# Question on QCD Feynman rules

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 $A_\mu(q)$, 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

dextercioby
Homework Helper
Yes, of course. "i" counts the flavor (different species of quarks), "a" counts the color (different types of gluons).

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.

Ok thank you very much. That's exactly what I was looking for!

tom.stoer