# Consistent and covariant anomalies in the abelian case

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Suppose the theory of left and right fermions, which interact with the abelian gauge field. Left and right sectors of the theory have the gauge anomaly: by defining the anomaly as the variation of the quantum effective action $\Gamma[A]$, obtained by integrating out the heavy fermions (the so-called consistent anomaly), we have
$$\partial_{\mu}J^{\mu}_{L/R} = \pm \frac{1}{48\pi^2}F_{\mu\nu}\tilde{F}^{\mu\nu}$$
I have met the statement that for the abelian case the above consistent anomaly is related to the so-called covariant anomaly,
$$\partial_{\mu}J^{\mu}_{L/R} =\pm\frac{1}{16\pi^2}F_{\mu\nu}\tilde{F}^{\mu\nu},$$
by the factor one third. See, precisely, remark [29] here. In one reference I've seen completely unclear manipulations with the Bardeen-Zumino polynomial, see section 2 here. What I don't understand is the formal difference between the covariant and consistent anomalies in the abelian case. In my opinion, for the abelian case the difference is absent, as I think, since the consistent anomaly in fact is the covariant one. So that I don't understand how to obtain the factor one third from formal thinking, and what in fact the covariant anomaly is in the abelian case. Could You explain, please?

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