New light mediators for the $R_K$ and $R_{K^*}$ puzzles

The measurements of $R_{K}$ and $R_K^{*}$ provide hints for the violation of lepton universality. However, it is generally difficult to explain the $R_{K^*}$ measurement in the low $q^2$ range, $0.045 \le q^2 \le 1.1$ GeV$^2$. Light mediators offer a solution by making the Wilson coefficients $q^2$ dependent. We check if new lepton nonuniversal interactions mediated by a scalar ($S$) or vector particle ($Z^\prime$) of mass between $10-200$ MeV can reproduce the data. We find that a 25 MeV $Z^\prime$ with a $q^2$-dependent $b-s$ coupling and that couples to the electron but not the muon can explain all three anomalies in conjunction with other measurements. A similar 25 MeV $S$ provides a good fit to all relevant data except $R_{K^*}$ in the low $q^2$ bin. A 25 MeV $Z^\prime$ with a $q^2$-dependent $b-s$ coupling and that couples to the muon but not the electron provides a good fit to the combination of the $R_K$ and $R_{K^*}$ data, but does not fit $R_{K^*}$ in the low $q^2$ bin well.