Table of contents

Nuclear structure theory is a domain of physics faced at present with great challenges and opportunities. A larger and larger body of high-precision experimental data has been and continues to be accumulated. Experiments on very exotic short-lived isotopes are the backbone of activity at numerous large-scale facilities. Over the years, tremendous progress has been made in understanding the basic features of nuclei. However, the theoretical description of nuclear systems is still far from being complete and is often not very precise. Many questions, both basic and practical, remain unanswered.

The goal of publishing this special focus issue of Journal of Physics G: Nuclear and Particle Physics on Open Problems in Nuclear Structure Theory (OPeNST) is to construct a fundamental inventory thereof, so that the tasks and available options become more clearly exposed and that this will help to stimulate a boost in theoretical activity, commensurate with the experimental progress.

The requested format and scope of the articles on OPeNST was quite flexible. The journal simply offered the possibility to provide a forum for the material, which is very often discussed at conferences during the coffee breaks but does not normally have sufficient substance to form regular publications. Nonetheless, very often formulating a problem provides a major step towards its solution, and it may constitute a scientific achievement on its own.

Prospective authors were therefore invited to find their own balance between the two extremes of very general problems on the one hand (for example, to solve exactly the many-body equations for a hundred particles) and very specific problems on the other hand (for example, those that one could put in one's own grant proposal). The authors were also asked not to cover results already obtained, nor to limit their presentations to giving a review of the subject, although some elements of those could be included to properly introduce the subject matter.

The focus of these collected articles is therefore on the discussion of topics that are not yet understood, or that are poorly understood. We very much welcomed presentations on:

(i) contradictory approaches, models, or theories that are, at present, difficult to reconcile,

(ii) unsolved theoretical problems that hamper applications of existing methods,

(iii) limitations of current approaches,

(iv) difficulties in deriving and justifying models and theories,

(v) generic problems in understanding or describing specific experimental data, and even

(vi) all possible, wildest speculations and/or conjectures.

The main idea behind the focus issue was to stimulate creative, unbounded thinking and provide young, but not only young, researchers with ideas that would promote further progress in this domain of science.

The community of nuclear structure theorists enthusiastically responded to the idea of publishing the volume on OPeNST. It seemed that the idea struck the right chord and many colleagues were willing to share their observations on what research directions to follow and which problems to attack.

The volume turned out to be a snapshot of the domain, revealing the burning questions that the community wants to address. All the articles also have a very interesting personal touch. They sometimes even present opposing or conflicting points of view, which is exactly what one would expect within a vibrant scientific discussion. All in all, the Editors of Journal of Physics G are very happy to offer you this unique collection, which will constitute very interesting reading for all those working in nuclear structure theory.

We address very briefly five critical points in the context of the Skyrme–Hartree–Fock (SHF) scheme: (1) the impossibility of considering it as an interaction, (2) a possible inconsistency of correlation corrections such as, e.g., the centre-of-mass correction, (3) problems of describing the giant dipole resonance (GDR) simultaneously in light and heavy nuclei, (4) deficiencies in the extrapolation of binding energies to super-heavy elements (SHE) and (5) a yet inappropriate trend in fission lifetimes when going to the heaviest SHE. While the first two points have a more formal bias, the other three points have practical implications and wait for solution.

This paper describes five questions of some interest for the density functional theory in nuclear physics. These are, respectively, (i) the need for concave functionals, (ii) the nature of the Kohn–Sham potential for the radial density theory, (iii) a proper implementation of a density functional for an 'intrinsic' rotational density, (iv) the possible existence of a potential driving the square root of the density and (v) the existence of many models where a density functional can be explicitly constructed.

In view of recent neutron–deuteron (nd) breakup data for neutron–neutron (nn) and neutron–proton (np) quasi-free-scattering (QFS) arrangements and the large discrepancy found between theoretical predictions and measured nn QFS cross sections, we analyze the sensitivity of the QFS cross sections to different partial wave components of the nucleon–nucleon (NN) interaction. We found that the QFS cross section is strongly dominated by the ¹S₀ and ³S₁–³D₁ contributions. Because the standard three-nucleon force (3NF) only weakly influences the QFS region, we conjecture that it must be the nn ¹S₀ force component which is responsible for the discrepancy in the nn QFS peak. A stronger ¹S₀ nn force is required to bring theory and data into agreement. Such an increased strength of the nn interaction will, however, not help to explain the nd breakup symmetric-space-star (SST) discrepancy. Further experimental cross-checkings are required.

