  We study the Landau gauge photon propagator in the three--dimensional Abelian
Higgs model with compact gauge field and fundamentally charged matter in the
London limit. The total gauge field is split into singular and regular parts.
On the confinement side of the string breaking crossover the momentum
dependence of the total propagator is characterized by an anomalous dimension
similarly to 3D compact QED. At the crossover and throughout the Higgs region
the anomalous dimension disappears. This result perfectly agrees with recent
observations that the monopole--antimonopole plasma leads to nonzero anomalous
dimension and the presence of the matter fields causes monopole pairing into
dipole bound states. The Yukawa mass characterizing the propagator part from
regular gauge fields is non-vanishing at the Higgs side and coincides with the
mass found for the total propagator. The regular gauge field without anomalous
dimension becomes massless at the crossover and in the confinement region.
