  In this paper we show that the diphoton mass spectrum in proton-proton
collisions at the LHC is sensitive to the top quark flavor changing neutral
current in the vertices of $tu\gamma$ and $tc\gamma$. The diphoton mass
spectrum measured by the CMS experiment at the LHC at a center-of-mass energy
of 8 TeV and an integrated luminosity of 19.5 fb$^{-1}$ is used as an example
to set limits on these FCNC couplings. It is also shown that the angular
distribution of the diphotons is sensitive to anomalous $tu\gamma$ and
$tc\gamma$ couplings and it is a powerful tool to probe any value of the
branching fraction of top quark rare decay to an up-type quark plus a photon
down to the order of $10^{-4}$. We also show that the $tu\gamma$ FCNC coupling
has a significant contribution to the neutron electric dipole moment (EDM) and
the upper bound on neutron EDM can be used to constrain the $tu\gamma$ FCNC
coupling.
