EFFECTIVE FIELD THEORIES FOR VECTOR PARTICLES AND ...

2.2 chiral effective field theory 13
experimental data in a quantum field theory. The aforementioned global
chiral symmetry leads to relations among different Green’s functions if
promoted to a local one. This was firstly discovered in QED with respect
to a U(1) symmetry and these relations are thus denoted shortly as Ward
identities [46, 47, 48]. Note that chiral Ward identities are still useful in
a modified form if the underlying chiral symmetry is explicitly broken,
i.e. in the physical case of non-vanishing quark masses [49].
Green’s functions can be obtained elegantly by functional derivatives
with respect to external fields in the path integral formalism. To that
end, the SU(2)-adapted Lagrangian in equation (2.10) is extended by
such external fields, which couple to the vector-, axial-vector-, scalarand
pseudoscalar quark currents as follows:
L = L 0 QCD+L ext = L 0 QCD+ ¯qγ µ (v µ + 1 3 v µ (s) +γ 5a µ )q− ¯q(s−iγ 5 p)q . (2.12)
The color-neutral external fields acting in flavor space are defined with
the help of the Pauli matrices 4 as
v µ =
s =
3
τ i
∑
i=1
3
∑
i=0
2 vµ i
, v µ (s) = τ 0v µ 0 , aµ =
τ i s i , p =
3
τ i
∑
i=1 2 aµ i
,
3
∑
i=0
τ i p i .
(2.13)
Note that the vector current possesses an isovectorial and isoscalar part. 5
The original QCD Lagrangian with finite quark masses can be obtained
by setting s = diag(m u , m d ) and v = a = p = 0. Finally, the generating
functional Z is given by
exp(iZ[v, a, s, p]) = ⟨0∣T exp[i ∫ d 4 xL ext (x)]∣0⟩ (chiral limit)
. (2.14)
This functional represents the crucial link between QCD in the lowenergy
limit and effective field theories for the strong interaction.
4 The definition is given in equation (A.5) on page 95.
5 The isoscalar vector current plays an important role in the SU(2) sector and is hence
included explicitly. The isoscalar axial-vector current has an anomaly and is hence
omitted [14, 50].