The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.1. Chiral symmetry 49
The left- and right-handed fields are the eigenstates of the chirality operator 1'5 to the eigenvalues
-1 and 1. From eq. (3.6) one immediately realizes that the Lagrangian .L:QCD has the global
symmetry
The axial U(1)A symmetry turns out to be broken at the quantum level due to the Abelian anomaly
[153], i. e. the corresponding Noether current is not conserved but receives a contribution arising
from quantum corrections. Thus, U(1)A is not a symmetry of QCD. The vector U(1)v symmetry
is connected with the baryon number conservation. Its conserved charge is the total number of
quarks minus antiquarks. The symmetry group SU(Nf)L x SU(Nf)R is called the chiral group.
The chiral group transformations of the quark fields can be expressed by
where the Nf x
Nf
(3.7)
(3.8)
matrix generators t i are fundamental representations of SU(Nf) and (Bdi,
(eR)i are the corresponding angles. Here, the index i varies from 1 to NJ -1. Note that one can
skip the indices L and R on the quark fields in eq. (3.8) without changing the result because of
the PR , L in the exponentials. Alternatively, one can parametrize the chiral symmetry group in
terms of SU(Nf)V x SU(Nf)A as follows:
q ---- -+ q' = exp (iOv . t)q, (3.9)
Let us now calculate Noether currents corresponding to an arbitrary symmetry of the Lagrangian
density. For that consider an infinitesimal group transformation
where
(3.10)
(3.11)
Here the constants cabe are the structure constants of the group and ra are the abstract group
generators. 1 Since we consider a global transformation, fa does not depend on space-time coordinates.
The field