Progress in Particle and Nuclear Physics 98 (2018) 119–206

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Progress in Particle and Nuclear Physics

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Review Baryons and baryon resonances in nuclear matter Horst Lenske a,*, Madhumita Dhar a,b , Theodoros Gaitanos a,c , Xu Cao a,d,e a Institut für Theoretische Physik, JLU Giessen, Heinrich-Buff-Ring 16, D-35392, Gießen, Germany b Balurghat College, Balurghat, India c Aristotle University of Thessaloniki, Thessaloniki, Greece d Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China e State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China article info a b s t r a c t

Article history: Theoretical approaches to the production of hyperons and baryon resonances in ele- Available online 7 October 2017 mentary hadronic reactions and heavy ion collisions are reviewed. The focus is on the production and interactions of baryons in the lowest SU(3) flavor octet and states from Keywords: the next higher SU(3) flavor decuplet. Approaches using the SU(3) formalism for inter- Hyperons actions of mesons and baryons and effective field theory for hyperons are discussed. Baryon resonances Strangeness production An overview of application to free space and in-medium baryon–baryon interactions is Transport theory given and the relation to a density functional theory is indicated. The intimate connection Nuclear matter between baryon resonances and strangeness production is shown first for reactions on the Hypernuclei nucleon. Pion-induced hypernuclear reactions are shown to proceed essentially through the excitation of intermediate nucleon resonances. Transport theory in conjunction with a statistical fragmentation model is an appropriate description of hypernuclear production in antiproton and heavy ion induced fragmentation reactions. The excitation of subnuclear degrees of freedom in peripheral heavy ion collisions at relativistic energies is reviewed. The status of in-medium resonance physics is discussed. © 2017 Elsevier B.V. All rights reserved.

Contents

1. Introduction...... 120 2. Interactions of SU(3) flavor octet baryons...... 124 2.1. General aspects of nuclear strangeness physics...... 124 2.2. Interactions in the baryon flavor octet...... 125 2.3. Baryon–baryon scattering amplitudes and cross sections...... 129 2.4. In-medium baryon–baryon vertices...... 134 2.5. Vertex functionals and self-energies...... 136 3. Covariant DFT approach to nuclear and hypernuclear physics...... 137 3.1. Achievements of the microscopic DDRH nuclear DFT...... 137 3.2. Covariant Lagrangian approach to in-medium baryon interactions...... 137 4. DBHF investigations of Λ hypernuclei and hypermatter...... 139 4.1. Global properties of single-Λ hypernuclei...... 139 4.2. Spectroscopic details of single-Λ hypernuclei...... 140 4.3. Interactions in multiple-strangeness nuclei...... 142 4.4. Hyperon interactions and hypernuclei by effective field theory...... 143 4.5. Brief overview on LQCD activities...... 144

* Corresponding author. E-mail address: [email protected] (H. Lenske). https://doi.org/10.1016/j.ppnp.2017.09.001 0146-6410/© 2017 Elsevier B.V. All rights reserved. 120 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

4.6. Infinite hypermatter...... 145 5. Strangeness and resonance excitation on the free nucleon...... 147 5.1. Coupled channels approach to nucleon resonances...... 147 5.2. The Giessen coupled channels model for baryon spectroscopy...... 147 5.3. Strangeness production on the nucleon...... 149 5.4. Probing nucleon resonances by η-meson production...... 151 5.5. Double-pion production on the nucleon through resonances...... 153 5.6. Discussion of strangeness and resonance physics on the nucleon...... 156 6. Strangeness production in coherent reactions with elementary probes...... 158 + + 6.1. Covariant model for the A(π , K )ΛA reaction...... 159 6.2. Interaction Lagrangians...... 159 6.3. Resonance propagators...... 161 6.4. Nuclear model...... 161 6.5. Hypernuclear spectra from (π +, K +) reactions...... 161 7. Gauge theory of high-spin fermionic fields...... 162 7.1. Gauge properties of spin-3/2 fields...... 162 7.2. Gauge properties of spin-5/2 fields...... 164 8. Heavy ion collisions as a probe for strangeness in nuclear matter...... 166 8.1. Transport theory for hadronic collisions...... 167 8.1.1. Transport theoretical description of particle production in hadronic reactions...... 167 8.1.2. Time evolution of the 1-body density...... 169 8.1.3. Fragmentation reactions...... 169 9. Hypernuclear production in fragmentation reactions...... 170 9.1. Results of the hybrid approach to fragmentation...... 170 9.2. Dynamics of strangeness and hypernuclei in heavy-ion collisions...... 173 9.3. Brief discussion of alternative approaches...... 174 10. Multi-strangeness dynamics in antiproton-induced reactions...... 175 10.1. Reaction dynamical aspects...... 175 10.2. Hyperon production...... 176 11. Production of Ω− hyperons in antiproton annihilation on nuclei...... 179 11.1. Hyperon–nucleon interaction in the S = −3 sector...... 179 11.2. Ω− production on nuclei...... 181 12. Brief theory of resonances in nuclear matter...... 183 12.1. Decuplet baryons as dynamically generated, composite states...... 183 12.2. Excitations of resonances in nuclear matter: the N∗N−1 resonance nucleon-hole model...... 185 12.3. ∆ mean-field dynamics and resonances in neutron stars...... 188 12.4. Response functions in local density approximation...... 189 13. Production and spectroscopy of baryon resonances in nuclear matter...... 191 13.1. Resonances as nuclear matter probes...... 191 13.2. Interaction effects in spectral distribution in peripheral reactions...... 192 13.3. Resonances in central heavy ion collisions...... 192 13.4. The delta resonance as pion source in heavy ion collisions...... 193 13.5. Perspectives of resonance studies by peripheral heavy ion reactions...... 195 14. Summary...... 198 Acknowledgments...... 200 References...... 200

1. Introduction

In the early days of nuclear and elementary particle physics the plethora of particles observed in high energy experiments casted doubts on their elementary particle nature. Obviously, a new approach was necessary allowing to bring order into the data. The group-theoretical approach introduced independently by Murray Gell-Mann and Yuval Ne’eman in the beginning of the sixties of the last century, leading finally to the quark model, was the long awaited for breakthrough toward a new understanding of hadrons in terms of a few elementary degrees of freedom given by quarks and gluon gauge fields as the force carrier of strong interactions. One of the central predictions of early QCD was the parton structure of hadrons. Once that conjecture was confirmed by experiment in the early 1970’s [1], Quantum Chromo Dynamics (QCD) has evolved into the nowadays accepted standard model of strong interaction physics, describing hadrons by quarks and gluons with color and flavor quantum numbers. Renormalization group techniques and regularization methods, asymptotic freedom on the one side and quark confinement on the other side are defining parts of strong interaction physics. The Higgs-mechanism is responsible for the non-vanishing masses of the current quarks. Gluon self-interactions and effects from the spontaneously broken chiral symmetry are important ingredients of constituent quark and hadron masses. The highly non-linear nature of QCD inhibits analytical or perturbative solutions for most of the accessible energy region, except for the highest energies as e.g. reachable at the Large Hadron Collider (LHC) at CERN. Since long, QCD theory has become part of the solid foundations of H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 121

Fig. 1. The first two baryon SU(3) flavor multiplets are given by an octet (left) and a decuplet (right). The valence quark content of the baryons is indicated explicitly. The vertical axes are representing the hypercharge Y = S + B, given by the strangeness S and baryon number B; the horizontal axes indicate = − 1 I3 Q 2 Y the third component of the isospin I, which includes also the charge number Q . The group theoretical background and construction of these kind of diagrams is discussed in depth in textbooks, see e.g. [2].

modern science. As indispensable part of the scientific narrative QCD gauge theory has entered into text books, for instance the one by Cheng and Li [2]. Today, more than 120 baryons and baryon resonances are known on firm grounds ranging from the proton and neutron, − − the hyperons including at least one strange quark and charm and bottom states up to the Ξb and Ωb baryons. However, that number is still far away from matching the quark model predictions. In this article, we consider specifically the production, interactions, and formation of states of the baryons composed of {u, d, s} valence quarks. In Fig.1 the SU(3) group diagrams of 1 + 3 + the lowest flavor octet and decuplet of 2 and 2 baryons, respectively, are shown. Phenomena at energy scales extending from several GeV per baryon down to nuclear separation energies will be considered. The focus of this article is on the non-perturbative region of strong interaction physics. One aim is to point to intersections of hadron physics and nuclear many-body physics and the large common research potential yet to be discovered. While the high energy sector of QCD is well understood by perturbative approaches, the low-energy limit is still a field with many open questions and this is even more so for nuclear systems. The description of hadrons which are the QCD mass eigenstates in the infrared region is still a highly nontrivial task. Lattice QCD (LQCD) provides the required powerful numerical methods. Given the necessary computing power, QCD is allowing, in principle, to study the baryon and meson spectrum on the level of the basic principles of a quantum gauge field theory. In 2008, the Budapest–Wuppertal collaboration obtained a first QCD description of the mass spectrum of the lightest hadrons of the SU(3) flavor octets and the baryon decuplet [3]. The mass spectrum is displayed in Fig.2. Recent QCD activities on mass spectra were reviewed e.g. in [4]. While the cited QCD investigations approach hadron spectroscopy from the inner sector of the quark core of hadrons, effective field theories and phenomenological reaction models put their focus on the properties reflecting the nature of hadrons as dynamically generated composite states. Compositeness is another aspect of the dualism hidden in baryons and meson as visible in the observed resonances in scattering and production experiments. Until today, phenomenological approaches based on an underlying covariant field theory are an important source of spectroscopic information on the quantum numbers of resonances, their formation and their decay. The models are apt for large scale numerical coupled channels calculations utilizing the partial wave formalism. A representative example for this class of approaches is the Giessen Model (GiM) [5–7] using a covariant Lagrangian formulation which accounts for the fundamental symmetries of QCD in the {uds} flavor sector. The meson production on the nucleon is described through the excitation of baryon resonances in photo- and meson-induced reactions on the nucleon. Baryon resonances are identified through their traces left as complex energy poles in the partial wave scattering matrix elements which in many cases are strongly affected by coupled channels effects. Similar approaches have been formulated independently by several other groups, among those the Bonn–Gatchina (BoGa) [8] and the SAID [9] coupled channels partial wave solutions are accessible online. A notorious problem of the QCD-inspired quark models is the overabundance of theoretically predicted baryons exceeding the number of the experimentally identified states considerably. In the 3-quark baryon sector the combined O(3) ⊗ SUflavor(3) ⊗ SUspin(2) group structures leads to a rather complex multiplet structure with a large number of states of which only a small fraction has been confirmed by experiment. The collection of the latest data is found and classified in the regularly updated listings of the Particle Data Group [10] (where, by the way, also a couple of GiM results have been entered). This mismatch between the theoretically predicted and experimentally observed spectroscopic densities is known in hadron physics as the missing resonance puzzle. To some extent, the overabundance problem may be related to the fact that physically the pure baryon QCD states are embedded into and coupled to a continuum (of resonant and non-resonant) meson–baryon scattering states, as exploited by the coupled channels models. Those interactions are inducing dispersive self-energies which are leading to a more complex pattern of the spectral distributions beyond the assumed O(3) ⊗ SU(6) scheme. By these polarization effects the bare QCD configurations are shifted in mass and their spectroscopic strength is distributed over the eigenstates of the coupled system. Thus, baryons heavier than the nucleon acquire rather short lifetimes because they may decay either by strong interactions or, as in the case for some of the octet baryon by weak interactions. 122 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 2. The light hadron spectrum of QCD as obtained by QCD. Horizontal lines and bands are the experimental values with their decay widths. QCD results are shown by solid circles. Vertical error bars represent combined statistical systematic error estimates. The proton (p), the S = −2 cascade baryon Ξ, and the S = −1 kaon (K) have no error bars, because they were used to set the light quark mass, the strange quark mass and the overall scale, respectively. Source: From Ref. [3].

+ P = 3 The multiplet of ∆33(1232) states, belonging to the J 2 - baryon decuplet, is a typical case: As a QCD state the Delta- = 3 3 resonance is explained as a spin–isospin excitation of the nucleon into a (J, I) ( 2 2 ) configuration. The state is coupled, however, strongly to the continuum of pion–nucleon p-wave scattering states which induces a rapid decay within a half- −23 life of about t1/2 ∼ 10 s. Energetically, this is corresponding to a spectral distribution of full width at half maximum (FWHM) Γ∆ ≃ 120 MeV. In contrast to other baryon resonances, the Delta-resonance is prominently excited in practically all types of reactions, from virtual and real photo-excitation by electrons and photons to neutral and charge current neutrino– nucleon and neutrino–nucleus reactions, and in inelastic hadronic scattering of protons, both on the elementary targets and on nuclear targets. The same kind of channel coupling effects are also present in the meson spectrum, as seen e.g. by the finite lifetime of the states in JP = 1− vector meson octet. The ρ(770) meson, for example, is embedded into the ππ p-wave continuum resulting in a decay width of Γρ ∼ 150 MeV. Our present understanding of the wave functions of baryon resonances is indicated pictorially in Fig.3: Baryons with masses above the pion–nucleon threshold should be considered as superpositions of molecular-like, loosely bound or unbound meson–nucleon configurations which are in competition with the genuine QCD-type quark core configurations where the latter are shielded from the exterior by a virtual qq¯ meson cloud. Very likely also glue-ball like configurations will contribute as indicated by the large differences between the current and the constituent quark masses. A given resonance will be composed by an arbitrary mixture of two building blocks. In some cases the quark-type configurations will dominate, in others the molecular configuration may prevail. A hot candidate for the latter type of states is the Λ∗(1405) S = −1 resonance, found to be a KN¯ composite [11]. Likewise, for quark–antiquark pairs corresponding configuration mixtures are found also in meson resonances. Interestingly, the molecular and the quarkonic components exist on quite different energy–momentum scales. The meson–baryon parts can be expected to cover four-momenta of a few times of the pion rest mass, mπ ∼ 140 MeV. The QCD-driven configurations, however, are covering momentum regions above the vector meson rest mass of mv ∼ 800 MeV. These scale differences may have important consequences for the response of hadrons under extreme environmental conditions as in cold or heated compressed nuclear matter, even before the phase transition in the heavily discussed Quark–Gluon–Plasma (QGP). Obviously, the molecular-type components of the hadron wave function are more easily to polarize than the quarkonic part. Searches for signals indicating modifications of the properties of baryons or mesons in compressed and heated nuclear matter have been a strong motivation of measuring the production and/or decay of hadron resonances in heavy ion collisions ever since their advent in the 1970’s. Relativistic heavy ion collisions with kinetic energies per nucleon at or well above of the proton rest mass were always considered as an ideal tool for studies of in-medium properties of hadron resonances. The initial phase of experimental high energy nuclear research was characterized by using former particle physics accelerator facilities like the BEVALAC at Berkeley, the SPS at CERN, and the Dubna Synchro-Phasetron as heavy ion accelerators. Soon after, the first dedicated heavy ion research facilities like the AGS at Brookhaven, and SIS at GSI, each with a couple of specialized detector systems, came into operation. As a follow-up to the AGS, the RHIC facility was set up at Brookhaven. The main task of the experiments at GSI and AGS/RHIC was to explore the nuclear equation of state and eventually explore the path toward the QGP as a new state of matter. The latter research area is continued nowadays under much improved and extended conditions by the ALICE experiment at CERN and will be the topic of the CBM experiment planned at the FAIR facility at GSI. Depending on the centrality of the reaction, characterized either by the value of the impact parameter or the rapidity, a relativistic heavy ion collision can be divided into the participant and the spectator matter. The spectator components are the sheared off parts of projectile and target nucleus outside the overlap region of the two ions. The participant component consists of the interacting fraction of projectile and target nucleons undergoing compression and heating. In H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 123

Fig. 3. Pictorial illustration of the structure of baryon resonances in Fock space as a superposition of molecular and quark configurations. Meson cloud, qq¯ and gluonic contributions are subsumed in the second component.

first approximation ion–ion collisions at relativistic energies (Tlab/N ≳ mp) correspond to a large number of simultaneously occurring nucleon–nucleon reactions. Hence, a sizable number of baryon resonances is likely to be produced in the participant matter. Such a scenario is utilized by transport theoretical approaches describing the reaction dynamics in terms of the semi-classical picture behind the Boltzmann–Uehling–Uhlenbeck (BUU) transport model. Already early BUU- applications [12–15] were able to describe quantitatively the particle yields measured in experiments at the BEVALAC, SATURNE, the AGS, and GSI. BUU results indicate that at Tlab ∼ 1...2 AGeV total baryon densities ρB of approximately 3 3 times normal nuclear matter density ρ0 = 0.16/fm are achieved in the center collision zone. The meson multiplicities measured in central heavy ion collisions indicate that at these energies 20%–30% of the nucleons are excited into baryon resonances N∗. Thus, in the central overlap region of colliding heavy ions the density of baryon resonances reaches values comparable to that of nuclear matter at the saturation point, i.e. ρN∗ ≃ ρ0. Therefore, once the bombarding energy per nucleon in heavy ion collisions becomes comparable to the nucleon rest mass nuclear matter may undergo for a short time interval a transition into state with a sizable content of resonance matter. Fragmentation reactions are the ideal tool for the production of all kinds of short lived nuclei of masses below the summed mass of the incident projectile–target configuration. The method has been used most prominently and highly successfully to produce and study exotic nuclei at the fragment separator facilities at GSI, NSCL at MSU, GANIL, and RIKEN. Fragmentation reactions are described successfully by supplementing the kinetic phase of the collision by a thermo- statistical approach like the Statistical Multi-Fragmentation Model (SMM) developed in the 1980ties and early 1990ties by Bondorf and collaborators [16]. A recent development is to apply the same techniques also for resonance and hypernuclear studies. In order to have sizable cross sections beam energies above Tlab > 1 AGeV are required. At present, such energies are only available at the fragment separator (FRS) at GSI where indeed these types of reactions are part of the experimental program. Hypernuclear physics is a longstanding field of research. Lambda-hypernuclei are being studied already for decades, thus providing information on the S = −1 sector of nuclear many-body physics. The status of the field was comprehensively reviewed by Hashimoto at al. [17] and quite recently by Gal, Hungerford and Millener [18]. The ‘‘hyperonization puzzle’’ heavily discussed for neutrons stars [19] is another aspect of the revived strong interest in-medium strangeness physics. In the past, (π, K) experiments were a major source of hypernuclear spectroscopy. More recently, those studies were complemented by electro-production experiments at JLab and, at present, at MAMI at Mainz. Hypernuclear physics has 6 gained new momentum by a series of spectacular observations including an exotic system like ΛH[20]. The STAR experiment at RHIC has filtered out of their data samples exciting results for an totally unexpected reduced lifetime of the Lambda- 3 hyperon bound in ΛH. Soon after, that result was observed also by the HypHI-experiment at the FRS@GSI [21]. Observations by the ALICE collaboration at the LHC confirm independently this unexpected and yet to be explained result. Recent observations on light hypernuclei and their antimatter counterparts at RHIC [22] and the LHC [23], respectively, seem to confirm the surprising life-time reduction and, moreover, point to a not yet understood reaction mechanism. The HypHI group also found strong indications for a nnΛ bound state [24] which – if confirmed – would a spectacular discovery of the first and hitherto only charge–neutral system bound by strong interactions. Resonance studies with peripheral light and heavy ion reactions wee initiated in the late 1970’s and thereafter continued at SATURNE and later at the Synchro-Phasetron at Dubna and at KEK. The major achievements were the observation of an apparent huge mass shift of the Delta-resonance by up to ∆M ∼ −70 MeV. As found in the review article of Osterfeld [25] detailed theoretical investigations, accounting for in-medium interaction and an appropriate reaction theory, showed that the observed shifts were in fact due to distortions of the shape of the spectral distribution induced mainly by reaction dynamics and residual interactions. The new experiments at GSI on the FRS have shown already a large potential for resonance studies under well controlled conditions and with hitherto unreached high energy resolution. Once the Super- FRS will come into operation resonance physics with beams of exotic nuclei will be possible, thus probing resonances in charge-asymmetric matter. Viewed from another perspective, investigations of nucleon resonances in nuclear matter are a natural extension of nuclear physics. On the nucleonic scale, the Delta-resonance, for example, appears as the natural partner of the corresponding spin–isospin changing ∆S = 1, ∆I = 1 nuclear excitation, well known as Gamow–Teller resonance (GTR). GTR charge exchange excitations and the Delta-resonance are both related to the action of στ± spin–isospin transition operators, in the one case on the nuclear medium, in the other case on the nucleon. Actually, a long standing problem of 124 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 nuclear structure physics is to understand the coupling of the nuclear GTR and the nucleonic ∆33 modes: The notorious (and never satisfactorily solved) problem of the quenching of the Gamow–Teller strength is related to the redistribution of transition strength due to the coupling of ∆ particle–nucleon hole (∆N−1) and purely nucleonic double excitations of particle– hole (NN−1) states. Similar mechanisms, although not that clearly seen, are present in the Fermi-type spectral sections, i.e. the non-spin flip charge exchange excitations mediated by the τ± operator alone. In that case the nucleonic particle–hole states + π = 1 = 1 may couple to the P11(1440) Roper resonance. Since that J 2 , I 2 state falls out of the group theoretical systematics it is a good example for a dynamically generated resonance. From a nuclear structure point of view the ∆33(1232) and other resonances are of central interest because they are an important source of induced three-body interactions among nucleons. Moreover, the same type of operators are also acting in weak interactions leading to nuclear beta-decay or by repeated action to the rather exotic double-beta decay with or, still hypothetical, without neutrino emission. These investigations will possibly also serve to add highly needed information on interactions of high energy neutrinos with matter and may give hints on nucleon resonances in neutron star matter. On the experimental side, concrete steps toward a new approach to resonance physics are under way. The production of nucleon resonances in peripheral heavy ion collisions, their implementation into stable and exotic nuclei in peripheral charge exchange reactions, and studies of their in-medium decay spectroscopy are the topics of the baryon resonance collaboration [26]. The strangeness physics, hypernuclear research, and resonance studies belong to the experimental program envisioned for the FAIR facility. One aim of this article is to point out common aspects of hypernuclear and the in-medium physics of baryon resonances. Both are allowing to investigate the connections, cross-talk, and dependencies of nuclear many-body dynamics and sub-nuclear degrees of freedom. Such interrelations can be expected to become increasingly important for a broader understanding of nuclear systems under the emerging results of effective field theory and lattice QCD. The unexpected observations of neutron stars heavier than two solar masses are a signal for the need to change the paradigm of nuclear physics. Already in low energy nuclear physics we have encountered ample signals for the entanglement of nuclear and sub- nuclear scales, e.g. in three-body forces and quenching phenomena. In Section2 we introduce the concepts of flavor SU(3) physics and discuss the application to in-medium physics of baryons in a covariant Lagrangian formulation. The theoretical results are recast into a density functional theory with dressed in-medium meson–baryon vertices in Section3. Results for hypermatter and hypernuclei are discussed in Section4, addressing also the investigations by modern effective field theories. An indispensable prerequisite for strangeness and resonance physics in nuclear matter is to understand the excitation mechanisms on the elementary level. For that purpose the production of strangeness and resonances in elementary reactions on the nucleon are the topics of Section5 and reactions with elementary probes on nuclei are discussed in Section6. Formal = 3 aspects of a gauge invariant theory of high spin elementary matter fields, as encountered in the physics of s 2 and = 5 s 2 baryon resonances, are discussed in Section7. Transport theory for heavy ion collisions is covered in Section8. The use of fragmentation reactions for hypernuclear production is considered in Section9. Hypernuclear production by antiproton induced fragmentation reactions is the topic of Section 10. The production of Ω− baryons in antiproton-induced fragmentation reactions is the issue of Section 11. A brief introduction into the theory of resonance configurations in nuclei is presented in Section 12. Section 13 is devoted to resonance production in heavy ion reactions and resonance physics in nuclei and nuclear matter. In Section 14 the article is summarized and conclusions are drawn before closing with an outlook.

2. Interactions of SU(3) flavor octet baryons

2.1. General aspects of nuclear strangeness physics

Hypernuclear and strangeness physics in general are of high actuality as seen from the many experiments in operation or preparation, respectively, and the increasing amount of theoretical work in that field. There are a number of excellent review articles available addressing the experimental and theoretical status, ranging from the review of Hashimoto and Tamura [17], the very useful collection of papers in [27] to the more recent review on experimental work by Feliciello et al. [28] to the article by Gal, Millener, and Hungerford [18]. The latest activities are also recorded in two topical issues: in Ref. [29] strangeness (and charm) physics are highlighted and in Ref. [30] the contributions of strangeness physics with respect to neutron star physics is discussed. On free space interactions of nucleons a wealth of experimental data exist which are supplemented by the large amount of nuclear data. Taken together, they allow to define rather narrow constraints on interactions and, with appropriate theoretical methods, to predict their modifications in nuclear matter. Nuclear reaction data have provided important information on the energy and momentum dependence on the one hand and the density dependence of interactions in nuclear matter on the other hand. For hyperons, however, the situation is much less well settled. All attempts to derive hyperon–nucleon (YN) interactions in the strangeness S = −1 channel are relying, in fact, on a small sample of data points obtained mainly in the 1960’s. By obvious reasons, direct experimental information on hyperon–hyperon (YY) interactions is completely lacking. A way out of that dilemma is expected to be given by studies of hypernuclei. Until now only single-Lambda hypernuclei are known as bound systems, supplemented by a few cases of S = −2 double-Lambda nuclei. While a considerable number of Λ-hypernuclei is known, safe signals for a particle-stable Σ or a S = −2 cascade hypernucleus have not been recorded yet, see e.g. [17,18]. On the theoretical side, large efforts are made to incorporate strange baryons into the nuclear agenda. The conventional non-relativistic single particle potential models, the involved few-body methods for light hypernuclei, and the many-body H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 125 shell model descriptions of hypernuclei were reviewed recently in the literature cited above and will not be repeated here. Approaches to nucleon and hyperon interactions based on the meson-exchange picture of nuclear forces have a long tradition. They are describing baryon–baryon interactions by one-boson exchange (OBE) potentials like the well known Nijmegen Soft Core model (NSC) [31], later improved to the Extended Soft Core (ESC) model, for which over the years a number of parameter sets were evaluated [32–34]. A specialty of the NSC models is the use of a scalar interaction component described by Pomeron exchange. The Jülich model [35–37] and also the more recently formulated Giessen Boson Exchange model (GiBE) [38,39] belongs to the OBE-class of approaches. In the Jülich model the JP = 0+ scalar interaction channel is generated dynamically by treating those mesonic states explicitly as correlations of pseudo-scalar mesons. In the GiBE and the early NSC models scalar mesons are considered as effective mesons with sharp masses. The extended Nijmegen soft-core model ESC04 [40,41] and ESC08 [32,34] includes two pseudo-scalar meson exchange and meson-pair exchange, in addition to the standard one-boson exchange and short-range diffractive Pomeron exchange potentials. The Niigata group is promoting by their fss- and fss2-approaches a quark–meson coupling model which is regularly updated [42,43]. The resonating group model (RGM) formalism is applied to the baryon–baryon interactions using the SU(6) quark model (QM) augmented by modifications like peripheral mesonic or (qq¯) exchange effects. Chiral effective field theory (χEFT) for NN and NY interactions are a more recent development in baryon–baryon interaction. The review by Epelbaum et al. [44] on these subjects is still highly recommendable. The connection to the principles of QCD are inherent. A very attractive feature of χEFT is the built-in order scheme allowing in principle to solve the complexities of baryon–baryon interactions systematically by a perturbative expansion in terms of well-ordered and properly defined classes of diagrams. In this way, higher order interaction diagrams are generated systematically, controlling and extending the convergence of the calculations in an increasingly larger energy range. In addition to interactions, studies of nuclei require appropriate few-body or many-body methods. In light nuclei Faddeev-methods allow an ab initio description by using free space baryon–baryon (BB) potentials directly as practiced e.g. in [45]. Stochastic methods like the Green’s function Monte Carlo approach are successful for light and medium mass nuclei [46]. A successful approach up to oxygen mass region, the so-called p-shell nuclei, is the hypernuclear shell model of Millener and collaborators [47,48]. Over the years, a high degree of sophistication and predictive power for hypernuclear spectroscopy has been reached. For heavier nuclei, density functional theory (DFT) is the method of choice because of its applicability over wide ranges of nuclear masses. The development of an universal nuclear energy density functional is the aim of the UNEDF initiative [49–51]. Already some time ago we have made first steps in such a direction [52] with the Giessen Density Dependent Hadron Field (DDRH) theory. The microscopic DDRH approach incorporates Dirac–Brueckner Hartree–Fock (DBHF) theory into covariant density functional theory (DFT) [53–58]. Since then, the approach is being used widely on a purely phenomenological level as e.g. in [59–62]. In the non-relativistic sector comparable attempts are being made, ranging from Brueckner theory for hypermatter [63,64] to phenomenological density functional theory extending the Skyrme-approach to hypernuclei [65,66]. In recent works energy density functionals have been derived also for the Nijmegen model [34] and the Jülich χEFT [67]. Relativistic mean-field (RMF) approaches have been used rather early for hypernuclear investigation, see e.g. [68,69]. A covariant DFT approach to hypernuclei and neutron star matter was used in a phenomenological RMF approach in Ref. [70] where constraints on the scalar coupling constants were derived by imposing the constraint of neutron star masses above two solar masses. In Refs. [71–73] and also [74,75] hyperons and nucleon resonances are included into the RMF treatment of infinite neutron star matter, also with the objective to obtain neutron stars heavier than two solar masses. A yet unexplained additional repulsive interaction at high densities is required. A covariant mean-field approach, including a non-linear realization of chiral symmetry, has been proposed by the Frankfurt group [76] and is being used mainly for neutron star studies.

2.2. Interactions in the baryon flavor octet

+ P = 1 In Fig.4 the SU(3) multiplets are shown which are considered in our approach. The lowest J 2 baryon octet, representing the baryonic ground state multiplet, is taken into account together with three meson multiplets, namely the pseudo-scalar nonet (P) with JP = 0−, the scalar nonet (S) with JP = 0+, and the vector nonet (V) with JP = 1−, which, in µ fact, consists of four subsets, V = {V }|µ=0...3, according to the four components of a Lorentz-vector. In addition, we include also the corresponding meson singlet states, represented physically by the η′, φ, σ ′ states but not shown in Fig.4. Masses and lifetimes are displayed in Table1. The mesons of each multiplet couple to the baryons by vertices of a typical, generic operator structure characterized the Lagrangian densities

′ − 1 BB ′ ¯ ′ µ LM(0 ) = − gBB MψB γ5γµψB∂ ΦM (1) mπ ′ − µ fBB′M µν BB ′ ¯ ′ ¯ ′ LM(1 ) = − gBB MψB γµψBVM + ψB σµν ψBFM (2) 2(MB + MB′ ) ′ + BB ′ ¯ ′ 1 LM(0 ) = gBB MψB ψBΦM (3) ¯ = † given by Lorentz-covariant bilinears of baryon field operators ψB and the Dirac-conjugated field operators ψB γ0ψB and Dirac γ -matrices, to which meson fields (V, S) or their derivatives (P) are attached, such that in total a Lorentz-scalar 126 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

1 + + − Fig. 4. The SU(3) multiplets considered in this section: the 2 baryon octet (upper left), the scalar 0 (upper right), the vector 1 (lower left), and the pseudo-scalar 0− (lower right). The octets have to be combined with the corresponding meson-singlets, which are not displayed, thus giving rise to meson nonets.

Table 1 + P = 1 Mass, lifetime, and valence quark configuration of the J 2 octet baryons. Source: From Ref. [10]. State Mass/MeV Life time/s Configuration p 938.27 >6.62 · 10+36 [uud] n 939.57 880.2 ± 1 [udd] Λ 1115.68 2.63 · 10−10 [uds] Σ+ 1189.37 0.80 · 10−10 [uus] Σ 0 1192.64 7.4 · 10−20 [uds] Σ− 1197.45 1.48 · 10−10 [dds] Ξ 0 1314.86 2.90 · 10−10 [uss] Ξ − 1321.71 1.64 · 10−10 [dss]

µν is obtained. Vector mesons may also couple to the baryons through their field strength tensor FM (see Eq. (11)) and the = i [ ] relativistic rank-2 tensor operator σµν 2 γµ, γν , given by the commutator of γ matrices. The tensor coupling involves a separate tensor coupling constant fBB′M. The mass factors in the P and the tensor part of the V vertices are serving to compensate the energy–momentum scales introduced by derivative operators for the sake of dimensionless coupling constants. The Bjorken-convention [77] is used for spinors and γ -matrices. We introduce the flavor spinor ΨB

T ΨB = (N, Λ, Σ, Ξ) (4) being composed of the isospin multiplets ⎛ +⎞ ( ) Σ ( 0 ) p Ξ N = , Σ = Σ 0 , Ξ = (5) n ⎝ ⎠ Ξ − Σ− and the iso-singlet Λ. Each baryon entry is given by a Dirac spinor. For later use, we also introduce the mesonic isospin doublets ( +) ( ) K K 0 K = , Kc = − , (6) K 0 −K and corresponding structures are defined in the vector and scalar sector involving the K ∗ and the κ mesons, respectively. The Σ hyperon and the π isovector-triplets are expressed in the basis defined by the spherical unit vectors e±,0 which leads to + − − + Σ ·π = Σ π + Σ 0π 0 + Σ π , (7) also serving to fix phases [78]. H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 127

Table 2 Intrinsic quantum numbers and the measured masses [10] of the mesons used in the calculations. Also the cut-off momenta used to regularize the high-momentum part of the tree-level interactions are shown. The mass of the isoscalar–scalar σ meson is chosen at the center of the f0(500) spectral distribution [10]. Channel Meson Mass [MeV] Cut-off [MeV/c] 0− π 138.03 1300 0− η 547.86 1300 0− K 0,+ 497.64 1300 0+ σ 500.00 1850 0+ δ 983.00 2000 0+ κ 700.00 2000 1− ω 782.65 1700 1− ρ 775.26 1700 1− K ∗ 891.66 1700

The full Lagrangian is given by

L = LB + LM + Lint (8) accounting for the free motion of baryons with mass matrix Mˆ ,

[ µ ˆ ] LB = Ψ B iγµ∂ − M ΨB , (9) and the Lagrangian density of massive mesons, ( ) 1 ∑ ( µ 2 2) 1 ∑ 1 2 2 2 LM = ∂µΦi∂ Φi − m Φ − F − m V (10) 2 Φi i 2 2 λ λ λ i∈{P,S,V} λ∈V where P, S, V denote summations over the lowest nonet pseudo-scalar, scalar, and vector mesons. The field strength tensor µ ∗ of the vector meson fields Vλ , λ ∈ {ω, ρ, K , φ, γ } is defined by

µν µ ν ν µ Fλ = ∂ Vλ − ∂ Vλ . (11) µ For the scattering of charged particles and in finite nuclei, the electromagnetic vector field Vγ of the photon is included. Of special interest for nuclear matter and nuclear structure research are the mean-field producing meson fields, given ′ by the isoscalar–scalar mesons σ , σ , the isovector-scalar δ meson, physically observed as the a0(980) meson, and their isoscalar–vector counterparts ω, φ and the isovector–vector ρ meson, respectively. While the pseudoscalar and the vector mesons are well identified as physical particles or as well located poles in the complex plane, the situation is less clear for = = = ∗ the scalar nonet. We identify σ f0(500), δ a0(980), and κ K0 (800) as found in the compilations of the Particle Data Group [10]. The so-called κ-meson is of particular uncertainty. It has been observed only rather recently as resonance-like structures in charmonium decay spectroscopy. A two-bump structure with maxima at about 640 and 800 MeV has been detected. In the recent PDG compilation a mean mass mκ = 682 ± 29 MeV is recommended [10]. We use mκ = 700 MeV . The octet baryons are listed in Table1, the meson parameters are summarized in Table2.

Last but not least we consider the interaction Lagrangian Lint . SU(3) octet physics is based on treating the eight baryons on equal footing as the genuine mass-carrying fields of the theory. Although SU(3) flavor symmetry is broken on the baryon mass scale by about ±20%, it is still meaningful to exploit the relations among coupling constants imposed by that symmetry, + P = 1 thus defining a guideline and reducing the number of free parameters considerably. The eight J 2 baryons are collected into a traceless matrix B, which is given by a superposition of the eight Gell-Mann matrices λi combined with the eight baryons Bi ∈ {N, Λ, Σ, Ξ}, leading to the familiar form

⎛ 0 ⎞ Σ Λ + √ + √ Σ p ⎜ 2 6 ⎟ ⎜ 0 ⎟ ∑ ⎜ − Σ Λ ⎟ B = λiBi = ⎜ Σ −√ + √ n ⎟ , (12) ⎜ ⎟ i=1···8 2 6 ⎜ 2Λ ⎟ ⎝ −Ξ − Ξ 0 −√ ⎠ 6 which is invariant under SU(3) transformations. The pseudo-scalar (P), vector (V), and the scalar (S) meson octet matrices P − are constructed correspondingly by replacing the baryons Bi by the appropriate mesons Mi. Taking the J = 0 pseudo-scalar 128 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 mesons as an example we obtain the (traceless) octet matrix ⎛ 0 ⎞ π η8 + + √ + √ π K ⎜ 2 6 ⎟ ⎜ 0 ⎟ ⎜ − π η8 0 ⎟ P8 = ⎜ π −√ + √ K ⎟ , (13) ⎜ ⎟ ⎜ 2 6 ⎟ − 2η8 ⎝ K K 0 − √ ⎠ 6 √ which, for the full nonet, has to be completed by the singlet matrix P1, given by the 3 × 3 unit matrix multiplied by η1/ 3. Thus, the full pseudo-scalar nonet is described by P = P8 +P1. We define the SU(3)-invariant baryon–baryon–meson vertex combinations [BBP] = Tr ({B, B} P ) , [BBP] = Tr ([B, B] P ) , D 8 F 8 [BBP] = Tr(BB)Tr(P ) (14) S 1 where F and D couplings correspond to anti-symmetric combinations, given by anti-commutators, {X, Y }, and symmetric combinations, given by commutators [X, Y ], respectively. The singlet interaction term is indexed by S. With these relations we obtain the pseudoscalar interaction Lagrangian √ P { [ ] [ ] } 1 [ ] Lint = − 2 gD BBP8 + gF BBP8 − gS √ BBP1 , (15) D F 3 S with the generic SU(3) coupling constants {gD, gF , gS }. The de Swart convention [78], underlying the Nijmegen and the Jülich approaches, is given by using g8 ≡ gD + gF , α ≡ gF /(gD + gF ), and g1 ≡ gS leading to the equivalent representation √ P { [ ] [ ] } 1 [ ] Lint = −g8 2 α BBP8 + (1 − α) BBP8 − g1 √ BBP1 . (16) F D 3 S µ In order to evaluate the couplings we define the pseudo-vector derivative vertex operator mπ ΓP = γ5γµ∂ , following Eq. (1). From the F- and D-type couplings, Eq. (15), we obtain the pseudo-scalar octet–meson interaction Lagrangian in an obvious, condensed short-hand notation, going back to de Swart [78]

P Lint = − gNNπ (NΓP τN)·π + igΣΣπ (Σ ×ΓP Σ)·π

− gΛΣπ (ΛΓP Σ + ΣΓP Λ)·π − gΞΞπ (ΞΓP τΞ)·π [ ] − gΛNK (NΓP K)Λ + ΛΓP (KN) [ ] − gΞΛK (ΞΓP Kc )Λ + ΛΓP (Kc Ξ) [ ] − gΣNK Σ · ΓP(KτN) + (NΓP τK)·Σ [ ] − gΞΣK Σ ·ΓP (Kc τΞ) + (ΞΓP τKc )·Σ − − gNNη8 (NΓP N)η8 gΛΛη8 (ΛΓP Λ)η8 − · − gΣΣη8 (Σ ΓP Σ)η8 gΞΞη8 (ΞΓP Ξ)η8. (17) ′ The – in total 16 – pseudo-scalar BB -meson vertices are completely fixed by the three nonet coupling constants (gD, gF , gS ) or, likewise, by (g8, g1, α). Corresponding relations exist also for interactions induced by the vector and the scalar meson nonets. As in the µ µ µ = µ + µ pseudoscalar case, they are given in terms of octet (V8 , S8) and singlet multiplets (V1 , S1), resulting in V V8 V1 and S = S8 + S1, respectively, and having their own sets of respective coupling constants (gD, gF , gS )S,V . The BBV coupling ∗ constants are obtained in analogy to Eq. (17) by the mapping {K, π, η8, η1} → {K , ρ, ω8, φ1}. Correspondingly, the scalar couplings BBS are obtained from Eq. (17) by replacing {K, π, η8, η1} → {κ, δ, σ8, σ1}. Thus, the baryon–baryon interactions as given by the exchange of particles from the three meson multiplets M ∈ {P, V, S} are of a common structure which allows to express the BBM coupling constants in generic manner. For that purpose, we denote the isoscalar octet meson 0,+ ∗0,+ by f ∈ {η8, ω8, σ8}, the isovector octet meson by a ∈ {π, ρ, δ}, and the iso-doublet mesons by K ∈ {K , K } and = { ∗,0 ∗,+} κ K0 , K0 . Irrespective of the particular interaction channel, the coupling constants are then given by the relations [78] √ 1 1 gNNa = gD + gF , gΛNK = − (gD + 2gF ), gNNf = √ (3gF − gD), 3 3 1 2 gΣΣa = 2gF , gΞΛK = √ (3gF − gD), gΛΛf = −√ gD, 3 3 (18) 2 2 gΛΣa = √ gD, gΣNK = (gD − gF ), gΣΣf = √ gD, 3 3 1 gΞΞa = −(gD − gF ), gΞΣK = −(gD + gF ), gΞΞf = −√ (3gF + gD). 3 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 129

Fig. 5. Hyperon–nucleon OBE interactions in the S = −1 sector. Interactions with and without strangeness-exchange are displayed in the upper and lower row, respectively. The interactions from S = 0 scalar (σ , δ) and vector mesons (ω, ρ) will contribute to the hypernuclear mean-field self-energies. The diagrams in the lower row correspond to graphs with a topology similar to the exchange diagrams encounter in antisymmetrized nucleon–nucleon matrix elements but here they are of dynamical origin.

where gNNa = g8a and, depending on the case, {gD, gF , gS } denote either the pseudo-scalar, vector, or scalar set of fundamental ′ SU(3) couplings, respectively. The interactions due to the exchange of the isoscalar-singlet mesons f ∈ {η1, φ1, σ1} are treated accordingly with the result

gNNf ′ = gΛΛf ′ = gΣΣf ′ = gΞΞf ′ = gS = g1, (19) where again the proper gS ≡ g1 coupling constant for the {P, V , S} multiplet under consideration has to be inserted. The complete interaction Lagrangian is given by the sum over the partial interaction components

= ∑ M Lint Lint . (20) M∈{P,S,V}

The coupling constants above define the tree-level interactions entering into calculations of T -matrices. The corresponding diagrams contributing to the NΛ and NΣ interactions in the S = −1 sector are shown in Fig.5.

2.3. Baryon–baryon scattering amplitudes and cross sections

In practice, the BB′ states are grouped into substructures which reflect the conservation laws of strong interaction physics. Assuming strict SU(3) symmetry, this can be done in terms of flavor SU(3) ⊗ SU(3) irreducible representations (irreps), as e.g. in [2] and for a practical application in [34]. However, as seen from Table1, SU(3) symmetry is obviously broken on the mass scale: The average octet mass is m¯ 8 = 1166.42 MeV, the octet mass splitting is found as ∆m8 = m¯ Ξ − m¯ N = 379.19 MeV and, thus, the mass symmetry is broken by about 32%. This implies differences in the kinematical channel threshold and an order scheme which takes into account those effects is of more practical and physical use. In Fig.6 the kinematics for the nucleonic S = 0 and the S = −1 nucleon–hyperon channels are illustrated in terms of the invariant channel momenta in a two-body channel with masses m1,2 and Mandelstam total energy s:

2 1 ( 2 2 ) q (s) = (s − (m1 − m2) )(s − (m1 + m2) ) , (21) 4s

2 2 where at the channel threshold s = sthres = (m1 + m2) and q (sthres) = 0. In observables like cross sections, the coupling to a channel opening at a certain energy produces typically a kink-like structure, seen as a sudden jump in both the date and numerical results. An example is found in Fig.7. The spike showing up in the elastic Λp cross√ section at about 0 plab ∼ 640 MeV/c is due to the coupling to the Σ p channel which crosses the kinematical threshold at the s = MΣ0 + Mp reached that channel momentum, namely at plab ≃ 642 MeV/c for a proton incident on a Λ hyperon. In scattering phase shifts such a channel coupling effect is typically seen as a sudden jump. The Lagrangian densities serve to define the tree-level interactions of the BB′ configurations built from the octet baryons. The derived potentials include in addition also vertex form factors. Formally, they are used to regulate momentum integrals, physically they define the momentum range for which the theory is supposed to be meaningful. The OBE models typically use hard cut-offs in the range of 1..2 GeV/c. The χEFT cut-offs are much softer with values around 600 MeV/c. 130 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

√ Fig. 6. The on-shell channel momentum q(s), Eq. (21), is shown as a function of the Mandelstam variable s for the proton–proton and the S = −1 proton–hyperon two-body channels. Below threshold q2(s) < 0 and the threshold is defined by q(s) = 0. Σ and Ξ masses have been averaged over the respective isospin multiplets.

± Fig. 7. Total cross sections as a function of plab for Λp and Σ p elastic scattering. The shaded band is the LO EFT result for varying the cutoff as Λ = 550...700 MeV/c. For comparison results are shown of the Jülich 04 model [35] (dashed), and the Nijmegen NSC97f model [31] (solid curve). Source: From Ref. [44].

In the NΛ/NΣ-system, for example, the tree-level interactions are given by

= (η) + (σ ) + (ω) VNΛ,NΛ VNΛ,NΛ VNΛ,NΛ VNΛ,NΛ (22) (I) = (η) + (σ ) + (ω) VNΣ,NΣ VNΣ,NΣ VNΣ,NΣ VNΣ,NΣ ( ) + (π) + (δ) + (ρ) ⟨ ∥ · ∥ ⟩ VNΣ,NΣ VNΣ,NΣ VNΣ,NΣ I τN τΣ I (23) ⟨ ⟩ (I) ( (π) (δ) (ρ) ) 1 1 V = V + V + V I = ∥τN · τY ∥I = . (24) NΛ,NΣ NΛ,NΣ NΛ,NΣ NΛ,NΣ 2 2 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 131

Fig. 8. The np s-wave interactions in the (S = 0, I = 1) singlet-even channel in Born approximation, U(q, q) = V (q, q), (upper curve, red) and the full K-matrix result, U(q, q) = K(q, q), (lower curve in blue). The K-matrix results reproduce the measured scattering phase shift. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The strengths of the interactions inhibit a perturbative treatment. Rather, the scattering series must be summed to all orders. For that purpose, the bare interactions are entering as reaction kernels into a set of coupled Bethe–Salpeter equations, connecting BB′ channels of the same total charge and strangeness. In a matrix notation, V indicates the Born terms, G denotes the diagonal matrix of channel Green’s functions, and the resulting T -matrix is defined by ∫ 4 ′ ′ d k ′ T (q , q|P) = V(q , q|P) + V(q , k|P)G(k, P)T (k, q|P) (25) (2π)4 describing the transition of the system for the (off-shell) four-momenta q′ → q and fixed center-of-mass four-momentum P with s = P 2. A numerically solvable system of equations is obtained by projection to a three-dimensional sub-space. Such a reduction depends necessarily on the choice of projection method as discussed in the literature, e.g. [79,80]. A widely used scheme is the so-called Blankenbecler–Sugar reduction [81] consisting in projection of the intermediate energy variable to a fixed value, typically chosen as k0 = 0. In this way, the full Bethe–Salpeter equation is reduced to an effective Lippmann– Schwinger equation in three spatial variables, but still obeying Lorentz invariance: ∫ 3 ′ ′ d k ′ T (q , q|s) = V(q , q|s) + V(q , k|s)gBbS (k, s)T (k, q|s) (26) (2π)3 where the propagator is now replaced by the Blankenbecler–Sugar propagator with (diagonal) elements 1 g (k, s) = (27) BbS 2 s − (E1(k) + E2(k)) + iη √ = 2 + 2 where E1,2(k) m1,2 k is the relativistic energy of the particles in that given channel. The T -matrix may be expressed by the K-matrix:

− T = (1 − iK) 1K (28) and very often the scattering problem is solved in terms of the (real-valued) K-matrix [79]. In practice, a decomposition into invariants and partial waves is performed which reduces the problem further to a set of linear integral equations in a single variable, namely the modulus of the three-momentum involved. Here, we refrain from going into these mathematically very involved details. They have been subject of many well written standard text books and review papers, e.g. [79]. An interesting question is to what extent the higher order terms of the scattering series are contributing to the scattering amplitude. That is quantified in Fig.8 where the s-wave scattering phase shift in the ( S = 0, I = 1) NN singlet-even channel are shown in Born approximation and for the fully summed K-matrix result. The higher order correlation effects are seen to be of overwhelming importance especially close to threshold and at low (on-shell) momenta. Representative results for nucleon–hyperon scattering are shown in Fig.7, comparing total cross sections obtained by EFT and the Jülich and Nijmegen OBE approaches, respectively. The latest χEFT account for interactions up to next-to-leading- order (NLO) [82]. The resulting phase shifts in the spin–singlet channel are shown in Fig.9 for a few baryon–baryon channels: the pp (S = 0), pΣ+ (S = −1), and Σ+Σ+ (S = −2). From the pp results it is seen that the NLO calculations are trustable up to about plab ∼ 300 MeV/c. 132 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

+ + + 1 Fig. 9. The pp, Σ p, and Σ Σ phase shifts in the S0 partial wave. The filled band represents the results at NLO [82]. The pp phase shifts of the GWU SAID-analysis [9] (circles) are shown for comparison. In the Σ+p case the circles indicate upper limits for the phase shifts as deduced from the measured sections. Source: From Ref. [82].

In nuclear matter, the scattering equations are changed by the fact that part of the intermediate channel space is unavailable by Pauli-blocking. Formally, that is taken into account by the Pauli-projector QF (p1, p2) = 1 − PF (p1, p2), projecting to the momentum region outside of the respective Fermi-spheres:

2 2 2 2 QF (p1, p2) = Θ(p1 − kF1)Θ(p2 − kF2). (29)

The particle momenta p1,2 are given in the nuclear matter rest frame. Together with the time-like energy variables p0 they = 0 T define the four-momenta p1,2 (p1,2, p1,2) which are related to the total two-particle four-momentum P and the relative momentum k by the Lorentz-invariant transformation

p1,2 = ±k + x1,2P (30) with − 2 + 2 − 2 + 2 s m2 m1 s m1 m2 x1 = ; x2 = (31) 2s 2s obeying x1 + x2 = 1 and which in the non-relativistic limit reduce to x1,2 = m1,2/(m1 + m2). Furthermore, in-medium self-energies must be taken into account in propagators and vertices. In total, the in-medium K-matrix for a reaction B1 + B2 → B3 + B4 is determined by a modified in-medium Lippmann–Schwinger equation, known as Brueckner G-matrix equation, given in full coupled channels form as ′ = ˜ ′ KB1B2,B3B4 (q, q ) VB1B2,B3B4 (q, q ) ∫ 3 ∑ d k 2 2 ′ + P VB B ,B B (q, k)GB B (k, qs)QF (p , p )KB B ,B B (k, q ) (32) (2π)3 1 2 5 6 5 6 5 6 5 6 3 4 B5B6 where the integration has to be performed as a Cauchy Principal Value integral and the propagator GB5B6 (k, qs) includes now vector and scalar baryon self-energies. If the background medium consists only of protons and neutrons, the Pauli projection affects only the nucleonic part of the intermediate states |B5B6⟩, i.e if B5 = N or B6 = N or both are nucleons. The coupled equations have to set up carefully with proper account of flavor exchange and antisymmetrization effects for the interactions in the channels of higher total strangeness, as discussed e.g. in [31,83]. The calculation of in-medium interactions is in fact a very involved self-consistency problem, see e.g. [57,84]. Interactions are entering in a nested manner to all orders into propagators, G-matrix equations, and self-energies. The (Dirac-) Brueckner– Hartree–Fock ((D)BHF) self-consistency cycle is indicated in Fig. 10 where the nested structure is recognized. However, as is easily found, self-energies contribute to the Brueckner G-matrix only in second and higher order. Thus, a perturbative approach becomes possible for the channels which are weakly coupled to the medium: To a good approximation it is sufficient to solve the Brueckner equations with bare intermediate propagators, but including the Pauli-projector [39]. For nuclear structure work, the low-energy behavior of scattering amplitudes is of particular importance. The low momentum behavior of the s-wave K-matrix is described by the effective range expansion for vanishing momentum q,

1 1 2 4 q cot δ|q→0 = − + q r + O(q ) (33) a 2 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 133

Fig. 10. The (D)BHF self-consistency problem for the G-matrix (upper row), connecting the single particle self-energy (middle row) and the one-body propagator (lower row). The two-body Pauli-project is denoted by QF , Eq. (29), while PF is the projector onto the single particle Fermi sphere.

Table 3 Low energy parameters for the indicated YN system. Results from the OBE- oriented Jülich 04 [35], NSC97f [31], the recent Giessen Boson Exchange (GiBE) [39] models are compared to the leading order chiral EFT results [86].

Channel as [fm] rs [fm] Model = 1 + − ΣN (I 2 ) 0.90 4.38 Jülich 04 = 3 − ΣN (I 2 ) 4.71 3.31 Jülich 04 −4.35 3.16 NSC97f 04 −1.80 1.76 χEFT LO −1.44 5.18 GiBE = 1 − ΛN (I 2 ) 2.56 2.75 Jülich 04 −2.60 2.74 NSC97f −1.01 1.40 χEFT LO −2.41 2.34 GiBE

where δ is the s-wave scattering phase shift and the expansion parameters a and r denote the s-wave scattering length and effective range, see e.g. [85]. In Table3 the low energy parameters are given for several OBE approaches and compared to leading order (LO) χEFT results. In Fig. 11 the medium effects are illustrated for the total cross sections of NN-scattering and Σ+p-scattering. Both for NN and Σ+p scattering one observes a rapid set in of a reduction with increasing density. Already at half the nuclear saturation = 1 1 + density, i.e. kF 209 MeV/c the cross sections are reduced by a factor of 3 (NN) and a factor of 2 (Σ p), respectively. At higher densities the reduction factors converge to an almost constant value close to the one in Fig. 11 seen for nuclear + saturation density and kF = 263 MeV/c: the NN cross section is down by a factor of 0.25, the Σ p cross section by a factor close to 0.35. Thus, in nuclear matter the NN and the YN interactions are evolving differently, as to be expected from Eq. (32). The main effect is due to the different structure of the Pauli-projector. While the intermediate NN channels experience blocking in both particle momenta, in the intermediate YN channels only the nucleon states are blocked.

The elastic scattering amplitudes for B1B2 → B1B2 we may separate into effective coupling constants g˜B B g˜B B and a ¯ 1 1 2 2 matrix element MB1B2 , amputated by all coupling constants. The cross sections are given as

σ ∼ g˜ 2 g˜ 2 σ¯ , (34) B1B2 B1B1 B2B2 B1B2 where 4π ¯ 2 σ¯B B = |MB B | . (35) 1 2 q2 1 2 The cross sections provide an estimate for the interaction strength of the iterated interactions, i.e. after the solution of the Bethe–Salpeter or Lippmann–Schwinger equations, respectively. Assuming that σ¯ is an universal quantity within a given flavor multiplet, we find ˜ 2 σB B gB B R2 = 1 2 = 2 2 . (36) B2B3 2 σB B g˜ 1 2 B3B3

From Fig. 11 we then obtain RpΣ+ ∼ 0.62...0.77 for ρ = 0 to ρ = ρsat and similar results are found also for other channels. The values range surprising well around the naïve quark model assumption, namely that BB interactions scale essentially with the number of u and d valence quarks in the baryons. 134 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 11. Free-space and in-medium total cross sections for pp (left) and Σ+p (right) elastic scattering. As indicated, the cross sections are evaluated in free 3 space (ρ = 0), at half (ρ = ρsat /2) and at full nuclear matter density ρ = ρsat = 0.16/fm , respectively. Source: From Ref. [39]

0 Fig. 12. The in-medium scattering lengths as for the coupled nΛ and nΣ channels are shown as a function of the Fermi momentum kFN of symmetric nuclear matter. Source: From Ref. [39].

By evaluating the low-energy parameters as function of the background density we gain further insight into the density dependence of interactions. In Fig. 12 the scattering lengths of the coupled nΛ and nΣ 0 channels are shown as a functions 2 1/3 of the Fermi momentum of symmetric nuclear matter, kF = (3π ρ/2) . For kF → 0 the free space scattering length is approached. With increasing density of the background medium the scattering lengths decrease and approach an asymptotically constant value. Thus, at least in ladder approximation a sudden increase of the vector repulsion as discussed for a solution of the hyperon-problem in neutron stars is not in sight.

2.4. In-medium baryon–baryon vertices

′ A Lagrangian of the type as defined above leads to a ladder kernel V BB (q′, q) given in momentum representation by the ′ superposition of one boson exchange (OBE) potentials V BB α(q′, q). The solution of the coupled equations, Eq. (32) is tedious and sometimes numerically cumbersome by occasionally occurring instabilities. For certain parameter sets, unphysical YN and YY deeply bound states may show up. While for free space interactions the problems may be overcome, an approach avoiding the necessity to repeat indefinitely many times the (D)BHF calculations may be of advantage for applications in nuclear matter, neutron star matter, and especially in medium and heavy mass finite nuclei. Since systematic applications of (D)BHF theory in finite nuclei is in fact still not feasible, despite longstanding attempts, see e.g. the article of Müther and Sauer in [87], effective interactions in medium and heavy mass nuclei strongly rely on results from infinite nuclear matter calculations. Density functional theory (DFT) provides in principle the appropriate alternative, as known from many applications to atomic, molecular, and nuclear systems. However, DFT does not include a method to derive the appropriate interaction energy density which is a particular problem for baryonic matter. In the nuclear sector Skyrme-type energy H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 135 density functionals (EDF), e.g. [88] for a hypernuclear Skyrme EDF, are a standard tool for nuclear structure research. The UNEDF initiative is trying to derive the universal nuclear EDF [51]. Finelli et al. started work on an EDF based on χ EFT. In [34] the ESC08 G-matrix was used to define an EDF. Relativistic mean-field (RMF) theory relies on a covariant formulation of DFT and the respective relativistic EDF (REDF) [89–92]. Similar to the Skyrme-case, many different REDF versions are on the market, without and with non- linear self-interactions of the meson fields. One of the first RMF studies of hypernuclei was our work in [69] and many others have followed, see e.g. [93–95]. The Giessen DDRH theory is a microscopic approach with the potential of a true ab initio DFT description of nuclear systems. In a series of papers [52–55,96] a covariant DFT was formulated with an REDF derived from DBHF G-matrix interactions. The density dependence of meson–baryon vertices as given by the ladder approximation are accounted for. Before turning to the discussion of the DDRH approach, we consider first a presentation of scattering amplitudes in terms of effective vertices, including the correlations generated by the solution of the Bethe– Salpeter equations. ′ ′ For formal reasons we prefer to work with the full BB T -matrix TBB′ . The ladder summation is done in the BB -rest frame, but for calculations of self-energies and other observables the interactions are required in the nuclear matter rest frame rather than in the 2-body c.m. system. For that purpose the standard approach is to project the (on-shell) scattering amplitudes on the standard set of scalar (S), vector (V), tensor (T), axial vector (A) and pseudo scalar (P) Lorentz invariants, see e.g. [84,97–99]. A more convenient representation, allowing also for an at least approximate treatment of off-shell effects, is obtained by representing the T -matrices in terms of matrix elements of OBE-type interactions, similar to the ′ construction of the tree-level interactions V BB but now using energy and/or density dependent effective vertex functionals

Γa and propagators Da for bosons with masses ma. A natural choice is to use the same boson masses as in the construction of the tree-level kernels. In the following, the expansion of a fully resummed interaction in terms of dressed vertices and boson propagators is ′ sketched. For the reaction amplitude for B = (B1B2) → B = (B3B4) we use the ansatz

′ ∑ † ′ TBB′ (q, q , kF ) = Γ (qs, kF )UBB′a(q, q )ΓB B a(qs, kF ) (37) B1B3a 2 4 a where

′ (a) ′ UBB′a(q, q ) = M Da(q, q ) (38) B1B3,B2B4 is the invariant matrix element containing the operator structure of the interaction of type a ∈ {A, P, S, T , V }. The Born-terms are given as

′ = ′ 2 VBB a UBB agBB′a (39)

2 where gBB′a denotes the bare coupling constants. Inserting this ansatz into the Brueckner equations we obtain an equation for the dressed vertex in two steps. In a symbolic notation, first we find the formal solution

− 1 ′ 2 = 1 ′ MBB aΓBB′a Da ∫ ′ ∗ VBB a (40) 1 − dq VaG QF | BB′

3 ′ 2 ′ d q ∗ where Γ ′ stands for the matrix of bilinears in the boson a-space and dq = . G is the two-particle propagator, Q the BB a (2π)3 F projector onto the complement of the combined Fermi-spheres of the intermediate baryon states. The inverse of the boson propagator has to be understood as ′ − δ δUBB′a(k, k ) ′ 1 → ; = 2 ′ − Da ′ gBB′aMBB aδ(p p ) (41) δDa δDa(q, q ) where p′ = k − k′ and p = q − q′ are the momentum transfers and δ(p − p′) indicates a Dirac delta-function. We obtain

2 1 2 Γ ′ = g ′ + · · · (42) BB a ∫ ′ ∗ 2 BB a (1 − dq VaG QF ) | BB′ plus higher order terms given by off-shell contributions and the derivatives of the self-energies contained in the dressed two-nucleon propagator G∗ which we neglect for the sake of a slim presentation. Neglecting in Eq. (42) the off-shell integral and evaluating the expression at the on-shell point, q = qs, the leading order result is 1 ′ ≃ ′ ΓBB a(qs, kF ) ∫ ′ ∗ gBB a. (43) 1 − dq VaG QF | BB′ 136 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 13. Diagrammatic structure of the dressed vertex functionals ΓBB′a (filled square) in terms of the bare coupling constant gBB′a (filled circle) and the interaction UBB′a, indicated by a wavy line. The integral over the complement of the combined Fermi spheres of the intermediate baryons is shown as a loop (see text).

The diagrammatic structure is depicted in Fig. 13. Inspecting Eq. (43), one finds that dressed and the bare vertices are related by a susceptibility matrix ( ∫ )−1 ′ ∗ χBB′a(qs, kF ) = 1 − dq VaG QF (44) | ′ √BB = { } depending on the center-of-mass energy s through qs and the set of Fermi momenta kF kFB |B=n,p.... 2 Two limiting cases are of particular interest. At vanishing density where QF → 1, and since Va ∼ ga we find that in free-space the dressed vertices retain their general structure as a fully resummed series of tree-level coupling constants. At density ρ → ∞, where QF → 0 over the full integration range, the tensor χ reduces to the unity matrix and we find Γa ≃ ga. Albeit for another reason, a similar result is found in the high energy limit: The√ most important contribution to the integral is coming from the region around the Green’s function pole. With increasing s the pole is shifted into the tail of the form factors regularizing the high momentum part of Ua. Thus, at large energies the residues are increasingly suppressed, as indicated by the decline of the matrix element shown in Fig.8. As a conclusion, the dressing effects are the strongest for low energies and densities.

2.5. Vertex functionals and self-energies

It is interesting to notice that a similar approach, but on a purely phenomenological level, was used long ago by Love and Franey to parameterize the NN T -matrix over a large energy range, Tlab = 50...1000 MeV. Also the widely used M3Y G-matrix parametrization of Bertsch et al. [100] is using comparable techniques, as also the work in [101]. In order to generalize the approach, a field-theoretical formulation is of advantage which is the line followed in DDRH theory. For ∼ 1/3 that purpose, the dependence on kF ρB is replaced by a functional dependence on the baryon four-current by means µ of the Lorentz-invariant operator relation 2 = j j leading to vertex functionals ˆ ( ¯ ). The C-numbered vertices are √ ρB Bµ B Γa ΨBΨB ˆ ¯ recovered as expectation values, Γa( s, kF ) = ⟨P, kF |Γa(ΨBΨB)|P, kF ⟩, under given kinematical conditions P and for a baryon configuration defined by kF . For studies of single particle properties it is sufficient to extract the vertices directly on the mean-field level. An efficient way is to use the baryon self-energies. For our present illustrating purposes it is enough to consider the Hartree tadpole-term. The self-energy ΣaB(kF ) due to the exchange of the boson a felt by the baryon B is given by

1 ∑ (a) ΣaB(kF ) = ΓBBa(kF ) ΓB′B′a(kF )ρ ′ . (45) m2 B a B′ For studies of single particle properties it is sufficient to extract the vertices directly on the mean-field level. An efficient way is to use the baryon self-energies. For our present illustrating purposes it is enough to consider the Hartree tadpole-term. The self-energy ΣaB(kF ) due to the exchange of the boson a felt by the baryon B is given by

1 ∑ (a) ΣaB(kF ) = ΓBBa(kF ) ΓB′B′a(kF )ρ ′ (46) m2 B a B′ where on the right side we have inserted the above decomposition. Thus, a set of quadratic forms is obtained, bilinear in the (a) = = vertex functions ΓBBa(kF ). ρB denotes either a scalar (a s) or a vector (a v) ground state density of baryons of type B. Using on the left hand side the microscopic self-energies, the quadratic form can be evaluated. The vertices are fixed in their (relative) phases and magnitudes by the solutions

ΓB B a(kF ) ΣaB (kF ) 1 1 = 1 (47) ΓB2B2a(kF ) ΣaB2 (kF ) 2 2 ΣaB(kF ) ΓaB(kF ) = . (48) ∑ ′ ′ B′ ρaB ΣaB (kF ) H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 137

Fig. 14. In-medium NNα isoscalar (α = σ , ω) and isovector (α = δ, ρ) vertices determined from the DBHF self-energies in asymmetric nuclear matter of Ref. [84].

Using DBHF self-energies as input the dressed vertices are obtained in ladder approximation, being appropriate for use in RMF theory. Although nuclear matter (D)BHF self-energies depend on the particle momentum [84], those dependencies are canceling in the above expressions to a large extent. As discussed in [96] the mild remaining state dependence can be included by a simple correction factor, ensuring the proper reproduction of the nuclear matter equation of state and binding energies of nuclei. Results of such a calculation using DBHF self-energies as input [84] are shown in Fig. 14.

3. Covariant DFT approach to nuclear and hypernuclear physics

3.1. Achievements of the microscopic DDRH nuclear DFT

The RMF scheme for the extraction of the vertices within the DBHF ladder approximation has proven to lead to a quite successful description of nuclear matter and nuclear properties. Out of that scheme, DDRH theory has emerged, that later gave rise to the development of a purely phenomenological approach [59] by fixing the density dependence of the vertex functionals by fits to data. Relativistic DFT approaches based on density dependent vertices, e.g. by the Beijing group [102,103], are now a standard tool for covariant nuclear structure mean-field theory, describing successfully nuclear ground states and excitations. While the DBHF realizations of DDRH theory are of a clear diagrammatic structure, namely including interactions in the ladder level only, the parameters of the phenomenological models are containing unavoidably already higher order effects from core polarization self-energies and other induced many-body interactions. For obvious reasons, in nuclear structure research the meson fields giving rise to condensed classical fields are of primary importance. The isoscalar and isovector self-energies produced by scalar (JP = 0+) and vector mesons (JP = 1−) are dominating nuclear mean-field dynamics. In this section, we therefore set the focus on the mean-field producing scalar and vector mesons. The formulation is kept general in the sense that the concepts are not necessarily relying on the use of a DBHF description of vertices as discussed in the previous section. For a clear distinction, here vertex functionals will be ∗ denoted by gBB′M (ρ). The concepts have been developed earlier in the context of the Giessen Density Dependent Relativistic Hadron (DDRH) theory. By a proper choice of vertex functionals an ab initio approach is obtained. Binding energies and root-mean-square radii of stable and unstable nuclei are well described within a few percent [53,54,96]. The equations of state for neutron matter, neutron star matter, and neutron stars are obtained without adjustments of parameters. In [58] the temperature dependence of the vertices was studied and the thermodynamical properties of nuclear matter up to about a temperature of T = 80 MeV were investigated, looking for the first time into the phase diagram of asymmetric nuclear matter as encountered in heavy ion collisions and neutron stars by microscopically derived interactions. For that purpose, the full control of the isovector interaction channel – which is not well under control in phenomenological approaches – was of decisive importance. Results for hypermatter and hypernuclei will be discussed below.

