The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

2.3. Going to higher energies: a toy model 25
where A is a momenturn cut-off which separates the low from the high moment um region. Its
precise value will be given below. Apparently, 7]2 = 7], .>..2 = .>.., 7]'>" = '>"7] = 0 and 7] + .>.. = 1. Using
the 7] and .>.. projectors, the Schrödinger equation takes the form
(2.62)
Obviously, the low and the high momentum components of the state \[f are coupled. Our aim is to
derive a Hamiltonian acting only on low moment um states and which furthermore incorporates aH
the physics related to the possible bound and scattering states with initial asymptotic momenta
from the 7] states. This can be accomplished by an unitary transformation U
H -+ H' = UtHU (2.63)
such that
7]H'.>.. = .>..H'7] = 0 . (2.64)
We choose a parametrization of U given by Okubo15 [51],
where A has to satisfy the condition
-At(1 + AAt)-lj2
'>"(1 + AAt)-lj2
(2.65)
A = '>"A7] .
(2.66)
Different transformations as weH as the relations between the resulting effective Hamiltonians are
discussed in ref. [52]. One can easily check that U given by eq. (2.65) is indeed unitary provided
that the operator A satisfy the condition (2.66):
t _
(
7](I +AtA)-lj2 (1 + AtA)-lj2At ) ( 7](I +AtA)-lj2 -At(I +AAt)-lj2 )
U U - -(1 + AAt)-lj2 A '>"(1 + AAt)-lj2 A(1 + At A)-lj2 '>"(1 + AAt)-lj2
( 7](1 + AtA)-lj2(1 + AtA)(1 + AtA)-lj2 0 )
o '>"(1 + AAt)-lj2(1 + AAt)(1 + AAt)-lj2
(2.67)
It is then straightforward to recast the conditions (2.64) in a different form,
.>.. (H - [A, H] - AHA) 7] = O . (2.68)
This is a nonlinear equation for the operator A, which takes the explicit form
V(p, ij) J d3 q , A( p, � q �')V(�' q ,q �) + / d3 p , V( p,p � �')A(�' p ,q �)
J d3 q , d3 p , A( p, � q �')V(�' q ,p �')A(�' p ,q �)
(Eq - Ep) A(p, ij) (2.69)
15Note that one obviously can perform additional unitary transformations in 1)- and A-subspaces. We do not
consider such transformations here. The condition (2.66) shows that we are looking for those transformations that
map 1)- and A-subspaces into each other. We refrain from a further discussion ofthe generality of the parametrization
(2.65)