Propagator of Chern-Simons Abelian gauge theory

I need to compute the "topologically massive photon" propagator.

I've started with :

$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{\mu}{4}\epsilon^{\mu\nu\lambda}A_\mu\partial_\nu A_\lambda$$ $$=A_\mu\underbrace{[\frac{1}{2}g^{\mu\lambda}\partial^2+\frac{\mu}{2}\epsilon^{\mu\nu\lambda}\partial_\nu]}_{(\Delta^{-1})^{\mu\lambda}}A_{\lambda}$$

So how can I invert The under braced part which will yield the topologically massive photon propagator?

Answer

The following is a rough calculation. If the operator,

$$\triangle^{\mu\lambda} = \eta^{\mu \lambda} \partial^2 + \mu \epsilon^{\mu\nu\lambda}\partial_{\nu},$$ is the one you wish to invert, then we must solve the differential equation

$$\triangle^{\mu\lambda} G = \left[\eta^{\mu \lambda} \partial^2 + \mu \epsilon^{\mu\nu\lambda}\partial_{\nu} \right] G = -i\delta^{(4)}(x-y)$$

where $(-i)$ is by convention. I think the momentum space equivalent is

$$\left[ \eta^{\mu \lambda}p^2 + i\mu \epsilon^{\mu\nu\lambda}p_{\nu}\right]\hat{G} = -i\mathrm{e}^{-ip\cdot y}.$$

To obtain $\hat{G}$ in position/physical space, you must perform an inverse fourier transform. Let me know what you get, I'm curious!