  There exist both continuum and lattice regularizations of gauge theories with
fermions which preserve chiral U(1) invariance ("fermion number"). Such
regularizations necessarily break gauge invariance but, in a covariant gauge,
one recovers gauge invariance to all orders in perturbation theory by including
suitable counterterms.
  At the non-perturbative level, an apparent conflict then arises between the
chiral U(1) symmetry of the regularized theory and the existence of 't Hooft
vertices in the renormalized theory. The only possible resolution of the
paradox is that the chiral U(1) symmetry is broken spontaneously in the
enlarged Hilbert space of the covariantly gauge-fixed theory. The corresponding
Goldstone boson is unphysical. The theory must therefore be defined by
introducing a small fermion-mass term that breaks explicitly the chiral U(1)
invariance, and is sent to zero after the infinite-volume limit has been taken.
Using this careful definition (and a lattice regularization) for the
calculation of correlation functions in the one-instanton sector, we show that
the 't Hooft vertices are recovered as expected.
