The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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