Fourth-order similarity renormalization of a model Hamiltonian

We study the similarity renormalization scheme for Hamiltonians to the fourth order in perturbation theory using a model Hamiltonian for fermions coupled to bosons. We demonstrate that the free finite parts of counterterms can be chosen in such a way that the $T$ matrix is covariant up to the fourth order and the eigenvalue equation for the physical fermion reduces to the Dirac equation. Through this choice, the systematic renormalization scheme reproduces the model solution originally proposed by Głazek and Perry.