Computation of theta-term from triangle diagram

Question

The chiral $U(1)$ anomaly in QCD can be calculated exactly by one-loop Feynman diagrams, for example by the famous triangle diagram. I am currently performing the computation to get a better understanding of the QCD $\theta$-term, $\mathcal{L}\propto\theta G\tilde{G}$, where $G$ is the gluon field strength tensor and $\tilde{G}_{\mu\nu}=\frac{1}{2}\varepsilon_{\mu\nu\rho\sigma}G^{\rho\sigma}$ is its Hodge dual.

However, I am stuck in the last step of the computation. After evaluating the triangle diagram with the usual approach of Feynman parametrization, shift of the momentum integral, exploiting symmetry properties of the momentum integral, ..., I finally obtain the expression

$$g^2 Tr(T^aT^b)\varepsilon_{\mu\nu\rho\sigma}q_1^\mu q_2^\nu \varepsilon_1^\rho \varepsilon_2^\sigma,$$

where $g$ is the QCD coupling strength, $T^a$ are the generators of the Lie group, $\varepsilon_{\mu\nu\rho\sigma}$ is the epsilon tensor, $q_1$ and $q_2$ are the incoming gluon momenta, and $\varepsilon_1$ and $\varepsilon_2$ are the gluon helicity states.

The trace $Tr(T^aT^b)$ is simply $\delta^{ab}$, but I don't know how to handle the momenta and the helicity states. How do I rewrite this expression into the final result,

$$Tr(G\tilde{G}),$$

to obtain the above-mentioned $\theta$-term?

Answer 1

You now want to go back to position space. I'll do that here very schematically which gives the answer without keeping track of the overall factor.

Essentially, under the Fourier transform, $q_{i\mu} \to \partial_\mu$ and $\epsilon_{i\mu} \to A_\mu$. Then

$$ g^2 \text{Tr}(T^aT^b) \varepsilon_{\mu\nu\rho\sigma} q_1^\mu q_2^\nu \epsilon_1^\rho \epsilon_2^\sigma \to g^2 \text{Tr}(T^aT^b) \varepsilon_{\mu\nu\rho\sigma} \partial^\mu A^\nu \partial^\rho A^\sigma \sim g^2 \text{Tr}(T^aT^b) \varepsilon_{\mu\nu\rho\sigma} F^{\mu\nu} F^{\rho\sigma} \sim g^2 \text{Tr}(T^aT^b) \ast ( F \wedge \ast F ) \sim g^2 \ast \text{Tr} (F \wedge \ast F ) $$