Spontaneously broken chiral symmetry is an established property of low-energy quantum chromodynamics, but finding direct evidence for it from nuclear structure data is a difficult challenge. Indeed, phenomenologically successful energy–density functional approaches do not even have explicit pions. Are there smoking guns for chiral symmetry in nuclei?

Nuclear observables such as binding energies and cross sections can be directly measured. Other physically useful quantities, such as spectroscopic factors, are related to measured quantities by a convolution whose decomposition is not unique. Can a framework for these nuclear structure 'non-observables' be formulated systematically so that they can be extracted from experiment with known uncertainties and calculated with consistent theory? Parton distribution functions in hadrons serve as an illustrative example of how this can be done. A systematic framework is also needed to address questions of interpretation, such as whether short-range correlations are important for nuclear structure.

At present there are two vastly different ab initio approaches to the description of the many-body dynamics: the density functional theory (DFT) and the functional integral (path integral) approaches. On one hand, if implemented exactly, the DFT approach can allow in principle the exact evaluation of arbitrary one-body densities. However, when applied to large amplitude collective motion (LACM), this approach needs to be extended in order to accommodate the phenomenon of surface-hopping, when adiabaticity is strongly violated and the description of a system using a single (generalized) Slater determinant is not valid anymore. The functional integral approach on the other hand does not appear to have such restrictions, but its implementation does not appear to be a straightforward endeavor. However, within a functional integral approach one seems to be able to evaluate in principle any kind of observable, such as the fragment mass and energy distributions in nuclear fission. These two radically different approaches can likely be brought together by formulating a stochastic time-dependent DFT approach to many-body dynamics.

A survey of our present understanding of absolute spectroscopic factors near stable closed-shell nuclei is given based on our current interpretation of the (e, e'p) reaction. Uncertainties associated with the imprecise knowledge of the degree of nonlocality of the optical potential of the knocked out proton are identified. Theoretical expectations for the dependence of spectroscopic factors on the nucleon asymmetry are outlined. Conflicting results obtained from various reactions concerning the asymmetry dependence of spectroscopic factors are surveyed. The possibility of obtaining reliable absolute spectroscopic factors with reactions initiated by hadrons is discussed with special emphasis on present and future radioactive beam experiments. The combined and integrated analysis of such reactions using a Green's function framework in the form of an extended dispersive optical model may provide a useful approach to resolve this theoretical open problem.

Today's ion trap technology has made it possible to measure ultra-low beta-decay Q values of a few hundred eV and less. Recent measurements of the ¹¹⁵In beta decay to the first excited state of ¹¹⁵Sn imply that the theory for such decay needs further developments. The atomic effects that are negligible for most beta decays and introduce minor corrections for the decays with low Q values seem to have dramatic consequences for the beta decays with an ultra-low Q value. We use the ultra-low-Q-value decay branch of ¹¹⁵In to illustrate the problem and point out areas where new theory needs to be developed.

We review the notion of symmetry breaking and restoration within the frame of nuclear energy density functional methods. We focus on key differences between wavefunction- and energy-functional-based methods. In particular, we point to difficulties in formulating the restoration of symmetries within the energy functional framework. The problems tackled recently in connection with particle-number restoration serve as a baseline to the present discussion. Reaching out to angular-momentum restoration, we identify an exact mathematical property of the energy density that could be used to constrain energy density functional kernels. Consequently, we suggest possible routes toward a better formulation of symmetry restorations within energy density functional methods.

We investigate three-α continuum states in the hyperspherical formalism for J = 0⁺ and J = 2⁺. Two different types of α + α interactions are used: the shallow Ali–Bodmer potential and the deep potential of Buck et al. We determine the 3α phase shifts up to E = 6 MeV, in parallel with an analysis of resonances in the framework of the complex scaling method. We show that shallow potentials provide additional narrow resonances, in contrast with experimental data. Deep potentials, however, only give rise to broad resonances, and are more consistent with the data.

