The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.2. Effective Lagrangians 65
As an illustration, let us now construct the leading Lagrangian for pions, i. e. the Lagrangian with
the minimum number of derivatives. We will first work out all chiral symmetrie terms and skip,
for simplicity, the external sources. It is clear from the above discussion that the most general
chiral invariant Lagrangian can be built up from the covariant derivatives of the pion fields. In the
matrix notation used above, such a covariant derivative of the first order is given by eq. (3.102).
One can easily obtain analogous expressions for higher derivatives. For instance, the quantity
8JLuv clearly does not transform in a covariant way as the ufl in eq. (3.105):
(3.114)
To form a covariant object we need again the chiral connection r fl defined in eq. (3.109): (3.115)
Let us now establish the properties of the building blocks (ufl' UJLV, . . . ) und er Lorentz and parity
transformations. To simplify the notation, we will work with the dimensionsless field c/J instead of
7r, which is defined as18
(3.116)
where T is the Pauli matrix. Clearly, observables do not depend on a partieular parametrization
of the quantity u. This follows from Haag's theorem [162], [67]. Using the explicit parametrization
eq. (3.116) of u, we can express u8JLu t and U t 8flu in terms of the pion field c/J:
(3.117)
where