Abstract
Propagation of particles with the emission of an arbitrary number of identical bosons all being at rest is considered. It is shown that in certain models the tree-level amplitudes for production of $n$ scalar bosons by two incoming particles are all equal to zero at the threshold starting from some small number $n$. In particular, this nullification occurs for production of massive scalars by two Goldstone bosons in the linear $\ensuremath{\sigma}$ model for $n>1$ and also for production of Higgs bosons in the standard model by gauge bosons and/or by fermions, provided that the ratios of their masses to that of the Higgs boson take special discrete values.
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