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 [itex]i \epsilon_{\alpha \mu \beta \nu} k'_{\alpha}k_{\beta}g^{\mu \nu}[/itex] 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..)
Symmetry of the metric ([itex]g^{\mu \nu}=g^{\nu \mu}[/itex]) and total antisymmetry of [itex] \epsilon_{\alpha \mu \beta \nu}[/itex] imply that [itex]\epsilon_{\alpha \mu \beta \nu}g^{\mu \nu}=0[/itex] (together with all possible permutations of indices). So all terms of that form vanish.
Ah, that makes quite a lot of sense. What if, on the other hand, you have a term like [itex]i \epsilon_{\alpha \mu \beta \nu} k'_{\alpha}k_{\beta}P^{\mu} P^{\nu}[/itex]? k, k' and P are all 4-vectors. How would you be able to evaluate this?
The product [itex]P^\mu P^\nu[/itex] is symmetric under [itex]\mu\leftrightarrow\nu[/itex], so that term vanishes by symmetry as well. If there were a prime on one of the [itex]P[/itex]s, it would not. To anticipate the next logical question, we can ask how to express [itex]i \epsilon_{\alpha \mu \beta \nu} k'^{\alpha}k^{\beta}P^{\mu} P'^{\nu}[/itex]. There really isn't any simpler form to reduce it to, but we can note that [itex]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).[/itex]