  The pion and quark annihilation mechanisms of dilepton production during
relativistic nuclear collisions are studied. We focus on the modifications
caused by the specific features of in-medium pion and quark states rather than
by medium modification of the $\rho $-meson spectral density. The main
ingredient emerging in our approach is a form-factor of the multi-pion
(multi-quark) system. Replacing the usual delta-function the form-factor plays
the role of distribution which, in some sense, "connects" the total 4-momenta
of the annihilating and outgoing particles. The difference between the c.m.s.
velocities attributed to annihilating and outgoing particles is a particular
consequence of this replacement and results in the appearance of a new factor
in the formula for the lepton pair production rate. We obtained that the
form-factor of the multi-pion (multi-quark) system causes broadening of the
rate which is most pronounced for small invariant masses, in particular, we
obtain a growth of the rate for the invariant masses below two masses of the
annihilating particles.
