ar X iv : n uc l - th / 9 91 10 35 v 1 9 N ov 1 99 9 A Sketch of Two and Three Bodies 1

A cartoon of the Effective Field Theory of many nucleon systems is drawn, concentrating on Compton scattering in the two nucleon system, and on nd scattering in the three body system. The purpose of this presentation is to give a concise introduction into the Effective Field Theory (EFT, for a review see e.g. [1]) of two and three nucleon systems as it emerged in the last three years. However, I can only give a “teaser” with a lot of words and figures and a few cheats in details, referring to the literature, esp. to the excellent proceedings of the INT-Caltech Workshops 1998 and 1999 [2]. I concentrate on work undertaken with J.-W. Chen, R.P. Springer and M.J. Savage in [3,4], P.F. Bedaque in [5,6], and F. Gabbiani in [6]. M. Birse’s talk at this Workshop provides a more formal investigation of the EFT of the two nucleon system, and U.-G. Meißner’s alternative approach he presented here follows Weinberg’s original suggestion [7] but needs to be studied further. Effective Field Theory methods are largely used in many branches of physics where a separation of scales exists. In low energy nuclear systems, the two well separated scales are, on one side, the low scales of the typical momentum of the process considered and the pion mass, and on the other side the higher scales associated with chiral symmetry and confinement. This separation of scales was explored with great success in the mesonic sector (Chiral Perturbation Theory [8]) and in the one baryon sector (Heavy Baryon Chiral Perturbation Theory [9]), producing a low energy expansion of a variety of observables (see also the Chiral Perturbation Theory section of this Workshop). It provided for the first time a description of strongly interacting particles which is systematic, rigorous and model independent (meaning, 1) Talk held at the International Workshop on Hadron Physics “Effective Theories of Low Energy QCD” in Coimbra, Portugal, 10th – 15th September 1999; to be published in the Proceedings; preprint numbers nucl-th/9911035, NT@UW-99-62, TUM-T39-99-25. 2) Email: [email protected] independent of assumptions about the non-perturbative QCD dynamics). Three main ingredients enter the construction of an EFT: The Lagrangean, the power counting and a regularisation scheme. First, the relevant degrees of freedom have to be identified. In his original suggestion how to extend EFT methods to systems containing two or more nucleons, Weinberg [7] noticed that below the ∆ production scale, only nucleons and pions need to be retained as the infrared relevant degrees of freedom of low energy QCD. Because at these scales the momenta of the nucleons are small compared to their rest mass, the theory becomes nonrelativistic at leading order in the velocity expansion, with relativistic corrections systematically included at higher orders. The most general chirally (and iso-spin) invariant Lagrangean consists hence of contact interactions between non-relativistic nucleons, and between nucleons and pions, with the first few terms of the form