Quenching of single-particle strength from direct reactions with stable and rare-isotope beams

Progress in Particle and Nuclear Physics 118 (2021) 103847 Contents lists available at ScienceDirect Progress in Particle and Nuclear Physics journal homepage: www.elsevier.com/locate/ppnp Review Quenching of single-particle strength from direct reactions with stable and rare-isotope beams T. Aumanna,b, C. Barbieric,d,e, D. Bazinf,g, C.A. Bertulanih, A. Bonaccorsoi, W.H. Dickhoffj, A. Gadef,g, M. Gómez-Ramosa,k, B.P. Kayl, A.M. Morok,m, T. Nakamuran, A. Obertellia,∗, K. Ogatao,p, S. Paschalisq, T. Uesakar aTechnische Universität Darmstadt, Fachbereich Physik, Institut für Kernphysik, 64289 Darmstadt, Germany bGSI Helmholtzzentrum für Schwerionenforschung, 64289 Darmstadt, Germany cDepartment of Physics, University of Surrey, Guildford GU2 7XH, United Kingdom dDipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16, I-20133 Milano, Italy eINFN, Sez. di Milano, Via Celoria 16, I-20133 Milano, Italy fNational Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA gDepartment of Physics & Astronomy, Michigan State University, East Lansing, MI 48824, USA hDepartment of Physics and Astronomy, Texas A&M University-Commerce, Commerce, TX 75429, USA iINFN, Sez. di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy jDepartment of Physics, Washington University, St. Louis, MO 63130, USA kDepartamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain lPhysics Division, Argonne National Laboratory, Argonne, IL 60439, USA mInstituto Interuniversitario Carlos I de Física Teórica y Computacional (iC1), Apdo. 1065, E-41080 Sevilla, Spain nDepartment of Physics, Tokyo Institute of Technology, 2-12-1 O-Okayama, Meguro, Tokyo, 152-8551, Japan oResearch Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan pDepartment of Physics, Osaka City University, Osaka 558-8585, Japan qDepartment of Physics, University of York, York, YO10 5DD, UK rRIKEN Nishina Center, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan article info Article history: Available online 7 January 2021 Keywords: Nuclear structure Direct reactions Spectroscopic factors Radioactive beams abstract In this review article we discuss the present status of direct nuclear reactions and the nuclear structure aspects one can study with them. We discuss the spectroscopic information we can assess in experiments involving transfer reactions, heavy-ion- induced knockout reactions and quasifree scattering with (p,2p), (p,pn), and (e,e′p) reactions. In particular, we focus on the proton-to-neutron asymmetry of the quenching of the spectroscopic strength. ©2021TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY license (http://creativecommons.org/licenses/by/4.0/). Contents 1. Introduction............................................................................................................................................................................................... 3 2. Single-particle spectral functions and correlations.............................................................................................................................. 6 ∗Corresponding author. E-mail address: [email protected] (A. Obertelli). https://doi.org/10.1016/j.ppnp.2021.103847 0146-6410/©2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/ licenses/by/4.0/). T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 2.1. Introduction.................................................................................................................................................................................. 6 2.2. Single-particle Green’s function and the self-energy.............................................................................................................. 6 2.3. Aspects of the dispersive optical model linking nuclear structure and reaction domains ................................................ 10 2.4. Analysis of the (e,e′p) reaction.................................................................................................................................................. 11 2.5. Calculations of spectroscopic factors and spectral distributions ........................................................................................... 14 2.5.1. Influence of long-range correlations.......................................................................................................................... 14 2.5.2. Influence of short-range and tensor force correlations........................................................................................... 15 2.5.3. Combined effects of LRC and SRC .............................................................................................................................. 18 2.5.4. Some results from the dispersive optical model...................................................................................................... 21 2.6. Reaction mechanism.................................................................................................................................................................... 22 2.7. Ab initio computations of correlations ...................................................................................................................................... 24 2.7.1. Role of shell-model and particle-vibration coupling in LRC................................................................................... 25 2.7.2. Isospin asymmetry dependence of spectroscopic factors ....................................................................................... 26 2.7.3. Observability of spectroscopic factors ....................................................................................................................... 27 2.7.4. Optical potentials ......................................................................................................................................................... 27 3. Nucleon transfer reactions ...................................................................................................................................................................... 28 3.1. Introduction.................................................................................................................................................................................. 28 3.2. Choice of experimental conditions............................................................................................................................................ 29 3.2.1. Beam energy and momentum matching................................................................................................................... 29 3.2.2. Detection angles........................................................................................................................................................... 32 3.2.3. Comment on heavy-ion transfer ................................................................................................................................ 32 3.3. Reaction theory for transfer reactions ...................................................................................................................................... 33 3.3.1. The distorted-wave Born approximation (DWBA) ................................................................................................... 33 3.3.2. Optical potential........................................................................................................................................................... 34 3.3.3. Overlap functions ......................................................................................................................................................... 35 3.4. Analyses of transfer-reaction data............................................................................................................................................. 36 3.4.1. Relative and absolute spectroscopic factors from transfer reactions .................................................................... 36 3.4.2. Nucleon-transfer sum rules ........................................................................................................................................ 37 3.5. Quenching of transfer reaction cross sections ......................................................................................................................... 37 3.6. Transfer reactions and ∆S.......................................................................................................................................................... 40 3.7. Conclusion..................................................................................................................................................................................... 40 4. Heavy-ion induced knockout .................................................................................................................................................................. 41 4.1. Introduction.................................................................................................................................................................................. 41 4.2. Ingredients of the Rsplot and their sensitivities..................................................................................................................... 42 4.2.1. Measured inclusive cross sections ............................................................................................................................. 42 4.2.2. Theoretical cross sections combining eikonal reaction theory and shell model.................................................. 43 4.3. Probing the reaction model........................................................................................................................................................ 46 4.3.1. Stripping vs. diffraction............................................................................................................................................... 46 4.3.2. Core survival and core excitations............................................................................................................................. 46 4.3.3. Kinematic limit for the removal of deeply-bound nucleons .................................................................................. 48 4.4. Putting structure and reactions onto a more similar footing ................................................................................................ 49 4.4.1. Incorporating ab initio densities and overlaps into reaction model ...................................................................... 49 4.4.2. Ab initio p-shell calculations at face value: 10Be and 10C....................................................................................... 49 4.4.3. The power of mirror reactions ................................................................................................................................... 49 4.5. Other theoretical methods.......................................................................................................................................................... 50 4.6. Conclusions................................................................................................................................................................................... 51 5. Quasifree (p,2p) and (p,pn) scattering ................................................................................................................................................. 52 5.1. Introduction.................................................................................................................................................................................. 52 5.2. Distorted wave impulse approximation.................................................................................................................................... 52 5.3. Cross section................................................................................................................................................................................. 54 5.4. Plane wave impulse approximation .......................................................................................................................................... 54 5.5. DWIA vs. PWIA ............................................................................................................................................................................ 55 5.6. The quantum transfer-to-the-continuum method................................................................................................................... 57 5.7. Comparison with the DWIA and Faddeev methods ................................................................................................................ 59 5.8. Application to lower energies: final-state interactions ......................................................................................................... 59 5.9. Application to R3Binclusive data .............................................................................................................................................. 61 5.10. Discussion on the DWIA description of nucleon knockout reactions by proton in normal and inverse kinematics...... 62 5.11. Conclusion..................................................................................................................................................................................... 66 6. Analysis of proton-removal reactions from 16O................................................................................................................................... 67 7. Conclusions and outlook.......................................................................................................................................................................... 68 Acknowledgments .................................................................................................................................................................................... 70 Appendix A. Analysis of the nucleon self-energy and some pedagogical illustrations ................................................................... 71 Appendix B. Momentum distributions in heavy-ion induced knockout ........................................................................................... 74 B.1. Differential stripping cross section............................................................................................................................................ 74 B.2. Differential diffraction dissociation cross section.................................................................................................................... 75 B.3. Longitudinal vs. transverse momentum distributions............................................................................................................. 76 2 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 B.4. Nuclear diffraction dissociation revisited ................................................................................................................................. 78 References ................................................................................................................................................................................................. 80 1. Introduction The structure of nuclei and its evolution with the number of constituent nucleons, in particular with neutron-to- proton asymmetry, has been a fascinating and multifaceted area in nuclear physics for a long time. It has become even more so recently due to the vast advances in new and planned accelerator facilities and experimental techniques as well as in nuclear theory. While these recent developments permitted already great advances in our understanding [1,2], a global picture of nuclear structure and the driving mechanisms responsible for the observed phenomena remain elusive. Major efforts are being made to realize the development of new or upgraded rare-isotope facilities such as the RIBF [3], FRIB [4], FAIR [5], ISOLDE [6], GANIL-SPIRAL2 [7], ARIEL [8], and SPES [9], and new experimental techniques that allow for spectroscopic studies of nuclei in unexplored regions of the nuclear landscape. In parallel, new theoretical techniques enabled the calculations of nuclear states from an ab initio point of view reaching medium-mass nuclei (see , for example, Ref. [10–12]). Such advances in both the laboratory and theoretical modeling have a direct impact on our understanding of astrophysical nucleosynthesis and compact objects of the universe. Direct reactions have been one of the key experimental probes to build up our understanding of the nuclear shell structure since the 1950s (see, e.g., Ref. [13]). At incident energies close to the Fermi momentum, nucleon-transfer reactions can populate particle and hole states via nucleon-adding and nucleon-removing reactions, respectively. At higher energies, nucleons, assumed to be quasifree, can be knocked out in nucleus–nucleus, proton–nucleus, or electron–nucleus collisions (a comprehensive review is published in [14]). The measured cross sections to individual final states give access to single-particle (s.p.) configuration mixing and identify the main configurations of the wavefunction quantified traditionally through spectroscopic factors and s.p. energies. Fig. 1 shows a summary of spectroscopic factors (SF) extracted from electron-induced (e,e′p) knockout reactions on different stable nuclei spanning a wide mass range [15]. A reduction of the spectroscopic s.p. strength to around 65% relative to the independent particle model has been found for all stable nuclei. The reduction is generally understood to originate about equally from long-range correlations (LRC) and short-range correlations (SRC) beyond the pairing correlations around the Fermi surface taken into account in the independent-particle shell model. The nuclear shell structure and the so-called s.p. energies, i.e., the energy orbital sequence at the mean-field level, can be extracted from the excitation energy of physical (correlated) states and the measurement of nucleon removal and addition cross sections, following the Baranger sum rule [16]. Although s.p. energies are framework dependent and not strictly observables [17,18], they are accepted as meaningful quantities in a specified theoretical framework [19]. Removal energies as encountered in high-resolution direct-reaction experiments to discrete final states are observable and are associated with the results of the (e,e′p) experiments shown in Fig. 1. Different reactions are sensitive to different parts of the radial overlap functions between the initial and final states, though. However, experiments using different reactions and/or beam energies should lead, after reaction-model-based interpretations, to consistent nuclear-structure information, and this for all nuclei, spanning over a wide range of nuclear binding energies, angular momentum, and neutron-to-proton asymmetries. Most of the reaction models are based on the assumption that the structure information necessary to predict cross sections is contained in an overlap function between projectile/target and residue, quantified either by Spectroscopic Factors (SFs), Asymptotic Normalization Coefficients (ANCs), or Electromagnetic Matrix (EM) elements. A reaction is called direct if it proceeds directly from the initial to the final state without the formation of an intermediate compound nucleus, and therefore can be factorized into a nuclear-structure term, quoted C2Sin the following (the notation will be explained later in the review), and a reaction term σsp corresponding to the cross section in case of s.p. states. The factorization can be expressed as σ=C2Sσsp.(1) Recent developments in reaction theory for direct reactions at high bombarding energies (Elab ≳10 MeV/nucleon) have been devoted to (a) the treatment of the many-body continuum in coupled-channels calculations using the continuum-discretized coupled-channels (CDCC) method [20–22], (b) the treatment of clusters with near exact solutions of high-energy few-body scattering such as the solution of the Alt–Sandhas–Grassberger (AGS) equations [23–25], or (c) the inclusion of short-range correlations [26–28]. In practice, in many reaction models, such as in most distorted-wave Born-approximation (DWBA) formulations, the theoretical cross section can be ‘‘satisfactorily’’ factorized into a product of two parts, one dependent on the reaction dynamics and another part involving form factors for an operator probing the nuclear structure. In some cases, when the interaction is amenable to an exact treatment such as electron-induced knockout reactions, the theoretical approach can be fully realized without a factorization so that the reaction mechanism and the nuclear-structure input are described on the same footing [29,30]. Ideally, measured cross sections involving hard probes should be interpreted directly by comparison to theoretical values computed by solving the full many-body problem of the collision, treating consistently structure and reaction in a unified framework and a unique Hamiltonian. 3 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 1. Normalized spectroscopic factors from the (e,e′p) reaction as a function of target mass. The independent-particle model predicts a normalized value of 1. Source: Reprinted from Ref. [15], with permission from Elsevier. This is today numerically impossible, with the exception of few-body systems at low incident energy (see, e.g., [31]). An intermediate step provided by the dispersive optical model (DOM) connects structure and reaction information [32–34], while nucleon–nucleus optical potentials from ab initio theory are being developed but have not yet reached the necessary sophistication for predictive power [35–38]. It is important to clarify the notation used in this review. PWBA (DWBA) in electron-induced proton knockout is used for plane (distorted) waves for the electron. On the other hand, the notation PWIA (DWIA) often refers to the plane (distorted) wave for the outgoing proton. The notations DWBA and DWIA are often used as synonyms in hadronic interactions, such (p,2p) reactions. Plane waves are rarely used in theoretical calculations of (p,2p) reactions or heavy-ion induced knockout reactions, although we show an example later for clarification of some physics aspects in the reaction. Additionally, eikonal methods often imply that the distorted waves in DWBA (or DWIA) are taken as eikonal waves. Whereas PWBA and DWBA models are perturbative, mostly first-order perturbation theory, CDCC methods include all orders. Two methods are usually called Transfer to the Continuum. One is the semiclassical model of Bonaccorso and Brink [39], in the following renamed STC. The other (cf. Section 5.6) is a quantum mechanical model QTC of breakup which considers the nucleon–target full states generated by a real potential. In the STC instead the final nucleon–target states are generated by an energy dependent, complex, optical potential. Finally, in the spirit of the Glauber lectures in physics [40] and the work of Akhiezer and Sitenko (see Ref. [41] and references therein),‘‘Glauber method" is often used to describe multi-step collisions using eikonal waves for initial, intermediary, and final waves. In Glauber approaches the eikonal waves are often disguised in the form of profile functions, which include all multiple nucleon–nucleon scattering effects for a nucleus–nucleus collision at impact parameter b. Eikonal, or Glauber methods are also amenable to a proper inclusion of relativistic kinematics (conservation laws) and of relativistic dynamical corrections (interaction modification) [42,43]. The present review emerges from the necessity to improve the interpretation of experimentally determined directreaction cross sections. The starting point is a 15-year old puzzle in the comparison of experimental one-nucleon removal cross sections obtained from experiments performed at the NSCL, USA, in inverse kinematics using light (9Be, 12C) reaction targets and beam energies of about 70–120 MeV/nucleon, together with theoretical calculations based on shell-model spectroscopic factors and a S-matrix reaction formalism based on the eikonal approximation [44,46,52]. While it is expected that the calculated cross sections are 30%–50% larger than experimental values due to missing correlations in the shell model [15] as discussed above (in relation to Fig. 1), a systematic trend has been observed as a function of the asymmetry of the projectile in terms of proton to neutron separation energy (∆S=ϵ|Sn−Sp|,ϵ= +1 for proton removal, ϵ= −1 for neutron removal). This trend is illustrated in Fig. 2. Note that this trend is not observed in the analysis of transfer data from 14,16,18O oxygen isotopes [53], although this conclusion relies on one data set and depends on the inputs of the model [54]. A previous transfer study on 34,46Ar data did not suggest a large ∆S dependency [55,56] either, while another analysis of the same data concluded that it is possible to corroborate the neutron–proton asymmetry dependence reported from knockout measurements [57]. Recent experimental developments at GSI [58] and RIKEN [59] allowed quasifree scattering experiments to be performed in inverse kinematics with high-energy radioactive beams. The method has been applied to proton-knockout reactions on oxygen isotopes in a wide range of asymmetry. None 4 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 2. Compilation of the ratios Rsof the measured and calculated inclusive one-nucleon-removal cross sections for each of the labeled projectile nuclei. Rsis displayed as a function of ∆S, a measure of the asymmetry of the neutron and proton Fermi surfaces. The red (blue) points are for neutron(proton)-removal cases. The solid (black) squares, deduced from electron-induced proton knockout data, are identical to the earlier compilations of [44,45]. Source: The figure is adapted and updated from Ref. [46] – courtesy of J.A. Tostevin (2016); the added data points for 24O, 30Ne, 33Na, 36S and 71Co were then preliminary based on the now-published Refs. [47–51]. of these two experiments confirm such a strong dependence in the AO(p,2p) reduction factors, although