  We consider a class of Hamiltonians describing a fermion field coupled to a
boson field. The interaction kernels are assumed bounded in the fermionic
momentum variable and decaying like $|q|^{-p}$ for large boson momenta $q$. A
realistic physical case would be $p=\frac12$. We impose a spatial cutoff and
UV-renormalise the resulting Hamiltonian by subtracting its ground state
energy. Our renormalisation procedure works for the physically realistic value
$p=\frac12$ in spatial dimensions $1$ and $2$ and for $p>\frac34$ in spatial
dimension $d=3$.
