  Electrons/positrons produced in a pulsar magnetosphere emit synchrotron
radiation, which is widely believed as the origin of the non-thermal X-ray
emission detected from pulsars. Particles are produced by curvature photons
emitted from accelerated particles in the magnetosphere. These curvature
photons are detected as pulsed $\gamma$-ray emissions from pulsars with age
$\lesssim10^6$ yr. Using $\gamma$-ray observations and analytical model, we
impose severe constraints on the synchrotron radiation as a mechanism of the
non-thermal X-ray emission. In most middle-aged pulsars ($\sim10^5-10^6$ yr)
which photon-photon pair production is less efficient in their magnetosphere,
we find that the synchrotron radiation model is difficult to explain the
observed non-thermal X-ray emission.
