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 $\chi(k^2) = i\int d^4x~e^{ikx}$, where Q = {\a_s\over8\pi} {\rm tr} G_{\m\n} \tilde G^{\m\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 $\eta'$ 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.Comment: Lecture at 1998 Zuoz Summer School, `Hidden Symmetries and Higgs Phenomena'. 22 pages, plain TeX, 2 ps or eps figure