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

It is shown that in a λφ theory of one real scalar field with spontaneous breaking of symmetry a calculation of the amplitudes of production by a virtual field φ of n onmass-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 → 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 → 3 should vanish at the threshold in all loops. It is also shown that the factor n! in the 1 → n amplitudes at the threshold is not eliminated by loop effects. The tree-level amplitudes of the process in which one virtual scalar field φ produces n on-mass-shell bosons of this field in a λφ theory displays a growth proportional to n! at large n [1]−[4]. If this growth is not eliminated by the loop effects this would imply that a strong interaction develops in a theory with weak coupling λ at energies where production of n > 1/λ interacting bosons is kinematically possible. It was recently found that in a λφ theory of a scalar field φ one can explicitly sum all tree graphs for the process 1 → n at the threshold, i.e. when all the produced particles have zero spatial momenta, both in the theory without spontaneous symmetry breaking[3] and in the case of the spontaneous breaking of the reflection symmetry φ → −φ [4]. An elegant development of these calculations was subsequently suggested by Brown[5] who had pointed out that the threshold amplitudes of the 1 → n processes are related to a complex vacuum solution of the field equations. An extension of Brown’s technique has enabled explicit calculation of these amplitudes at the one-loop level[6, 7, 8]. In particular it was found that the relative magnitude of the loop corrections is described by the parameter nλ so that the quantum effects may indeed completely modify the behavior of the amplitudes at n ≥ λ. A surprising result of this study was that the on-mass-shell scattering amplitudes for the processes 2 → n at the tree level are vanishing at the n particle threshold for all n greater than 4 in unbroken λφ theory[6] and for n > 2 in the theory with spontaneously broken symmetry[7]. One of the consequences of the latter behavior is that the threshold amplitudes for the 1 → n processes do not develop absorptive part at the one-loop level in the theory with the spontaneous symmetry breaking. In this paper we consider the threshold production of particles in the theory of one real scalar field with spontaneously broken symmetry beyond the one-loop approximation. This is possible due to the fact that the loop expansion around the Brown’s solution can be mapped into the loop expansion in the Euclidean space around a kink-type configuration of the field. As a result it will be shown that the threshold amplitudes of the 1 → n processes have no absorptive part to any order in this expansion. Through the unitarity relation this implies that the on-mass-shell amplitude of the process 2 → 3 should vanish at the threshold in all orders of perturbation theory, while for higher final state multiplicities in the on-mass-shell processes k → n this implies existence of non-trivial relations between amplitudes with different k. Also we will use the analytical properties of the time dependence of the Brown type solution to prove that the generating function for the amplitudes of 1 → n has a singularity at a finite distance in the complex plane, so that the n! growth survives all the loop effects.