  For microwave-dressed polar molecules, we analytically derive an
intermolecular potential composed of an anisotropic van der Waals shielding
core and a long-range dipolar interaction. We validate this effective potential
by comparing its scattering properties with those calculated using the full
multi-channel interaction potential. It is shown that scattering resonances can
be induced by a sufficiently strong microwave field. We also show the power of
the effective potential in the study of many-body physics by calculating the
critical temperature of the Bardeen-Cooper-Schrieffer pairing in the
microwave-dressed NaK gas. It turns out that the effective potential is
well-behaved and extremely suitable for studying the many-body physics of the
molecular gases. Our results pave the way for the studies of the many-body
physics of the ultracold microwave-dressed molecular gases.
