The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
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32 2. Low-momentum effective theories for two nucleons<br />
2.3.3 Numerical realization of the projection formalism<br />
Let us now consider <strong>in</strong> more detail the numerical realization of the projection formalism. To<br />
simplify the notation we will omit all <strong>in</strong>dices l, s and j and consider the uncoupled case. <strong>The</strong><br />
generalization to a coupled case is straightforward. <strong>The</strong> nonl<strong>in</strong>ear equation (2.95) can only be<br />
solved numerically. We do this by iteration start<strong>in</strong>g with the value of A(p, q) given by17<br />
p,q V(p, q)<br />
E - E .<br />
A( ) =<br />
q p<br />
(2.105)<br />
Certa<strong>in</strong>ly, the iteration method does not always lead to a convergent solution. We have found<br />
that one can achieve a better convergence if one slightly modifies the usual iteration method:<br />
after each four iterations we perform an averag<strong>in</strong>g over the values of the operator A with suitably<br />
chosen weight factors. <strong>The</strong>se weight factors are found numerically for each particular value of<br />
the cut-off A requir<strong>in</strong>g that the number of iterations, needed to calculate the function A(p, q)<br />
with so me given precision, is m<strong>in</strong>imal. This scheme allows to obta<strong>in</strong> the operator A(p, q) even <strong>in</strong><br />
those cases, when the usual iteration method fails to provide the convergence. Of course, also this<br />
algorithm does not work <strong>in</strong> all cases.<br />
Alternatively, we have derived the operator A(p, q) from the l<strong>in</strong>ear equation (2.96). For that<br />
we have first calculated the usual half-shell T-matrix by solv<strong>in</strong>g the correspond<strong>in</strong>g LS-equation<br />
(2.100), which can be rewritten for the S-channel as<br />
(2.106)<br />
where we have omitted the <strong>in</strong>dices l, s and j and have not shown explicitly the regulariz<strong>in</strong>g<br />
function f. <strong>The</strong> term -mJooodkq�V(ql, q2)T(q2,q2)/(q� - k2) which equals zero is added to the<br />
right-hand side of this equation <strong>in</strong> order to replace the pr<strong>in</strong>cipal value <strong>in</strong>tegration by the ord<strong>in</strong>ary<br />
one, which can easily be handled numerically. We have solved this equation us<strong>in</strong>g the standard<br />
methods, which are described <strong>in</strong> ref. [39]. In particular, we discretize it us<strong>in</strong>g the ord<strong>in</strong>ary Gauss<br />
Legendre quadrature rules. Choos<strong>in</strong>g n quadrat ure po<strong>in</strong>ts {q{} leads to n + 1 coupled equations<br />
for the matrix elements T(qL q2) and T(q2, q2), which can also be expressed <strong>in</strong> a matrix form. We<br />
have solved this matrix equation by <strong>in</strong>version us<strong>in</strong>g standard FORTRAN rout<strong>in</strong>es. Note that the<br />
on-shell po<strong>in</strong>t q2 should, clearly, not belong to the set of quadrat ure po<strong>in</strong>ts {qD.<br />
Once the half-shell T-matrix is calculated, one can obta<strong>in</strong> the operator A(p, q) for each fixed<br />
value of p from the l<strong>in</strong>ear equation (2.96) . As already po<strong>in</strong>ted out before, this equation has a socalled<br />
mov<strong>in</strong>g s<strong>in</strong>gularity, which makes it more difficult to handle than the Lippmann-Schw<strong>in</strong>ger<br />
equation. Indeed, one has to discretize both q and q' po<strong>in</strong>ts <strong>in</strong> this equation. But this does not<br />
allow to solve this equation as <strong>in</strong> the last case. <strong>The</strong> difference Eq - Eql can be exactly zero, s<strong>in</strong>ce q<br />
and q' belongs now to the same set of quadrat ure po<strong>in</strong>ts. Thus, one cannot calculate the pr<strong>in</strong>cipal<br />
value <strong>in</strong>tegral <strong>in</strong> the same manner as for the LS equation. To solve this equation we have used<br />
a method proposed by Glöckle et al. [144] . <strong>The</strong> idea is to expand the function A(p, q) <strong>in</strong>to cubic<br />
spl<strong>in</strong>es Si(q)<br />
(2.107)<br />
I 7Rere and <strong>in</strong> what follows we will always consider the regularized potential vreg(qI, eh) = !(qI) V(qI , q2 ) !(q2)<br />
and suppress the <strong>in</strong>dex "reg" for brevity.