The second Chern form $\epsilon^{\mu\nu\rho\sigma} F_{\mu\nu}F_{\rho\sigma}$ is topological in 4-dimensional spacetime. However, we usually only consider this term in non-Abelian gauge theory, but not in Abelian gauge theory. Is this term vanishing identically for Abelian gauge field? Somehow, I cannot see it. Or actually, we do consider it, e.g. in QED. But I never see any discussion on it.
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$\begingroup$ It couples to the neutral pion in the WZW effective action, driving its decay. $\endgroup$– Cosmas ZachosCommented Apr 3, 2020 at 10:34
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$\begingroup$ @CosmasZachos Thanks for your comment. I am not familiar with the WZW effective action, have to take a look. $\endgroup$– Wein EldCommented Apr 3, 2020 at 15:46
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$\begingroup$ You must; the gauged variety. Look for the term $\pi^0 \tilde F \cdot F /f_\pi$ in Witten's paper. Without it, the neutral pion would be stable. $\endgroup$– Cosmas ZachosCommented Apr 3, 2020 at 15:49
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1 Answer
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For Abelian gauge theory $$\epsilon^{\mu\nu\lambda\rho}F_{\mu\nu} F_{\lambda\rho}=\epsilon^{\mu\nu\lambda\rho}\partial_\mu(A_\nu F_{\lambda\rho}).$$ Thus, the term in the action coming from this term can be converted to a surface integral of $A_\nu F_{\lambda\rho}$, which vanishes since $F_{\lambda\rho}$ vanishes on the surface at infinity. But for nonabelian gauge theories with gauge coupling $g$, this term is $$\sim \epsilon^{\mu\nu\lambda\rho} \partial_\mu(A_\nu^a\partial_\lambda A_\rho^a-\frac{g}{3}f_{bca}A_\nu^a A_\lambda^b A_\rho^c)$$ which does not vanish at infinity because $A^a_\mu$ need not vanish at inifinity.
