Loops with heavy particles in production amplitudes for multiple Higgs bosons

In view of a recently renewed interest to the production of multiple Higgs bosons, the amplitude for such process at the threshold of n particles is considered. An explicit calculation is done for the loop corrections to the amplitude arising from the interaction with a heavy fermion (e.g., top quark) and also with a heavy scalar. It is shown that such corrections generally break the scaling dependence on the number n and the Higgs self-coupling, which dependence is known from the past studies of models with one field. The correction due to the top quark loop is also found to be numerically large and exceeding that from the self-coupling up to very high n.