Resolving the Hubble tension in a U(1)Lμ – Lτ model with Majoron

In this paper, we explore possibilities of resolving the Hubble tension and (g - 2)μ anomaly simultaneously in a U(1)Lμ-Lτ model with Majoron. We only focus on a case where the Majoron ϕ does not exist at the beginning of the universe and is created by neutrino inverse decay νν → ϕ after electron-positron annihilation. In this case, contributions of the new gauge boson Z′ and Majoron ϕ to the effective number of neutrino species Neff can be calculated in separate periods. These contribution are labelled N′eff for the U(1)Lμ-Lτ gauge boson and ΔN′eff for the Majoron. The effective number Neff = N′eff + ΔN′eff 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′ mass mZ′ results in the smaller N′eff and requires the larger ΔN′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 mZ′.