Remarks on double Higgs boson production by gluon fusion at threshold

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.