analyzed by using different reaction models, as will be discussed in Section 5. Several recent developments have been made towards analyzing quasifree scattering data [60–62]. The above puzzle emerged from the study of isotopes far from stability; it may be seen as a proton-to-neutron asymmetry or binding-energy effect in nuclear structure models or the reaction dynamics, noting that both domains involve uncharted territory. A dependence of SRC to the proton-to-neutron asymmetry, as predicted by ab initio nuclearmatter theory [63,64], has been extracted from large-momentum-transfer (e,e′pN) data [28,65–67] but its amplitude does not account for the effect of Fig. 2. Consistently, modern ab initio structure theories predict a correlation dependence with nucleon separation energy [68,69], which reflects the trend but does not account for the magnitude of the slope and cannot explain the discrepancies among the conclusions obtained in the different experimental probes, suggesting the need for improved reaction descriptions. Part of the community therefore believes that the approximations in the modeling of the reactions should be refined for the reaction models involved. Solving this puzzle in the near future is crucial: with the current state-of-the-art reaction models, structure information extracted from direct reactions at different incident energies or with different targets will differ, clouding our ability to understand the underlying nuclear structure. It is thus essential to understand this discrepancy in view of the goal of obtaining consistent nuclear-structure information for neutron-rich nuclei from reactions performed at the existing and under-construction rare-isotope facilities. The interplay of structure and reaction mechanism in the interpretation of cross sections should also be further treated [19,70] and has been raised as a key issue in nuclear physics to be solved in the coming years in the NUPECC European Long Range Plan (2017) [71]. In this review, we propose to give up-to-date answers to the following questions: •What is the expected effect of correlations on the s.p. strength in stable and radioactive nuclei? •What are the main assumptions and model-dependent inputs to calculate direct-reaction cross sections? •What accuracy and precision can be reached from direct-reaction data to obtain structure information? •What are the possible explanations for the discrepancy between theory and experiment for light-ion induced deeply-bound-nucleon removal at incident energies of about 100 MeV/nucleon? •What is the consensus within the community, and which particular points are still a source of controversy? •What new experimental data are called for to solve this problem and to guide theory? •What theoretical developments should be put forward towards a consistent treatment of the reaction mechanism and structure? With these objectives in mind, a theoretical background and an overview of past works on short- and long-range correlations in nuclei is given in Section 2. In particular, the concepts of s.p. Green’s function, self energy, spectroscopic factors, and spectral distributions are detailed. Analysis of the nucleon self-energy and qualitative comparison with results from the (e,e′p) reaction are presented. In the following sections, the most common direct-reaction mechanisms used for nuclear-structure studies — namely low-energy nucleon transfer (Section 3), heavy-ion induced nucleon removal (Section 4), and proton-induced quasifree scattering (Section 5) — are introduced together with their main approximations. Other direct reactions used to probe weakly-bound nuclei, such as Coulomb breakup [72] and resonant elastic 5 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 scattering [73], are not discussed in the following since they are not applicable to a wide range of ∆S, which is the focus of the review. The consistency of the studied reaction mechanisms in terms of extracting structure information is tested against a unique benchmark (Section 6): the proton removal from 16O. The comparison of inclusive cross-section data with predictions as a function of the proton-to-neutron asymmetry ∆S is addressed for all the above-mentioned reaction mechanisms. Conclusions and still-open questions from past works are emphasized, as well as possible interpretations for the origin of the so-called ‘‘reduction factor’’ at the heart of the present review. Perspectives for both theory and experiment are given in Section 7. 2. Single-particle spectral functions and correlations 2.1. Introduction In this section, a summary is compiled of some of the theoretical concepts that play a role in extracting information from direct reactions that intend to elucidate the behavior of individual nucleons in nuclei. The chosen language here is that of the Green’s function method [74] as it simultaneously covers the domain of the continuum relevant for nucleon scattering as well as structure information related to nucleon removal. In addition it is a suitable method that can be utilized to perform ab initio calculations of relevant quantities [75]. In Section 2.2 the nucleon Green’s function or propagator is introduced which contains at negative energy the nucleon removal amplitudes both to discrete as well as continuum final states. It is often referred to as the s.p. spectral function (for removal). While such a quantity is also available for nucleon addition both to discrete and continuum states, it is more practical in this energy domain to consider the equivalent information encoded in the nucleon self-energy which in earlier practical incarnations has been referred to as the optical potential. The link between the energy domains related to nuclear structure and reactions is briefly emphasized in Section 2.3 by discussing the dispersion relation connecting them. This approach forms the basis of the dispersive optical model (DOM) which provides an ideal framework to simultaneously treat reactions and structure and can also perform an intermediate role between experiment and ab initio theory. With this theoretical background (pedagogically illustrated in Appendix A) it is possible to understand how to employ removal amplitudes to valence hole states as probed in (e,e′p) reactions to determine the extraction of their normalization or so-called spectroscopic factors which are the main topic of this review. Calculations of spectroscopic factors and the physical ingredients that determine their quantitative value are presented in Section 2.5 with attention to the role of both long-range and short-range correlations. Recent results from the DOM are discussed as they provide a framework to utilize the Green’s function method to let experimental data constrain the nucleon propagator and self-energy and thereby generate implied quantitative results for spectroscopic factors. This efficacy is illustrated in Section 2.6 which provides an essential confirmation of the analysis of the (e,e′p) by the NIKHEF efforts [15] as well as provides insight into the nucleon asymmetry dependence of proton spectroscopic factors in Ca isotopes. We will complete our discussion of the theoretical aspects of correlations by discussing the most recent results from ab initio theory, in Section 2.7. 2.2. Single-particle Green’s function and the self-energy We start with a brief summary of relevant results from the Green’s function formulation of the many-body problem [74]. The s.p. propagator in a many-particle system is defined as G(α, β;t−t′)= −i ¯ h⟨ΨA 0|T[aαH(t)a† βH(t′)]|ΨA 0⟩.(2) The expectation value with respect to the exact ground state of the system of Aparticles, samples an operator that represents both particle as well as hole propagation. The state |ΨA 0⟩is the normalized, nondegenerate Heisenberg ground state for the A-particle system and EA 0the corresponding eigenvalue. The particle addition and removal operators in the definition of the sp propagator are given in the Heisenberg picture and the labels αor βrefer to an appropriate complete set of quantum numbers associated with a s.p. basis. The time-ordering operation Tis defined here to include a sign change when two fermion operators are interchanged. The propagator depends only on the time difference t−t′. In the following we employ the completeness of the exact eigenstates of ˆ Hfor both the A+1 as well as the A−1 system, together with the corresponding eigenvalues of the Hamiltonian. Using the integral representation of the step function, the Fourier transform of the propagator can be expressed as: G(α, β;E)=∑ m ⟨ΨA 0|aα|ΨA+1 m⟩⟨ΨA+1 m|a† β|ΨA 0⟩ E−(EA+1 m−EA 0)+iη+∑ n ⟨ΨA 0|a† β|ΨA−1 n⟩⟨ΨA−1 n|aα|ΨA 0⟩ E−(EA 0−EA−1 n)−iη.(3) The expression is known as the Lehmann representation [76] of the sp propagator. Note that any s.p. basis can be used in this formulation of the propagator. Continuum solutions in the A±1 systems are also implied in the completeness relations but are not explicitly included to simplify the notation. At this point one assumes that a meaningful nuclear Hamiltonian ˆ H 6 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 exists that contains a two-body component that describes nucleon–nucleon scattering up to a chosen energy, usually the pion production threshold, as well as relevant bound-state data. There is sufficient experimental evidence and theoretical insight suggesting that at least a three-body component should also be present in the Hamiltonian. Maintaining the present general notation with regard to the s.p. quantum numbers, we introduce the spectral functions associated with particle and hole propagation. At energy E, the hole spectral function represents the combined probability density for removing a particle with quantum numbers αfrom the ground state, while leaving the remaining A−1-system at an energy EA−1 n=EA 0−E. The relation to the imaginary part of the diagonal element of the s.p. propagator is given by Sh(α, E)=1 πIm G(α, α;E)=∑ n⏐⏐⏐⟨ΨA−1 n|aα|ΨA 0⟩⏐⏐⏐ 2 δ(E−(EA 0−EA−1 n)),(4) for E≤ε− Fidentifying the Fermi energy for the removal of a particle. The probability density for the addition of a particle with quantum numbers α, leaving the A+1-system at energy EA+1 m=EA 0+Esimilarly reads Sp(α, E)= −1 πIm G(α, α;E)=∑ m⏐⏐⏐⟨ΨA+1 m|a† α|ΨA 0⟩⏐⏐⏐ 2 δ(E−(EA+1 m−EA 0)) (5) for E≥ε+ F. Eq. (5) defines the particle spectral function. The Fermi energies introduced in Eqs. (4) and (5) are given by ε− F=EA 0−EA−1 0(6) and ε+ F=EA+1 0−EA 0,(7) respectively. The occupation number of a s.p. state αcan be generated from the hole part of the spectral function by evaluating n(α)= ⟨ΨA 0|a† αaα|ΨA 0⟩ = ∑ n⏐⏐⏐⟨ΨA−1 n|aα|ΨA 0⟩⏐⏐⏐ 2=∫ε− F −∞ dE Sh(α, E).(8) The depletion (or emptiness) number is determined by the particle part of the spectral function d(α)= ⟨ΨA 0|aαa† α|ΨA 0⟩ = ∑ m⏐⏐⏐⟨ΨA+1 m|a† α|ΨA 0⟩⏐⏐⏐ 2=∫∞ ε+ F dE Sp(α, E).(9) An important sum rule exists for n(α) and d(α) which can be deduced by employing the anticommutation relation for aα and a† α n(α)+d(α)= ⟨ΨA 0|a† αaα|ΨA 0⟩+⟨ΨA 0|aαa† α|ΨA 0⟩=⟨ΨA 0|ΨA 0⟩ = 1.(10) The distribution between occupation and emptiness of a sp orbital in the correlated ground state is a sensitive measure of the strength of correlations, provided a suitable s.p. basis is chosen. The s.p. propagator generates the expectation value of any one-body operator in the ground state ⟨ΨA 0|ˆ O|ΨA 0⟩ = ∑ α,β ⟨α|O|β⟩⟨ΨA 0|a† αaβ|ΨA 0⟩ = ∑ α,β ⟨α|O|β⟩nαβ .(11) Here, nαβ is the one-body density matrix element which can be obtained from the s.p. propagator using the Lehmann representation nβα =∫dE 2πieiEηG(α, β;E).(12) The convergence factor ensures that only removal amplitudes contribute. Knowledge of Gin terms of nβα, therefore yields the expectation value of any one-body operator in the correlated ground state according to Eq. (11). An important recent application of this result concerns the nuclear charge density which is measured in detail for stable closed-shell nuclei, providing important constraints on the properties of the s.p. propagator. If neutron properties related to scattering are constrained and isospin symmetry is invoked, it is also be possible to make predictions for the neutron distribution and as a consequence the neutron skin. The perturbation expansion of the s.p. propagator is discussed in various textbooks, e.g., Refs. [74,77]. It is necessary to order the expansion into the so-called Dyson equation to obtain a meaningful nonperturbative link between the inmedium potential experienced by a nucleon, the so-called irreducible self-energy, and the propagator. The resulting equation is G(α, β;E)=G(0)(α, β;E)+∑ γ ,δ G(0)(α, γ ;E)Σ(γ , δ;E)G(δ, β;E).(13) 7 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 The noninteracting propagator G(0) can be chosen depending on the particular problem [74,75] and most often incorporates global conservation laws especially those associated with rotational symmetry and parity in nuclear applications. We note that for the purpose of performing approximate calculations of the nucleon self-energy often a noninteracting propagator is chosen that corresponds to localized nucleons [75] which is accomplished by a corresponding term in the irreducible self-energy so that its effect ultimately cancels out. Ab initio calculations in infinite systems naturally proceed from the noninteracting Fermi gas starting point. For applications of the DOM, it is convenient to work in a coordinate or momentum-space s.p. basis suitably accompanied by conserved quantum numbers. The corresponding quantum numbers are the orbital and total angular momentum of the nucleon, which can therefore also be employed to label the propagator and the irreducible self-energy. When calculating the nucleon propagator with respect to the A-body ground state in the s.p. basis with good radial position (or momentum), orbital angular momentum (parity) and total angular momentum, the numerators of the particle and hole components of the propagator in the Lehmann representation in Eq. (3) include the products of overlap functions associated with adding or removing a nucleon from the A-body ground state. For the present discussion the noninteracting propagator is assumed to involve only kinetic energy contributions. The nucleon self-energy contains all linked diagrammatic contributions that are irreducible with respect to propagation represented by G(0). All contributions to the propagator are then generated by the Dyson equation itself. The solution of the Dyson equation generates all discrete poles corresponding to bound A±1 states explicitly given by Eq. (3) that can be reached by adding or removing a particle with quantum numbers rℓj. The hole spectral function is obtained from Sℓj(r;E)=1 πIm Gℓj(r,r;E) (14) for energies in the A−1 continuum. The total spectral strength at Efor a given ℓjcombination, Sℓj(E)=∫∞ 0 dr r2Sℓj(r;E),(15) yields the spectroscopic strength per unit of energy. For discrete energies as well as all continuum ones, overlap functions for the addition or removal of a particle are generated as well. For discrete states in the A−1 system one can show that the overlap function obeys a Schrödinger-like equation [74]. Introducing the notation ψn ℓj(r)= ⟨ΨA−1 n|arℓj|ΨA 0⟩,(16) for the overlap function for the removal of a nucleon at rwith discrete quantum numbers ℓand j, one finds [p2 r 2m+¯ h2ℓ(ℓ+1) 2mr2]ψn ℓj(r)+∫dr′r′2Σℓj(r,r′;ε− n)ψn ℓj(r′)=ε− nψn ℓj(r),(17) where ε− n=EA 0−EA−1 nand in coordinate space the radial momentum operator is given by pr= −i¯ h(∂/∂r+1/r). Discrete solutions to Eq. (17) exist in the domain where the self-energy has no imaginary part and these are normalized by utilizing the inhomogeneous term in the Dyson equation. For an eigenstate of the Schrödinger-like equation [Eq. (17)], the so-called quasihole state labeled by αqh, the corresponding normalization or spectroscopic factor is given by [74] Zn ℓj=(1−∂Σℓj(αqh, αqh;E) ∂E⏐⏐⏐⏐ε− n)−1 ,(18) which is the discrete equivalent of Eq. (15). The particle spectral function for a finite system can be generated by the calculation of the reducible self-energy T which also yields direct access to all elastic-scattering observables in the case of neutrons. For protons a direct solution of the relevant differential equation is usually employed to generate scattering observables. In some applications relevant for elucidating correlation effects, a momentum-space calculation [78] can be employed. In an angular-momentum basis, iterating the irreducible self-energy Σto all orders, then yields Tℓj(k,k′;E)=Σℓj(k,k′;E)+∫dq q2Σℓj(k,q;E)G(0)(q;E)Tℓj(q,k′;E),(19) where G(0)(q;E)=(E−¯ h2q2/2m+iη)−1is the free propagator. The propagator is then obtained from an alternative form of the Dyson equation in the following form [74] Gℓj(k,k′;E)=δ(k−k′) k2G(0)(k;E)+G(0)(k;E)Tℓj(k,k′;E)G(0)(k′;E).(20) The on-shell matrix elements of the reducible self-energy in Eq. (19) are sufficient to describe all aspects of elastic scattering like differential cross sections, reaction cross sections, and total cross sections as well as polarization data [78]. This connection between the nucleon propagator and elastic-scattering data identifies the nucleon elastic-scattering T-matrix with the reducible self-energy obtained by iterating the irreducible one to all orders with G(0) [74,79–81]. 8 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 The spectral representation of the particle part of the propagator, referring to the A+1 system, appropriate for a treatment of the continuum and possible open channels is given by [82] Gp ℓj(k,k′;E)=∑ n φn+ ℓj(k)[φn+ ℓj(k′)]∗ E−E∗A+1 n+iη+∑ c∫∞ Tc dE′χcE′ ℓj(k)[χcE′ ℓj(k′)]∗ E−E′+iη,(21) generalizing the discrete formulation of Eq. (3). Overlap functions for bound A+1 states are given by φn+ ℓj(k)= ⟨ΨA 0|akℓj|ΨA+1 n⟩, whereas those in the continuum are given by χcE ℓj(k)= ⟨ΨA 0|akℓj|ΨA+1 cE ⟩, indicating the relevant channel by c and the energy by E. Excitation energies in the A+1 system are with respect to the A-body ground state E∗A+1 n=EA+1 n−EA 0. Each channel chas an appropriate threshold indicated by Tcwhich is the experimental threshold with respect to the ground-state energy of the A-body system. The overlap function for the elastic channel can be explicitly calculated by solving the Dyson equation while it is also possible to obtain the complete spectral density for E>0 Sp ℓj(k,k′;E)=∑ c χcE ℓj(k)[χcE ℓj(k′)]∗.(22) In practice, this requires solving the scattering problem twice at each energy so that one may employ Sp ℓj(k,k′;E)=i 2π[Gp ℓj(k,k′;E+)−Gp ℓj(k,k′;E−)](23) with E±=E±iη, and only the elastic-channel contribution to Eq. (22) is explicitly known. Equivalent expressions pertain to the hole part of the propagator Gh ℓj[82]. Calculations are performed in momentum space according to Eq. (19) to generate the off-shell reducible self-energy and thus the spectral density by employing Eqs. (20) and (23). Because the momentum-space spectral density contains a deltafunction associated with the free propagator, it is convenient to also consider the Fourier transform back to coordinate space Sp ℓj(r,r′;E)=2 π∫dkk2∫dk′k′2jℓ(kr)Sp ℓj(k,k′;E)jℓ(k′r′),(24) which has the physical interpretation for r=r′as the probability density Sℓj(r;E) for adding a nucleon with energy Eat a distance rfrom the origin for a given ℓjcombination. By employing the asymptotic analysis to the propagator in coordinate space following, e.g., Ref. [74], one may express the elastic-scattering wavefunction that contributes to Eq. (24) in terms of the half on-shell reducible self-energy obtained according to χelE ℓj(r)=[2mk0 π¯ h2]1/2{jℓ(k0r)+∫dkk2jℓ(kr)G(0)(k;E)Tℓj(k,k0;E)},(25) where k0=√2mE/¯ his related to the scattering energy in the usual way. The constraints imposed on the nucleon self-energy by elastic scattering data also indirectly determine the presence of strength in the continuum associated with mostly-occupied orbits (or mostly empty but E<0 orbits). This strength is obtained by double folding the spectral density in Eq. (24) in the following way Sn+ ℓj(E)=∫drr2 ∫dr′r′2φn− ℓj(r)Sp ℓj(r,r′;E)φn− ℓj(r′),(26) using an overlap function √Sn ℓjφn− ℓj(r)= ⟨ΨA−1 n|arℓj|ΨA 0⟩,(27) with φn− ℓj(r) normalized to 1 and for valence orbits, Sn ℓjtheir spectroscopic factor [78]. In practice, any normalized bound-state wavefunction can be utilized. In the case of an orbit below the Fermi energy, this strength identifies where the depleted strength resides in the continuum. The occupation number of this orbit is given by an integral over a corresponding folding of the hole spectral density Sn− ℓj(E)=∫drr2 ∫dr′r′2φn− ℓj(r)Sh ℓj(r,r′;E)φn− ℓj(r′),(28) where Sh ℓj(r,r′;E) provides equivalent information below the Fermi energy as Sp ℓj(r,r′;E) above. An important sum rule is valid for the sum of the occupation number nnℓjfor the orbit characterized by nℓj nnℓj=∫εF −∞ dE Sn− ℓj(E) (29) 9 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 7. The momentum distribution of 16O adapted from Ref. [131]. Shown are also the quasihole contribution and the results obtained with various energy cutoffs in the integration of the spectral functions. The units for n(k) are in fm and therefore contain already the factor k2. can generate sensible results. An example is provided by the Green’s function method. Recent work on a direct calculation of the nucleon self-energy for 40Ca with an emphasis on SRC was reported in Ref. [78]. Such a microscopic calculation of the nucleon self-energy proceeds in two steps, as employed in Refs. [130–132]. A diagrammatic treatment of SRC always involves the summation of ladder diagrams. When only particle–particle (pp) intermediate states are included, the resulting effective interaction is the so-called G-matrix. Employing such an interaction calculated in nuclear matter, it is possible to iterate the difference between this result and the appropriate finite nucleus quantity thereby generating a sensible approximation to the finite nucleus self-energy that adequately treats SRC. The momentum distribution from such a calculation for 16O is shown in Fig. 7. The quasihole part corresponds to the energy domain for (e,e′p) cross sections with small energy transfer, i.e., leading to the ground state of the final nucleus and excited states up to ≈20 MeV. The curve denoted by E>−100 MeV reflects the momentum distribution, including all states of the final nucleus up to around 80 MeV excitation energy, etc. As a consequence, the high-momentum components of the momentum distribution due to short-range correlations including the effect of the tensor force (inclusively denoted by SRC in the following) can be observed mainly in knockout experiments with an energy transfer of the order of 100 MeV or more. Such high-momentum components were indeed not found experimentally near the Fermi energy [133]. The momentum distribution for 16O calculated with different approaches and different interactions, all of which properly include the effects of SRC, yield quantitatively similar results. The other approaches include: local density approximation (LDA) [134], Fermi hypernetted chain (FHNC) [135] and Variational Monte Carlo (VMC) [136]. To understand Fig. 7, it is important to recall that the appearance of high-momentum components at a certain energy in the A−1 system is related to the self-energy contribution containing 1p2h states at that energy. From energy conservation it is then clear that at low energy it is less likely to find such states with a high-momentum particle, than at high energy. The same feature is observed in nuclear matter where the peak of the s.p. spectral function for momenta above kFincreases in energy as k2. Hence, the hole strength in nuclear matter as a function of momentum, exhibits the tendency that higher momenta become more dominant at higher excitation energy. The location of high-momentum components was confirmed in [137] with an (e,e′p) experiment on 12C while also demonstrating that about 10% of the protons exhibited this high-momentum behavior. It is noteworthy that this also implies that the reduction of s.p. strength due to SRC associated with removing strength also roughly corresponds to this value. Experimental NIKHEF data suggest that a substantial additional reduction is required to describe the valence hole distributions of 16O [138]. Such an additional reduction must therefore be associated with LRC. A complete treatment of SRC can actually be accomplished in nuclear matter calculations. The self-energy diagrams of Fig. 58(a) of Appendix A can be calculated completely when Γ4pt is approximated by the sum of ladder diagrams which ensures a proper treatment of SRC. In turn the ladder diagrams are obtained by including the fully off-shell propagating particles in the medium. This incorporates the important physics that particles propagate with knowledge of the correlated ground state in which occupation below the Fermi momentum is depleted and compensated by the occupation of high-momentum states. The self-consistent implementation of such a scheme has been fully realized in recent finite temperature calculations that avoid the technical complications of having to deal with possible pairing instabilities as 16 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 8. Occupation probability for nuclear matter at equilibrium density calculated by integrating hole spectral functions obtained with mean-field propagators as input (dotted line). Self-consistent determination of the s.p. propagators yields the solid line [140]. discussed, e.g., in Refs. [64,139]). An approximate result at zero temperature was obtained in [140] employing the Reid softcore interaction [120]. The resulting momentum distribution is shown in Fig. 8 and compared with an earlier calculation that propagated mean-field particles in the summation of ladder diagrams. While these two distributions are quite similar, the corresponding spectral function exhibits substantial differences that influence the saturation properties of nuclear matter [141]. On the basis of such calculations it was predicted that a similar depletion of orbits in finite nuclei would be observed [142] which was later confirmed in Ref. [105]. Near k=0, n(k) becomes fairly constant with a value of 0.85. Investigating the influence of the tensor force, it may be concluded that roughly 1/3 of the 15% depletion is due to the effect of tensor correlations in the ladder equation. Another 