The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

28 2. Low-momentum efIective theories for two nucleons
which contains only a small portion of V and, thus, can not support bound states at all. The
transformed bound states have therefore to be solutions of eq. (2.72) and the 'x-components of
the transformed bound states have to be exactly zero. It is not known to us whether this remains
true choosing sm aller and sm aller cut-off values. For the physically reasonable choices used in
next seetions this turns out to be true. Trivially there can not be solutions to the same bound
state energy for both equations (2.72) and (2.73), since this would contradict the non-degeneracy
assumption for the bound states of H.
One can argue just in the same way to see the validity of the equation (2.71) for the states
Iwq(-)) == limHo ifG(E q - if) Iq) and Iwif(-)) == limt----to ifG'(E q - if) Iq). Therefore, the Smatrices
in the original and transformed problem are the same:
(2.86)
As a consequence, the on-shell T-matrix element evaluated by means of the Lippman-Schwinger
(LS) equation
T' = V' + V'GoT' , (2.87)
yields exactly the same matrix elements as gained via the LS equation of the original problem
T = V + VGoT . (2.88)
Note that in eq. (2.88) one integrates over the whole (infinite) moment um range whereas in
eq. (2.87) only momenta up to the cut-off A are involved.
An important observation is that after performing the unitary transformation most local operators
become non-Iocal. For an arbitrary local operator O(ih,ih) = O(pd 8(Pl -P2) one obtains in the
transformed space
(2.89)
which, in general, contains the usual 8-function part but in addition also a strong non-Iocal piece.
These non-Iocalities, which are easy to handle, are nothing but the trace of the high momentum
components from the full space. Note, however, that the momentum operator P of a particle
and, as a consequence, the free particle Hamiltonian Ho are required to be unchanged, in order
to define free asymptotic states. Certainly, one could also unitarily transform the operators Ho
and p. This would not yield any new aspects since the original and the unitarily transformed
problems would be trivially identical, the unitary transformation only leads to change of basis.
We shall not consider this trivial case any furt her. Physical observables such as the deuteron
binding energy and phase shifts are identical in the original and transformed problems as shown
before. We remark that for instance the average momentum in the deuteron will change in the
transformed problem because the moment um operator is not unitarily transformed. This is not
a problem since it is not direct observable. From the other side, all current operators have to be
unitarily transformed and this guarantees the equivalence of all observables.
Let us now discuss an alternative way of determining the operator A, as exhibited in ref. [52J.
That method uses the knowledge of the scattering states to the original Hamiltonian H. Let
us now look in some more detail at this formalism. As it was already pointed out ab ove , the
connection between the scattering states in the original and transformed problem is given by
u Iwif(+))
(2.90)
((1J + ,XA1J) (1 +AtA)- 1 /2 1J+ ('x- 1JAt,X) (1+AAt)-1 /2 ,X) Iwif(+))