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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166 5. Summary and outlook<br />
• We have also considered an alternative way of determ<strong>in</strong><strong>in</strong>g the operator A from the<br />
half-shell two-nucleon T-matrix. This leads to the l<strong>in</strong>ear equation (2.94), which has<br />
been solved numericaIly. For all selected values of the cut-off A the solutions of the<br />
l<strong>in</strong>ear and nonl<strong>in</strong>ear equations for A agree perfectly with each other.<br />
• We have exam<strong>in</strong>ed a low-momentum expansion of the potential. For that we have<br />
expanded the heavy meson exchange term <strong>in</strong> a str<strong>in</strong>g of local operators with <strong>in</strong>creas<strong>in</strong>g<br />
dimension but kept the light meson exchange unchanged, see eqs. (2.115), (2.116).<br />
<strong>The</strong> correspond<strong>in</strong>g coupl<strong>in</strong>g constants accompany<strong>in</strong>g these local operators, which are<br />
monomials of even power <strong>in</strong> the momenta, can be determ<strong>in</strong>ed precisely from the exact<br />
solution. For A =300 (400) MeV their values are given <strong>in</strong> table 2.1. We have<br />
shown that they are of "natural" size, i.e. of order one, with respect to the mass scale<br />
Ascale = 600 MeV. We have also discussed the relation of this scale to the mass of the<br />
heavy meson, which is <strong>in</strong>tegrated out, and the convergence properties of such type of<br />
expansion. In particular, to recover the b<strong>in</strong>d<strong>in</strong>g energy with<strong>in</strong> a few percent, one has<br />
to reta<strong>in</strong> terms of rather high order <strong>in</strong> this expansion, cf. eq. (2.116). This is to be<br />
expected due to the unnatural smallness of this energy on any hadronic mass scale. <strong>The</strong><br />
3 SI scatter<strong>in</strong>g phase shift can be weIl reproduced up to k<strong>in</strong>etic energies llab ':::' 120 Me V<br />
with the first three terms <strong>in</strong> the contact term expansion.<br />
• Based on the expanded heavy meson exchange term, we have also determ<strong>in</strong>ed the<br />
constants Ci directly from a fit to the phase shifts. This is equivalent to the procedure<br />
performed <strong>in</strong> an effective field theory approach. We could show that as long as one does<br />
not <strong>in</strong>clude polynomials of order six (or higher), the result<strong>in</strong>g values of these constants<br />
are close to their exact ones. Furthermore, the b<strong>in</strong>d<strong>in</strong>g energy is reproduced with<strong>in</strong><br />
2%. Includ<strong>in</strong>g dimension six terms, the fits become unstable. This can be traced back<br />
to the fact that the contribution of such terms to the phase shifts are very small (at<br />
low and moderate energies) and thus can not really be p<strong>in</strong>ned down.<br />
• We have also studied the quantum averages of the expanded potential <strong>in</strong> the bound and<br />
scatter<strong>in</strong>g states. For A = 300 MeV, the expansion parameter is of the order of 1/2 and<br />
we f<strong>in</strong>d fast convergence for the bound and the low-Iy<strong>in</strong>g scatter<strong>in</strong>g states, as shown<br />
<strong>in</strong> table (2.3). As expected, for scatter<strong>in</strong>g states with higher energy, the convergence<br />
becomes slower.<br />
• We have determ<strong>in</strong>ed the effective range parameters us<strong>in</strong>g the expansion (2.116) for<br />
the potential and demonstrated the predictive power of such an effective theory. In<br />
particular, hav<strong>in</strong>g fixed the free parameters <strong>in</strong> the potential from the first n terms <strong>in</strong><br />
the effective range expansion one obta<strong>in</strong>s a prediction for the next coefficient, which has<br />
not been used <strong>in</strong> the fit. This is different from the pionless effective theory considered<br />
<strong>in</strong> the seetion 2.2, where no predictions are possible.<br />
• In the model space of small momenta only, one can also study the non-Iocalities <strong>in</strong><br />
the coord<strong>in</strong>ate space representation. We have shown that for typical cut-off values, the<br />
effective potential V(x, x') is highly non-Iocal and looks very different from the orig<strong>in</strong>al<br />
one. For very large values of the cut-off, one recovers the orig<strong>in</strong>al local potential.<br />
While this thesis was be<strong>in</strong>g written down, a similar work done by Bogner et al. [214] has<br />
been reported, show<strong>in</strong>g a considerable research activity <strong>in</strong> this field. In this work, effective<br />
low-momentum potentials def<strong>in</strong>ed below a given moment um space cut-off A are obta<strong>in</strong>ed<br />
from the Bonn-A and Paris potentials us<strong>in</strong>g the folded-diagram method of Kuo, Lee and<br />
Ratcliff [215]. <strong>The</strong> method leads to non-hermitian potentials and preserves the half-shell<br />
NN T-matrix.