The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

2.3. Going to higher energies: a toy model 33
based on a set of suitably chosen grid points {qi} in the interval [0, A]. Inserting this expression
for the unknown function A(p, q) in eq. (2.96) and choosing q = qi one obtains for each i
A( .) - T(p,qi) _" A( .)1000 '2 d ,Sj(q')T(q',qi)
p, qt -m 2 _ 2 � m p, % q q 2 _ + qi P j 0 qi q '2 ZE
' .
(2.108)
The integral can easily be performed, since the q-dependence of A(p, q) is now given by the known
functions Si(q). For each interval qj � q � qj+l these functions are expressed by cubic polynomials
(splines), which satisfy Si(qj) = Oij. Their precise form can be found in the reference [144]. This
procedure leads to a system of linear equations for the expansion coefficients A(p, qi), which can
be solved by the standard methods.
It is important to note that for each p the second derivative of the function A(p, q) has to vanish
at the ends of the interval of interpolation:
d2A(p, q)
d 2 .
(2.109)
I =0
q q=A
These conditions are necessary for the spline method to work. Clearly, only one of these conditions
is satisfied, namely those for q = A. This is guaranteed by the regularization of the potential, the
details of which are described in the last section. Practically, if the function A(p, q) is smooth
enough, one can simply ignore these requirements and obtain a good approximate solution taking
a large number of grid points.
Having calculated the operator A, we are now in the position to consider observables. First, we
need the transformed Hamiltonian H'. The determination of H' according to eq. (2.70) requires
the calculation of (TI + At A) -1/2. This is done in the following manner: as already described in
section 2.3.2, we first define the function B(q, q') as given in eq. (2.80). Putting the projector TI
onto the left-hand side of this equation and taking the square of it we end up with the following
equation:
(2.110)
This can be rewritten (for the S-channel) as the following nonlinear integral equation for the
function B(q, q'):
B(q, q') = -1100 p2 dp A(p, q)A(p, q') -1 fo A
- 100 p2 dp fo A
;p dij B(ij, q)B(ij, q') (2.111)
ij2 dij A(p, q)A(p, ij) (B(ij, q') + fo A
ij, 2 dij' B(ij, ij')B(ij', q'))
Note that q,q' E [O,A]. The function B(q,q') defined in eq. (2.80) is obviously symmetrie in the
arguments q, q': B(q, q') = B(q', q). This is because the function A(p, q) is real, which can be seen
from eq. (2.69). We have solved the equation (2.111) by iteration, using the same Gauss-Legendre
quadrat ure points as discussed before and starting with B(q, q') = -(1/2) If p2 dp A(p, q)A(p, q').
We have found a very fast convergence of the iteration method in this case. The integrations
present in eq. (2.70) to determine H' are performed by standard Gauss-Legendre quadratures
and we end up with an effective potential V' (q', q) defined for q, q' � A.
2.3.4 Unitary transformation for a potential of the Malfliet-Tjon type
We will now present the results obtained using the formalism introduced in the last sections. Our
starting point is a model potential which captures the essential features of the nucleon-nucleon