The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

2.3. Going to higher energies: a toy model 29
For momenta ij from the 1]-space this equation takes a simpler form
(2.91)
since the resolvent operator of the transformed Hamiltonian H' is block-diagonal as already
pointed out before, cf eq. (2.77). Now we act with the operator (1] + )'A1]) on both sides of
this equation. Because of the trivial relation (1] + )'A1]) (1] + )'A1]) = 1] + )'A1] we conclude from
eq. (2.91) for all momenta ij E 1] that
), Iw�+) ) =
q E 1] " q E Ti
)'A'l1 Iw�+) )
Projecting this equation onto the state (pi, p > A, and making use of the relation
(2.92)
(2.93)
where T(Pl ,P 2 , z) is the ordinary off-shell T-matrix, we end up with the following linear integral
equation for the operator A:
A(� ;1\ _ T(p, ij, Eq) _ ! d3 , A(p, ij') T(ij',ij,Eq)
.
p, q ) - E E q E E
q - p q - q' + Zf
(2.94)
Here the integration over q ' goes from 0 to A. Consequently, the dynamical input in this method
is the T-matrix T(ql,q""2,Eq2 ). Note that this is not a usual Lippmann-Schwinger equation, since
the position of the pole, Eq, in the integration over q' is not fixed but moves with q. It is the
second argument in A which varies, the first one is a parameter for the integral equation.
To end this section we would like to rewrite the decoupling equation (2.69), which is given in the
three-dimensional vector space of momenta, in a more convenient partial wave decomposed form.
For two particles with spin 1/2 one obtains, using the standard Ilsj) representation, the following
equation
Vi:! (p, q)
� 0
InA [00 . . .
'2 ' '2 ' SJ , sJ " sJ ,
q dq P dp All (p, q )Vil' (q ,p )Al'l'(p ,q)
A
11'
(Eq - Ep) A:f, (p, q )
(2.95)
where Vi:!(p,q ) == (lsj,plVll'sj,q) and A:f,(p,q) == (lsj,pIAll'sj,q). In the uncoupled case 1 is
conserved and equals j. In the coupled cases it takes the values 1 = j ± 1. Analogously, we can
partial wave decompose the linear equation (2.94):
sj ( ) A AS2( ') T!j( , E)
ASj ( ) - Tl!' p,q, Eq _ " In '2 d ' 11 p,q 11' q , q, q
q p _
11' p, q -
E - E � q q E - E +'
I
0 q q' Zf
2.3.2 Regularization of the decoupling equation
(2.96)
The factor (Eq - Ep) in eq. (2.69) multiplying A(p, ij) requires some caution when p == IPI and
q == l