The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.8. Two-nucleon potential 91
3.8 Two-nucleon potential
3.8.1 Expressions and discussion
In the last section we have applied the method of unitary transformation to the most general chiral
invariant Hamiltonian for nucleons and pions and obtained the formal expressions (3.253)-(3.255)
for the effective potential. We will now explicitly evaluate all these contributions inserting the
vertices from eqs. (3.231)-(3.238) and reshufRing the operators into normal order. For that, we
will switch to the interaction picture. The free Hamiltonian is chosen according to eqs.
(3.200).
(3.199),
The pion and nucleon field operators in the interaction picture satisfy the free-field
equations of motion:
(0 + m;) 7r 0,
(WO + :�) N o.
Correspondingly, one can decompose the pion field operators 7r via
7r +(t, x)
7r-(t, x)
7r0 (t, x)
! d3k 1 1 [e-ik'Xa (k) + eikoxat (k)]
(27r)3/2 v'2W
! d3k 1 _1_ [e-ikoXa_(k)
(27r )3/2 v'2W
+ +
+ eikoXa�(k)]
! d3k (27 r�3/2 vk- [e-ikoXao(k) + eikoXab(k)] ,
where w = ko =
and neutral pions. The cartesian components of the pion field are given by
(3.256)
(3.257)
(3.258)
Vk2 + m; and ato (a±,o) are the creation (destruction) operators of charged
7r2
= --=
/n-
The equal-time commutation relations of the pion fields and their conjugate require
[a(k), a(k')] = [at(k), at(k')] = 0 ,
v2i
[a(k), at(k')] = o3(k - k') .
For the nucleon field N one has the decomposition:
(3.259)
(3.260)
(3.261)
(3.262)
Here, v is a Pauli spinor, Eis an isospinor and Po = p2j(2m) = E. Further, bt(p, s) (bt(p, s))
is the destruction (creation) operator of a nucleon with the spin and isospin quantum numbers s
and t and moment um p. v and E are normalized via
vt(s)v(s)
Et(t)E(t)
1 ,
1.
(3.263)
The creation and destruction operators b1 (p, s) and bt(p, s) satisfy the anti-commutation relations:
(3.264)