The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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