The Colour of Quarks

It is shown that colour can only be defined on gauge invariant states. Since the ability to associate colour with constituent quarks is an integral part of the constituent quark model, this means that, if we want to extract constituent quarks from QCD, we need to dress Lagrangian quarks with gluons so that the result is gauge invariant. We further prove that gauge fixings can be used to construct such dressings. Gauge invariant dressed quark states are presented and a direct approach to the interquark potential is discussed. Some further aspects of dressing quarks are briefly discussed. In the naive quark model colour had to be introduced to preserve Fermi statistics for the quarks. As such quarks were identified as spin 1/2 particles with an SU(3) colour index. This successful, phenomenological approach has now been subsumed into Quantum Chromodynamics (QCD). Here the Lagrangian fermion fields also carry an SU(3) index which it is natural to view as colour and hence the QCD quarks are generally identified with those of the quark model. Of course it is usually understood that this identification needs some refinement — the Lagrangian fermions are not gauge invariant and the masses of the (light) Lagrangian quarks are about one hundredth of their constituent counterparts. However, we will see in this letter that the Lagrangian fermions do not even have a well defined colour. Hence the construction of constituent quarks from QCD must start with a better understanding of the colour of quarks in QCD. The structure of this letter is as follows. We first recall that the QCD colour charge is not gauge invariant. We then demonstrate the important property that its action on gauge invariant states is gauge invariant. It is then shown that gauge fixings may be used to construct dressings with whose help gauge invariant dressed quarks may be built up. We will thus show that dressed quarks with well defined colour quantum numbers can be constructed in QCD. The rest of the letter is given over to describing some applications of such dressed fields. In particular we sketch a new, simple approach to the QCD static interquark potential which is based upon dressed quarks. Let us now demonstrate our claim that the Lagrangian quarks of QCD have an illdefined colour. The colour charge, Q = QT , corresponds to the generator of rigid gauge transformations: