We present the recent measurements of very low transverse momentum ($p_{\rm T}$) dilepton and J/$\psi$ production in peripheral Au+Au collisions via the $\mu^+\mu^-$ channel by the STAR experiment. Significant J/$\psi$ and $\mu^{+}\mu^{-}$ enhancements are observed at very low $p_{\rm T}$ compared to the expectation of hadronic production. The comparison to model calculations is presented, and its physics implications are discussed.