A continuum order parameter for deconfinement Page: 3 of 7
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Figure 1: Dyson-Schwinger equation for the quark self energy (QCD gap equa-
tion): D is the dressed gluon propagator; I is the dressed quark-gluon vertex;
the quark propagator S(p) = 1/[iy -p+ E(p)], E(p) = iy -p[A(p2) -1]+ B(p2),
is obtained as the solution of this nonlinear integral equation.
The phenomenological success of the approach is founded on the important
qualitative observation that the gluon vacuum polarisation diagram, tied to
the existence of the 3-gluon vertex, generates a significant enhancement of the
gluon propagator for q2 < 1 GeV2 with an integrable singularity at q2 = 0.
Without fine-tuning, this ensures quark confinement and DCSB, because the
gluon propagator is the primary element of the kernel in the DSE for the quark
self energy, represented diagrammatically in Fig. 1.
2. Dynamical Chiral Symmetry Breaking. The quark condensate is
defined via: (qq), = - f' 24tr [S(p)]. One aspect of DCSB is the statement
that, when the current-quark mass is zero, one nevertheless has (q), 0 0. In
terms of the dressed quark mass function, M(p2) = B(p2)/A(p2), this is equiv-
alent to the statement that, when the current-quark mass is zero, the quark
DSE in Fig. 1 yields M(p2) # 0, Fig. 2. DCSB is more than simply a nonzero
quark condensate, however. It is also a mass-enhancement mechanism with
observable consequences in QCD. One means of quantifying this is the ratio
Mf /mf( ), where mf(g) is the current-quark mass and Mf, the Euclidean
constituent quark mass, is the solution of p2 = Vf (p2).
flavour u/d s c b t
400 20 5 2.5 -+1 (1)
Eq. (1) indicates that the dynamical enhancement of the mass is extremely
important for the light quarks and, although it diminishes with increasing
current-quark mass, it remains significant even for the b-quark. The magnitude
of (gq), and this ratio are sensitive to details of the gluon propagator.
3. Quark Dyson-Schwinger Equation. The Matsubara formalism is
the natural framework for nonperturbative studies at finite-T. In this case the
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Roberts, C.D. A continuum order parameter for deconfinement, article, March 1, 1997; Illinois. (https://digital.library.unt.edu/ark:/67531/metadc681899/m1/3/: accessed May 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.