# MCS Lagrangian and Euler-Lagrange

I'm trying to solve the Euler-Lagrange equation for the MCS Lagrangian density as given by Kharzeev in this article (Eqn. 7):

$$\mathcal{L}_{\textrm{MCS}} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-A_\mu J^{\mu} - \frac{c}{4}\theta\tilde{F}^{\mu\nu}F_{\mu\nu}$$

I already know that the first two terms yield Maxwell's equations,

$$\partial_\mu F^{\mu\nu} = \mu_0{J^\nu}$$

Now, I am trying to solve the Euler-Lagrange equation for the third term. In principle, $$\theta = \theta(\textbf{x},t)$$, but I first would like to focus on the case where $$\theta$$ is constant - for this, the equations of motion should not be altered by this term.

So far, for this third term, I've calculated

$$\mathcal{L}_3 \propto \tilde{F}^{\mu\nu}F_{\mu\nu} = 2\varepsilon^ {\mu\nu\rho\sigma}\partial_\mu A_\nu \partial_\rho A_\sigma$$

This can be plugged in into the Euler-Lagrange equation,

$$\partial_\mu\left(\frac{\partial \mathcal{L}_3}{\partial(\partial_\mu A_\nu)}\right)-\frac{\partial \mathcal{L}_3}{\partial A_\mu} = 0$$

The latter term is zero, and when I focus on the part in brackets of the first term, I find

$$\frac{\partial \mathcal{L}_3}{\partial(\partial_\alpha A_\beta)} = 2\varepsilon^{\alpha\beta\rho\sigma}\partial_\rho A_\sigma + 2\varepsilon^{\mu\nu\alpha\beta}\partial_\mu A_\nu$$ $$= 2\varepsilon^{\alpha\beta\rho\sigma}\partial_\rho A_\sigma - 2\varepsilon^{\alpha\beta \nu\mu}\partial_\mu A_\nu$$ $$= 2\varepsilon^{\alpha\beta\rho\sigma}(\partial_\rho A_\sigma - \partial_\sigma A_\rho) = 2\varepsilon^{\alpha\beta\rho\sigma}F_{\rho\sigma} = 4\tilde{F}^{\alpha\beta}$$

such that the result of the EL equation would be

$$4\partial_\alpha \tilde{F}^{\alpha\beta}=0$$

Recently, I found a document of a colleague of mine that also did similar calculations and he ended up with

$$-8\Box A^{\nu} = 0$$

For what follows in his calculations on a varying $$\theta(\textbf{x},t)$$ this term is reoccurring, and because I did not get his result I was wondering:

Is my calculation correct, or is his - and if so, why?