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Question on QCD Feynman rules

  1. Feb 26, 2013 #1
    Hi guys! A serious doubt just passed through my mind. It is probably a silly question. In writing the Feynman rules for QCD, we know that the quark-gluon vertex is given by:

    $$-ig\gamma_\mu T^a_{ij}$$

    where T is the SU(3) generator and i and j are the colors of the incoming and outgoing quarks. My question is: does the quark spinors also brigs indices? If for example I would like to write down the matrix element for a quark interacting with a certain color field [itex]A_\mu(q)[/itex], do I have to write:

    $$\bar{u}^i(k)(-ig)T^a_{ij}\gamma^\mu A_\mu^a(q) u^j(k')$$

    Thank you very much
  2. jcsd
  3. Feb 26, 2013 #2


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    Yes, of course. "i" counts the flavor (different species of quarks), "a" counts the color (different types of gluons).
  4. Feb 26, 2013 #3
    I think in Einj's notation, i and j are quark colors (SU(3) color triplet indices), not flavors.

    But yes, you need to handle spinor indices properly too. It's a sad fact that spinor indices are almost always suppressed, but ##\gamma_\mu## has some. Writing them explictly, the quark-quark-gluon vertex is

    ##-ig \gamma_\mu^{\alpha \beta} T^a_{i j}##

    where here ##\alpha## and ##\beta## are Dirac spinor indices that will get contracted with the Dirac indices of the ingoing and outgoing quarks.
  5. Feb 27, 2013 #4
    Ok thank you very much. That's exactly what I was looking for!
  6. Feb 27, 2013 #5


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    Einj, instead of reconstructing the complete term based on the vertex it's better to start with the Lagrangian which already contains this term and from which the vertex is derived.
  7. Feb 27, 2013 #6
    there is no terms involving Aμ(q) there.that just ruins everything.
  8. Feb 27, 2013 #7
    Aμ is a term is inserted as a classical color field. It the analogous of the term used in the amplitude for the scattering by an external field in QED.
  9. Feb 27, 2013 #8


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