  It is shown that in a $\lambda \phi^4$ theory of one real scalar field with
spontaneous breaking of symmetry a calculation of the amplitudes of production
by a virtual field $\phi$ of $n$ on-mass-shell bosons all being exactly at rest
is equivalent in any order of the loop expansion to a Euclidean space
calculation of the mean field of a kink-type configuration. Using this
equivalence it is found that all the $1 \to n$ amplitudes have no absorptive
part at the thresholds to any order of perturbation theory. This implies
non-trivial relations between multi-boson threshold production amplitudes. In
particular the on-mass-shell amplitude of the process $2 \to 3$ should vanish
at the threshold in all loops. It is also shown that the factor $n!$ in the $1
\to n$ amplitudes at the threshold is not eliminated by loop effects.

High Energy Physics - Phenomenology