Nuclear forces and the quantum many-body problem

Abstract

1. Challenges for the nuclear many-body problemIntricate nuclear forces, whichhave yet to be completely determined, two different fermionic species (protons and neutrons) and the lack of an external force, generate arange and diversity of behaviours that make the nucleus a truly unique quantum many-body system. One major goal of the physics of nuclei is todevelop a unified predictive theory of nuclei and nuclear matter that can describe the diverse phenomena found in nuclei. Furthermore, physical properties, such as masses and lifetimes,of very short-lived, and hence very rare, nuclei are importantingredients that determine element production mechanisms inthe universe. While nuclear scientists have developed many excellent descriptionsthat embody various properties of the nucleus, a full microscopicunderstanding is lacking. The challenge is therefore to use the properties of unstable and short-lived nuclei to obtain thatfully microscopic description. How to deal with weaklybound systems and coupling to resonant states is an unresolved problem innuclear spectroscopy. Similarly, a description of larger nuclei using afully microscopic first principles (ab initio) approach, starting from thefundamental laws of quantum theory, is another unresolved problemin nuclear physics that awaits a satisfactory, computationally tractable,solution.The aim of the Workshop on Nuclear Forces and the Quantum Many-Body Problem, held at the Institute of Nuclear Theory, Seattle, on 4–8 October 2004,and its pertinence to the INT Program for the fall of 2004, was to shed light on such challenges and particularly on several open questions facing the nuclear few- and many-body problems. In particular, in order to stimulate possible new directions of research, we singled out five major topics that we think encompass a number of the open problems in nuclear many-body theory. The program of this meeting hopefully reflects these choices, aslisted in the following subsections. In this issue of Journal of Physics G: Nuclear and Particle Physics, which constitutesthe proceedings of the Workshop, the reader will find 16 articlesthat cover much of the content of the Workshop, from a total of 25 presented talks. We mention all of the speakers in our outlineof the Workshop below. 1.1. From free interactions to few- and many-body systems The connection between QCD and the free two-body and three-body interactionsis still an open and unresolved problem in nuclear physics. It is crucial to the dialectics of all ab initio methods, which start typically with a two-body interaction and eventually a three-body interaction fitted to reproduce low-energy scattering data and propertiesof selected light nuclei. The contributions from R Machleidt and D R Entem, by U van Kolck, by H-W Hammer and by D R Phillips,pursued many of the underlying issues connected with derivingfrom effective field theory a nucleon–nucleon interaction. A Nogga described how various bare interactions behave in few-bodycalculations. A Schwenk described how a three-body force could be obtained in the Vlow k approach, while A Shirokov et al described theconstruction of a two-body potential using inversescattering theory. How a two-body and/or three-body interaction evolves in a many-body medium is an open problem in nuclear physics, since, for example, a two-body interaction fitted to reproduce phase shifts introduces an off-shell dependence in the medium. 1.2. Ab initiomany-body methodsIn order to move beyond the standard shell model, and beyond thelightest of nuclei, it will be necessary toidentify methods which can complement the shell model for heavier systems and preserve an ab initio philosophy. Presently, the Green's function Monte Carlo approach (presented by S Pieper) using realisticand bare two- and three-body potentials, and the no-core shell-model approaches (presented by W E Ormand and by P Navratil) offer very precise few- and many-body calculations of systems with A ≤ 16. The coupled cluster approach (discussed by D J Dean and by Wloch et al), inspired by the success in quantum chemistry, have shown a potential for performing similar calculationsfor nuclei with A ≤ 16 and A > 16. Other methods based on Green's function approaches and Block–Horowitz are interesting candidates for ab initio studies, asdescribed in the contributions of C Barbieri and W H Dickhoff and of T Luu. An interestingapplication of low-energy no-core shell-model ideas to quantum field theory was presented by J P Vary. 1.3. Methods for unstable systems It is important to identify and investigate methods that will extend to unstable systems, where we especially face the problem of an increasing single-particle level density and likely resonant states.Presently, there are several attempts at combining shell-model technologies with studies of weakly bound systems. One of the promising approaches is the so-called Gamow shell model as discussedby N Michel et al, R Id Betan et al and G Hagen et al. This methodis based on complex scaling techniques developed in quantum chemistry and atomic physics, as discussed by N Moiseyev. 1.4. Shell model and effective interactions For heavier systems, one will most likely need to complement the shell model with methods which allow for precise derivations of effectiveinteractions including both two-body and three-body effective interactions.Presently, the shell model with effective two-body interactions based on, for example, perturbative many-body methods,offers a very good description of the excited spectra of several nuclei. However, this approach faces a number of challenges, as pointed out in thecontribution by B R Barrett, and fails in reproducing binding energies, single-particle energies and basically all known shell closures. Recent Green's function Monte Carlo and no-core shell-model calculations demonstrate the need of three-body interactions. But for heavier nucleisuch calculations are not feasible due to the large dimensionalities involved.One needs, therefore, in parallel more phenomenologically based effective interactions. These play a crucial rolesince, being fitted to reproduce the available body of data, they may be of great help in clarifyingwhich matrix elements are of importance for shell closures and other features of excited states. The talks by B A Brown and by T Otsuka demonstrate the latter. Another interesting topic is how to match the above ab initio methods with extensionsof density functional theories, as discussed by R J Furnstahl and by M Bender and P-H Heenen. 