Renormalized feynman amplitudes for fields with fixed masses

One of the ways of regularizing causal functions in field theory is to add an imaginary quantity to the squares of the masses of the corresponding fields. We consider the question of the existence of a limit of the renormalized Feynman amplitudes on the transition to the true masses of the fields. Existence of this limit is proved both for theories with nonzero masses and for a large class of renormalized amplitudes in theories with zero masses.