  This paper presents a~way to regularize the $t$-channel singularity (which
appears when a~massive, stable $t$-channel mediator of a~given process is
allowed to be on-shell, making the cross section infinite) in a~general case of
particles of any spin (0,~\nicefrac12,~1) interacting within a~thermal medium.
Those interactions result in a~finite lifetime of the mediator and allow to
introduce an~effective momentum- and temperature-dependent width. As a result,
the would-be-singular cross section becomes finite. A~complete derivation and
an analytical result for the width are provided. For an illustration, the
method is used to calculate the thermal widths and cross sections within
the~Vector-Fermion Dark Matter model.
