Covariant three-body equations in φ 3 field theory

Abstract

We derive four-dimensional relativistic three-body equations for the case of a field theory with a three-point interaction vertex. These equations describe the coupled 2 → 2, 2 → 3, and 3 → 3 processes, and provide the means of calculating the kernel of the 2 → 2 Bethe-Salpeter equation. Our equations differ from all previous formulations in two essential ways. Firstly, we have overcome the overcounting problems inherent in earlier works. Secondly, we have retained all possible two-body forces when one particle is a spectator. In this respect, we show how it is necessary to also retain certain three-body forces as these can give rise to (previously overlooked) two-body forces when used in a 2 → 3 process. The revealing of such hidden two-body forces gives rise to a further novel feature of our equations, namely, to the appearance of a number of subtraction terms. In the case of the πNN system, for example, the NN potential involves a subtraction term where two pions, exchanged between the nucleons, interact with each other through the ππ t-matrix. The necessity of an input ππ interaction is surprising and contrasts markedly with the corresponding three-dimensional description of the πNN system where no such interaction explicitly appears. This illustrates the somewhat unexpected result that the four-dimensional equations differ from the three-dimensional ones even at the operator level.