3.2. Covariant Lagrangian approach to in-medium baryon interactions

A quantum field theory is typically formulated in terms of bare coupling constants which are plain numbers and the field theoretical degrees of freedom are contained completely in the hadronic fields. The preferred choice is a Lagrangian of the simplest structure, realized by bilinears of matter fields which are coupled to the meson fields, as discussed in Section 2. That apparent simplicity on the Lagrangian level, however, requires additional theoretical and numerical efforts for a 138 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 theory with coupling constants which are too large for a perturbative treatment. Hence, the theoretical complexities are only shifted to the treatment of the complete resummation of scattering series. DDRH theory attempts to incorporate the resummed higher order effects already into the Lagrangian with the advantage of a much simpler treatment of interactions. ∗ ′ This is achieved by replacing the coupling constants gBB M by meson–baryon vertices gBB′M(ΨB) which are Lorentz-invariant functionals of bilinears of the matter field operators ΨB. The derivation of those structures from Dirac–Brueckner theory has been discussed intensively in the literature [52–54,57,84]. The mapping of a complex system of coupled equations to a field theory with density dependent vertex functionals is leading to a formally highly non-linear theory. However, the inherent complexities are of a similar nature as known from quantum many-body theory in general. In practice, approximations are necessary. Here, we discuss the mean-field limit. The essential difference of the Lagrangian discussed here to the one of Eq. (8) lies in the definition of the interaction vertices. Overall, their structure is constrained by the requirements of maintaining the relevant symmetries. Thus, at least the functionals must be Lorentz-scalars and scalars under SU(3) flavor transformations. The simplest choice fulfilling these µ constraints is to postulate a dependence on the invariant density operator ρˆ = jµj of the baryon 4-current operator µ = ¯ µ ∗ = ∗ ˆ j ΨBγ ΨB. Thus, we use gBB′M(ΨB) gBB′M(ρ). Applying the SU(3) relations, this implies immediately a corresponding { } → { ∗ ˆ ∗ ˆ ∗ ˆ } structure for the fundamental interaction vertices, gD, gF , gS gD(ρ), gF (ρ), gS (ρ) . Thus, the interaction Lagrangian with density functional (DF) vertices is described by √ { } ∑ ∗(M) ∗(M) ∗(M) 1 LDF = − 2 g (ρˆ )[BBP ] + g v[BBP ] − g (ρˆ ) √ [BBP ] . (49) int D 8 D F 8 F S 1 S M∈{P,S,V} 6 ∗ ˆ By the same relations as in Eq. (17) we obtain baryon–meson vertices gBB′M (ρ) but with an intrinsic functional structures. With the standard methods of field theory we obtain the meson field equations which are of the form

′ ( µ + 2 ) s = ∑ ∗ ˆ BB s ∂µ∂ mM ΦM gBB′M(ρ)ρ , (50) BB′ ′ ( µ + 2 ) λ = ∑ ∗ ˆ BB λ ∂µ∂ mM VM gBB′M(ρ)ρ , (51) BB′ where Lorentz-scalar and Lorentz-vector fields are described by the first and second equation, respectively. The baryons are obeying the field equations

( ( µ µ ) (s) ) γµ p − ΣB (ρˆ ) − MB + ΣB (ρˆ ) ΨB = 0 . (52)

The general structure of the self-energies is given by a folding part, defined as the sum over meson field multiplied by coupling constants, and an additional rearrangement terms resulting from the variation of the intrinsic functional structure of the vertices [53,54]. As discussed above, the density dependence of the in-medium meson–baryon vertices is chosen ˆ 2 = µ by a functional structures given by ρ jµjB which is the Lorentz-invariant baryon density operator. Since higher order interaction effects are accounted for by the density-dependent vertex functionals, they must be used in a first order treatment only, i.e. on the level of effective Born-diagrams. With our choice of density dependence, only the vector self-energies are modified by rearrangement effects. Thus, the s scalar self-energy ΣB is obtained as a diagonal matrix with the elements of pure Hartree-structure (s) ˆ = ∑ ˆ ∗ ˆ ΣB (ρ) ΦM(ρ)gBBM(ρ). (53) M∈S µ The vector self-energies, contained in the matrix ΣB , are consisting of the direct meson field contributions (d)µ ˆ = ∑ µ ˆ ∗ ˆ ΣB (ρ) VM(ρ)gBBM(ρ) (54) M∈V and the rearrangement self-energies ∗ ∂g ′ ′′ (ρˆ ) δ (r)µ ˆ = ∑ B B M DF ΣB (ρ) ∗ Lint , (55) ∂jBµ δg ′ ′ ′ B′B′′M B B f M where the latter are generic for a field theory with functional vertices. Physically, the Σ (r) are accounting for changes of the interactions under variation of the density of the system as resulting from static polarization effects. They are indispensable for the thermodynamical consistency of the theory [53,54,58], because their neglection would violate the Hugenholtz–van Hove theorem. Similar conclusion have been drawn before by Negele for non-relativistic BHF theory [104]. The structure of the rearrangement self-energies is discussed in much detail in [57,104]. In Fig. 15, the diagrammatic structure of the mean-field self-energies is depicted. A substantial simplification is obtained in mean-field approximation. As discussed in [58], the vertex functionals are replaced by ordinary functions of the expectation value of their argument. The meson fields are replaced by static classical H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 139

Fig. 15. Derivation of the DFT mean-field self-energies by first variation of the energy density with respect to the density, leading to tadpole and rearrangement contributions, indicated by the first and second graphs on the right hand side of the figure, respectively. The density dependent vertices are indicated by shaded squares, the derivative of the vertices by a triangle. The DFT tadpole self-energy accounts for the full Hartree–Fock self-energy shown in Fig. 10.

fields and the rearrangement self-energies reduce to derivatives of the vertex functions with respect to the density. The mean-field constraint also implies that in a uniform system we can neglect in the average the space-like vector self-energy components. Thus, we are left with the scalar and the time-like vector self-energies. The scalar and the direct, Hartree-type vector mean-field self-energies, respectively, are obtained as

(s) ∑ 1 ∗ ∗ (s) Σ (ρB) = g (ρB)g ′ ′ (ρB)ρ ′ (56) B m2 BBM B B M B B′,M∈S M

(d) ∑ 1 ∗ ∗ Σ (ρB) = g (ρB)g ′ ′ (ρB)ρB′ (57) B m2 BBM B B M B′,M∈V M and the time-like vector rearrangement self-energies are given by the derivatives of the vertex functions with respect to the total baryon density of nucleons N and hyperons Y , ρB = ρN + ρY , ∗ DF ∂g ′ ′ (ρ ) ∂L (r) = ∑ B B M B int ΣB (ρB) ∗ , (58) ∂ρB ∂g ′ ′ B′,M B B M

(s) where the usual number and scalar densities are denoted by ρB and ρB , respectively. For later use we define the total time- like baryon vector self-energies

(v) = (d) + (r) Σ (ρB) ΣB (ρB) ΣB (ρB). (59)

By means of the SU(3) relations also the in-medium vertices in the other baryon-meson interaction channels can be derived [105], leading to a multitude of vertices which are not shown here. In magnitude (and sign) the BB′ vertices are of course different from the one shown in Fig. 14. But a common feature is that the relative variation with density is similar to the NN-vertices in Fig. 14.

4. DBHF investigations of Λ hypernuclei and hypermatter

4.1. Global properties of single-Λ hypernuclei

An important result of the following investigations is that at densities found in the nuclear interior the Λ and nucleon ∗ ≃ ∗ vertices are to a good approximation related by scaling factors, gΛΛσ ,ω Rσ ,ωgNNσ ,ω. Microscopic calculations show that these ratios vary in the density regions of interest for nuclear structure investigations only slightly. Averaged over the −3 ¯ ¯ densities up to the nuclear saturation density ρeq = 0.16 fm , we find Rσ = 0.42 and Rω = 0.66 with variation on the level of 5%. Hence, as far as interactions of the Λ hyperon are concerned it is possible to describe their properties with interactions following the scaling approach. In fact, the scaling approach was used widely before in an ad hoc manner, e.g. in [68,69]. Occasionally, the quark model is used as a justification by arguing that non-strange mesons couple only to the non-strange ¯ valence quarks of a baryon which for the Λ gives scaling factors Rσ ,ω = 2/3. Surprisingly, the cited value of Rω is extremely close to the one expected by the quark counting hypothesis. In this section, we investigate the scaling hypothesis by using self-consistent Hartree-type calculations for single-Lambda hypernuclei. In view of the persisting uncertainties on YN interactions, we treat Rσ ,ω as free constants which are adjusted in fits to Λ separation energies. Since wave functions, their densities, and energies are calculated self-consistently we account simultaneously for effects also in the nucleonic sector induced by the presence of hyperons. The nucleons (B = n, p) and the Lambda-hyperon (B = Λ) single particle states are described by stationary Dirac equations ( ) ˆ + (v) + 0 ∗ − = α·p ΣB (r) γ MB (r) εnℓj ψnℓjm 0 (60) 140 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

0 with pˆ = −i∇ and α = γ γ. We assume spherical symmetry. The Dirac spinors ψnℓjm with radial quantum number n and total angular momentum and projection jm are eigenfunctions to the eigenenergy εnℓj. The orbital angular momentum of the upper component is indicated by ℓ. The direct vector self-energies are

(d) = ∗ 0 + ⟨ ⟩ ∗ 0 + 0 ΣB (r) gBBω(ρB)Vω(r) τ3 BgBBρ (ρB)Vρ (r) qBVγ (r). (61)

ρB(r) = ρp(r) + ρn(r) + ρΛ(r) is the radial dependent total baryon density with a volume integral normalized to the total baryon number AB = An + Ap + Aλ where in a single Λ nucleus AΛ = 1. qB denotes the electric charge of the baryon. We use ⟨τ3⟩B = ±1 for the proton and neutron, respectively, and ⟨τ3⟩B = 0 for the Λ. The scalar self-energies, contained in the ∗ relativistic effective mass MB , are given by

(s) ∗ ∗ Σ (r) = g (ρ )Φ (r) + ⟨τ ⟩ g (ρ )Φ (r). (62) B BBσ B σ 3 B BBδ B δ The condensed meson fields are determined by the classical field equations

(−∇2 + 2 ) 0 = ∗ ( + ) + ∗ mω Vω gNNω(ρB) ρp ρn gΛΛω(ρB)ρΛ (63) (−∇2 + 2 ) 0 = ∗ ( − ) mρ Vρ gNNω(ρB) ρp ρn (64) − ∇2 0 = (c) Vγ epρp (65) (−∇2 + 2 ) = ∗ ( (s) + (s)) + ∗ (s) mσ Φσ gNNσ (ρB) ρp ρn gΛΛσ (ρB)ρΛ (66) (−∇2 + 2) = ∗ ( (s) − (s)) mδ Φδ gNNδ(ρB) ρp ρn , (67)

(c) where ep is the proton charge and ρp denotes the proton charge density, including the proton charge form factor. The vector (s) and scalar densities ρB and ρB are given in terms of the sum and the difference, respectively, of the densities of upper and lower components of the wave functions of occupied states. For further details we refer to our previous work [54,55,57]. From the field-theoretical energy–momentum tensor the energy density is obtained as the ground state expectation value 00 EA(r; Z, N, Y ) = ⟨T ⟩ for a nucleus with Ap protons, An neutrons and AΛ hyperons [54,52]. Of particular interest are the nuclear binding energies defined by [∫ ] 1 3 BA = d rEA(r) − ApMp − AnMn − AΛMΛ , (68) A from which the baryon separation energies are obtained as usual. We treat the Λ interaction vertices in the scaling approach and determine the renormalization constants Rσ ,ω by a fit to the available single-Λ hypernuclear data, as discussed in [52]. The lightest nuclei were left out of the fitting procedure because they are not well suited for a mean-field description. From a χ 2 procedure, we find that the two scaling constants are correlated linearly, leading to a χ 2 distribution with a steep, but long stretched valley. Two solutions of comparable quality are (R1σ , R1ω) = (0.506, 0.518) and (R2σ , R2ω) = (0.490, 0.553). On a 20% level both sets are compatible with the quark model hypothesis and with the SU(3)-symmetry based results [105]. Taking into account also the light nuclei, the values show a considerably large spread. In Fig. 16 the DDRH results are compared to known measured single-Λ separation energies. It is seen that the global two-parameter fit leads to a surprisingly good description of the observed Λ separation energies. The remaining theoretical uncertainties are indicated. Extrapolating the separation energies shown in Fig. 16 to (physical unaccessible) large mass numbers, the limiting value ∞ SΛ ≃ 28 MeV is asymptotically approached for A → ∞ which we identify with the separation energy of a single Λ-hyperon in infinite nuclear matter. Thus, we predict for the in-medium Λ-potential in ordinary nuclear matter a value of ∞ UΛ ∼ −28 MeV. In Fig. 17 results of the Nijmegen group are shown for comparison. The ESC08 interaction was used as input for non- relativistic Brueckner G-matrix calculations which then was used in a folding approach to generate the Lambda mean-field potentials. The relativistic DDRH and the non-relativistic ESC08-results agree rather well for the Lambda separation energies over the full known mass range of core nuclei. The agreement may be taken as an indication that a common understanding of single particle dynamics for single-Λ hypernuclei is obtained, at least within the presently available data base.

4.2. Spectroscopic details of single-Λ hypernuclei

41 89 For the results shown in Fig. 16 the KEK-data of Hotchi et al. [106] for Λ V and λ Y have been especially important because of their good energy resolution and the resolution of a large number of Λ bound states. This wealth of spectroscopic information did help to constrain further the dynamics of medium and heavy hypernuclei. A caveat for those nuclei is that the hyperon is attached to a high-spin core, 40V(6−) and 88Y(4−). Hence, the Λ spectral distributions are additionally broadened by core-particle spin–spin interactions. A consistent description of the spectra could only be achieved by including those interactions into the analysis. A phenomenological approach was chosen by adding the H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 141

= − = − 2 Fig. 16. Separation energies of the known S 1 single Λ hypernuclei as a function of the mass number A to power γ 3 . Results of two sets of scaling parameter sets are compared to measured separation energies. For the ℓ > 0 levels the spin–orbit splitting is indicated. The theoretical uncertainties are ∞ marked by colored/shaded bands. For A → ∞ the limiting value SΛ ≃ 28 MeV is asymptotically approached as indicated in the figure. Thus, we predict for the in-medium Λ-potential in ordinary nuclear matter a depth of 28 MeV. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Source: The data are from Refs. [106–110].

Fig. 17. Single Λ binding energies obtained with the Nijmegen ESC08 G-matrix interaction [33] for the known S = −1 single Λ hypernuclei. The energies = − 2 → ∞ ∞ ≃ − are shown as a function of the mass number A to power γ 3 . For A the limiting value EΛ 28 MeV is asymptotically approached as indicated in the figure. Thus, we predict for the in-medium Λ-potential in ordinary nuclear matter a depth of 28 MeV. Source: From Ref. [33].

core-particle spin–spin energy to the Λ eigenenergies

= RMF + ⟨ | · | ⟩ e(jJC )JΛ εjΛ EjJc (jJC )J jλ J C (jJC )J (69) giving rise to a multiplet of states with total angular momentum J = Jc + j. The multiplet-spreading is found to account for about half of the spectral line widths. Hence, if neglected, badly wrong conclusions would be drawn on an extraordinary large 142 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Table 4 DDRH results for Λ single particle energies. 89 41 Level Λ Y Λ V

1s1/2 −22.94 ± 0.64 MeV −19.8 ± 1.4 MeV 1p3/2 −17.02 ± 0.07 MeV −11.8 ± 1.3 MeV 1p1/2 −16.68 ± 0.07 MeV −11.4 ± 1.3 MeV 1d5/2 −10.26 ± 0.07 MeV −2.7 ± 1.2 MeV 1d3/2 −9.71 ± 0.07 MeV −1.9 ± 1.2 MeV 1f7/2 −3.04 ± 0.11 MeV − 1f5/2 −3.04 ± 0.11 MeV −

89 Fig. 18. Single Λ binding energies in Λ Y. The DDRH results (dotted, dashed, and full lines) are compared to the data of Hotchi et al. [106], shown as histogram. The extracted single particle levels, obtained after defolding the spectrum and taken into account core polarization effects (see text), are indicated at the top of the figure.

spin–orbit splitting, too large by about a factor 2. Including the spin–spin effect leads to a spin–orbit energy fully compatible with the values known from light nuclei. The analysis includes also the contributions from the relativistic tensor vertex [111], modifying the effective Λ-spin–orbit potential to [( ∗ ) ] (so) 1 MB fΛΛω Λ Λ UΛ = r · ∇ 2 ∗ + 1 Σω + Σσ . (70) r MB gΛΛω

Here, the tensor strength f appears as an additional parameter. For the NN case, fNNω is known to be weak and usually it is set to zero. The small spin–orbit splitting observed in hypernuclei have led to speculations that the tensor part may be non-zero, partly canceling the conventional spin–orbit potential, given by the sum of vector and scalar self-energies. so ∗ This should happen for f /g ∼ −1 as seen by considering that UΛ is a nuclear surface effect where M ∼ M and also the self-energies are about the same. The KEK-spectra are described the best with vanishing Λ tensor coupling, fΛλω/gΛλω = 0, thus agreeing with the NN-case. Our results for the Λ single particle spectra in the two nuclei are found in Table4. The = averaged spin–orbit splitting is about 223 keV and 283 keV and the spin–spin interaction amounts to EjJc 106 keV and = 89 EjJc 61.3 keV in Vanadium and Yttrium, respectively. The experimentally obtained and the theoretical spectra of Λ Y are compared in Fig. 18.

4.3. Interactions in multiple-strangeness nuclei

In contrast to the YN data – scarce as they are – for hyperon–hyperon systems like ΛΛ no direct scattering data are available. The only source of (indirect) experimental information at hand comes from double-Lambda hypernuclei. The 6 probably best recorded case is the so-called Nagara event [112], a safely identified ΛΛHe hypernucleus produced at KEK by a − + 12 (K , K ) reaction at plab = 1.66 GeV/c on a C target. The KEK-E373 hybrid emulsion experiment [112] traced the stopping − 6 4 of an initially produced Ξ hyperon, captured by a second carbon nucleus, which then was decaying into ΛΛHe plus an He 6 5 − nucleus and a triton. The ΛΛHe nucleus was identified by it is decay into the known ΛHe and a proton and a π . The data were used to deduce the total two-Lambda separation energy BΛΛ and the Lambda–Lambda interaction energy ∆BΛΛ which is a particular highly wanted quantity. A re-analysis in 2013 [113] led to the nowadays accepted values BΛΛ = 6.91 ± 0.16 MeV and ∆BΛΛ = 0.67 ± 0.17 MeV while the original value was larger by about 50% [112]. H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 143

Fig. 19. Leading order (LO) and next-to-leading order (NLO) diagrams used in the χEFT descriptions of hypermatter and hypernuclei. The full lines indicate either a nucleon or a hyperon, preferentially a Λ-hyperon. The LO contact interaction accounts for unresolved short range interactions. NLO interactions by pions are indicated by dashed lines.

These data are an important proof that double-Lambda hypernuclei are indeed highly useful for putting firm constraints 1 6 on the ΛΛ S0 scattering length. Theoretical studies for the ΛΛHe hypernucleus have been performed by a variety of approaches such as three-body Faddeev cluster model, in Brueckner theory, or with stochastic variational methods. In order to reproduce the separation energies obtained from the Nagara event the theoretical results suggest a ΛΛ scattering length aΛΛ = −1.3... − 0.5 fm, including cluster-type descriptions [41,114–116], calculations with the NSC interactions [117,118], and variational results [119]. Analyzes of hyperon final state interactions in strangeness production reactions also allow to 1 estimate aΛΛ. A recent theoretical analysis of STAR data [120] led to aΛΛ = −1.25... − 0.56 fm [121]. The theoretical results on ∆BΛΛ agree within the cited uncertainty ranges which should be considered as an optimistic signal indicating a basic understanding of such a complicated many-body system for the sake of the extraction of a much wanted data as ∆BΛΛ. The experimental results on the ΛΛ interaction energy have initiated on the theoretical side considerable activities, see e.g. [17,18]. In [34] recent ESC results are discussed. In the ESC model the attraction in the ΛΛ channel can only be changed by modifying the scalar exchange potential. The authors argue, that if the scalar mesons are viewed as being mainly qq¯ states, the (attractive) scalar-exchange part of the interaction in the various channels satisfies |VΛΛ| < |VNΛ| < |VNN |, implying indeed a rather weak ΛΛ potential. The ESC fits to the NY scattering data give values for the scalar mixing angle, which seem to point to almost ideal mixing for the scalar mesons. This is also found for the former Nijmegen OBE models NSC89/NSC97. In these models an increased attraction in the ΛΛ channel gives rise to (experimentally unobserved) deeply bound states in the NΛ channel. In the ESC08c model, however, the apparently required ΛΛ attraction is obtained without giving rise to unphysical NΛ bound states.

4.4. Hyperon interactions and hypernuclei by effective field theory

As mentioned afore, a promising and successful approach to nuclear forces is chiral effective field theory which describes the few available NY scattering data quite well, see Fig.7 and Ref. [44]. Already rather early the Munich and the Jülich groups have applied χEFT also to hypernuclei. The early applications as in [122] were based on the leading order (LO) and next-to-leading order (NLO) diagrams shown in Fig. 19. The results obtained by Finelli et al. in [122] in their FKVW approach are in fact quite close to the OBE-oriented approaches of the covariant DDRH-theory and the non-relativistic ESC-model. The covariant FKVW density functional was used, incorporating SU(3) flavor symmetry, supplemented by constraints from QCD sum rules serving to estimate the scalar and vector coupling constants. In Fig. 20 the single-Λ separation energies are shown. The LO and NLO nucleon–hyperon 13 16 40 89 139 208 interactions were derived by fits to the spectra of Λ C, Λ O, Λ Ca, Λ Y, Λ La, and Λ Pb. As discussed above, a scaling description was used to adjust the hyperon interaction vertices. The Λ-nuclear surface term, appearing in the gradient expansion of a density functional for finite systems, was generated model-independently from in-medium chiral SU(3) perturbation theory at the two-pion exchange level. The authors found that term to be important in obtaining good overall agreement with Λ single particle spectra throughout the hypernuclear mass table. It is quite interesting to follow their explanation of the small spin–orbit splitting seen in Λ hypernuclei. An important part of the Λ-nuclear spin–orbit force was obtained from the chiral two-pion exchange ΛN interaction which in the presence of the nuclear core generates a (genuinely non-relativistic, model-independent) contribution. This longer range contribution

1 Note the different sign convention used by the STAR group. 144 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 20. χEFT results for Λ separation energies over the hypernuclear mass table shown as a function of A−2/3. The shaded areas indicate the theoretical uncertainties for the range of hyperon vertex scaling parameters ζ as indicated in the box. Also shown are results (dashed lines) obtained in calculations where the relativistic mean-field self-energies were fitted by potentials with Wood–Saxon form factors. The particle threshold is indicated by a dotted line. Source: From Ref. [122].

counterbalances the short-distance spin–orbit terms that emerge from scalar and vector mean-fields, in exactly such a way that the resulting spin–orbit splitting of Λ single particle orbits is extremely small. A three-body spin–orbit term of Fujita– Miyazawa type that figures prominently in the overall large spin–orbit splitting observed in ordinary nuclei, is absent for a Λ attached to a nuclear core because there is no Fermi sea of hyperons. The confrontation of that highly constrained approach with empirical Λ single-particle spectroscopy turns out to be quantitatively successful, at a level of accuracy comparable to that of the best existing hypernuclear many-body calculations discussed before. Also the χEFT approach predicts a Λ-nuclear single-particle potential with a dominant Hartree term of a central depth of about −30 MeV, consistent with phenomenology. Since then, the work on SU(3)-χEFT was been intensified by several groups and in several directions. The Munich–Jülich collaboration [123,124,67,125], for example, has derived in-medium baryon–baryon interactions. A density-dependent effective potential for the baryon–baryon interaction in the presence of the (hyper)nuclear medium has been constructed. That work incorporates the leading (irreducible) three-baryon forces derived within SU(3) chiral effective field theory, accounting for contact terms, one-pion exchange and two-pion exchange. In the strangeness-zero sector the known result for the in-medium nucleon–nucleon interaction are recovered. In [67] explicit expressions for the hyperon–nucleon in- medium potential in (asymmetric) nuclear matter are presented. In order to estimate the low-energy constants of the leading three-baryon forces also the decuplet baryons were introduced as explicit degrees of freedom. That allowed to construct the relevant terms in the minimal non-relativistic Lagrangian and the constants could be estimated through decuplet saturation. Utilizing this approximation, numerical results for three-body force effects in symmetric nuclear matter and pure neutron matter were provided. Interestingly, a moderate repulsion is found, increasing with density. The latter effect is going in the direction of the much wanted repulsion expected to solve the hyperonization puzzle in neutron star matter. A different aspect of hypernuclear physics is considered by the Darmstadt group of Roth and collaborators. In [125] light finite hypernuclei are investigated by no core shell model (NCSM) methods. In that paper, ab initio calculations for p-shell hypernuclei were presented including for the first time hyperon–nucleon–nucleon (YNN) contributions induced by a similarity renormalization group (SRG) transformation of the initial hyperon–nucleon interaction. The transformation including the YNN terms conserves the spectrum of the Hamiltonian while drastically improving model-space convergence of the importance-truncated no-core model. In that way a precise extraction of binding and excitation energies was achieved. Results using a hyperon–nucleon interaction at leading order in chiral effective field theory for lower- to mid-p-shell hypernuclei showed a good reproduction of experimental excitation energies but hyperon separation energies are typically overestimated as seen in Fig. 21. The induced YNN contributions are strongly repulsive, explained by a decoupling of the Σ hyperons from the hypernuclear system, corresponding to a suppression of the Λ − Σ conversion terms in the Hamiltonian. Thus, a highly interesting link to the so-called hyperonization puzzle in neutron star physics is found which provides a basic mechanism for the explanation of strong ΛNN three-baryon forces.

4.5. Brief overview on LQCD activities

On the QCD-side the lattice groups in Japan (HALQCD) and the Seattle-Barcelona (NPLQCD) collaborations are making strong progress in computing baryon–baryon interactions numerically. The HALQCD method [126] relies on recasting the H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 145

13 Fig. 21. Absolute and excitation energies of Λ C. (a) Nucleonic parent absolute and excitation energies, (b) hypernucleus with bare (dashed line) and SRG-evolved (solid line) YN interaction, (c) hypernucleus with added YNN terms for cutoffs 700 MeV/c (solid line) and 600 MeV/c (dotted line). Source: From Ref. [125].

lattice results into a Schrödinger-type wave equation by which binding and scattering observables of baryonic systems are extracted. Baryon–baryon interactions in three-flavor SU(3) symmetric full QCD simulations are investigated with degenerate quark masses for all flavors. The BB potentials in the orbital S-wave are extracted from the Nambu–Bethe– Salpeter wave functions measured on the lattice. A strong flavor-spin dependence of the BB potentials at short distances is observed, in particular, a strong repulsive core exists in the flavor-octet and spin-singlet 8s channel, while an attractive core appears in the flavor singlet channel, i.e. the SU(3singlet-irrep. In recent calculation, the HALQCD group achieved to approach the region of physical masses, obtaining results for various NN, YN, and YY channels, see e.g. [127–129]. A somewhat different approach is used by the NPLQCD collaboration [130–133]. The effects of a finite lattice spacing is systematically removed by combining calculations of correlation functions at several lattice spacings with the low-energy effective field theory (EFT) which explicitly includes the discretization effects. Thus, NPLQCD combines LQCD methods with the methods of chiral EFT which a particular appealing approach because it allows to match the χEFT results obtained from hadronic studies. Performing calculations specifically to match LQCD results to low-energy effective field theories will provide the means for first predictions at the physical quark mass limit. This allows also to predict quantities beyond those calculated with LQCD. In [133], for example, the NPLQCD collaboration report the results of calculations of nucleon–nucleon 3 3 1 interactions in the S1− D1 coupled channels and the S0 channel at a pion mass mπ = 450 MeV. For that pion mass, the n-p system is overbound and even the di-neutron becomes a bound state. However, extrapolations indicate that at the physical pion mass the observed properties of the two-nucleon systems will be approached.

4.6. Infinite hypermatter

Calculations in infinite matter are simplified because of translational invariance. By that reason, the baryons are in plane wave states and the meson mean-fields become independent of spatial coordinates. Under these conditions the field equations reduce to algebraic equations and many observables can be evaluated in closed form. The total baryon number density becomes ∑ ∫ d3k ρB = trs nsB(k, kF ) (71) (2π)3 B B 146 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 22. Binding energy per baryon of (n, p, Λ) matter. The Λ fraction is defined as ξΛ = ρΛ/ρ and the background medium is chosen as symmetric (p, n) matter, ξn = ξp. The absolute minimum is marked by a filled circle. The line ε = 0 is indicated by a red line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

where the trace is to be evaluated with respect to spin s. In cold spin saturated matter the occupation numbers nsB are 2 2 independent of s and are given by nB = Θ(k − k ) resulting in FB

Ns 3 ρB = k (72) 3π 2 FB and Ns = 2 is the spin multiplicity for a spin-1/2 particle. Frequently we use ρB = ξBρB where the fractional baryon numbers ξB = ρB/ρB add up to unity. The scalar densities are defined by ∫ d3k M∗ ρ(s) = N B (73) B s 3 ∗ (2π) EB (k) √ ∗ = 2 + ∗2 (s) = where EB (k) k MB . The integral is easily evaluated in closed form and one finds ρB ρBfs(zB) where 3 ( ) √ 2 √ 2 fs(z) = z 1 + z − log(z + 1 + z ) (74) 2z3 ≤ = ∗ ∗ (s) is a positive transcendental function with fs 1 depending on zB kFB /MB . Since MB depends on ρB via the scalar fields, the scalar densities are actually defined through a system of coupled algebraic equations which has to be solved iteratively. Thus, already on the mean-field level a theoretically and numerically demanding complex structure has to be handled. In infinite matter, also the energy–momentum tensor can be evaluated explicitly in mean-field approximation. The energy density in the mean-field sector is

00 ∑ 1 [ ∗ (s)] E(ρB) = ⟨T ⟩ = 3EF ρB + m ρ 4 B B B B

1 [ 2 2 2 2 2 2 2 02 2 02 2 02] + m Φ + m Φ + m ′ Φ ′ + m V + m V + m V , (75) 2 σ σ δ δ σ σ ω ω ρ ρ φ φ

(s)B where the sum runs over all baryons and their energies are weighted by the partial vector and scalar densities ρB and ρ , respectively. The (classical) field energies of the condensed meson mean-fields are indicated. For completeness the field energy of the SU(3)-singlet scalar and vector mesons σ ′ and φ, respectively, are also included. An important observable is the binding energy per particle ∑ ε(ρB) = E(ρB)/ρB − ξBMB. (76) B

In Fig. 22 results of DDRH calculation for ϵ(ρB) are shown for (p, n, Λ)-matter. A varying fraction of Λ-hyperons is embedded into a background of symmetric (p, n)-matter. Hence, we fix ξp = ξn and ξΛ = 1 − 2ξp. H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 147

The saturation properties of pure symmetric (p, n)-matter are very satisfactorily described: The saturation point is located −3 within the experimentally allowed region at ρsat = 0.166 fm and ε(ρsat ) = −15.95 MeV with an incompressibility K∞ = 268 MeV which is at the upper end of the accepted range of values. Adding Λ hyperons the binding energy first increases −3 until a new minimum for 10% Λ-content is reached at ρmin = 0.21 fm with a binding energy of ε(ρmin) = −18 MeV. Increasing either ξΛ and/or the density, the binding energy approaches eventually zero, as marked by the red line in Fig. 22. The minimum, in fact, is located in a rather wide valley, albeit with comparatively steep slopes, thus indicating the possibility of a large variety of bound single and even multiple-Λ hypernuclei. Note, however, that the binding energy per particle considerably weakens at high densities as the Λ-fraction increases: At high values of the density and the Λ fraction p, n, Λ matter becomes unbound, finally.

5. Strangeness and resonance excitation on the free nucleon

5.1. Coupled channels approach to nucleon resonances

The investigations of resonances and strangeness in nuclear systems rely obviously on a good understanding of the underlying elementary processes. Thus, before turning to the discussion of production and properties of strangeness and resonances on nuclear systems, in this section we give first an overview of that kind of physics on the free nucleon. As mentioned above, the discovery of nucleon resonances in the early pion–nucleon scattering experiments provided first indications for a complicated intrinsic structure of the nucleon. With establishing the quark picture of hadrons and developments of the constituent quark models the interest in the study of the nucleon excitation spectra was renewed. Soon after, it was realized that there was an obvious discrepancy between the number of resonances predicted by theory and those identified experimentally. Since then, the problem of missing resonances is a major issue of baryon spectroscopy. Final answers about the number of excited states of the nucleon and their properties are still pending. Solutions are searched for both experimentally and theoretically. On the theory side constituent quark models (CQM), lattice QCD (LQCD) and Dyson–Schwinger (DSE) approaches have been developed to describe and predict the nucleon resonance spectra. As the main problem remains, however, the serious disagreement between the number (and properties) of the theoretically found states and the experimentally observed spectrum of baryons. The main information about the hadron spectra comes from the analysis of scattering data. Coupled-channel approaches have proven to be an efficient tool to extract baryon properties from experiment. The Giessen coupled-channel model (GiM) is taken here as an example representing the whole class of field-theoretically based phenomenological models. The GiM is obeying the elementary symmetries of hadron physics and conserving unitarity. Meson production proceeds through the excitation of s-channel resonances and t- and u-channel processes. Parallel to the work at Giessen, the Jülich group has been formulating a coupled channels model [134]. That approach, in fact, is based largely on the work of Haberzettl et al. [135]. The Jülich πN–ηN model was used for the hadronic final state interactions comprising stable hadrons as well as the effective ππN channels as given by π∆, σ N, and ρN. In this respect, it has some overlap with early GiM-versions where double-pion channels have been accounted for by introducing effective single-mass mesons [136]. As the GiM, also the Jülich hadronic model has been quite successful in describing πN → πN scattering for center-of-mass energies up to about 2 GeV. A long-term project on the dynamical description of resonances has also been monitored by the former EBAC group at JLab [137]. As the GiM, the EBAC group emphasizes the importance of an unified, complete coupled-channels analysis of the whole set of meson and photo-production data on the nucleon, including not only single-meson but also the ππN channels. The expectation is that the multi-meson channels will serve to detect N∗ states which couple only weakly to the single-meson production or decay channels. The coupled channels scheme is derived from effective Lagrangian of a structure comparable to the one used in the GiM. Background and resonance terms are determined consistently within the model. The resonant amplitudes are generated from a number of pre-chosen bare excited nucleon states that are dressed by the non-resonant interactions as constrained by the unitarity condition. Similar to the GiM (and Jülich) results, also the EBAC group finds in certain channels strong interference effects produced by the channel coupling among the varies single- and multiple-meson channels contributions. Also, large interference between the resonant and non-resonant amplitudes is also demonstrated. As discussed below, interfering amplitudes may produce in cross sections signals of fake resonance structures. The Bonn–Gatchina model [8] and the GWU SAID package [9], both being used heavily for data analyzes, are using comparable concepts. The GiM has been developed for a combined analysis of pion- and photon-induced reactions on the nucleon, (π/γ )+N, for extracting properties of nucleon resonances. The applications range from investigations of the elastic and inelastic πN and πN∗ channels [138] to ωN [5], ηN [139,140] production as well as the strangeness channels KΛ [141] and KΣ [6]. The 2πN channels were investigated recently in [7]. Here we review central issues of the Giessen approach and present results for selected reactions. As an illustrative overview we present already here in Fig. 23 our results on the total cross sections in the various hadronic reaction channels.

