The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

32 2. Low-momentum effective theories for two nucleons
2.3.3 Numerical realization of the projection formalism
Let us now consider in more detail the numerical realization of the projection formalism. To
simplify the notation we will omit all indices l, s and j and consider the uncoupled case. The
generalization to a coupled case is straightforward. The nonlinear equation (2.95) can only be
solved numerically. We do this by iteration starting with the value of A(p, q) given by17
p,q V(p, q)
E - E .
A( ) =
q p
(2.105)
Certainly, the iteration method does not always lead to a convergent solution. We have found
that one can achieve a better convergence if one slightly modifies the usual iteration method:
after each four iterations we perform an averaging over the values of the operator A with suitably
chosen weight factors. These weight factors are found numerically for each particular value of
the cut-off A requiring that the number of iterations, needed to calculate the function A(p, q)
with so me given precision, is minimal. This scheme allows to obtain the operator A(p, q) even in
those cases, when the usual iteration method fails to provide the convergence. Of course, also this
algorithm does not work in all cases.
Alternatively, we have derived the operator A(p, q) from the linear equation (2.96). For that
we have first calculated the usual half-shell T-matrix by solving the corresponding LS-equation
(2.100), which can be rewritten for the S-channel as
(2.106)
where we have omitted the indices l, s and j and have not shown explicitly the regularizing
function f. The term -mJooodkq�V(ql, q2)T(q2,q2)/(q� - k2) which equals zero is added to the
right-hand side of this equation in order to replace the principal value integration by the ordinary
one, which can easily be handled numerically. We have solved this equation using the standard
methods, which are described in ref. [39]. In particular, we discretize it using the ordinary Gauss­
Legendre quadrature rules. Choosing n quadrat ure points {q{} leads to n + 1 coupled equations
for the matrix elements T(qL q2) and T(q2, q2), which can also be expressed in a matrix form. We
have solved this matrix equation by inversion using standard FORTRAN routines. Note that the
on-shell point q2 should, clearly, not belong to the set of quadrat ure points {qD.
Once the half-shell T-matrix is calculated, one can obtain the operator A(p, q) for each fixed
value of p from the linear equation (2.96) . As already pointed out before, this equation has a socalled
moving singularity, which makes it more difficult to handle than the Lippmann-Schwinger
equation. Indeed, one has to discretize both q and q' points in this equation. But this does not
allow to solve this equation as in the last case. The difference Eq - Eql can be exactly zero, since q
and q' belongs now to the same set of quadrat ure points. Thus, one cannot calculate the principal
value integral in the same manner as for the LS equation. To solve this equation we have used
a method proposed by Glöckle et al. [144] . The idea is to expand the function A(p, q) into cubic
splines Si(q)
(2.107)
I 7Rere and in what follows we will always consider the regularized potential vreg(qI, eh) = !(qI) V(qI , q2 ) !(q2)
and suppress the index "reg" for brevity.