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There exist both continuum and lattice regularizations of gauge theories withfermions which preserve chiral U(1) invariance ("fermion number"). Suchregularizations necessarily break gauge invariance but, in a covariant gauge,one recovers gauge invariance to all orders in perturbation theory by includingsuitable counterterms.  At the non-perturbative level, an apparent conflict then arises between thechiral U(1) symmetry of the regularized theory and the existence of 't Hooftvertices in the renormalized theory. The only possible resolution of theparadox is that the chiral U(1) symmetry is broken spontaneously in theenlarged Hilbert space of the covariantly gauge-fixed theory. The correspondingGoldstone boson is unphysical. The theory must therefore be defined byintroducing 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 thecalculation of correlation functions in the one-instanton sector, we show thatthe 't Hooft vertices are recovered as expected.Comment: Improvements in presentation; no change in conclusions. Version to  appear in Phys. Rev.

Fermion-number violation in regularizations that preserve fermion-number symmetry