Zeros of Tree-Level Amplitudes at Multi-Boson Thresholds

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Propagation of particles with emission of 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 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 ratia of their masses to that of the Higgs boson take special discrete values.