Integral transform approaches are numerous in many fields of physics, but in most cases limited to the use of the Laplace kernel. However, it is well known that the inversion of the Laplace transform is very problematic, so that the function related to the physical observable is in most cases unaccessible. The great advantage of kernels of the bell-shaped form has been demonstrated in few-body nuclear systems. In fact the use of the Lorentz kernel has allowed us to overcome the stumbling block of the ab initio description of reactions to the full continuum of systems of more than three particles. The problem of finding kernels of similar form, applicable to many-body problems, deserves particular attention. If this search were successful, the integral transform approach might represent the only viable ab initio access to many observables that are not calculable directly.

α particle condensation is a novel state in nuclear systems. We briefly review the present status on the study of α particle condensation and address the open problems in this research field: α particle condensation in heavier systems other than the Hoyle state, linear chain and α particle rings, Hoyle-analog states with extra neutrons, α particle condensation related to astrophysics, etc.

We discuss different approaches to the problem of reproducing the observed features of nuclear single-particle spectra. In particular, we analyze the dominant energy peaks and the single-particle strength fragmentation, using the example of neutron states in ²⁰⁸Pb. Our main emphasis is the interpretation of that fragmentation due to the particle–vibration coupling (PVC). We compare the results from PVC with recent energy density functional (EDF) approaches, and try to present a critical perspective.

This paper provides an insight into several open problems in the quest for novel modes of excitation in nuclei with isospin asymmetry, deformation and finite-temperature characteristics in stellar environments. Major unsolved problems include the nature of pygmy dipole resonances, the quest for various multipole and spin–isospin excitations both in neutron-rich and proton drip-line nuclei mainly driven by loosely bound nucleons, excitations in unstable deformed nuclei and evolution of their properties with the shape phase transition. Exotic modes of excitation in nuclei at finite temperatures characteristic of supernova evolution present open problems with a possible impact in modeling astrophysically relevant weak interaction rates. All these issues challenge self-consistent many-body theory frameworks at the frontiers of on-going research, including nuclear energy density functionals, both phenomenological and constrained by the strong interaction physics of QCD, models based on low-momentum two-nucleon interaction V_low-k and correlated realistic nucleon–nucleon interaction V_UCOM, supplemented by three-body force, as well as two-nucleon and three-nucleon interactions derived from the chiral effective field theory. Joined theoretical and experimental efforts, including research with radioactive isotope beams, are needed to provide insight into dynamical properties of nuclei away from the valley of stability, involving the interplay of isospin asymmetry, deformation and finite temperature.

In recent years impressive progress has been made in the development of highly accurate energy density functionals, which allow us to treat medium–heavy nuclei. In this approach one tries to describe not only the ground state but also the first relevant excited states. In general, higher accuracy requires a larger set of parameters, which must be carefully chosen to avoid redundancy. Following this line of development, it is unavoidable that the connection of the functional with the bare nucleon–nucleon interaction becomes more and more elusive. In principle, the construction of a density functional from a density matrix expansion based on the effective nucleon–nucleon interaction is possible, and indeed the approach has been followed by few authors. However, to what extent a density functional based on such a microscopic approach can reach the accuracy of the fully phenomenological ones remains an open question. A related question is to establish which part of a functional can be actually derived by a microscopic approach and which part, in contrast, must be left as purely phenomenological. In this paper we discuss the main problems that are encountered when the microscopic approach is followed. To this purpose we will use the method we have recently introduced to illustrate the different aspects of these problems. In particular we will discuss the possible connection of the density functional with the nuclear matter equation of state and the distinct features of finite-size effect typical of nuclei.

We discuss the relevant progress that has been made in the last few years on the microscopic theory of the pairing correlation in nuclei and the open problems that still must be solved in order to reach a satisfactory description and understanding of the nuclear pairing. The similarities and differences with the nuclear matter case are emphasized and described by a few illustrative examples. The comparison of calculations of different groups on the same set of nuclei, besides agreements, also shows discrepancies that remain to be clarified. The role of the many-body correlations, like screening, that go beyond the BCS scheme, is still uncertain and requires further investigation.

Physics related to weakly bound nuclei and low-density asymmetric nuclear matter is discussed from a theoretical point of view. Especially, we focus our discussion on new correlations in nuclei near the drip lines and on open issues of the density functional theory with a proper account of the continuum.

