The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

2.3. Going to higher energies: a toy model
2
o
-2
-4
-6
-8
-10
o
q
0.6
0.4
q ' [GeV]
Figure 2.3: Modification of the potential due to the regularization.
which follows from eq. (2.103) is that the difference between the regularized and unregularized
T -matrix is linear in a.
This means that for any fixed q < A - a, q > A + a and any finite E > 0 one can choose a such
that ITreg(q, q) - T(q, q)1 < E.
Similarly, the leading correction to the binding energy E of a bound state \)1 due to the regularization
of the potential can be obtained by cakulating the expectation value of (\)1 IV - V reg I \)1):
ll'
31
(2.104)
where we have used the partial wave decomposed form. Again, for any fixed q < A - a, q> A + a
and any finite E > 0 one can choose a such that IEreg - EI < E. Later we will give numerical
examples of the effects caused by that potential modification in some specific cases. We will show
numerically that the effective potential V' (q, q') is affected only within the width a for q, q' ---+ A.
There both V' and V go to zero.