Ultraviolet Renormalisation of a Quantum Field Toy Model II

Abstract

We consider a class of toy models describing a fermion field coupled with a boson field. The model can be viewed as a Yukawa model but with scalar fermions. As in our first paper, the interaction kernels are assumed bounded in the fermionic momentum variable and decaying like $|q|^{-p}$ for large boson momenta $q$. With no restrictions on the coupling strength, we prove norm resolvent convergence to an ultraviolet renormalized Hamiltonian, when the ultraviolet cutoff is removed. We do this by subtracting a sufficiently large, but finite, number of recursively defined self-energy counter-terms, which may be interpreted as arising from a perturbation expansion of the ground state energy. The renormalization procedure requires a spatial cutoff and works in three dimensions provided $p>\frac12$, which is as close as one may expect to the physically natural exponent $p = \frac12$.