Construction of the microscopic theory of large-amplitude collective motion, capable of describing a wide variety of quantum collective phenomena in nuclei, is a long-standing and fundamental subject in the study of nuclear many-body systems. The present status of the challenge toward this goal is discussed taking the shape coexistence/mixing phenomena as typical manifestations of the large-amplitude collective motion at zero temperature. Some open problems in rapidly rotating cold nuclei are also briefly discussed in this connection.

We discuss parity-violating elastic electron scattering as a complementary tool in the race for more precise determinations of neutron densities in nuclei. Isovector and isoscalar densities and form factors in N > Z and N = Z stable nuclei are discussed taking ²⁰⁸Pb and ²⁸Si as examples. Distorted wave calculations of parity-violating asymmetries are shown and are compared to the plane wave impulse approximation. The extraction of the ratio between neutron and proton monopole form factors is discussed. The isospin mixing produced by Coulomb interaction in the ground state of N = Z nuclei with Skyrme selfconsistent mean fields is also discussed.

I discuss the inadequacy of the 'projected density' prescription to be used in density-dependent forces/functionals when calculations beyond mean field are pursued. The case of calculations aimed at the symmetry restoration of mean fields obtained with effective realistic forces of the Skyrme or Gogny type is considered in detail. It is shown that, at least for the restoration of spatial symmetries like rotations, translations or parity, the above prescription yields catastrophic results for the energy that drive the intrinsic wave-function to configurations with infinite deformation, thereby preventing its use both in projection after and before variation.

Coexistence of two very different types of nuclear structure, mean-field-type structure and cluster structure, is discussed. We briefly review actual features of the coexistence of these two different types of structure in some selected nuclei including neutron-rich nuclei. In each of the selected nuclei, we discuss respective problems of the coexistence by utilizing theoretical calculations of coexistence phenomena. Common characteristic problems of the coexistence include dynamical mechanism of the formation of cluster states from the ground state with mean-field-type structure, the mixture of mean-field-type character and clustering character in individual states, and so forth. It is realized that unified description of the mean-field-type structure and cluster structure is a basic open problem for nuclear structure theory.

The ¹S₀ phase shift is large and positive at low densities (relative momenta), while it vanishes and eventually becomes negative at densities of the order of the saturation nuclear density. The bare NN-potential, parametrized so as to reproduce these phase shifts, leads to a sizable Cooper pair binding energy in nuclei along the stability valley. It is a much debated matter whether this value accounts for the 'empirical' value of the pairing gap or whether a similarly important contribution arises from the exchange of collective vibrations between Cooper pair partners. In keeping with the fact that two-particle transfer reactions are the specific probe of pairing in nuclei, and that exotic halo nuclei like ¹¹Li are extremely polarizable (representing, as far as this property is concerned, almost a caricature of stable nuclei), we find that the recently studied reaction, namely ¹¹Li+p → ⁹Li+t, provides, for the first time, direct evidence of phonon-mediated pairing in nuclei.

It is generally accepted that the inner crust of neutron stars is formed by a Coulomb lattice of nuclei immersed in a gas of quasi-free neutrons. We discuss the implications of the inhomogeneity of the crust and of nuclear shell effects on the linear response and on the superfluid properties of the system, as well as on the structure of vortices.

Dynamics of a many-body quantum system of strongly interacting constituents is traditionally considered in a hierarchy of approximations: mean field—quasiparticles—collective excitations—large amplitude motion. Incoherent collision-like processes can be included on the level of kinetics as the mechanism for thermalization. On the other hand, starting from some excitation energy, local spectral properties can always be practically described by random matrix theory. Deep interrelation between two seemingly opposite pictures is not fully understood. This is discussed below, including unresolved problems and possible ways to better theory. Along the road I show how the ideas of quantum chaos can serve as a powerful conceptual, theoretical, experimental and computational tool.

Open problems in the interpretation of the observed pair of near-degenerate ΔI = 1 bands with the same parity as the chiral doublet bands are discussed. The ambiguities for the existing fingerprints of the chirality in atomic nuclei and problems in existing theory are discussed, including the description of quantum tunneling in the mean field approximation as well as the deformation, core polarization and configuration of the particle rotor model (PRM). Future developments of the theoretical approach are anticipated.

