Question on QCD Feynman rules

  • Thread starter Einj
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  • #1
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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
 

Answers and Replies

  • #2
dextercioby
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Yes, of course. "i" counts the flavor (different species of quarks), "a" counts the color (different types of gluons).
 
  • #3
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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.
 
  • #4
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Ok thank you very much. That's exactly what I was looking for!
 
  • #5
tom.stoer
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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.
 
  • #6
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there is no terms involving Aμ(q) there.that just ruins everything.
 
  • #7
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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.
 
  • #8
tom.stoer
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