U(1) problem: Current algebra and the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>θ</mml:mi></mml:math>vacuum

Abstract

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 ${\ensuremath{\eta}}^{\ensuremath{'}}$ mass and the $\ensuremath{\theta}$ 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.