A quarter of a century's concerted international work in halo physics has resulted in an extended nuclear paradigm encompassing the limits of existence of cold nuclei and also structures beyond—continuum structures of open (nuclear) quantum systems. Realistic working models, based on cluster constituents, have sprung out of the very nature of halo phenomena, in particular from the three-body Borromean property of two-neutron halos, the lack of low-lying binary breakup channels. This has provided transparency and possibility for insight into new quantum behaviour, also in continua beyond driplines—a focus of this status assessment. Breakup spectra and progressively exclusive correlation cross sections can be computed and show, where relevant data exist, not only that general agreement is encouraging but also that some exclusive observables exhibit significant disagreement that has to be clarified. Progress in studies of two-proton emitters has provided another pathway beyond driplines, where again few-body theory appears promising.

Three-body decays of many-body nuclear resonances are processes where N particles in a quasi-stable configuration are divided into three fragments. The momentum distributions of the fragments carry the information of the resonance and the decay process. Calculated results should be compared to accurate and complete measurements with the purpose of extracting such information. Two almost independent problems must be solved before fully reliable results become available. First, contraction of the N-body degrees of freedom to those of three particles has to be consistently achieved. This presents a conceptual problem since it implies matching of rather incompatible models and the related effective interactions. The second problem is that the resonance structure often furthermore undergoes major changes from small to large distances. The couplings causing these changes may in principle be known within a given three-body model, but even under this assumption, the accuracy requirements are very hard to meet in difficult cases. Different reasons apply to different cases. One example is when very small couplings extend over large distances as for prominent substructures. We illustrate these two open problems with a number of nuclear three-body decays. We emphasize that these problems are the simplest of a much more advanced series of multi-body decays and reaction processes proceeding from N particles to three-body clusters.

Recent studies of the stable Cd isotopes, using inelastic neutron scattering, radioactive decay and nucleon-transfer reactions, have resulted in the need to fundamentally reassess the concept of low-energy vibrational modes in these isotopes. We present a synopsis of the nuclear physics that has led to this situation, followed by the present status of spectroscopy for and against vibrational modes in nuclei neighbouring the Cd isotopes. We continue with some details of the spectroscopic techniques used and examples of the data obtained which lead to the above statements. We provide further perspective with some details of octupole degrees of freedom. We conclude with an outlook of where spectroscopic data are needed to further assess the presence or absence of low-energy vibrations in nuclei; and we provide some reflections on where future theoretical investigations should be directed.

The quest to build a mass formula which has in it the most relevant microscopic contributions is analyzed. Inspired by the successful Duflo–Zuker mass description, the challenges to describe the shell closures in a more transparent but equally powerful formalism are discussed.

This contribution to the focus issue on Open Problems in Nuclear Structure Theory looks at some issues with using time-dependent Hartree–Fock (TDHF) and related techniques to study structural phenomena in nuclear physics. We limit the discussion to structures like giant resonances and discuss some open questions regarding the interpretation of TDHF calculations.

In this paper we introduce a subjective notion of the predictive power of nuclear Hamiltonians (an objective one does not exist) and examine it in the particular context of the single-nucleon energy spectra. We consider various types of uncertainties originating both from the experiment and theory stressing the dominating character of the theoretical errors. The latter originate from the complexity of the nuclear many-body systems that is not matched adequately by the formalism behind the present day nuclear Hamiltonians. The related inverse problem is formulated and the presence of errors (ignorance, lack of knowledge) is parametrized in terms of the associated probability distributions. Various hypotheses concerning the input uncertainties ('numerical noise') are formulated and the impact of the input-uncertainties in the adjustment procedures down to the final parameter values and theoretical spectra is illustrated and discussed. A number of open problems are formulated and listed at the end of the paper.

