Using the pinch technique, we compute the one-loop vertices of weak interactions in the B-LSSM and incorporate their pinch contributions into the gauge boson self-energies. Compared to the definitions of the $S$, $T$, and $U$ parameters in the Standard Model based on the $SU(2)_L\otimes U(1)_Y$ group, the corresponding parameters in the B-LSSM are modified. We provide these redefined $S$, $T$, and $U$ parameters and demonstrate the convergence of the results. In the framework of the low-energy effective Lagrangian for weak interactions, the $S$, $T$, and $U$ parameters can be expressed as functions of certain parameters in the B-LSSM. The updated experimental and fitting results constrain the parameter space of the B-LSSM strongly.

We analyze the lepton flavor violating decays $\tau\rightarrow Pl$ ($P=\pi,\eta,\eta';\;l=e,\mu$) in the scenario of the minimal R-symmetric supersymmetric standard model. The prediction on the branching ratios BR$(\tau\rightarrow P e)$ and BR$(\tau\rightarrow P \mu)$ is affected by the mass insertion parameters $\delta^{13}$ and $\delta^{23}$, respectively. These parameters are constrained by the experimental bounds on the branching ratios BR($\tau\rightarrow e (\mu) \gamma$) and BR($\tau\rightarrow 3e(\mu)$). The result shows $Z$ penguin dominates the prediction on BR($\tau\rightarrow Pl$) in a large region of the parameter space. The branching ratios for BR($\tau\rightarrow Pl$) are predicted to be, at least, five orders of magnitude smaller than present experimental bounds and three orders of magnitude smaller than future experimental sensitivities.