1/3 is due the to high-energy tail in the particle spectral function at energies above 500 MeV [143,144]. Other many-body methods such as Brueckner theory [145] and [146] and correlated basis functions (CBF) theory [147], using other realistic interactions, produce very similar occupation near k=0. In the work of [145] and [146], 0.82 is reported for the Paris potential. All these calculations for different interactions, using different methods, give similar values for n(0) which appears to be mostly determined by the strength of the repulsive core and the inclusion of a realistic tensor force. This is encouraging, since it implies that nonrelativistic many-body calculations yield stable results in the region, where one would like to compare to finite nuclei. In contrast to n(0), the occupation at kFvaries significantly between methods. The extra depletion in n(k) as k→kFarises from the enhanced ability of the s.p. state to couple to low lying 2p1h excitations, as its energy approaches these states and therefore reflects the effect of LRC in nuclear matter. These conclusions are essentially maintained when more recently developed interactions that contain a substantial repulsion like AV18 [122] and CD-Bonn [123] are employed. These interactions generate depletions at k=0 of 13% and 11%, respectively [64]. This information is displayed in Fig. 9. The figure displays the occupation of the k=0 momentum state in matter at a density of 0.16 fm−3and temperature T=5 MeV as a function of nucleon asymmetry α=(N−Z)/(N+Z) for protons and neutrons, as well as for the CD-Bonn (dotted lines) and AV18 (solid lines) interactions obtained from a fully self-consistent calculations as outlined above. The inclusion of the Fermi gas result indicates that the proton results become temperature dependent at asymmetries beyond α=0.6 clarifying the earlier result of Ref. [63]. For the correlated depletion, the occupation of the zero momentum state is an increasing (decreasing) function of the asymmetry for the more (less) abundant component. The behavior is very similar for both nucleon–nucleon interactions, although the occupations for AV18 are systematically smaller than those for CD-Bonn. The physical effect of this increasing difference between neutron and proton depletion and the implied occupation of high-momentum states must be sought in the increasing (decreasing) importance of the 3S1-3D1channel for protons (neutrons) with increasing nucleon asymmetry. The effect is therefore associated with the tensor force. In spite of the differences observed in both symmetric nuclear and pure neutron matter for the AV18 and CD-Bonn results, the asymmetry dependence of the k=0 occupation is very similar at all densities. This is surprising because both forces have a rather different short-range behavior and tensor structure. Given the almost linear dependence of the depletion with asymmetry, a better insight into these differences can be gained by plotting the difference between the k=0 occupation of neutrons and protons. In Fig. 10, this iso-depletion is shown for different interactions as a function of the asymmetry again at T=5 MeV. This difference is the same for a wide variety of modern nucleon–nucleon potentials, independent of 17 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 9. Isospin asymmetry dependence of the neutron (filled circles) and proton (squares) lowest momentum occupation at T=5 MeV and density of 0.16 fm−3. Correlated results for the AV18 (solid lines) and CD-Bonn (dotted lines) interactions are compared to the free Fermi gas (dashed lines) predictions. Source: Reprinted figure with permission from Ref. [64]. ©2009 The American Physical Society. Fig. 10. Difference of neutron and proton occupation of the lowest momentum state as a function of isospin asymmetry for different interactions at T=5 MeV for a density of 0.16 fm−3. The curve labeled FFG refers to the free Fermi gas results while the other curves refer to fully self-consistent results for various nucleon–nucleon interactions. Source: Reprinted figure with permission from Ref. [64]. ©2009 The American Physical Society. their short-range or operatorial structure. This seems to suggest that the iso-depletion is fixed by the phase shifts, most probably via their isospin dependence. This is corroborated at higher densities by comparing the very similar AV18 and CD-Bonn predictions [64]. Two results in Fig. 10 fall below the main iso-depletion line. Immediately below most of the interactions, one finds the results corresponding to the AV4′potential. This interaction has an extremely simplified operatorial structure, with only a spin–isospin part and no tensor components, and is fitted to reproduce the binding energy of the deuteron [148]. The fact that it lies significantly below the other results shows the importance of tensor effects for isospin asymmetric systems. It appears that the tensor force tends to increase the difference between neutron and proton momentum distributions as asymmetry increases. Moreover, the comparison with AV6′and AV8′suggests that, once the tensor components are included in a force, the iso-depletion will remain almost the same independently of the extra spin–orbit terms. This fact is very surprising, particularly if one considers the fact that the momentum distributions of neutrons and protons can be different for each potential. These results appear to be relevant for the discussion of the presence of high-momentum components of protons and neutrons when there is a neutron majority as in 48Ca and 208Pb [66]. In addition, this increased correlation for protons with increasing nucleon asymmetry must ultimately play a role in the removal probability of valence orbits when large nucleon asymmetries are encountered. The importance of the nuclear tensor force is also well documented in the dominance of proton–neutron events in experiments where two nucleons are removed [65]. 2.5.3. Combined effects of LRC and SRC The role of SRC is important but not sufficient to account for the extracted spectroscopic factors from the (e,e′p) reaction. The effects of SRC on the spectroscopic factors of 16O have been computed in Refs. [130,149,150]. All these 18 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 11. Imaginary volume integrals Jℓ Wof the 40Ca self-energy for neutrons. The dashed curves represent the FRPA results. The results of the nonlocal DOM fit of Ref. [32] are shown by the solid lines. Source: Figure adapted from Fig. 11 of Ref. [33]. approaches treat essentially only SRC and yield spectroscopic factors of about 0.90 for the knockout of a proton for the p1/2 and p3/2shells in the (e,e′p) reaction. These results disagree with the extracted experimental value of about 0.65 for these orbitals employing the standard NIKHEF analysis [138]. In Ref. [151] a successful attempt to combine the treatment of lowenergy long-range and short-range (and tensor) correlations was achieved using the energy dependence of the G-matrix to account for the depleting effect of SRC. The final result yielded a p3/2spectroscopic factor of 76%. This points to the importance of LRC which were included by taking into account the interaction between the hole and particles propagating in the system at the level of the TDA and therefore cannot go beyond the 2h1p level. Calculations reported in [115] can account for the coupling of quasiparticles to RPA collective modes and therefore go beyond 2h1p contributions, although it is limited by the difficulties in including the coupling to collective excitations in both the pp(hh) and ph channels. A procedure to include both ph and pp(hh) excitations at the RPA level employing a Faddeev technique (FRPA) in the construction of the self-energy was proposed and implemented in Refs. [152,153]. The results of these calculations only slightly reduce the above mentioned spectroscopic factors but provide an excellent strategy for accurate calculations in finite electronic systems [154,155] and other nuclei (see, e.g., Ref. [156]). An important observation is that the configuration space needed for the incorporation of long-range (surface) correlations, including the coupling to giant resonances, is much larger than the space that can be utilized in large-scale shell-model diagonalizations. FRPA calculations for 40Ca and 48Ca and 60Ca were performed to shed light on the ab initio self-energy properties in medium-mass nuclei including the nucleon asymmetry dependence [157]. The main goal of this work was to clarify whether substantial nonlocal contributions should be expected when optical potentials for elastic scattering are considered. In particular, one may expect to extract useful information regarding the functional form of the DOM potentials from a study of the self-energy for a sequence of calcium isotopes. The resulting analysis was intended to provide a microscopic underpinning of the qualitative features of empirical optical potentials. Additional information concerning the degree and form of the nonlocality of both the real and imaginary parts of the self-energy were also addressed. Early implementations of the DOM employed local potentials and an energy dependence of its imaginary part symmetrically centered around εF[82]. Such features are not obtained in the FRPA as illustrated in Fig. 11, where for ℓ-values up to 5, the volume integrals of the FRPA calculation are displayed by the dashed lines. The domain of the DOM fit extends beyond those shown in Fig. 11, but is limited to those energies for which a meaningful comparison with FRPA results is possible. We therefore also include the result of the nonlocal fit of Ref. [32] shown by the solid lines in Fig. 11 which confirms this assessment. Since the absorption above the Fermi energy is strongly constrained by elastic-scattering data, it is encouraging that the ℓ-dependent FRPA result is reasonably close to the DOM fit in the domain where the FRPA is expected to be relevant on account of the size of the chosen configuration space. Note that the calculated JWvalues decrease quickly at energies E−εF>100 MeV due to the truncation the model space. Instead, it correctly remains sizeable even at higher energies 19 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 12. The distribution of s.p. strength in a nucleus like 208Pb. The present summary is a synthesis of experimental and theoretical work discussed in this section. A slight reduction (from 15% to 10%) of the depletion effect due to SRC must be considered for light nuclei like 16O. Source: Reprinted figure with permission from [75]. ©2004 Elsevier. in the DOM. Also at negative energies, the FRPA results do not adequately describe the admixture of high-momentum components that occur at large missing energies. An important conclusion is therefore that ab initio calculations most likely will not succeed in generating accurate optical potentials that can be used to analyze nuclear reactions. The conclusion also holds when the coupled-cluster approach is employed [36,37]. It is for this reason that the DOM provides a suitable interface between experimental data and ab initio theory. Theory informs the functional forms employed in the DOM analysis but cannot describe elastic scattering data in detail. The potentials generated by the DOM can however be employed to analyze nuclear reactions as will be clarified in Section 2.6. As discussed in Ref. [75], knowledge obtained from the (e,e′p) reaction and elastic electron scattering combined with some theoretical ingredients shed light on the properties of protons in the nucleus and may be summarized as follows. The consequences of SRC in nuclear systems appear to be theoretically well understood, while they have also become available for experimental scrutiny [137]. These results demonstrate that the effect of SRC is two-fold. First, it involves the depletion of spectroscopic strength from the mean-field domain. The data in from Ref. [105] indicate that the depletion in heavy nuclei corresponds to a little over 15% for all the deeply bound proton levels. This was predicted quite some time ago based on nuclear-matter calculations as discussed above [142]. For lighter nuclei all theoretical work suggests that this amount may be closer to 10%, also discussed earlier. Accompanying this information is the realization that valence shells near the Fermi energy will not contain substantial amounts of high-momentum components. This has been experimentally confirmed and clarifies the other role played by SRC in nuclei, i.e., the admixtures of highmomentum components at high missing energy that account for the missing protons removed from the mean-field location. The location of these high-momentum components [137] broadly conforms with the mechanism that admixes these correlations with 1p2h states at large missing energies. Being able to identify high-momentum components in addition to locating all the s.p. strength, associated with the mean-field orbits [105], completes the identification of the properties of protons in the ground state of double closed-shell nuclei like 16O, 40Ca, 48Ca, and 208Pb. The latter understanding is illustrated in Fig. 12, where several generic diagrams are identified that have unique physical consequences for the redistribution of the s.p. strength. The middle column characterizes the mean-field picture that is used as a starting point for the theoretical description. The right column identifies the location of the s.p. strength of the orbits, just below the Fermi energy, when correlations are included. One may apply this picture, for example, to the 2s1 2proton orbit in 208Pb. The physical mechanisms responsible for the correlated strength distribution are also identified. The strength of this orbit, remaining at the quasihole energy, is about 20 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 65%. Long-range correlations are responsible for the loss of 20% of the strength due to the coupling to nearby 2p1h and 1p2h and more complicated states. This loss is roughly symmetrically distributed above and below the Fermi energy and is physically represented by the coupling to low-lying surface modes and higher-lying giant resonances. The resulting occupation number of the orbit therefore corresponds to about 75%. More deeply bound nucleons have higher occupation numbers corresponding to about 85%. This is true for all the deep-lying orbits, and is consistent with a global depletion due to SRC of 15%. The corresponding location of this strength is identified at very high energy in the particle domain and is due to the SRC induced by a realistic nucleon–nucleon interaction with a substantial repulsive core. The left column depicts the generic diagram that is responsible for the admixture of high-momentum components in the ground state. The energy domain of these high-momentum nucleons is at large missing energies as discussed for Fig. 7. This rather complete picture of the properties of a proton in the nucleus is unique to the field of nuclear physics. Indeed, unlike other fields with strong correlation effects, like particle or condensed matter physics, it is possible in nuclear physics to state that the properties of the constituent protons inside the nucleus are identified experimentally and understood in global theoretical terms. While an update of these considerations will be presented in Section 2.6, the crucial question how these correlations for protons change when neutrons are added or removed, and, conversely, what happens with neutrons when they are in the majority remains the main concern of the present work. Some glimpses have been provided by the study of SRC as discussed for Figs. 9 and 10. 2.5.4. Some results from the dispersive optical model A recent, more detailed overview of the DOM work described in this section can be found in Ref. [33] while a more general review of the optical model is provided in Ref. [34]. The Washington University group in St. Louis first performed local dispersive optical-model fits to understand the N−Zasymmetry dependence of correlations in nuclei [158–160]. The forms of the potential were largely consistent with those used by Mahaux and Sartor [82]. Fits for neutron and proton elastic-scattering angular distributions, total and absorption cross sections, s.p. energies and spectroscopic factors deduced from (e,e′p) reactions were generated. For protons, the long-range correlations associated with the surface imaginary potential showed a strong N−Zasymmetry dependence, with the proton correlations increasing in neutron-rich systems. This was consistent with the assumed asymmetry dependence of this potential in most global optical-model fits. On the other hand, neutrons seemed to have very little, if any, dependence. The optical model parametrizations were later used to provide distorted waves and overlap functions for the analysis of transfer reactions [161]. The connection of the local DOM optical potential and the self-energy in the Green’s-function formalism can be made if the energy-dependent real potential in the local optical model fits is replaced by an energy-independent nonlocal one. Efforts in this regard were made for the case of 40Ca [162] where the fitted energy-dependent real potential was replaced with such a nonlocal potential. This limited the binding of the lowest s1/2in better agreement with experiment. However, it was not possible to describe the charge distribution generating too much charge at small radii, as well as particle number. These problems were cured when a fully nonlocal fit to n,p+40Ca data was performed with nonlocality in both the real and imaginary potentials [32]. As well as the positive-energy reaction data, this fit also included the experimental s.p. energies, the experimental charge distribution and the neutron and proton particle numbers. The introduction of nonlocality in the imaginary potential changes the absorption profile as a function of angular momentum and was essential in reproducing the particle numbers and the experimental charge distribution. This fit allowed for the reproduction of the yield of high-momentum nucleons determined by data from Jefferson Lab [32]. We note that these high-momentum components involve the single-nucleon spectral function as extracted from measurements reported in Refs. [137,163]. The DOM is therefore not yet capable of addressing the relative momentum distribution as probed by knockout experiments involving two removed nucleons [65]. Nonlocal DOM potentials allow for a calculation of the complete off-shell elastic nucleon-scattering matrix in momentum space, as discussed in Section 2.2. Using Eqs. (26) and the appropriate complementary information below the Fermi energy given by Eq. (28), it is possible to generate the complete spectral function for suitably chosen bound s.p. wavefunctions. By identifying the wavefunctions associated with the main peaks of the spectral distribution for each ℓj-combination, including those that correspond to valence hole states near the Fermi energy, it is possible to map out the complete distribution of these states as a function of energy. From the nonlocal DOM potentials it is thus possible to generate the spectral function for bound orbitals at both negative energies [32] and at positive energies [164], as shown in Fig. 13 for neutron orbits in 40Ca. For plotting purposes the small imaginary part near the Fermi energy was employed giving the peaks a small width. It is particularly important to note that the strength of bound orbits that resides in the particle continuum is constrained by the fits to experimental data. While the distributions follow the expected behavior observed in (e,e′p) reactions, the particle strength displays considerable sensitivity to the location of the main peak relative to the continuum. The deeply bound 0s1 2exhibits a depletion of a few percent (with about 5% above 200 MeV) implying an occupation of about 90%. The valence 1s1 2orbit exhibits an additional 5% depletion and is therefore 85% occupied in this analysis. For mostly empty orbits with main peaks near the continuum threshold considerably more strength is identified in the continuum as shown in Fig. 13. This behavior suggests that considerable s.p. strength may be located in the immediate vicinity of the main fragment of the distribution when this occurs near the continuum. This observation may be of paramount importance for properties of the s.p. strength in exotic nuclei. The nonlocal DOM treatment has also been extended to the n,p+48Ca reactions and used to deduce the neutronmatter and weak-charge density distributions. The results imply a neutron skin of 0.249 ±0.023 fm [165] which is larger than the prediction of 0.12–0.15 fm from the coupled-clusters formalism [166]. Future experiments at Jefferson Lab with parity-violating electron scattering should be able to resolve this disagreement [167]. 21 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 13. Spectral function for neutron s.p. orbitals in 40Ca obtained by the St. Louis group. The sequence of peaks follows the standard order of the independent particle model. Source: Reprinted figure with permission from [164]. ©2014 The American Physical Society. 2.6. Reaction mechanism Quite recently, an updated version of the St.-Louis nonlocal DOM parametrization for 40Ca was used to calculate the cross sections for the 40Ca(e,e′p)39K reaction employing the DWIA [94]. In this case the proton distorted waves, the radial-dependence of the overlap functions, their normalization (spectroscopic factor) all were calculated from the DOM potentials and therefore no additional fit parameters were employed. To accurately calculate the (e,e′p) cross section in DWIA, it is imperative that the DOM self-energy describe not only scattering data but bound-state information as well. This is due to the fact that the shape of the cross section is primarily determined by the bound-state overlap function [90]. Thus, not only should the experimental charge radius be reproduced, but the charge density should match the experimental data. The present DOM self-energy leads to the spectral strength distributions with large peaks for the main fragments. The experimental distributions from Ref. [168] for ℓ=0 clearly show that the strength is already strongly fragmented at low energies. The DOM does not yet include the details of the lowenergy fragmentation of the valence hole states which requires the introduction of pole structure in the self-energy [75]. The spectroscopic factor of Eq. (18) corresponds to the main peak of each distribution. It is calculated directly from the 40Ca DOM self-energy resulting in values of 0.71 and 0.74 for the 0d3 2and 1s1 2peaks, respectively. Agreement with data obtained for the electron-induced knockout of the 0d3 2and 1s1 2orbitals with outgoing proton energy to 100 MeV shown in Fig. 14, is as good as, or better than, previous descriptions employing the Schwand et al.’s local global OM potential [92] and overlap functions from Wood-Saxon potentials where the radius and normalization factor are adjusted to fit the data. The net effect of nonlocality is in the opposite direction to the transfer study of Ross et al. [169], and allows for slightly larger spectroscopic factors. Presumably this is related to the role of knockout from the interior of the nucleus in the (e,e′p) case. Fig. 14 shows the agreement of these DWIA calculation with the distorted spectral functions obtained from the (e,e′p) data. A slight increase of the spectroscopic factors obtained from the DOM description compared to the values of the standard NIKHEF analysis [15] is obtained otherwise validating the procedure of the NIKHEF group [94]. The agreement with the ℓ=0 cross section is achieved by taking into account the experimental fragmentation at low energy which generates a spectroscopic factor of 0.60 for the main fragment. In order to estimate the uncertainty for the DOM spectroscopic factors, the bootstrap method from Ref. [170] was followed, which was also employed in Ref. [165] to assess the uncertainty for the neutron skin in 48Ca. New modified data sets were created from the original data by randomly renormalizing each angular distribution or excitation function within the experimental error to incorporate fluctuations from the systematic errors. Twenty such modified data sets were generated and refit. The resulting uncertainties are listed in Table 1. The DOM results also generate the complete spectral distribution for the 0d3 2and 1s1 2orbits according to Eqs. (26) and (28). These distributions are displayed in Fig. 15 from −100 to 100 MeV. The energy axis refers to the A−1 system below the Fermi energy and the A+1 system above. As in Fig. 13, a small imaginary part near the Fermi energy was employed to give the main peaks a small width. The occupation probabilities are obtained from Eq. (29) and correspond to 0.80 and 0.82 for the 0d3 2and 1s1 2orbits, respectively. The strength at negative energy not residing in the DOM peak therefore corresponds to 9 and 7%, respectively. This information is constrained by the proton particle number and the charge density. The strength above the Fermi energy is constrained by the elastic-scattering data and generates 0.17 and 0.15 for the 0d3 2and 1s1 2orbits, respectively, when Eq. (30) is employed up to 200 MeV. The sum rule given by Eq. (31), 22 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 14. Comparison of the spectral functions measured at NIKHEF for outgoing proton energies of 100 MeV to DWIA calculations using the proton distorted waves, overlap function and its normalization from a nonlocal DOM parametrization. Results are shown for the knockout of a 0d3 2and 1s1 2proton in the upper and lower panels respectively. Source: Reprinted figure with permission from [94]. ©2018 The American Physical Society. Table 1 Comparison of spectroscopic factors deduced from the previous analysis [90] using the Schwandt optical potential [92] to the normalization of the corresponding overlap functions obtained in the present analysis from the DOM including an error estimate as described in the text. Source: Reprinted table with permission from [94]. ©2018 The American Physical Society. Z0d3 21s1 2 Ref. [90] 0.65 ±0.06 0.51 ±0.05 DOM 0.71 ±0.04 0.60 ±0.03 associated with the anticommutation relation of the fermion operators, therefore suggests that an additional 3% of the strength resides above 200 MeV, similar to what was found in Ref. [164]. Strength above the energy where surface physics dominates can be ascribed to the effects of short-range and tensor correlations. The main characterizations of the strength distribution shown in Fig. 12 [75] are therefore confirmed for 40Ca. Application of this strategy to 48Ca has also generated excellent results for the corresponding (e,e′p) cross sections to valence hole states. The importance of proton reaction cross sections is confirmed as they were essential for the success of Ref. [94]. The 48Ca results were published in Ref. [171] and demonstrate that a non negligible nucleon asymmetry dependence is obtained for the spectroscopic strength of valence hole states when 8 neutrons are added to 40Ca. The smaller spectroscopic factors in 48Ca shown in Table 2 are consistent with the experimental cross sections of the 48Ca(e,e′p)47K reaction. The comparison of Z48 and Z40 in Table 2 reveals that both orbitals experience a reduction. This indicates that strength from the spectroscopic factors is pulled to the continuum in S(E) when eight neutrons are added to 40Ca. The stronger coupling to surface excitations in 48Ca, documented by the larger proton reaction cross section when compared to 40Ca [171], contributes to the quenching of the proton spectroscopic factor and is also consistent with the 23 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 15. Spectral distribution of the 0d3 2and 1s1 2orbits in 40Ca as a function of energy. Additional strength outside this domain is not shown. Source: Reprinted figure with permission from [94]. ©2018 The American Physical Society. Table 2 Comparison of DOM spectroscopic factors in 48Ca and 40Ca. These factors have been renormalized to account for low-energy fragmentation as observed experimentally [171]. Z0d3 21s1 2 40Ca 0.71 ±0.04 0.60 ±0.03 48Ca 0.58 ±0.03 0.55 ±0.03 np-dominance observed in the nucleon pair momentum distribution [28,65] which is associated with the nucleon–nucleon tensor force as discussed in Section 2.5.2. The results presented in Table 2 document a nucleon asymmetry dependence that is weaker than suggested by Fig. 2 but stronger than the (p,2p) results on Oxygen isotopes [58,59] and transfer reactions. The present results thus suggest that it is possible to generate a consistent picture of the strength distributions of valence orbits in Ca isotopes employing all the available experimental constraints. Recent DOM results for 208Pb confirm this observation [172]. We therefore conclude that it is indeed quite meaningful to employ concepts like spectroscopic factors and occupation probabilities when discussing correlations in nuclei. 