We formulate the principles of the mean-field theory of nuclear stability employing the point-group and group-representation theories. The related point-group hierarchy of importance in the context of nuclear stability is constructed and discussed. We introduce the notion of the magic-number chains associated with each symmetry—in analogy with the spherical-symmetry nuclear magic numbers. To prepare the criteria for the experimental search of introduced symmetries, we examine the simplified collective rotation-vibration model whose Hamiltonian is invariant under the symmetries in question. We illustrate the construction of the solutions that form at the same time irreducible representations of the point groups in question—in view of formulating the experimental symmetry criteria through the application of the branching-ratio techniques. Since the criteria may involve very weak transitions whose experimental research may be at the limit of the present-day experiments, the desires may arise, as it was the case in the past, to replace the difficult experiments by an inadequate modelling. In this context, we present an alert: the use of oversimplified quantum mechanics exercises in place of experiments and/or microscopic theories is likely to produce meaningless results.

One of the central open problems in nuclear physics is the construction of effective interactions suitable for many-body calculations. We discuss a recently developed approach to this problem, where one starts with an effective field theory containing only fermion fields and formulated directly in a no-core shell-model space. We present applications to light nuclei and to systems of a few atoms in a harmonic-oscillator trap. Future applications and extensions, as well as challenges, are also considered.

Despite a great success of the Skyrme mean-field approach in the exploration of nuclear dynamics, it seems to fail in the description of the spin-flip M1 giant resonance. The results for different Skyrme parameterizations are contradictory and poorly agree with experiment. In particular, there is no parameterization which simultaneously describes the one-peak gross structure of M1 strength in doubly magic nuclei and two-peak structure in heavy deformed nuclei. The reason of this mismatch could lie in an unsatisfactory treatment of spin correlations and spin–orbit interaction. We discuss the present status of the problem and possible ways of its solution. In particular, we inspect (i) the interplay of the collective shift and spin–orbit splitting, (ii) the isovector M1 response versus isospin-mixed responses and (iii) the role of tensor and isovector spin–orbit interaction.

This paper presents several challenges to nuclear many-body theory and our understanding of the stability of nuclear matter. In order to achieve this, we present five different cases, starting with an idealized toy model. These cases expose problems that need to be understood in order to match recent advances in nuclear theory with current experimental programs in low-energy nuclear physics. In particular, we focus on our current understanding, or lack thereof, of many-body forces, and how they evolve as functions of the number of particles. We provide examples of discrepancies between theory and experiment and outline some selected perspectives for future research directions.

Shape coexistence and mixing, isospin mixing, competition between neutron–proton and like-nucleon pairing correlations have been identified as the main characteristic features of nuclei near the N = Z line in the A ≃ 70 mass region. The self-consistent treatment of exotic phenomena dominated by their interplay represents a challenge for nuclear many-body models. Currently, the realistic description of tiny effects in this mass region aiming to test the fundamental interactions and symmetries, and the required theoretical predictions concerning the nuclear properties relevant for astrophysical scenarios, are still open problems.

We discuss possible avenues to study fission dynamics starting from a time-dependent mean-field approach. Previous attempts to study fission dynamics using the time-dependent Hartree–Fock (TDHF) theory are analyzed. We argue that different initial conditions may be needed to describe fission dynamics depending on the specifics of the fission phenomenon and propose various approaches toward this goal. In particular, we provide preliminary calculations for studying fission following a heavy-ion reaction using TDHF with a density constraint. Regarding prompt muon-induced fission, we also suggest a new approach for combining the time evolution of the muonic wavefunction with a microscopic treatment of fission dynamics via TDHF.

The 'breathing mode' of neutron-rich nuclei is our window into the incompressibility of neutron-rich matter. After much confusion on the interpretation of the experimental data, consistency was finally reached between different models that predicted both the distribution of isoscalar monopole strength in finite nuclei and the compression modulus of infinite matter. However, a very recent experiment on tin isotopes at the Research Center for Nuclear Physics (RCNP) in Japan has again muddied the waters. Self-consistent models that were successful in reproducing the energy of the giant monopole resonance (GMR) in nuclei with various nucleon asymmetries (such as ⁹⁰Zr, ¹⁴⁴Sm and ²⁰⁸Pb) overestimate the GMR energies in the tin isotopes. Just as important, the discrepancy between theory and experiment appears to grow with neutron excess. This is particularly problematic as models artificially tuned to reproduce the rapid softening of the GMR in the tin isotopes become inconsistent with the behavior of dilute neutron matter. Thus, we regard the question of 'why is tin so soft?' as an important open problem in nuclear structure.

