Construction of energy-independent potentials above inelastic thresholds in quantum field theories

We construct energy independent but non-local potentials above inelastic thresholds, in terms of Nambu-Bethe-Salpeter wave functions defined in quantum field theories such as QCD. As an explicit example, we consider NN --> NN + n pi scattering processes for n=0,1,2,.... We show an existence of energy-independent coupled channel potentials with a non-relativistic approximation, where momenta of all particles are small compared with their own masses. In the case of two-body inelastic scatterings such as Lambda Lambda --> Lambda Lambda, N Xi, Sigma Sigma, on the other hand, we show that energy-independent potentials can be constructed without relying on non-relativistic approximations. We also propose a method to extract these potentials using time-dependence of general correlation functions.