Neutrino-neutrino to WW amplitude via Z-exchange

1. Nov 10, 2015

blankvin

1. The problem statement, all variables and given/known data

I am working through an example in Chapter 6 of Quigg's Gauge Theories. I have it mostly figured out, with the exception of how to work out the $S^{\mu}S^{\nu}$ term. All he writes is "...the term is impotent between massless spinors."

2. Relevant equations
I begin with:

What I want to know is how to obtain the factors that include $S$:

3. The attempt at a solution
I have all of the terms except those which include $S$. An explicit calculation or explanation would be extremely appreciated!

 I will show my work to point out where I am stuck.

I worked out the term involving $g^{\mu\nu}$. After the contraction of $\gamma_\nu g^{\mu\nu}$, the polarization vectors contract with the terms in square brackets to give:

$\epsilon_+^{*\alpha}\epsilon_-^{*\beta}[...] = \epsilon_+^{*} \cdot \epsilon_-^{*} (k_- - k_+)_{\nu} + \epsilon_-^{*} \cdot k_+ \epsilon_{+\nu}^* - \epsilon_+^* \cdot k_- \epsilon_{-\nu}^*$ [1]

My understanding is that the $S^{\mu}S^{\nu}$ will act on [1] above, but I do not see how to get the desired result. I thought that the contravariant $S^{\nu}$ term would contract with the covariants, but instead somehow the $k_+$ and $k_-$ in the second and third terms of [1] above are replaced by $S$. Either this is something I do not quite get, or I am being foolish.

blankvin

Last edited: Nov 10, 2015
2. Nov 10, 2015

saybrook1

Are you just looking for the amplitude?

3. Nov 10, 2015

blankvin

I know what the amplitude is. It is how to deal with the $S^{\mu}S^{\nu}$ that I do not know.

blankvin

4. Nov 10, 2015

saybrook1

Yeah sorry, the first time I saw your post I was on mobile. I will check back on it when I have time if you haven't received a response yet.

5. Nov 11, 2015

blankvin

I figured it out.

Working out the $S^{\mu}S^{\nu}$ terms lead to zero contribution to the amplitude.

This blunder will be blamed on fatigue.

blankvin