The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3. 6. Application to chiral invariant Hamiltonians
3.6 Application to chiral invariant Hamiltonians
We first want to recall the structure of the most general chiral invariant Hamilton density for
pions and nucleons,
(3.198)
As already noted before, the nucleons are treated nonrelativistically, as it has also been done in
[73], [76]. Consequently, the purely nucleonic part of 1-l0 is nothing but the kinetic energy
V2
1-lNO = -Nt -N
(3.199)
2m
At higher orders in small momenta, which we will not treat here, relativistic corrections to the
kinetic energy (3.199) must be taken into account.
The free Hamilton density for the pion fields (in the interaction picture) is given by
1-l = !ir2 + ! (V1t' )2 + !m2 1t'2
11"0 2 2 2 (3.200)
11"
where the ,., denotes the time derivative and m1l" the pion mass. We split the interaction Hamilton
density into three parts:
(3.201)
The first piece describes self-interactions of pions and contains an even number of derivatives and
pion field operators and any number of M;-factors. The terms with two derivatives (or one M;)
have at least four pion fields. The second piece in eq. (3.201) contains terms with four or more
nucleon fields and any number of derivatives. The terms denoted by 1-l1l"N have any number of
pion fields and at least two nucleon fields and one derivative or factor M1I" ' The structure of the
effective Hamiltonian is now clarified. Later we will explicitly give the leading terms in the effective
Hamiltonian, that we will need to calculate the potential. We now show how to eliminate pions
and to obtain the effective potential for nucleons for the most general chiral invariant Hamiltonian
(3.201) .
Let us first note that because of the baryon number conservation and the absence of anti-nucleons,
the subspaces of the Fock space with different number of nucleons are automatically decoupled.
That is why we define the state I