2.7. Ab initio computations of correlations Several aspects of nuclear correlations, and the quenching of s.p. strength in particular, have been addressed through first principles computations in recent years. In practical applications one seeks for a nearly exact (or the most accurate possible) solution of the many-nucleon Schrödinger equation, while exploiting a realistic nuclear Hamiltonian which should be derived as closely as possible from the underlying QCD. The quality of such predictions is in constant evolution as new theory and computational advances are introduced. However, it is generally expected for an ab initio approach that the uncertainties due to both the many-body computations and the Hamiltonian can be estimated, so that a direct comparison between theory and experiment (each with their own uncertainties) is meaningful. To better grasp the discussion below, it should be clarified that almost all modern ab initio applications exploit nuclear Hamiltonians obtained through some effective field theory (EFT). The latter can be seen as a low-energy and low-momentum expansions of QCD, hence limiting phenomenological inputs. The implicit pre-diagonalization of shortdistance degrees of freedom leads to very soft nuclear Hamiltonians, which have made it relatively easy for a class of post-HF many-body approaches to reach ab initio predictions with mass numbers up to A∼140 [166,173,174]. This approach does not affect the low-energy structure, which is the topic of this review, because of a clear separation of scales between long- and short-range physics. From the discussion of Section 2.5 we know that direct signatures of SRC should be seen through experiments at large momentum transfer, that is outside the realm of applicability of EFT. Due to the separation of scales, the SRC effects for low-energy states are limited to an overall partial reduction of occupations and SFs that turns out to be about 10-15%—one should also keep in mind that SRC do not account for all of the quenching. Since EFTs wash out short-range physics by construction, this reduction effect disappears from typical ab initio computations. 24 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 16. Single particle spectral function, S(r, ω), for the removal of addition of a neutron from/to 56Ni at a distance rfrom its center, from SCGF-FRPA computations. Quasiparticle states of the f7/2hole and the p3/2,p1/2and f5/2particle are clearly seen. Source: Adapted form Ref [156]. Table 3 Spectroscopic factors (given as a fraction of the IPM) for valence orbits around 56Ni. For the SC-FRPA calculation in the large harmonic oscillator space, the values shown are obtained by including only SRC, SRC and LRC from particlevibration couplings (full FRPA), and by SRC, particle-vibration couplings and extra correlations due to configuration mixing (FRPA+∆Zα). The last three columns give the results of SC-FRPA and SM in the restricted 1p0fmodel space. The ∆Zαare the differences between the last two results and are taken as corrections for the SM correlations that are not already included in the FRPA formalism. 10 h.o. shells Exp. [178] 1p0fspace FRPA full FRPA FRPA SM ∆Zα (SRC) FRPA +∆Zα 57Ni: ν1p1/20.96 0.63 0.61 0.79 0.77 −0.02 ν0f5/20.95 0.59 0.55 0.79 0.75 −0.04 ν1p3/20.95 0.65 0.62 0.58(11) 0.82 0.79 −0.03 55Ni: ν0f7/20.95 0.72 0.69 0.89 0.86 −0.03 57Cu: π1p1/20.96 0.66 0.62 0.80 0.76 −0.04 π0f5/20.96 0.60 0.58 0.80 0.78 −0.02 π1p3/20.96 0.67 0.65 0.81 0.79 −0.02 55Co: π0f7/20.95 0.73 0.71 0.89 0.87 −0.02 However, EFT Hamiltonians are all expected to reproduce the same, experimentally observed, cross sections. Thus, any difference in the SFs must be compensated through changes in the final state interactions and reaction mechanism. 2.7.1. Role of shell-model and particle-vibration coupling in LRC First principle computations of the nuclear spectral function are possible through the self-consistent Green’s function (SCGF) approach, as already introduced at the beginning of this section and in Appendix A. One normally truncates the many-body expansion using the most complete Faddeev random phase approximation (FRPA) introduced in Refs. [152,154, 155] or the third-order algebraic diagrammatic construction [ADC(3)] [175–177]. The two approaches are closely related and both compute the nuclear self-energy including particle-vibration couplings systematically in all possible channels. While model spaces include several oscillator shells and are large enough to account for coupling to all giant resonances (GRs), the configuration mixing effects are limited to 2p1h and 2h1p states and cover only part of a standard shell-model computation. Fig. 16 displays the computed particle and hole spectral functions, of Eqs. (4) and (5) together, for a neutron on 56Ni and for the case were the radial coordinate ris chosen instead of a general basis {α}. This plot is interpreted as the probability distributions of adding or removing a neutron at distance rfrom the center while transferring an energy ±ωto the system. It could be compared to the schematic representation of Fig. 12. The spikes near the Fermi surface are the (square of the) quasiparticle overlap functions for the pf -shell orbits, the heavy fragmentation of sd and sdg shells is visible further away in both directions, while for energies ω > 0 the n+56Ni system is in the continuum and S(r, ω) extends to infinity as expected for an elastic-scattering wavefunction. Ref. [156] used the computations from Fig. 16 to disentangle the different effects of nuclear correlations. The outcome of this analysis for 56Ni is shown in Table 3. The computation was performed in FRPA, covering 10 major nuclear shells and using the chiral N3LO interaction of Ref. [123]. This early EFT Hamiltonian is only partially soft and contains mild short-range repulsion effects. Moreover, it was augmented by a phenomenological correction, set to constrain the particle– hole (p-h) gap, that corrected for the then missing three-nucleon forces (see Ref. [179] for details). As shown in the table, we first estimated the SRC reduction of SFs by dynamically diagonalizing two-body ladders, as demonstrated in 25 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 23. Examples of (a) neutron-removing (p,d) and (3He, α) reactions [200] at 23 and 26 MeV, respectively and (b) neutron-adding (d,p) and (α, 3He) reactions [201] at 10 and 38 MeV, respectively, demonstrating the role of momentum matching in nucleon transfer reactions. Source: Reprinted figures with permission from Refs. [200] and [201]. ©2013 The American Physical Society. ratios of cross sections between the two reactions for ℓ=1 or 4 differ by about two orders of magnitude depending on their ℓvalue. This is an astonishing demonstration of momentum matching. The cross sections from the respective reactions are low (high) because of poor (good) matching. When the momentum matching is poor, the contributions from more complicated, indirect (multi-step) pathways may contribute more significantly, and the interpretation of the cross section in terms of a simple one-step process becomes more questionable—and as a result the spectroscopic factors will not be reliable. 3.2.2. Detection angles Strongly coupled to the choice of beam energy and the role of momentum matching (Section 3.2.1), is the angular range or singular angle at which the cross section was measured, and thus the spectroscopic factors are determined. An example of the kinematics of various reactions in ‘normal’ and inverse kinematics can be seen in Fig. 20. The preceding discussion points to the reliable cross sections being determined from forward peaked (in the c.m. frame), surfacedominated reactions. In this regime, reliable spectroscopic factors are determined from the peak cross sections, or as close to the peak as is practical. While a general reproducibility of the angular distributions gives some faith in the distorting potential used, the details of the shape are sensitive to the model. Attempting to extract spectroscopic factors from the second maxima or angles near minima is likely to result in misleading results. It is noted that the angle of the peak is related to momentum transfer (QR ≃kcm√2(1 −cos θ)), which naturally leads to the peaks being more forward-focused at higher energies, as evidenced by plots in Fig. 22. For studies in inverse kinematics, particularly for neutron-rich sd-shell nuclei, where many of the systems with large ∆S lie, the challenge is to determine cross sections for ℓ=0 transfers. Often, detector systems are highly capable of detecting outgoing ions over 10–40◦and to larger c.m. angles. To measure close to 0◦is something that can be challenging even in normal kinematics, where detector systems can become rate limited at angles less than ∼5◦in the lab frame (similar to the c.m. frame in stable beam measurements at these modest energies). It is possible in some circumstances. For example, the outstanding measurement by Margerin et al. [202], who measured the 26Al(d,p)27Al reaction with the luxury of a very intense beam, allowing for the placement of annular Si detectors at large distances from the target. This resulted in an angular coverage of around 2◦< θc.m.<15◦, essentially capturing the peak ℓ=0 cross section. This was a system with small ∆S. Other examples of measurements carried out with radioactive-ion beams that captured relatively far-forward c.m. angles are Flavigny et al. [203] for ℓ=1 transfer and Schmitt et al. [204] for ℓ=0. 3.2.3. Comment on heavy-ion transfer The availability of heavy ion beams opened up the possibility to explore heavy ion transfer. It was soon realized that in comparing different heavy-ion transfer reactions, a notable jdependence, or selectivity, was observed at energies both above and below the Coulomb barrier. It was Brink [205] who looked at the kinematics of heavy-ion transfer and commented on both the Q-value dependence and the jdependence. Changes to both the reaction choice and thus Q- value significantly impact the yields to given final states. A beautiful demonstration is shown via heavy-ion transfer on 54Fe carried out at ∼5 MeV/u [206] to final states in 55Co. The (16O, 15N) and (14N, 13C) reactions both transfer a p1/2 proton which has ji=ℓi−sand therefore one would expect strong population of jf=ℓf+sstates (Ref. [205]) relative to the jf=ℓf−s(if seen at all). These are contrasted with the (12C, 11B) reaction, where a p3/2proton is transferred. For the (16O, 15N) reaction, the f7/2ground state (j=ℓ+s) is strongly populated, and relatively little yield is seen the f5/2 state (j=ℓ−s). It is, however, less pronounced an effect for the (14 N, 13C) reaction, which has smaller Qvalue. 32 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 With careful choices of pairs of reactions, robust spin assignments can be made with heavy-ion transfer reactions. This has particular value for high-jstates, where the matching conditions are more favorable in heavy-ion transfer. In this vain, the Oak Ridge group has demonstrated the advantages of sub-barrier heavy-ion transfer in inverse kinematics. They used complementary (13C, 12C) and (9Be, 8Be) reactions on isotopes of 136Xe, 134Te [207–209], and 132Sn [210] to help determine the excitation of high-jstates, in particular the (j>) 13/2+and (j<) 9/2−states, and extract useful spectroscopic information. In most instances, such measurements are coupled with coincidence gamma-ray measurements, which can support the assignments. While similar techniques are likely to be a powerful technique both at current and next generation facilities, there remains unresolved discrepancies in describing the absolute cross sections for heavy-ion transfer using DWBA. Pieper et al. [211] and Olmer et al. [212] explored this in some detail via the (16O, 15N) and (16O, 17O) reactions on 208Pb, noting that while relative cross sections (and spectroscopic factors) at different energies, both above and below barrier, appear in quite good agreement with light-ion transfer, the absolute magnitude of the cross sections does not scale well with energy. The discrepancies are as much as a factor of a 2–3 when compared to light-ion transfer. Both works [211,212] noted that the discrepancies could not be corrected by simple changes to optical-model or bound-state parameters, which is a finding corroborated by Winfield et al. [213] in a study of the (9Be, 10B) reaction on isotopes of O, Mg, Fe, Ca, and Cu at energies around 5 MeV/u. Further studies by Winfield et al. [214], at higher energies, found similar discrepancies with respect to light-ion transfer overlaps. The issue appears not to have been resolved. 3.3. Reaction theory for transfer reactions 3.3.1. The distorted-wave Born approximation (DWBA) Given the nearly sixty years of history behind the DWBA formalism, and its extensive use during this period, only a very brief description of this model is given here. Refs. [215,216] contain detailed descriptions of the model. Significant effort has been spent on the model uncertainties and parameterizations, much of it leading to better understandings (a recent example, relevant for this work is Ref. [54]), but the importance of carrying out the measurement in a regime that is suitable for the model and the parameters remains essential, as discussed in the previous section. There is a strong connection between the experimental data uncertainties and the analyses uncertainties. Using the description in Satchler’s book [215], the scattering amplitude for a reaction A(a,b)B[in the center-of-mass frame, where normal and inverse kinematics can be treated in the same way], can be written as fDWBA(θ, φ)= − µβ 2π¯ h2∫ ∫ χ(−) β(kβ,rβ)∗⟨b,B⏐⏐V′⏐⏐a,A⟩χ(+) α(kα,rα)drαdrβ,(40) where the functions χαand χβare distorted waves describing the elastic scattering of the particles in the entrance (α=a+A) and exit (β=b+B) channels, with momentum and relative coordinates kα,βand rα,βrespectively. V′ is the interaction inducing the transition, specific to the reaction, and µβis the reduced mass in the exit channel. It follows that the cross section for a s.p. state, with a given angular momentum and orbital angular-momentum transfer is dσ(θ) dΩ=vβ vα|fDWBA(θ)|2,(41) where vαand vβare the c.m. velocities in the entrance and outgoing exit channels. In Eq. (40), the matrix element ⟨b,B⏐⏐V′⏐⏐a,A⟩involves the integral over the internal coordinates of the many-body wavefunctions of the incident and outgoing particles. It is usually assumed that V′does not depend on these internal coordinates. For example, if Bresults from the addition of one neutron to the target A, one needs to perform the overlap integral ∫dξΨ∗ B(ξ, r)ΨA(ξ)≡ψℓ,j AB (r),(42) where ξstands for the internal coordinates of Aand rthat of the additional neutron. This overlap integral can be identified with that in Eq. (16) of Section 2.2, and is proportional to the probability amplitude of finding the state Awhen a nucleon is removed from B. In general ψℓ,j AB (r) is not normalized to one. In fact, its normalization gives the so-called spectroscopic factor,∫dr|ψℓ,j AB (r)|2=Sℓ,j AB (43) Very often, in practical calculations of the DWBA method the overlap function is approximated by the a s.p. wavefunction, obtained as a solution of a Schrödinger equation with some mean-field potential (typically of Woods–Saxon type), with the appropriate quantum numbers ℓ, jand separation energy. Since the s.p. wavefunction is unit normalized, one then writes ψℓ,j AB (r)≈√Sℓ,j AB ψℓ,j sp (r) (44) where ψℓ,j sp (r) is the s.p. wavefunction. 33 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 When angular momentum is explicitly introduced, additional Clebsh–Gordan coefficients appear. Furthermore, if the isospin formalism is used to express the states Aand B, and additional isosopin coefficient (C) appears as well which is sometimes singled out from the definition of the spectroscopic factor and hence written explicitly (C2S). The use of Clebsch–Gordan coefficients in the context of transfer reactions in discussed in Ref. [217] and an explicit example is given in Ref. [218]. C2is often 1 and not discussed at length, or sometimes intentionally or unintentionally ignored. If the s.p. overlap Eq. (44) is used in the scattering amplitude Eq. (40), one may express the differential cross section as a s.p. cross section multiplied by the corresponding spectroscopic factor dσ(θ) dΩ=gC2Si dσ(θ) dΩ⏐⏐⏐⏐sp (45) where Siis the spectroscopic factor for a specific state i. Note that, if both the projectile and target overlaps are expressed in terms of s.p. overlaps, a product of the corresponding spectroscopic factors will appear in Eq. (45). A statistical factor gis needed, which is (2j+1) for adding and 1 for removing. Many codes have been written to perform the calculations (the full integration of Eq. (40)) and are generally available. Examples include Ptolemy [219], FRESCO [220], TWOFNR [221], etc. Finite-range calculations do not approximate the integration and if the bound state overlaps are known, the resulting cross section should reflect the experimental cross section without normalization. A number of the available codes allow for modifications to simple DWBA calculations, such as the performance of coupled-channels Born approximation (CCBA) and coupled-reaction-channels (CRC) calculations, or the use of the adiabatic distorted wave approximation (ADWA) [222], which is able to include the breakup of the weaklybound deuteron in (p,d) and (d,p) reactions in a very computationally-efficient way. The ADWA approximation has been compared to the more sophisticated continuum-discretized coupled-channel (CDCC) [223] and Faddeev/AGS [57,224,225], finding differences of less than ∼20% at energies of ∼10 MeV/nucleon, for which the cross sections of (p,d) and (d,p) reactions are highest, although the differences increase with increasing energy and decreasing cross section [225]. As stressed above, the current discussion concerns analyses using the DWBA framework, although for (d,p) and (p,d) reactions the similar ADWA framework is also commonly used. In the 1960s there was considerable work done with stable systems (modest ∆S) in validating the DWBA framework. Perhaps one of the earliest and most cited being the study of the 40Ca(d,p) reaction by Lee et al. [226]. As is so often concluded, that work summarized that so-called ‘absolute’ spectroscopic factors could be extracted with an accuracy of around 20% with numerous caveats, such as the choice of bound-state potentials and optical-model parameterizations. It is these choices that are discussed below. It is only recently that compelling efforts have been made towards an ab initio approach to nuclear reactions and nuclear structure, both aspects of which are considered in the same framework [227]. This approach is likely to be the crowning achievement of reaction theory in the coming decade or so, and in particular its application to very weakly or tightly bound systems. 3.3.2. Optical potential Optical potentials1are key ingredients of any reaction calculation and transfer reactions are not an exception. In a DWBA calculation, optical potentials are meant to describe the elastic scattering of the a+Aand b+Bsystems. Flux is removed from the elastic-scattering channel as the reaction occurs, and this is modeled using imaginary components in the optical-model potential. As described in Section 2, optical potentials are also intrinsically nonlocal, as well as energy and angular-momentum dependent but, most commonly, they are approximated by more simple local representations, U(r). Moreover, they are assumed to be described by a central potential, U(r), which is frequently parameterized as follows [228]: U(r)=VC(r)−Vf (r,R0,a0)−iWf (r,Rw,aw)−iWDg(r,Rd,ad) (46) where VC(r) is the Coulomb potential, f(r,R,a) is the Woods–Saxon form factor, given by f(r,R,a)=1/[1+exp((r−R)/a)], and g(r,R,a)=4a(d/dr)f(r,R,a). The constants V,Wand WDdefine the strengths of the real volume, imaginary volume, and imaginary surface potentials, respectively, whereas Rxand axare the radius and diffuseness parameters defining their geometry. When either the projectile or target have nonzero spin, a spin–orbit component is usually added. For that, the Thomas form is commonly adopted: (¯ h mπc)21 r df (r,Rso,aso) dr 2L·S(47) where (¯ h/mπc)2≈2 fm2is introduced for convenience. Optical-model parameterizations can be specific or global. The concept of global parameterizations was established very early in the progression of the field as a natural extension of understanding the systematics of scattering reactions. It was clear that for nuclei near stability and particularly those greater than somewhere between around mass say A∼16, the trends in terms of mass, volume, and N−Zshow a smooth variation, with a classic analysis performed in Ref. [229]. 1The name optical model arose from the analogy of light scattering from a cloudy crystal ball. 34 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 24. Calculated angular distributions (left) for the five different pairs of optical-model potential parameterizations for protons and deuterons in the ℓ=1 (d,p) reaction at 5 MeV/u on 62Ni and (right) for ten different pairings of parameters for the ℓ=0 (d,p) reaction at 10 MeV/u on 14C. Over the last fifty or so years, a wealth of elastic scattering data has led to very well evolved global optical-model parameterizations, particularly for protons [170,230] (and neutrons) and deuterons [231]. The situation is improving for 3He (e.g., [232]) and 4He, were fewer data have been collected. For light nuclei, the situation is poorer. Many global potentials are defined for heavier nuclei, typically above C or O. Further, for exotic systems, such as halo nuclei or other very weakly bound systems, there is still work to be done with regard to optical-model potentials (or parameterizations). This is likely to go hand in hand with new and numerous transfer-reaction studies at future radioactive-ion beam facilities. For heavier systems, global optical-model potentials are indispensable. They are particularly useful when exploring a range of nearby systems or a chain or isotopes or isotones. A rigorous discussion of this was presented in Ref. [201] and is presented in more detail below. The optical-model parameters can modify the magnitude and particularly the shape of the angular distribution. The parameters used for the bound-state form factors can have a large impact on the magnitude of the calculated cross section. However, these can typically be well constrained. Fig. 24(a) shows calculated angular distributions for the (d,p) reaction on 62Ni to a hypothetical ℓ=1 state at 1 MeV using five different combinations of optical-model parameters (for both proton and deuteron) that are well defined in this mass and energy region. The rms spread in the magnitude of the cross section at the peak cross section is <11% . Fig. 24(b) shows calculated angular distributions for ten different combinations (again, for both proton and deuteron) of optical-model parameters for a system that has a relatively large ∆S, and one for which the optical-model parameters are not necessarily well defined, the 14C(d,p)15Cg.s.reaction. Here the rms spread at the peak is <8%. In each case, the bound-state potentials were kept the same. This gives some sense of the systematic uncertainties in the optical model parameterizations. There are, however, detailed studies that explore the propagation of uncertainties from the optical potentials obtained from elastic scattering to transfer cross sections, covering all angles, that suggest larger uncertainties. See, for example, Refs. [233–235]. In the study of (d,p) and (p,d) reactions, the deuteron wavefunction has typically used the Reid potential [120], or more recently the Argonne ν18 potential [122]. For the more complex projectiles such as t,3He, and 4He, there was the recent development of Green’s Function Monte Carlo derived parameterizations [236]. In general, transfer reactions at low energies depend mostly on volume properties of the overlaps ⟨p|d⟩,⟨d|3He⟩, so they are rather insensitive to the models used to describe them, as long as they are sensible. 3.3.3. Overlap functions For the overlap function of the target and residual nuclei, as discussed before, this is usually approximated by a s.p. wavefunction, computed in some mean-field potential, whose depth is varied to match the experimental separation energy of the transferred nucleon for the given final state. The wavefunction is often generated in a Woods–Saxon potential, which can be defined by radial parameters that are consistent with, for example, the (e,e′p) data discussed at length in Section 2.1. From Ref. [237], for most nuclei above mass 20, the radius parameter is r0∼1.28 fm and the diffuseness a=0.65 fm. A spin–orbit term is essential, with the depth often around 6 MeV and the radius term around 6/7 of the volume term, or rso0 ∼1.1 fm. In analyses of both (e,e′p) and transfer reaction data involving stable nuclei, parameterizations similar to this have proven consistent [237,238]. However, it is likely not the case far from stability and for removal of nucleons from deeply-bound orbitals. The sensitivities of these various parameters have been explored in detail by several groups (for recent examples related to the ∆Sdependence, see Ref. [54,57]). In many cases, variations based on ‘sensible’ input parameters are of the order of 20%. In low-energy transfer reactions, the absolute value of the 35 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 25. Various comparisons of spectroscopic factors for (a) different reactions to the same final states for ℓ=3 transfer on the stable Ni isotopes modified with permission from Ref. [201]c 2013 by the American Physical Society, (b) the (d,p) reaction from, and (p,d) to, the same final state for many nuclei studied in Ref. [109], adapted with permission from Ref. [109]c 2007 by the American Physical Society, and (c) different proton knockout and transfer reactions to the same final states (see e.g., data summarized in [237,239]) for nuclei between Li and Pb. cross section is mostly proportional to the asymptotic normalization coefficient (ANC) so the whole dependence on the bound state can be assigned to the ANC. The following sections approach the analyses of these data in the context of the parameters and analysis types. 3.4. Analyses of transfer-reaction data 3.4.1. Relative and absolute spectroscopic factors from transfer reactions Relative spectroscopic factors are not important to the topic of this review. In general, for a given set of parameters for the form factors and the optical-model potentials, the relative strength of states with a given jπare reliable to better than 5% or so [240]. This is in a large part due to the fact that the DWBA is handling the kinematics and distortions, resulting in only a modest dependence on reaction Qvalue. Determining the centroid (effective s.p. energy) of an orbital that is fragmented into several states is thus likely to be quite insensitive to the choice of DWBA parameters (in fact, it can be shown that there is often only a small difference between the spectroscopic-factor weighted centroid and the cross-section weighted centroid). The focus here is on absolute spectroscopic factors. That is, assuming absolute cross sections have been determined in a suitable experiment in terms of beam energy, momentum matching (choice of reaction), and at a suitable angle, and suitable parameterizations have been used in a DWBA (or alternative) calculation, then Eq. (45) gives the absolute spectroscopic factor for a given state with respect to a pure independent s.p. configuration (S=1). It is common for individual states to be scrutinized, typically certain low-lying excitations, often the ground state. This is particularly the case for knockout reactions that are central to the topic of this review. As such, it can be instructive to perform a shell-model calculation, or similar, to determine the extent to which the state of interest is a pure (S=1) configuration, or some fraction thereof. Computing the ratio between the shell-model spectroscopic factor and the absolute spectroscopic factor from the DWBA calculation can define the extent to which the experimental cross section relates to that of a simple ‘‘shell-model’’ state—if we understand the motion of nucleons in the mean-field, this should be unity. This is not the case, and central to the topic of this paper, discussed shortly. The accuracy of spectroscopic factors was previously mentioned to be around 20%. Fig. 25 shows various ways in which one might deduce this in a somewhat empirical manner. Fig. 25(a) from Ref. [201] shows the spectroscopic factors deduced from the population of identical 0f5/2final states outside of the stable, even Ni isotopes with different reactions. The figure suggests that for reasonably strong states, greater than about 10% of the total s.p. strength, the correlations between different reaction mechanisms (and thus different bound-state and optical-model parameterizations) is robust. Below this fraction, the scatter is large and thus so are the uncertainties. Fig. 25(b) from Ref. [109] shows a robust agreement between complementary adding and removing reactions probing the same final state, subtlety different from Fig. 25(a). The scale of the axes is slightly different, but highlighting that for a state with S>10%, the agreements are good, certainly of the order of 20% or better. Finally, Fig. 25(c) shows the comparison of the proton-removing (d,3He) transfer reaction to the same final states as populated in the (e,e′p) and (p,2p) proton knockout reactions (with data summarized in Refs. [237,239]). Again, here there is an excellent agreement. Essentially all of the examples shown here are for stable nuclei, with modest ∆S. This is intentional to set the scene. As mentioned in the preamble to this section, it reflects the dearth of transfer reaction data from radioactive-ion beams. In many of the cases described throughout this review, spectroscopic factors are derived from single states. Unique to transfer reactions is the ability to carry out both adding and removing reactions and use the nucleon-transfer sum rules. 36 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 26. (left) Showing the strength of low-lying ℓ=1 (p3/2and p1/2) states populated in the neutron-removing (p,d) reaction (negative energy) and the neutron-adding (d,p) reaction (positive energy) and (right) a histogram of sums of the adding and removing strengths in 0.5-MeV bins, which is fitted with a Lorentzian shape. Source: Reprinted figures with permission from Ref. [240]. ©2012 The American Physical Society. 3.4.2. Nucleon-transfer sum rules A natural extension to the concept of spectroscopic factors, describing the extent to which a given state represents as.p. excitation, is that the sum of all s.p. excitations with the same quantum configuration should represent the total degeneracy of that orbital. These are the nucleon-transfer sum rules, introduced by Macfarlane and French [241]. There have been many works in the past that have invoked the sum rules in the analyses of nucleon-transfer cross sections (e.g., [242]). Many of the recent results have come from the group of John Schiffer, triggered by detailed studies of nuclei involved in neutrinoless double beta decay [243]. These include works on Ge and Se isotopes [200], which spawned a detailed work on the interpretation of nucleon-transfer cross sections and the sum rules in the Ni isotopes [201,240], for which examples of the adding and removing strength, and the summed strength are shown in Figs. 26 and 27. These works also explored quenching in some detail. In the simplest sense, Eq. (45) can be written explicitly in terms of the spectroscopic factors for adding (S+) and removing (S−) reactions for transfer to final states of a given jπ, and naively the sum should equal (2j+1), or (2j+1)Nj=∑(2j+1)C2S+ j+∑C2S− j,(48) where Njis a seemingly ad hoc normalization factor in this equation. Were our understanding of s.p. motion in nuclei described by the independent s.p. model, then Nj≡1.0. It should be noted that Eq. (48) can be seen as a simple reexpression of Eq. (31). Below is a simple example taken from Ref. [200], where the neutron-removing (p,d) and neutron-adding (d,p) reactions were carried out on several isotopes of Ge and Se. Taking the ℓ=1 (to both p3/2and p1/2states, where the jvalues were previously known) spectroscopic strength for each reaction and adding them up and dividing by the total degeneracy of the p3/2and p1/2orbitals, (4+2), one gets a value of Nj=0.53. That is, the normalization factor derived from the sum rules reveals quenching. 3.5. Quenching of transfer reaction cross sections As highlighted throughout the paper, it was the (e,e′p) reaction studies of NIKHEF group led by Lapikás that shone light on this matter. That led to a reanalysis transfer reaction data, presenting in the famous paper of Kramer, Blok and Lapikás [237]. A summary of their findings and of much of the (e,e′p) work is given there and summarized in Fig. 28, showing the results before the reanalysis, and afterwards. This work came at a time when transfer reactions as a tool for exploring nuclear structure were not used as frequently as in the 60 s and 70 s, though it did coincide with the time when people were starting to consider transfer reactions in inverse kinematics in anticipation of future radioactive-ion beam facilities [244]—the era we are now in. It was in the early 2000s that significant interest in quenching came about, prompted by the intermediate energy knockout at NSCL, discussed in Section 4. While there was much activity on this front, and by definition on unstable systems, other groups started to re-explore transfer reactions on stable systems in the context of (a) detailed determinations of centroids across isotopic and isotonic chains to scrutinize the evolution of s.p. states and the role of the tensor force, and (b) to constrain the calculations of neutrinoless double beta decay (0ν2β) nuclear matrix elements. 37 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 27. Neutron occupancies and proton vacancies outside of the stable, even Ni isotopes extracted using transfer-reaction data. Source: The data are from Ref. [240]. Fig. 28. (a) For nuclei between 12 <A<208, the degree to which the proton knockout strength derived from the (e,e′p) reaction totals that of the independent-particle shell-model (IPSM) limit, compared to the same properties determined from the proton-removal (d,3He) reaction, analyzed independently, and (b) the same data with careful attention paid to the radial parameters used in the analysis of the transfer-reaction data. Source: Reprinted from Ref. [237], with permission from Elsevier. In Ref. [245], much of the data from Schiffer’s group and also some past works where reliable cross sections had been published, was analyzed consistently using modern global optical-model parameterizations and treatments of the bound-state wavefunctions. The data fell into two broad categories. The first considered studies that had used both the adding and removing reactions, such that the normalization, or quenching factor, could be derived in the way that is described by Eq. (48). In each of these cases, the data were derived from studies of several nearby nuclei and common normalizations applied to all. These include the (d,p), (p,d), (α,3He), and (3He, α) reactions on 58,60,62,64Ni [201,240], 74,76Ge, 76,78Se [200], and 130Te [245]. The second category was for studies where only adding or only removing data were available for a given nucleus— typically when looking at trends in particle or hole states outside of closed shell nuclei. This analysis is essentially the same as what was done for the (e,e′p) data [237], requiring that the total strength adds up to the number of vacancies in the closed shell, or the number of particles outside it, such that a simplified version of Eq. (48) can be expressed as Nj≡1 (2j+1) ∑(2j+1)gC2Sj,(49) where Njis used to denote the normalization, which is the quenching factor (similar to the reduction factor) and ghides the statistical factor. For these analyses, a mix of old and new data were considered. These include: 16O(d,p) [246], 40Ca(d, p) [226], 48Ca(d,p) [247], the (d,p) and (α,3He) reactions on 88Sr, 90Zr, and 92Mo [248], the (d,p), (p,d), (3He, α), and (3He, d) reactions on 98,100Mo and 100,102Ru [249], the 112−124Sn(3He, d) and (α,t) reactions [250], the (p,d), (3He, α), and (α,3He) reactions on the stable even N=82 isotones [251,252], and the (d,p) [199] and (α,3He) [253] reactions on 208Pb (this list and further details are available in Ref. [238]). 38 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 29. Observed s.p. strength compared to that of the independent-particle shell-model (IPSM) limit as a function of mass A. The (e,e′p) data in Panel (a) are from Refs. [237]. The gray band represents the mean ±2σof the (e,e′p) data to guide the eye. The data in Panels (b), (c), and (d), are from the analysis presented in Ref. [238]. Solid symbols are from adding and removing reactions while the empty ones are from just adding or just removing. Source: Reprinted figure with permission from Ref. [238]. ©2013 The American Physical Society. Fig. 30. (a) Average of the quenching factor for different ℓtransfer. The error bars shown represent the rms spread in values. Figure reprinted with permission from Ref. [238]c 2013 by the American Physical Society. (b) The sum of the 3/2−strength below 12C as probed via the (d,3He) and (t, α) proton-removal reactions, 9Be-induced knockout, proton-induced (p, 2p) knockout in both ‘normal’ and inverse kinematics, and lepton-induced (e,e′p) proton knockout, as a fraction of the simple shell-model sum rule limit of 4—that is to say, the quenching factor. The average, 0.53(10) (excluding the 9Be-induced knockout as the excited state is missing), is indicated by the horizontal dashed line. Source: The data were taken from the Evaluated Nuclear Structure Data File [254]. Fig. 29 shows a summary of these results, compared to those from the (e,e′p) work. The transfer data are 124 unique data points from 16 <A<208 using proton and neutron adding and removing reactions to final states with 0 ≤ℓ≤7 and a large range of binding energies. Perhaps the most impressive of these results are those of Ref. [249], which was acquired using the high-resolution Munich Q3Dspectrograph. Given the outstanding statistics and resolving power, hundreds of states contributed to the sums with quenching factors of 0.618(20), 0.614(48), 0.564(18), and 0.647(25) from different adding and removing reactions on four isotopes, being 100,102Ru and 98,100Mo. The uncertainties are considerably smaller than other studies. It is, however, clear from Fig. 29 that in general the spread is modest, around ±10% rms. This is true for transfer studies and comparable to the (e,e′p) work. It is likely dominated by unknown systematic uncertainties in the experimental cross sections. Fig. 30 emphasizes several additional aspects of the data. The first is clear from Fig. 29, and that is that the quenching factor seems to be ubiquitous not only in Aand independent adding or removing reactions involving protons or neutrons, but also independent of the ℓvalue of the final state. Further, for a light nucleus where many different reactions have been carried out, 12C, there is excellent agreement in the strength of the low-lying p3/2excitations between the (d,3He), (t,α), 9Be-induced knockout, (p,2p), and (e,e′p) reactions—all of which remove a proton from 12C. What is clear from these stable-beam studies is that while the quenching appears to ubiquitous, these systems typically have −12 ≲∆S≲12 MeV. The final part of this section discusses quenching of transfer reactions in the context of ∆S and other probes. 39 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 31. Transfer-reaction studies with radioactive-ion beams with large ∆S. (Left) A study of the (p,d) reaction on isotopes of Ar [55,57] compared to results from intermediate-energy knockout reactions and (right) a simultaneous measurement of the (d,3He) and (d,t) reaction on neutron-deficient 14O [203]. The top panel is obtained with overlap functions computed from a Woods–Saxon and shell-model spectroscopic factors (WS+SM), while the bottom panel considered overlap functions computed within the self-consistent Green’s Function (SCGF) theory. Source: Reprinted figure with permission from Ref. [203]. ©2013 The American Physical Society. 3.6. Transfer reactions and ∆S Fig. 56 attempts to contrast the available transfer reaction data, this time including transfer reactions on systems that have a large ∆S, with data from other probes. There are only a small number of quantitative experimental studies of transfer reactions on nuclei with large ∆S. These include a study of the 34,36,46Ar(p,d)33,35,45Ar reactions [55], directly mirroring those studied with intermediate-energy knockout, and the simultaneous measurements of the 14O(d,t/3He)13O/13N reactions [203]. Both of these were milestone measurements in response to the observed trends seen in Ref. [44]. As can be seen from Fig. 56 (and the original published results are shown in Fig. 31), these results neither firmly corroborate nor contradict the knockout data. The work of Ref. [55] studied the (p,d) reaction at an ideal beam energy of 33 MeV/u on neutron-deficient, stable (N=Z), and neutron-rich Ar isotopes spanning −10 ≲∆S≲12 MeV, which on the negative (weakly bound) side of the ∆Splot (Fig. 56) is similar to the stable beam data but pushes further on the deeper bound (positive) side. In a simple analysis using the ADWA model, it was not obvious that a trend was seen (Fig. 31) to support the knockout data. Further analyses using more refined models [57] led to a similar systematics of reduction factor for the analyzed three Ar isotopes (see Fig. 31) but concluded that from this data set, a slope similar to that derived from heavy-ion-induced knockout data cannot be excluded. The work of Ref. [203] used the same incident beam, neutron-deficient 14O at 18.1 MeV/u, to determine the neutron and proton removal cross sections via the (d,t) and (d,3He) reactions, respectively. The measurement probes a large range in ∆S, comparable to that of the majority of the intermediate-energy knockout data. The simultaneous measurement allowed for an absolute comparison of two reaction channels. Using different analyses, both a traditional approach and a more detailed microscopic analysis, as shown in Fig. 31, suggested that there was a weak or negligible dependence of reduction factor on ∆S[203]. This was corroborated by later studies, as discussed earlier [54]. 3.7. Conclusion All probes of s.p. structure quantitatively agree that there is a quenching of s.p. strength for nuclei with −12 ≤∆S≤ 12 MeV, and that the value is ∼40%–70% of the independent s.p. model. A large body of nucleon-transfer reaction data corroborate this, and it appears to be independent of target mass, choice of transfer reaction, and angular momentum transfer. Data from transfer reactions on systems with large ∆Sare very few at present. With a consistent analysis of the available data for oxygen isotopes 14,16,18O, no strong ∆Sdependence of the quenching factor was found so far. The published analyses for 34,36,46Ar did not reproduce the reduction obtained from heavy-ion-induced knockout but could not conclude on the presence of absence of a significant slope with ∆Seither. Optical potentials for light ions (p,d,eHe, tmostly) with rare isotopes remain the main source of uncertainties in the cross section analysis. This challenge can be partly mitigated by the measurement of precision elastic scattering, accessible at least for the entrance channel, and transfer angular distribution covering small angles in the center of mass. 40 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 32. The sensitivity to the s.p. degree of freedom shown for the example of the one-neutron knockout reaction 9Be(34Ar,33Ar+γ)X measured at NSCL [52]. (left) Momentum distribution for events leading to the knockout residue 33Ar in the ground state. (right) Momentum distribution in coincidence with de-excitation γrays. The calculated momentum distributions for knockout from dand the sorbitals confirm the simple picture for 34Ar (middle), with neutron removal from the s1/2orbital leading to the ground state and the knockout from dorbitals populating excited states in 33Ar. Today, only few high-quality transfer data exist at large (≥15 MeV) ∆S. More data are called for to further assess the slope of Rswith ∆Sfor transfer reactions. 4. Heavy-ion induced knockout 4.1. Introduction Direct nuclear reactions have long been used as important tools for the spectroscopy – and, to a lesser degree the quantification – of the proton and neutron s.p. degrees of freedom in the nuclear wavefunction. On the quest to study the structural evolution of rare isotopes near the nucleon driplines, often only available for experiments as low-intensity, fast beams of ions with energies exceeding 50 MeV/nucleon, 9Be- or 12C-induced one-nucleon removal reactions have been introduced and developed by Gregers Hansen and the NSCL team for spectroscopy as a fast-beam, inverse-kinematics alternative to light-ion-induced transfer reactions that remove a nucleon from the nucleus of interest [194,255–257]. For these one-nucleon removal reactions, the channel of interest is one where, in a single step, one proton or neutron is removed from the projectile and the projectile-like residue with A−1 nucleons survives. This reaction channel is dominated by fast, surface grazing collisions of the projectile (with Anucleons) and the target nucleus. From a large body of experiments performed at energies of 50–1600 MeV/nucleon at (rare-isotope) facilities around the world, it has been established that, with large cross sections, the dominant single-hole states relative to the projectile ground state are populated in the projectile-like reaction residue, demonstrating the sensitivity to the s.p. degree of freedom. The residue longitudinal momentum distributions measure the momentum distribution on the nuclear surface [258], and encode in their shape and width the information of the orbital angular momentum ℓand separation energy of the removed nucleon [194,259–261], the result of the predominant single-step direct reaction mechanism. The selectively populated single-hole state cross sections scale with the respective spectroscopic factor or wave-function overlap (see Fig. 32 for an example). Statistical descriptions of such a reaction will not capture these features. These characteristic features of the reaction observables have been used extensively to explore halo nuclei (e.g., see [255,256,262–267]) and details of the shell evolution towards the driplines (e.g., see [47,268–274]). Comparisons of such one-nucleon removal data with structure model calculations have been enabled by a direct reaction model description [194,257,275] that uses the sudden (short interaction time) and the spectator-core approximation to many-body eikonal (forward scattering) theory. The s.p. nuclear structure information then enters the calculations through spectroscopic factors, or wave-function overlaps, that scale the calculated cross sections for the removal of one nucleon from the corresponding orbital, as in light-ion-induced transfer reactions. In 2004, the measured cross section to the 5/2+ground state of 31Ar, the only bound state, extracted from the 9Be(32Ar,31Ar)X reaction performed at NSCL, was found to be only 24(3)% of that predicted using the many-body shell-model spectroscopy [276]. This case was the first example of the removal of a strongly-bound neutron (Sn=22 MeV) leading to a reaction product that is proton-bound by less than 400 keV, and prompted systematic studies of a large number of nuclei covering essentially all scenarios of nuclear binding realized over the nuclear chart. With remarkable consistency, a strong correlation was observed for the cross-section ratio Rs=σexp/σth with ∆S, where σexp is the measured, final-state inclusive cross section for the nucleon removal to all bound states in the residue and σth is the calculated cross section, combining eikonal reaction 41 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 37. (left and middle) The left panel shows the 3611-keV γ-ray peak detected in 35Si in one-neutron knockout, with the inset indicating the gates used on the outgoing parallel momentum distribution. The middle panel shows the γ-ray spectrum gated on the main peak in the momentum distribution (blue hatches) and the low-momentum tail (solid red). The 3611-keV peak seems clearly associated with the tail of the distribution. (right) Exploration of the momentum distribution of (in)elastically scattered 36Si projectiles. Violent events that may destroy the target nucleus with the debris causing a prompt, high-energy γ-ray background, is clearly associated with the low-momentum tail of the 36Si momentum distribution. Source: Figure adapted with permission from Ref. [272]. ©2014 The American Physical Society. Fig. 38. Kinematic cutoff at 100 MeV/c (dashed) and 150 MeV/c (full) plotted as a function of incident energy and binding energy of the removed nucleon. Depending of the width of the residue’s momentum distribution, reactions performed below the cutoff lines are subject to a kinematical cut that lowers the measured cross section. Several reactions from performed experiments are indicated (red filled symbols), including those from the figure’s original publication [53]. Source: Reprinted with permission from Ref. [53]. ©2012 The American Physical Society. energy [53]. Although the effect of higher-order contributions appears rather minor in the cases studied, core excitations ending up in the particle continuum and leading to the speculated decrease in core survival should be investigated in future experiments. 