The equivalence theorem and the Bethe–Salpeter equation

We solve the Bethe–Salpeter equation for two-particle scattering in a field-theoretical model using two Lagrangians related by a field transformation. The kernel of the equation consists of the sum of all tree-level diagrams for each lagrangian. The solutions differ even if all four external particles are put on the mass shell, which implies that observables calculated by solving the Bethe–Salpeter equation depend on the representation of the theory. We point out that this violation of the equivalence theorem has a simple explanation and should be expected for any Bethe–Salpeter equation with a tree-level kernel. Implications for dynamical models of hadronic interactions are discussed.