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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Chapter 3<br />
<strong>The</strong> derivation of nuclear forces from<br />
chiral Lagrangians<br />
3.1 <strong>Chiral</strong> symmetry<br />
Let us start with the QCD Lagrangian for Nf quarks q<br />
LQCD<br />
1<br />
qif/Jq - 4 G�vGQ' JlV - qMq (3.1)<br />
L�CD + L�CD '<br />
where L�CD is the Lagrangian for massless quarks given by the first two terms <strong>in</strong> the equation<br />
(3.1) and L�CD denotes the quark mass term. To simplify the notation we do not show explicitly<br />
the color <strong>in</strong>dices of quark fields. Note that the Nf x Nf fiavor matrix M can be brought <strong>in</strong>to a<br />
diagonal form by an appropriate unitary transformation on quark fields <strong>in</strong> the fiavor space. <strong>The</strong><br />
covariant derivative of the quark field DJlq is def<strong>in</strong>ed via<br />
D Jlq = (OJl + igA� >'<br />
2 Q) q (3.2)<br />
where >' Q are 3 x 3 matrices <strong>in</strong> the color space. <strong>The</strong>y satisfy the SU(3) Lie algebra<br />
Furt her , 9 is the strong coupl<strong>in</strong>g constant and AJl is the gluon field. Whereas quarks belong to<br />
the fundamental representation of the SU(3)col group (color triplet), gluon fields form the adjo<strong>in</strong>t<br />
representation of that gauge group (color octet). <strong>The</strong> field strength tensor G JlV is def<strong>in</strong>ed as<br />
GQ _ !Cl AQ !Cl AQ jQ ß 'YA ß A'Y<br />
JlV - UJl v - Uv Jl - 9 Jl v '<br />
where jQ ß 'Y are the SU(3) structure constants <strong>in</strong>troduced <strong>in</strong> eq. (3.3).<br />
<strong>The</strong> QCD Lagrangian for massless quarks L�CD can be rewritten <strong>in</strong> terms of left- and righthanded<br />
quark fields qL == PLq, qR == PL q, where PR,L are the correspond<strong>in</strong>g projectors<br />
as follows:<br />
48<br />
(3.3)<br />
(3.4)<br />
(3.5)<br />
(3.6)