The effective Lagrangian which gives a full solution of the U(1) problem is obtained from the $\frac{1}{N}$ expansion of quantum chromodynamics. This Lagrangian satisfies all the anomalous $n$-point Ward identities for arbitrary $q^2$, and includes chiral and SU(3) breaking. The relationship between the ${\eta }^{\prime }$ mass and the $\theta$ dependence of the vacuum energy in the absence of quarks and the form of the strong $\mathrm{CP}$ violation in the effective-Lagrangian framework are deduced.

• Received 7 January 1980

DOI:https://doi.org/10.1103/PhysRevD.23.473