Contribution of Majoron to Hubble Tension in Gauged U(1)Lμ – Lτ Model

In this paper, we analyze parameter regions that can alleviate the Hubble tension in the U(1)L_μ - L_τ model with broken lepton number U(1)L symmetry. As new particles, this model has a U(1)L_μ - L_τ gauge boson Z and a Majoron φ, which can affect the early universe and the effective number of neutrino species Nₑff. If Z and φ simultaneously exist in the early universe, Z\,–\, φ direct reaction processes such as Zν _α ↔ φ ν ̄_β and Z φ ↔ ν ̄_α ν ̄_β occur. The comparison of Nₑff between the cases with and without the Z\,–\, φ direct reaction processes shows that these processes make a small contribution of O(10⁻⁴) to Nₑff, and it does not need to be considered for the alleviation of the Hubble tension. Based on these facts, we calculated Nₑff for various Majoron parameters without the Z\,–\, φ direct reaction processes to search for parameters that could alleviate the Hubble tension. As a result, we found that the U(1)L_μ - L_τ gauge boson and Majoron can alleviate the Hubble tension in some parameter regions, and there is a nontrivial indirect synergy contribution between Z and φ, although the contribution from Z\,–\, φ direct reaction processes is negligible.