  Dynamical chiral symmetry breaking (DCSB) in QED$_{3}$ with finite gauge
boson mass is
  studied in the framework of the rainbow approximation of Dyson-Schwinger
equations.
  By adopting a simple gauge boson propagator ansatz at finite temperature, we
first numerically solve the
  Dyson-Schwinger equation for the fermion self-energy to
  determine the chiral phase diagram of QED$_3$ with finite gauge boson mass
  at finite chemical potential and finite temperature, then we study the
  effect of the finite gauge mass on the phase diagram of QED$_3$. It is found
  that the gauge boson mass $m_{a}$ suppresses the occurrence of
  DCSB. The area of the region in the chiral phase diagram corresponding to
DCSB phase decreases as
  the gauge boson mass $m_{a}$ increases. In
  particular, chiral symmetry gets restored when $m_{a}$ is above a
  certain critical value. In this paper, we use DCSB to describe the
  antiferromagnetic order and use the gauge boson mass to describe the
  superconducting order. Our results give qualitatively a physical
  picture on the competition and coexistence between antiferromagnetic
  order and superconducting orders in high temperature cuprate superconductors.
