The U(1) problem

A detailed analysis of the problems associated with the conserved U(1) axial-vector current in quark-gluon models is presented. It is shown that such models involve a light isoscalar pseudoscalar boson, with a mass less than $\sqrt{3}{m}_{\pi }$. The existence of this boson would produce a strong off-shell variation in the $\eta \to 3\pi$ matrix element, thus invalidating the usual conclusions about the rate and energy dependence of this decay. Following Kogut and Susskind, it is proposed that the light Goldstone boson is actually a dipole, with positive- and negative-metric parts, which cancel in matrix elements of gluon-gauge-invariant operators but not in operators such as the U(1) current. It is shown that the masses of the observable pseudoscalar bosons and the $\eta$ decay rate are then just as they would be in a theory without the U(1) symmetry, and in fair agreement with experiment. The application of current algebra to theories with charmed quarks is briefly discussed.