In case of the gauge-fixed Faddeev-Popov Lagrangian:
$$ \mathcal{L}=-\frac{1}{4}F_{\mu\nu}\,^{a}F^{\mu\nu a}+\bar{\psi}\left(i\gamma^{\mu}D_{\mu}-m\right)\psi-\frac{\xi}{2}B^{a}B^{a}+B^{a}\partial^{\mu}A_{\mu}\,^{a}+\bar{c}^{a}\left(-\partial^{\mu}D_{\mu}\,^{ac}\right)c^{c} $$

(for example in Peskin and Schrder equation 16.44)

If you expand the last term (for the ghost fields) you get:
$$ \bar{c}^{a}\left(-\partial^{\mu}D_{\mu}\,^{ac}\right)c^{c} = -\bar{c}^{a}\partial^{2}c^{a}-gf^{abc}\bar{c}^{a}\left(\partial^{\mu}A_{\mu}\,^{b}\right)c^{c}-gf^{abc}\bar{c}^{a}A_{\mu}\,^{b}\partial^{\mu}c^{c} $$

And so, the Lagrangian has a term proportional to the second derivative of $c^a$.

In this case, how does one find the classical equations of motion for the various ghost fields and their adjoints?

I found the following equations of motion so far:
$$ D_{\beta}\,^{dc}F^{\beta\sigma}\,^{c}=-g\bar{\psi}\gamma^{\sigma}t^{d}\psi+\partial^{\sigma}B^{d}+gf^{dac}\left(\partial^{\sigma}\bar{c}^{a}\right)c^{c} = 0 $$
$$ \partial_{\sigma}\bar{\psi}_{\alpha,\, i}i\gamma^{\sigma}-\sum_{\beta}\sum_{j}\bar{\psi}_{\beta,\, j}\left(gA_{\mu}\,^{a}\gamma^{\mu}\,_{ji}t^{a}\,_{\beta\alpha}-m\delta_{ji}\delta_{\beta\alpha}\right)=0 $$
$$ \left(i\gamma^{\mu}D_{\mu}-m\right)\psi=0 $$
$$ B^{b}=\frac{1}{\xi}\partial^{\mu}A_{\mu}\,^{b} $$
$$ \partial^{\mu}\left(D_{\mu}\,^{dc}c^{c}\right)=0 $$
$$ f^{abd}\left(\partial_{\sigma}\bar{c}^{a}\right)A^{\sigma}\,^{b}=0 $$

But it is the last equation that I suspect is false (I saw the equation $ D_\mu\,^{ad} \partial^\mu \bar{c}^d = 0 $ in some exercise sheet and I also saw the equation $D^\mu\,^{ad}\partial_\mu B^d = igf^{dbc}(\partial^\mu\bar{c}^b)D_\mu\,^{dc} c^c$ which I don't understand how they were derived.)

Any help about what are the proper equations of motions and how to get them would be much appreciated.

This post imported from StackExchange Physics at 2014-07-03 18:16 (UCT), posted by SE-user PPR