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

Abstract

Abstract In this paper, we analyze parameter regions that can alleviate the Hubble tension in the U(1)$_{L_\mu - L_\tau }$ model with broken lepton number U(1)$_L$ symmetry. As new particles, this model has a U(1)$_{L_\mu - L_\tau }$ gauge boson $Z^{\prime }$ and a Majoron $\phi$, which can affect the early universe and the effective number of neutrino species $N_\textrm{eff}$. If $Z^{\prime }$ and $\phi$ simultaneously exist in the early universe, $Z^{\prime }\,$–$\, \phi$ direct reaction processes such as $Z^{\prime }\nu _\alpha \leftrightarrow \phi \bar{\nu }_\beta$ and $Z^{\prime } \phi \leftrightarrow \bar{\nu }_\alpha \bar{\nu }_\beta$ occur. The comparison of $N_\textrm{eff}$ between the cases with and without the $Z^{\prime }\,$–$\, \phi$ direct reaction processes shows that these processes make a small contribution of $\mathcal {O}(10^{-4})$ to $N_\textrm{eff}$, and it does not need to be considered for the alleviation of the Hubble tension. Based on these facts, we calculated $N_\textrm{eff}$ for various Majoron parameters without the $Z^{\prime }\,$–$\, \phi$ direct reaction processes to search for parameters that could alleviate the Hubble tension. As a result, we found that the U(1)$_{L_\mu - L_\tau }$ gauge boson and Majoron can alleviate the Hubble tension in some parameter regions, and there is a nontrivial indirect synergy contribution between $Z^{\prime }$ and $\phi$, although the contribution from $Z^{\prime }\,$–$\, \phi$ direct reaction processes is negligible.