The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

Appendix E
Anti-symmetrization of the contact
interactions
In this appendix we will entirely concentrate on contact interactions and their contributions to
the efIective N N potential. Let us start with the simplest case of the contact interactions without
derivatives. Only two such terms enter the Hamiltonian (3.234). In principle, one can construct
two additional interactions that satisfy all required symmetry principles by insert ions of the isospin
matrices T'S. Weinberg pointed out [73] that only two of these four contact terms are independent.
The remaining ones can be eliminated from the Hamiltonian applying the Fierz transformation,
see, for instance, [132]. Another way to see that is to perform anti-symmetrization ofthe potential.
We will now explain this in detail following the logic of the reference [108].
The contribution of all four contact interactions to the efIective potential is
Vo = Ü1 + Ü2 0\ .
The anti-symmetrized potential v� is given by
where the exchange operator A is defined via
eh + Ü3 Tl ' T2 + Ü4 (0\ . (2) (Tl ' T2) .
Vo - A[vo]
2
and where m i (mD denotes symbolically spin, isospin and momentum quantum numbers of the
nucleon. According to [108], in the two nucleon center-of-mass system the exchange operator A involves
a left-multiplication with the isospin exchange operator (1 + Tl ' T2) /2, a left-multiplication
with the spin exchange operator (1 + 51 . (2)/2 and a substitution p' -+ _p'.1 Using these rules
one obtains the following identities:
A[l] 1 (1 + (51 . 52) + (Tl ' T2) + (51 ' (2)(T1 ' T2)) ,
1 (3 - (51 . 52) + 3(T1 ' T2) - (51 ' (2)(T1 ' T2)) ,
1 (3 + 3(51 . 52) - (Tl ' T2) - (51 ' (2)(T1 ' T2)) ,
1 (9 - 3(51 . 52) - 3(T1 . T2) + (51 ' (2) (T1 . T2)) .
I p and pi are the nucleon initial and final ems momenta, respeetively.
181
(E.1)
(E.2)
(E.3)
(E.4)