Some properties of amplitudes at multiboson thresholds in spontaneously broken scalar theory

It is shown that in a 4 theory of one real scalar field with spontaneous breaking of symmetry a calculation of the amplitudes of production by a virtual field 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 1n amplitudes have no absorptive part at the thresholds to any order of perturbation theory. This implies nontrivial relations between multiboson threshold production amplitudes. In particular, the on-mass-shell amplitude of the process 23 should vanish at the threshold in all loops. It is also shown that the factor n! in the 1n amplitudes at the threshold is not eliminated by loop effects.