5.2. The Giessen coupled channels model for baryon spectroscopy

Here we briefly outline the main ingredients of the model. More detail can be found in [7,6,138–141]. We need to solve the Bethe–Salpeter (BS) equation for the scattering amplitude: √ ∫ 4 √ ′ ′ d q ′ M(p , p; s) = V (p , p; w) + V (p , q; w)GBS (q; s)M(q, p; w), (77) (2π)4 148 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 23. Total cross sections for pion-induced reactions. Results of the Giessen model (solid curves) are compared to experimental data (symbols). Source: Taken from [142].

√ where w = s is the available center-of-mass energy. Here, p (k) and p′ (k′) are the incoming and outgoing baryon (meson) four-momenta. After splitting the two-particle BS propagator GBS into its real and imaginary parts, we introduce the K-matrix given schematically by ∫ K = V + V Re [GBS ] K, (78) where V denotes the tree level interactions shown in Fig. 24, and obtain the full scattering amplitude from ∫ M = K + i MIm [GBS ] K . (79)

Since the imaginary part of GBS is given by the on-shell part, the reaction matrix T , defined via the scattering matrix S = 1 + 2iT , can now be calculated from K after a partial wave decomposition (PWD) into total spin J, parity P, and isospin I via matrix inversion: ′ ′ −1 ′ T (p , p; w) = (1 − iρ(w)K(p , p; w)) K(p , p; w), (80) where ρ(w) is an appropriately chosen phase space factor. Hence unitarity is fulfilled as long as K is Hermitian. For simplicity we apply the so-called K-matrix Born approximation, which means that we neglect the real part of GBS and thus K reduces to K = V . The validity of this approximation was tested a long time ago by Pearce and Jennings [143]. The potential V is built by a sum of s-, u-, and t-channel Feynman diagrams by means of effective Lagrangians which are not shown here but can be found in very detail in [136] and the previously cited references. Certain aspects of the more 3 5 involved cases of spin- 2 and spin- 2 resonances are discussed below in Section7. In all reaction channels the non-resonant background contributions to the scattering amplitudes are consistently derived from the u- and t-channel diagrams, thus reducing the number of free parameters greatly. In addition, each vertex is multiplied by a cutoff form factor: 4 Λq F(q2) = , (81) 4 2 2 2 Λq + (q − mq) 2 where mq (q ) denotes the mass (four-momentum squared) of the off-shell particle. To reduce the number of parameters the cutoff value Λq is chosen to be identical for all final states. We only distinguish between the nucleon cutoff (ΛN ), the resonance cutoffs (ΛR), a common one for all resonances in a given spin-channel as indicated by the indices, and the t-channel cutoff (Λt ). A considerable numerical simplification is obtained by the afore mentioned K−matrix approximation. Using the partial wave decomposition the integral over dΩq can be calculated analytically. Then Eq. (77) reduces to a linear system of coupled equations for the partial wave scattering amplitudes: JP = JP Tf i (w) Kf i (w) ∫ ∞ + ∑ 2 JP JP i dµj Aj(µj)Tf j (µj)Kj i (µj) (82) 2 j µj0 where K = V , f , i, j denote the final, initial, and intermediate meson − baryon channels, respectively. The photo-production reaction channels are treated perturbatively in leading order of the γ N and γ N∗ vertices, respectively. The spectral functions H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 149

Fig. 24. The structure of the tree-level interaction potential V . s−, u−, and t-channel interactions defining the non-resonant background contributions are shown in the first line, including contact terms which are chosen such that gauge invariance is assured. The so-called z-diagrams, displayed in the second line are generic for the double-pion channels. s-channel resonance interactions are depicted in the last line. Time is running from left to right.

2 in the meson–meson (π, K, σ , ρ...) and the meson–baryon isobar channels, as e.g the ∆ resonance, are denoted by Aj(µj ), being integrated over the energy µj available in the isobar subsystem starting at the isobar threshold energy µj0. For stable particles the spectral functions reduce to delta-functions, projecting the integrand on the particles’ mass. Since the two-pion discontinuities are taken into account the three-body unitarity in the form of the optical theorem if fulfilled up to interference terms between different isobar subchannels [7].

5.3. Strangeness production on the nucleon

Strangeness production on the nucleon by excitation of resonances which decay into kaon–hyperon channels is an important spectroscopic tool giving access to the SU(3) flavor structure of baryons. Moreover, such exotic channels like the kaon–hyperon final states are expected to play a key role in identifying hitherto undetected excited states of the nucleon, thus addressing the notorious problem of missing resonances. In [141] we have performed a study of the pion- and photon- induced KΛ reactions within our unitary coupled-channel effective Lagrangian approach. We have analyzed the latest data sets from the CLAS- and the CBELSA-experiment and reanalyzed the earlier SAPHIR data on photoproduction of kaons on the nucleon. Using the GiM coupled channels K-matrix approach kaon production was described in parallel to the full set of all other meson–baryon channels. Thus, a major revision of the complete parameter set was performed. A major goal of those investigations was to address the at that time still open question on the major contributions to the associated strangeness production channels. Since KΛ photoproduction data [144,145] gave an indication for missing resonance contributions, a combined analysis of the (π, γ )N → KΛ reactions was expected to identify clearly these states. Assuming small couplings to πN, these hidden states should not exhibit themselves in the pion-induced reactions and, consequently, in the πN → KΛ reaction. The aim of our calculations was to explore to what extent the data available at that time can be explained by known reaction mechanisms without introducing new resonances. Our results for total cross sections are displayed in Fig. 25 and further results on differential cross sections, polarization observables and angular distributions are found in [141]. As discussed in [141] the SAPHIR [144] and the CLAS [145] data sets, in fact, are leading to two slightly different sets of interaction parameters, reflecting and emphasizing the differences among the two measurements. Below, that point is discussed again. More recent CLAS-data on KΣ production by polarized beams initiated an updated large scale coupled-channels analysis of associated strange production on the nucleon. Based on the coupled-channel effective Lagrangian formalism underlying the Giessen model (GiM) a combined analysis of (π, γ )N → KΣ hadro- and photo-production reactions were performed. The analysis covered a center of mass energy range up to 2 GeV. The central aim was to extract the resonance couplings to the KΣ state. In [6] the Giessen model was used to reanalyze newly released data from various experimental groups for KΣ production on the nucleon. Both s-channel resonances and t, u-channel background contributions are found to be important for an accurate description of angular distributions and polarization observables, assuring a high quality description of the data. The extracted properties of isospin I = 3/2 resonances were discussed in detail. We found that the I = 1/2 resonances are largely determined by the non-strangeness channels. Our calculations included 11 isospin I = 1/2 resonances and 9 isospin I = 3/2 resonances, respectively. In this work we continued the investigations of the I = 1/2 and 3/2 sectors with the parameters fitted to newly√ published KΣ photoproduction data together with the previous πN → KΣ measurements in the energy region s ≤ 2.0 GeV. The included KΣ photoproduction data are those of the γ p → K +Σ 0 published by the LEPS [149–151], CLAS [152,153] and GRAAL [154] group, and those of γ p → K 0Σ+ released by the CLAS [155] and CBELSA [156] collaboration, respectively. The SAPHIR data have been left out here because of the known inconsistencies of the K +Σ 0 data [144] with the corresponding 150 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 25. π −p → K 0Λ total partial wave cross sections, predicted by parameter set C of Ref. [141], obtained from a fit to the CLAS data [145]. Source: The experimental cross section data are taken from [146–148].

Fig. 26. Total cross sections for kaon production on the nucleon. Results of the Giessen model [6] are compared to CLAS, CBELSA, and SAPHIR data. Results of the full model calculation are shown in the left panel. Results using only the Born-amplitudes and t-channel meson exchange are displayed in the right panel.

CLAS and GRAAL data (for the details, see Ref. [153]). Also, the K 0Σ+ SAPHIR data [144] have much bigger error bars than those of the CBELSA and CLAS group. The data before 2002√ are also no longer used. Results for total cross sections are shown in Fig. 26. Up to a total center-of-mass energy of about s = 2 GeV the data are well described. The analysis included all charge channels, K 0Σ∓ and K ±Σ 0. We achieved a quite satisfactory description of the γ p → K +Σ 0 data (χ 2 = 1.8) and the γ p → K 0Σ+ data (χ 2 = 2.0). However, the pion-induced strangeness production reactions are described slightly less accurate as indicated by the corresponding χ 2 values of χ 2 = 4.1, 3.2 and 2.8 for the π +p → K +Σ+, π −p → K 0Σ 0 and π −p → K +Σ− reactions, respectively. The parameters have been varied in our fit simultaneously to the I = 1/2 and 3/2 sectors. Although the new data are available with reduced total errors the refitted model parameters were changed only very little. A typical result is displayed in Fig. 27, illustrating the quality of the description on the example of π −p → K 0Σ 0 reaction. The complete set of results, including partial wave cross sections, angular distributions of cross sections and polarization observables for the full set of KΣ exit channels are found in [6]. H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 151

Fig. 27. (Color online) The differential cross section of π −p → K 0Σ 0 reaction. The solid (green), dashed (blue) and dotted (magenta) lines are the full model calculation, the model calculation with the S11(1650) and F15(1680) turned off, respectively. The numerical labels denote the center of mass energies in units of GeV. Source: From Ref. [6].

5.4. Probing nucleon resonances by η-meson production

Understanding the dynamics of eta-meson production and, vice versa, the decay of nucleon resonances into the nucleon- eta exit channel is of ongoing interest in hadron spectroscopy. The η-meson photoproduction on the proton has been measured with high precision by the Crystal Ball collaboration at MAMI [157]. These high-resolution data provide a new step forward in understanding the reaction dynamics and in the search for a signal from the ‘weak’ resonance states. The main result reported in [157] is a very clean signal for a dip structure around W = 1.68 GeV, seemingly confirming older data [158–161]. This raised the question on the origin of that structure, eventually indicating the appearance of a new narrow, possibly exotic, resonance state. The aim of the study was to extend our previous coupled-channels analysis of the γ p → ηp reaction by including the data from the new high-precision measurements [157]. The main question is whether the ηp reaction dynamics can be understood in terms of the established resonance states or whether a new state has to be introduced, thus conforming previous conjectures. A major issue for the analysis is unitarity and a consistent treatment of self-energy effects as visible in the total decay width of resonances. Since the latter are driven by hadronic interactions the analysis of photo-production data requires the knowledge of the hadronic transition amplitudes as well. Hence, a coupled-channels description as in the Giessen model (GiM) is an indispensable necessity. As discussed in every detail in [140] various relevant meson–baryon coupling constants were newly determined at the occasion of this work in large scale coupled-channels calculations. This gave rise to improved constraints on the interaction parameters and the derived resonance parameters, i.e. masses and widths. As an representative example we mention here the mass and width of the D13(1520) resonance. Our results confirm the values obtained by Arndt et al. [162]: mass M = 1516 ± 10 MeV and width Γ = 106 ± 4 MeV. It is interesting to note that the mass of this resonance deduced from pion photoproduction tends to be 10 MeV lower than the values derived from the pion-induced reactions [10]. The second D13(1900) state has a very large decay width. We associate this state with the D13(2080) two-star state, proposed by PDG. The results of the calculation of the η-photo production channel are shown in Fig. 28 in comparison with the experimental data. Our calculations demonstrate a very satisfactory agreement with the experimental data in the whole kinematical region. − The first peak is related to the S11(1535) resonance contribution. Similar to the π p → ηn reaction the S11(1650) and S11(1650) states interfere destructively producing a dip around W = 1.68 GeV. The coherent sum of all partial waves leads to the more pronounced effect from the dip at forward angles. We also corroborate our previous findings [139] where a small effect from the ωN threshold was found. We also do not find any strong indication for contributions from a hypothetic narrow P11 state with a width of 15–20 MeV around W = 1.68 GeV. It is natural to assume that the contribution from this state would induce a strong modification of the beam asymmetry for energies close to the mass of this state. While the beam asymmetry is less sensitive to the absolute magnitude of the various partial wave contributions it is strongly affected by the relative phases between different partial 152 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 28. Differential ηp cross section compared to MAMI data [157]. Source: From Ref. [139].

Fig. 29. Photon-beam asymmetry Σ compared to GRAAL data. Source: From Ref. [139].

waves. Thus even a small admixture of a contribution from a narrow state might result into a strong modification of the beam asymmetry in the energy region of W = 1.68 GeV. In Fig. 29 we show results for the photon-beam asymmetry Σ in comparison with the GRAAL data. One can see that even close to the ηN threshold where our calculations exhibit a dominant S11 production mechanism the beam asymmetry is non-vanishing for angles cos(θ) ≥ −0.2. This shows that this observable is very sensitive to very small contributions from higher partial waves. At W = 1.68 GeV and forward angles the GRAAL measurements show a rapid change of the asymmetry behavior. We explain this effect by a destructive interference between the S11(1535) and S11(1650) resonances which induces H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 153

Fig. 30. Diagrammatic structure of the tree-level interactions contributing to double-pion production on the nucleon.

the dip at W = 1.68 GeV in the S11 partial wave. The strong drop in the S11 partial wave modifies the interference between S11 and other partial waves and changes the asymmetry behavior. Note that the interference between S11(1535) and S11(1650) and the interference between different partial waves are of different nature. The overlapping of the S11(1535) and S11(1650) resonances does not simply mean a coherent sum of two independent contributions, but also includes rescattering (coupled- channel effects). Such interplay is hard to simulate by the simple sum of two Breit–Wigner forms since it does not take into account rescattering due to the coupled-channel treatment.

5.5. Double-pion production on the nucleon through resonances

In certain energy regions the πN → 2πN reaction accounts for up to 50% of the πN inelasticity as seen from Fig. 23. Therefore, this production channel had been included from the very beginning into the GiM approach. An improved and considerably extended description of double-pion production within our coupled channels scheme was started recently and first results are found in [7]. The inclusion of multi-meson configurations into a coupled channels approach is a highly non- trivial exercise in three-body dynamics. In view of the complexities physically meaningful approximations are necessary, retaining the essential dynamical aspects but making numerical calculations feasible. For that goal the ansatz used in [7] relies on an isobar description of intermediate two-pion configurations and their decay into the final double pion states on the mass shell. In Fig. 30 the tree-level interactions for two-pion production are displayed diagrammatically. The derived processes contributing to the T -matrix of double-pion production on the nucleon in that energy region are depicted in Fig. 31. This approach allows for the direct analysis of the 2πN experimental data. Since the corresponding Dalitz plots are found to be strongly non-uniform it is natural to assume that the main effect to the reaction comes from the resonance decays into isobar subchannels [163]. The most important contributions are expected to be from the intermediate σ N, π∆(1232), and 154 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 31. The processes contributing to double-pion production T -matrix are depicted diagrammatically: (a) and (b) production through the σ -isobar, (c) and (d) production through the ∆0-isobar. Symmetrization is indicated.

ρN states. The analysis of the πN → 2πN reaction would therefore provide very important information about the resonance decay modes into different isobar final states. The much richer baryon spectrum found in LQCD simulations [164,165], the functional DSE approaches [166], and the CQM results [167,168] that observed in scattering experiments indicates the necessity for broader investigations, including a larger class of reaction channels. Experimentally, most of the non-strange baryonic states have been identified from the analysis of the elastic πN data [162,169]. As pointed out in [167] the signal of excited states with a small πN coupling will be suppressed in the elastic πN scattering. As a solution to this problem a series of photoproduction experiments has been done to accumulate enough data for study of the nucleon excitation spectra. However, the results from the photoproduction reactions are still controversial. While recent investigations of the photoproduction reactions presented by the BoGa group [170] reported indications for some new resonances, not all of these states are found in other calculations [10]. This raises a question about independent confirmation for the existence of such states from the investigations of other reactions. Because of the smallness of the electromagnetic couplings, the largest contribution to the resonance self-energy comes from the hadronic decays. If the N∗ → πN transition is small, one can expect sizeable resonance contribution into remaining hadronic decay channels. As a result the effect from the resonance with a small πN coupling could still be significant in the inelastic pion–nucleon scattering: here the smallness of resonance coupling to the initial πN states could be compensated by the potentially large decay branching ratio to other different inelastic final states. Such a scenario is realized e.g. in the case of the well known N∗(1535) state. While the effect from this resonance to the elastic πN scattering is only moderate at the level of total cross section its contribution to the πN → ηN channel turns out to be dominant [140]. Since the πN → 2πN reaction could account for up to 50% of the total πN inelasticity this channel becomes very important not only for the investigation of the properties of already known resonances but also for the search for the signals of possibly unresolved states. Another important issue in studies of the 2πN channel is related to the possibility to investigate cascade transitions like ′ ′ N∗ → πN∗ → ππN, where a massive state N∗ decays via intermediate excited N∗ or ∆∗. It is interesting to check whether such decay modes are responsible for the large decay width of higher lying mass states. So far only the πN∗(1440) isobar channel has been considered in a partial wave analysis (PWA) of the πN → 2πN experimental data [163]. There are several complications in the coupled-channel analysis of 2 → 3 transitions. The first one is the difficulty to perform the partial-wave decomposition of the three-particle state. The second complication is related to the issue of three-body unitarity. For a full dynamical treatment of the 2 → 3 reaction the Faddeev equations have to be solved. This makes the whole problem quite difficult for practical implementations. Here we address both issues and present a coupled- channel approach for solving the πN → 2πN scattering problem in the isobar approximation. In this formulation the (π/ππ)N → (π/ππ)N coupled-channel equations are reduced to the two-body scattering equations for isobar production. Such a description accounts by construction for the full spectroscopic strength of intermediate channels and, in addition provides a considerable numerical simplification. Three-body unitarity leads to a relation between the imaginary part of the elastic scattering amplitude and the sum of the total elastic and inelastic cross sections by the well known optical theorem. Since in the isobar approximation the pions in the ππN channel are produced from the isobar subchannels, all contributions to the total πN → ππN cross section are driven by the isobar production. The optical theorem can be fulfilled, if all discontinuities in isobar subchannels are taken into account. In the present work the three-body unitarity is maintained up to interference term between the isobar subchannels. As a first application of our model we apply the developed approach for the study of the π −p → π 0π 0n data in the first resonance energy region assuming the dominant S11 and P11 partial wave contributions in the σ N and π∆ reaction subchannels. The main purpose here is to introduce the model and demonstrate the feasibility of treating two-pion dynamics in the framework of a large-scale coupled channels approach. For this aim, we restrict the calculations to the π 0π 0n channel, taking advantage of the fact that only isoscalar two-pion and π∆ isobar channels are contributing to the process. We emphasize that this restriction is not a matter of principle but is only for the sake of a feasibility study. In particular, this means that at this stage we do not consider the ρN state but postpone its inclusion into the numerical scheme to a later stage. Naturally, the results presented in the following are most meaningful for the energy region of the N∗(1440) Roper resonance. H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 155

− 0 0 2 Fig. 32. Reaction π p → π π n: differential cross section and bare phase space distribution (short dashed) as a function of mππ at fixed c.m. energies are compared to the Crystal Ball data (dashed) [176]. Source: From Ref. [7].

The first resonance energy region is of particular interest because of the sizable effect from N∗(1440). The dynamics of the Roper resonance turns out to be rich because of the two-pole structure reported in earlier studies [171,172], (see [162,173,174] for the recent status of the problem). The origin of the Roper resonance is also controversial. For example the calculations in the Jülich model explain this state as a dynamically generated pole due to the strong attraction in the σ N subchannel. At the same time the Crystal Ball collaboration finds no evidence of strong t-channel sigma-meson production in their π 0π 0 data [175]. From the further analysis of the π 0π 0 production the effect of the sigma meson was found to be small [176]. On the other hand, the pp → ppπ 0π 0 scattering experiment by the CELSIUS-WASA collaboration [177] finds the σ N decay mode of the Roper resonance to be dominant. The difference between the σ N and the π∆ production mechanism is seen in the invariant mass distributions, Fig. 32. Close to threshold the Crystal Ball data demonstrate a shift to the higher invariant masses for all energies up to 1.5 GeV whereas the three-body phase space tends to have a maximum at lower m2 . In the present calculations π 0π 0 the main contributions to the π −p → π 0π 0n reaction close to threshold are driven by t-channel pion exchange. This mechanism produces the invariant distributions which are shifted to the higher π 0π 0 invariant masses. However, the present calculations do not completely follow the experimental data at 1.303 and 1.349 GeV. In the region of the Roper resonance our calculations are able to describe the mass distributions rather satisfactorily. Also in this region the production strength is shifted to higher invariant masses m2 . At the same time a peak at small m2 π 0π 0 π 0π 0 becomes also visible. In the present calculations the fit tends to decrease the magnitude of the π∆(1232) production and compensate it by enhancing the strength into σ N. The obtained decay branching ratio of N∗(1440) for the σ N channel is about twice as large as for the π∆(1232). Both the small peak at small and the broad structure at large invariant masses are well reproduced indicating an important interplay between the σ N and π∆(1232) production mechanism. It is interesting that the isoscalar correlations in the ππ rescattering are also found to be necessary in order to reproduce the asymmetric shape of the mass distributions. Though the π∆(1232) production gives rise to a two-peak structure only the first one at small m2 is visible at energies 1.4–1.468 GeV. π 0π 0 Within the present calculation the second peak at high m2 is not seen because of the large σ N contributions. In the present π 0π 0 study π 0π 0n production is calculated as a coherent sum of isobar contributions. Though the interference effect are important they are found to be very small at the level of the total cross sections. We briefly discuss the reaction data base used in the calculations. To simplify the analysis the S11 and P11 πN partial waves are directly constrained by the single energy solutions (SES) derived by GWU(SAID) [162]. The experimental data on the π −p → π 0π 0n reaction are taken from [176]. These measurements provide high statistics data on the angular distributions dσ /dΩππ where Ωππ is the scattering angle of the ππ pair (or the final nucleon in c.m.). These data are accompanied by the corresponding statistical and systematical errors. No such information is available for the mass distributions in [176]. These observables are provided in a form of weighted events without systematic and statistical uncertainties. In the data analysis we impose the constraint that the integrated distributions must reproduce the total cross section of the π −p → 2π 0n reaction. We also have assigned about 10% error bars to each mass bin to perform the χ 2 minimization. From 1.46 GeV on ∗ = 3 the excitation of N (1520) starts to be important. Already at this energy a small contribution from the spin J 2 partial wave could modify the angular and mass distributions. Because of this reason we do not try to fit the data above 1.46 GeV. The calculated π 0π 0 differential cross sections are shown in Fig. 33 and compared to the Crystal Ball data as a function of the c.m. energy. The measurements demonstrate a rapid rise of the cross sections at the energies 1.3–1.46 GeV. We 156 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

0 0 − 0 0 Fig. 33. π π differential cross sections for the reaction π p → π π n at fixed tπ 0π 0 = cos θπ 0π 0 , shown in the upper left corner of the panels. Energy distributions for −0.95 ≤ tπ 0π 0 ≤ +0.95 are shown. The experimental data are from [176]. Source: From Ref. [7].

identify this behavior as an indication for the strong contribution coming from the Roper resonance. Indeed, the resulting πN inelasticities from the GWU(SAID) [162] analysis indicate that the P11 partial wave dominates the inelastic transitions at these energies. The inelasticity from the S31 channel is about three times less than that from P11. At the same time the ∆(1620) is strongly coupled to the 2πN final state through the π∆(1232) decay [10]. Since the contribution from the σ N subchannel is found to be about twice as large than that of π∆(1232), possible effects from the decay of the ∆(1620) resonance can safely be neglected in first approximation. We also allow the N∗(1535) resonance decays to the π∆(1232) and σ N isobar final states which are however found to be negligible. At energies close to 1.5 GeV the obtained cross section slightly overestimates the experimental data at backward and underestimates them at forward scattering angles. This is a region where the N∗(1520) starts to play a dominant role. We conclude that the contribution from the D13 partial wave should be included for the successful description of the data at 1.5 GeV. The invariant π 0π 0 mass distributions play a crucial role in the separation of the isobar contributions. The π −p → π 0π 0n reaction close to threshold is dominated by the σ N production due to the t−channel pion exchange. The nucleon Born term contribution to the π∆(1232) channel is found to be less significant. For the decay branching ratios of N∗(1440) we obtain N(1440) = +4 N(1440) = +5 Rσ N 27−9 % and Rπ∆(1232) 12−3 %. N(1440) = +7 N(1440) = +8 The comparison of our results with the parameters extracted by the BoGa group Rσ N 17−7% and Rπ∆(1232) 21−8% demonstrates that in spite of the visible difference in the central values these quantities could still coincide within their error bars. The extended analysis of the ππN, which includes higher partial waves, would help to reduce the uncertainties of the extracted resonance properties. Already some time ago, also the EBAC group analyzed the same data, but failed to describe the mass distribution. As seen in [137], the final result resembles still very closely the phase space distribution of the non-interacting particle, indicating a lack of interactions in that earlier attempt. The total two-pion production cross section, however, are described rather well by the EBAC analysis, as seen in Fig. 34. An interesting side aspect seen in that figure is to note that the direct πN → ππN production channel contributes quite differently to the cross sections of the various reactions. The double-pion channels are also of interest for χEFT. In [178] the Jülich group has been studying pion production off nucleons in the heavy baryon chiral perturbation theory to third order in the chiral expansion aiming at the determination of the low-energy constants. Most of the at that time available differential cross sections and angular correlation functions at low pion incident energies could be described together with total cross sections at higher energies. The contributions from the one loop graphs were found to be essentially negligible once the dominant terms at second and third order are related to pion–nucleon scattering graphs with one pion were added. An interesting aspect is that the ππN channels provide the possibility of extracting the pion–pion S-wave scattering lengths which otherwise is hard to access.

5.6. Discussion of strangeness and resonance physics on the nucleon

Coupled channels approaches like the one discussed in this section are a valuable tool for understanding excited states of the nucleon. The structures observed in πN and γ N reactions have been described here in the GiM coupled channels H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 157

Fig. 34. πN → ππN total cross sections obtained by the EBAC approach. The dotted line indicates the results if the direct πN → ππN interaction kernel is omitted in the coupled channels analysis. Source: From Ref. [137].

Fig. 35. (Color online) Comparison of the GiM N∗ resonance level scheme in various partial waves to the recent PDG compilation [10].

approach as an representative example. The GiM approach is based on a phenomenological field theory describing the baryons, mesons, and their interactions by a Lagrangian density. By construction the fundamental symmetries of QCD, including also chiral symmetry, are conserved. Resonance and background contributions are generated consistently out of the tree-level interactions obtained from the Lagrangian. The scattering amplitudes are determined by linear system of coupled equations which is solved numerically in K-matrix approximation and partial wave representation. The gauge- invariant description of high-spin resonances was discussed in detail. Applications to selected reaction channels have been presented, ranging from single pion, eta, and kaon production to double-pion production. The spectroscopic results are summarized in Fig. 35 where the GiM spectrum of resonances is compared to the recent PDG resonance compilation. Many-particle decay channels will be an important source for additional spectroscopic information, possibly revealing features of baryon resonances which are not seen in single-meson spectroscopy. Nucleon resonance studies in free space are pursued at various laboratories with recent major activities at ELSA (Bonn) and MAMI (Mainz) in Germany and JLab in the US. A large number of experimental and theoretical groups are contributing to this key activity of modern hadron physics. Over the years, ample evidence on the spectroscopy of hadron resonances in the (u, d, s) sector has been collected. The identification of resonances requires large scale meson–baryon partial wave coupled channels calculations as e.g. practiced by the covariant field theoretical Giessen Model (GiM) [5–7]. With its strong phenomenological components through adjusting interactions and masses of the underlying Lagrangians by fits to data the GiM is representative for the existing coupled channels approaches which are capable of large scale investigations. The 158 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 36. Identification of baryon resonances from free space meson–nucleon scattering data. The iterative approach eventually connecting the extracted resonances to (L)QCD is indicated schematically.

connection of the results to (L)QCD requires an iterative approach, eventually converging to a (self-) consistent picture. The principle of such a scheme is indicated in Fig. 36. Resonance physics is the central topic also in the charm (c) and bottom (b) mass regions. A prominent recent example for the surprises of resonance physics is the unexpected appearance of the so-called (X, Y , Z) states in the charmonium spectrum, observed independently at BELLE and BESS, which are pending a safe theoretical explanation. Resonance physics will continue to be a central goal also for the research program at the future facilities. For example, perspectives and the physics potential of resonance physics by antiproton annihilation reactions at the coming up PANDA experiment at FAIR was discussed recently, as summarized in Ref. [179].

6. Strangeness production in coherent reactions with elementary probes

The resonances excited in reactions on the nucleon are an important source for any other particle production processes, either in reactions with elementary probes or with heavy ions. Investigations with elementary probes are an important test ground for production reactions with complex particles because in those cases we do not have direct access to the dynamics of the elementary strangeness changing vertices. In order to describe quantitatively the hypernuclear cross section obviously full control on the contributing processes is required. In this section coherent reactions are considered in which a single nucleon is converted into a bound hyperon and, due to strangeness conservation of strong interactions, a K + meson leaves the nucleus. Thus, a d quark is substituted by a s quark. Such processes have been studied in a series of papers investigating hypernuclear production by various elementary probes. In [180,181] proton-induced (p, K +) reactions with have been considered. The photo-production by (γ , K +) reactions was the topic in [182,183]. The work in Ref. [184] was + + devoted to A(π , K )ΛA reactions on nuclear targets A. Besides the interest on strangeness production on nuclei, in this section we illustrate the intimate connection of baryon resonances and hypernuclear production experiment on the example of (π +, K +) reactions. Together with complementary (K −, π −) reactions, that type of reaction has been a standard method for experimental research on hypernuclei, as found in the review of Gal et al. [18]. A not unimportant side aspect of this section is to illustrate on a physically relevant example the intimate connection between the in-medium physics discussed in Section2 and the resonance physics on the free nucleon, Section5. A fully quantum mechanical approach to coherent strangeness production was developed a few years ago in [184]. The kinematical properties of the (K −, π −) reaction allow only a small momentum transfer to the nucleus (at forward angles), thus there is a large probability of populating Λ-substitutional states, in which the Λ assumes the same orbital angular momentum as that of the replaced neutron. On the other hand, in the (π +, K +) reaction the momentum transfer is larger than the nuclear Fermi momentum. Therefore, this reaction can populate states with the configuration of an outer neutron hole and a Λ hyperon in a series of orbits covering all the bound states supported by the binding potential. The richness of the spectroscopic information on Λ bound states in the (π +, K +) reaction was demonstrated in the experiments performed at the Brookhaven National Lab and National Laboratory for High Energy Physics (KEK) (see, e.g., Ref. [17] for a comprehensive review). Furthermore, although the reaction cross section of the strangeness production, via the (π +, K +) process, is smaller than that of the strangeness exchange reaction (K −, π −), the higher luminosity of pion beams makes experiments more feasible. H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 159

Fig. 37. Graphical representation of our model to describe the (π +, K +) reaction. The elliptic shaded area represents the optical model interactions in the incoming and outgoing channels.

In the experimental studies reported in Refs. [108,106], this reaction has been used to carry out the spectroscopic 12 51 89 investigations of hypernuclei ranging from light mass Λ C, to medium mass Λ V and Λ Y with the best resolution (∼1.6 − −1.7 MeV) achieved with the spectrometer at KEK. This experiment has succeeded in clearly observing a characteristic fine structure in heavy systems by precisely obtaining a series of Λ single-particle states in a wide range of excitation energies. Most of the theoretical models have been describing the (π +, K +) reaction by employing a non-relativistic distorted wave impulse approximation (DWIA) framework [185] (see also Ref. [186] for a comprehensive review of these models). In these calculations, the Λ bound states are generated by solving the Schrödinger equation with Woods–Saxon or harmonic oscillator potentials. However, for processes involving momentum transfers of typically 300 MeV/c or more, a non-relativistic treatment of the corresponding wave functions may not be adequate as in this region the lower component of the Dirac spinor is no longer negligible in comparison to its upper component as pointed out, e.g., in Ref. [182]. + + In [184] the A(π , K )ΛA reaction was studied within a fully covariant model by retaining the field theoretical structure of the interaction vertices and by treating the baryons as Dirac particles. In this model, the kaon production proceeds via the collision of the projectile pion with one of the target nucleons. This excites intermediate baryon resonance states (N∗) decay into a kaon and a Λ hyperon. The hyperon is captured in the respective nuclear orbit while the kaon rescatters onto its mass ∗ 1 − ∗ 1 + shell (see Fig. 37). The intermediate resonance states included are the previously cited N (1650)[ 2 ], N (1710)[ 2 ], and ∗ 3 + + N (1720)[ 2 ] states which have dominant branching ratios for the decay to the K Λ channel [10]. Terms corresponding to the interference among various resonance excitations are included in the total reaction amplitude.

