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

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