The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

26 2. Low-momentum effective theories for two nucleons
Here we denoted the momenta from the TJ- and ),-spaces by if and p, respectively, and Eq, Ep stand
for the corresponding kinetic energies. Once A and thus U have been determined, the effective
Hamiltonian in the TJ-space takes the form
This interaction is by its very construction energy-independent and hermitian [51], [53] .
(2.70)
After this block-diagonalization, the Schrödinger equation separates into two uncoupled equations
in the respective subspaces. According to eq. (2.63) the connection between the eigenstates of H
and H' is
w' = utw ,
(2.71)
so that the transformed problem separates into
TJH'TJW'
),H' ),w'
ETJW' ,
E),w' .
Let us now define the potential after performing the unitary transformation via
v' == H' -Ho .
(2.72)
(2.73)
(2.74)
Note that in this definition we have used the free Hamiltonian Ho, which is not unitarily transformed.16
That is why
v' i= U t VU . (2.75)
In what follows, we will nevertheless refer, for brevity, to V' as to a unitarily transformed potential.
The inequality (2.75) should always be kept in mind.
Let us now consider the transformed scattering states corresponding to an asymptotic momentum
if:
IWq'
� (+) ) = lim iEG'(Eq + iE) lif) ,
(2.76)
f--+O
where Iq) is a momentum eigenstate, Ho Iq) = Eq Iq) and G'(z) is the transformed resolvent
operator
o
(z - ),H' ),)-1 )
. (2.77)
Obviously, z should not be in the spectrum of H. Note that the resolvent operator of the transformed
Hamiltonian H' is block-diagonal. As a consequence, the scattering states IW:Z(+)) lie in
the TJ- (),-) space, if the corresponding asymptotic momentum if belongs to the TJ- (),-) space,
respectively. Immediately, the crucial question arises about the connection between the scattering
states Iw�+)) == limHo iEG(Eq + iE) lif) and Iw:z(+)). It is not obvious that eq. (2.71) holds also
for these scattering states, which are defined through specific boundary conditions, see eq. (2.76).
Note that if the transformed and original scattering states are not connected via an unitary operator,
then the S-matrix would change after performing the unitary transformation. We sketch
now a proof, showing that eq. (2.71) is indeed valid in this case and that the relation
(2.78)
16The unitary operator U does, in general, not commute with the free Hamiltonian Ho . In this sense, equation
(2.74) corresponds to a non-trivial definition of the potential V' . We will comment more on that later on.