1.5. Computational and algorithmic issuesHere one needs to single out important computational and algorithmic developments. The factorization scheme described byT Papenbrock and D J Dean is one such candidatewhich offers a reduction by more than an order of magnitude in dimensionality for shell-model studies. Similarly, the coupled cluster approachesallow one to study ground- and excited-state properties of nucleiwith dimensionalities beyond the capability of present shell-modelapproaches, with a much smaller numerical effort when compared tothe more traditional shell-model methods aimed at similar accuracies.Furthermore, we feel it is important to stress recent advances in computational science.Parallel technology offers, for example, a totally new paradigmfor many-body theories. It is imperative, therefore, to develop methods which are capableof utilizing these advances fully. 2. Experimental challenges and many-body methods In this section we wish to link the above with present and future challengesfrom experiments, in order to emphasize our requirements and demands from many-body theories in nuclear physics.Twelve years ago, two of us (BRB and JVP) organized a similar three-month programat the INT in Seattle. Back in 1992 we were not in a situation where one could performalmost exact calculations of nuclear systems.At this time, we feel that nuclear many-body theory is now in the positionwhere precise and benchmark calculations can be performed with a giventwo-body and/or three-body Hamiltonian. For mass A ≤ 16, the Green's function Monte Carloapproach, the no-core shell model and recently the coupled cluster approaches,represent in principle ab initio methods. This offers wide perspectives for studiesof various Hamiltonians used in solving the non-relativistic Schrödingerequation. An eventual disagreement with data can then be retraced to our Hamiltonian.The calculations with three-body interactions clearly demonstrate this point. Unless a three-body interaction is included, one cannot reproduce the binding energy or some excited states for nuclei with A ≤ 16. For stable nuclei, important for the s-process and Big Bang nucleosynthesis,we have typically proton and neutron separation energies Sp and Sn of the order of Sp ∼ Sn ∼ 8 MeV,and the excited states are dominated by an interplay between collective and single-particle degrees of freedom. However, for stable nuclei with A > 16 we are not able to describe properties such as shell closures or the binding energy based on a microscopic approach,unless we employ a fitted two-body interaction in shell-model studies. Well-known cases are the shell-closures in 48Ca or the chain of oxygen isotopes.We may, however, be able to reproduce several excited states of stable nuclei starting from a perturbative many-body approach. It is, on the other hand, difficult to extend such approaches in order to include three-body interactions or go beyond a certain order in perturbation theory. This poses serious challenges to nuclear many-body theories when we move to proton-richor neutron-rich nuclei. Such nuclei have been produced recently or will be studied in the near future. As an example, neutron-rich nuclei, with Sp ∼ 15 MeV and Sn ∼ 0 MeV and crucial for the r-process, supernovae and star formation, have been studied extensively in the past few years. They exhibit features like halos and resonances which put a strong pressure on existing many-body techniques. How to deal with weakly bound states and an increased number of single-particle degrees of freedom is not easy to account for in present shell-model approaches. Other features like the predictions of new shell-closures and their originare also properties we would expect a many-body approach to deal with. Summarizing, it is our firm belief that new developments in many-body theoriesfor nuclear problems should contain as many as possible of the following ingredients:• It should be fully microscopic and start with present two- and three-bodyinteractions derived from, for example, effective field theory.• It can be improved upon systematically, for example, by inclusion ofthree-body interactions and more complicated correlations.• It allows for description of both closed-shell systems and valence systems.• For nuclear systems where shell-model studies are the only feasible ones,such as a small model space requiring an effective interaction, one should be able toderive effective two and three-body equations and interactions for the shellmodel.• It is amenable to parallel computing.• It can be used to generate excited spectra for nuclei in which many shells are involved (It is hard for the traditional shell modelto go beyond one major shell. The inclusion of several shells may imply the need for complex effective interactionsrequired in studies of weakly bound systems).• Finally, nuclear structure results should be used in marrying microscopic many-body results with reaction studies. This will be another hot topicof future ab initio research. AcknowledgmentsWe are much indebted to the Institute of Nuclear Theory (INT) in Seattle for all their supportin organizing this meeting. We are especially indebted to Laura Lee, Lesley Reece and Linda Vilett at the INT for their kind assistance and help in all practical matters. BRB wishes to acknowledge partial financial support by NSF grantPHY0244389; MHJ thanks the Research Council of Norway; JPV acknowledges support form US DOE Grant No DE-FG-02-87ER-40371, and DJD acknowledges US DOE Contract No DE-AC05-00OR22725 with UT-Battelle,LLC (Oak Ridge National Laboratory).