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