Operator regularization and the chiral anomaly

Abstract

Operator regularization is used to compute anomalies in axial gauge theories to one- and two-loop order. To one-loop order, we consider the decay of a U(1) axial current into two, three, and four SU(N) vector currents. This is done first by directly computing the relevant Green functions, and then by a functional approach. The one-loop anomaly is found to be proportional to trF*F. We note that the quadrilinear term in this expression is zero. The Schwinger expansion, employed in operator regularization to generate Green functions, is then used to derive the Schwinger–de Witt WKB (Wentzel–Kramers–Brillouin) expansion of the heat kernel, recovering the known diagonal elements and also giving the previously unknown off-diagonal elements. For the case of a constant background electromagnetic field, these off-diagonal terms can also be obtained from an exact expression for the heat kernel given by Schwinger. We then use this result, in conjunction with the functional technique originally introduced in one-loop calculations, to compute exactly the amplitude for the decay of the U(1) axial current into both U(1) and O(3) constant gauge fields to two-loop order. With the appropriate choice of mass scale parameters, a vanishing result is obtained, in agreement with the Adler–Bardeen result.

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