The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
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3.5. Bloch-Horowitz scheme and the method of unitary transformation 79<br />
apply the standard methods of few-body physics to obta<strong>in</strong> the S-matrix. We will denote the<br />
rema<strong>in</strong><strong>in</strong>g part of the Fock space by 1'ljJ): I\J!) = 11» + 1'ljJ). Let 'rJ and A be projection operators on<br />
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<br />
now be written <strong>in</strong> the form<br />
( 'rJH'rJ 'rJHA ) ( 11» ) = E ( 11» )<br />
AH'rJ AHA 1'ljJ) 1'ljJ)<br />
One can express the state 1'ljJ) from the second l<strong>in</strong>e of this matrix equation as<br />
1<br />
1'ljJ) = E _ AHA AH'rJI \J!) .<br />
(3.182)<br />
(3.183)<br />
Putt<strong>in</strong>g this <strong>in</strong>to the first l<strong>in</strong>e of eq. (3.182) leads immediately to a s<strong>in</strong>gle equation of Schröd<strong>in</strong>ger<br />
type for the state 11»<br />
with an effective potential Veff(E) given by<br />
(Ho + Veff(E)) 11» = EI1» (3.184)<br />
(3.185)<br />
This decoupl<strong>in</strong>g scheme has been first proposed by Bloch and Horowitz [183]. Expand<strong>in</strong>g the<br />
denom<strong>in</strong>ator <strong>in</strong> eq. (3.185) <strong>in</strong> powers of H[ leads to<br />
(3.186)<br />
In this form it is obviously identical with the result obta<strong>in</strong>ed from old-fashioned perturbation<br />
theory [187, 11]. Note that the states 11» are not orthonormal:<br />
(3.187)<br />
Let us now take a look <strong>in</strong>to the method of unitary transformation. We first <strong>in</strong>troduce new states<br />
Ix ) and Icp) , which are related to 11» and 1'ljJ) by a unitary transformation<br />
( ) ( )<br />
Ix )<br />
= ut 11» (3.188)<br />
Icp) 1'ljJ)<br />
<strong>The</strong>n one can rewrite eq. (3.182) <strong>in</strong> an equivalent form<br />
ut HU ( Ix ) ) = E ( I x ) )<br />
Icp) Icp)<br />
(3.189)<br />
<strong>The</strong> two subspaces for I X) and I cp) can be decoupled by choos<strong>in</strong>g U such that the operator ut HU<br />
is diagonal with respect to the two subspaces. We aga<strong>in</strong> adopt the ansatz of Okubo [51], as <strong>in</strong><br />
chapter 2, <strong>in</strong> which the unitary operator U is parametrized <strong>in</strong> terms of a s<strong>in</strong>gle operator A as<br />
follows<br />
(3.190)