  We compute the contributions to continuum photon pair production at hadron
colliders from processes initiated by gluon-gluon and gluon-quark scattering
into two photons through a four-leg virtual quark loop. Complete two-loop cross
sections in perturbative quantum chromodynamics are combined with contributions
from soft parton radiation resummed to all orders in the strong coupling
strength. The structure of the resummed cross section is examined in detail,
including a new type of unintegrated parton distribution function affecting
azimuthal angle distributions of photons in the pair's rest frame. As a result
of this analysis, we predict diphoton transverse momentum distributions in
gluon-gluon scattering in wide ranges of kinematic parameters at the Fermilab
  Tevatron collider and the CERN Large Hadron Collider.