+ + 6.1. Covariant model for the A(π , K )ΛA reaction

The structure of the resonance model for (π +, K +) reactions is described in Fig. 37. The N∗ corresponds to the − + + ∗ [ 1 ] ∗ [ 1 ] ∗ [ 3 ] N (1650) 2 , N (1710) 2 , and N (1720) 2 baryon resonance intermediate states. Terms corresponding to interference between various amplitudes are retained. The elementary process involved in this reaction is shown in Fig. 38. The model has only s-channel resonance contributions. In principle, Born terms and the resonance contributions in u- and t-channels should also be included in description of both the processes depicted by Figs. 37 and 38. These graphs constitute the non-resonant background terms. Their magnitudes depend on particular models used to calculate them and also the parameters used in those models [136]. The contributions of the background terms are about 15%–20% of the resonance terms within a coupled-channel K matrix model for the energies of interest (≈1 GeV/nucleon; the corresponding invariant mass is about 1.7 GeV, see [136]). The effect of the background terms on the total production cross sections of the π −p → K 0Λ reaction at the beam energies can be indirectly inferred from the calculations reported in Ref. [141], where the total production cross sections at invariant mass around 1.7 GeV are dominated by the contributions of S11 and P11 resonance terms. Therefore, the magnitudes of the background terms are likely to be limited to about 10%–20% of those of the resonance terms. Our results may be uncertain by 10%–20% due to omission of the background contributions.

6.2. Interaction Lagrangians

For the interaction terms of the spin-1/2 resonances, vertices of pseudoscalar (PS) or pseudovector (PV) form have been used. The pseudovector coupling is consistent with the chiral symmetry requirement of QCD. In contrast to that, the pseudoscalar one does not have this property, but it is easier to calculate. The couplings are in both cases fixed in such a 160 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 38. Tree diagram for the elementary process of pion-induced strangeness production via resonance excitation and decay on a single nucleon.

Table 5 Coupling constants for various vertices used in the calculations. Vertex Coupling constant (g) N∗(1650)Nπ 0.81 N∗(1650)ΛK + 0.76 N∗(1710)Nπ 1.04 N∗(1710)ΛK + 6.12 N∗(1720)Nπ 0.21 N∗(1720)ΛK + 0.87

way that they are equal on-shell; for off-shell cases, their difference is suppressed due to the denominator of the resonance propagator. It is, therefore, arguable which Lagrangian to use. The best approach would be to introduce a mixing parameter. To avoid the introduction of additional model parameters due to a PS–PV mixing for the interaction Lagrangians, we use the convention of either choosing the PS or the PV couplings for these vertices as done in Ref. [136]. The pseudoscalar interaction Lagrangians for the spin-1/2 resonances are given by

PS ∗ = − ∗ ¯ ∗ ⃗ · ⃗ + L NN gπNN ψN Γ (τ φπ )ψN h. c. , (83a) π 1/2 PS ∗ = − ∗ ¯ ∗ + LN K gN ΛK ψN Γ φK ψΛ h. c. , (83b) 1/2 Λ where the Γ takes care of parity conservation. We use

{1 for odd parity Γ = iγ 5 for even parity , and h. c. in Eqs. (83) denotes the Hermitian conjugate. The pseudovector Lagrangians involve the derivative of the pion wave function rather than the wave function itself. This introduces an additional mass dimension, which is taken care of by a ‘‘rescaling’’ of the coupling constant. It also ensures the matching of the on-shell behavior the two types of Lagrangians. The pseudovector Lagrangians are given by

∗ PV gπNN µ ∗ = − ¯ ∗ ⃗ · ⃗ + LπNN ψN γ Γ ∂µ(τ φπ )ψN h. c. , (84a) 1/2 mN∗ ± mN ∗ PV gN KΛ µ ∗ = − ¯ ∗ + LN KΛ ψN γ Γ ∂µφK ψΛ h. c. , (84b) 1/2 mN∗ ± mΛ where Γ is given by

{i for odd parity Γ = γ 5 for even parity , and the upper and lower signs are used for even and odd parity resonances, respectively. The spin-3/2 resonance Lagrangians are given by

gπNN∗ µ ∗ ¯ ⃗ L NN = ψ ∗ ∂µ(τ⃗ · φπ )ψN + h. c. , (85a) π 3/2 N mπ gN∗KΛ µ ∗ ¯ LN K = ψ ∗ ∂µφK ψΛ + h. c. (85b) 3/2 Λ N mK The values and signs of the various coupling constants have been taken from Ref. [180] and are shown in Table5. These parameters describe well the associated K +Λ production in proton–proton collisions within a similar resonance picture. All the pion-resonance-kaon vertices that are of interest in this paper are involved in this reaction. Thus the vertex parameters used by us inherently describe the elementary process shown in Fig. 38. H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 161

6.3. Resonance propagators

The two interaction vertices of Fig. 38 are connected by a resonance propagator. For the spin-1/2 and spin-3/2 resonances the propagators are given by µ γµp + m D = i (86) 1/2 2 2 p − (m − iΓN∗ /2) and, as discussed below in Section7, a tensorial structure is used for the spin-3/2 Green’s functions, λ µν γλp + m D = −i Pµν , (87) 3/2 2 2 p − (m − iΓN∗ /2) respectively. In Eq. (87) we have defined 1 2 1 Pµν = ηµν − γ µγ ν − pµpν + (pµγ ν − pν γ µ) . (88) 3 3m2 3m

ΓN∗ in Eqs. (86) and (87) is the total width of the resonance. It is introduced in the denominator term to account for the finite lifetime of the resonances for decays into various channels. This is a function of the center of mass momentum of the decay channel, and it is taken to be the sum of the widths for pion and rho decay (the other decay channels are considered only implicitly by adding their branching ratios to that of the pion channel). We have not introduced any correction in the resonance propagators to account for the nuclear medium effects as no major change is expected in the results due to these effects. The medium correction effects on the widths of the s- and p-wave resonances, which are the dominant contribution to the cross sections investigated here, are not substantial. The reason for this is that resonances occur only as intermediate states which implies an integration over their respective spectral distributions.

6.4. Nuclear model

The spinors for the final bound hypernuclear state (corresponding to momentum pΛ) and for the intermediate nucleonic state (corresponding to momenta pN ) are required to perform numerical calculations of various amplitudes. We assume these states to be of pure-single particle or single-hole configurations with the core remaining inert. In experimental measurements, however, core excited states have also been detected (see, e.g., [17]). A covariant description of the core polarization can, in principle, be achieved by following the method discussed, e.g., in Ref. [187]. This procedure is somewhat tedious and was out of the scope of the present review. We concentrate on those transitions which involve pure single- particle and single-hole states. The spinors in momentum space are obtained by the Fourier transformation of the corresponding coordinate space spinors which are solutions of the Dirac equation with potential fields consisting of an attractive scalar part (Vs) and a repulsive vector part (Vv) as discussed in Section3. For simplicity, the self-consistent scalar and vector mean-field self-energy potentials were replaced for these studies by Woods–Saxon form factors which are a suitable parametrization. In turn, Dirac equations are solved numerically leading to the eigenenergies and wave functions of the nucleon and hyperon bound states. In order to obtain a quantitative description, a realistic treatment of the pion–nucleus initial state interactions is of primary importance. The strong coupling to N∗ modes introduces a strong absorption in pion–nucleus scattering as discussed in detail in the textbook of Feshbach [85]. At the energies at question, the pion–nucleus scattering waves are well described in eikonal approximation [184]. In contrast, the kaon–nucleus weak final state interactions are of minor relevance and can safely be neglected and the outgoing kaon waves are chosen conveniently as plane waves. For details we refer to [184].

6.5. Hypernuclear spectra from (π +, K +) reactions

In [184] an elaborate discussion on the close connection of nuclear structure and reaction physics is found. The major 41 89 result is the rather good description of the Λ V and Λ Y spectra on a quantitative level. The results are shown in Fig. 39. The spectral distributions are reproduced very satisfactory even by magnitude. Slight deviations are seen between the major Λ single particle peaks because the core excitation contributions – see Section4 – were neglected for that analysis. The data were produced in the (π +, K +) reaction on 51V and 89Y targets at the beam momentum of 1.05 GeV. In Fig. 39 the experimental energy resolution was taken into account by a convolution with a Gaussian of the appropriate width. 51 [ −1 ] In ΛV the series of levels are obtained by the following configurations and binding energies (Ebind): (f7/2 )N , (s1/2)Λ − = [ −1 ] + = [ −1 ] − = (3 , Ebind 19.75 MeV), (f7/2 )N , pΛ (4 , Ebind 11.75 MeV), and (f7/2 )N , dΛ (5 , Ebind 3.75 MeV). These states 89 make the largest contributions to the corresponding cross sections. On the other hand, the levels in ΛY are obtained with the configuration where the neutron hole state corresponds to the g9/2 orbit with the Λ in the 0s, 0p, 0d and 0f orbitals. [ −1 ] + = [ −1 ] − = The configurations of these levels are: (g9/2)N , (1s)Λ (4 , Ebind 23.6 MeV), (g9/2)N , (1p)Λ (5 , Ebind 16.5 MeV), [ −1 ] + = [ −1 ] − = (g9/2)N , (1d)Λ (6 , Ebind 10.0 MeV), and (g9/2)N , (1f)Λ (7 , Ebind 2.3 MeV). 162 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

+ + 41 89 Fig. 39. Total cross section for kaon production in a (π , K ) reaction on V at Plab = 1.05 GeV/c (upper panel) and Y at plab = 1.05 GeV/c. Results of 41 89 the Giessen resonance model are compared to KEK data [106]. As indicated, s-, p-, and d-orbitals in Λ V and up to f-orbitals in Λ Y are resolved. Source: From Ref. [184].

For both nuclei the calculated spectral distributions reproduce the overall global trends of the data of Ref. [106] reasonably 51 89 well. The calculations reproduce the experimentally observed three and four clear peaks in the ΛV and ΛY spectra, respectively. However, looking more closely we note that the theory overestimates the experimental cross sections for the (0s)Λ orbitals somewhat in both cases. For these states, the corresponding binding energies (Ebind) are quite large and for these strongly mismatched cases, distortion effects could play a more significant role. In both cases, one observed some minor peaks which fill up the gaps between the major peaks. These peaks correspond to core excited states, discussed in Section4, and the states corresponding to other, less dominant, members of the configurations mentioned above. Clearly, a quantitative description of the spectral shapes of the heavier mass Λ hypernuclei requires proper consideration of the core excitation and mixing of states of different parity.

7. Gauge theory of high-spin fermionic fields

7.1. Gauge properties of spin-3/2 fields

The description of pion- and photon-induced reactions in the resonance energy region is faced with the problem of a proper treatment of high spin states. In 1941 Rarita and Schwinger (R–S) suggested a set of equations which a high spin field operator should obey [188]. Another formulation has been developed by Fierz and Pauli [189] where an auxiliary field concept is used to derive subsidiary constraints on the field function. Regardless of the procedure used the obtained 3 Lagrangians for free higher-spin fields turn out to be always dependent on arbitrary free parameters. For the spin- 2 fields this issue is widely discussed in the literature (c.f. [190–192]). 3 It is well known that the wave equation for the free spin- 2 field [188] depends on one free parameter A (see e.g. [193]). The commonly used Rarita–Schwinger theory [188] corresponds to the special choice A = −1. While the so-called Pascalutsa- coupling removes the unwanted degrees of freedom from the Rarita–Schwinger propagator it leaves the problem unsolved in the more general case A ̸= −1 resulting in the appearance of ‘off-shell’ components, for example in the πN scattering 3 amplitude. Hence, further investigations of the general properties of the interacting spin- 2 fields are of great importance. H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 163

In [190] we have discussed the origin of this problem. For a solution, two alternative approaches were proposed: • 3 to construct a coupling which includes higher derivatives of the spin- 2 field. • to generalize the gauge invariant interaction to arbitrary (real) values of the gauge parameter A.

In the latter case the obtained Lagrangian depends on one free parameter which also appears in the free field formalism. However, the physical observables should not depend on this parameter. Hence, the matrix element corresponding to the πN scattering at tree level does not contain an off-shell background. The Rarita and Schwinger constraints [188] on the free 3 spin- 2 field are ν γ ψν (x) = 0, ν ∂ ψν (x) = 0, (89) provided that also the Dirac equation (p/ − m)ψν (p) = 0 is fulfilled. In a consistent theory the set of Eqs. (89) should follow 3 from the equation of motion obtained from the corresponding Lagrangian. The Lagrangian of the free spin- 2 field can be written in a general form as follows (see, e.g., [193] and references therein)

3 2 = ¯ µν L0 ∆µ(x) Λ ∆ν (x), (90) 3 µν where ∆ν (x) stands for the spin- 2 field and the Λ -operator is

Λµν = (i/∂ − m)gµν + iA(γ µ∂ν + γ ν ∂µ) i + (3A2 + 2A + 1)γ µ/∂γ ν 2 + m (3A2 + 3A + 1)γ µγ ν , (91) ̸= − 1 where A is the aforementioned arbitrary free parameter, subject only to the restriction A 2 . The propagator of the free 3 spin- 2 field can be obtained as a solution of the equation, e.g. in momentum space,

ρσ Λµρ (p) g Gσ ν (p) = gµν . (92)

3 1 1 2 2 2 The propagator Gσ ν (p) can be written as an expansion in terms of the spin projection operators Pµν (p), P11; µν , P22; µν , 1 1 2 2 P12; µν (p), and P21; µν (p)[190]. The first three operators correspond to different irreducible representations of spin-vector 1 whereas the last two account for a mixing between two spin- 2 representations. Without going too deep into the mathematical details, we limit the discussion here to the consequences for interaction ¯ νµ µν µν µ ν vertices. A commonly used ∆Nπ-coupling is Lint ∼ ψN θ(z) ∆µ∂ν π with θ (z) = g + z γ γ . The free parameter z is used to control the off-shell contributions to the interaction vertex but does not affect the pole term. In order to remove the dependence on z (or, likewise, A) we eliminate the unwanted degrees of freedom by using a gauge-invariant coupling to the 3 2 spin- 2 field as explained in [190]. The modified ∆Nπ interaction Lagrangian can be written as follows g∆Nπ ¯ µν LP = ψN (x)γ5γµT∆ (x)∂ν π(x) + h. c., mπ mN µν 1 µνρσ ( ) T (x) = ϵ ∂ρ ∆σ (x) − ∂σ ∆ρ (x) , (93) ∆ 2 µνρσ where ϵ is the fully antisymmetric Levi-Civita tensor. The tensor Tµν (x) is invariant under the gauge-transformations ∆ν (x) → ∆ν (x) + ∂ν ξ(x) where ξ(x) is an arbitrary spinor field. Hence, Tµν (x) behaves like a conserved current with the µν constraint ∂µ T∆ (x) = 0. The coupling defined in Eq. (93) guarantees that the so-called off-shell background does not 3 contribute to the physical observables provided that the free spin- 2 propagator is chosen in the special form corresponding to A = −1. This, however, does not hold in the general case for arbitrary values of A. The problem reported above can be solved in different ways. The straightforward solution is to use a coupling with higher 3 3 P 2 order derivatives of the spin- 2 field which explicitly involves the µν (p) projection operator:

[ 3 ] g∆Nπ ¯ (x) P 2 ( )∆ν (x) µ (x) + h. c. (94) 4 ψN □ µν ∂ ∂ π mπ mN 3 P 2 3 The use of µν (∂) ensures that only the spin- 2 part of the propagator contributes and the d’Alembert-operator guarantees that no other singularities except the mass pole term (p2 − m2)−1 appear in the matrix element. Note, that the coupling written in the form of Eq. (94) restores the invariance of the full Lagrangian under the point-like transformations ∆µ → ν ∆µ + z γµγ ∆ν .

2 We omit isospin indices. 164 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

To keep the interaction term in the full Lagrangian as simple as possible another coupling is of advantage

g∆Nπ LI = mπ mN ¯ [ η ] ν × ψN (x) Γνη (A, ∂) ∆ (x) ∂ π(x) + h. c., (95) with the modified vertex operator Γνη (A, ∂) depending on the parameter A:

µ σ ρ Γνη (A, ∂) = γ5 γ ϵµνρσ θ η (A) ∂ , (96)

A + 1 θσ η(A) = gσ η − γσ γη. (97) 2

In momentum space at A = −1 the vertex function Eq. (96) reduces to that suggested by Pascalutsa. The θµν (A)-operator has a simple physical meaning: it relates the Rarita–Schwinger theory to the general case of arbitrary A. Hence, the R–S RS propagator, can be obtained from the general propagator by means of the transformation Gµν (p) = θµρ (A) Gρσ (p, A) θσ ν (A). Using the coupling Eq. (95) the final result for the matrix element of πN scattering is independent on the none-pole 1 spin- 2 terms in the full propagator

[ ρσ † ] Γµρ (A, p∆) G (p∆, A) Γσ ν (A, p∆) + 2 3 p/ m p∆ 2 = Pµν (p ) (98) 2 − 2 2 ∆ p m mN and coincides with that obtained for the case A = −1. The coupling Eq. (95) can be written in a more compact form which does not contain the Levi-Civita tensor explicitly

ig∆Nπ LI = 4 mπ mN ¯ [ σ ρν ρ η ] ν × ψN (x) γ θσ η(A)∂ ∆ (x) ∂ π(x) + h. c., (99) σ ρν σ ρ ν σ ρ σ ρ where γ = {γ , γ } and γ = [γ , γ ] and θσ η(A) is defined in Eq. (97). The full Lagrangian for the interacting ∆Nπ fields can be written in the form

3 3 π N 2 = 2 + + + L L0 LI L0 L0 , (100) π = + 2 N = ¯ − where L0 π(□ m )π and L0 ψN (i/∂ m)ψN stand for the free Lagrangians of pion and nucleon fields, respectively. 3 3 L 2 L The free spin- 2 Lagrangian 0 and ∆Nπ coupling I are given by expressions Eqs. (90) and (95). The Lagrangian equation (100) depends on one arbitrary parameter A which points to the freedom in choosing the ‘off-shell’ content of the theory. 3 2 Although L0 contains one free parameter the physical observables should not depend on it. Without going into the details, we mention that similar conclusion can be made for the electromagnetic coupling and refer to Ref. [190] for details. Summarizing, it can be said that we could show that the gauge-invariant ∆Nπ coupling, originally suggested by 3 3 Pascalutsa for spin- 2 fields, removes the off-shell degrees of freedom only for a specific choice of the spin- 2 propagator but 3 [ 1 ⊗ ] not in the general case. In the general case the spin- 2 propagator contains a term associated with the 2 1 1 irreducible 2 representation. We have shown that the problem can be solved by introducing higher order derivatives to the interaction Lagrangian or by generalizing the original ∆Nπ coupling suggested by Pascalutsa. In the latter case the full Lagrangian of the interacting ∆Nπ fields depends on one free parameter which reflects the freedom in choosing an off-shell content of the theory.

7.2. Gauge properties of spin-5/2 fields

5 The spin- 2 fields are rarely studied. First attempts were made in [194,195]. The authors of [195] deduced an equation of motion as a decomposition in terms of corresponding projection operators with additional algebraic constraints on parameters of the decomposition. The free particle propagator is a central quantity in most of the calculations in quantum field theory. In [195] the authors 5 deduced a spin- 2 propagator written in operator form. In practical calculations, however, one needs an explicit expression of 5 the propagator. An attempt to construct a propagator only from the spin- 2 projection operator has been made in [196,197]. 5 We demonstrated that such a quantity is not consistent with the equation of motions for the spin- 2 field. In addition, Hermiticity can be violated, as was pointed in [198]. Clearly, it is important to derive the propagator and investigate its properties in detail. To the best of our knowledge our study was the first attempt in that direction. Hence, the aim of the 5 work was to deduce an explicit expression for the spin- 2 propagator and study its properties. Guided by the properties of 3 5 the free spin- 2 Rarita–Schwinger theory one would expect the equation of motion for the spin- 2 field has two arbitrary free 3 1 5 parameters which define the non-pole spin- 2 and- 2 contributions to the full propagator. The coupling of the spin- 2 field to H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 165 the (e.g.) pion–nucleon final state is therefore defined up to two off-shell parameters which scale the non-pole contributions to the physical observables. Hence, one can ask whether such an arbitrariness can be removed from the theory. The possibility to construct consistent higher-spin massless theories has already been pointed out by Weinberg and Witten a while ago [199]. As we demonstrated in [190] the demand for gauge-invariance may not be enough to eliminate the extra degrees of freedom at the interaction vertex. The problem appears when the theory does not have a massless limit. 5 However, a coupling which removes non-pole terms from the spin- 2 propagator can be easily constructed by using higher 3 order derivatives. A corresponding interaction Lagrangian has been deduced in [190] for the case of spin- 2 fields and can be easily extended to higher spins too, as exercised in [198]. The field function of higher spins in a spinor–tensor representation is a solution of the set of equations suggested by Rarita and Schwinger in [188]. In a consistent theory the description of the free field is specified by setting up an appropriate L 5 Lagrange function (ψµν , ∂ρ ψµν ). The spin- 2 Lagrangian in the presence of the auxiliary spinor field ξ(x) can be written in the form

L = L(1) + L(2) + L(aux), (101) where the lengthy and mathematically involved expressions for the three pieces are found in Ref. [198]. An important observation is that the Lagrangian in Eq. (101) in general depends on only three independent real parameters a, b, and c. By variations with respect to ψµν and ξ two equations of motion are obtained which in momentum space are given as ( ) (1) + (2) ρσ + µν = Λµν;ρσ (p) Λµν;ρσ (p) ψ (p) c m g ξ(p) 0, (102)

ρσ m c g ψρσ (p) + B(a, b, c) (p/ + 3m) ξ(p) = 0, (103) (1) (2) where the explicit forms of the operators Λµν;ρσ (p), Λµν;ρσ (p) are found in [198]. Here, it is of interest that the operator (1) (1) µν = 5 µν = Λµν;ρσ (p) would give an equation of motion Λµν;ρσ (p)ψ 0 for the spin- 2 fields provided g ψµν 0, where the later [ (1) ]−1 property is assumed a priori. However, the corresponding inverse operator Λµν;ρσ (p) has additional non-physical poles 1 µν = in the spin- 2 sector. This indicates that the constraint g ψµν 0 should also follow from the equation of motion and cannot (2) 1 be assumed a priori. The second operator Λµν;ρσ (p) acts only in the spin- 2 sector of the spin–tensor representation. This can be checked by a direct decomposition of the operator in terms of projection operators [198]. The same conclusion can be 5 3 (2) 2 2 = drawn from the observation that Λµν;ρσ (p) is orthogonal to all Pρσ ;τδ(p), Pij;ρσ ;τδ(p) projection operators, where i, j 1, 2. 1 3 1 Hence the parameter b is related only to the spin- 2 degrees of freedom whereas a scales both spin- 2 and - 2 ones. Of particular interest for spectroscopic research is the coupling of resonances to meson–nucleon channels. In the case 5 of the spin- 2 field in the spinor–tensor representation we deal with a system (ψµν , ξ) which contains auxiliary degrees of freedom. The question arises whether the nonphysical degrees of freedom could be eliminated from physical observables. 5 ∗ Here we consider a simple case of spin- 2 resonance contribution to πN scattering. The corresponding πNN 5 coupling can 2 be chosen as follows

gπNN∗ L = I 2 4mπ [ (ψρσ )] × (ψ¯ (x), 0) Γ P ∂µ∂ν π(x) N µν;ρσ ξ + h. c., (104) ( ¯ ) where the nucleon field is written as ψN (x), 0 which implies the absence of auxiliary fields in the final state. The operator

(1 0) P = 0 0

5 5 projects out the spin- 2 field and ensures that there is no coupling to ξ. Hence, only the spin- 2 component of the propagator 5 2 Gµν;ρσ (p) contributes to physical observables at any order of perturbation theory. In [198] we could demonstrate that the inclusion of auxiliary degrees of freedom in the vector field does not affect the physical observables. To the best of our knowledge this statement is not generally proven for the (ψµν , ξ) system beyond the perturbation expansion. The reason is 5 that the equation of the motion for massive spin- 2 field in the spinor–tensor representation is defined only in the presence of an auxiliary field. This is unlike the case of the vector field where auxiliary degrees of freedom can be removed by proper field transformations. These degrees of freedom contribute due to ψµν − ξ mixing. The mixing takes place only between the 1 spin- 2 sector of the spinor–tensor and the auxiliary spinor fields. One may therefore hope that the use of a coupling which 1 suppresses the spin- 2 contributions would also prevent the appearance of the auxiliary degrees of freedom in the physical observables in the non-perturbative regime. The solution to the problem is following closely the results for spin-3/2 fields, presented in [190]. According to our previous findings the interaction vertex fulfills the condition γ · Γ = Γ · γ = 0. With this constraint one finds that only 166 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

5 P 2 5 the µν;ρσ (q) projector fulfills the desired property [198], ensuring that only the spin- 2 part of the propagator contributes. The formalism also guarantees that no other singularities except the mass pole term (p2 − m2)−1 appear in the amplitude. Thus, the physical observables no longer depend on the arbitrary parameters a and b of the free Lagrangian. Finally, the πN scattering amplitude reads ( )2 gπNN∗ ′ M = u¯ (p ) 2 N mπ [ 4 ] ( 2 ) 5 q ′ ′ P 2 (q) u (p) k µ k ν kλ kτ (105) 2 µν;λτ N . mR

5 Thus, the spinor–tensor representation is the appropriate formalism for a gauge invariant description of (free) spin- 2 fields. We have shown that the Lagrangian in general depends on three arbitrary parameters; two of them are associated 3 1 with the lower spin- 2 and - 2 sector of the theory whereas the third one is linked to the auxiliary field ξ. We have deduced a free propagator of the system in form of a 2×2 matrix in the (ψµν , ξ) space. The diagonal elements stand for the propagation 5 − of the spin- 2 and ξ fields whereas the non-diagonal ones correspond to ψµν ξ mixing. The mixing takes place between the 1 spin- 2 sector of the spinor–tensor representation and an auxiliary spinor field. An important result was that the auxiliary degrees of freedom do not contribute to the physical observables calculated within the perturbation theory provided there is ∗ no coupling to ξ. As an application to hadron spectroscopy, the πNN 5 interaction vertex was discussed. Gauge invariance was 2 obtained by constructing a coupling with higher order derivatives. In the latter case the amplitude of the πN scattering does not depend on the arbitrary parameters of the free Lagrangian. The suggested coupling is generalized to the Rarita–Schwinger fields of any half-integer spin.

8. Heavy ion collisions as a probe for strangeness in nuclear matter

Research on the equation of state (EoS) of strongly interacting matter belongs to central topics in nuclear and hadron physics ever since the advent of high energy beams of heavy ions. A recent extended review on these topics is found in [200]. Concrete investigations on the nuclear EoS were initiated in the early 1970s with the realization of the first relativistic ion beams at BEVALAC in Berkeley [201–203], followed by more selective heavy-ion experiments in centrality, particle identification and phase space reconstruction [204–210]. These measurements gave first hints on collective particle flow [211–215]. They allowed first interpretations on the nuclear EoS at different density regions beyond saturation [216–220]. High-precision heavy-ion experiments on pion and kaon dynamics [217,221] revealed more details of the in- medium hadronic properties [222–226]. A soft nuclear EoS at high baryon densities up to (2−3)-times the saturation density was predicted from transport theoretical strangeness production analyzes [222–229]. The rise and fall of collective flow with increasing energy was a hint for sudden changes in the softness of the nuclear EoS at high densities [230–232]. More than one decade passed to obtain more constraints on the EoS of highly compressed matter. The recent astrophysical measurements of neutron stars with masses of 1.97 ± 0.04 M⊙ [233] and 2.01 ± 0.04 M⊙ [234] brought more controversial insights on the high-density EoS (see also the recent work by Fonseca et al. [235] and the review article be Oertel et al. [236]). These observations provide lower bounds for the maximum neutron star mass, excluding a soft EoS at high baryon densities. Neutron stars exhibit a complex internal structure [237–241]. For instance, strangeness carrying mesons (kaons) and baryons (hyperons) can appear in the neutron star interior. In fact, the presence of hyperons (Λ, Σ, Ξ- and Ω) inside neutron stars is in principle energetically allowed, since their chemical potentials are sufficiently large at high baryon densities. Nuclear matter with hyperons weakens the EoS largely at high baryon densities. Several nuclear models, successfully applied to nuclear systems (finite nuclei, heavy-ion collisions), cannot explain the observed data of neutron star masses, if they include hyperons in their descriptions. This is known as the ‘‘hyperon puzzle’’ issue, see e.g. [242,243] for a discussion of the recent status. Investigations of the strangeness part of the hadronic EoS are hampered by the lack of detailed experimental information on YN and YY interactions. In fact, in the nucleon–nucleon (NN) S = 0 sector high precision 4300 scattering data are available allowing an accurate determination of the in-medium NN-interaction, at least for densities up to saturation. Adding strangeness to the system (S = −1) the situation changes largely. In the S = −1-sector (hyperon–nucleon) one has so far had access to 38 scattering data only. They still allow a reasonable determination of the S = −1 hyperon–nucleon (YN) interactions, however, with remaining ambiguities particularly for the in-medium YN-potentials. The next higher strangeness domain with total strangeness S = −2, involving interactions either of Λ and Σ hyperons among themselves or ΞN interactions, is still an unexplored experimental region, relying completely on theoretical predictions. The same is true for the S = −3 channels given by ΛΞ and ΣΞ configurations, respectively, as well as ΞΞ interactions with total S = −4. Studies of those interaction within chiral effective field theory are found e.g. in [244,82]. QCD studies of that channel are reported in [127]. There is little to nothing known for interactions of mixed octet–decuplet states given by combinations of Σ∗, Ξ ∗ and the S = −3 Ω−-baryon with octet baryons. Under QCD aspects, interactions with the Ω− baryon are of special interest because the [sss] quark configuration of that baryon poses close constraints on reaction and production dynamics. As discussed before, YN-interactions are formulated on a group-theoretical basis, using the SU(3) flavor symmetry group. Often the SU(2) spin-group is taken into account, leading to a description in terms of an SU(6) spin-flavor symmetry group. As H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 167 far as in-medium interactions are concerned the whole spectrum of approaches known from NN-interactions is employed also to the full octet sector. Phenomenological Skyrme-like approaches [88] and covariant models are in use. The latter are based on the Relativistic Mean-Field (RMF) theory, firstly introduced by Dürr [89] and Walecka [90], considerably improved by Bodmer and Boguta [91] and further developed by Serot and Walecka [92]. This RMF framework has been extended to the description of nuclear matter with hyperons [245,246,95]. A microscopically oriented alternative was discussed in Section3 by using density dependent coupling constants to mimic the microscopic non-linear structure of the interaction. The Density Dependent Hadronic (DDH) approaches [54,59,247] have been applied to matter with hyperons too [52,96]. The microscopic approaches consider higher order correlations in the spirit of the ladder approximation. They are known in the literature as non-relativistic Brueckner–Hartree– Fock [248,249] or covariant Dirac–Brueckner–Hartree– Fock models [250]. They are based on the One-Boson-Exchange (OBE) picture of the baryon–baryon interaction [251,79]. Further theoretical works toward a description of the full octet and decuplet baryons exist. Prominent examples for the S = −1 YN-interaction are the models of the Nijmegen [31,40,41,252], Jülich [35–37] and Kyoto–Niigata [253] groups. Recently, QCD inspired approaches have been also developed and lattice QCD simulations have been applied to the YN-interactions [132,254,255]. Chiral effective field (χEFT) theory has been also used for the construction of realistic in- medium YN-potentials [256,257,124,258]. For the purpose of this section, the theoretical models for the YN-interactions, in free space and inside the hadronic medium, provide the essential physical input for transport studies of in-medium reactions induced by heavy-ions or hadrons impinging on nuclear targets. In relativistic collisions between heavy-ions or hadrons and nuclei one can probe the in- medium dynamics of produced hyperons and hyperfragments. Thus, with a proper theoretical analysis such reaction studies are giving useful information on the in-medium YN interaction by a systematic comparison with experimental data of produced hyperons and bound hypernuclei. Hypermatter is here of particular importance. While in a single-Λ hypernucleus the S = −1 sector of the YN-interaction can be investigated, hypernuclei with higher strangeness content are the ideal laboratories for exploring the higher strangeness part of the in-medium hyperonic interactions. Moreover, they are the only known way to access interactions in the |S| ≥ 2 sectors. Nuclei with more than one bound hyperons are usually referred to as multi-strangeness hypernuclei or, occasionally, also as superstrange nuclei as introduced by Kerman and Weiss in their pioneering work [259] on multi-Λ hyperfragment production.

8.1. Transport theory for hadronic collisions

Kinetic theory, introduced in the pre-quantum era of physics by Ludwig Boltzmann in the year 1872 [260], has been found to provide the proper basis for the description of violent nuclear interactions. Boltzmann formulated a transport equation for classical many-body systems based on Liouville’s theorem. Vlasov extended the Boltzmann equation by including a mean- field potential, followed by the famous work of Uehling and Uhlenbeck taking into account Pauli blocking effects in binary collisions [261]. Since then different theoretical aspects of transport dynamics were investigated. Kadanoff and Baym derived the kinetic equations from non-relativistic quantum statistics [262]. Danielewicz introduced another microscopic derivation of the kinetic equations and applied them for the first time to the dynamics of heavy-ion collisions [263]. Bertsch and Das Gupta continued the transport theoretical studies with more details concerning mean-field and collision dynamics [264]. A modern covariant derivation of relativistic kinetic equations was formulated by Botermans and Malfliet [265], based on the microscopic Dirac–Brueckner–Hartree–Fock formalism. A more pedagogical introduction to relativistic kinetic theory can be found in De Groot [266]. Since then various models based on (relativistic) kinetic theory have been applied in in-medium hadronic reactions. The various versions of transport theoretical approaches in strangeness and multi-strangeness producing reaction were reviewed recently by Gaitanos [200]. The renewed interest in theoretical studies has been motivated by a series of ongoing and forthcoming experimental activities, comprehensively reviewed in Ref. [28]. The Hypernuclear-Heavy-Ion (HypHI) collaboration [21,267,24] has recently reported longitudinal momentum spectra of low-mass single-Λ hypernuclei in intermediate energy collisions between low-mass nuclei [267]. Within the J-PARC experimental project [268] high energy proton beams will be used for the production of bound hypersystems. We emphasize further experimental activities concerning hypernuclear studies, such as STAR (RHIC) [269], ALICE (LHC) [270], FOPI and HADES at CBM [271,272] and NICA [273]. The forthcoming PANDA experiment at FAIR [274,275] is of great interest concerning multi-strangeness hypernuclear physics. Indeed, in-medium collisions with antiproton-beams at intermediate energies of few GeV can overcome the high production thresholds of hyperons. The high annihilation cross sections at low incident energies into multiple meson production (antikaons) and the formation of strangeness resonances can accumulate energy and strangeness content via secondary scattering. Thus, a copious production of heavy hyperons through a multi-step collision process is possible.

