The Euroschool on Exotic Beams - Vol. 5 pp 161-253 | Cite as
Hyperons and Resonances in Nuclear Matter
Abstract
Theoretical approaches to interactions of hyperons and resonances in nuclear matter and their production in elementary hadronic reactions and heavy ion collisions are discussed. The focus is on baryons in the lowest SU(3) flavor octet and states from the SU(3) flavor decuplet. Approaches using the SU(3) formalism for interactions of mesons and baryons and effective field theory for hyperons are discussed. An overview of application to free space and in-medium baryon-baryon interactions is given and the relation to a density functional theory is indicated. SU(3) symmetry breaking is discussed for the Lambda hyperon. The symmetry conserving Lambda-Sigma mixing is investigated. In asymmetric nuclear matter a mixing potential, driven by the rho- and delta-meson mean-fields, is obtained. 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.
5.1 Introduction
In 1947, Rochester and Butler observed a strange pattern of tracks on a photographic emulsion plate which was exposed in a high altitude balloon mission to cosmic rays [1]. That event marks the inauguration of strangeness physics, indicating that there might be matter beyond nucleons and nuclei. Seven years later, that conjecture was confirmed by Danysz and Pniewski with their first observation of a hypernucleus [2], produced also in a cosmic ray event. These observations had and are still having a large impact on elementary particle and nuclear physics. In recent years, a series of spectacular observations on hypernuclear systems were made, giving new momentum to hypernuclear research activities, see e.g. [3, 4, 5, 6, 7].
Lambda-hypernuclei are being studied already for decades. They are the major source of information on the S = −1 sector of nuclear many-body physics. The status of the field was comprehensively reviewed quite recently by Gal et al. [10]. The “hyperonization puzzle” heavily discussed for neutrons stars [11] is another aspect of the revived strong interest in 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. The FINUDA collaboration at the Frascati ϕ-meson factory observed for the first time the exotic superstrange system \(^6_\varLambda H\) [3]. The STAR experiment at RHIC has filtered out of their data samples exciting results for a totally unexpected reduced lifetime of the Lambda-hyperon bound in \(^3_{\varLambda }H\). Soon after, that result was observed also by the HypHI-experiment at the FRS@GSI [5]. 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 [6] and the LHC [7], 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 [4] which—if confirmed—would be 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 were initiated in the late 1970s and thereafter continued at SATURNE and later at the Synchrophasotron at Dubna and at KEK. The major achievements were the observation of an apparent huge mass shift of the Delta-resonance by up to ΔM ∼−70 MeV. Detailed theoretical investigations, however, have shown 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. The Delta-resonance, for example, appears as the natural partner of the corresponding spin-isospin changing ΔS = 1, ΔI = 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 nuclear structure physics is to understand the coupling of the nuclear GTR and the nucleonic Δ_{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 Δ particle-nucleon hole (ΔN^{−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 may couple to the P_{11}(1440) Roper resonance. Since that \(J^\pi =\frac {1}{2}^+,I=\frac {1}{2}\) 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 Δ_{33}(1232) and other resonances are of central interest because they are important sources 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 towards 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 [12]. The strangeness physics, hypernuclear research, and resonance studies belong to the experimental program envisioned for the FAIR facility.
In this article we intend to point out common aspects of hypernuclear and nucleon in-medium resonance physics. 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 Sect. 5.2 we introduce the concepts of flavor SU(3) physics and discuss the application to in-medium physics of the baryons for a covariant Lagrangian approach. The theoretical results are recast into a density functional theory with dressed in-medium meson-baryon vertices in Sect. 5.3. Results for hypermatter and hypernuclei are discussed in Sect. 5.4, addressing also the investigations by modern effective field theories. In Sect. 5.5 we derive hyperon interactions in nuclear matter by exploiting the constraints imposed by SU(3) symmetry. Investigations of baryon resonances in nuclei are discussed in Sects. 5.6 and 5.7. In Sect. 5.8 the article is summarized and conclusions are drawn before closing with an outlook.
