Anomalous Short-Distance Behavior of Quantum Field Theory: A Massive Thirring Model

The anomalous short-distance behavior of the products of quantum field operators, of the sort found in the Thirring model, is investigated. This is done by studying a modified Thirring model, based on a massive Fermi field $\psi \mathrm{}\left(x\right)\mathrm{}$, from several points of view in perturbation theory. This model has the same scaling short-distance behavior as the massless Thirring model. A modification is also constructed in which $\psi \mathrm{}\left(x\right)\mathrm{}$ is an isotopic-spin doublet in two dimensions. This model does not scale at short distance. Our main conclusion is that the massive Thirring model scales at short distance because the special form of the interaction yields no charge renormalization. Further, this interaction allows a kind of gauge transformation under which $S$-matrix elements and current matrix elements are unchanged but which changes the scaling dimension. Thus, no physical significance can be attributed to this anomalous dimension.