The Schwinger-Dyson equation and dynamical mass generation in two-dimensional field theories

A truncated Schwinger-Dyson equation is used to investigate the dynamical generation of mass in (1+1)-dimensional field theories with a four-fermion interaction. An effective potential formalism is developed to clarify the nature of the approximations made to the Schwinger-Dyson kernel. The gap equation for the Gross-Neveu model is found to have solutions of the 'collapse of the wave function' type, which must be renormalised to keep the dynamically-generated mass scale of the theory finite. The corresponding flow of the coupling indicates that the model is asymptotically free if N, the number of fermion flavours, is greater than one. The large-N limit of the β-function is found to agree with the result of previous work. The Thirring model is treated in a similar fashion; a variational argument for the existence of the massive phase of the theory is given and the renormalisation group flow near the UV fixed point, g=-π/2, is obtained. The results are compared with those of bosonisation and other methods. Similarities between these phenomena and the case of quenched, planar QED are discussed. (orig.)