  We construct the massive scalar propagator for planar gravitational wave
backgrounds propagating on Minkowski space. We represent the propagator in
terms of the Bessel's function of suitably deformed nonlocal distance
functions, the deformation being caused by gravitational waves. We calculate
the propagator both for nonpolarized, for the plus ($+$) and cross ($\times$)
polarized plane waves, as well as for more general planar waves both in $D$
space-time dimensions. The propagator is useful for studying interactions of
scalar fields on planar gravitational wave backgrounds in the context of
interacting quantum fields, in which renormalization plays a crucial role. As
simple applications of the propagator we calculate the one-loop effective
action, the scalar mass generated by the scalar self-interaction term and the
one-loop energy-momentum tensor, all renormalized by using dimensional
regularization and renormalization. While the effective action and scalar mass
remain unaffected by gravitational waves, the energy-momentum tensor exhibits
flow of energy density in the direction of gravitational waves which grows
quadratically with the scalar field mass, gravitational wave amplitude and its
frequency.
