Resolving the Hubble tension in a U(1)(L mu-L tau) model with the Majoron

We explore the possibility 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 the case where the Majoron phi does not exist at the beginning of the universe and is created by neutrino inverse decay vv -> phi after electron-positron annihilation. In this case, the contributions of the new gauge boson Z' and the Majoron phi to the effective number of neutrino species N-eff can be calculated in separate periods. These contributions are labelled N-eff for the U(1)(L mu-L tau) gauge boson and Delta N-eff' for the Majoron. The effective number N-eff = N-eff' + Delta N-eff' is evaluated by the evolution equations of the temperatures and the chemical potentials of light particles in each period. As a result, we find that the heavier Z' mass m(Z') results in a smaller N-eff' and requires a larger Delta N-eff' 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').