Gauge invariance of gg->gg scattering amplitude?

I'm trying to calculate the spin and color averaged gg->gg cross section, and I am stumbling upon gauge invariance:

Must the amplitude not be invariant under replacements $\epsilon_i \to \epsilon_i + \kappa_i p_i$ where $p_i\cdot\epsilon_i=0$ and $\kappa_i$ is arbitrary? For me it is not :/

I have summed the 4-gluon diagram and s,t,u-channel diagrams with two 3-gluon vertices each.

When I square the amplitude and carry out the spin averaged polarization sum in axial gauge, the extra "n"-vector does not vanish.

Am I missing something trivial? Can someone point me to some resource? I do not want to calculate the amplitude with spinor helicity formalism. I am thankful for any hints because I've been sitting on this problem with a collegue for quite a while.

Thanks, Tobias

• It should just be invariant under $\epsilon_\mu \rightarrow \epsilon_\mu + p_\mu$ with no additional factors. – DJBunk Jul 17 '12 at 14:47
• I think it must be with an arbitrary prefactor, because you can compute $p_\mu \mathcal{A}^\mu$ and this must vanish (with an arbitrary prefactor). – Tobias Jul 17 '12 at 15:23
• OK, first just to clarify, you have to take $\epsilon(p)_\mu \rightarrow \epsilon(p)_\mu +p_\mu$, so its the $p_\mu$ associated with that same $\epsilon(p)_\mu$. Second, I wasn't sure if you meant to have a contraction $\kappa_\mu p^\mu$ in your above expression so that $\epsilon(p)_\mu \rightarrow \epsilon(p)_\mu +\kappa^\mu p_\mu$ which doesn't make any sense index-wise. If you want to take $\epsilon(p)_\mu \rightarrow \epsilon(p)_\mu +\kappa p_\mu$ for arbitrary $\kappa$ thats fine. – DJBunk Jul 17 '12 at 15:42
• If you still have an error, I would take a guess that you might not be assigning the momentum properly at the vertices - momentum dependent Feynman vertices are tricky since you have to be careful which way the momentum is going in the diagram (in or out). – DJBunk Jul 17 '12 at 15:44
• Ok, I figured it out :) I did not simplify enough. I had to apply momentum conservation in FORM several times, replacing each momentum vector by momentum conservation and also mandelstam variable replacements. The gauge check with $\epsilon\to\epsilon+p$ doesn't work yet, but I think I need some $\mathrm{SU}(3)$ algebra simplifications for that. – Tobias Jul 17 '12 at 18:14