  We explore how the dilepton production rate is modified near the critical
temperature of color superconductivity and QCD critical point by the soft modes
inherently associated with the phase transitions of second-order. It is shown
that the soft modes affect the photon self-energy significantly through so
called the Aslamasov-Larkin, Maki-Thompson and density of states terms, which
are known responsible for the paraconductivity in the metalic
superconductivity, and cause an anomalous enhancement of the production rate in
the low energy/momentum region.
