Renormalization of overlapping transverse divergences in a model light-front Hamiltonian

We construct a simplified model of a light-front Hamiltonian and describe its renormalization. Our construction starts from a Hamiltonian which acts in a space spanned by free states of two fermions and states of two fermions and one scalar boson. For the purpose of this paper the starting Hamiltonian is regularized by chopping factors in the interaction vertices. We derive an effective Hamiltonian acting in the space of two fermions. Then we make ad hoc simplifications in the effective Hamiltonian to produce a model that we can analyze. We drop fermion self-interactions and replace the eigenvalue in the effective one-boson-exchange term by a constant. Our model Hamiltonian acts in the space of two fermions only. The model involves logarithmic ultraviolet transverse divergences, analogous to overlapping divergences in perturbative Lagrangian $S$-matrix calculations. We describe the construction of a Hamiltonian counterterm that removes the divergences to all orders in Hamiltonian perturbation theory. The counterterm is local in the transverse direction and contains an arbitrary function of the longitudinal momenta of fermions. We suggest that a suitable choice of the arbitrary function may partly remove the violation of rotational invariance in the spectrum of the model Hamiltonian. The renormalization group transformation for the renormalized two-fermion interaction, $V_{\lambda }$, is determined by a nonlinear integro-differential equation of the form $\frac{d{V}_{\lambda }}{d\lambda }=-V_{\lambda }{K}_{\lambda }{V}_{\lambda }$, where $K_{\lambda }$ is a known kernel and $\lambda$ is the scale of the relative transverse momenta of fermions at which the interactions are chopped off.