Understanding gluons and SU(3)

[franoisbelfor]

How are gluons related to the generators of SU(3),
the Gell-Mann matrices?

I do not understand how the structure constants f and d
describe how, for example, a red-antigreen gluon transforms
into a red-antiblue and a blue-antigreen one.
Do the f or the d factors describe the three-gluon vertices?
Or both? And where do 4-gluon vertices come in?

Thank you in advance for any help!

François

 

[hamster143]

QCD lagrangian contains a term that is proportional to F^a_{\mu\nu} F^{a\mu\nu} where F^a_{\mu\nu} = \delta_{\mu}A^a_{\nu} - \delta_{\nu} A^a_{\mu} - gf^{abc} A^b_{\mu} A^c_{\nu}, A^a_{\mu} are gluon vector fields ('a' goes from 1 to 3), and f's are structure constants.

You can expand this formula and see some terms that contain one 'f' and three 'A's (three-gluon vertices), and a term that contains two 'f's and four 'A's (four-gluon vertex).

Does that help?

p.s. it seems that LaTeX generator is down.

 

[franoisbelfor]

QCD lagrangian contains a term that is proportional to F^a_{\mu\nu} F^{a\mu\nu} where F^a_{\mu\nu} = \delta_{\mu}A^a_{\nu} - \delta_{\nu} A^a_{\mu} - gf^{abc} A^b_{\mu} A^c_{\nu}, A^a_{\mu} are gluon vector fields ('a' goes from 1 to 3), and f's are structure constants.

You can expand this formula and see some terms that contain one 'f' and three 'A's (three-gluon vertices), and a term that contains two 'f's and four 'A's (four-gluon vertex).

Ah, the f constants describe both 3 and 4 gluon vertices. The three rows and columns
of the generators are rgb and anti-r, anti-g and anti-b, am I right?
And the first generator (see http://en.wikipedia.org/wiki/Special_unitary_group#SU.283.29")
"is" the red-antigreen gluon?

François

 

[jdstokes]

In SU(3) Yang-Mills theory, the associated field lives in the SU(3) Lie algebra which is an 8-dimensional vector space spanned by (eg) the 8 Gell-Mann matrices. The general field can thus be written as a linear combination of the Gell-Mann matrices. You can think of the gluons as the linear combination coefficients, or better as the linear combination coefficients multiplied by the corresponding generator.

This is all just semantics of course, and ultimately you should think of the gluons not individually but as part of a unified 8-dimensional entity.