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.
