Vector gauge boson radiation from compact binary systems in a gauged $L_\mu-L_\tau$ scenario

The orbital period of a compact binary system decays mainly due to quadrupole gravitational radiation, which agrees with the observation to within one percent. Other types of radiation such as ultralight scalar or pseudoscalar radiation, massive vector boson radiation also contribute to the decay of orbital period as long as the mass of the emitted particle is less than the orbital frequency of the compact binary system. We obtain an expression of the energy loss due to the radiation of massive vector field from the neutron star-neutron star and neutron star-white dwarf binaries. Due to large chemical potential of the degenerate electrons, neutron stars have large muon charge. We derive the energy loss due to $U(1)_{L_\mu-L_\tau}$ gauge boson radiation from the binaries. For the radiation of vector boson, the mass is restricted by $M_{Z^\prime}<\Omega \simeq10^{-19}eV$ are the orbital frequencies of the compact star binaries. Using the formula of orbital period decay, we obtain constraints on the coupling constant of the gauge boson in the gauged $L_\mu-L_\tau$ theory for the four compact binary systems. For vector gauge boson muon coupling we find that for $M_{Z^\prime}<10^{-19}eV$, constraints on the coupling constant is $g<\mathcal{O}(10^{-20})$. We also obtain the exclusion plots of the massive vector proca field and the gauge field which can couple to muons.