8.1.1. Transport theoretical description of particle production in hadronic reactions There are several ways to derive the kinetic equations for quantal systems. A field-theoretical covariant description of strongly interacting many-body systems is found in [276,277] using the non-equilibrium Green’s function approach for an interacting many-body system. The hierarchy of many-particle Green’s functions are connected in principle through Dyson equations. In practice, however, one uses typically a system of equation constrained to up to 2-particle correlations. The Schwinger–Keldysh formalism serves then to derive relativistic kinetic equations for a correlated Green’s function, which is related to a single-particle phase-space density. From a theoretical point of view, the advantage of this derivation is a 168 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 direct connection to the microscopic Dirac–Brueckner theory [265] with a natural interpretation in terms of (semi-classical) quasi-particles. Another, more practicable, derivation is based on Quantum Hadrodynamics (QHD), which we follow here [278]. The starting point is a QHD-type baryon–meson Lagrangian density. In principle the formalism includes baryons of arbitrary strangeness content. The openness and flexibility of that formalism easily allows for extensions also beyond the octet sector. In fact, baryon resonances are playing a central role in understanding strangeness production in central heavy ion collisions. Also the meson sector is extended to include at least the full set of flavor octet 0− pseudo-scalar, JP = 1− vector and JP = 0+ scalar mesons, respectively, as discussed in Section2. In the mean-field or Hartree approximation, the meson field operators are replaced by their classical fields given by the expectation values of the quantum fields. The energy and pressure densities are defined through the conserved energy–momentum tensor, thus defining the connection to the EoS. By means of the Euler–Lagrange equations the Klein–Gordon and Proca equations of motion for the virtual mesons and the Dirac equation 1 + for the spinors of 2 baryons are derived, obtained as ( µ − µ) − − = γµ i∂ VB Ψ (m SB)ΨB 0 (106) µ for a baryon of type B moving in an effective vector potential VB and an effective scalar potential SB. Higher-spin baryon states are in principle described by the formalism discussed in Section5. The key quantity for the derivation of the transport equation is the Wigner function

1 ∫ µ 1 1 4 −ikµR ˆ ˆ Fαβ (x, k) = d Re ⟨Ψ β (x + R)Ψα(x − R)⟩. (107) (2π)4 2 2 This is the so-called Wigner-transformation of the correlated Green’s function, which is related to the 1-body phase-space distribution f (x, k). The matrix notation introduced above will be omitted in the following. As seen from Eq. (107), the Wigner function is in fact determined by the expectation value of the quantal one-body density operator of the baryons propagated in the system. In the following, we use the identity for a differentiable function

1 1 µ − Rµ∂x g(x2) = g(x − R) = e 2 g(x) (108) 2 and take advantage of the Dirac equation to obtain for the Wigner function F(x, k) the following equation of motion (we omit the indices): ( ) ( ) µ µ − 1 ih¯∆ µ − + = 2 − γµ(h¯∂ 2ik ) 2imB F(x, k) 2ie SB(x) VB (x) F(x, k) , (109)

k µ where the triangle-operator ∆ ≡ ∂µ∂x describes the shift in position and momentum in phase space. Eq. (109) is still a quantal relation as seen from the presence of Planck’s constant h¯ . The semi-classical approximation is manifested by a Taylor expansion of the exponential function up to first order in h¯ , which requires a smooth, non-fluctuating behavior of the baryon fields and the Wigner function in phase space. This approach is semi-classical in a two-fold sense, namely by neglecting terms of O(h¯ 2) and that the meson-field dynamics is treated classically. Quantal effects, however, are included in the collision integral. The classical treatment is justified for reaction energies close to the Fermi energy and above, where the de Broglie wave length is small compared to the considered scales of a few fm. A Taylor expansion up to first order of Eq. (109) leads to the following expressions

∗µ ∗ [γµk − mB]F(x, k) = 0 (110) and ( ) µ − − µ = γµ∂ ∆(SB(x) γµVB (x)) F(x, k) 0 (111) for the imaginary and real parts of the operators of Eq. (109), respectively. Here in-medium self-energies were introduced, which contain the mean-field potential. The self-energies are understood to be given by tensorial structures in the baryon– µ baryon flavor space. The Lorentz-vector (VB ) and the Lorentz-scalar (SB) parts define effective masses and kinetic momenta according ∗µ = µ − µ k k VB (112) ∗ mB = MB − SB . (113) The imaginary part, Eq. (110), includes already the quasi-particle approximation. That is, the in-medium on-shell constraint for quasiparticles, which are characterized by an effective mass m∗ and a kinetic momentum k∗µ. The real part, Eq. (111), is used to derive the transport equation (without collisions) for the phase-space distribution function. The expression (111) is still a matrix equation in spinor space. With a decomposition of the Wigner matrix Fαβ into the elements of the Clifford-Algebra

µ F(x, k) = F(x, k) · 11 + Vµ(x, k)γ + P(x, k)γ5 + A(x, k)γµγ5 + T µν (x, k)σ µν (114) H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 169 we obtain the desired transport equation for the scalar part F(x, k), which can be identified with the 1-body phase-space distribution function f (x, k∗). The number of binary collisions is proportional to the corresponding 1-particle phase-space distributions. Taking into account the Pauli principle and inserting damping effects due to binary collisions we arrive at the transport equation known as the Relativistic Boltzmann–Uehling–Uhlenbeck (RBUU) equation: ( ) ∗ ∗ µ + ∗ µν + ∗ µ ∗ k ∗ kµ∂x (kν FB mB(∂ mB))∂µ f (x, k )

∫ 3 ∗ 3 ∗ 3 ∗ 1 d k d k d k ∗ ∗ ∗ ∗ = 2 3 4 W (k k |k k ) ∗ 3 ∗ 3 ∗ 3 2 3 4 2 E2 (2π) E3 (2π) E4 (2π) [ ∗ ∗ ( ∗ )( ∗ ) × f (x, k3)f (x, k4) 1 − f (x, k ) 1 − f (x, k2) ∗ ∗ ( ∗ )( ∗ ) ] − f (x, k )f (x, k2) 1 − f (x, k3) 1 − f (x, k4) , (115) √ ∗ = ∗ 2 + ∗ 2 = with Ej mj kj for j 2, 3, 4. The right-hand side gives the temporal changes of the phase-space distribution f (x, k∗) due to binary collisions with other particles, if allowed by the Pauli principle. The physical quantity for a scattering ∗ ∗| ∗ ∗ process is given by the transition probability W (k k2 k3 k4), which is proportional to the scattering cross section and to a δ-function. The latter ensures energy–momentum conservation for each binary process. The practical use of transport theory, the definition of the mean-field part, the construction of the collision term and suitable numerical methods for solving Eq. (115) are found in the literature, e.g. [200,279]. Here, we only point out that depending on the reaction type and the beam energy, several exclusive channels must be taken into account in the numerical treatment of the collision integral. For the simulation of heavy-ion collisions and proton-induced reactions up to incident energies of 1 − −2 GeV per nucleon the inelastic channels up to the ∆(1232) resonance including secondary scattering with pion production and absorption are sufficient. The formation of strangeness at these intermediate energies is a rare process. However, it must be included if one intends to study strangeness production. A more precise treatment of the collision term is possible by considering all baryonic resonances up to 2 GeV, e.g. taken from the compilation of the Particle Data Group [10]. With increasing number of primary processes, the number of secondary scattering increases too. Re-scattering is a very important mechanism for the formation of hypernuclei, as we will see later on. These aspects are included into recent realizations of transport theory, such as the UrQMD [280], HSD [281] and the Giessen-BUU (GiBUU) [279]. Selected GiBUU results will be discussed below.

8.1.2. Time evolution of the 1-body density The transport equations give the full information on single-particle dynamics of nucleons and produced particles only. From the knowledge of the phase-space distribution one can determine thermodynamical properties such as particle densities, energy densities, pressure and the temperature and the related excitation energy at each phase-space point of the dynamical evolution. This is very useful to gain information on (local) equilibration and the onset of instabilities. The degree of equilibration can be obtained by comparing the transverse and longitudinal components of the local energy–momentum tensor, describing the transverse and longitudinal pressure, respectively. The onset of an instability can be determined by calculating the pressure versus the density at a specific space-point as function of time. Fig. 40 shows the dynamics of thermodynamical quantities, extracted from RBUU calculations. They were obtained at the center of a heavy-ion reaction as function of time. The typical compression/expansion stages appear at intermediate times, as it can be seen from the time dependence of the central density and temperature. The two pressure components (longitudinal and transversal) differ from each other. That is, the participant system is out of local equilibrium. Freeze-out sets in when the particles do not collide any more. Just before freeze- out appears, the pressures are isotropic and local equilibrium occurs. In fact, as shown in Ref. [282], spinodal instabilities appear at the center of participant and spectators at time stages close to freeze-out for heavy-ion collisions at intermediate incident energies. Thus, transport calculations provide the onset of the fragmentation process, as the consequence of pressure instabilities. Physically this is explained as follows. The matter at densities below saturation intends to reach again the ground state, which is possible only by clusterization and formation of bound fragments.

8.1.3. Fragmentation reactions The fragmentation process cannot be described by transport. There are attempts to go beyond the single-particle dynamics, see for instance Refs. [283–285]. The mean-field approach for the nuclear potential does not include clusterization processes for dilute matter. Thus, propagation of the physical fluctuations is missing, except for the stochastic fluctuations in the collision integral. There exist recent developments toward this direction. Typel re-formulated the mean-field theory by considering in-medium clusters explicitly in the theoretical framework [286]. In any case, with the knowledge of instabilities and equilibration from transport one can determine when fragmentation sets in and use, as an effective method, more sophisticated statistical models for the clusterization process. The Statistical Multi-fragmentation Model (SMM) [16,287] is a well-established approach to describe statistical de- excitation of residual systems. A grand-canonical description is used to describe the formation of fragments out of a collection of protons and neutrons. A central ingredient are the nuclear binding energies which are taken from data. The SMM includes 170 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 40. Time evolution of the density, longitudinal and transverse pressures (red and black curves in the second graph, respectively), temperature and number of collisions (from top to the bottom) at the central shell of a Au+Au heavy-ion collision at 0.6 GeV beam energy per nucleon. The vertical line marks the onset of freeze-out. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Source: From Ref. [200].

the relevant mechanisms for the formation of fragments, i.e. evaporation, fission and multi-fragmentation as well as de- excitation of primary fragments. It has been used in hybrid simulations of in-medium hadronic reactions successfully, see for instance Refs. [288–290]. The connection between pre-equilibrium transport and statistical fragmentation models should be done with care. A critical quantity here is the excitation energy, which is obtained from the transport description of an excited source of hadronic matter, such as spectators in heavy-ion collisions or the residual nucleus in hadron-induced reactions. The excitation energy usually takes values of a few MeV per nucleon only. That is, it is comparable with the binding energy per nucleon or even less. On the other hand, the statistical models are strongly dependent on this quantity [287,16], most likely due to the non-linear functional dependence on level densities. Thus, a very accurate determination of the excitation energy is required in transport studies for a reliable application of statistical fragmentation models. This task is closely related with a precise treatment of the initial stage in transport studies. The nuclei used for simulations should be initialized as precise as possible to avoid numerical noise in the temporal evolution of the binding energy and artificial particle emission. The binding energy enters into the calculation of the excitation of the residual source. The particle emission due to numerical fluctuations can affect the extracted mass and charge numbers of the source. This importance of an accurate initialization has been discussed in detail in Ref. [291] and was reviewed more recently in [200]. The most important insight from those studies is that the common practice of using empirical ground state density profiles for the construction of initial configurations will lead to instabilities seen in observables like e.g. the total nuclear binding energies of the colliding ions. The reason is the incompatibility of the empirical input and the theoretical densities. During the initial stages of the numerical simulations the system is driven toward the self-consistent solution for the (non- interacting) nuclear ground states of the colliding ions. The convergence process is of an asymptotic nature requiring a large number of time steps before stability is reached. The convergence toward numerical stability, however, is considerably enforced and accelerated by orders of magnitude by using densities from a self-consistent static ground state calculation with the same energy density functional as used later in the transport simulation.

9. Hypernuclear production in fragmentation reactions

9.1. Results of the hybrid approach to fragmentation

The production of hypernuclei in heavy ion collisions demands two indispensable ingredients: the presence of hyperons, of course, and the presence of secondary nuclei by which the hyperons can be captured. The first issue is a matter of appropriate elementary strangeness production reactions. They must occur in the hot and compressed medium of the overlap H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 171

Fig. 41. Pictorial presentation of the heavy ion fragmentation reactions at incident energies Tlab ≳ 1 AGeV.

zone with a sufficiently high probability and under conditions which are allowing to populate that part of the phase space appropriate for the formation of a bound system with the nuclear fragments. The fragments, on the other hand, are produced by a freeze-out process in the late stages of the collision after the expansion and cooling of initial fireball. At that time the cloud of participant matter has reached a state of equilibrium and expands mainly as a free gas of baryons. In this stage, coalescence-type capture processes will set in and protons and neutrons may form bound nuclei. In the grand-canonical picture underlying the SMM the formation probability for a nucleus of mass number A = N + Z, composed of N neutrons and Z protons, is essentially determined by binding energy [16]. A schematic picture of the scenario of a fragmentation reaction is shown in Fig. 41. Before turning to the strangeness exit channels we first discuss the general method how pre-equilibrium transport and statistical approaches can be combined into a hybrid model. Proton-induced reactions are very well suited to test such hybrid models. In such a reaction, the target remains close to its ground state, gaining only a moderate amount of excitation because collective effects are suppressed, in contrast to the violent compression and expansion dynamics in ion– ion collisions. In a central proton–nucleus reaction the nucleus gets excited by the absorption of the incoming proton. In our energy range of interest, the excitation energy will exceed various particle emission thresholds. Hence, the nucleus starts to emit nucleons (pre-equilibrium emission) and evolves toward an thermodynamically equilibrated compound system. The compound system acts as residual source for the following stages of the reaction. The source consists of all particles inside the nuclear radius by excluding the emitted nucleons. There are several methods to determine the residual source in transport simulations. One can use either the binding energy of the particles as criterion or apply a density constraint at each particle’s position. Assuming that all nucleons inside the nuclear radius belong to the compound system, we define a residual (or fragmenting) source by the density constraint of ρcut = 0.01 × ρsat . The particles with a density greater than ρcut belong to the residual source. Fig. 42 shows the time evolution of the characteristic properties of the residual source as obtained from a transport calculation. We estimate the onset of the SMM model at the freeze-out time, i.e. at the time when a stable configuration has been reached by the transport calculation. This is indicated in Fig. 42 by the vertical lines. Performing exclusive reactions for all the impact parameter range from central up to most peripheral collisions and applying at each simulation event the SMM model, one arrives at the results of Figs. 43 and 44. There the charge distribution of fragments and differential energy spectra of free neutrons are shown. In particular, the charge distribution reproduces the experimental data fairly well. The evaporation peak close to the initial target charge number Zinit = 79 and the wide fission peak at around Z = 79/2 are clearly visible. These calculations predict also a multi-fragmentation region at low Z-values. This part of the distribution originates mostly from central events. For the formation of hypernuclei not only the mass and charge multiplicities, but also momentum distributions are crucial. The comparison of the hybrid calculations with data on differential energy spectra is shown in Fig. 44, for a similar reaction system as in Fig. 43. These spectra are selected at various polar angles and for free neutrons only. In this figure one realizes the importance of having both, pre-equilibrium and statistical emission. The high energy part of the spectrum with the clearly visible quasi-elastic peaks at forward angles results from the particle emission of the transport calculations. The low energy spectrum of emitted particles is then the result of the SMM step due to de-excitation. The combination of both approaches, GiBUU and SMM, is required to explain the energy spectrum over the entire range. The application of the combined approach in the dynamics of heavy-ion collisions follows in principle the same scheme. The difference with the case of proton-induced reactions are the collective effects. The participant region, which is formed during the pre-equilibrium stage, exhibits a violent compression/expansion dynamics showing up in a strong radial flow component [296,297]. Spectator dynamics, on the other hand, exhibits better controlled conditions. As discussed in detail in Ref. [200] the degree of excitation is relatively high for semi-central collisions, while with increasing centrality the excitation decreases because of the reduced mixing between spectator and highly excited participant matter with rising impact parameter. At about a time t ∼ 50 fm/c fluctuations set in and the fragmentation process starts. The question appears at which time step the SMM model should be applied. It is found that after roughly a time of again t ∼ 50 fm/c the ratio of longitudinal and transversal pressure approaches unity indicating the onset of local equilibration inside spectator matter [200]. That is, instabilities and equilibration occur at almost the same freeze-out time with a density of around ρsat /3 and a central temperature of T ≃ 5 MeV [282]. 172 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 42. Mass number, charge number and total energy per nucleon (less the nucleon rest mass) in the panels on the top-left, top-right and bottom-left, respectively, as function of time for the residual target nucleus. The vertical marks in each graph indicate the onset of the SMM model.

Fig. 43. Charge distribution for the reaction as indicated. Hybrid calculations (solid curve) are compared with experimental data (open symbols) taken from [292,293]. Source: From Ref. [294].

Having determined the mass, charge numbers and the excitation energy of the spectator at freeze-out, one can apply the SMM model for spectator fragmentation. An example of this procedure is shown in Fig. 45. There the velocity distributions of various spectator fragments for a Xe+Pb heavy-ion collision at TLab = 1 AGeV are shown. As one sees, the theoretical calculations reproduce the experimental data of Ref. [298] satisfactorily well. More detailed comparisons including fragment multiplicities can be found in Ref. [299]. There exist also other recent studies on spectator fragmentation. In Refs. [288,289] and [290] the SMM model has been applied to the fragmentation of projectile-like residues in intermediate energy collisions between Sn-isotopes. It is shown that the SMM approach reproduces the experimental isotope distributions fairly well. H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 173

Fig. 44. Kinetic energy spectra of emitted neutrons at different polar angles (as indicated on the right) for p+Pb reactions at a beam energy of TLab = 0.8 AGeV. Hybrid calculations (solid curves) are compared with experimental data (open symbols) taken from [295]. Source: From Ref. [200].

Fig. 45. Velocity distributions for various fragments at the rest-frame of projectile spectator, as indicated. The hybrid calculations (solid curves) are compared with experimental data (open symbols), taken from Ref. [298]. Source: From Ref. [299].

9.2. Dynamics of strangeness and hypernuclei in heavy-ion collisions

For a reliable prediction of hypermatter production, two features should be described as precise as a model allows: fragmentation and strangeness dynamics. We have shown that fragmentation phenomena are well described by a com- bination of non-equilibrium transport and statistical approaches. Strangeness dynamics in heavy-ion collisions was studied in detail in [279], resulting in very satisfactory description of kaon and hyperon data. We continue the discussion with the production of hypernuclei. Fig. 46 shows the rapidity spectra of produced fragments in projectile and target spectators as well as of hyperons. The fragment distributions are the result of the hybrid simulations, while the hyperons result from the transport calculations only. This figure shows the idea of coalescence for hypermatter production. In fact, a part of the hyperon spectrum overlaps with the fragment longitudinal momentum distributions close to projectile and target rapidities. The Λ-particles are mainly created in primary BB → BYK-collisions and in secondary πB → YK-scattering. Elastic and quasi-elastic scattering, i.e., with strangeness exchange, are included too. Secondary scattering involving fast pions is here important to create such a wide spectrum in beam momentum. Thus, some of the produced hyperons with a velocity close to that of spectators can be captured and form hyperfragments. This is realized by a coalescence in coordinate and momentum space between the hyperons and cold SMM fragments. A typical result of hypermatter formation is shown in Fig. 47. There the rapidity spectra of light-mass spectator fragments with the corresponding hyperfragments are displayed. The considered C+C reaction at 2 GeV beam energy per nucleon has been chosen, since for a similar colliding system (Li-beam on C-target) ongoing experimental activities are 174 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 46. Rapidity distributions of projectile and target fragments as well as of Λ-hyperons, as indicated, for Li+C heavy-ion collisions at 2 GeV incident energy per nucleon. Source: From Ref. [200].

Fig. 47. Rapidity spectra of produced spectator fragments (dashed curves with green area) and the corresponding hyperfragments (solid curves with yellow area), as indicated. The considered reactions are a C+C collisions at 2 GeV incident energy per nucleon. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Source: From Ref. [200].

in progress [267,24]. One sees in Fig. 47, that the light-mass hypernuclei are produced with relatively low cross sections. Their production rates are in the range of few µb only. The main reason for the low production cross sections is the small size of the colliding systems. The smaller the nucleus, the less secondary scattering. The latter feature is, however, important to de-accelerate the hyperons inside the spectators, so that they can be captured. The predicted values for the light hypernuclei are very close to the preliminary experimental data [267]. We have used here the symmetric C+C system for the determination of hypernuclei, in order to obtain a better statistics. In any case, it is preferable to use colliding systems as heavy as experimentally possible, in order to obtain large hypernuclear cross sections.

9.3. Brief discussion of alternative approaches

At present the study of hypermatter formation in intermediate energy heavy-ion collisions is an active field of research. There exist recent investigations by other groups. They use alternative approaches of transport dynamics combined with potential, coalescence and statistical prescriptions for the description of the fragmentation process. In Ref. [300] the well- established isospin-QMD transport model in combination with a newly developed fragmentation algorithm [301] has H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 175

Fig. 48. Time evolution of the average mass number, charge number (upper panel, as indicated) and the average excitation energy per nucleon (lower panel) of the residual nuclei. GiBUU calculations for proton-induced (open diamonds) and antiproton-induced (filled stars) reactions at an incident energy of 5 GeV and impact parameter of b = 3.4 fm are shown. Source: From Ref. [200].

been applied in collisions between heavy-ions. They have studied the formation of light-mass hypermatter in heavy-ion reactions at energies just above the kaon production threshold. As an important outcome of their analysis, it was found that rescattering strongly affects the hypernuclear formation. In Refs. [302] the Dubna–Cascade and the Ultra-relativistic QMD (UrQMD) kinetic approaches have been adopted for the dynamical treatment of heavy-ion collisions, in combination with potential and coalescence prescriptions for the formation of hypermatter. Their studies show the possibility of hypernuclear formation in heavy-ion reactions at intermediate energies. Even the production of multi-strangeness hypersystems is favorable with rather high cross sections, however, using heavy-mass projectile and target nuclei. For instance, single- and double-hypernuclei in spectator fragmentation are produced with cross sections of few mb and µb, respectively, in Pb+Pb collisions around 1 GeV incident energy per particle. Bound hypermatter with S = −3 is also predicted with cross section in the µb-region at higher beam energies above 2 GeV per nucleon. The authors of Ref. [303] have used the UrQMD and HSD transport models in combination with a coalescence of baryons (CB) for the description of the in-medium hyperonic capture. In particular, they have applied the hybrid UrQMD+CB and HSD+CB approaches to collisions between heavy-ions of different size at projectile energies per particle above 2 GeV per nucleon. In their analysis the formation mechanism of hyperfragments originating not only from residual spectators, but also from the participant region, has been studied in detail.

10. Multi-strangeness dynamics in antiproton-induced reactions

10.1. Reaction dynamical aspects

A clear advantage of antiproton-induced annihilation reactions is that they allow to access the full spectrum of flavor and angular and parity states for arbitrary hadron–antihadron configurations. In p¯-reactions on nuclei strange particles are produced in primary annihilation processes localized in the low-density surface region of the target nucleus. Depending on the centrality, the initial perturbation can penetrate deep into the nucleus through multi-step binary processes. The average excitation per nucleon is comparable to that of the proton–nucleus reactions. This is shown in Fig. 48 in terms of the time evolution of average mass, charge and excitation energy per particle for both type of reactions. However, the main difference between p- and p¯-induced reactions shows up in the abundance of produced particles, as it can be seen in Fig. 49. Generally, the multiplicity of all produced particles increases in the p¯-case. In particular, the multiplicity of antikaons and Λ-hyperons increases largely in the p¯-induced reactions and the heavy Ξ-baryon appears. This is due to the strong annihi- lation cross sections mainly into multi-mesonic final states [304]. Furthermore, the annihilation into hyperon–antihyperon pairs becomes less and decreases by an order of magnitude with increasing mass of the produced (anti)hyperons. On the other hand, secondary scattering involving the cascade particles will be important for the formation of superstrange matter, as we will see. The fragmentation process here (see Fig. 50) is very similar to the p-induced reactions, as it can be seen in Fig. 43. Again, evaporation, fission and multi-fragmentation regions are visible going from most peripheral to most central reactions. 176 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 49. GiBUU results for the particle yields as function of time for the same reactions as in Fig. 48. The vertical arrows indicate the change of particle yields going from p-induced (dashed curves) to p¯-induced reactions (solid curves). The different colored curves denote the various particles, as indicated. The black and red curves, which drop fast in time, correspond to ∆ and higher resonances, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 50. Fragment charge distribution for centrality-inclusive p¯-induced reactions, as indicated. The curve is the result of the GiBUU+SMM hybrid approach.

10.2. Hyperon production

Fig. 51 summarizes the results of the pre-equilibrium transport and SMM calculations in terms of the rapidity dis- tributions of produced residual fragments and hyperons. The Λ-rapidity spectrum is rather broad. Secondary scattering is responsible for the low energy part of produced Λ-particles. This supports the formation of Λ-hypernuclei, as in the case of spectator fragmentation in heavy-ion collisions. The most interesting part here is the Ξ-production, which will be responsible for the production of multi-strangeness hypermatter. Ξ-particles are created with rather high probability, even if their production cross sections from annihilation are very low. In fact, while ΛΛ¯ -pairs are produced with cross sections of several hundred µb, the antiproton annihilation cross section for Ξ-production is very low with orders of few µb only [306]. The Ξ-nucleon cross sections used in the transport simulations in [307] were derived from the fss quark model interactions [114,253]. They are shown in Fig. 52. Most of the Ξ-hyperons escape the nucleus, but there is a small fraction of particles with rapidities close to those of the residual fragments. Due to the high production threshold of the heavy Ξ(1315)-particles one would naively expect that they escape the nucleus with high rapidities. However, secondary scattering of produced Ξ-hyperons with the hadronic environment is crucial for low energy cascade particles. This is manifested in Fig. 53 in terms of momentum spectra of the total Ξ-yield including the exclusive contribution channels. Indeed, secondary scattering processes involving (anti)kaons, and kaonic/hyperonic resonances contribute largely to the low energy tail of the Ξ-momentum distribution. The situation H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 177

Fig. 51. Rapidity spectra of different fragments, Λ- and Ξ-hyperons for the same reaction as in Fig. 50. The curves result from GiBUU+SMM hybrid calculations. Source: From Ref. [305].

Fig. 52. The elementary Ξ-nucleon total elastic and production cross sections entering into the transport theoretical collision term. The (polynomial) parametrizations used in the calculations are shown by the solid lines.

is similar for the momentum spectra of produced Λ and Σ hyperons in p¯-induced reactions, where experimental data exist [308]. The formation of not only single-Λ hypernuclei, but also of double-Λ hypermatter is possible in p¯-induced reactions. However, the probabilities of S = −2-hypermatter are still very low with respect to Λ-hypernuclear yields [305]. An alternative method has been proposed by the PANDA -collaboration to enhance the production of superstrange bound matter [274]. That is a two-step reaction with primary and secondary targets. An antiproton beam interacts with a first target. The low energy part of the produced Ξ-hyperons can be used as a secondary beam for Ξ-induced reactions on the secondary target. The interaction between the Ξ-particles with the particles of the secondary target can create multiple captured hyperons and, thus, multi-strangeness hypernuclei. Indeed, our transport-theoretical studies [305] support this scenario. Fig. 54 shows the strangeness dynamics for Ξ-induced reactions at three different low energies of the Ξ-beam. Two aspects are visible here. At first, a significant capture of Λ-particles inside the matter is observed. Secondly, this feature shows a strong energy dependence. The main mechanism of Λ-production here is the inelastic ΞN → ΛΛ channel. According to microscopic calculations [309,310] this cross section can increase largely at low Ξ-energies with respect to elastic ΞN → ΞN-scattering. Thus, double Λ production drops with increasing energy of the cascade particles. This energy dependence in transport calculations is very strong and reflects just the strong energy dependence of the corresponding cross section [309,310]. Note that for these important channels no experimental data exist, in contrast to S = −1-scattering. Formation of double-strangeness ΛΛ hypernuclei can thus occur in the secondary reaction. This is shown in Fig. 55, where 178 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 53. GiBUU results for the momentum spectrum of inclusive Ξ-production (solid curve) together with the partial contributions to the total Ξ-production for the antiproton-induced reaction, as indicated. Source: From Ref. [304].

Fig. 54. GiBUU results for Ξ-induced reactions at three beam energies on a Cu target. The total yields, normalized to unity at t = 0 fm/c, of Λ (black lines) and Ξ (green lines) hyperons are shown as a function of time. For comparison, also the yield of unbound Λ hyperons are shown (red lines). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

the charge distributions of fragments (upper panel) and ΛΛ-hyperfragments (lower panel) at different Ξ-energies are shown. At first, the fragment distribution becomes broader with increasing beam-energy of the cascade particles. Higher incident energy is associated with increasing excitation of the residual target nucleus. Thus, the fission region at around half the initial target charge and multi-fragmentation show up with rising beam energy. As an important result, an abundant production of S = −2-hypernuclei is predicted by these transport calculations. This is visible in Fig. 55 by comparing corresponding curves for fragments (upper panel) and hyperfragments (lower panel) for each incident Ξ-energy. With increasing beam energy the production of hypermatter around the evaporation peak drops significantly. However, the production yields of double-strangeness ΛΛ-hypernuclei are in the mb-region at these low Ξ-energies. Therefore, the proposed PANDA -experiment can be very well suited to explore in more detail not only the nucleon– hyperon interactions, but also the still less understood regions of the higher strangeness sectors. Recent theoretical activities H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 179

Fig. 55. Charge distributions of fragments (panel on the top) and double-Λ hyperfragments (panel on the bottom) for Ξ-induced reactions at incident energies as indicated. Source: From Ref. [305].

have investigated the in-medium hyperon interactions for |S| > 1 [252,309,310]. These are based on the microscopic meson- exchange picture or using quark–cluster approaches. The predictions between the theoretical models differ to a large extend. Therefore, a parameter-free theoretical framework would be obviously desired. This is possible only if one goes beyond the meson-exchange picture and considers the internal hadron structure. Recent attempts in this direction have been started within Lattice QCD-simulation by the HALQCD collaboration [126,129] and the NPLQCD collaboration [311,132,133] and, in the last decade, χEFT has extended the attention to strangeness as well [82,124] and, very recent, even to a derivation of an in-medium energy density functional [67]. The formation of multi-strangeness bound matter in PANDA -type reactions can be used as a probe to test the microscopic predictions for the superstrange sector of the in-medium hyperon potential. This issue has been studied recently in detail [312]. An example is shown in Fig. 56, where the mass number distributions of all fragments, double-ΛΛ hyperfragments and, in particular, Ξ-hyperfragments are displayed using two different approaches for the ΞN-scattering. The transport results on the left are based on the extended-soft-core (ESC) approach [309], while those on the right are performed within the quark-cluster model (fss) [310]. Both models lead essentially to different results for ΞN-elastic and inelastic ΞN → ΛΛ-scattering. In particular, the fss ΞN → ΛΛ-cross section is strongly reduced relative to the ESC-predictions. Thus, the Ξ-multiplicity increases in the transport calculations using the fss model. The consequence is a higher production yield of Ξ-hypernuclei, as clearly shown in Fig. 56. Note that the formation of Ξ-hypernuclei beyond the conventional evaporation region is possible depending, however, on the microscopic model applied. The PANDA -proposed scenario could thus be used to better constrain the higher strangeness sector of the in-medium interaction.

11. Production of Ω− hyperons in antiproton annihilation on nuclei

11.1. Hyperon–nucleon interaction in the S = −3 sector

The Ω− hyperon with S = −3 falls out of the flavor octet systematics considered in this section up to here because it 3 + belongs to the 2 SU(3) decuplet. However, we append the discussion at this point by virtue of strangeness. As elaborated in detail in [313], baryons belonging to the high strangeness sector are hardly explored. For reasons which will become obvious below, we distinguish between primary and secondary production processes of the Ω-baryon. For the primary binary processes we focus the discussion to antiproton–proton collisions. Various possibilities exist for the secondary Ω-baryon production, which will be discussed below. The p + p¯ → Ω + Ω¯ reaction requires the creation of 3 ss¯ pairs out of the vacuum and is at least a three gluon process from which we expect a small cross section. In the literature a single theoretical work is found on this subject. Kaidalov et al. [306] have considered the pp¯ → ΩΩ¯ process with a reggeon-like calculation starting at the production threshold. In Fig. 57 the predicted total production cross section is shown together with the polynomial√ parametrization of the theoretical results which is used in the transport calculations. Above the production threshold of s ≥ 3.344 GeV (corresponding to a kinetic energy of Tlab = 4 GeV or to a beam momentum of plab = 4.9 GeV /c in the laboratory frame) the cross section increases to a maximum of several nb only before decreasing again. Remarkably, these scales are approximately one (two) order of magnitude less with respect to the antiproton–proton production cross section of the cascade (Λ) particles, as also discussed in Ref. [306]. Consequently, the question arises whether the production of heavy multi-strangeness hyperons is possible in the energy domain planned at FAIR. In this context one should remind, that the PANDA -experiment consists of a two-step process: 180 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 56. Mass distributions of fragments (dashed curves), ΛΛ-hyperfragments (dot-dashed curves) and Ξ-hyperfragments (thick solid curves) for Ξ-induced reactions at TLab = 0.3 GeV. Both panels show the results of the GiBUU+SMM model using two different microscopic approaches for the ΞN-interaction. Source: From Ref. [312].

Fig. 57. Antiproton–proton cross section for direct ΩΩ¯ production. The symbols refer to theoretical estimations from Ref. [306] and the solid curve is a parametrization derived and used in the transport calculations. As an insert the minimal QCD process is depicted, given by at least three gluon radiations, materializing into three ss¯ pairs which rearrange into the Ω baryon and its antiparticle.

