Loops with heavy particles in multi Higgs production amplitudes

In view of a recently renewed interest to 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 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 until very high $n$.