Minus sign on the chiral anomaly

Question

I've been going through various derivations of the chiral anomaly for using the Fujikawa method, particularly that in Srednicki's QFT textbook (see chpt. 77 in particular).

A lot of literature quotes the result as, \begin{equation} \partial_\mu J_A^\mu = - \frac{g^2}{16\pi^2}T(R)\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}^aF_{\rho\sigma}^a \end{equation} where $\text{Tr}(T_R^a T_R^b) = T(R)\delta^{ab}$. Critically, this includes a minus sign. However I have found other resources where the minus sign is absent, in particular here for equation (1).

I cannot figure out whether this sign difference is simply a result of different choices for the metric, different conventions for the normalization of the generators or some misunderstanding on my part.

Any help and insight would be appreciated, thanks.

Answer 1

There are many sign conventions but the physics should be independent of them. What is true is that for a right-handed Weyl field immersed in uniform electric and magnetic fields, the rate at which particles emerge from the vacuum, is $$ \frac{d\rho_{\rm RH}}{dt}= \frac{1}{4\pi^2} \left(\frac {e^2}{\hbar^2}\right) {\bf E}\cdot {\bf B}. $$ For Dirac particles this becomes $$ \frac{d}{dt}(\rho_{\rm RH}-\rho_{\rm LH})= \frac{1}{2\pi^2} \left(\frac {e^2}{\hbar^2}\right) {\bf E}\cdot {\bf B} $$

With suitable conventions, and with $\hbar=c=1$, we have
$$ \frac{e^2}{4\pi^2} {\bf E}\cdot {\bf B}= \frac {e^2}{32\pi^2} \epsilon_{\alpha\beta\gamma\delta} F^{\alpha\beta}F^{\gamma\delta}. $$ Other conventions for the metric or the epsilon tensor can change the sign.

For other flavours etc you need the trace.

This is the covariant physical anomaly that relates to particle non-conservation. There is another version called the consistent anomaly which describes a failure of gauge invariance when the left and right-hand components couple to different gauge fields. Even in the abelian case this differs from the above by being a factor of three smaller. The consistent anomaly is only relevent when theory is inconsistent in the absence of a higher dimensional "bulk."