In the field of energy density functionals (EDFs) used in nuclear structure and dynamics, one of the unsolved issues is the stability of the functional. Numerical issues aside, some EDFs are unstable with respect to particular perturbations of the nuclear ground-state density. The aim of this contribution is to raise questions about the origin and nature of these instabilities, the techniques used to diagnose and prevent them, and the domain of density functions in which one should expect a nuclear EDF to be stable.

We point out a strong influence of the pairing force on the size of the two-neutron Cooper pair in ¹¹Li, and to a lesser extent also in ⁶He. It seems that these are quite unique situations, since Cooper pair sizes of stable superfluid nuclei are very little influenced by the intensity of pairing, as recently reported. We explore the difference between ¹¹Li and heavier superfluid nuclei, and discuss reasons for the exceptional situation in ¹¹Li.

In spite of the great progress we have seen in recent years in the derivation of nuclear forces from chiral effective field theory (EFT), some important issues are still unresolved. In this contribution, we discuss the open problems which have particular relevance for microscopic nuclear structure, namely, the proper renormalization of chiral nuclear potentials and sub-leading many-body forces.

Is there a connection between the branch point singularity at the particle emission threshold and the appearance of cluster states which reveal the structure of a corresponding reaction channel? Which nuclear states are most impacted by the coupling to the scattering continuum? What should be the most important steps in developing the theory that will truly unify nuclear structure and nuclear reactions? The common denominator of these questions is the continuum shell-model approach to bound and unbound nuclear states, nuclear decays and reactions.

One of the open problems in nuclear structure is how to predict properties of finite nuclei from the knowledge of a bare nucleon–nucleon interaction of the meson-exchange type. We point out that a promising starting point consists in Dirac–Brueckner–Hartree–Fock (DBHF) calculations using realistic nucleon–nucleon interactions like the Bonn potentials, which are able to reproduce satisfactorily the properties of symmetric nuclear matter without the need for three-body forces, as is necessary in non-relativistic BHF calculations. However, the DBHF formalism is still too complicated to be used directly for finite nuclei. We argue that a possible route is to define effective Lagrangians with density-dependent nucleon–meson coupling vertices, which can be used in the relativistic Hartree (or relativistic mean field (RMF)) or preferably in the relativistic Hartree–Fock (RHF) approach. The density dependence is matched to the nuclear matter DBHF results. We review the present status of nuclear matter DBHF calculations and discuss the various schemes to construct the self-energy, which lead to differences in the predictions. We also discuss how effective Lagrangians have been constructed and are used in actual calculations. We point out that completely consistent calculations in this scheme still have to be performed.

The main steps involved in realistic shell-model calculations employing two-body low-momentum interactions are briefly reviewed. The practical value of this approach is exemplified by the results of recent calculations and some remaining open questions and directions for future research are discussed.

This article was published online on 9 April 2010 but the captions to figures 4 and 5 must be corrected as follows. This corrigendum also applies to the print version published as article 064028 in this issue.

Figure 4.B(E2) values, relative to the B(E2; 2₁⁺ → 0₁⁺) value, observed for ¹¹²Cd. The uncertainties for the last digit are listed in parentheses. Dashed transitions indicate an unobserved transition. Underlined numbers indicate a B(E2) value relative to the strongest transition (defined as 100 units) from the level. (The upper limit for the E2, 2121 → 1312 transition is dictated by the δ(E2/M1) value for the 2121 → 1312 transition.)

Figure 5. Systematics of B(E2) values for the even–even Cd isotopes ^110–116Cd. Transitions are labelled with their B(E2) values in W.u. with 1σ uncertainties on the last digit in the parentheses; the listing of two numbers reflects asymmetric uncertainties with +1σ and −1σ, respectively. Values without uncertainties are relative B(E2) values, or upper limits. Dashed arrows indicate unobserved transitions where upper limits have been established. Of particular note are the greatly disparate values in ¹¹⁴Cd between results of Coulomb excitation and lifetime measurements for some levels, the most serious of which occurs for the 1842 keV 2⁺ level; the values obtained from Coulomb excitation are listed above the transitions and are up to a factor of ∼35 greater than those derived from the most stringent lifetime limits.

Figures 4 and 5 with their corrected captions are presented in the associated PDF file.