Tight constraint on photon mass from pulsar spindown

Pulsars are magnetized rotating compact objects. They spin down due to magnetic dipole radiation and wind emission. If photon has a nonzero mass, the spin down rate would be smaller than the zero mass case. We show that an upper limit of the photon mass, i.e. $m_\gamma\lesssim h/Pc^2$, may be placed if a pulsar with period $P$ is observed to spin down. Recently, a white dwarf (WD) --- M dwarf binary, AR Scorpii was discovered to emit pulsed broadband emission with pulses. The spin-down luminosity of the WD can comfortably power the non-thermal radiation from the system. Applying our results to the WD pulsar with $P=117~\rm{s}$, we obtain a stringent upper limit of the photon mass between $m_\gamma<6.3\times10^{-50}~\rm{g}$ assuming a vacuum dipole spindown, and $m_\gamma<9.6\times10^{-50}~\rm{g}$ assuming a spindown due to a fully developed pulsar wind.