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

Abstract

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 $\nu \nu \to \varphi$ after electron-positron annihilation. In this case, contributions of the new gauge boson $Z^{\prime }$ and Majoron $\varphi$ to the effective number of neutrino species $N_{\mathrm{eff}}$ can be calculated in separate periods. These contribution are labelled $N_{\mathrm{eff}}^{\prime }$ for the U(1)${}_{L_{\mu }-L_{\tau }}$ gauge boson and $\Delta N_{\mathrm{eff}}^{\prime }$ for the Majoron. The effective number $N_{\mathrm{eff}}=N_{\mathrm{eff}}^{\prime }+\Delta N_{\mathrm{eff}}^{\prime }$ 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 $Z^{\prime }$ mass $m_{Z^{\prime }}$ results in the smaller $N_{\mathrm{eff}}^{\prime }$ and requires the larger $\Delta N_{\mathrm{eff}}^{\prime }$ 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 $m_{Z^{\prime }}$.

Comments: 19 pages, 7 figures, published in PTEP