4.3.3. Kinematic limit for the removal of deeply-bound nucleons The limits at which the knockout picture of the removal reaction breaks down were explored experimentally by removing deeply-bound nucleons from projectiles at lower energy than the usual 100 MeV/nucleon or so customarily used. The removal of a proton from 16C and a neutron from 14O at energies of 75 MeV/nucleon and 53 MeV/nucleon, respectively, clearly exhibits a cutoff in the upper parts of their respective residues momentum distributions [53]. This cutoff is easily explained from the kinematical limit for removing the valence proton and neutron in these projectiles, that are bound by 22.5 and 23.2 MeV, respectively, as reproduced in Fig. 38. It can be noted in Fig. 38 that the cutoff depends on both the binding energy of the removed nucleon and the incident beam energy. The effect of the cutoff is to lower the measured cross section, thus a reaction model not-taking into account this kinematical cutoff would get larger cross sections and smaller reduction factors than a model containing the kinematical cutoff. However, this kinematical limit is unable to explain the trend of low reduction factors observed in knockout 48 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 reaction, as reactions performed well above the limit fall well within the systematics. Two examples were recently measured at energies above 200 MeV/nucleon: proton knockout from 30Ne at 230 MeV/nucleon [50] and from 33Na at 221 MeV/nucleon [47]. The measured cross sections from these two experiments were compared to calculated ones following the same procedure outlined in Section 4.2.2 and have been added to the recent update of the quenching plot (Fig. 2). While these nuclei present challenges to the nuclear structure description since they are located in the N=20 island of inversion, a very recent measurement of one-proton knockout from 16C at 240 MeV/u reports a value for Rsconsistent with the reduction observed in the same reaction at 75 MeV/u [306]. 16C was shown to exhibit no exotic behavior that challenges shell model [307]. The fact that so far all of these measurements follow the systematics obtained from lower incident energies indicates that the kinematical limit is likely not a predominant cause for the large quenching observed for removal of deeply-bound nucleons. Systematic studies of the energy dependence for a set of nucleon removal reactions from projectiles whose structure is well described by shell model would be useful to solidify the above, initial conclusions. Moreover, reduction factors deduced from neutron knockout from various carbon isotopes on a 9Be target at energies well above 1 GeV/nucleon also fall well into the systematics established at lower energies [308]. These observations assert the robustness of the approximations used in the eikonal reaction model, especially when taking care of avoiding the kinematical limit set by the separation energy of the removed valence nucleon. 4.4. Putting structure and reactions onto a more similar footing A long standing issue in using nuclear reactions to probe nuclear structure properties is the disconnect between reaction and structure models. Reaction models tend to use global description of nuclei such as Hartree–Fock densities, with forces fitted to a large sample of (mostly) stable nuclei, whereas wavefunctions used to describe the static properties of the reacting nuclei are often based on locally adjusted interactions. This unfortunate situation is starting to change, with the introduction of new methods such as ab initio Monte-Carlo, no-core shell model, or dispersive optical model (see Section 2). In the case of the knockout reactions discussed here, an ongoing effort is being pursued to investigate the origin of the observed quenching with experimental data on p-shell nuclei for which microscopic wavefunctions from Variational Monte-Carlo (VMC) as well as No-Core Shell Model (NCSM) calculations are now available. 4.4.1. Incorporating ab initio densities and overlaps into reaction model These new ab initio microscopic description of nuclei provide more accurate models of the densities and overlaps that are used in the reaction model framework. They can be incorporated on a case-by-case basis, by fitting them with a Wood–Saxon line-shape for instance, and use the resulting shapes in the eikonal calculations. An example of such a procedure is shown in Fig. 39 for the cases of the one-neutron knockout from 10Be and 10C [281]. The core densities calculated from these ab initio methods are also directly used in the eikonal calculations to compute the Smatrices relevant for the reaction. This self-consistent method is a lot more robust and can actually be used to probe the effects of the reaction model on the observed strength reduction in knockout reactions. 4.4.2. Ab initio p-shell calculations at face value: 10Be and 10C The neutron knockout reactions from 10Be and 10C were selected as a first attempt to benchmark the eikonal reaction model, because (i) their ab initio wavefunctions as well as that of their residues 9Be and 9C are available, (ii) they have only one final state since both 9Be and 9C are only bound in their ground states, therefore only the inclusive cross section needed to be measured, and (iii) the binding energy of their valence neutrons differs greatly, from 6.8 MeV to 21.3 MeV for 10Be and 10C, respectively. The corresponding values of the asymmetry energy ∆Sare −12.8 MeV and 17.3 MeV, respectively. Incorporating the ab initio densities and overlaps as described in Section 4.4.1, and computing the theoretical cross sections using the ab initio spectroscopic factors shows that the VMC-deduced cross sections agree the most with the experimental values [281]. The reduction factors relative to the VMC spectroscopic factors are 1.01(6) and 0.75(5) for 10Be and 10C, respectively. Taking the VMC calculations at face value reveals the following: i) no quenching due to the reaction model is observed in the removal of the (relatively) weakly bound neutron of 10Be, and (ii) a 25% quenching is observed for the removal of the deeply bound neutron of 10C. This effect may be attributed to the core survival approximation, where in this case it is assumed that the 9C core, which is bound by only 1.3 MeV, survives after the removal of a 21.3 MeV bound neutron in 10C. This first attempt at testing the limitations of the reaction model should be expanded to other cases, but can also guide the improvements of the reaction theory. 4.4.3. The power of mirror reactions The isospin symmetry of the nuclear force provides an opportunity to disentangle the effects of the reaction model on the observed quenching from the quenching stemming from the limitations of the independent particle model. In so-called mirror reactions, where the initial and final states are identical except for the projection of the isospin TZ, the spectroscopic factors describing the overlap between the initial and final wavefunctions are identical. However, because the number of protons and neutrons in the initial and final states differ within a given mirror reaction pair, the separation energy asymmetry ∆Sand the separation energies of the cores also vary. Examples of such reactions in the p-shell region are shown in Table 4. Measuring the knockout cross sections between the initial and final states of these reactions requires 49 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 39. Example of fitted Wood–Saxon line-shapes on VMC and NCSM overlaps for the neutron removal from 10Be and 10C. The insets show the integrations of the overlaps that converge to the spectroscopic factors. Source: Reprinted figure with permission from Ref. [281]. ©2011 The American Physical Society. identification of the final state of the residues since in some cases they are either excited states, or the residual nucleus has more than one bound state that can be populated by the reaction. Although an experimental program to study these reactions at various incident energies is under way for Be- and C-induced knockout, they should also be probed with other methods currently used to extract spectroscopic factors, such as transfer reactions or quasifree scattering off proton targets. 4.5. Other theoretical methods The above discussion is based on the analysis with the eikonal reaction theory [257,260,283,309]. However, there are alternative methods that can be used. For example, the CDCC [310] method can be used to calculate elastic breakup (diffraction) or the Ichimura–Autern–Vincent (IAV) method, recently implemented numerically by [311], can be used for both stripping and diffraction. Both these methods are fully quantum mechanical and are based on the core–spectator picture plus a description of the breakup particle final state interaction with the initial core and the target. These methods contain energy conservation and the IAV present implementation allows for the use of energydependent nucleon–target and core–target optical potentials. The numerical implementation of the CDCC and IAV methods are heavy and the calculations are lengthy but feasible on a super-computer. An intermediate method, the semiclassical transfer-to-the continuum (STC) exists [39,312]. It provides analytical formulae which are straightforward to implement numerically, thanks to the use of analytical Hankel functions for initial and final states. This method is based on a semiclassical treatment of the relative motion within a time dependent ‘‘semiclassical transition’’ amplitude. Thus energy conservation is preserved. One of the most important characteristics of this method is that it uses an optical-model final-state wavefunction for the breakup particle interacting with the target. Then, from the properties of the energy averaged S-matrix, the total probability is obtained naturally as a sum of a diffraction (elastic breakup) and stripping (inelastic breakup) terms. It allows to use energy-dependent optical potentials and it can include spin couplings. In the 50 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Table 4 Examples of mirror reactions listed as pairs. The initial and final states are identical except for the projection of the isospin. The separation energy differences ∆Svary within each pair because of the different binding of the initial and final nuclei. Reaction Initial state Final state ∆S (MeV) (7Li,6He) 7Li (3/2−)6He (0+) 2.72 (7Li,6Li) 7Li (3/2−)6Li (0+)−2.72 (9Li,8Li) 9Li (3/2−)8Li (2+)−9.88 (9C,8B) 9C (3/2−)8B (2+)−12.95 (10Be,9Li) 10Be (0+)9Li (3/2−) 12.83 (10C,9C) 10C (0+)9C (3/2−) 17.28 (11B,10Be) 11B (3/2−)10Be (0+)−0.22 (11C,10C) 11C (3/2−)10C (0+) 4.43 (12C,11B) 12C (0+)11B (3/2−)−2.76 (12C,11C) 12C (0+)11C (3/2−) 2.76 high-energy limit, the method provides an eikonal formalism consistent with the [283,309] formalism for the stripping part. The diffraction part will be further discussed in Appendix A. IAV and STC model have been benchmarked and found consistent in the situations in which both can be applied [313]. They can calculate the same observables but the P⊥or core angular distributions which are possible only in the IAV model. STC and eikonal have been benchmarked [314] and it has been shown that phase space effects which arise from energy conservation can be introduced in the eikonal, thus improving, in principle, its accuracy, which is particularly relevant at low incident energies. In general, it is found that the STC gives somewhat smaller cross sections than the eikonal predictions, because it calculates the n-target S-matrices by the optical model which is more accurate for the low partial waves. Formally, the core–target treatment is the same in the two methods, apart from the choice of the optical potential. The eikonal uses the double folding via a tρρ treatment. The accuracy of the STC has been enhanced for the 9Be target by the careful fitting of two optical potentials on a large range of energies for n-9Be scattering data [315]. One such potential is derived with the DOM method [83,316]. Furthermore, it has been shown that core–target potentials can be obtained by single-folding appropriate core densities with the n-9Be potential [317–319]. Such potentials have been used in the benchmark of IAV and STC and shown to provide a satisfying description of the momentum distribution and cross sections obtained in [53]. More comparisons to data are necessary to benchmark the predictive power of both the IAV and STC models. In order to discuss the quenching and the dependence with the asymmetry, a systematic comparison still has to be done and is the objective of an ongoing project [320]. 4.6. Conclusions Single-nucleon removal reactions induced by light nuclear targets have now been used successfully for approximately two decades to probe the s.p. structure of nuclei far from stability. The trend of the experimental to theoretical cross sections ratio Rsobserved with cross sections deduced from configuration-interaction models such as the shell model (see Fig. 2), is now well documented, comprising numerous data points spanning a wide range of separation-energy difference as well as incident beam energy of the projectile. The theoretical framework used to describe these reactions quantitatively has been investigated experimentally in various ways, and continues to be a focus of ongoing research projects. From these studies, to date, the eikonal reaction model appears robust. The least-explored issue at this time is the extent to which the core survival is accurately described within the effective three-body reaction model in cases where the removal is of a strongly-bound nucleon and where the core is weakly bound. Higher order core excitations into the continuum that lead to core breakup may make a small contribution to the quenching in these cases, but beyond-shell-model structure effects are also hinted at by the particular case of a 25% quenching in neutron removal from 10C when analyzed in a self-consistent manner using VMC calculations as input to the eikonal model [281] (this assumes that the VMC model provides the correct description of 10C). In contrast, there has been little discussion of the lack of quenching for the removal of a weakly-bound nucleon, as compared to the 30%–40% or so observed on nuclei located in the valley of stability, in agreement with observations reported from (e,e′p) reactions. Together with new data from Jefferson Laboratory [66], one may speculate that the cross sections ratio trend observed from the removal reactions discussed here could be an indication of a dependence of Short Range Correlations (SRC) on binding energy. In the extreme cases of halo nuclei for instance, the loosely-bound valence nucleon is largely spatially removed from the bulk of the nucleus, and, therefore, may indeed experience less SRC (and other correlation) effects than those occupying orbitals within the nuclear volume. In this context, a promising prospect of disentangling the effects of the reaction model approximations on the observed cross sections is offered by the use of mirror reactions, as discussed above. 51 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 40. Energy spectrum for the reaction 16O(p,2p)15N with the binding energy of the removed nucleon EBincreasing from right to left. The dashed curve is a fit to the experimental data [323]. 5. Quasifree (p,2p) and (p,pn) scattering 5.1. Introduction In quasifree (p,pN) scattering a proton with energy of several hundred MeV knocks out a bound nucleon N. The energy and momentum of the outgoing nucleons provide information on the quantum numbers of the removed nucleon in its initial state in the nucleus. The shape of the angular distributions of the outgoing nucleons, the nuclear recoil momentum, and the momentum distribution are all important observables, connected with the initial state of the knocked-out nucleon. In the past five decades quasifree scattering (QFS) has been a great experimental probe of the nucleus and much of this work has been reviewed in seminal papers by Maris and Jacob [14,321], who fled Europe and settled at the Federal University of Rio Grande do Sul, Brazil. Maris participated in some of the first experiments on quasifree scattering [322], for the reaction 12C(p,2p)11B, and is also the creator of a method, the ‘‘Maris effect’’, to obtain additional information in nuclear structure using a small detail of QFS scattering with polarized protons [323–326] (described below). The first experiments on (p,2p) reactions were performed at the Berkeley laboratory using a 350 MeV proton beam [327,328] incident on d, Li, C and O targets. In Fig. 40 we show a typical spectrum of one of the first (p,2p) experiments for the reaction 16O(p,2p)16N [323]. It is clear that the reaction is an excellent probe of nuclear structure with the identification of the expected energy locations of the nuclear levels. Early on it was also noticed that ‘‘absorption’’ of the incoming and outgoing particles, due to multiple collisions, results in an energy loss. However, the energy losses are smeared out over the whole particle energies and angles, yielding a smooth background in the reconstructed missing-energy spectrum of the residual nucleus, starting at its ground state and rising with excitation energy. Peaks are superimposed to this spectrum caused by clean quasifree collisions. Therefore, the energy spectrum contains the signature of quasi-elastic collisions even on the presence of a background due to absorption or multiple collisions. It is thus clear that in order to extract meaningful results for spectroscopic amplitudes, a treatment of multiple collisions needs to be incorporated in the theory. This fact has been known for a long time, e.g., in Refs. [323] a reduction of (p,2p) cross sections due to absorption is shown to be as large as 1/10 for 40Ca nuclei, increasing drastically with the atomic number A. Following a picture based on the shell model, a hole created by the knockout is filled by the frequency of collision between the outgoing proton and nucleons within upper shells. As an example, for 1pshell nuclei (Li, Be, B, C, N, and O isotopes) one expects the width of the 1s hole state to be directly proportional to the number of proton–nucleon pairs in the 1pshell and inversely proportional to the nuclear volume. As summarized in Refs. [14,321], this is actually observed in experiments. 5.2. Distorted wave impulse approximation As with other probes in direct reactions, the distorted wave impulse approximation (DWIA) is the most widely used theoretical method in the study of (p,pN) reactions [329,330]. The DWIA assumes that the knockout reaction occurs by a single interaction of the incident proton with the struck nucleon. Multiple scattering with the other nucleons is only accounted for by means of an optical potential, which can also include absorption due to the nuclear excitations. In medium modifications of pN interaction are sometimes accounted for [294,331–339]. Ambiguous treatments of in-medium interactions and in the reaction mechanism still exist, and multi-step processes are not negligible [323,324, 340–342]. 52 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 41. The coordinates used in the text are shown. Assuming the validity of the DWIA, the quasifree cross section can be shown to factorize as [321,326] d3σ dTNdΩ′ pdΩN=K′dσpN dΩ|F(Q)|2,(55) where the momentum distribution of the knocked-out nucleon Nwithin the nucleus is given by |F(Q)|2,dσpN /dΩdenotes the free pN cross section and K′is a kinematic factor. An off-shell pN t-matrix is needed in Eq. (55) and the off-shellness is only treated as a kinematical correction. Therefore, it is only valid if off-shell effects are not relevant, which seems to be the case for high-energy collisions [321,326]. For the bulk of quasi-elastic scattering, since one is almost on-shell, it does not make too much difference which cross section one uses. On the other hand, for extracting the smaller, more interesting, parts of the spectral function, the high-momentum and large-missing-energy components, the off-shell ambiguity can be quite large. To clarify how several approximations can be used, we start from the basic DWIA formulation where the transition amplitude for the A(p,pN)B reaction is [321] Tp,pN =√S(lj)⟨χ(−) k′ pχ(−) kN⏐⏐τpN ⏐⏐χ(+) kpψjlm⟩.(56) Here, χ(−) k′ p(χ(−) kN) denotes the distorted wave for the outgoing proton (knocked-out nucleon) interacting with the residual nucleus B, whereas χ(+) kpdenotes the incoming proton distorted wave interacting with the target nucleus A. ψjlm represents the bound-state wavefunction of the struck nucleon, and √S(lj) is the spectroscopic amplitude of the ejected nucleon, with angular momentum quantum numbers (lj). The energy Eentering the two-body pN scattering matrix τpN is described below. Eq. (56) means Tp,pN =√S(lj)∫d3r′ pBd3r′ NBd3rpAd3rNBτ(rpB,rNB;r′ pA,r′ NB) ×χ(−)∗ k′ p(r′ pB)χ(−)∗ kN(r′ NB)χ(+) kp(rpA)ψjlm(rNB),(57) with the normalization in the continuum as ∫d3rχ∗ k(r)χ′ k(r)=δ(k−k′) and the normalization of the bound-state wavefunction ψnlj as ∫d3r|ψnlj(r)|2=1.The coordinates used above are shown in Fig. 41. Further approximations have been used in the literature and proved to be numerically accurate. They have been used to reduce the number of integrations in Eq. (57). Notice that the proton and the struck nucleon Ncoordinates are related by rpA =rpN +rNA and, since the pN interaction has a smaller range than the nuclear size, Eq. (57) will only include small values of rpN for which rpN ≪rNA, Eq. (57) for the T-matrix becomes an integral over rNB [321,326], Tp,pN =√S(lj)τ(k′ pN ,kpN ;E)∫d3rNBχ(−)∗ k′ p(rNB)χ(−)∗ kN(rNB)χ(+) kp(αrNB)ψjlm(rNB),(58) 53 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 with α=(A−1)/Aand τ(k′ pN ,kpN ;E) denoting the Fourier transform of the pN t-matrix appearing in Eq. (57). This equation has been used as the standard for numerical calculations of quasifree reactions [329,330]. Several corrections have been introduced in the literature to study the validity of approximations used (see, e.g. [321,326,343]). 5.3. Cross section Using the T-matrix of Eq. (58), the differential cross section becomes [321,326,343] d3σ dE′ pdΩ′ pdΩN=K (2sp+1)(2JA+1) ∑ γ⏐⏐Tp,pN (γ)⏐⏐2(59) where the kinematical factor Kis (with ki=pi/¯ h) [323] K=m2 pmNc6 (¯ hc)6(2π)5 k′ pkN kp⏐⏐⏐⏐1+mN MB[1−kp kN cos θN+k′ p kN cos(θ′ p+θN)]⏐⏐⏐⏐ −1 (60) and the prime indices denote the proton variables in the final channel. The summation of the transition matrix Tp,pN (γ) in Eq. (59) over γ=(µp, µ′ p, µN,MA,MB),(61) involves the spin quantum numbers of the incident proton, spand µp, the target nucleus, JAand MA, the final proton, sp and µ′ p, the struck nucleon, sNand µN, and the residual nucleus, JBand MB. For single particle states, a simpler notation jlm is often used for the angular momentum quantum numbers of the ejected nucleon. The missing momentum, pm, missing energy, ϵm, and missing mass, mm, are defined by ϵm=EN+E′ p−Ep−mNc2 pm=¯ h(kN+k′ p−αkp) m2 mc4=ϵ2 m−p2 mc2,(62) where α=(A−1)/Aaccounts for the c.m. correction. They are the momentum and energy taken over by the residual nucleus in the final state. 5.4. Plane wave impulse approximation The pN scattering amplitude (in the pN c.m.) relates to the pN t-matrix by means of [344] fpN (θ;E)= −2π2mN ¯ h2τ(k′ pN ,kpN ;E),(63) and the elastic scattering cross section is given by dσpN dΩ=⏐⏐fpN (θ;E)⏐⏐2.(64) Using plane waves in Eq. (58) yields T(PWIA) p,pN =√S(lj)τ(k′ pN ,kpN ;E)∫d3re−iQ.rψjlm(r),(65) where Q=pmis the missing momentum defined above. Therefore, Eq. (55) is reproduced where F(Q)=∫d3re−iQ.rψjlm(r) (66) and the kinematic factor in Eq. (55) is related to the kinematic factor of Eq. (59) through the relation K′=(¯ h2 2π2mN)2 K.(67) The distortions due to absorption and elastic scattering are not accounted for in the PWIA, but can easily be accommodated in the DWIA. The PWIA is still useful for physical interpretation of experimental data. 54 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 42. Ratio of DWIA and PWIA results for the cross section of (p,2p) reactions on several nuclei. 5.5. DWIA vs. PWIA In Fig. 42 we show the ratio of PWIA and DWIA calculated cross section of (p,2p) reactions on several nuclei. It highlights the importance of distortion and absorption effects. In Ref. [342], eikonal wavefunctions were used to describe the distorted waves. The distorted waves were separated in initial and final channels allowing for a treatment of collisions in which the outgoing nucleons can scatter into large angles, by assuming that the outgoing proton and ejected nucleon suffer minor deflections on their way out of the residual nucleus. In terms of the momentum transfer Q, the momentum distributions are given by (dσ d3Q)DWIA =1 (2π)3 S(lj) 2j+1∑ m⟨dσpN dΩ⟩Q⏐⏐⏐⏐∫d3re−iQ.r⟨S(b)⟩Qψjlm(r)⏐⏐⏐⏐ 2 ,(68) where ⟨S(b)⟩Qis the product of the incoming proton and outgoing nucleons S-matrices averaged over all scattering angles leading to the same momentum transfer Q. The absorption is included in terms of the in-medium nucleon–nucleon cross sections [294]. Further integration of this equation allows one to obtain the longitudinal, dσ/dQz, and transverse, dσ/dQt momentum distributions [342]. One of the advantages of using eikonal distorted waves is the interpretation of the classical impact parameters sampled by heavy-ion knockout reactions and in (p,2p) reactions. For heavy-ion knockout reactions, one obtains from Eq. ((53)) for the probability to remove a nucleon in orbital (lj) located at distance b(perpendicular to the collision axis) from the center of the projectile nucleus as Pko(b)=S(lj) 2j+1|Sc(b)|2∑ m∫d3r⏐⏐ψjlm(r)⏐⏐2[1−⏐⏐⏐⏐Sn(√r2sin2θ+b2−2rb sin θcos φ)⏐⏐⏐⏐ 2].(69) Here Sn(Sc) is the eikonal matrix amplitude for the scattering of the nucleon N(core c) on the target, and r≡(r, θ, φ). In the case of (p,2p) reactions, the nucleon-knockout probability is given by integrating Eq. (68) over the momentum transfer. The integrand of the resulting cross-section formula yields the quasifree scattering probability [342] Pqfs(b)=S(lj) 2j+1∑ m⟨dσpN dΩ⟩o.s.|Clm|2⏐⏐⟨S(b)⟩o.s.