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Threshold amplitudes are considered for multi-particle production in $\phi^k$and $O(N) ~\phi^4$ theories. It is found that the disappearance of tree-levelthreshold amplitudes of $2$ on-shell particles producing a large number ofparticles occurs in $\phi^k$ theory only for $k=3$ and $k=4$. The one-loopcorrection to the threshold amplitude for a highly virtual scalar particledecaying into $n$ particles in an $O(N)$ model is derived.Comment: 7 pages, LaTex, TPI-MINN-92/60-

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

Properties of perturbative multiparticle amplitudes in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi>φ</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="normal">O</mml:mi><mml:mn /><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo><mml:mn /></mml:math>theories

Properties of perturbative multiparticle amplitudes inφkandO(N)theories