The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

3.5. Bloch-Horowitz scheme and the method of unitary transformation 79
apply the standard methods of few-body physics to obtain the S-matrix. We will denote the
remaining part of the Fock space by 1'ljJ): I\J!) = 11» + 1'ljJ). Let 'rJ and A be projection operators on
the states 11» and 1'ljJ) which satisfy 'rJ 2 = 'rJ, A 2 = ,\, 'rJA = A'rJ = 0 and A + 'rJ = 1. Eq. (3.181) can
now be written in the form
( 'rJH'rJ 'rJHA ) ( 11» ) = E ( 11» )
AH'rJ AHA 1'ljJ) 1'ljJ)
One can express the state 1'ljJ) from the second line of this matrix equation as
1
1'ljJ) = E _ AHA AH'rJI \J!) .
(3.182)
(3.183)
Putting this into the first line of eq. (3.182) leads immediately to a single equation of Schrödinger
type for the state 11»
with an effective potential Veff(E) given by
(Ho + Veff(E)) 11» = EI1» (3.184)
(3.185)
This decoupling scheme has been first proposed by Bloch and Horowitz [183]. Expanding the
denominator in eq. (3.185) in powers of H[ leads to
(3.186)
In this form it is obviously identical with the result obtained from old-fashioned perturbation
theory [187, 11]. Note that the states 11» are not orthonormal:
(3.187)
Let us now take a look into the method of unitary transformation. We first introduce new states
Ix ) and Icp) , which are related to 11» and 1'ljJ) by a unitary transformation
( ) ( )
Ix )
= ut 11» (3.188)
Icp) 1'ljJ)
Then one can rewrite eq. (3.182) in an equivalent form
ut HU ( Ix ) ) = E ( I x ) )
Icp) Icp)
(3.189)
The two subspaces for I X) and I cp) can be decoupled by choosing U such that the operator ut HU
is diagonal with respect to the two subspaces. We again adopt the ansatz of Okubo [51], as in
chapter 2, in which the unitary operator U is parametrized in terms of a single operator A as
follows
(3.190)