EFFECTIVE FIELD THEORIES FOR VECTOR PARTICLES AND ...

2.1 quantum chromodynamics 11
There exists an accidental global symmetry due to the numerical
values of the so-called current quark masses, see table 2.1. One can
divide the six quark flavors into the „light“ quarks up, down, strange,
and into the „heavy“ quarks charm, bottom, and top. The splitting scale
of about 1 GeV is justified by the mass of the lightest 2 strongly interacting
particles, i.e. the rho meson with a mass of 770 MeV, and by the scale
of spontaneous symmetry breaking 4πF ≈ 1170 MeV. This leads to a
Lagrangian approximately describing low-energy processes of the strong
interaction
L 0 QCD =
∑ ¯q f iγ µ D µ q f − 1
f =u,d,s
2 Tr(G µνG µν ) , (2.8)
where the light quark masses are set to zero and the heavy quarks are
omitted in comparison to equation (2.1). This approximation is called
chiral limit. By introducing the projection operators,
P R = 1 2 (1 + γ 5) and P L = 1 2 (1 − γ 5) , (2.9)
which project the quark fields q f onto their right-handed q R f
= P R q f and
left-handed q L f
= P L q f components, respectively, the Lagrangian can be
rewritten as
L 0 QCD =
∑ ( ¯q R f iγµ D µ q R f
+ ¯q L f iγµ D µ q L f ) − 1
f =u,d,s
2 Tr(G µνG µν ) . (2.10)
Since the covariant derivative D µ is flavor independent, it follows that
L 0 QCD is invariant under a transformation associated with a U(3) R ×
U(3) L symmetry group in flavor space, which is isomorphic to the symmetry
group SU(3) R × SU(3) L × U(1) V × U(1) A . Here and henceforth,
the basis 3 V = R + L and A = R − L with well-defined parity +1 and −1,
respectively, is used. Naïvely, one would expect 2 × 8 + 2 = 18 conserved
currents according to Noether’s theorem [37]. However, an anomaly in
QCD due to quantum corrections breaks the conservation of the singlet
axial-vector current associated with U(1) A [38, 39, 40], so that QCD in
the chiral limit possesses the symmetry group
SU(3) R × SU(3) L × U(1) V , (2.11)
2 The pseudoscalar pions and kaons are regarded as pseudo-Goldstone bosons and are
therefore treated specially, see section 2.2.
3 Note that this notation is rather symbolic, in some cases a prefactor 1/2 or 1/ √ 2 is
inserted by convention.