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

Chapter 2
Chiral Symmetry Breaking in QCD
2.1 Chiral symmetry of the QCD Lagrangian
Together with gauge invariance chiral symmetry is a further important symmetry of QCD
in the limit of zero current quark mass. In this section we will discuss this feature and
its connection to the phenomenology of hadrons and strong interaction. Starting with the
general form of a Lagrangian density of a SU(3) gauge theory of quarks and gluons one
can apply the Faddeev-Popov procedure in order to show that it can always be written
as 1 [Ynd, PS, Pok]
N f
∑
L QCD = Ψ f (iγ µ D µ − m f )Ψ f − 1 2 tr(F µνF µν ) + L ghost + L g.f. . (2.1)
f=1
In this expression the variables Ψ f are the quark fields, which occur in different flavours
f=u,d,s,... . The masses m f in the Lagrangian density are the current quark masses of a
quark with flavour f. L g.f. is the gauge fixing term and L ghost contains the Faddeev-Popov
ghosts, which are required in quantised gauge theories to cancel spurious gluon degrees of
freedom. The gauge covariant derivative is given as
D µ = ∂ µ − igA µ , (2.2)
where g is the strong coupling constant, and the field strength tensor is defined in terms
of the gluon fields as
F µν = ∂A ν − ∂ ν A µ − ig[A µ , A ν ] . (2.3)
The gauge covariant derivative and the field strength tensor are elements of the Lie algebra
of SU(3) and are therefore matrices. Usually the gluon fields are expanded in terms of the
1 In this chapter we work in Minkowski space, cf. A.1 .
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