Operator-Product Expansions and Anomalous Dimensions in the Thirring Model

An example of an operator-product expansion is worked out for the Thirring model. The Thirring model involves a two-dimensional zero-mass Dirac field $\ensuremath{\psi}$ interacting via the Fermi interaction. The model is scale invariant but the dimensions of local fields in the model vary with the coupling constant $\ensuremath{\lambda}$. It is shown that $\ensuremath{\psi}$ has dimension $\frac{1}{2}+(\frac{{\ensuremath{\lambda}}^{2}}{4{\ensuremath{\pi}}^{2}}){(1\ensuremath{-}\frac{{\ensuremath{\lambda}}^{2}}{4{\ensuremath{\pi}}^{2}})}^{\ensuremath{-}1}$, while the composite fields $\overline{\ensuremath{\psi}}\ensuremath{\psi}$ and $\ensuremath{\psi}{\ensuremath{\gamma}}_{5}\ensuremath{\psi}$, appropriately defined, have the dimension $(1\ensuremath{-}\frac{\ensuremath{\lambda}}{2\ensuremath{\pi}}){(1+\frac{\ensuremath{\lambda}}{2\ensuremath{\pi}})}^{\ensuremath{-}1}$.