The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.2. Effective Lagrangians 61
The representation (3.78) of the 8U(2)v group generators Vi is preferable because we will apply
the formalism to the nucleon fields, which form an isospin doublet, but this is not necessary in
general.13 Note that a particular choice of I in eq. (3.78) is not relevant, since this quantity will
not enter the explicit expressions for the effective Lagrangian, as will be shown below. It is easy
to verify that the Lie algebra (3.52), (3.75) is satisfied with fijk = Eijk if one defines V and A via
eq. (3.78). Using eq. (3.78), one can express the quantity e2�.A in the form:
cos � - ilnm sin � , (3.80)
e2�.A =
where � == lei and ni = �d�. The chiral 8U(2) x 8U(2) group is isomorph to 80(4). It can be
shown, see e.g. ref. [161], that the coordinates ( 0', *) defined via
0'
- == - cos (�),
(3.81)
fn
where fn is some constant, transform as the cartesian components of a 80(4) vector.14 Clearly,
the 80(4) group describes rotations on the four dimensional sphere
(3.82)
Therefore and as it is also obvious from eq. (3.81), the coordinates 0' and * are not independent
from each other. Note furt her that the �i and, therefore, the 7ri are pseudoscalar quantities:
(3.83)
This is because the Noether charges of the 8U(2)A are pseudoscalars, as can be seen from eq. (3.21)
in the case of the QCD Lagrangian (3.1). Furt her , the parameters € in eq. (3.32) should have the
same properties under parity transformation Pas the Noether charges Q, since otherwise the fields
cjJ' in eq. (3.34) would transform under P differently than the cjJ's. Therefore, the transformation
parameters � related to 8U(2)A must be pseudoscalar quantities.
The so-called stereographical coordinates (0', 'Fr ) proposed by Weinberg [160] are defined as
2* (3.84)
Für that particular parametrization one finds the covariant derivative D p, of the pion field (up to
some irrelevant numerical factor) [160], [75], see also [161]:
D _ -
_ � ßp,'Fr P, fn D
'
(3.85)
with
D == 1 + (�)2 2fn .
(3.86)
For the nucleon field \[I, which is an isospin or, equivalently, 8U(2)v doublet, one obtains the
covariant derivative in the form
(3.87)
13 In particular, one has to choose another representation far the V's if one considers matter fields, that form a
different isospin multiplet.
14 Since we consider the group SU(2)v x SU(2)A in some neighborhood of the unity, the mapping � .;==} ir is
unique.