  The $t$-channel singularity of a cross-section for a binary $2 \to 2$
scattering process occurs when a particle exchanged in the $t$ channel is
kinematically allowed to be on its mass-shell, that is when the process can be
viewed as a sequence of two physical subprocesses: a two-body decay $1\to 2$
and an inverse decay $2 \to 1$. We derive conditions for the singularity to be
present in a binary process. A class of divergent cross sections for Standard
Model processes has been determined and illustrated by the weak analog of the
Compton scattering $Ze^{-}\to Ze^{-}$. After critically reviewing
regularization methods proposed in literature, we discuss singular processes
which occur in a medium composed of particles in thermal equilibrium. The
medium is shown to regulate the singularities naturally as particles acquire a
finite width. We demonstrate a possible cosmological application by calculating
thermally averaged cross section for an elastic scattering within a simple
scalar model. The cross section, which is divergent in vacuum, becomes finite
when the process occurs in a thermal bath due to the imaginary self-energy of
the $t$-channel mediator computed within the Keldysh-Schwinger formalism.
