Understanding Quark-Gluon Vertex in QCD Feynman Rules

[Einj]

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]

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

 

[The_Duck]

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.

 

[Einj]

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

 

[tom.stoer]

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.

 

[andrien]

there is no terms involving A_μ(q) there.that just ruins everything.

 

[Einj]

A_μ 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.

 

[tom.stoer]

Have a look at http://en.wikipedia.org/wiki/Quantum_chromodynamics#Lagrangian

What is missing? The vertices follow directly from the Lagrangian, especially from $\bar{\psi}\,i\gamma^\mu\,D_\mu\,\psi$