
According to the latest experimental data [3], We take M Q = 2TeV, A Q = 1.5TeV.And for the slepton sector, we take m l = m ẽ = 2 TeV, T e = Y e diag(A e , A e , A e ) and A e = 1.5TeV.Then we take µ = 1 TeV, tan β = 8, tan β ′ = 10, λ = 0.95,κ = 0.9, χ t = 0.4, T Λ T = 1.5 TeV, T κ = 700 GeV, T λ = −700 GeV and v 2 T + v2 T = 2 GeV.We employ the following parameters as variable parameters in the numerical analysis And in our next numerical analysis we keep the lightest chargino always more than 800 GeV, all the mass of sleptons and squarks are more than 1900 GeV.
In this subsection we calculate the signal strengths for process h → γγ, h → V V * and h → Zγ.Some relevant formulas of h → γγ and h → V V * can be found in the works [27,28].At first we take parameters M 2 = 1500 Gev, Λ T = 0.8, A t = 1500 GeV and T χ d = −800 GeV.And we paint the signal strength of the h → γγ varying with χ d in Fig. 3(a), for T χt = −800 GeV (solid line), T χt = −900 GeV (dashed line) and T χt = −1000 GeV (dot dashed line).In order to keep the SM-like Higgs mass satisfy the 3σ error of experimental constraints, we let the χ d vary from 0.5 to 1.In Fig. 3(a), all the three curves are 1.03 < µ ggF γγ < 1.16 and they they behave the same way.These curves tend to be decreases with the increase of the χ d .The solid line varies from 1.16 to 1.05, the dashed line varies from 1.15 to 1.045 and the dot dashed line varies from 1.14 to 1.035.Our results for process h → γγ satisfy the experiment constraints [3].Then, the signal strengths for processes h → ZZ * and h → W W * are very close.So we take µ ggF V V * = µ ggF W W * = µ ggF ZZ * for simplicity, and we just paint the signal strength of h → ZZ * .We take parameters M 2 = 1500 GeV, T χ d = −800 GeV, T χt = −800 GeV and A t = 1500 GeV.And we paint the signal strength of the h → V V * varying with χ d in Fig. 3(b), for Λ T = 0.7 (solid line), Λ T = 0.8 (dashed line) and Λ T = 0.9 (dot dashed line).In Fig. 3(b) the signal strength of the h → V V * decreases with the increase of χ d .
These curves are above 1.073 and below 1.171, and their behaviors are similar to each other.
The experiment constraints [3] µ W W * = 1.19 ± 0.12.So our calculate result µ ggF ZZ * satisfies the experimental constraint that the error is 1σ, and µ ggF W W * satisfies the experimental constraint that the error is 2σ.
The new physics contributions to the decay h → M Z come from the effective coupling of hZγ.So we study the process h → Zγ at this subsection.In the numerical calculation of process h → Zγ, we take the parameters M 2 = 1500 GeV, T χ d = −800 GeV, T χt = −800 GeV and A t = 1500 GeV.In Fig 3(c), these curves are close to each other.And all the curves are varies from 1.164 to 1.248.When 0.6 ≤ Λ T ≤ 0.9, all the lines here have a smaller slope.When 0.9 ≤ Λ T ≤ 1, all the lines here have a bigger slope.And the result agrees with the observed signal strength with 1.5σ.

B. The processes h → M Z
In this subsection we will study the processes h → M Z.The vector mesons decay constants for ω, ρ, ϕ, J/ψ and Υ can be found in Table I.The NP contribution of the process h → M Z comes from the effective coupling hZγ.So our calculated results of decay h → M Z for different mesons should be similar.Now we study the signal strengths of process h → M Z. First, we take the parameter T χ d = −800 GeV, T χt = −800 GeV and A t = 1500 GeV.We paint the signal strengths of processes h → M Z in Fig. 4.And in Fig. 4 the solid line is obtained with χ d = 0.7, Λ T = 0.9 the dashed line is obtained with χ d = 0.8, Λ T = 0.8, and the dot dashed line is obtained with χ d = 0.9, Λ T = 0.7.We can see from Fig. 4 the signal strengths increase with the increase of strengths of processes h → ϕZ are in region 1.141-1.172,the signal strengths of processes h → JψZ are in region 1.147-1.179,the signal strengths of processes h → ΥZ are in region 1.108-1.138.
The new contributions to the decay h → M Z come from the effective coupling of hγZ.
So we can infer that our results are consistent with the process h → γZ.As we can see, in Fig. 3(c from 0.6 to 1 with χ d = 0.7, 0.8, 0.9.The results for signal strengths of µ ggF M Z versus Λ T are plotted in Fig. 5.The µ ggF ωZ are in region 1.133-1.238,the µ ggF ρZ are in region 1.138-1.249,the µ ggF ϕ Z are in region 1.109-1.191,the µ ggF J/ψZ are in region 1.112-1.198and the µ ggF ΥZ are in region 1.07-1.155.And we can see that the signal strengths of h → M Z are similar to the signal strength of h → Zγ.In Fig. 5, these solid lines all the curves are varies from 1.164 to 1.248.When 0.6 ≤ Λ T ≤ 0.9, all the lines here have a smaller slope.When 0.9 ≤ Λ T ≤ 1, all the lines here have a bigger slope.And we can see the signal strengths of h → M Z are decrease as χ d increase.Then we study the effect of the T χ d on signal strengths of h → M Z.The parameters we take as M 2 = 1500 GeV, Λ T = 0.8, χ d = 0.7 and T χt = −800 GeV.And in order to keep the SM-like Higgs mass satisfy the 3σ error of experimental constraints, we let the T χ d vary from −1000 GeV to 1000 GeV with A t = 1000, 1200, 1500 GeV.We paint the signal strengths of processes h → M Z in Fig. 6.And in Fig. 6 the solid lines are obtained with A t = 1000 GeV, the dashed lines are obtained with A t = 1200 GeV and the dot dashed lines are obtained with A t = 1500 GeV.In Fig. 6, we can see tha the signal strengths increase with T χ d increase.In Fig. 6(a strength of processes h → ϕZ and h → J/ψZ are region 1.138-1.178and 1.145-1.186.The meson Υ is the heaviest meson we've studied.So that the signal strength of process h → ΥZ is obviously less than the other results in Fig. 6.The signal strength of process h → ΥZ is in region 1.105-1.142.We can see from Fig. 6, the A t have a great influence on the signal strengths of h → M Z.The signal strengths of h → M Z increase as the A t increases. At last we study the effect of the A t on signal strengths of processes h → M Z.The coupling of higgs and third generation squarks include the A t .We take the parameters as M 2 = 1500 GeV, T χ d = −800 GeV and T χt = −800 GeV.In order to keep the SM-like Higgs mass satisfy the 3σ error of experimental constrains, we let the A t vary from −1000 GeV to 1000 GeV with (Λ T = 0.5, χ d = 0.8), (Λ T = 0.6, χ d = 0.7) and (Λ T = 0.7, χ d = 0.6).
We paint signal strengths of process h → M Z in Fig. 7.In Fig. 7 the solid lines are obtained with Λ T = 0.5, χ d = 0.8, the dashed lines are obtained with Λ T = 0.6, χ d = 0.7 and the dot dashed lines are obtained with Λ T = 0.7, χ d = 0.6.In Fig. 7(a) and Fig. 7(b), the signal strengths of the processes h → ωZ and h → ρZ are in region 1.165-1.253and 1.174-1.265.
The signal strengths of processes h → ϕZ and h → J/ψZ are in region 1.125-1.205and 1.132-1.213.For the heaviest meson Υ we're studied, the signal strength of h → ΥZ is obviously less than the other results in Fig. 7.The lines in Fig. 7(e) are in region 1.095-1.169.

