# Does Ward Identity in QCD has origin of U(1) or SU(3) symmetry?

1. Nov 7, 2011

### ndung200790

Can we deduce Ward Identity in QCD from U(1) symmetry of QED?Because QCD is a theory of quarks and quarks have electric charge.So we need not deduce the Ward Identity from SU(3) symmetry,but we can be able to demontrate the Ward Identity( considering gluons)with U(1) symmetry.

2. Nov 8, 2011

### tom.stoer

The Ward identity for U(1) in QED and the Slavnov-Taylor identities for SU(3) in QCD are the counterparts to the classical Noether theorem for the path integral; they represent the gauge symmetries on the level of the effective action.

In QCD the gluons live in the adjoint rep. of SU(3), they are color-octets, but they are not charged w.r.t. U(1), i.e. carry no electric charge. The electric charge of the quarks is usually not taken into account, that means in 'pure textbook QCD' you will not find a U(1) Ward identity. The reason is that higher loops with U(1) coupling are suppressed by powers of alpha = 1/137 whereas higher loops with SU(3) coupling come with alpha_s of order one; that's why in many calculations U(1) i.e. el.-mag. interaction is studied at tree level only, that means no loops, that means no need to worry about Ward identities (the effective action in the el.-mag sector is idetical with the classical action, so to speak).

If you want to couple quarks to photons and if you want to calculate el.-mag. quantum correction as well you have to introduce U(1) gauge fields, the U(1) covariant derivative, the total gauge symmetry becomes U(1)*SU(3) where the U(1) does not act on gluons.

You have to derive the U(1) Ward identities seperately, like in QED. At one-loop U(1) and SU(3) decouple and it's like doing QED in QCD in parallel w/o any interference; but at higher loops you can have 'intersecting' U(1) and SU(3) contributions. Think about a q-qbar-loop to which you can attach both external photons and external gluons; in addition you can have both internal photons and internal gluons exchanged between q and qbar.

So I expect that at higher order you can no longer disentangle el.-mag. Ward identities and SU(3) Slavnov-Tayler identities. But to be honest I have never done such a calculation - nor have seen something like that.