Lepton-flavor violation and two loop electroweak corrections to $(g-2)_\mu$ in the triplets

In the Standard Model (SM), charged lepton flavor-violating (LFV) processes are strictly forbidden, and thus observation of LFV would signal the presence of new physics. Recently, the Muon $g{-}2$ collaboration at Fermilab reported their final result for the muon magnetic dipole moment (MDM), which is now consistent with the latest SM prediction at the $1\sigma$ level. This imposes a significant constraint on new-physics contributions to $(g{-}2)_\mu$. In this work, we study the LFV processes $\ell_j^- \to \ell_i^-\gamma$ and $\ell_j^- \to \ell_i^- \ell_i^- \ell_i^+$ within the framework of the triplet next-to-minimal supersymmetric standard model (TNMSSM). We include two-loop contributions to the muon MDM and investigate the resulting constraints on the model parameter space. Our numerical analysis shows that, in the TNMSSM, doubly charged Higgs bosons and the couplings $Y_L$ and $Y_D$ associated with the Type-I+II seesaw mechanism play a crucial role in both LFV processes and the muon MDM. Moreover, two-loop electroweak corrections have a significant impact on the muon MDM in this framework.