V. CONCLUSION
In this work, we study the decays h → γZ and h → M Z in the TNMSSM, with M = ω, ρ, ϕ, J/ψ, Υ.There are two types of contributions to decay h → M Z: the direct contributions and indirect contributions.For indirect contributions, there is a process h → Zγ * → M Z, where γ * is off-shell and changes into the final state vector meson.There is no hγZ coupling at tree level, but it can be contributed by loop diagram.In the models beyond SM, the coupling constant can be divided into two parts: CP-even coupling constant C γZ and CP-odd coupling constant CγZ .The CP-even coupling constant C γZ is more important than the CP-odd coupling constant CγZ .
The experiment results of the signal strengths µ ggF γγ and µ ggF ZZ are µ ggF γγ = 1.10 ± 0.07 and µ ggF ZZ = 1.01 ± 0.07.Our numerical results of the signal strengths µ ggF γγ and µ ggF ZZ are in region 1.035-1.16and 1.073-1.171which satisfy the error of 1σ.Our numerical results of the signal strength in region 1.164-1.248.The result agrees with the observed signal strength with 1.5σ.The numerical results show that the TNMSSM contributions to the processes h → ωZ and h → ρZ are more considerable.The signal strengths µ ggF ωZ,ρZ are about 1.13-1.26.The TNMSSM corrections to the processes h → ϕZ and h → J/ψZ are during 1.11-1.21,and h → ΥZ about 1.07-1.17.The decays h → M Z may be accessible at future high energy colliders.

Appendix B: The mass of Higgs and Charginos
In the basis (ϕ d , ϕ u , ϕ s , ϕ T , ϕ T ), the definition of mass squared matrix for neutral Higgs is given by where This matrix is diagonalized by Z H : In the basis ( W − , H− d , T − ), ( W + , H+ u , T + ), the definition of mass matrix for charginos is given by Appendix C: Tadpole equation and some corresponding vertexes The CP-even tree level part of tadpole are given by ∂V ∂ϕ Then we can identify the m 2 H d , m 2 Hu , m 2 S , m 2 T and m 2 T by the minimum conditions of the scalar potential.
Here, we show some corresponding vertexes in this model.Their concrete forms are shown as For the coupling between Higgs particles and charged scalar particles is too complicated, we calculate it by computer.

FIG. 1 :
FIG.1:The diagrams contributing to the decay h → M Z.The crossed circle in the last graph represents the effective vertex h → Zγ * from the one loop diagrams.

⊥
and F M Z ⊥ could be divided into direct and indirect parts.The indirect contributions are shown as follows:

2
IV. NUMERICAL ANALYSISIn this section, we discuss the numerical results of the Higgs boson decays h → M Z in the TNMSSM are present.The results are constrained by the SM-like Higgs boson mass in the TNMSSM with 124.74 GeV ≤ m h ≤ 125.76 GeV, where a 3σ experimental error is considered.For the SM parameters, we take m W = 80.385GeV, m Z = 91.1876GeV,m u = 2.16MeV, m d = 4.67MeV, m s = 93.4MeV,m c = 1.27GeV, m b = 4.18GeV, m t = 172.69GeV.For the squark sector, we take m Q

FIG. 6 :
FIG.6:The signal strengths versus T χ d are plotted by the solid line (A t = 1000 GeV), dashed line (A t = 1200 GeV) and dot dashed line (A t = 1500 GeV).

TABLE I :
The mesons decay constants f M , Q M , v M will be used in the numerical analysis, where f ⊥ M and f q⊥ M represent the transverse decay constants and the flavor-specific transverse decay constants.

TABLE II :
The contributions for the decay width of h → M Z in SM, with C SM γZ ≃ −2.43.