5.2 Interactions of SU(3) Flavor Octet Baryons
5.2.1 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 [13], the very useful collection of papers in [14] to the more recent review on experimental work by Feliciello et al. [15] to the article by Gal et al. [10]. The latest activities are also recorded in two topical issues: in Ref. [16] strangeness (and charm) physics are highlighted and in Ref. [17] the contributions of strangeness physics with respect to neutron star physics is discussed. In [18] we have reviewed the status of in-medium baryon and baryon resonance physics, also covering production reactions on the free nucleon and on nuclei.
On free space interactions of nucleons a wealth of experimental data exists which are supplemented by the large amount of data on nuclear spectroscopy and reactions. 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 on the other hand of interactions in nuclear matter. For the 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 1960s. 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, no safe signal for a particle-stable Σ or a S = −2 Cascade hypernucleus has been recorded, see e.g. [10, 13].
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 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) [19], later improved to the Extended Soft Core (ESC) model [20, 21], for which over the years a number of parameter sets were evaluated [22, 23, 24]. The Jülich model [25, 26, 27] and also the more recently formulated Giessen Boson Exchange model (GiBE) [28, 29] belongs to the OBE-class of approaches. In the Jülich model the J^{ P } = 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 [20, 21] and ESC08 [22, 24] 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 being updated regularly [30, 31]. 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 \((q\bar q)\) exchange effects.
Chiral effective field theory (χ EFT) for NN and NY interactions are a more recent development in baryon-baryon interaction. The review by Epelbaum et al. [32] on these subjects is still highly recommendable. The connection to the principles of QCD are inherent. A very attractive feature of χ EFT 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 over 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 [33]. Stochastic methods like the Green’s function Monte Carlo approach are successful for light and medium mass nuclei [34]. A successful approach up to oxygen mass region, the so-called p-shell nuclei, is the hypernuclear shell model of Millener and collaborators [35, 36]. Over the years, a high degree of sophistication and predictive power for hypernuclear spectroscopy has been achieved by these methods for light nuclei. 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 [37, 38, 39]. Already some time ago we have made first steps in such a direction [40] within the Giessen Density Dependent Hadron Field (DDRH) theory. The DDRH approach incorporates Dirac-Brueckner Hartree-Fock (DBHF) theory into covariant density functional theory [41, 42, 43, 44, 45, 46]. Since then, the approach is being used widely on a purely phenomenological level as e.g. in [47, 48, 49, 50]. In the non-relativistic sector comparable attempts are being made, ranging from Brueckner theory for hypermatter [51, 52] to phenomenological density functional theory extending the Skyrme-approach to hypernuclei [53, 54]. In recent works energy density functionals have been derived also for the Nijmegen model [24] and the Jülich χ EFT [55]. Relativistic mean-field (RMF) approaches have been used rather early for hypernuclear investigation, see e.g. [56, 57]. A covariant DFT approach to hypernuclei and neutron star matter was used in a phenomenological RMF approach in Ref. [58] where constraints on the scalar coupling constants were derived by imposing the constraint of neutron star masses above two solar masses. In Refs. [59, 60, 61] and also [62, 63] 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 under debate. A covariant mean-field approach, including a non-linear realization of chiral symmetry, has been proposed by the Frankfurt group [64] and is being used mainly for neutron star studies.
5.2.2 Interactions in the Baryon Flavor Octet
Mass, lifetime, and valence quark configuration of the \(J^P= \frac {1}{2}^+\) octet baryons
State | Mass (MeV) | Lifetime (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] |
Intrinsic quantum numbers and the measured masses [65] of the mesons used in the calculations
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 advantage of referring to SU(3) symmetry is obvious: For each type of interaction (pseudo-scalar, vector, scalar) only four independent parameters are required to characterize the respective interaction strengths with all possible baryons. These are the singlet coupling constant g_{ S }, the octet coupling constants g_{ D }, g_{ F }, g_{ S }, and eventually the three mixing angles θ_{P,V,S}, one for each meson multiplet, which relate the physical, dressed isoscalar mesons to their bare octet and singlet counterparts.
