The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

2.3. Going to higher energies: a toy model 35
In what follows, we will restrict our considerations to N N S-waves. Because of no spin dependence
one can work out the corresponding S-wave potential simply by integrating over angles. This leads
to
(2.113)
with ql,2 = ItZi,21. Correspondingly, we have to set s = [ = [' = j = 0 in the decoupling equation
(2.95).
As already pointed out before, we first need to regularize the equation (2.95) by multiplying
the potential V(ql, q2) by functions f(ql), f(q2)' The presice form of the function f is given in
eq. (2.98). The nonlinear integral equation (2.95) is solved by the iteration method as described
in the last section. This leads to a matrix equation. Typically, we choose a =
20 keV, b = 10 keV
and 100 Gauss-Legendre points to discretize eq. (2.95). The shift in the binding energy due to the
regularization is 0.012 keV, which is a 0.01 permille effect. Thus, the effects of the regularization
are quite small and can be made smaller if so desired. For the value of the cut-off A = 400 MeV
we show the resulting function (Eq - Ep)A(p, q) in fig. 2.4. Here we have multiplied the function
A(p, q) by (Eq - Ep) in order to avoid a peak at p, q ---+ A. A typical total number of iterations is
40-100 to achieve an accuracy of 0.0001 Ge V-3 for the function A(p, q) related to typical values
of A 0f the order 1 GeV -3.
Having calculated the operator A, we are now in the position to consider observables. After the
function B(q, q') is evaluated from equation (2.111), as described in the last section, we perform
the integrations present in eq. (2.70) to determine H' using again Gau�s-Legendre quadratures
and end up with an effective potential V' (q', q) defined for q, q' ::; A. It is displayed in the left
panel of fig. 2.5 in comparison to the original underlying potential for A = 400 MeV. Note that
the very small region of the regularization is again not shown to keep the presentation elearer.
One finds that the effective and the original potentials have a similar shape for momenta below
the cut-off. The main effect of integrating out of high moment um components at the level of the
potential seems to be given in this case just by an overall shift. Indeed, from the right panel of
fig. (2.5) one can see that the variation of the difference between the two potentials is only about
8% for all values of momenta q, q'. The solution of the effective LS equation (2.87) is now very
simple since the integration is confined to q ::; A. The effective bound state wave function obeys
a corresponding homogeneous integral equation
2 A
�'lt(q) + r iP dijV'(q,ij)'lt(ij) = E'lt(q) , (2.114)
mN Ja
with mN = 938.9 MeV the nueleon mass. Using 40 quadrature points the resulting binding energy
agrees within 9 digits with the result gained from the corresponding homogeneous equation driven
by the original potential V and defined in the whole moment um range. Furthermore, the S-wave
phase shifts agree perfectly solving either eq. (2.88) in the full moment um space or eq. (2.87) in the
space of only low momenta. This is shown in fig. 2.6. Note that due to the regularization the phase
shifts go to zero for q ---+ A. Further, the phase shifts are shown as a function of the kinetic energy
in the lab frame and the zero occurs at 7lab = 2A2/mN = 341 (85) MeV for A = 400 (200) MeV.
One can repeat this numerical exercise choosing different cut-off values. Setting for instance A = 2
GeV, the corresponding function (Eq - Ep)A(p, q) is shown in fig. 2.7 and the effective potential
V'(q',q) turnes out to be rather elose to the original one, [V(q',q) - V'(q',q)l/V(O,O) rv 0.02.
Here we have choosen a =200 keV and b =100 keV, which leads to the same shift in the deuteron