Antiproton-beams on a nuclear target as the origin of cascade particles (Ξ −), which will serve as a Ξ −-beam for collisions with a secondary target. Thus, secondary re-scattering may contribute to Ω-production as well. Fig. 58 shows the cross sections for some particular secondary processes. Obviously we select such processes with the highest strangeness degree of freedom in the initial channel. That is, re-scattering between antikaons and other hyperons with S = −1 (panels on top in Fig. 58) or between the cascade particles with other non-strange baryons or mesons (panels on the bottom in Fig. 58). In principle, these cross sections can be evaluated within χEFT approaches. However, in view of the limited energy range accessible in the EFT approach and the persisting sizeable bands of uncertainty, in the present study we prefer to use phenomenological cross sections extracted directly from elementary events by means of the PYTHIA program [314]. At first, the particle energies should√ lie above the corresponding Ω-production thresholds. With the Ω-mass of 1.672 GeV this results in a threshold of s ≥ 3.107 GeV for the ΞN → ΩKN-channel (corresponding to Elab ≥ 3.75 GeV kinetic energy or to plab ≥ 3.5 GeV beam momentum). According to these calculations high energy Ξ-beams are necessary to enforce the formation of Ω-baryons from second chance binary collisions, as shown in the bottom-left panel of Fig. 58. Note H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 181

Fig. 58. Total cross sections for various channels (as indicated) for Ω-baryon production. Source: From Ref. [313].

that the order of magnitude of these cross sections ( mb-region) is comparable with the typical strangeness production cross sections (NN → NYK and πN → YK with Y = Λ, Σ and K for the hyperons and kaons, respectively). Furthermore the Ω-production cross sections for primary pp¯ → ΩΩ¯ -channels is several orders of magnitude less with respect to the re-scattering cross sections. Obviously, high energy Ξ-beams with rather heavy-mass nuclear targets might be necessary to produce Ω-particles with high probability. This is the topic of the next section.

11.2. Ω− production on nuclei

We focus√ now on the formation of the heavy Ω-hyperon. In antiproton–proton reactions the Ω-production threshold opens at s ≃ 3.344 GeV . From the small total cross section displayed in Fig. 57 it is seen that ΩΩ¯ pair production is a very rare process. We emphasize that we have analyzed around 4 million transport-theoretical events for each incident energy in order to identify a sizable number of production data. The main reason for the low Ω production yields is the extremely low annihilation cross section of a few nb only, see Fig. 57. This value is far below the annihilation cross section of other exclusive processes. Thus, the direct production is suppressed. The major contribution to Ω− production is coming from strangeness-accumulation processes taking advantage of the abundantly available mesons and resonances. It is important to note that the origin of the produced Ω-particles is not pp¯-annihilation directly into that channel, but secondary processes involving re-scattering between antikaons, antikaonic resonances with hyperonic resonances (Y ⋆(S = −1)). For instance, for the reaction p¯ + 93Nb at 4 GeV incident energy these secondary scattering processes contribute with a cross section of 1.148 nb to the total Ω-production yield with a cross section of σΩ = 1.15 nb. The cascade particles and their resonances, which carry already S = −2 and thus would preferably contribute to the Ω-formation, mainly escape the target nucleus because of their longer lifetime. For this reason they do not contribute here. The realization of a second target using the produced cascade particles as a secondary beam is important. At first, for the copious production of multi-strangeness hyperons and multi-strangeness hypernuclei, as proposed by the PANDA - experiment [274,275]. According to the PANDA -proposal low-energy Ξ-beams will be used for the production of ΛΛ-hypernuclei. First theoretical predictions on such exotic hypermatter in low-energy Ξ-induced reactions have been indeed reported in Ref. [312]. Not only ΛΛ-hypernuclei, but also the direct formation of Ξ-hypermatter is accessible depending on the cascade–nucleon interaction [312]. We show now that the same experiment can be used to explore the formation of Ω baryons. As discussed above, secondary re-scattering including intermediate production of high-mass hyperon resonances can be a more favorable possibility for Ω production. The production of a baryon with such a high strangeness value is favored by entrance channels containing already particles with an as high as possible strangeness content. Thus, the secondary Ξ-beam envisioned for the PANDA -experiment will be a good candidate for our purpose. This is illustrated in Fig. 59 where the rapidity spectra for Ξ-induced reactions at incident energies just above the Ω-production threshold are shown. Overall, similar dynamical effects are observed as in the p¯-induced Λ production reactions. The spectra show the expected broad distribution in rapidity 182 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 59. Rapidity distributions of different hyperons for Ξ-induced reactions on 64Cu-targets at four beam-energies, as indicated in each panel. Dotted curve: Ξ-, dashed curve: Λ-, dot-dashed curve: bound Λ-, dot–dot-dashed curve: bound Ξ-, thick solid curve: Ω-rapidity spectra. Source: From Ref. [313].

because of the enhanced multiple re-scattering. The rescattering effects cause abundant capture events inside the target nucleus. The rapidity distributions of the cascade particles are seen to be peaked around the beam-value, but there is also a significant fraction of events at lower rapidities due to secondary scattering. This mechanism enhances the production of bound cascade hyperons and the formation of exotic Ξ-hypernuclei, as discussed in more details in Ref. [312]). The most interesting results obtained in [313] are the rapidity yields of the produced Ω-baryons shown in Fig. 59. The formation dynamics of the Ω decuplet particle here is similar to the dynamical production of the cascade octet particle in p¯-induced reactions. However, the peak of the Ω rapidity spectra is now located at much higher energies. In particular, the probability of bound Ω-particles inside the residual nucleus is very low. These different dynamical formation scenarios of the Ξ- and Ω-particles have physical reasons beyond the trivial ones of slightly different target masses and beam-energies. The decuplet particles are much heavier and carry one additional unit of strangeness. The latter property causes multi-particle final states in many secondary processes of Ω-production due to strangeness conservation. For instance, in binary collisions between the cascade-beam with nucleons three particles in the final state are required to conserve strangeness and baryon number. This leads to rather high threshold energies for such processes. The Ω-production thresholds are also high in other secondary processes between the abundantly produced antikaons K −(S = −1) with Λ, Σ hyperons or Y ⋆(S = −1) hyperon resonances. Thus, the Ω-particles are produced with relatively high energies. The probability of secondary Ω-scattering is low and most likely they will escape the nucleus. Another interesting result of [313] is, that, in comparison to antiproton-induced reactions, the Ω-formation is signifi- cantly enhanced in Ξ-nucleus collisions. In fact, the Ω-production cross sections are in the range between 0.7−−3.5 mb for the incident Ξ-energies under consideration. This arises from the rather high cross section values of secondary scattering ranging in the mb-region. This is illustrated in Fig. 60. It shows the contributions of various channels to the total Ω-yield for the Ξ − + 64Cu-reaction at an incident Ξ-energy of 12 GeV. The high momentum part of the produced Ω-particles comes from primary collisions between the Ξ-beam and target nucleons. However, as one can see in Fig. 60, the second- chance collisions involving antikaonic and hyperonic resonances contribute considerably to the Ω-production over a broad longitudinal momentum, even for the intermediate mass number A = 64 of the target nucleus. A more complete picture is given in Fig. 61, where the total Ω-production yield and the most important contri- butions to it are displayed as function of the incident energy of the secondary cascade beam. The Ξ-nucleon binary collisions dominate over this energy range, as expected. The other secondary processes with the major contribution to the Ω-multiplicity involve scattering between cascade resonances (Ξ ∗) with nucleons and scattering between antikaons (K −, K ⋆−) with hyperonic resonances (Y ⋆). Furthermore, with increasing beam energy channels with higher-mass reso- nances open, in particular, those channels with the hyperonic resonances. Therefore, the contributions from ΞR-scattering (with R being non-strange resonances) drops as the energy increases. Finally, the secondary scattering between cascade particles and non-strange mesons (Ξπ-channel) opens at higher energies only. In general, secondary processes with a hyperonic resonance Y ⋆ in the initial channel together with the ΞN-collisions mainly contribute to the production of H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 183

Fig. 60. Rapidity distributions for Ω-hyperons in the reaction Ξ − + 64Cu at 12 GeV incident energy. Rapidity spectra for the total Ω-yield (thick-solid black curve) and some particular contributions (thick-solid red: ΞN, dashed blue: Ξ ⋆N, thick-dashed green: K ⋆−Y ⋆, solid-violet: Y ⋆) to the total spectrum are shown. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Source: From Ref. [313].

Fig. 61. Ω-production yields as function of the incident energy (in units of GeV) for Ξ − + 64Cu-reactions. (circles) total yield, (squares) ΞN-contribution, (diamonds) Ξ ⋆N-contribution, (stars) Y ⋆-contribution, (left triangles) K −Y ⋆-contribution, (right triangles) Ξπ-contribution, (bottom triangles) K ⋆−Y ⋆-contribution, (top triangles) ΞR-contribution. Source: From Ref. [313]

Ω-baryons. We predict total Ω-production yields of several mb in the second-step Ξ-induced reactions and conclude the importance of the PANDA -proposal toward the investigation of multi-strange in-medium hadronic interactions. In particular, many theoretical approaches, like for instance the chiral EFT calculations, provide us with elastic and quasi- elastic cross sections for scattering processes between the cascade baryon (Ξ) and the lighter Λ- and Σ-hyperons. Due to the importance of these secondary binary collisions to the dynamical formation of Ω-baryons, the Ξ-nucleus reactions of the PANDA -experiment may serve to better constrain the still existing uncertainties in these theoretical approaches.

12. Brief theory of resonances in nuclear matter

12.1. Decuplet baryons as dynamically generated, composite states

Almost all of the decuplet baryons, Fig.1 and Table6, are decaying by strong interactions to octet states [10], thus giving −23 − them lifetimes of the order of t 1 ∼ 10 s. An exception is the Ω state with its seminal S = −3 [sss] valence quark 2 − −10 structure. The Ω baryon decays by weak interaction with a probability of ∼68% into the ΛK channel with t 1 ∼ 0.8·10 s. 2 In Table6 masses, lifetimes, and valence quark configurations of the decuplet baryons are listed. 184 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Table 6 + P = 3 Mass, width, lifetime, and valence quark configuration of the J 2 decu- plet baryons. Source: From Ref. [10]. State Mass/MeV Width/MeV Life time/s Configuration ∆++ 1230 120 10−23 [uuu] ∆+ 1232 120 10−23 [uud] ∆0 1234 120 10−23 [udd] ∆− 1237 120 10−23 [ddd] Σ∗+ 1385 100 10−23 [uus] Σ∗0 1385 100 10−23 [uds] Σ∗− 1385 100 10−23 [dds] Ξ ∗0 1530 50 10−23 [uss] Ξ ∗− 1530 50 10−23 [dss] Ω− 1672 9.9 0.810−10 [sss]

The large decay widths of the decuplet baryons indicate a strong coupling to the final meson–nucleon decay channels, thus pointing to wave functions with a considerable amount of virtual meson–nucleon admixtures. However, as discussed below, there is theoretical evidence that the amount of mixing varies over the multiplet with the tendency to decrease with increasing mass. At present, QCD-inspired effective models are still highly useful approaches to understand baryons at least until LQCD [4] and functional methods, e.g. [315], will be able to treat of decay channels quantitatively. The coupling to meson–baryon configurations has been exploited in a number of theoretical investigations, among others especially by the Valencia group. Aceti and Oset [316,317] are describing in their chiral unitary formalism the decuplet states and hadronic states above the ground state octets as dynamically generated, composite states in terms of meson–baryon or meson–meson scattering configurations. They apply an extension of the Weinberg compositeness condition on L = 1 partial waves and resonant states to determine the weight of the meson–baryon component in the ∆(1232) resonance and the other members + P = 3 of the J 2 baryon decuplet. The calculations predict an appreciable πN fraction in the ∆(1232) wave function, as large as 60%. However, this at first sight surprising result looks more acceptable when one recalls that experiments on deep inelastic and Drell–Yan reactions are indicating that already the nucleon contains admixtures of virtual below-threshold pion-like uuN¯ and ddN¯ components on a level of up to 30% [318,319]. The wave functions of the larger mass decuplet baryons contain smaller meson–baryon components, steadily decreasing with mass. Thus, the Σ∗, Ξ ∗ and especially the Ω− baryons acquire wave functions in which the meson–baryon components are suppressed and genuine QCD-like configurations start to dominate. A rather diverse picture is emerging from those studies, indicating the necessity for case-by-case studies, assigning a large pion– nucleon component to the ∆(1232) but leading to different conclusions about the decuplet baryons with non-vanishing strangeness. These differences have a natural explanation by considering particle thresholds: S = −1 baryons should couple preferentially to the KN¯ channel but that threshold is much higher than the pion–nucleon one. The S = −2 baryons would couple preferentially to K¯ Λ or K¯ Σ channels with even higher thresholds and so on. The Aceti–Oset approach was further 3 + extended by investigating the formation of resonances by interactions of 2 decuplet baryons with pseudo-scalar mesons from the lowest 0− octet [320] and vector mesons from the lowest 1− octet [321], respectively, thus investigating even higher resonances. The coupling to meson–baryon channels will also affect states below the particle emission threshold by virtual admixtures of the meson–baryon continuum. Those effects are found not only for the aforementioned Λ(1405) state [11] but also the 1 + Λ(1520) [322] resonances. A compelling insight from those and similar studies is that the baryons above the lowest 2 octet have much richer structure than expected from a pure quark model with valence quarks only. The same features, by the way, are also found in mesonic systems. The best studied case is probably ρ(770) JP = 1− vector meson which is known to be a pronounced ππ p-wave resonance [10]. Also the other members of the 1− vector meson octet contain strong substructures given by p-wave resonances of mesons from the 0− pseudo-scalar octet. For example, in [323] the Aceti–Oset approach was used to investigate the Kπ-component of the K ∗(800) vector meson. Prominent examples are also the scalar mesons. All members of the 0+ meson octet are dominated by meson-scattering configurations of the 0− multiplet, as discussed in the previous sections. Besides spectral studies there is a general interest in meson–baryon interactions as an attempt to generalize the work from NN- and YN-interaction to higher lying multiplets. The chiral SU(3) quark cluster model was used in [324] to derive interactions among decuplet baryons, neglecting, however, the coupling to the decay channels. In the framework of the resonating-group method, the interactions of decuplet baryon–baryon systems with strangeness S = −1 and S = −5 were investigated within the chiral SU(3) quark model. The effective baryon–baryon interactions deduced from quark– quark interactions and scattering cross sections of the Σ∗∆ and Ξ ∗Ω systems were calculated. The so restricted study led to rather strongly attractive decuplet interaction, producing deeply bound Σ∆ and ΞΩ dibaryons with large binding energies, exceeding that of the deuteron by at least an order of magnitude. These results resemble the deeply bound S = −2 H-dibaryon predicted by Jaffe [325]. Here, we are less ambitious and consider mainly interactions of the ∆ baryon and few other resonances in nuclear matter. H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 185

Fig. 62. Diagrams entering into the self-consistent description of the dressed pion propagators (upper row), the ∆ propagator (second row), and the resulting dressed vertex (third row). Nucleon propagators are given by lines, the ∆ propagator is shown as a double-line. Bare vertices are indicated by filled circles, the dressed vertices are denoted by Γ . V is the ∆N−1 residual interaction. Note that the ∆-resonance obtains a self-energy due to its decay into intermediate πN configurations.

12.2. Excitations of resonances in nuclear matter: the N∗N−1 resonance nucleon-hole model

The Delta resonance is taken here as a representative example but the results can be generalized essentially unchanged also to other resonances N∗ after the proper adjustments of vertices and propagators as required by spin, isospin, and parity. The creation of a resonances in a nucleus amount to transform a nucleon into an excited intrinsic states, Thus, the nucleon is removed from the pre-existing Fermi-sea, leaving the target in a N∗N−1 configuration. That state is not an eigenstate of the many-body system but starts to interact with the background medium through residual interactions VNN∗ . The appropriate theoretical frame work for that process is given by the polarization propagator formalism [326], also underlying, for example, the approaches in [327–329]. In brief, the Delta-hole approach consists of calculating simultaneously the pion self-energies and effective vertices by the coupling to the ∆N−1 excitations of the nuclear medium. These requirements are illustrated in Fig. 62, where the diagrams representing the approach are shown. As seen in that figure, the Dyson equations for the propagation of pions and ∆’s in nuclear matter have to be solved self-consistently. The ultraviolet divergences of the loop integrals are regularized by using properly defined hadronic vertex functions. An elegant and transparent formulation of the ∆N−1 problem is obtained by the polarization propagator method [326]. We consider first the Green’s function of the interacting system. Here, we limit the investigations to the coupling of NN−1 and ∆N−1 modes. For the non-interacting system the propagator is given by a diagonal matrix of block structure

( (0) ) G − (ω, q) 0 G(0)(ω, q) = NN 1 , (116) 0 G(0) (ω, q) ∆N−1 where G(0) and G(0) describe the unperturbed propagation of two-quasiparticle NN−1 and ∆N−1 states through the NN−1 ∆N−1 nuclear medium, including, however, their self-energies ΣN,N∗ of Fig. 63. In momentum representation, these propagators are given by the Lindhard functions [326] ∫ d4p φN (k) = i GN (p)GN (p + k) , (117) (2π)4 ∫ d4p φ∆(±k) = i GN (p)G∆(p ± k), (118) (2π)4 which is a particular convenient for infinite matter. Here, p denotes the 4-momentum and GN (p) and G∆(p) are the nucleon and ∆ propagators: 1 G (p) = + 2πi n(p)δ(p0 − ε(p) − Σ (p2)) , (119) N 0 N p − ε(p) − ΣN (p) + i0 µν 1 G (p) = δµν , (120) ∆ 0 2 p − ε∆(p) − Σ∆(p ) with the single particle (reduced kinetic) energies εN,∆, and n(p) is the nucleon occupation number. We follow the general 1 practice and approximate the Delta-propagator by the leading order term resembling a spin- 2 Green’s function, thus leaving away the complexities of a Rarita–Schwinger propagator discussed in Section7. This is an acceptable approximation in the low-energy limit used below where the neglected terms will be suppressed, anyway. Since Σ∆ is generically of non- Hermitian character, we can omit the infinitesimal shift into the complex plane. 186 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 63. In-medium interactions of a baryon resonance N∗ via the static mean-field (left) and the dispersive polarization self-energies (center) indicated here by the decay into intermediate nucleon–meson configurations. Moreover, in nuclear matter the coupling to NN−1 excitations contributes a spreading width (right).

In the non-relativistic limit of cold infinite matter with nucleons filling up the Fermi sea up to the Fermi momentum kF , = 2 − 2 ∝ we have n(p) θ(kF p ). The nucleon propagator Eq. (119) consists of the vacuum part and the in-medium part ( n(p)). The ∆ propagator Eq. (120) includes the vacuum part only, since we have neglected the presence of preformed ∆ excitations in nuclear matter. Both propagators take into account effective mass corrections if present. After the contour integration over p0 (c.f. [326]) the nucleon-hole Lindhard function, Eq. (117), takes the following form: ∫ d3p ( n(p + k)(1 − n(p)) n(p)(1 − n(p + k)) ) φN (k) = − + , (121) (2π)3 ε∗(p + k) − k0 − ε∗(p) + i0 ε∗(p) + k0 − ε∗(p + k) + i0 ∗ = + where for simplicity, we have introduced the quasiparticle energies εN (p) εN (p) ΣN (p). Corresponding expressions are found for the Delta-hole Lindhard function, Eq. (118): ∫ d3p n(p) φ (±k) = − . (122) ∆ 3 ∗ − ∗ ± ± 0 (2π) εN (p) ε∆(p k) k Replacing the dependence of the self-energies on the integration variable by a conveniently chosen external value, for example by replacing the argument by the pole value, the integration can be performed in closed form in the zero temperature limit. Analytic formulas are found in Ref. [326]. Including the residual NN−1 and ∆N−1 interactions by (V V ) V = NN N∆ (123) V∆N V∆∆ the Green function of the interacting system is given by the Dyson equation for the 4-point function

G(ω, q) = G(0)(ω, q) + G(0)(ω, q)VG(w, q) (124) truncated to the two-quasiparticle sector, i.e. evaluated in Random Phase Approximation (RPA). Actually, the approach discussed below corresponds to a projection, not a truncation, to the 4-point function because the coupling to the hierarchy of higher order propagators is taken into account effectively by induced self-energies and interactions. The coherent response of the many-body system with ground state |A⟩ to an external perturbation described by an iq·r Sa Ta iq·r operator Oa(q) ∼ e σ τ where a = (S, T ) denotes spin (Sa = 0, 1), isospin (TA = 0, 1) and momentum (∼e ) transfer, is described by the polarization propagators of the non-interacting system (0) = ⟨ | † | ⟩ Πab (ω, q) A Ob G(0)Oa A (125) and the interacting system = ⟨ | † | ⟩ Πab(ω, q) A Ob GOa A (126) which – by definition – has the same functional structure as the propagator G. For a single resonance the polarization tensor is given by a 2-by-2 tensorial structure ( ) ( (0) (0) ) ( (0) (0) ) ( )( ) Π Π ΠNN ΠN∆ ΠNN ΠN∆ V V Π Π NN N∆ = + NN N∆ NN N∆ , (127) Π∆N Π∆∆ (0) (0) (0) (0) V∆N V∆∆ Π∆N Π∆∆ Π∆N Π∆∆ Π∆N Π∆∆ where the reference to the transition operators Oa,b is implicit. The diagrammatic structure is shown in Fig. 64. The mixing of the NN−1 and the ∆N−1 configurations by the residual interactions displayed in Fig. 65, is resulting in the mixed polarization tensors ΠN∆ and Π∆N , respectively. The polarization tensor contains the full multipole structure as supported by the nuclear system, the interaction V, and the external transition operators Oa,b. Thus, a decomposition into irreducible tensor H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 187

Fig. 64. The RPA polarization propagator. The N−1N → N−1N (left), the mixed N−1N → N−1∆ and the N−1∆ → N−1∆ components are displayed. Also the bare particle–hole type propagators are indicated. External fields are shown by wavy lines, the residual interactions are denoted by dashed lines. Only part of the infinite RPA series is shown.

components may be performed. Alternatively, a decomposition into longitudinal and transversal components is frequently invoked. In practice, one often evaluates the tensor elements in infinite nuclear matter and applies the result in local density approximation by mapping the particle densities and Fermi momenta to the corresponding radial-dependent quantities of a finite nucleus, e.g. ρp,n → ρp,n(r), see e.g. [330–332]. Once the polarization propagator is known, observables are easily calculated. Spectral distributions and response function are of particular importance because they are entering directly into cross sections. The response functions are defined by 1 R (ω, q) = − Im [Π (ω, q)] . (128) ab π ab The response function techniques are frequently used in lepton induced reactions like (e, e′p) or neutrino-induced pion production (ν, µπ). In [25,333] similar methods have been applied in light ion-induced charge exchange reactions up to the Delta-region including also the pion-production channel. A recent application to N∗ excitations in heavy ion charge exchange reaction with Sn-projectiles is found in [334,335]. A widely used choice for V is a combination of pion-exchange and contact interactions of Landau–Migdal type cor- responding to the afore mentioned OPEM approach, see e.g. [327–329]. Inclusion of other mesons, e.g. the ρ-meson, is easily obtained. The pion exchange part is taken care of the long-range interaction component. In the frequently used non- relativistic reduction (c.f. [336]), but using relativistic kinematics, the πNN and πN∆ interactions, the Lagrangians are:

f † LπNN = ψ στψ · ∇π , (129) mπ f∆ † LπN∆ = ψ∆STψ · ∇π + h. c., (130) mπ where ψ, ψ∆ and π are the nucleon, ∆ resonance and pion field respectively. The dot-product indicates the contraction of the spin and gradient operators. Typically values for the coupling constants are f = 1.008 and f∆ = 2.202, see e.g. [329,337]. σ and τ are the spin and isospin Pauli matrices. The (1/2 → 3/2) spin and isospin transition operators are given by S and T, defined according to Ref. [338]. Pion-exchange is seen to be of spin-longitudinal structure. Occasionally, also ρ- meson exchange is treated explicitly, e.g. [333,330] introducing an explicit spin-transversal interaction component into V which, however, can be decomposed into spin–spin and spin-longitudinal terms [339]. The short range pieces are subsumed into the contact interactions, defined by the following Lagrangian:

2 f ′ L = − g (ψ †στψ) · (ψ †στψ) SRC 2 NN 2mπ [ ] f∆ ′ − g (ψ †στψ) · (ψ † STψ) + h. c. 2 N∆ ∆ mπ 2 f ′ − ∆ g (ψ † STψ) · (ψ †S†T†ψ ) 2 ∆∆ ∆ ∆ mπ [ 2 ] f ′ − ∆ g (ψ † STψ) · (ψ † STψ) + h. c. . (131) 2 ∆∆ ∆ ∆ 2mπ ′ ′ ′ gNN , gN∆ and g∆∆ are the Landau–Migdal parameters. The spin–isospin scalar products are indicated as a dot-product. In the literature the values of the Landau–Migdal parameters are not fixed unambiguously by theory but must be ′ = ′ = ′ ≡ ′ determined on phenomenological grounds. Within a simple universality assumption gNN gN∆ g∆∆ gBW , which is ′ = ± the so-called Bäckmann–Weise choice (see Ref. [340] and Refs. therein), one gets gBW 0.7 0.1 from the best description of the unnatural parity isovector states in 4He,16O and 40Ca. However, the same calculations within the Migdal model [341] 188 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 65. Direct (a) and exchange (b) diagrams of the OPEM approach for the process N1N2 → N3∆4. The wavy line denotes either π (and ρ) exchange or ∝ ′ the contact interaction gN∆.

′ = ′ = ′ = assumption gN∆ g∆∆ 0 produce gNN 0.9...1. The description of the quenching of the Gamow–Teller matrix elements ′ = ′ = ′ requires, on the other hand, g∆∆ 0.6...0.7 (assuming gN∆ g∆∆)[342]. The real part of the pion optical potential ′ = ′ = in π-atoms implies gN∆ 0.2 and g∆∆ 0.8 [341]. The pion induced two-proton emission is the best described with ′ = gN∆ 0.25...0.35. The Delta self-energies are a heavily studied subject because they are of particular importance for pion–nucleus interactions. c.f. [339,343]. The general conclusion is that the modifications of the decay width by the medium can be expressed to a good approximation as a superposition of Pauli-blocking terms from the occupation of the Fermi-sea and an absorption or spreading term due to the coupling to NN−1 excitations [344,345]

Γ∆(ω, ρ) = −2ImΣ∆(ω, ρ) ∼ Γfree(ω) + ΓPauli(ω, ρ) + Γabs(ω, ρ) . (132)

Since the self-energies are not known over the large energy and momentum regions necessary for theoretical applications, parametrizations are introduced and used for extrapolations. A frequently used parametrization of the in-medium width is ( ∗ )3 ∗ 2 + 2 ∗ ∗ ρ 0 q(ω, mN , mπ ) m∆ β0 q (m∆, mN , mπ ) Γ∆(ω) = Γabs(ω) + Γ , (133) ∆ 2 + 2 ∗ ρsat q(m∆, mN , mπ ) ω β0 q (ω, mN , mπ ) √ −3 where β0 is an adjustable parameter and ρsat = 0.16 fm is the nuclear saturation density. ω = s∆ is the total relativistic energy of the Delta resonance in the medium as defined in the π+N system . The spreading width due to the coupling to NN−1 modes is denoted by Γabs and a simple scaling law is used for the density dependence. The Lorentz-invariant center-of-mass momentum is defined as usual,

2 2 2 2 2 2 q (ω, m1, m2) = (ω − (m1 + m2) )(ω − (m1 − m2) )/4ω . (134) ∗ The free and effective nucleon and Delta in-medium masses are denoted by mN,∆ and mN,∆, respectively. Using a (subtracted) dispersion relation the real part can be reconstructed by a Cauchy Principal Value integral ω − m ∫ Γ (ω′) = − ∆ ′ ∆ Re(Σ∆(ω)) P dω ′ ′ , (135) π (ω − m∆)(ω − ω) by which the dispersive self-energy is completely determined.

12.3. ∆ mean-field dynamics and resonances in neutron stars

While the dispersive Delta self-energies have obtained large attention, the static mean-field part is typically neglected or treated rather schematically. Some authors use the universality assumption stating that the Delta mean-field should agree with the nucleon one. In the relativistic Hartree scheme this amounts to use the same scalar and vector coupling constants for nucleons and resonances. Obviously, that assumption comes to an end in charge-asymmetric matter by the fact that the Delta resonance comes in four charge states. A simple but meaningful extension is to introduce also an isovector potential, thus allowing for interactions of the resonance and the background medium through exchange of isovector scalar and vector ∆ mesons. In the non-relativistic limit, the Delta Hartree potential becomes a sum of isoscalar and isovector potentials U0 and ∆ U1 , respectively: 2 U (H) = U∆ + U∆τ∆ · τN , (136) ∆ 0 A 1 ∆ N where A is the nucleon number and τ = 2T∆ and τ denote the Delta and nucleon isospin operators, respectively, with the known properties of isospin

N N τ3 |p⟩ = +|p⟩; τ3 |n⟩ = −|n⟩ (137) H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 189 and

++ 3 ++ + 1 + T ∆|∆ ⟩ = + |∆ ⟩; T ∆|∆ ⟩ = + |∆ ⟩ 3 2 3 2 1 − 3 − T ∆|∆0⟩ = − |∆0⟩; T ∆|∆ ⟩ = − |∆ ⟩. (138) 3 2 3 2 In a nucleus, the resonance is moving in background medium with Z protons and N neutrons and A = N + Z. Thus, integrating out the nucleons, our simple model potential becomes N − Z U (H) ≃ U∆ − U∆τ ∆ . (139) ∆ 0 1 z A Thus, we find − (H) ∆ ∆ N Z U ++ = U − 3U (140) ∆ 0 1 A − (H) ∆ ∆ N Z U + = U − U (141) ∆ 0 1 A N − Z U (H) = U∆ + U∆ (142) ∆0 0 1 A − (H) ∆ ∆ N Z U − = U + 3U (143) ∆ 0 1 A ∆ = N N = − − and the universality assumption would mean to use U0,1 U0,1. At the nuclear center, typical values are U0 40 − N = + + 3 60 MeV and U1 20... 30 MeV , depending on the chosen form factor. These estimates agree quite well with those of the involved many-body calculations in [346]. Actually, also spin–orbit potentials will contribute about which nothing is known. As for nucleons and hyperons the RMF approach is also being used to describe the N∗ mean-field dynamics. In Refs. [71–73] and also [74,75] the Delta resonance was included into the RMF treatment. In [72] ∆ dynamics in nuclear matter is described by the mean-field Lagrangian density

= [ µ − − − µ − µ] ν L∆ ψ ∆ ν iγµ∂ (m∆ gσ ∆σ ) gω∆γµω gρ∆γµI3ρ3 ψ∆ , (144) ν ++ + 0 − where ψ∆ is the Rarita–Schwinger spinor for the full set of ∆(1232)-isobars (∆ , ∆ , ∆ , ∆ ) and I3 = diag(3/2, 1/2, −1/2, −3/2) is the matrix containing the isospin charges of the ∆s. Assuming SU(6) universality, the same coupling constants as for nucleons may be used. However, contributions from dispersive self-energies will surely spoil SU(6) symmetry and deviations from that rule will occur. Interestingly, neutron stars may be useful systems to study N∗ mean-field dynamics [71,72,74] and also in [75]. In [71,72] nucleons, hyperons and Deltas are described within the same RMF approach, used to investigate the composition of neutron star matter. In Fig. 66 particle fractions as a function of the baryon density nB = ρB of ρsat are shown. With the ∆-resonance included the particle mixtures are changed considerably. Furthermore, the onset of the Delta appearance depends on the gω∆ RMF coupling constant. That effect is illustrated in Fig. 66 by varying xω∆ = . gωN Moreover, the investigations in [71,72] lead to the important conclusion that the onset of ∆-isobars is strongly correlated with the value of the slope parameter L of the density dependence of the symmetry energy. For the accepted range of values of about 40 < L < 120 MeV [347], the additional Delta degrees of freedom influence the appearance of hyperons and cannot be neglected in the EoS of beta-stable neutron star matter. This correlation of the ∆ onset and the symmetry energy slope are indicating also another interesting interrelation between nuclear and sub-nuclear degrees of freedom. These findings are leading immediately to the question to what extent the higher N∗ resonances will influence the nuclear and neutron star equations of state.

12.4. Response functions in local density approximation

An application of the response function formalism is shown in Fig. 67 where the spectra for the Fermi-transition operator 58,64,78 Oa = στ+ for Ni are shown. The response functions are normalized to the nucleon numbers A = 58, 64, 78. As discussed above, the polarization tensor, Eq. (127), was evaluated in infinite matter at a dense mesh of proton and neutron ′ densities, thus leading to ΠBB′ (ρp, ρn) for B, B = N, ∆. By mapping the nucleon densities to the radial densities ρp,n(r) of the Ni-isotopes, we obtain in local density approximation ΠBB′ (r). The response function shown in Fig. 67 is obtained finally by integration ∫ 1 1 3 R(ω, q) = − d rρA(r)Im [ΠNN (ω, q, r) + Π∆∆(ω, q, r)] , (145) π A

3 More meaningful values are in fact the volume integrals per nucleon. 190 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 66. Particle fractions as functions of the baryon density within the SFHo model: only hyperons (panel (a)), hyperons and ∆s (panel (b)) for − xσ ∆ = xω∆ = xρ∆ = 1. The red line indicates the fraction of the ∆ which among the four ∆s are the first to appear. The blue and the green vertical − lines indicate the onset of the formations of ∆ for xω∆ = 0.9 and xω∆ = 1.1, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Source: From Ref. [72].

58,64,78 Fig. 67. RPA response function per nucleon for the operator στ+ in the isotopes Ni [348] obtained by an energy density functional as in [101]. Note the apparent shift of the ∆N−1 peak to higher energy which is introduced by mixing with the NN−1 component due to the residual interactions.

