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Weak interaction and helicity

  1. Aug 31, 2008 #1
    Hi...

    Consider a neutrino with a Dirac mass [tex] m_\nu [/tex] and the weak interaction

    [tex]{\cal{L}}=\frac{g}{2 \sqrt{2}} \sum_l[{W_{\mu}^+ \cdot \bar{\psi}_{\nu_l} \gamma^{\mu}(1-\gamma_5)\psi_l + W_{\mu}^- \cdot \bar{\psi}_{l} \gamma^{\mu}(1-\gamma_5)\psi_{\nu_l} }\right{]} + \frac{g}{4 \cos(\theta_w)}
    \sum_l Z_{\mu}[ \bar{\psi}_{\nu_l} \gamma^{\mu}(1-\gamma_5)\psi_{\nu_l} +\bar{\psi}_{l} \gamma^{\mu}(a+b\gamma_5)\psi_{l} ] [/tex]
    Why this interaction doesn't change the helicity of the neutrino? It is true?
     
  2. jcsd
  3. Aug 31, 2008 #2

    Vanadium 50

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    Do you have any elements in that interaction that have a left-handed neutrino on one side of the operator and a right-handed one on the other?
     
  4. Aug 31, 2008 #3
    For massless particle ok, because the elicity is the chirality projector [tex]\frac{1 \pm \gamma^5}{2} [/tex]... But for a massive neutrino? It's the same?
     
  5. Aug 31, 2008 #4

    Vanadium 50

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    I don't see any [tex]1+\gamma^5[/tex] there, do you? Which leads me back to my original point: do you see anything in the Lagrangian which has a left-handed neutrino on one side of the operator and a right-handed one on the other?
     
  6. Aug 31, 2008 #5
    Yes... For example [tex] Z_{\mu} \bar{\psi}_{\nu_l} \gamma^{\mu}(1-\gamma_5)\psi_{\nu_l} [/tex]. You may take [tex] \nu \rightarrow \nu +Z [/tex] with the first neutrino left-handed and the second right-handed.
    The amplitude is [tex] {\cal{M}}_{fi} \approx \bar{u}' \gamma^{\mu}(1-\gamma_5)u \epsilon_{\mu} [/tex]
    with [tex]u^t=\sqrt{\epsilon+m}(\omega_+,\frac{\vec{p}\cdot\vec{\sigma}}{\epsilon+m}\omega_+ )[/tex] and [tex]u'^t=\sqrt{\epsilon'+m}(\omega_-,\frac{\vec{p'}\cdot\vec{\sigma}}{\epsilon+m}\omega_- )[/tex] where [tex] \omega_{\pm} [/tex] are the eigenstates of the elicity...
     
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