Chiral Theory Applied to Nucleon-Nucleon Systems

E. Epelbaum

Forschungszentrum Jülich, Institut für Kernphysik (Th)

Chiral effective field theory offers a systematic and controlled method to study the dynamics of few-nucleon systems. In the approach proposed by Weinberg, one starts from an effective Lagrangian for nucleon and pion fields as well as external sources, in harmony with chiral and gauge invariance. Based on a systematic power counting and using the method of unitary transformation one constructs energy independent and hermitean nucleon-nucleon potential [1]. To leading order, one has the one-pion exchange together with two four-nucleon interaction terms accompanied by low-energy constants (LECs). At next-to-leading order, renormalizations of the OPE, the leading two-pion exchange diagrams and seven more 4N operators appear. At NNLO, we have in addition dimension two pion-nucleon operators, whose LECs can in principle be determined from the chiral analysis of $\pi N$ scattering. The potential is then used in a properly regularized Lippmann-Schwinger equation (e.g. by a sharp or exponential momentum cut-off) to generate the bound and scattering states as detailed in ref. [2]. The iteration of the potential leads to a non-perturbative treatment of the pion exchange which is of major importance to properly describe the NN tensor force. The nine LECs can be determined from a fit to the low $np$ partial waves. Most partial waves are well reproduced, with some exceptions in the D- and F-waves. Various static deuteron properties are mostly well described without any fine tuning. In the same framework, one can also include charge symmetry breaking and charge dependence of the nuclear force by including the light quark mass difference and elecromagnetic corrections [3]. Phase shifts for $pp$ and $nn$ scattering can be compared to what is obtained using the Nijmegen, Argonne or CD-Bonn potentials. Elastic electron-deuteron scattering has also been investigated to NNLO [4], leading to a good description of the deuteron form factors and structure function up to photon virtualities of 0.2 GeV$^2$. The extension to three- and four-nucleon systems has also been started, see [5,6]. In addition, we have shown, that the numerical values of the LECs corresponding to the four-nucleon contact operators can be understood on the basis of phenomenological one-boson-exchange models [7]. We also extract these values from various modern high accuracy nucleon-nucleon potentials and demonstrate their consistency and remarkable agreement with the values in the chiral effective field theory approach. This paves the way for estimating the low-energy constants of operators with more nucleon fields and/or external probes. Furthermore, it can be demonstrated that the highly singular chiral EFT potential exhibits a so-called limit cycle behaviour in the limit of letting the cut-off go to infinity [8]. This allows to perform exact (cut-off independent) renormalization. A further outlook and discussion of outstanding problems will also ge given.


[1] E. Epelbaoum, W. Glöckle and Ulf-G. Meißner, Nucl. Phys. A637 (1998) 107.
[2] E. Epelbaum, W. Glöckle and Ulf-G. Meißner, Nucl. Phys. A671 (2000) 295.
[3] M. Walzl, Ulf-G. Meißner and E. Epelbaum, Nucl. Phys. A (2001) in press.
[4] M. Walzl and Ulf-G. Meißner , Phys. Lett. B (2001) in press.
[5] E. Epelbaum et al., Phys. Rev. Lett. 86 (2001) 4787.
[6] E. Epelbaum et al., in preparation.
[7] E. Epelbaum et al., nucl-th/0106007.
[8] E. Epelbaum and Ulf-G. Meißner, in preparation.