Contribution of Majoron to Hubble tension in gauged U(1)${}_{L_{\mu }-L_{\tau }}$ Model

In this paper, we analyze parameter regions that can alleviate the Hubble tension in the U(1)${}_{L_{\mu }-L_{\tau }}$ model with the broken lepton number U(1)${}_L$ symmetry. As new particles, this model has a U(1)${}_{L_{\mu }-L_{\tau }}$ gauge boson $Z^{\mathrm{\prime }}$ and a Majoron $\phi$, which can affect the early universe and the effective number of neutrino species $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$. If $Z^{\mathrm{\prime }}$ and $\phi$ simultaneously exist in the early universe, $Z^{\mathrm{\prime }}$-$\phi$ interaction processes such as $Z^{\mathrm{\prime }}{\nu }_{\alpha }\leftrightarrow \phi {\bar{\nu }}_{\beta }$ occur. The comparison of $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ between the cases with and without the $Z^{\mathrm{\prime }}$-$\phi$ interaction processes shows that these processes make a small contribution of $\mathcal{O}({10}^{-4})$ to $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$, and it does not need to be considered for the alleviation of the Hubble tension. Based on these facts, we calculated $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ for various Majoron parameters without the $Z^{\mathrm{\prime }}$-$\phi$ interaction processes to search 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 non-trivial synergy contribution between $Z^{\mathrm{\prime }}$ and $\phi$. Moreover, the parameter region with a lighter mass $m_{\phi }\lesssim 2$ MeV and a larger coupling $\lambda \gtrsim {10}^{-8}$ is excluded because it predicts too large $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$, i.e. $N_{\mathrm{e}\mathrm{f}\mathrm{f}}\gtrsim 3.5$. The favored and restricted regions of the Majoron parameters depend on the $Z^{\mathrm{\prime }}$ parameters because of the presence of the $Z^{\mathrm{\prime }}$ contribution and synergy one.