Consistent and covariant anomalies in the abelian case

Question

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?

Comments

If the right-hand side is zero, then no problem exists between these two anomalies.