Gauge-symmetric approach to effective Lagrangians: the $\eta$' meson from QCD

We present a general scheme for extracting effective degrees of freedom from an underlying fundamental Lagrangian, through a series of well-defined transformations in the functional integral of the cut-off theory. This is done by introducing collective fields in a gauge-symmetric manner. Through appropriate gauge fixings of this symmetry one can remove long-distance degrees of freedom from the underlying theory, replacing them by the collective fields. Applying this technique to QCD, we set out to extract the long-distance dynamics in the pseudoscalar flavour singlet sector through a gauging (and subsequent gauge fixing) of the $U(1)_A$ flavour symmetry which is broken by the anomaly. By this series of exact transformations of a cut-off generating functional for QCD, we arrive at a theory describing the long-distance physics of a pseudoscalar flavour singlet meson coupled to the residual quark-gluon degrees of freedom. As examples of how known low-energy physics can be reproduced in this formulation, we rederive the Witten-Veneziano relation between the $\eta'$ mass and the topological susceptibility, now for any value of the number of colours $N_c$.