Quenched massive Schwinger model in the infrared approximation

In the quenched, continuum version of the Schwinger model, a gauge-invariant summation over soft photons exchanged across a fermion loop is performed for the order parameter 〈ψ¯ ψ〉 and the correlation function 〈ψ¯ ψ(x) ψ¯ ψ(y)〉 with a fermion mass m≠0 and photon momentum k<m. The limit m→0 leads to a finite, nonzero value for 〈ψ¯ ψ〉; in the correlation function the leading terms, proportional to 〈ψ¯ ψ${〉}^2$, cancel between the two types of diagrams leaving free massless propagation.