  The amplitude of double Higgs boson production by the gluon fusion, $gg \to
hh$, is known to be small due to cancellation between the graphs with the boson
trilinear coupling and those with the coupling to the top quark. For this
reason a study of this process was suggested as a sensitive probe of the Higgs
sector nonlinearity. We calculate in a closed analytical form this amplitude at
the threshold of the two bosons, where the cancellation is the strongest, and
discuss the origin of the small value of the amplitude. We also note that the
cancellation in the double boson production is in fact a part of a more general
phenomenon of suppression of similar threshold amplitudes for multiple boson
production, which, although not directly relevant to the actual top quark and
the Higgs boson, can be useful in other studies.
