Neutrino-neutrino to WW amplitude via Z-exchange

[blankvin]

Homework Statement

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."

Homework Equations

I begin with:

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

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!

[Edit] 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

 

[saybrook1]

Are you just looking for the amplitude?

 

[blankvin]

Are you just looking for the amplitude?

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

 

[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.

 

[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