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Neutrino-neutrino to WW amplitude via Z-exchange

  1. Nov 10, 2015 #1
    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 [itex]S^{\mu}S^{\nu}[/itex] term. All he writes is "...the term is impotent between massless spinors."

    2. Relevant equations
    I begin with:

    nunu_WW_amplitude.png

    What I want to know is how to obtain the factors that include [itex]S[/itex]:

    result.png

    3. The attempt at a solution
    I have all of the terms except those which include [itex]S[/itex]. 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 [itex]g^{\mu\nu}[/itex]. After the contraction of [itex]\gamma_\nu g^{\mu\nu}[/itex], the polarization vectors contract with the terms in square brackets to give:

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

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


    blankvin
     
    Last edited: Nov 10, 2015
  2. jcsd
  3. Nov 10, 2015 #2
    Are you just looking for the amplitude?
     
  4. Nov 10, 2015 #3
    I know what the amplitude is. It is how to deal with the [itex]S^{\mu}S^{\nu}[/itex] that I do not know.


    blankvin
     
  5. Nov 10, 2015 #4
    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.
     
  6. Nov 11, 2015 #5
    I figured it out.

    Working out the [itex]S^{\mu}S^{\nu}[/itex] terms lead to zero contribution to the amplitude.

    This blunder will be blamed on fatigue.


    blankvin
     
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