The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

2.3. Going to higher energies: a toy model 27
is satisfied. The following steps appear highly plausible but do not replace a mathematically
rigorous proof. Consider the left hand side of this equation:
lim iEG'(Eq + iE) Iv = lim iEUt G(Eq + iE) U Iq')
€-+o €-+o
Ut !� iEG(Eq +iE) ((77 + >'A77) (1 + AtA)- 1/2
+ (>' - 77At>.) (1 + AAt) - 1/2 ) Icf) .
In the last line we used eq. (2.65). Let us define the operators B and C by
Then eq. (2.79) takes the form
with
B
C
( ) - 77B77 = 77 1 + At A 1/2 - 'TJ ,
>'C>' = >. (1 + AAt) - 1/2 _ >. .
(2.79)
(2.80)
(2.81)
(2.82)
(2.83)
Note that the operator A and, as a consequence, B and C depend on the interaction V and are at
least linear in V to lowest order. This is obvious from eq. (2.69). Further it can be excluded that
the matrix element (q'IFIq') is infinitely large. This is because we ass urne that the operator F can
be represented in terms of a (convergent) expansion in powers of the smooth potential V and that
is why its matrix element (plFlcf) does not contain terms like 8(p - cf). Stated differently, it is
not possible that Flq') = alq') + ... , where a is some finite constant and the ellipses correspond to
other states. Therefore, no pole cx 1 j (iE) can be generated in the application of G (Eq + iE). This
can be seen using a perturbative argument and expanding the full resolvent operator as follows:
lim itG(Eq + iE) F Icf) = lim iE (f (Go(Eq + iE)V)i) Go(Eq + iE) F Iq')
(2.84)
€-+o €-+o
i=O
Because of the above mentioned property of F it is clear that (p lFlq') does not contain a term
proportional to 8(p - cf) and therefore one can not generate a pole term 1j(iE) through the
application of Go(Eq + iE) onto FIif). That is why each single term of the series on the right hand
side of this equation equals zero. As a consequence, one obtains
lim iEG(Eq + iE) F Iq') = 0 ,
€-+o
(2.85)
and thus eq. (2.78) is indeed satisfied. We thus have shown that the scattering states \ji�+) initiated
by momenta cf from the low momentum 77-space are connected to the transformed states \ji:r(+)
via the unitary operator U. The corresponding >.-components of \ji:r(+) are strictly zero.
After considering the scattering states, we now turn our attention to the bound states. Again,
the obvious question is: Where do the transformed bound states of H reside? If the cut-off A is
sufficiently large then A goes to zero. This is a simple consequence of the fact that V is assumed
to fall off sufficiently fast for high momenta. Consequently, >.H' >. is approximately equal to >.H >.