⏐⏐2∫∞ −∞ dz ⏐⏐⏐⏐ ulj(r) rPlm(b,z)⏐⏐⏐⏐ 2 ,(70) In Fig. 43, upper panel, the dotted curve shows the probability for removal of a neutron in the reaction 12C(68Ni, 67Ni) at 500 MeV/nucleon as a function of the distance to the c.m. of 68Ni. The dashed curve represents the removal probability in a 68Ni(p,pn) reaction at the same energy. For comparison the square of the radial wavefunction u2(r) is also shown (solid curve). We assume a neutron in the 0f7/2orbital in 68Ni, bound by 15.68 MeV. The figures in the middle panel are for the reactions 12C(11Be, 10Be) and 11Be(p,pn) at 500 MeV/nucleon. We assume a neutron in the 1s1/2orbital in 11Be, bound by 0.54 MeV. The figures in the lower panel are for the reactions 12C(8B, 7Be) and 8B(p,2p) at 500 MeV/nucleon. We assume a proton in the 0p3/2orbital in 8B, bound by 0.14 MeV. One observes that the removal cross sections for both knockout and (p,pn) reactions probe the surface part of the wavefunction. This is due to the fact that in both cases, the absorption is very strong for small impact parameters. For proton or neutron removal from even deeper bound states, with a concentration of the wavefunction closer to the origin, both reaction mechanisms will probe an even smaller part of the wavefunction tail. 55 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 43. Upper panel: The dashed curve shows the probability for removal of a neutron in the reaction 12C(68Ni, 67Ni) at 500 MeV/nucleon as a function of the distance to the c.m. of 68Ni. The dotted curve represents the removal probability in a 68Ni(p,pn) reaction at the same energy. For comparison the square of the radial wavefunction u2(r) is also shown (solid curve). Middle panel: Same as the upper panel, but for the reactions 12C(11Be, 10Be) and 11Be(p,pn) at 500 MeV/nucleon. Lower Panel: Same as upper panel, but for the reactions 12C(8B, 7Be) and 8B(p,2p) at 500 MeV/nucleon. Source: Figure adapted with permission from [60]. ©2013 The American Physical Society. In the middle panel of Fig. 43 one sees that the part of the wavefunction probed in the 12C(11Be, 10Be) reaction is again limited to the surface of the nucleus, beyond the orbital maximum density. On the other hand, the (p,pn) reaction has a much larger probability of accessing information on the inner part of the wavefunction, as seen by the dashed curve. These results are in agreement with the conclusions drawn in Ref. [345] for stable nuclei where it has been shown that for light nuclei the average density probed in (e,e′p) is comparable to the one probed in (p,2p). There is a strong A dependence, though, and for medium-heavy and heavy nuclei one is rather probing the surface region in (p,pN) reactions. It is thus clear that knockout reactions with heavy ions and (p,pN) reactions yield complementary nuclear spectroscopic information. For deep bound states the first reaction is only accessible to the tail of the nuclear wavefunction, whereas the (p,pN) reaction process probes the largest part of the wavefunction for loosely bound nuclei. In recent years, with the advent of radioactive nuclear beams, it is necessary to perform (p,2p) reactions in inverse kinematics. (p,2p) experiments using this technique were measured for the first time with 9−16C beams at 250 MeV/nucleon at HIMAC [346]. The missing mass spectra obtained in these measurements show distinct peak structures for p-shell and s-shell proton knockout. The first exclusive and kinematically complete (p,2p) experiment in inverse kinematics was performed using a 12C beam at an energy of 400 MeV/nucleon as a benchmark [347]. This new technique has been developed to study the s.p. structure of exotic nuclei in experiments with radioactive-ion beams. The outgoing pair of protons and the fragments were measured simultaneously, enabling an unambiguous identification of the reaction channels and a redundant measurement of the kinematic observables. Both valence and deeply-bound nucleon orbits are probed, including those leading to unbound states of the daughter nucleus. Fig. 44 shows measurements of this 12C(p,2p)11B reaction with transverse (left) and total (right) recoil momentum of 11B in the rest frame of the incident 12C [347]. The experimental momentum distributions (circles) are compared with theoretical calculations (curves) for 0p-shell QFS knockout which take into account nuclear absorption effects. The theoretical curves are normalized to the experimental data with the scaling factor of 0.64. The agreement between this (p,2p) experiment and the spectroscopic factors deduced from electron-induced knockout reactions demonstrates the validity of the proposed method, which has become a powerful tool for studies with radioactive beams to investigate the s.p. structure of exotic nuclei. In Fig. 45 we plot the cross sections for neutron removal in (p,pn) reactions on 23O from [0d5/2]6and [1s1/2]1orbitals as a function of the separation energy. The cross sections for neutron removal from the [0d5/2]6orbital is divided by the number of the neutrons in the orbital (6). As expected, the cross sections are strongly energy dependent close to the threshold and steadily decrease with increasing separation energy. Close to threshold (Sn=0) a large fraction of the wavefunction lies in a region where absorption, or multiple scattering, is smaller, thus increasing the removal probability at larger impact parameters, consequently increasing the knockout cross section. As the separation energy increases it becomes less likely to knockout a neutron without rescattering effects. In Fig. 46 we show the full width at half maximum (FWHM) of the transverse momentum distributions in (p,2p) and (p,pn) reactions of 20O at 500 MeV/nucleon as a function of the proton and neutron separation energies. The upper curve is for (p,pn) and lower curve for (p,2p) reactions. As expected, the widths for d-states are larger than those for p-states. 56 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 44. Measurements of 12C(p,2p)11B reactions with transverse (left) and total (right) recoil momentum of 11B in the rest frame of the incident 12C. The experimental momentum distributions (circles) are compared with theoretical calculations (curves) for 0p-shell QFS knockout which take into account nuclear absorption effects. The theoretical curves are normalized to the experimental data with the scaling factor of 0.64. Copyright by Elsevier. Source: Figure adapted with permission from [347]. Fig. 45. Cross sections for neutron removal in (p,pn) reactions on 23O from [0d5/2]6and [1s1/2]orbitals as a function of the separation energy. The cross sections for neutron removal from the [0d5/2]6orbital is divided by the number of the neutrons in the orbital (6). The separation energies are varied artificially. The dashed and dotted curves are guide to the eyes. Source: Figure reprinted with permission from [60]. ©2013 The American Physical Society. 5.6. The quantum transfer-to-the-continuum method Standard analyses of (p,pN) reactions have been performed within the DWIA approximation. However, in recent years, some alternative methods, which provide additional insights as well as an independent extraction of spectroscopic factors and momentum distributions, have been proposed. One of these methods is the so-called transfer to the continuum formalism. This is a fully quantum-mechanical approach based on the direct application of the prior-form transition amplitude for the (p,pN) reaction. For a process of the form A(p,pN)C, this transition amplitude can be written as [348] T3b if = ⟨Ψ3b(−) f|VpN +UpC −UpA|φCAχ(+) pA ⟩,(71) where VpN is the nucleon–nucleon interaction, UpC and UpA are effective nucleon–nucleus interactions, χ(+) pA is the distorted wave generated by the entrance channel auxiliary potential UpA( R) and φCA is the overlap function between the initial and residual nuclei Aand C. This function could be in principle directly obtained from ab initio calculations of the Aand C systems, as discussed in Section 4.4.1 in the context of heavy-ion knockout. Since in many cases these are not available, it is a common practice to approximate this overlap by the product of a s.p. wavefunction ϕℓ,j CA for a s.p. configuration ℓ, j (obtained with some mean-field potential) and the square root of the spectroscopic factor SCA,ℓ,j,i.e.,φCA ≈√SCA,ℓ,jϕℓ,j CA . 57 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 53. TDX and Ayfor 12C(p,2p)11Bg.s.at 392 MeV calculated with DWIA (solid lines). The results calculated with the plane-wave impulse approximation (PWIA) are also shown by the dashed lines for comparison. Source: Figure adapted with permission from [239]. ©2020 Elsevier. deduced from the comparison with the data is 1.82(3), which well agrees with that deduced from the (e,e′p), that is, S(e,e′p)=1.72(11) [90,113]. This is the case also with the components for the 1/2−and 3/2−excited states of 11B; S(p,2p) and S(e,e′p) are shown in Table 5. A systematic (re)analysis of the existing (p,2p) data has been done in Ref. [239] and it was concluded that the (p,2p) reactions work as a spectroscopic tool; S(p,2p) were found to be consistent with S(e,e′p) within 15% at energies above 200 MeV and within 30% in the vicinity of 100 MeV. Based on the results of [58], a phenomenological approach to examine the role of short- and long-range nucleon– nucleon correlations in the quenching of s.p. strength and in particular how they evolve in asymmetric nuclei has been proposed [358]. It has been shown that the (p,2p) data analyzed with the eikonal method is in agreement with an isospin dependence predicted in this simple model assuming a 20% contribution of SRC as extracted from the JLab data [67]. However, the systematics of quenching in the (p,2p) depends slightly on the reaction-model used to analyze the data (see Fig. 52). As emphasized in Ref. [239], the success in reproducing not only the shape of the observables but also the absolute value of the TDX relies on the use of a reliable global optical potential that is applicable to various nuclei and in a wide range of energies; the global Dirac-type parameter, developed by Cooper et al. [353] was employed: the parametrization EDAD1 for the exit channel and the parametrization EDAI for the incident one. Another important feature of the DWIA analysis is that the same proton s.p. wavefunction as in the (e,e′p) analysis was adopted. Furthermore, the correction to the bound-state wavefunction and the distorted waves due to the nonlocality of the nucleon–nucleus potential is crucial to obtain a quantitatively reliable TDX. Details of the correction for nonlocality and its outcome are given in Ref. [239]. It should be noted, however, that the prescription for the correction used so far is based on a phenomenological approach, which can be insufficient for fully accounting for the effect of nonlocality. Recently, the dispersive optical model (DOM) [33], which explicitly computes the wavefunctions with directly treating the nonlocality, was applied to the 40Ca(e,e′p) reaction and a larger S(e,e′p) than that obtained in the preceding analysis by about 10% was obtained [94]. It will be interesting and important to use the DOM wavefunctions in the DWIA analysis of the (p,2p) reactions. The last piece of the ingredients important for the quantitative calculation is the Møller factor [359] that transforms the nucleon–nucleon (NN) transition matrix from the NN c.m. frame to that in the c.m. frame of the (p,2p) reaction system. The importance of the Møller factor has been discussed in the literature, e.g. in the multiple-scattering-theory paper by Kerman, McManus and Thaler [331], and included also in the preceding studies on the charge exchange (p,n) reactions [360]. It is known to somewhat reduce the cross section at higher energies; we will return to this point below. For further investigation on the nucleon knockout reactions from a theoretical point of view, we carried out a benchmark study on the 15C(p,pn)14Cg.s.reaction at 420 MeV [349], with three different reaction frameworks, that is, the DWIA, the quantum transfer to the continuum model (QTC) [348], and the transition amplitude formulation of the 64 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Table 5 S(p,2p) obtained from the DWIA analysis of the 12C(p,2p)11B data for the specific spin-parity Jπand the excitation energy Eex of 11B. For the analysis of the GSI data taken in inverse kinematics, the results calculated with the DWIA are shown; w/o NL and w/o Møller mean the calculation neglecting the effect of nonlocality and the Møller factor, respectively, and both are neglected in the w/o NL&Møller case. S(p,2p) determined with the analysis of the RCNP data taken in normal kinematics [239] and S(e,e′p) [361] are also displayed. Jπ(Eex (MeV)) GSI RCNP (e,e′p) DWIA w/o NL w/o Møller w/o NL&Møller DWIA DWIA 3/2−(0) 3.04 2.66 2.41 2.11 1.82(3) 1.72(11) 1/2−(2.13) 0.32 0.29 0.26 0.23 0.30(2) 0.26(2) 3/2−(5.02) 0.28 0.25 0.22 0.20 0.23(3) 0.20(2) Faddeev equations due to Alt, Grassberger, and Sandhas (FAGS) [23]. As shown in Fig. 47, the momentum distribution of 14C calculated with these three are in excellent agreement with each other, if the same ingredients are adopted. A remark on this benchmark study is that the QTC with relativistic kinematics includes the effect of the Møller factor, though it does not appear in the formalism. This is because the QTC does not use the NN transition matrix but solves a three-body equation with a Hamiltonian that contains a bare NN interaction. It will be very important that the DWIA with the Møller factor gives the same numerical result as that obtained with the QTC with a different prescription of the relativistic kinematics, when the same inputs are used and the non-locality corrections are ignored. Further benchmark studies on (p,pN) reactions in a wide range of energies are ongoing. Recently, the (p,2p) reactions in inverse kinematics have been utilized to extract s.p. structure of unstable nuclei. To establish such measurements as a reliable spectroscopic tool as the (p,2p) in normal kinematics, a measurement of 12C(p,2p)11B at around 400 MeV/nucleon has been performed at GSI [347] in inverse kinematics. Because the same reaction in normal kinematics at almost the same energy has been successfully described by the DWIA as discussed above, we applied the same framework to the 12C(p,2p)11B in inverse kinematics. In Table 5, we show S(p,2p) extracted by taking the ratio of the measured (p,2p) cross sections for the 3/2− ground state and the 1/2−and 3/2−excited states of 11B [347], to the values obtained with the DWIA that adopts a normalized proton s.p. wavefunction. The same ingredients as in the calculation shown in Fig. 53 are used. The results of S(p,2p) based on the analysis of the data taken at RCNP in normal kinematics [239] are also listed, as well as S(e,e′p) [361]. In what follows, we denote the S(p,2p) based on the DWIA analysis of the GSI and RCNP data by S(p,2p)GSI and S(p,2p)RCNP, respectively. One sees that S(p,2p)GSI are larger than S(p,2p)RCNP for all the three states of 11B. In particular, S(p,2p)GSI/S(p,2p)RCNP is even 1.67 for the ground state component of 11B. For each component, neglect of the effect of nonlocality and the Møller factor increases the cross section, hence reduces the S(p,2p), by about 10% and 20%, respectively. If we disregard both, about 30% reduction of the S(p,2p) is obtained. Although the ‘‘w/o NL&Møller’’ calculation seems to result in much better consistency with S(p,2p)RCNP and S(e,e′p), obviously, such a calculation alters also S(p,2p)RCNP. Therefore it will be difficult for the DWIA framework, which was successfully applied to various (p,2p) processes in Ref. [239], to consistently explain the cross sections of the 12C(p,2p)11B reaction measured in normal and inverse kinematics. Recently, a systematic analysis of the (p,2p) reactions on oxygen isotopes at around 250 MeV has been accomplished [59]. The results are summarized in Fig. 54. An important finding is that the reduction factor Rshas a weaker dependence on ∆S=Sp−Sn, where Sp(Sn) is the proton (neutron) separation energy, than in Ref. [46]. However, we here focus on another aspect of the result. In Fig. 54, the results for (p,2p) reactions in normal kinematics (for 16O and 18O) and those in inverse kinematics (for 14O, 22O, and 24O) are both included, and the same DWIA as in Ref. [239] was employed. Even though the Rsfor the p3/2proton in 14O, 14Op3/2, will be still not conclusive, because |∆S|is not so large, one can expect that its Rswill not so deviate from that for 16O. Then, if the DWIA calculation for some reasons would give a significantly smaller cross section in case of the inverse kinematics measurement, Rsfor 14Op3/2would be affected accordingly. In reality, as shown in Fig. 54,Rsdeduced from the (p,2p) data for 16O in normal kinematics is almost the same as that for 14Op3/2in inverse kinematics. A key feature of the measurement in Ref. [59] is that the kinematics of the two outgoing protons are restricted rather severely, that is, the polar and azimuthal angles, θand φ, respectively, of each proton are limited as 20◦< θ < 65◦and |φ|<15◦, and its kinetic energy is required to be larger than 30 MeV. Such a selection can be a reason for the consistency between the results of the DWIA analysis on the (p,2p) data in normal and inverse kinematics in Ref. [59]. To draw a conclusion, however, further investigations will be necessary. The (p,2p) cross sections on oxygen isotopes systematically measured at GSI [58] will need to be analyzed with the same DWIA framework. Very recently, a systematic DWIA analysis has been carried out [363] on the (p,pN) data for carbon, oxygen and nitrogen. The ∆Sdependence of Rswas found to be very similar to that evaluated with QTC. On the other hand, the absolute value of Rsobtained by the DWIA analysis was somewhat larger than the values by QTC and eikonal DWIA. The difference between DWIA and QTC, which appears to contradict the agreement in Fig. 43, may be due to the nonlocality correction and/or the difference in the treatment of the NN interaction. Further benchmark study is expected to reveal the source of the disagreement. There is currently a collaboration plan for pinning down the effect of the kinematical cut on the comparison between the DWIA calculation and measured value [364]. 65 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 54. Reduction factor Rs, the ratio of S(p,2p) to the spectroscopic factor calculated with the shell model, for oxygen isotopes. The horizontal axis is the difference between the proton and neutron separation energies. The triangles (inverted triangles) correspond to the results of the DWIA with the optical potential of the Dirac phenomenology [353] (the g-matrix folding model [362]). For 16O, Rsbased on S(e,e′p) is shown by the asterisk. The circle shows the result obtained by a proton removal cross section for 14O with a carbon target. Source: Figure adapted with permission from Ref. [59]. ©2018 The Physical Society of Japan. Table 6 Single-particle cross sections for the 54Ca(p,2p)53K reaction at 250 MeV/nucleon for the ground and excited states of 53K, computed using the DWIA, eikonal DWIA formalism and QTC formalisms. Jπ(Eex (MeV)) σs.p. DWIA (mb) σs.p. Eik. DWIA (mb) σs.p. QTC (mb) 3/2+(0) 1.72 1.95 1.44 1/2+(0.837) 1.87 2.18 1.64 5/2+(1.143) 1.66 2.08 1.38 Table 7 Single-particle cross sections for the 54Ca(p,pn)53Ca reaction at 200 MeV/nucleon for the ground and excited states of 53Ca, computed using the DWIA, eikonal DWIA formalism and QTC formalisms. Jπ(Eex (MeV)) σs.p. DWIA (mb) σs.p. Eik. DWIA (mb) σs.p. QTC (mb) 1/2−(0) 7.25 10.67 8.48 3/2−(2.220) 6.21 8.62 7.43 5/2−(1.738) 3.38 4.99 5.29 As a further evaluation of consistency of (p,2p) calculations, we present in Table 6 the s.p. cross sections for the 54Ca(p,2p)53K reaction at 250 MeV/nucleon, as in [365], for the ground and first two excited states of 53K. We must note that in [365], (p,2p), the DWIA cross sections were averaged over the beam energy along the thick target. The results of DWIA shown in Table 6 are therefore different from the s.p. cross sections in Table 1 of [365]. One sees from Table 6 that the DWIA result is in between the results of eikonal DWIA and QTC for each state and the standard deviation is about 15%. This value is consistent with the uncertainty of the DWIA calculation estimated in [239] at energies higher than 200 MeV. Although there exist some differences in the inputs and prescriptions in the three calculations, for 54Ca(p,2p)53K at 250 MeV/nucleon, the results are reasonably consistent with each other, though the difference in the agreement in this case and that in the 12C(p,2p) case points to a dependence on the beam energy of this agreement. In [365], the Rswas found to be 0.62–0.81. Because ∆S=16.3 MeV for the proton knockout from 54Ca, one sees from Fig. 48 that this result of Rsis consistent with the weaker ∆Sdependence obtained for the (p,pN) reactions than for the heavy-ion-induced-nucleon-removal reactions. In Table 7, a similar comparison is presented for the 54Ca(p,pn) reaction at 200 MeV/nucleon, as in [366]. The agreement between the models deteriorates in this case with differences of ∼30% between DWIA and eikonal DWIA and QTC and ∼20% between QTC and eikonal DWIA. 5.11. Conclusion The analysis of (p,2p) and (p,pn) data with stable nuclei and rare-isotope beams seem to indicate a weak or even no dependence on the proton–neutron asymmetry. This finding is compatible with the ab initio Green’s function and 66 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 coupled-cluster calculations but contradicts the trend derived from intermediate-energy one-nucleon removal cross section measurements. In the future, quasifree knockout reactions in inverse kinematics will allow for a systematic investigation of proton and neutron knockout from exotic nuclei covering a wider range of neutron-to-proton asymmetry, which will be important to corroborate the observed trend and to improve our understanding on the evolution of the s.p. structure as a function of neutron-to-proton asymmetry. On the theory side, the challenges with quasifree reactions are still mounting. The most difficult one is to justify quasifree scattering in an environment with many nucleons. It is well known that the main effect of final-state interactions (FSI) in (e,e′p) is a reduction of the corresponding cross section due to attenuation caused by multiple collisions of the nucleon in the nuclear matter. In (p,2p) and (p,pn) reactions this challenge is augmented by similar effects in the entrance channel. On the other hand, since the effect of attenuation is thus larger in (p,2p) compared to (e,e′p), where two protons have to leave the nucleus, the consistency of the reduction factors for both reactions as discussed in this section provides evidence, that the eikonal DWIA used to calculate the FSI of the outgoing protons in both cases, is a reasonably-good approximation at high energies. In DWIA, the multiple collisions are commonly treated by introducing optical potentials. A relativistic treatment of these potentials is necessary because covariance requires not only a scalar but also a vector potential. Dirac phenomenology can fix some of these issues, but the scalar and vector part of the optical potentials are often treated phenomenologically by adjusting free parameters. Three-body models suffer from the same issues and they are also simplified by treating the daughter nucleus as a third particle, despite its complex compositeness. Glauber, or eikonal, methods cure some of these deficiencies by resorting to nucleon–nucleon cross sections as inputs, bypassing nucleus– nucleus potentials and requirements set by covariance. Dealing with the medium modifications of the nucleon–nucleon cross sections is also a formidable theoretical problem, with different models being sought. It is worthwhile noticing that some of these theoretical challenges are similarly encountered in heavy-ion knockout reactions. At present, nuclear reaction theory with high-energy probes is not amenable to an ab initio treatment, starting with a nucleon–nucleon interaction and the ensuing development of a numerical ‘‘nearly exact" solution for a reaction channel in a fully covariant form. Therefore, the results presented here for quasifree and heavy-ion knockout reactions are clearly dependent on the adopted theoretical model and thus require further investigations and comparisons for consistency. 6. Analysis of proton-removal reactions from 16O Transfer, heavy-ion induced knockout, and quasifree scattering are reviewed in the previous sections with emphasis on their comparison to data obtained with stable and radioactive beams. Here, we compare the ratio of predictions to data for these three reaction mechanisms for one specific case benchmarked with (e,e′p). In Fig. 55 we show the quenching factors (Rs) obtained from proton removal from 16O through different reactions: (e,e′p), (p,2p), proton removal from a carbon target, and (d,3He) transfer. The considered data are inclusive cross sections, i.e., they include the 1/2−ground state of 15N and 3/2−bound excited states among which the first one at 6.32 MeV carries the main spectroscopic strength. Note that (d,3He) data only consider the ground state and 6.32 MeV excited state. The quenching factors are computed with respect to the independent-particle model (IPM) occupation number (6 in this case). Note that large-scale shell model calculations would lead to an integrated spectroscopic strength for the porbitals very close to the IMP value of 6 when summed up to the proton separation energy of 10.2 MeV. Special care is taken to assure the consistency of the structure inputs (i.e. the one-proton overlap) in all performed calculations, so as to highlight the effect of the reaction description on the extracted quenching factors. The quenching factors from (e,e′p) (circles) have not been computed, but instead have been taken from the corresponding references of Lapikás [15] and Leuschner [138] (at a beam energy of ∼450 MeV). We note that both references present different analyses of the same set of experimental data. The error bars for [15] correspond to the quoted values while for [138], since multiple values are presented, the chosen error bar represents the spread of quoted values in the reference. The calculations for heavy-ion knockout (triangles) and transfer (squares) have been performed using the sudden eikonal [194] and coupled-reaction channel (CRC) [375] formalisms, respectively, and are compared to experimental data from [367] (at 2100 MeV/nucleon) and [368] (at 26 MeV/nucleon). For (p,2p) the data from [58] (at 451 MeV/nucleon) is analyzed using three different formalisms: Quantum Transfer to the Continuum (QTC) [348] (diamonds with a grid), eikonal DWIA [60] (dashed horizontal diamonds) and DWIA (dashed vertical diamonds) [329]. The Rspresented in [58] is displayed as a purple diamond. Rsfor the spectroscopic factors reported in [369–374] are also presented. The shaded areas correspond to calculations using the same one-proton overlap. From left to right, the blue symbols correspond to an overlap with rms radius of 2.943 and 2.719 fm for the p1/2and p3/2states respectively, taken from (e,e′p) measurements [138], while the red symbols represent calculations with rms radii computed from Hartree–Fock calculations using the SkX interaction [290] (2.902 and 2.767 fm for p1/2and p3/2). The error bars have been computed through the quadratic sum of the experimental errors and theoretical errors. For (p,2p) QTC calculations theoretical errors have been computed as the difference in the calculations using Dirac [353] and PH [354,355] parametrizations, while for eikonal DWIA and DWIA only experimental errors have been considered. Theoretical errors have been computed as the difference in results using Hartree–Fock and Gaussian [297] densities in heavy-ion knockout and Köning-Delaroche [230] and CH89 [170] potentials in transfer. The Rsfactors in the figure correspond to the choice of potentials mentioned first for each reaction. 