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

Understand

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 $\phi$ 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 $Z'$ and Majoron $\phi$ to the effective number of neutrino species $N_{\rm eff}$ can be calculated in separate periods.
• These contribution are labelled $N'_{\rm eff}$ for the U(1)$_{L_\mu - L_\tau}$ gauge boson and $\Delta N_{\rm eff}^\prime$ for the Majoron.