Casimir's Trick/Evaluating Cross Sections

[Spriteling]

Hi all.

I'm working on a project at the moment, and I've run into some questions regarding the 4-d Levi Civita tensor contracted with the metric tensor.

I'm working on finding the cross-section for neutrino-proton scattering. While ccontracting the hadronic and leptonic tensors, I end up with a term like i \epsilon_{\alpha \mu \beta \nu} k'_{\alpha}k_{\beta}g^{\mu \nu} but I am uncertain as to how to evaluate this term. I've looked all through Griffiths and Halzen and Marten and I've googled but I can't find a definitive answer. Can anyone help?
(I'm sorry if this is in the wrong section; I wasn't sure if it should go here, as it is related to HEP, or int he homework help thing, which it isn't really, but it is a question..)

 

[fzero]

Symmetry of the metric (g^{\mu \nu}=g^{\nu \mu}) and total antisymmetry of \epsilon_{\alpha \mu \beta \nu} imply that \epsilon_{\alpha \mu \beta \nu}g^{\mu \nu}=0 (together with all possible permutations of indices). So all terms of that form vanish.

 

[Spriteling]

Ah, that makes quite a lot of sense.

What if, on the other hand, you have a term like i \epsilon_{\alpha \mu \beta \nu} k'_{\alpha}k_{\beta}P^{\mu} P^{\nu}? k, k' and P are all 4-vectors. How would you be able to evaluate this?

 

[fzero]

The product P^\mu P^\nu is symmetric under \mu\leftrightarrow\nu, so that term vanishes by symmetry as well. If there were a prime on one of the Ps, it would not.

To anticipate the next logical question, we can ask how to express i \epsilon_{\alpha \mu \beta \nu} k'^{\alpha}k^{\beta}P^{\mu} P'^{\nu}. There really isn't any simpler form to reduce it to, but we can note that

i \epsilon_{\alpha \mu \beta \nu} k'^{\alpha}k^{\beta}P^{\mu} P'^{\nu} = i\left( k'^0k^1P^2 P'^3 \pm \text{permutations} \right).