67 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 55. Comparison of the quenching factors Rsfor one-proton removal from 16O through different reactions: (e,e′p) (circles), (p,2p) (diamonds), heavy-ion knockout (triangles) and transfer (squares). Dashed diagonal diamonds are meant to represent (p,2p) reactions in which the measurements were performed fixing the outgoing angles of the outgoing protons. (e,e′p)Rshave been taken from [15] and [138]. Experimental data are taken from [58] (p,2p), [367] (heavy-ion knockout) and [368] (d,3He). The Rsfactor from [58,369–374] are presented in purple. Calculations using the same one-proton overlaps are presented within shaded bands and with the same color: blue correspond to overlaps with rms radius from [138] and red to rms radius computed from Hartree–Fock using SkX interaction. (More details in text). It can be seen that the Rsfactors obtained from (p,2p) using QTC (∼0.7) and DWIA (∼1.0) are significantly larger than the ones from heavy-ion knockout (∼0.5–0.6) and transfer (∼0.4–0.5), while the result of the eikonal DWIA ( ∼0.65) is closer to them, and the (p,2p) results from eikonal DWIA calculations and the results from heavy-ion knockout are more compatible with the (e,e′p) result from [15,138], while the Rsfrom transfer and (p,2p) using QTC are in worse agreement, and those from DWIA are clearly incompatible. The different proton overlaps increase or decrease consistently the Rsfactors for the three reactions, with no significant change in their differences. Given the only moderate sensitivity to the structure input, we may compare quenching factors (when referred to the IPM) from previous publications: 0.7 (p,2p) [58], 0.59 (heavy-ion knockout) [297] and 0.46 (transfer) [203], which are consistent with the present results and trend: using the same structure inputs the quenching factor is larger for (p,2p) and smaller for (d,3He) in the considered frameworks. Although the scatter of published results of spectroscopic factors vary substantially as shown in the multiple references presented in Fig. 55, most of the values are in agreement with the scatter of data from (e,e′p) (indicated by the horizontal band), in particular if we compare only the results using the same overlap function (Leuschner), with the exception of the DWIA analysis, which results in a significantly larger Rsvalue consistent with the discussion of the 12C case discussed in Section 5. 7. Conclusions and outlook It is well established and understood that the single-particle strength distribution is reduced by 30%–40% compared to shell-model predictions due to correlations beyond that accounted for in the model. This picture is well established for stable nuclei and consistent with experimental nucleon-removal cross sections from (e,e′p) and from other reactions like transfer, knockout, and quasifree scattering when compared to predictions of cross sections based on single-particle wave functions and shell-model correlations. For the nuclear probes, however, a scatter of up to ∼25% is observed (see for example Section 6for 16O, and Section 5.10 for 12C). In this review, we discuss an extension of this picture which results from a large body of data using rare-isotope beams of very asymmetric nuclei, suggesting a strong dependence of this reduction as a function of the proton-to-neutron binding-energy asymmetry. This asymmetry dependence has been deduced from measurements of Be- and C-induced nucleon-removal reactions at intermediate energies of ∼100 MeV/nucleon (see Fig. 2). The trend could originate from additional correlation effects in asymmetric nuclei, beyond our current understanding of nuclear structure. For example, a stronger spread of the single-particle strength for deeply-bound orbitals, or an increase of short-range correlations for the minority nucleon species would result in reduced spectroscopic factors. Under the assumption that the discussed direct reaction mechanisms probe the same nuclear structure information, it is a consensus that if they were understood properly, they should lead to the same conclusions on the asymmetry dependence of correlations. In recent years, different direct reactions, namely transfer and quasifree scattering, have been employed to investigate this effect, although the data are still scarce for very asymmetric nuclei. So far, these measurements could not confirm such 68 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 56. The four panels of this plot show the quenching (reduction) factors for (a) electron-induced knockout reactions [87,172,237,376], (b) transfer reactions with radioactive ion beams [55,57,203], (c) quasifree (p,2p) proton knockout on stable nuclei (from the compilation in [239]) and radioactive nuclei [58,59], and (d) the inclusive intermediate-energy knockout data [46]. The measurements are compared to predictions based on effective-interaction shell-model SFs while, in the case of (e,e′p), the integrated strength is compared to the independent-particle expectation. a strong asymmetry dependence as observed in Be- or C-induced nucleon removal at intermediate energies. Fig. 56 shows a summary of various data from the different probes discussed in this review. For modest values of −12 ≲∆S≲12 MeV, all probes quantitatively agree that there is a quenching of single-particle strength, reduced to around 40%–70% of the total. This is arguably true for −15 ≤∆S≤15 MeV, within the moderately-large experimental and analytical uncertainties. The analyses of neutron-pickup transfer data with neutron-deficient 34Ar (∆S= MeV) and neutron-rich 46Ar (∆S= MeV) isotopes [55] using different methods and optical potentials lead to different conclusions regarding the presence or absence of a strong dependence of the data-to-prediction ratio with ∆S[55,57]. Transfer studies based on three data sets of oxygen isotopes [54] and analyzed within the coupled-channel formalism did not observe the strong trend from Be- and C-induced nucleon-removal reactions. Recent (p,2p) [58,59], covering essentially the full range in ∆S, do not confirm the strong trend from Be- and C-induced nucleon-removal reactions. These data have been meanwhile analyzed using different reaction models arriving at similar conclusions, although predictions of such reaction models can differ. Although Be- or C-induced nucleon-removal cross sections have been analyzed with different models, the systematics of the quenching factor as a function of ∆Shas been investigated with only one model so far. A recent analysis of the (e,e′p) reaction for both 40Ca (∆S= −7.3 MeV) and 48Ca (∆S=5.8 MeV) employing the DOM predicts a reduction of the spectroscopic strength of 0.71 and 0.58, respectively (see Fig. 56). The results are consistent with earlier analyses. Two recent DOM analysis for 208Pb give consistent results of 0.69 [172] and 0.64 ±0.06 [376], in agreement with the value of Ref. [87], a re-analysis of the initial work of Ref. [237] which led to a now-considered- too-low value for the quenching. All values are superimposed in the left panel of Fig. 56. The DOM links both structure and reaction quantities and relies on experimental data to constrain removal probabilities as well as the optical potential for these isotopes. It therefore simultaneously allows for a change in the structure properties as a function of nucleon asymmetry but importantly also covers the change in the way continuum nucleons experience nuclei with different asymmetry. This approach provides a distinct advantage over methods that rely on ingredients that are derived from free nucleon–nucleon scattering data or uncertain extrapolations of phenomenological optical potentials which are not constrained by experimental data. A continued exploration of the DOM to generate results from data-driven extrapolations to the respective drip lines is therefore a promising approach to provide further clarification of the issues discussed in the review. The DOM can also provide a liaison between ab initio nuclear-structure calculations and experimental results by providing nonlocal optical potentials or conversely provide overlap functions to combine with ab initio optical potentials that have become a focus of recent efforts some of which have been reviewed in Ref. [34]. Independently of the origin of the observed trend, we can conclude that there are inconsistencies between the directreaction model conclusions. Reviewing the state-of-the-art reaction studies, we conclude that the problem cannot be resolved at this stage. It is of utmost importance to further understand the different reaction mechanisms by dedicated key experiments hand in hand with theory developments in the near future. So far, most of the investigations on reaction mechanisms with very asymmetric nuclei have been performed using Beinduced reactions. Some selected nuclei should be investigated with transfer and quasifree scattering at different energies as well. Fig. 57 shows the ∆Svalues for ground-state to ground-state transitions for nuclei across the chart of nuclides, for nuclei where both Snand Spare known. Those with |∆S|≳15 MeV are naturally confined to lighter (below A∼40) 69 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 57. A plot of ∆Svalues on the chart of nuclides for ground-state to ground-state transitions for all nuclei with known values of Snand Spin the 2016 Atomic Mass Evaluation [377]. About 20% have |∆S|≳12 MeV. neutron-rich or neutron-deficient systems. Many of these, and even those with |∆S|≳18 MeV, will be available at new or upcoming facilities and at suitable energies for different types of direct reactions. A systematic study of proton removal cross sections along the proton-closed shell of Ca and Ni isotopic chains would be of special interest. Of particular relevance is the program started at the NSCL measuring mirror nuclei, which offers a promising prospect of disentangling the effects of the reaction-model approximations on the observed cross sections and resulting reduction factors. In particular, measurements on p-shell nuclei for which increasingly sophisticated ab initio wave functions are now available would be of interest (see Section 4.4.3). This method should also be tested using other probes and reaction mechanisms used to deduce spectroscopic strength, such as the transfer reactions and quasifree proton scattering discussed in this review. Measurements of the target excitation and breakup in Be- or C-induced nucleon-removal reactions are necessary to further understand the missing ingredients in the eikonal calculations. Specific studies in quasifree scattering could be helpful to understand the re-interaction probability of scattered nucleons with the residual nucleus, as discussed in [378] and [379]. A direct measurement of the excitation-energy distribution of the target after knockout with identified A−1 fragments has not been performed so far. A refined measurement of the energy dependence from ∼60 MeV/nucleon to ∼1 GeV/nucleon for both heavy-ion induced and quasifree (p,pN) nucleon knockout on reference stable and asymmetric nuclei would provide a unique benchmark for reaction models. The 16O and 14O oxygen isotopes appear ideal cases. Different incident energies should result in a difference in the peripherality of the reaction. Since the strengths of the nuclear correlations depend on the density, the difference in peripherality can lead to differences in the obtained reduction factor. As discussed above, there are still significant discrepancies among quasifree scattering models. In particular, we note that they occur for integrated cross sections while for limited-acceptance measurements they disappear. A comparison of calculated angular distributions of the different models, and to measurements, should shed light on these differences. Along the same line, the detection of the knocked-out particle in heavy-ion-induced reactions, as benchmarked with 12C in [378], over a broad range of neutron-to-proton asymmetries would provide further insight in the reaction mechanism and would allow a direct comparison to the (p,pN) studies. Another crucial type of measurement would be (e,e′p) quasifree scattering off rare isotopes. Electron scattering experiments at the SCRIT facility of RIKEN will start soon. Although the present luminosity does not allow (e,e′p) experiments with sufficient statistics, the opportunity will be opened by future upgrades, SCRIT-like experiments of new generation or by the realization of an electron-RI collider [380]. During the last two decades, direct reactions using rare-isotope beams have seen a fast development. In this review, we summarized the findings concerning the single-particle strength from direct reactions with rare isotopes. On the basis of these newly available data, together with the ongoing theoretical efforts and new experimental opportunities, we are confident that the remaining challenges will be addressed and will bring nuclear physics to a new level towards the understanding of nuclear structure from dripline to dripline. Acknowledgments We thank the ExtreMe Matter Institute EMMI at GSI, Darmstadt, for support in the framework of a dedicated EMMI Rapid Reaction Task Force meeting in Darmstadt, Germany, from July 30th to August 4th, 2018, during which this work has been initiated. The authors thank H. Feldmeier, T. Neff, R. Roth and S. Typel for their participation to the meeting and discussions. C. B. would like to thank A. Polls, V. Somà and T. Duguet for useful discussions. This work was also 70 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 supported by the U.S. National Science Foundation under grant PHY-1912643, 1613362, 1415656 and the by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, grants DE-FG02-08ER41533 and DE-SC0020451 (MSU) and under Contract Number DE-AC02-06CH11357 (ANL). A. M. M. is partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades and FEDER funds under project FIS2017-88410-P and RTI2018-098117-B-C21 and by the European Union’s Horizon 2020 research and innovation program under Grant Agreement No. 654002. K. O. is supported in part by Grant-in-Aid of the Japan Society for the Promotion of Science (Grant No. JP16K05352). T. N. acknowledges support from JSPS KAKENHI No. JP18H05404 and JP16H02179. C. A. B. acknowledges funding contributed through the LANL Collaborative Research Program by the Texas A&M System National Laboratory Office and Los Alamos National Laboratory, USA. T. A. acknowledges support by the German Federal Ministry of Education and Research (BMBF project 05P2015RDFN1), and through the GSI-TU Darmstadt cooperation agreement. C. B. acknowledges funding from the UK Science and Technology Facilities Council (STFC) through grants ST/P005314/1 and ST/L005516/1. First principle calculations discussed in Section 2.7 used HPC resources at the DiRAC DiAL system at the University of Leicester, UK, (BIS National E-infrastructure Capital, United Kingdom Grant No. ST/K000373/1 and STFC, United Kingdom Grant No. ST/K0003259/1). T. A. and A. O. acknowledge partial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 279384907 - SFB 1245. A. O. and M. G.-R. acknowledge support by the Alexander von Humboldt foundation. Appendix A. Analysis of the nucleon self-energy and some pedagogical illustrations Calculations of the nucleon self-energy can proceed from a diagrammatic form for the self-energy. Several possibilities are illustrated in Fig. 58 that are exact when only two-body interactions are present in the Hamiltonian. Substantial modifications are involved when three-body interactions are included [381]. Using the equation of motion for the s.p. propagator [74] one obtains Σ⋆(γ , δ;E)= −⟨γ|U|δ⟩+⟨γ|ΣHF |δ⟩1 2∫dE1 2π∫dE2 2π∑ µ,ν,λ ∑ α,β,ε⟨γ ε|V|αβ⟩ ×G(α, µ;E1)G(β, ν;E2)⟨µν|Γ4pt (E1,E2;E,E1+E2−E)|δλ⟩G(λ, ε;E1+E2−E) (80) where the term −⟨γ|ˆ U|δ⟩subtracts the auxiliary potential which was added in the unperturbed Hamiltonian ˆ H0=ˆ T+ˆ U. This result is shown in part (a) of Fig. 58), The second term in Eq. (80) is the HF contribution to the self-energy, ⟨α|ΣHF |β⟩ = ∑ γ δ ∫dE 2πie+iEη+⟨αγ |V|βδ⟩G(γ , δ;E).(81) which represent the (energy-independent) interaction of the nucleon with the quasihole excitations inside the system and requires knowledge of the one-body density matrix given in Eq. (12). When all the higher-order terms of the self-energy are neglected, the solution of the Dyson equation Gtakes the simple form as Eq. (82), which is equivalent to a Slater determinant description of the ground state filling the lowest-energy orbits. The corresponding unperturbed propagator associated with a Fermi-sea Slater determinant |ΦA 0⟩is given by G(0)(α, β;E)=δα,β {θ(α−F) E−εα+iη+θ(F−α) E−εα−iη},(82) where θ(α−F) ( θ(F−α) ) is equal to 0 (1) if αis an occupied state and it is 1 (0) otherwise. In this case the energies εα contained in G(0) refer s.p. energies corresponding to completely filled or empty orbitals in |ΦA 0⟩. The iterative solution of Eqs. (13) and (81) generates the standard HF approximation. All qualitative changes when higher-order contributions to the self-energy are incorporated, can already be illustrated with the introduction of the second-order term. We illustrate this by forgoing a self-consistent formulation and employ HF propagators to evaluate the second-order diagram that can be obtained by replacing Γ4pt in Eq. (80) by the bare interaction V. The result is given by Σ(2)(γ , δ;E)=1 2⎡ ⎣∑ p1p2h3 ⟨γh3|V|p1p2⟩⟨p1p2|V|δh3⟩ E−(εp1+εp2−εh3)+iη+∑ h1h2p3 ⟨γp3|V|h1h2⟩⟨h1h2|V|δp3⟩ E+(εp3−εh1−εh2)−iη⎤ ⎦,(83) where labels identifying particle (p) and hole (h) states in the HF approximation have been introduced. We will proceed to examine the solution Gof the equation G(α, β;E)=GHF (α, β;E)+∑ γ δ GHF (α, γ ;E)Σ(2)(γ , δ;E)G(δ, β;E).(84) It is clear from Eq. (83) that the second-order self-energy in principle has non-diagonal contributions, even when evaluated with the diagonal HF s.p. propagator. However, in some cases it is a good approximation to neglect the off-diagonal terms. 71 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 58. Various possible expansions of the irreducible self-energy in terms of higher-order Green’s functions. In all cases the first two terms represent the mean-field contributions −ˆ Uand the HF self-energy ΣHF . The dynamic part of the self-energy can be expressed in terms of the four-points vertex function Γ4pt . Part (a) represents the relation given in Eq. (80), while the part (b) corresponds to an alternative derivation involving the time derivative of Gw.r.t. t′. If Γ4pt is approximated in such a way that the corresponding two diagrams in (a) and (b) are equivalent, the Dyson equation will satisfy appropriate sum rules. Part (c) gives the expansion in terms of the one-particle irreducible two-particle–one-hole (2p1h) / two-hole–one-particle (2h1p) propagator R(ω) [75]. Note that the full four-time dependence of Γ4pt is needed in principle, while in the R(ω) formulation one can specialize to a two-time quantity. Source: Reprinted figure with permission from [75]. ©2004 Elsevier. This happens e.g. in closed-shell nuclei, where off-diagonal elements would require mixing between major shells having a large energy separation. Within this diagonal approximation, the self-energy (83) reads Σ(2)(α;E)=1 2⎡ ⎣∑ p1p2h3 |⟨αh3|V|p1p2⟩|2 E−(εp1+εp2−εh3)+iη+∑ h1h2p3 |⟨αp3|V|h1h2⟩|2 E+(εp3−εh1−εh2)−iη⎤ ⎦,(85) and the Dyson equation (84) becomes G(α;E)=GHF (α;E)+G(α;E)Σ(2)(α;E)GHF (α;E).(86) The latter has a simple algebraic solution, G(α;E)=1 1 GHF (α;E)−Σ(2)(α;E)=1 E−εα−Σ(2)(α;E),(87) using the inverse of the HF propagator [GHF (α;E)]−1=E−εα. Extracting physical information from the s.p. propagator in general requires the knowledge of its poles and residues. We assume for simplicity that the self-energy Σ(2) has poles at a set of discrete energies (i.e., a set of isolated simple poles), while an extension to treat the case when branch-cuts are present can be included straightforwardly. Most realistic finite systems have branch-cuts, but since practical calculations are usually performed by introducing a finite and discrete s.p. basis, the self-energy is then automatically restricted to a discrete pole structure. For the propagator G(α;E) given by the formal solution in Eq. (87), the discrete poles Enαobviously correspond to the roots of the nonlinear equation Enα=εα+Σ(2)(α;Enα),(88) 72 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Fig. 59. Graphical solution of Eq. (88). The second-order self-energy Σ(2)(α;E) of Eq. (85) is indicated by the dashed line. The roots of Eq. (88) are given by the intersections points with the straight line E−εα, drawn here as dots, for two values of εα. Source: Figure adapted from Ref. [74]. with Σ(2)(α;E) defined in Eq. (85). The residue Rnαat the pole Enαof the propagator follows from Rnα=lim E→Enα (E−Enα)G(α;E)== (1−dΣ(2)(α;E) dE ⏐⏐⏐⏐E=Enα)−1 .(89) To gain insight into the location of the roots of Eq. (88), a graphical solution of the Dyson equation is often helpful. In Fig. 59 the energy-dependence of the self-energy Σ(2)(α;E) of Eq. (85) is shown. The case on display is for a typical confined finite system, having a discrete HF s.p. spectrum. The hole and particle HF energies are separated by the particle–hole gap, which has a width ∆=εmin p−εmax hand is centered on the HF Fermi energy εF=1 2(εmin p+εmax h).(90) Since the poles in Eq. (85) all have positive residues, Σ(2)(α;E) is monotonically decreasing where defined. There is a sequence of simple poles in the addition domain, located at the unperturbed HF 2p1h energies, and another sequence in the removal domain, located at (minus) the unperturbed HF 1p2h energies. The poles of the addition and removal sequence are separated by a gap of (at least) three times the HF particle–hole gap. The roots of Eq. (88) are simply the intersection points of the self-energy Σ(2)(α;E) with the straight line E−εα. It is obvious from the graph in Fig. 59 that between any two successive poles of the self-energy a root is located. In addition, there is a root to the left and right of the sequence of self-energy poles. When a finite s.p. basis set is used, this implies that a self-energy having Dpoles leads to a s.p. propagator with D+1 poles. The interpretation of these roots should by now be straightforward. The poles Enα in the removal domain (below the Fermi energy) must be interpreted as approximate energies of the eigenstates in the A−1 system Enα≈EA 0−EA−1 n, that can be obtained by removing a particle in the s.p. state αfrom the A-particle ground state. The residue then corresponds to the (squared) removal amplitude, Rnα≈ |⟨ΨA−1 n|aα|ΨA 0⟩|2.(91) Similarly, the poles Enαin the addition domain (above the Fermi energy) correspond to eigenstates in the A+1 system Enα≈EA+1 n−EA 0, having addition probabilities Rnα≈ |⟨ΨA+1 n|a† α|ΨA 0⟩|2.(92) Note that since dΣ(2)(α;E)/dE <0, the residues Rnαthat follow from Eq. (89) obey 0≤Rnα≤1,(93) in accordance with their relation to the physical addition or removal probabilities. The sum of the removal probabilities then corresponds to the occupation of the state αas in Eq. (8). The sum of the addition probabilities Eq. (9) then complements the sum rule of Eq. (10). 73 T. Aumann, C. Barbieri, D. Bazin et al. Progress in Particle and Nuclear Physics 118 (2021) 103847 Table 8 Diffraction dissociation cross sections for the indicated initial states in a number of reactions, first column; incident energy, second column; initial state separation energy, and cross sections values as indicated. Reaction & s.p. orbital Elab SnEq. (53) Eq. (119) Eq. (53) no χC (A.MeV) (MeV) (mb) (mb) (mb) 11Be(2s1/2)+9Be 40 0.54 84.94 87.06 78.41 70 0.54 65.44 65.49 61.21 150 0.54 36.97 36.01 35.07 1000 0.54 24.78 24.42 23.9 11Be(2s1/2)+208Pb 40 0.54 1311 308.7 229.3 70 0.54 925.4 277.3 208.2 150 0.54 603.4 269.9 196.0 1000 0.54 258.4 178.1 129.8 14O(1p3/2)+9Be 53 23.18 3.431 3.321 1.988 32Ar(1d5/2)+9Be 70 21.56 2.942 1.681 1.212 32Ar(1d5/2)+208Pb 70 21.56 397.5 5.747 3.815 24Si(1d5/2)+9Be 85.3 21.09 2.962 2.33 1.494 clear that the Coulomb phase cannot be used in Eq. (53) in particular for the heavy targets. The results are different from those of Eq. (119) because of the interference between the nuclear and recoil amplitude in Eq. (53). Eq. (119) can be used for any type of target because it is free from ambiguities on its implementation, furthermore it has the same structure as the stripping formula Eq. (52) and as we said above, from its derivation one can also obtain momentum distribution. 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