Gauge-symmetric approach to effective lagrangians: The η′ meson from QCD

Abstract

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 η ′ mass and the topological susceptibility, now for any value of the number of colours N c . The resulting effective Lagrangian contains an axial vector field, which shares the relevant features with the Veneziano ghost. This field is responsible for removing the η ′ degree of freedom from the fundamental QCD Lagrangian.

Smooth non-Abelian bosonization

“…The structure of these additional terms is similar to the Abelian case, and in both (1+1) and (3+1) dimensions [6,12]. The higher derivatives acting on U(x) can thus be interpreted as an induced "regularization" on these bosonic fields.…”

“…1 Finally, obtaining the correct prescription for smooth non-Abelian bosonization in (1+1)-dimensions is a much needed ingredient in defining suitable semi-bosonized versions of lowenergy effective Lagrangians in (3+1) dimensions. Just as one can derive an effective longdistance action for the η ′ degree of freedom in QCD [12], one can apply the techniques of smooth non-Abelian bosonization to derive an effective quark-meson Lagrangian for the SU(N f ) pseudoscalar multiplets of N f -flavour QCD. But such a phenomenological application of our formalism, partial bosonization of regularized QCD, obviously lies much beyond the scope of the present paper.…”

BRST gauge fixing and regularization

“…It arises in the 2-dimensional context of "smooth bosonization" if one follows the path of ref. [5] and in the general discussion of an effective U (1) flavour-singlet Lagrangian using collective fields [6]. In both of these examples, one essentially has to "gauge fix on the chiral anomaly", and for this reason the solution to that particular gauge-fixing problem was called "anomalous gauge fixing".…”

“…The cut-off itself is related to the Pauli-Villars regulator masses as defined in (6). The idea of ref.…”

“…In refs. [5,6,7] various valid solutions to this rather unusual gauge-fixing problems were presented, but no general procedure was outlined. We emphasize again that modifications to the naive rules of (BRST) gauge fixing are required in all of these cases, and that these modifications are inescapable if one wishes to correctly describe the quantum mechanical phenomena in those gauges.…”