Renormalization of an effective model Hamiltonian by a counter term

It has two parameters α, m. From a physical point of view the equation is a QCD-inspired effective one-body bound-state equation for modeling mesons with different constituent quark flavors [1]. M are the invariant mass squares of the physical mesons, while m = m1 = m2 is the effective mass of the quark and anti-quark. It takes this explicit form due to an over-simplification by the ↑↓-model [1]. If one Fourier tranforms the kernel U to configuration space, the interaction potential consists of a long-ranged Coulomb-interaction and a shortranged delta-interaction. It is this latter part, which generates all the well known trouble. In order to get reasonable solutions one has to regulate the high momentum transfers Q = (~k′−~k)2. Therefore we substitute the number 1 by a regulating function, 1 → R(Λ, Q), for which the soft cut-off