Baryon-baryon channels for fixed strangeness S and total charge Q
Q = −2 | Q = −1 | Q = 0 | Q = 1 | Q = 2 | |
---|---|---|---|---|---|
S = 0 | nn | np | pp | ||
S = −1 | Σ ^{−} n | Λn Σ ^{0} n Σ ^{−} p | Λp Σ ^{+} n Σ ^{0} p | Σ ^{+} p | |
S = −2 | Σ ^{−} Σ ^{−} | Ξ ^{−} n Σ ^{−} Λ Σ ^{−} Σ ^{0} | ΛΛ Ξ^{0}n Ξ^{−}p Σ^{+}Λ Σ^{+}Σ^{0} | Ξ^{0}p Σ^{+}Λ Σ^{+}Σ^{0} | Σ ^{+} Σ ^{+} |
S = −3 | Ξ ^{−} Σ ^{−} | Ξ ^{−} Λ Ξ ^{0} Σ ^{−} Ξ ^{−} Σ ^{0} | Ξ ^{0} Λ Ξ ^{0} Σ ^{0} Ξ ^{−} Σ ^{+} | Ξ ^{0} Σ ^{+} | |
S = −4 | Ξ ^{−} Ξ ^{−} | Ξ ^{−} Ξ ^{0} | Ξ ^{0} Ξ ^{0} |
5.2.3 Baryon-Baryon Scattering Amplitudes and Cross Sections
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 χ EFT cut-offs are much softer with values around 600 MeV/c.
Low energy parameters for the indicated YN system
Channel | a (fm) | r (fm) | Model |
---|---|---|---|
ΣN (\(I=\frac {1}{2}\)) | +0.90 | -4.38 | Jülich 04 |
ΣN (\(I=\frac {3}{2}\)) | -4.71 | 3.31 | Jülich 04 |
-4.35 | 3.16 | NSC97f 04 | |
-1.80 | 1.76 | χ EFT LO | |
-1.44 | 5.18 | GiBE | |
ΛN (\(I=\frac {1}{2}\)) | -2.56 | 2.75 | Jülich 04 |
-2.60 | 2.74 | NSC97f | |
-1.01 | 1.40 | χ EFT LO | |
-2.41 | 2.34 | GiBE |
5.2.4 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 \(V^{BB'\alpha }(q',q)\). The solution of the coupled equations, Eq. (5.33) 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 [78], 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 density functionals (EDF), e.g. [79] for a hypernuclear Skyrme EDF, are a standard tool for nuclear structure research. The UNEDF initiative is trying to derive the universal nuclear EDF [39]. Finelli et al. started work on an EDF based on χ EFT. In [24] 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) [80, 81, 82, 83]. 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 [57] and many others have followed, see e.g. [84, 85, 86]. 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 [40, 41, 42, 43, 87] 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 \( T_{\mathcal {BB}'}\). 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. [74, 88, 89, 90]. A more convenient representation, allowing also for at least an 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 Γ_{ a } and propagators D_{ a } for bosons with masses m_{ a }. A natural choice is to use the same boson masses as in the construction of the tree-level kernels.
Two limiting cases are of particular interest. At vanishing density where Q_{ F } → 1, and since \(V_a\sim g^2_a\) we find that in free-space the dressed vertices retain their general structure as a fully summed series of tree-level coupling constants. At density ρ →∞, where Q_{ F } → 0 over the full integration range, we find Γ_{ a } ≃ g_{ a }. Albeit for another reason 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 \(\sqrt {s}\) the pole is shifted into the tail of the form factors regularizing the high momentum part of U_{ a }. Thus, at large energies the residues are increasingly suppressed, as indicated by the decline of the matrix element shown in Fig. 5.6. As a conclusion, the dressing effects are the strongest for low energies and densities.
