Four-point Green functions in the Schwinger Model

Abstract

The evaluation of the 4-point Green functions in the 1+1 Schwinger model is presented both in momentum and coordinate space representations. The crucial role in our calculations play two Ward identities: i) the standard one, and ii) the chiral one. We demonstrate how the infinite set of Dyson-Schwinger equations is simplified, and is so reduced, that a given n-point Green function is expressed only through itself and lower ones. For the 4-point Green function, with two bosonic and two fermionic external `legs', a compact solution is given both in momentum and coordinate space representations. For the 4-fermion Green function a selfconsistent equation is written down in the momentum representation and a concrete solution is given in the coordinate space. This exact solution is further analyzed and we show that it contains a pole corresponding to the Schwinger boson. All detailed considerations given for various 4-point Green functions are easily generizable to higher functions.