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

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