Lattice Fermions and the Chiral Anomaly

We show in the Wilson model that the contribution of the regular mass term to the four-divergence of the axial vector current in weak coupling perturbation theory is not zero in the chiral limit and precisely is the axial anomaly. Explicit breaking of chiral symmetry in the Wilson term is not relevant for the result. The ABJ anomaly is generated by the fermion mass term also with a chirally symmetric irrelevant term. Introduction The proliferation of species known as species doubling is a major problem in formulating a lattice action for fermions. On a 3 + 1 dimensional hypercubic euclidean lattice, there are in the spectrum 15 unwanted doublers in addition to the physical fermion. The doublers can be removed from the spectrum in the continuum limit (i.e., as the lattice spacing a → 0), but only at a price. This, in essence, is the message of the no-go theorem of Nielsen and Ninomiya [1]. The price to be paid is to necessarily abandon one or more of the properties of (i) locality, (ii) hypercubic symmetry, (iii) reflection positivity, and (iv) chiral symmetry, in the lattice action for fermions [2]. The most popular model for lattice fermions, the Wilson model [3], solves the problem of doubling through an irrelevant term, the Wilson term, which breaks chiral symmetry explicitly. This specific choice, viz., breaking chiral symmetry, turns out to have a more profound reason. It is generally accepted that in order to obtain the Adler-Bell-Jackiew (ABJ) anomaly in perturbation theory in the continuum limit, the irrelevant term in the lattice fermion action, needed to remove the spurious fermion doublers, should break chiral symmetry explicitly. Indeed, the contribution of the Wilson term to the four-divergence of the axial current is treated as the driving term for the ABJ anomaly [4,5]. This has been a major hurdle for lattice formulation of chiral gauge theories, because the regulator breaks the gauge symmetry explicitly. We thus arrive at an impasse. Whereas, a lattice regulator preserving chiral symmetry is desirable for construction of a chiral gauge theory, it appears that precisely this symmetry needs to be broken explicitly in the irrelevant term of the lattice fermion action in order to produce the ABJ anomaly in the continuum limit. For a way out of this apparent impasse