The QCD Quark Propagator in Coulomb Gauge and - Institut für Physik

Chapter 3. Remarks on QCD in Coulomb Gauge 33
In this approximation lattice calculations have shown that the Wilson potential signals
confinement of two static quarks by rising linearly at large R,
V W (R) = σR , (3.91)
where σ is the well-known string tension. The linearly rising energy is stored in the
colour electric flux string connecting the two quarks. If dynamical quarks are present,
which means Det[K] ≠ 1, the creation of a pair of dynamical quarks from the vacuum
is energetically more favourable than the expansion of a string between the two static
sources. Consequently a pair of mesons is formed at separation R. In this situation the
Wilson potential describes no longer the interaction of two external quarks in the vacuum
but the potential of two mesons at separation R and can be regarded as an analogue of
a Van der Waals potential. A different quantity should be found, which does not lose its
confining property in the presence of dynamical quarks.
Let us review the A 0 A 0 correlator (3.76), which contains besides the Coulomb potential
the quantity
〈
P 0 := −g0
2 (Vρgl ) a (x)(Vρ gl ) b (y) 〉 . (3.92)
It describes the vacuum polarisation induced by the dynamical quarks, where the minus
sign signals that it corresponds to screening. As we have seen the first term in (3.76) is
confining in the absence of dynamical quarks. Allowing for those V W constitutes no longer
a confining potential, which indicates that the screening polarisation term dominates in
(3.76). The Coulomb potential might still be confining, because the long range nature of
V C could be the reason that makes the creation of a dynamical quark pair preferable to
an expansion of the flux tube [CZ02c]. The Coulomb potential is thus a candidate for
an order-parameter of confinement in the presence of dynamical quarks. Recent lattice
studies [NNST07], however, show that the Coulomb string tension does not vanish for
temperatures far beyond the deconfinement transition, which suggests that the Coulomb
potential cannot serve as an order parameter.
3.6.4 Confinement in Coulomb gauge
When Gribov wrote his famous paper [Gri78] about the problem of gauge copies and
suggested the possible restriction of the gauge field integration to the Gribov region Ω, he
already pointed out the possible relation between the restriction to Ω and the confinement
problem by examining the infrared behaviour of the ghost propagator. This is given by
the vacuum expectation value of the Faddeev-Popov operator −∇ · D:
D ab (x − y) = 〈M −1 [A]| ab
(x,y) 〉 . (3.93)