  A new resummation of the $S$-factor of a composite system of two relativistic
spin-1/2 particles of arbitrary masses interacting via a Coulomb-like
chromodynamical potential is presented. The analysis is performed in the
framework of a relativistic quasipotential approach based on the Hamiltonian
formulation of the covariant quantum field theory in the relativistic
configuration representation. The pseudoscalar, vector, and pseudovector
systems are considered and the behaviour of the $S$-factor near the threshold
and in the relativistic limit is investigated in detail. The spin dependence of
the $S$-factors is discussed as well. It is argued that at the threshold the
contribution of spins significantly reduces the Sommerfeld effect, while at
ultrarelativistic velocities their role diminishes and the $S$-factor becomes
basically the same as for the spinless systems. A connection between the new
and previously obtained S-factors for spinless particles of arbitrary masses
and for relativistic spinor particles of equal masses is established.
