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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170 5. Summary and outlook<br />
<strong>The</strong> isosp<strong>in</strong> violat<strong>in</strong>g Lagrangian <strong>in</strong> the two-nucleon sector has been worked out by<br />
van Kolck <strong>in</strong> ref. [75]. We have reconsidered the correspond<strong>in</strong>g terms us<strong>in</strong>g a different<br />
approach, namely the method of external fields .<br />
• After a brief <strong>in</strong>troduction <strong>in</strong>to the KSW formalism, we have discussed the lead<strong>in</strong>g<br />
isosp<strong>in</strong> violat<strong>in</strong>g effects <strong>in</strong> the 1 So channel. In particular, the lead<strong>in</strong>g charge <strong>in</strong>dependence<br />
break<strong>in</strong>g effect is due to a comb<strong>in</strong>ation of the neutral to charged pion mass<br />
difference <strong>in</strong> one-pion exchange diagrams together with an electromagnetic N N contact<br />
term. Its correspond<strong>in</strong>g coupl<strong>in</strong>g constant scales as Q-2 but is numerically suppressed<br />
by the explicit appearance of the f<strong>in</strong>e structure constant a rv 1/137. We have shown<br />
how the KSW power count<strong>in</strong>g has to be modified <strong>in</strong> the presence of isosp<strong>in</strong> violat<strong>in</strong>g<br />
operators .<br />
• We explicitely evaluated the 1 So phase shifts for the np, nn and Coulomb-subtracted<br />
pp systems at next-to-Iead<strong>in</strong>g order. In addition, we have given a general classification<br />
of the various CIB and CSB corrections. This allows to understand so me phenomenologically<br />
found results.<br />
To the best of our knowledge the exact moment um space projection of the nucleon-nucleon <strong>in</strong>teraction<br />
discussed and performed <strong>in</strong> sec. 2.3 has never been done before. In this thesis we have<br />
applied this formalism to some problems aris<strong>in</strong>g <strong>in</strong> the context of an effective field theory for the<br />
N N system. <strong>The</strong> method, however, might also provide new <strong>in</strong>sights <strong>in</strong>to many other <strong>in</strong>terest<strong>in</strong>g<br />
quest ions <strong>in</strong> nuclear physics. In particular, it opens the possibility of study<strong>in</strong>g relativistic effects<br />
<strong>in</strong> a consistent and convergent manner, s<strong>in</strong>ce the moment um components of the order or higher<br />
than the nucleon mass can be <strong>in</strong>tegrated out. Furthermore, it would be <strong>in</strong>terest<strong>in</strong>g to generalize<br />
the above formalism to three- and more-nucleon system.<br />
Further, the method of unitary transformation (projection formalism) has never been applied<br />
<strong>in</strong> the context of the chiral effective field theory. Previous calculations of the 2N <strong>in</strong>teraction<br />
[73], [74], [76], [78] from chiral effective Lagrangians are based on time-ordered perturbation<br />
theory and lead to an energy-dependent potential. As already stressed above, the most important<br />
advantages of our formalism aga<strong>in</strong>st time-ordered perturbation theory is energy-<strong>in</strong>dependence of<br />
the correspond<strong>in</strong>g potential and the orthonormality of the related wave functions. Appropriate<br />
power count<strong>in</strong>g rules, which allow to perform calculations with<strong>in</strong> the projection formalism to any<br />
required order <strong>in</strong> the low-momentum expansion, have been worked out <strong>in</strong> sec. 3.6 and appendices<br />
A, B and C.<br />
Our results for nucleon-nucleon <strong>in</strong>teractions derived from the most general chiral <strong>in</strong>variant Hamiltonian<br />
do not only show that the scheme orig<strong>in</strong>ally proposed by We<strong>in</strong>berg works qualitatively, it<br />
even works much better than it was expected, namely quantitatively. It extends the successful<br />
applications of effective field theory (chiral perturbation theory) <strong>in</strong> the pion and pion-nucleon<br />
sectors to systems with more than one nucleon. Clearly, one should now reconsider processes,<br />
which have been evaluated us<strong>in</strong>g We<strong>in</strong>berg's hybrid approach [114] (?f - d scatter<strong>in</strong>g [114], [217],<br />
"(d -+ ?f o d [115], "(d -+ "(d [116]) and extend these considerations to systems with more than two<br />
nucleons. In addition, a fresh look at charge symmetry and charge <strong>in</strong>dependence break<strong>in</strong>g <strong>in</strong> the<br />
Hamiltonian formalism is called for (for earlier studies, see e.g. refs. [218], [99]).<br />
A very important and actual research field comprises the application of CHPT to 3N <strong>in</strong>teractions.<br />
Whereas the first results <strong>in</strong> the KSW scheme and <strong>in</strong> the pionless effective theory were already<br />
presented for the three-body system, see e.g. refs. [119], [120], [121], [122], no calculations with<strong>in</strong><br />
the potential approach <strong>in</strong>clud<strong>in</strong>g the complete lead<strong>in</strong>g 3N force predicted by CHPT have yet been<br />
performed. It would be <strong>in</strong>terest<strong>in</strong>g to see whether the <strong>in</strong>clusion of the 3N forces of such type