Resolving the Hubble tension in a U(1)${}_{L_{\mu }-L_{\tau }}$ model with Majoron

In this paper, we explore possibilities of resolving the Hubble tension and $(g-2{)}_{\mu }$ anomaly simultaneously in a U(1)${}_{L_{\mu }-L_{\tau }}$ model with Majoron. We only focus on a case where the Majoron $\varphi$ does not exist at the beginning of the universe and is created by neutrino inverse decay $\nu\nu\to \phi$ after electron-positron annihilation. In this case, contributions of the new gauge boson $Z^{\mathrm{\prime }}$ and Majoron $\varphi$ to the effective number of neutrino species $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ can be calculated in separate periods. These contribution are labelled $N_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{\prime }}$ for the U(1)${}_{L_{\mu }-L_{\tau }}$ gauge boson and $\mathrm{\Delta }{N}_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{\prime }}$ for the Majoron. The effective number $N_{\rm eff} = N’{\rm eff} + \Delta N{\rm eff}^\prime$ is evaluated by the evolution equations of the temperatures and the chemical potentials of light particles in each period. As a result, we found that the heavier $Z^{\mathrm{\prime }}$ mass $m_{Z^{\mathrm{\prime }}}$ results in the smaller $N_{\mathrm{e}\mathrm{f}\mathrm{f}}^{\mathrm{\prime }}$ and requires the larger $\Delta N_{$\mathrm{e}\mathrm{f}\mathrm{f}$}^\prime$ to resolve the Hubble tension. Therefore, compared to previous studies, the parameter region where the Hubble tension can be resolved is slightly shifted toward the larger value of $m_{Z^{\mathrm{\prime }}}$.