We generalize the large $N_c$ Di Vecchia–Veneziano-Witten (VVW) effective chiral Lagrangian to the case of finite $N_c$ by constructing the anomalous effective Lagrangian for QCD. The latter is similar to its supersymmetric counterpart and has a holomorphic structure. The VVW construction is then recovered, along with ${1/N}_c$ corrections, after integrating out the heavy “glueball” fields. A new mass formula for the ${\eta }^{\prime }$ meson in terms of QCD condensates is obtained. The picture of $\theta$ dependence in QCD for finite $N_c$ is more complicated than that predicted by the large $N_c$ approach.

• Received 20 March 1998

DOI:https://doi.org/10.1103/PhysRevLett.81.4071