U_A(1) Problems and Gluon Topology - Anomalous Symmetry in QCD

Abstract

Many of the distinctive and subtle features of the dynamics in the $U_A(1)$ channel in QCD can be related to gluon topology, more precisely to the topological susceptibility $\chi (k^2)=i\int d^{4}x{e}^{ikx}<0|TQ(x)Q(0)|0>$, where $Q=\frac{{\text{a}}_s}{8\pi }\mathrm{tr}{G}_{\text{m}\text{n}}{\tilde{G}}^{\text{m}\text{n}}$ is the gluon topological charge density. The link is the $U_A(1)$ axial (ABJ) anomaly. In this lecture, we describe the anomalous $U_A(1)$ chiral Ward identities in a functional formalism and show how two apparently unrelated `$U_A(1)$ problems' -- the mass of the ${\eta }^{\prime }$ and the violation of the Ellis-Jaffe sum rule in polarised deep-inelastic scattering -- can be explained in terms of the gluon topological susceptibility. They are related through a $U_A(1)$ extension of the Goldberger-Treiman formula, which is derived here for QCD with both massless and massive quarks.

Comments: Lecture at 1998 Zuoz Summer School, `Hidden Symmetries and Higgs Phenomena'. 22 pages, plain TeX, 2 ps or eps figures