Bound state equation for 4 or more relativistic particles

Abstract

We apply the 3D reduction method we recently proposed for the N-particle Bethe–Salpeter equation to the 4-particle case, neglecting the radiative corrections in a first time. We find that the writing of the N⩾4 Bethe–Salpeter equation is not a straightforward task, owing to the presence of mutually unconnected interactions, which could lead to an overcounting of some diagrams in the resulting full propagator. This difficulty can be overcomed by including counterterms in the Bethe–Salpeter kernel. In the N=4 case we must include three counterterms. We show that, despite these counterterms, the 3D potential is simply the sum of six two-body potentials, in the popular approximation of instantaneous two-body kernels with positive-energy propagators. We also show that these complications can be avoided by using a modified 3D reduction method, which gives directly the 3D potential, without the need of writing the Bethe–Salpeter kernel explicitly. This modified reduction method is usable for all N. Adding the radiative corrections introduces new unconnected terms into the Bethe–Salpeter kernel even for N=2, leading again to overcounting difficulties which are also avoided when using our reduction method.