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