Resummations in the Bloch-Nordsieck model

We studied different levels of resummations of the exactly solvable Bloch-Nordsieck model in order to be able to compare the approximations with an exact result. We studied one-loop perturbation theory, two-particle-irreducible resummation and Schwinger-Dyson equations truncated in a way to maintain Ward identities. At all levels we carefully performed renormalization. We found that although the two-particle-irreducible resummation does not exhibit infrared sensitivity at the mass shell, as the one-loop perturbation theory does, it is still far from the exact solution. The method of truncated renormalized Schwinger-Dyson equations, however, is exact in this model, so it provides a new way of solving the Bloch-Nordsieck model. This method may also be generalized to other, more complicated theories.