5.2.5 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, T_{ lab } = 50…1000 MeV. Also the widely used M3Y G-matrix parametrization of Bertsch et al. [93] is using comparable techniques, as also the work in [94]. In order to generalize the approach a field-theoretical formulation is of advantage which is the line followed in DDRH theory. For that purpose, the dependence on \(k_F\sim \rho ^{1/3}_B\) is replaced by a functional dependence on the baryon four-current by means of the Lorentz-invariant operator relation \(\rho ^2_B=j_{B\mu }j^\mu _B\) leading to vertex functionals \(\hat {\varGamma }_a(\bar \varPsi _B \varPsi _B)\). The C-numbered vertices are recovered as expectation values, \(\varGamma _a(\sqrt {s},k_F)=\langle P,k_F|\hat {\varGamma }_a(\bar \varPsi _B \varPsi _B)|P,k_F\rangle \), under given kinematical conditions P and for a baryon configuration defined by k_{ F }.
5.3 Covariant DFT Approach to Nuclear and Hypernuclear Physics
5.3.1 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, which later gave rise to the development of a purely phenomenological approach [47] 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 [95, 96], 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 (J^{ P } = 0^{+}) and vector mesons (J^{ P } = 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 clear distinction, here vertex functionals will be denoted by g^{∗}BB′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 [41, 42, 87]. The equations of state for neutron matter, neutron star matter, and neutron stars are obtained without adjustments of parameters. In [46] the temperature dependence of the vertices was studied and the thermodynamical properties of nuclear matter up to about a temperature of T = 50 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.
5.3.2 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, e.g. bilinears of matter field for the interaction vertices. That apparent simplicity on the Lagrangian level, however, requires additional theoretical and numerical efforts for a 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 \(g_{BB'\mathcal {M}}\) by meson-baryon vertices \(g^*_{BB'\mathcal {M}}(\varPsi _{\mathcal {F}})\) which are Lorentz-invariant functionals of bilinears of the matter field operators \(\varPsi _{\mathcal {B}}\). The derivation of those structures from Dirac-Brueckner theory has been discussed intensively in the literature [40, 41, 42, 45, 74]. 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.
By means of the SU(3) relations also the in-medium vertices in the other baryon-meson interaction channels can be derived [98], leading to a multitude of vertices which are not shown here. In magnitude (and sign) the BB′ vertices are of course different form the one shown in Fig. 5.12. But a common feature is that the relative variation with density is similar to the NN-vertices in Fig. 5.12.
5.4 DBHF Investigations of Λ Hypernuclei and Hypermatter
5.4.1 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, \(g^*_{\varLambda \varLambda \sigma ,\omega }\simeq R_{\sigma ,\omega }g^*_{NN\sigma ,\omega }\). Microscopic calculations show that these ratios vary in the density regions of interest for nuclear structure investigations only slightly. Averaged over the densities up to the nuclear saturation density ρ_{ eq } = 0.16 fm^{−3}, we find \(\bar {R}_\sigma =0.42\) and \(\bar {R}_\omega =0.66\) 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 [56, 57]. 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 R_{σ,ω} = 2/3. Surprisingly, the cited value of \(\bar {R}_\omega \) 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 R_{σ,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.
Extrapolating the separation energies shown in Fig. 5.14 to (physical unaccessible) large mass number, the limiting value \(S^{\infty }_\varLambda \simeq 28\) 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 \(U^\infty _\varLambda \sim -28\) MeV.
5.4.2 Spectroscopic Details of Single-Λ Hypernuclei
For the results shown in Fig. 5.14 the KEK-data of Hotchi et al. [99] for \(^{41}_\varLambda V\) and \(^{89}_\varLambda Y\) 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.