−1 where q = |q| and ρA = ρp + ρn is the total nuclear density. In the response functions of Fig. 67, the quasi-elastic NN and the deep-inelastic ∆N−1 are clearly seen. The two components are mixed by the residual interactions, chosen in that calculation according to the (non-relativistic) density functional of Ref. [101]. The configuration mixing induces an upward shift in energy of the Delta-component and a downward shift in energy of the quasi-elastic nucleon component. In this respect, the system behaves in a manner as typical for a coupled two-by-two system. There is a large body of data available on inclusive (e, e′p) cross sections [349–352] which are the perfect test case to the response function formalism. The cross sections are given by the superposition of longitudinal (RLL) transversal (RTT ) and interference terms (RTL) and (RLT ), respectively, weighted with the proper kinematical factors. In the high energy limit, the electron scattering waves are approximated sufficiently well by plane wave. Results of such a calculation [348] are shown in Fig. 68 where the double differential cross section for the reaction 40Ca(e, e′p) at an electron incident energy Tlab = 500 MeV at fixed momentum transfer q = 300 MeV/c are shown. The data are surprisingly well described by our standard choice of self-energies and interactions although no attempt was made to optimize parameters. The energy gap between the quasi-elastic and the resonance spectral components seems to be slightly too large, indicating that the configuration mixing interaction VN∆ was chosen slightly too strong. In Xia et al. [353], the close connection of in-medium pion interactions and Delta-hole excitations on the one side and nuclear charge exchange reactions and photo-absorption on the other side, were considered in detail. In that work it is emphasized that, unlike the conventional picture of level mixing and level repulsion for the pionic and ∆N−1 states, the real H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 191

′ Fig. 68. Double differential cross section for Ca(e, e p) at Tlab = 500 MeV [348]. The underlying longitudinal and transversal response functions were obtained by an energy density functional as in [101]. Note the apparent shift of the ∆N−1 peak to higher energy which is introduced by mixing with the quasi-elastic NN−1 component due to the residual interactions.

part of the pion inverse propagator vanishes at only one energy for each momentum because of the width of the Delta-hole excitations. The results of this self-consistent approach has been compared successfully to data on (p, n) charge exchange reactions and photo-absorption on nuclei in the ∆-resonance region. Moreover, the interesting result is obtained that the baryonic vertex form factors obtained for pionic and electromagnetic probes agrees with their interpretation as effective hadronic structure functions.

13. Production and spectroscopy of baryon resonances in nuclear matter

13.1. Resonances as nuclear matter probes

Exploring the spectral properties of resonances and their dynamics in the nuclear medium is the genuine task of nuclear physics. In Fig. 69 interactions are indicated which in heavy ion reactions are leading to excitation of nucleon resonances. In fact, already in the 1970’s first experiments were performed at the AGS at Brookhaven with proton-induced reaction on a series of targets between Carbon and Uranium [354]. It was recognized soon that charge exchange reactions would be an ideal tool for resonance studies. As early as 1976 high energy (p, n) reactions were used at LAMPF to investigate resonance excitation in heavy target [355,356]. About the same time also (n, p) reactions were measured at Los Alamos [357,358]. A few years later, similar experiments utilizing (p, n) charge exchange were started also at Saclay [359]. A broader range of phenomena can be accessed by heavy ion charge exchange reactions with their particular high potential for resonance studies on nuclei by observing the final ions with well defined charge numbers. This implies peripheral reactions corresponding to a gentle perturbation by rearranging of the initial mass and charge distributions by one or a few units. Thus, the colliding ions are left essentially intact and the reaction corresponds to a coherent process in which the mass numbers of projectile and target are conserved but the arrangement of protons and neutrons is modified. Experimental groups at SATURNE at Saclay took the first-time chance to initiate dedicated experiments on in-medium resonance physics especially with (3He, 3H) reactions on heavy targets [360,361]. In a series of experiments, the excitation of the Delta-resonance was observed. Due to the experimental limitations at that time only fixed target experiments on stable nuclei were possible. A few years later, similar experiments were started at the Synchro-Phasetron at JINR Dubna, making use of the higher energy at that facility to extend the spectral studies up to the region of the nucleon–pion s-wave and d-wave resonances [362,363]. While initially those experiments were concentrating on inclusive reactions, a more detailed picture is obviously obtained by observing also the decay of the excited states. Such measurements were indeed performed in the early 1990’s with the DIOGENE detector at SATURNE [364,365], at KEK using the FANCY detector [366]. However, with respect to the first experiments it took another decade or so before those exclusive data were studied theoretically [367]. A couple of years later, corresponding experiments were done at Dubna, taking advantage of the higher energies of up to plab = 4.2 AGeV/c reached at the Synchro- Phasetron [368–371]. In [372] the measurements were extended to the detection of up to N∗ → pπ +π − three particle decay channels in coincidence allowing to identify also the N∗(1440) Roper resonance and even higher resonances. A spectral distribution is shown in Fig. 77. The generic interaction processes shown in Fig. 69 are using a meson exchange picture which describes successfully the dynamics of N∗ production in ion–ion reactions. The ∆(1232) is produced mainly in NN → ∆N reactions and the Delta is subsequently decaying into Nπ, thus producing in total a NN → NNπ transition. The pion yield from the Delta source, coming from a p-wave process, competes with direct s-wave pion production, NN → NNπ. As discussed in Section5 the intermediate population of higher resonances like P11(144) will lead to NN → NNππ processes. With increasing energy baryons will be excited decaying into channels with higher pion multiplicities. Already the early theoretical studies lead 192 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 69. Resonance production by charge–neutral meson exchange (left) and charged meson exchange (right).

to the conclusion that in heavy-ion collisions at around 1 AGeV up to 30% of the participating nucleons will be excited into resonances. Thus, a kind of short-lived resonance matter is formed before decaying back into nucleons and mesons [373,374]. That figure is largely confirmed by the work of Ref. [371] on deuteron induced resonance production at the Synchro- = ± +4 Phasetron at plab 4.2 AGeV/c, although there a somewhat lower resonance excitation rate of 15 2−3% was found which ± +4 is in the same bulk as the excitation probability of 16 3−3% derived from proton induced reactions at the same beam momentum.

13.2. Interaction effects in spectral distribution in peripheral reactions

The excitation of the Delta resonance in proton and light ion induced peripheral reactions at intermediate energies was studied theoretically in very detail by Osterfeld and Udagawa and collaborators [25,333,375,376]. The first round of experimental data from 1980ties and 1990ties were investigated by microscopic theoretical approaches covering inclusive and semi-inclusive reactions. The conclusion from those studies are still valuable and are worth to be recalled. In the Osterfeld–Udagawa approach initial and final state interactions were taken into account by distorted wave methods. A microscopic approach was used to describe the excitation and intrinsic nuclear correlations of ∆N−1 states. This combination of – at that time – very involved theoretical methods was able to explain the observed puzzling shift of the Delta-peak by ∆M ∼ −70 MeV. In Fig. 70 conclusive results on that issue are shown. The calculations did not include the quasi-elastic component, produced by single and multiple excitations of NN−1 states and knock-out reactions. In the Delta region the theoretical 12C(He, t) cross section matches the experimental data almost perfectly well. Distorted wave effects, i.e. initial and final state interactions of the colliding ions, are of central importance for that kind agreement. They alone provide a shift of about ∆MDW ∼ −50 MeV [333]. Finite size and a detailed treatment of particle–hole correlations −1 within the ∆N configurations contribute the remaining ∆Mc ∼ −20 MeV . The polarization tensor may be decomposed into pion-like longitudinal contributions, the complementary transversal components, representative of vector–meson interactions, and mixed terms, see e.g. [376]. An interesting results, shown in Fig. 70, is that the longitudinal (LO) partial cross section appears to be shifted down to a peak values of ωL ∼ 240 MeV , while the transversal partial cross section 3 peaks at ωL ∼ 285 MeV. This is an effect of the He → t transition form factor which reduces the magnitude of the TR spectrum at high excitation energies because of its exponential falloff at large four-momentum transfer. The shape of the LO spectrum is less strongly affected by this effect. It is remarkable that in contrast to (p, n) reactions the full calculation reproduces the higher energy part of the spectrum so well. This is due to the fact that the high-energy flank of the resonance is practically background-free, since the probability that the excited projectile decays to the triton ground state plus a pion is extremely small. Also a negligible amount of tritons is expected to be contributed from the quasi-free decay of the target. The cross section in the resonance region shows an interesting scaling behavior: A proportionality following a (3Z + N) -law is found where Z and N are the proton and neutron number of the target. This dependence of the cross section reflects that the probability for the p + p → n + ∆++ process is three times larger than that for the p + n → n + ∆+ process. An even more detailed picture emerges from semi-inclusive reactions observing also decay products. For the reactions discussed in [364–366,377] at incident energies of about 2 AGeV, the pπ correlations were successfully analyzed and the mass distribution of the ∆(1232) resonance could be reconstructed. In these peripheral reactions on various targets, the resonance mass was found to be shifted by up to ∆M ≃ −70 MeV toward lower masses compared to those on protons. In reactions on various nuclei at incident energies around 1 AGeV the mass reduction of the ∆(1232) resonance was traced back to Fermi motion, NN scattering effects, and pion reabsorption in nuclear matter. These findings are in rough agreement with detailed theoretical studies of in-medium properties of the ∆-resonance by the Valencia group [344,378], considering also the decrease of the Delta-width because of the reduction of the available Nπ decay phase space by Pauli-blocking effects of nucleons in nuclear matter.

13.3. Resonances in central heavy ion collisions

Different aspects of resonance dynamics are probed in central heavy ion collisions. The process responsible for meson production in central heavy-ion collisions at energies of the order of several hundred MeV/nucleon to a few GeV/nucleon is believed to be predominantly driven by the excitation of baryon resonances during the early compression phase of the H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 193

12 3 Fig. 70. Zero-degree triton spectra for the reaction C( He, t) at Tlab = 2 GeV. The data are taken from Ref. [360]. The full curve represents the final result including initial and final state interactions, finite size effects, and particle–hole correlations. In addition, the longitudinal (LO) and transverse (TR) partial cross sections are shown. Source: From Ref. [333].

collision [15,14,373,374,12,379,380]. In the later expansion phase these resonances decay into lower mass baryon states and a number of mesons. The influence of the medium is expected to modify mass and width of the resonances by induced self-energies. In high density and heated matter, however, the genuine self-energy effects may be buried behind kinematical effects. The best studies case is the ∆33(1232) resonance which, in fact, is a 16-plet formed by four isospin and four spin sub- states. In nucleon–nucleon one observes a centroid mass M∆ ≃ 1232 MeV and the width Γ∆ = 115–−120 MeV which is in good agreement with the direct observation in pion–nucleon scattering [10]. Modifications of mass and width of the ∆(1232) resonances have been observed in central heavy-ion collisions leading to dense and heated hadron matter, e.g. at the BEVALAC by the EOS collaboration [381] and at GSI by the FOPI collaboration [380]. In [381], for example, the mass shift and the width were determined as functions of the centrality, both showing a substantial reduction with decreasing impact parameter. The modification of ∆33(1232) properties has been interpreted in terms of hadronic density, temperature, and various non-nucleon degrees of freedom in nuclear matter [382–384]. The invariant mass distributions of correlated nucleon and pion pairs provide, in principle, a direct proof for resonance excitation. As discussed in [380], in the early heavy ion collision experiments a major obstacle for the reconstruction of the resonance spectral distribution was the large background of uncorrelated pπ pairs coming from other sources. Only after their elimination from the data the resonance spectral distributions could be recovered. Results of a first successful resonance reconstruction in central heavy ion collisions are shown in Fig. 71.

13.4. The delta resonance as pion source in heavy ion collisions

As discussed in Section8 transport calculations are describing accurately most of the particle production channels in heavy-ion collisions already in the early days of transport theory [264,14,379,385]. However, surprisingly the pion yield from heavy-ion collisions at SIS energies (Tlab ∼ 1 AGeV) could not be reproduced properly by the transport-theoretical description. For a long time the pertinent overprediction of the pion multiplicity [386–393] was a disturbing problem. At the beam energies of a few AGeV the dominating source for pion production is the excitation of the ∆(1232) resonance in a NN collision NN → N∆ followed by the decay ∆ → Nπ. In transport calculations, the pion multiplicity, therefore, depends crucially on the value of the in-medium NN → N∆ cross section. A first attempt to solve that issue was undertaken by Bertsch et al. [327]. In the 1990ties Helgesson and Randrup [328,394] took up that issue anew. In their microscopic π +NN−1 +∆N−1 model [328] they considered the excitation of ∆N−1 modes in nuclear matter by RPA theory. The coupling to the purely nuclear Gamow–Teller-like NN−1 spin–isospin modes and the corresponding pion modes was taken into account. They point out that sufficiently energetic nucleon–nucleon collisions may agitate one or both of the colliding nucleons to a nucleon resonance with especial importance of ∆(1232), N∗(1440), and N∗(1535). Resonances propagate in their own mean-field and may collide with nucleons or other nucleon resonances as well. Moreover, the nucleon resonances may decay by meson emission and these decay processes constitute the main mechanisms for the production of energetic mesons. The derived in-medium properties of pions and ∆ isobars were later introduced into transport calculations by means of a local density approximation as discussed in the previous section, but for example also used in [389,390]. Special emphasis was laid on in-medium pion dispersion relations, the ∆ width, pion reabsorption cross sections, the NN → ∆N cross sections and the in-medium ∆ spectral function. Although the medium-modified simulations showed strong effects on in-medium properties in the early stages of the transport description the detailed in-medium treatment had only little effect on the final pion and other particle production cross sections. This is a rather reasonable result since in their calculations most of the emitted 194 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 71. The invariant mass spectrum of baryon resonances excited by the reactions stated in each panel. The filled areas correspond to the analysis of the measured transverse momentum spectra of π ±, the full points to the analysis of the measured pπ ± pairs. Traces of higher resonances are visible in the high energy tails of the spectral distributions. The arrows point to the maximum of the free ∆(1232) mass distribution. Source: From Ref. [380].

pions were produced at the surface at low densities where the in-medium effects are still quite small. Actually, in order to account also for the heating of the matter in the interaction zone, a description incorporating temperature should be used. Such a thermo-field theoretical approach was proposed independently by Henning and Umezawa [395], and by Korpa and Malfliet [396]. The approach was intentionally formulated for pion–nucleus scattering, where the coupling to the Delta resonance plays a major role, but it does not seem to have been applied afterward. Years later, the problem was reconsidered by the Giessen group. Initially, a purely phenomenological quenching prescription was used for fitting the data [393]. The breakthrough was achieved in [329] when the in-medium NN → N∆(1232) cross section were calculated within a one pion exchange model (OPEM), taking into account the exchange pion collectivity and vertex corrections by contact nuclear interactions. Also, the (relativistic) effective masses of the nucleon and ∆ resonance were considered. The ∆N−1 and the corresponding nuclear NN−1 modes, discussed above, were calculated again by RPA theory. In infinite matter the Lindhard functions [326], representing the particle–hole propagators, can be evaluated analytically. It was found that even without the effective mass modifications the cross section decreases with the nuclear matter density at high densities already alone by the in-medium ∆ width and includes the NN−1 Lindhard function (see below) in the calculations. The inclusion of the effective mass modifications for the nucleons and ∆’s leads to an additional strong reduction of the cross section. Altogether, the total pion multiplicity data [392] measured by the FOPI collaboration on the systems Ca+Ca, Ru+Ru and Au+Au at Tlab = 0.400, 1.000 and 1.500 AGeV, respectively, could be described by introducing a dropping effective mass with increasing baryon density. The results were found to depend to some extent on the in-medium value of the ∆-spreading width for which the prescription of the Valencia group was used: Γsp = 80ρ/ρ0 MeV [397,346]. The effect of the medium modifications of the NN ↔ N∆ cross sections on the pion multiplicity depends also on the assumption about other channels of the pion production/absorption in NN collisions, most importantly, on the s-wave interaction in the direct channel NN ↔ NNπ. In [329] it was found that including the effective mass modifications in the NN ↔ N∆ channel only, does not reduce pion multiplicity sufficiently, since then more pions are produced in the s-wave channel. An important conclusion for future work is that the in-medium modifications of the higher resonance cross sections do not influence the pion production at 1–2 AGeV collision energy sensitively: other particles like η and ρ mesons are, probably, more sensitive to higher resonance in-medium modifications. A subtle test for the transport description of pion production is given by (π ±, π ∓) double charge exchange (DCX) reactions on nuclear targets. In Ref. [398] such reactions were investigated by GiBUU transport calculations. The pionic double charge exchange processes were studied for a series of nuclear targets, including (16O, 40Ca and 208Pb), for pion incident energies Tlab = 120, 150, 180 MeV covering the Delta-region. As a side aspects, the results could confirm the validity of the so- called parallel ensemble scheme for those reactions in comparison to the more precise but time consuming full ensemble ± ∓ method [398]. GiBUU results for the DCX reaction π O → π X at Tlab = 120, 150 and 180 MeV are shown Fig. 72. A good agreement with data was achieved for the total cross section and also for angular distributions and double differential cross H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 195

± ∓ Fig. 72. Double differential cross sections for the DCX process π O → π X at Tlab = 120, 150 and 180 MeV . The results at different angles are shown as function of the kinetic energies of the produced pions. Data are taken from [399], only statistical errors are shown. The GiBUU results are shown as histograms, where the fluctuations indicate the degree of statistical uncertainty. Source: From Ref. [398].

sections. Some strength at backward angles and rather low pion energies below Tlab ≈ 30 MeV is still missed. A striking sensitivity on the thickness of neutron skins was found, indicating that such reactions may be of potential advantage for studies of nuclear density profiles.

13.5. Perspectives of resonance studies by peripheral heavy ion reactions

The large future potential of resonance physics with heavy ions was demonstrated by recent FRS experiments measuring the excitation of the Delta and higher resonances in peripheral heavy ion charge exchange reactions with stable and exotic secondary beams as heavy as Sn on targets ranging from hydrogen and 12C to 208Pb [400,26,335]. With these reactions, exceeding considerably the mass range accessed by of former heavy ion studies, a new territory is explored. A distinct advantage of the FRS and even more so, of the future Super-FRS facility is the high energy of secondary beams, allowing the unique experimental access to sub-nuclear excitations. This allows to perform spectroscopy in the quasi-elastic nucleonic NN−1 and the resonance N∗N−1 regions at and even beyond the Delta resonance. The elementary excitation mechanisms contributing to peripheral heavy ion charge exchange reactions are shown diagrammatically in Fig. 73. These outstanding experimental conditions open new perspectives for broadening the traditionally strong branch of nuclear structure physics at GSI/FAIR to the new territory of in-medium resonance physics. The most important prerequisites are the high energies and intensities of secondary beams available at the SUPER-FRS. In many cases, inelastic, charge exchange, and breakup or transfer reactions could be done in a similar manner at other laboratories like RIKEN, FRIB, or GANIL, only the combination of SIS18/SIS100 and, in perspective, the Super-FRS provides access beyond the quasi-elastic region allowing to explore sub-nuclear degrees of freedom. The experimental conditions at the FRS are providing a stand-alone environment of resonance studies in nuclear matter at large isospin. Reactions at the FRS will focus on peripheral processes. The states of the interacting nuclei will only be changed 196 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

Fig. 73. The excitation mechanisms of peripheral heavy ion charge exchange reactions induced by the exchange of charged mesons: quasi-elastic NN−1 excitations in target and/or projectile (a), RN−1 resonance-hole excitations in the target (b), and RN−1 excitations in the projectile (c), where R denotes the Delta P33(1232), the Roper P11(1440) or any other higher nucleon resonance.

Fig. 74. Results of recent heavy ion charge exchange reactions at the FRS at Tlab = 1 AGeV. Both p → n and n → p projectile branches were measured on the indicated targets. The excitations reached in the target are shown on the left and right of the data panels. Spectral distributions obtained from a peak-fitting procedure are also shown exposing the Delta resonance and higher resonance-like structures like the P11(1440) Roper resonance. Source: The data taken from Ref. [335].

gently in a well controlled manner. Resonances can be excited in inelastic and charge exchange reactions. In the notation of neutrino physics those reactions are probing neutral current (NC) and charge current (CC) events and the corresponding nuclear response functions. Here, obviously strong interaction vertices are involved but it is worthwhile to point out that the type of nuclear response functions are the same as in the weak interaction processes. By a proper choice of experimental conditions the following reaction scenarios will be accessible in either inelastic or charge exchange reactions with resonance excitation in coherent inclusive reactions or in semi -exclusive coherent reactions with pion detection. In the first type of reaction the energy–momentum distribution of the outgoing beam-like ejectile is observed. Since the charge and mass numbers of that particle are known it must result from a reaction in which it was produced in a bound state. Such reactions primarily record resonances in the target nuclei, folded with the spectrum of bound inelastic or charge exchange excitations, respectively, in the beam-like nucleus. Hence, the reaction is coherent with respect to the beam particles. Results of a recent experiment at the FRS, proving the feasibility of such investigations, are shown in Fig. 74. In the spectra, the quasi-elastic and the resonances regions, discussed in Fig. 73, are clearly seen and energetically well resolved. The second scenario is different by the detection of pions emitted by the highly excited intermediate nuclei. By tagging on the pions from the beam-like nuclei one obtains direct information on the spectral distributions of the pion sources, i.e. the nucleon resonances. This scenario, illustrated in Fig. 75, will provide access to resonance studies in nuclei with exotic charge-to-mass ratios. Obviously, also pions from the target nuclei can be observed which corresponds to similar early experiments at SATURNE [364,365], the Dubna Synchro-Phasetron [372] and, at slightly lower energies, at the FANCY detector at KEK [366]. In the Dubna experiments single and double pion channels have been measured. The gain in spectroscopic information is already visible in the pπ singles spectra, Fig. 76, and even more so in the pππ spectra in Fig. 77. In coincidence experiments measuring the decay particles of in-medium resonances are obviously complementary to the issues discussed in Section5 by establishing a connection of meson production on the free nucleon and on nuclei. Heavy ions and pions are strongly absorbed particles. Therefore, resonances will be excited mainly at the nuclear surface. Also pions from grazing reactions will carry signals mainly from the nuclear periphery. However, the high energies allow H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 197

Fig. 75. Exclusive resonance production processes by observation of the decay products.

12 12 Fig. 76. Proton–single pion yields measured at the Synchro-Phasetron in the reaction C + C at plab = 4.2 GeV/c per nucleon. Source: From Ref. [363].

resonance excitation also in deeper density layers of the involved nuclei. In order to overcome those limitations at least on the decay side it might be worthwhile to consider as a complementary branch to record also dilepton signals. The scientific perspectives of resonance physics at a high-energy nuclear facility like the Super-FRS at FAIR is tremendous. Pion emission will serve as indicator for resonance excitation and record the resonance properties by their spectral distribution. In the past, theoretically as well as experimentally the Delta resonance has obtained the largest attention. The work, however, was almost exclusively focused to nuclei close to stability, i.e. in symmetric nuclear matter. On the theoretical side, the main reason for that self-imposed constraint is our lack of knowledge on resonance dynamics in nuclei far off stability, although in principle theory is well aware of the complexities and changes of resonance properties in nuclear matter. Despite the multitude of published work, until today we do know surprisingly little about the isospin dependence of resonance self-energies. There is an intimate interplay between in-medium meson physics and resonance self-energies. Since the width and the mass location of resonances is closely determined by the coupling to meson–nucleon decay channels modifications in those sectors affect immediately resonance properties. At the Super-FRS such dependencies can be studied over wide ranges of neutron–proton asymmetries and densities of the background medium. Since such effects are likely to be assigned selectively to the various channels, a variation of the charge content will allow to explore different aspects of resonance dynamics, e.g. distinguishing charge states of the Delta resonance by the different in-medium interactions of positively and negatively charged pions. Last but not least, resonance physics at fragment separators will also add new figures to the astrophysical studies. In supernova explosions and neutron star mergers high energy neutrinos are generated. Their interaction with matter proceeds through quasi-elastic and, to a large extent, through resonance excitation. The assumed neutrino reheating of the shock wave relies on the knowledge of neutrino–nuclear interactions. Neutrino experiments themselves lack the resolving power for detailed spectroscopic studies. However, the same type of nuclear matrix elements is encounter in inelastic and charge exchange excitations of resonances in secondary beam experiments thus testing the nuclear input to neutral and charged current neutrino interactions. A large potential is foreseen for studying nucleon resonances in exotic nuclei which never was possible in the past and will not be possible by any other facility worldwide in the foreseeable future. The results obtained until now from the FRS-experiment are very interesting and stimulating [26,335]. Super-FRS will be an unique device to access resonance physics in a completely new context giving the opportunity to extend nuclear structure physics into a new direction. 198 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206

∗ + − Fig. 77. Observation of higher resonances in C + C collisions for a beam momentum plab = 4.2 GeV/c at the Synchro-Phasetron by N → pπ π two-pion decay spectroscopy. Expected N∗ states are indicated for the two lower structures. Source: The data are from Ref. [372].

14. Summary

Strangeness and resonance physics are fields of particular interest for our understanding of baryon dynamics in the very general context of low-energy flavor physics. Although SU(3) symmetry is not a perfect symmetry, the group-theoretical relations are exploited successfully as a scheme bringing order into to multitude of possible baryon–baryon interactions. The SU(3) scheme allows to connect the various interaction vertices of octet baryons and meson multiplets, thus reducing the number of free parameters significantly by relating coupling constants to a few elementary parameters. Since single hyperons and resonances, emersed into a nuclear medium, are not subject of the Pauli exclusion principle, their implementation into nuclei is revealing new aspects of nuclear dynamics. In that sense, hyperons and resonances may serve as probes for the nuclear many-body system. At first, hyperons, their interactions in free-space and nuclear matter was discussed. As an appropriate tool for the description of hypernuclei, density functional theory was introduced. Except for the lightest nuclei, the DFT approach is applicable practically over the whole nuclear chart and beyond to nuclear matter and neutrons star matter. The DFT discussion was following closely the content of the Giessen DDRH theory. As an appropriate method to describe the density dependence of dressed meson–baryon vertices in field-theoretical approach, nucleon–meson vertices were introduced which are given as Lorentz-invariant functionals on the matter fields. The theory was evaluated in the relativistic mean-field limit. The Lambda separation energies of the known S = −1 single- Λ hypernuclei are described satisfactory well, however, with the caveat that the experimental uncertainties lead to a spread in the derived parameters of about 20%. In hypermatter the minimum of the binding energy per particle was found to be shifted to larger −3 density (ρhyp ∼ 0.21 fm ) and stronger binding (ε(ρhyp) ∼ −18 MeV ) by adding Λ hyperons. The minimum is reached for a Λ -content of about 10% as shown in Section 4.6. The DFT results are found to be in good agreement with other theoretical calculations. Overall, on the theory side convergence seems to be achieved for single Λ- hypernuclei. However, open issues remain about the nature and mass dependence of the crucial Lambda spin–orbit strength. The existence of bound Σ hypernuclei is still undecided although the latest theoretical results are in clear favor of a weak or repulsive Σ potential. S = −2 double-Lambda hypernuclei would be an important – if not only – source of information on ΛΛ 6 interactions. Until now, the results rely essentially on a single case, the famous ΛΛHe Nagara event observed years ago in an emulsion experiment at KEK. Future research on the production and spectroscopy of those systems – as planned e.g. for the PANDA experiment at FAIR – is of crucial importance for the fields of hyperon and hypernuclear physics. The lack of detailed knowledge on interactions is also part of the problem of our persisting ignorance on the notorious hyperon puzzle in neutron stars. Nuclear reactions are the source of information on strangeness and resonances in nuclear matter. Their interpretation relies on the safe handling of the elementary processes on the nucleon. Therefore, an extended discussion was spent on the excitation of baryon resonances without and with strangeness in hadron-induced reactions on the free nucleon. The elementary amplitudes for production and decay of N∗ are essential for the quantitative description of nucleonic excitations and strangeness-carrying baryons in all kinds of nuclear reactions, either with elementary probes like pions, (anti-)nucleons, H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 199 and photo-production or in peripheral and central heavy ion collisions. In Section6 the important role of a few N∗ resonances for the production of hypernuclei in pion-induced reactions was discussed in detail. Formal aspects of high-spin resonances were the subject of Section7. Gauge invariant couplings of high-spin matter fields to their decay channels have been rarely investigated but they will be of importance for further spectroscopic work, both in free space and in matter. For investigations of central heavy ion collisions transport theory is the well established theoretical tool. The transport- theoretical description of ion–ion reactions is a highly valuable and successful approach especially for hypernuclear production studies. Microscopically developed approaches for in-medium interactions can be probed in complex situations of the reaction dynamics within kinetic approaches. We also point out that transport simulations are highly useful to simulate the event structure of proposed experiments. A combination of pre-equilibrium dynamics and statistical fragmentation is a perfect tool to improve our understanding of the complete dynamics in such reactions, i.e., pre-equilibrium propagation and dynamical particle production as well as statistical fragmentation. After a brief introduction into the theoretical formulation, we paid broad attention to the application of transport theory to a wide range of particle production processes, ranging from proton and antiproton induced reactions to the collision of massive ions at relativistic energies. Again, strangeness and resonance production is driven by the excitation of a sequence of intermediate N∗ states. For the production of hypernuclei, fragmentation reactions are playing the key role. The fragmentation scenario was applied to various types of hypernuclear production processes. In particular, it was pointed out that secondary processes as the capture of a primary Ξ-baryon by a second nucleus are indeed a feasible way to produce an abundant number of S = −2 hypernuclei, as planned at PANDA. Even the production of the exotic Ω− baryon with S = −3 should be possible, although the cross section is small. The production rate depends crucially on the accumulation of strangeness in a sequence of reactions by intermediate resonance excitation, involving also Σ∗ and Ξ ∗ states from the 3 + 2 decuplet. As a concluding remark on strangeness and hypernuclear production, we emphasize that in-medium reactions induced by heavy-ions and antiproton-beams represent an excellent tool to study in detail the multi-strangeness sector of the hadronic equation of state. The knowledge of the in-medium multi-strange interactions is essential for a deeper understanding of baryon–baryon interactions in nuclear media over a large range of densities and isospin. It is crucial also for nuclear astrophysics by giving access to the high density region of the EoS of hypermatter, at least for a certain amount of hyperon fractions. In addition, reaction studies on bound superstrange hypermatter offer unprecedented opportunities to explore the hitherto unobserved regions of exotic bound hypernuclear systems. Resonance physics is obviously of utmost importance for nuclear strangeness physics because they are the initial source for hyperon production. But the physics of N∗ states in the nuclear medium is an important field of research by its own. From the side of hadron spectroscopy, there is large interest to use the nuclear medium as a probe for the intrinsic structure of resonances. As pointed out repeatedly, the intrinsic configurations of resonances are mixtures of meson–baryon and 3-quark components, the latter surrounded by a polarization cloud of virtual mesons and qq¯ states. The various components are expected to react differently on the polarizing forces of nuclear matter thus offering a more differentiating access to N∗ spectroscopy. From the nuclear physics side, resonances are ideal probes for various aspects of nuclear dynamics which are not so easy to access by nucleons alone. They are emphasizing certain excitation modes as the spin–isospin response of nuclear matter. For that purpose, studies of the ∆ resonance are the perfect tool. Resonances are also thought to play a key role in many-body forces among nucleons, implying that nucleons in nuclear matter are in fact part of their time in (virtually) excited states. To the best of our knowledge stable nuclear matter exists only because three-body (or multi-body) forces, contributing the correct amount of repulsion already around the saturation point and especially at higher densities. Peripheral heavy ion reactions are the method of choice to produce excited nucleons under controlled conditions in cold nuclear matter below and close to the saturation density. It was pointed, that in central collisions the density of N∗ states will increase for a short time to values comparable to the density in the center of a nucleus. The existing data confirm that in peripheral reactions the excitation probability is sizable. Physically, the production of a resonance in a nucleus corresponds to the creation of N∗N−1 particle–hole configuration. The description of such configurations was discussed for the case of N∗ = ∆. Extensions to higher resonances are possible and in fact necessary for the description of the nuclear response already observed at high excitation energies. In future experiments a particular role will be played by the decay spectroscopy which is a demanding task for nuclear theory. In any case, the nuclear structure and reaction theory are asked to extend their tool box considerably for a quantitative description of hyperons and resonances in nuclear matter. In this respect, neutrons star physics is a step ahead: As discussed, there is an urgent demand to investigate also resonances in neutron star matter. In beta-equilibrated matter, resonances will appear at the same densities as hyperons. Thus, in addition to the hyperon puzzle there is also a resonance puzzle in neutron stars. Overall, in-medium hadronic reactions offer ample possibilities of studying sub-nuclear degrees of freedom. By using beams of the heaviest possible nuclei at beam energies well above the strangeness production thresholds, one can probe definitely superstrange and resonance matter at baryon densities far beyond saturation, e.g. coming eventually close to the conditions in the deeper layers of a neutron star. Theoretically, such a task is of course possible and the experimental feasibility will come in reach at the Compressed-Baryonic-Experiment (CBM) at FAIR which is devoted specifically to investigations of baryonic matter. The LHC experiments are covering already a sector of much higher energy density but their primary layout is for physics at much smaller scales. On the side of hadron and nuclear theory, LQCD and QCD-oriented effective field theories may bring substantial progress in the not so far future, supporting a new understanding of sub-nuclear degrees of freedom and in-medium baryon physics 200 H. Lenske et al. / Progress in Particle and Nuclear Physics 98 (2018) 119–206 in a unified manner. We have mentioned their achievements and merits on a few places. However, both LQCD and EFT approaches, would deserve a much deeper discussion as could be done here. In conjunction with appropriate many-body methods, such as provided by density functional theory, Green’s function Monte Carlo techniques, and modern shell model approaches they are apt to redirect nuclear and hypernuclear physics into the direction of ab initio descriptions. The explicit treatment of resonances will be a new demanding step for nuclear theory (and experiment!) adding to the breadth of the field.

Acknowledgments

Many members and guests of the Giessen group have been contributing to the research discussed in this review article. Contributions especially by C. Keil, A. Fedoseew, S. Bender, R. Shyam (Saha Institute, Kolkatta), and V. Shklyar are gratefully acknowledged. Supported by DFG, contract Le439/9 and SFB/TR16, BMBF, contract 05P12RGFTE, the GSI Darmstadt-JLU Giessen collaboration contract, and Helmholtz International Center for FAIR. T.G. gratefully acknowledges support from DAAD between the universities of Tübingen and Thessaloniki (grant ID 57340132) and the COST action ‘‘Theory of hot matter and relativistic heavy-ion collisions’’ (THOR) CA15213. X. C. was supported by the National Natural Science Foundation of China (Grant No. 11405222).

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