We present a detailed analysis of the Higgs boson decays $$h\rightarrow Z \gamma $$ h → Z γ and $$h\rightarrow m_V Z$$ h → m V Z in the $$U(1)_X$$ U ( 1 ) X SSM, with $$m_V$$ m V denoting one of the mesons ($$\omega ,\rho ,\phi ,J/{\psi },\Upsilon $$ ω , ρ , ϕ , J / ψ , Υ ). Using the effective Lagrangian method, we calculate the effective constants $$C_{\gamma Z}$$ C γ Z (CP-even) and $${\tilde{C}}_{\gamma Z}$$ C ~ γ Z (CP-odd) for the vertex $$h \gamma Z$$ h γ Z , which are corrected by the new particle loop diagrams. Numerically, the ratio $${\Gamma }_{U(1)_X}(h\rightarrow Z \gamma )/{\Gamma }_{SM}(h\rightarrow Z \gamma )$$ Γ U ( 1 ) X ( h → Z γ ) / Γ SM ( h → Z γ ) is between 1.02 and 1.35. For the vector mesons $$\omega ,\rho ,\phi $$ ω , ρ , ϕ and $$J/{\psi }$$ J / ψ , the ratios $${\Gamma }_{U(1)_X}(h\rightarrow m_V Z )/{\Gamma }_{SM}(h\rightarrow m_V Z)$$ Γ U ( 1 ) X ( h → m V Z ) / Γ SM ( h → m V Z ) are mainly distributed in the range of (1.01–1.45). When $$m_V$$ m V represents $$\Upsilon $$ Υ , the ratio $${\Gamma }_{U(1)_X}(h\rightarrow \Upsilon Z )/{\Gamma }_{SM}(h\rightarrow \Upsilon Z)$$ Γ U ( 1 ) X ( h → Υ Z ) / Γ SM ( h → Υ Z ) is mostly located in the range of (1.002–1.30). The aim of this work is to provide a reference for probing the $$U(1)_X$$ U ( 1 ) X SSM model via the Higgs decays $$h\rightarrow Z \gamma $$ h → Z γ and $$h\rightarrow m_V Z$$ h → m V Z .

U(1)XSSM is the extension of the minimal supersymmetric standard model (MSSM) and its local gauge group is SU(3)C × SU(2)L × U(1)Y × U(1)X. To obtain this model, three singlet new Higgs superfields and right-handed neutrinos are added to MSSM. In the framework of U(1)XSSM, we study lepton flavor violating decays τ → Pl( P = π, η, η′; l = μ, e). According to the latest experimental data of τ → Pe and τ → Pμ, the influence of different sensitive parameters on the results is analyzed to make a reasonable prediction for future experiments. From the numerical analysis, the non-diagonal elements which correspond to the generations of the initial lepton and final lepton are main sensitive parameters and lepton flavor violation (LFV) sources. This work can provide a basis for finding the existence of new physics (NP).

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