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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168 5. Summary and outlook<br />
• In sec. 3.8.1 we have also discussed the structure of the NNLO potential <strong>in</strong> the method<br />
of unitary transformation. It turns out that <strong>in</strong> that case one obta<strong>in</strong>s the same result<br />
<strong>in</strong> both the projection formalism and time-ordered perturbation theory. <strong>The</strong> NNLO<br />
potential has been calculated with<strong>in</strong> a different (but lead<strong>in</strong>g to the same result) scheme<br />
by the Munich group [108].<br />
• In sec. 3.8.3 we have generalized the projection formalism to <strong>in</strong>clude effects of the<br />
virtual Ll-isobar excitation with<strong>in</strong> the "small scale expansion" and discussed the lead<strong>in</strong>g<br />
contribution to the N N potential from the <strong>in</strong>termediate Ll 'so In that case, both the<br />
method of unitary transformation and time-ordered perturbation theory lead to the<br />
same result.<br />
• Furthermore, we have considered the lead<strong>in</strong>g contributions to the three-nucleon potential.<br />
As <strong>in</strong> time-ordered perturbation theory, we f<strong>in</strong>d that these sum up to zero.<br />
However, the mechanism for the cancelation between so me of the graphs is dist<strong>in</strong>ctively<br />
different <strong>in</strong> the projection formalism s<strong>in</strong>ce it is not related to an <strong>in</strong>tricate cancelation<br />
of terms generated by the iteration of the energy-dependent two-nucleon potential <strong>in</strong><br />
old-fashioned time-ordered perturbation theory. Rather, <strong>in</strong> the approach here, these<br />
cancelations can be traced back to the appearance of "reducible" graphs whose precise<br />
mean<strong>in</strong>g is expla<strong>in</strong>ed <strong>in</strong> section 3.8.1. <strong>The</strong>se diagrams are <strong>in</strong> fact responsible for the<br />
orthonormality of the wave functions and are thus sometimes called "wave function<br />
re-orthonormalization" graphs.<br />
• We have also constructed the most general reparametrization <strong>in</strong>variant effective Lagrangian<br />
for the contact <strong>in</strong>teractions with four nucleon legs up to order Lli = 3, L}J;;5�3)<br />
(reparametrization <strong>in</strong>variance is a consequence of Lorentz <strong>in</strong>variance of the underly<strong>in</strong>g<br />
theory, see ref. [182]). <strong>The</strong> Lagrangian used <strong>in</strong> previous calculations, see e.g. [74], [76],<br />
[78], [127], conta<strong>in</strong>s fourteen free parameters Ci,...,14 and leads to two-nucleon forces,<br />
which depend on the total moment um P of two nucleons. Whereas such P-dependent<br />
forces do not affect calculations of the two-nucleon system <strong>in</strong> the c.m.s., they would be<br />
very important for processes <strong>in</strong>clud<strong>in</strong>g other particles and for three and more nucleons.<br />
We have shown that requir<strong>in</strong>g the reparametrization <strong>in</strong>variance for the contact terms<br />
<strong>in</strong> the effective Lagrangian yields several constra<strong>in</strong>ts on the parameters Ci. Only seven<br />
of these fourteen parameters are really <strong>in</strong>dependent. <strong>The</strong> result<strong>in</strong>g NLO and NNLO<br />
two-nucleon potentials do not depend on the total moment um P.<br />
3. Thirdly, <strong>in</strong> chapter 4 we have calculated nuclear forces and properties of the two-nucleon<br />
system based on a chiral effective field theory and the projection formalism. <strong>The</strong> results of<br />
this <strong>in</strong>vestigation can be summarized as follows:<br />
• We considered the two-nucleon potential at NNLO result<strong>in</strong>g from the projection formalism.<br />
It consists of one- and two-pion exchange diagrams, <strong>in</strong>clud<strong>in</strong>g dimension two<br />
(Lli = 2) <strong>in</strong>sertions from the pion-nucleon Hamiltonian. <strong>The</strong> correspond<strong>in</strong>g LEes have<br />
been taken from an <strong>in</strong>vestigation of rrN scatter<strong>in</strong>g [195]. In addition, there are two and<br />
seven contact <strong>in</strong>teractions without and with two derivatives, respectively. <strong>The</strong> coupl<strong>in</strong>g<br />
constants of these terms must be fixed by a fit to data.<br />
• For large momenta, the potential becomes unphysical and has to be regularized. We<br />
performed this regularization on the level of the Lippmann-Schw<strong>in</strong>ger equation, as<br />
expla<strong>in</strong>ed <strong>in</strong> sec. 4.1 us<strong>in</strong>g either a sharp or an exponential regulator function. At<br />
NLO, physics does not depend on the cut-off <strong>in</strong> the range between 400 and 650 Me V. At