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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50 3. <strong>The</strong> derivation of nuc1ear forces from chiral Lagrangians<br />
<strong>The</strong> derivative of the Noether current (3.13) can be expressed as<br />
öJlJ� = -i ((öJl ö(�;(Pi)) fia(cP) + Ö(�;cPi) öJlfi(cP))<br />
-i (��ft(cP) + Ö(�;cPi) öJlfi(cP))<br />
where we have used the Euler-Lagrange equation<br />
(3.15)<br />
(3.16)<br />
Compar<strong>in</strong>g equations (3.14) and (3.15) we conclude that the Noether current is conserved if the<br />
Lagrangian density is <strong>in</strong>variant under symmetry transformations, s<strong>in</strong>ce then:<br />
(3.17)<br />
One can now calculate the Noether currents correspond<strong>in</strong>g to the transformations (3.8) and (3.9).<br />
<strong>The</strong> functions fia( q) are<br />
SU(Nf)v :<br />
SU(Nf)A :<br />
SU(Nf)L:<br />
SU(Nf)R:<br />
fi(q) = tijqj ,<br />
fia(q) = 'Y5tijqj ,<br />
fia(q) = tijPL qj ,<br />
fia(q) = tijPRqj .<br />
(3.18)<br />
(3.19)<br />
In all these cases the quark fields form the basis of the fundamental representations of the correspond<strong>in</strong>g<br />
symmetry groups.2 For the symmetry transformations (3.8) and (3.9) we f<strong>in</strong>d, us<strong>in</strong>g<br />
eq. (3.13)<br />
and<br />
respectively.<br />
JaJl A - = q'Y5'Y Jlta q ,<br />
<strong>The</strong> charges associated to the Noether currents can be found via<br />
Note that the charge is a constant of motion<br />
s<strong>in</strong>ce<br />
dQ<br />
dt =0,<br />
1 d3 X öJl J Jl - 1 d3 X V . J<br />
-I dS ·J<br />
O.<br />
(3.20)<br />
(3.21)<br />
(3.22)<br />
(3.23)<br />
(3.24)<br />
2 A representation of the algebra is simply its l<strong>in</strong>ear realization, i. e. the functions li (rp) are l<strong>in</strong>ear <strong>in</strong> the fields<br />
rp.