DDRH results for Λ single particle energies
Level | \(^{89}_{\varLambda }Y\) (MeV) | \(^{41}_{\varLambda }V\) (MeV) |
---|---|---|
1s_{1/2} | − 22.94 ± 0.64 | − 19.8 ± 1.4 |
1p_{3/2} | − 17.02 ± 0.07 | − 11.8 ± 1.3 |
1p_{1/2} | − 16.68 ± 0.07 | − 11.4 ± 1.3 |
1d_{5/2} | − 10.26 ± 0.07 | − 2.7 ± 1.2 |
1d_{3/2} | − 9.71 ± 0.07 | − 1.9 ± 1.2 |
1f_{7/2} | − 3.04 ± 0.11 | – |
1f_{5/2} | − 3.04 ± 0.11 | – |
5.4.3 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 first observation of a double-Lambda hypernuclear event, assigned in an emulsion experiment as either \(^{10}_{\varLambda \varLambda }Be\) or \(^{11}_{\varLambda \varLambda }Be\), was reported as early as 1963 by Danysz et al. [105]. The probably best recorded case is the so-called Nagara event [106], a safely identified \(^6_{\varLambda \varLambda }He\) hypernucleus produced at KEK by a (K^{−}, K^{+}) reaction at p_{ lab } = 1.66 GeV/c on a ^{12}C target. The KEK-E373 hybrid emulsion experiment [106] traced the stopping of an initially produced Ξ^{−} hyperon, captured by a second carbon nucleus, which then was decaying into \(^6_{\varLambda \varLambda }He\) plus an ^{4}He nucleus and a triton. The \(^6_{\varLambda \varLambda }He\) nucleus was identified by it’s decay into the known \(^5_{\varLambda }He\) and a proton and a π^{−}. The data were used to deduce the total two-Lambda separation energy B_{ ΛΛ } and the Lambda-Lambda interaction energy ΔB_{ ΛΛ } which is a particular highly wanted quantity. A re-analysis in 2013 [107] led to the nowadays accepted values B_{ ΛΛ } = 6.91 ± 0.16 MeV and ΔB_{ ΛΛ } = 0.67 ± 0.17 MeV while the original value was larger by about 50% [106].
These data are an important proof that double-Lambda hypernuclei are indeed highly useful for putting firm constraints on the ΛΛ^{1}S_{0} scattering length. Theoretical studies for the \(^6_{\varLambda \varLambda }He\) 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 a_{ ΛΛ } = −1.3… − 0.5 fm, including cluster-type descriptions [21, 108, 109, 110], calculations with the NSC interactions [111, 112], and variational results [113]. Analyses of hyperon final state interactions in strangeness production reactions also allow to estimate a_{ ΛΛ }. A recent theoretical analysis of STAR data [114] led to a_{ ΛΛ } = −1.25… − 0.56 fm [115] (note the change of sign in order to comply with our convention).
The theoretical results on ΔB_{ ΛΛ } agree within the cited uncertainty ranges which should be considered 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 ΔB_{ ΛΛ }.
The experimental results on the ΛΛ interaction energy have initiated on the theoretical side considerable activities, see e.g. [10, 13]. In [24] 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 \(q\bar q\) states, the (attractive) scalar-exchange part of the interaction in the various channels satisfies |V_{ ΛΛ }| < |V_{ NΛ }| < |V_{ NN }|, implying indeed a rather weak ΛΛ potential. The ESC fits to the NY scattering data give values for the scalar-meson 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, however, 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.
5.4.4 Hyperon Interactions and Hypernuclei by Effective Field Theory
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 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 χ EFT 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)-χ EFT was been intensified by several groups and in several directions. The Munich-Jülich collaboration [55, 117, 118, 119], 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 [55] 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.
5.4.5 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 [120] relies on recasting the lattice results into a Schroedinger-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 8_{ s } channel, while an attractive core appears in the flavor singlet channel, i.e. the 1 SU(3) representation. 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. [121, 122, 123].
A somewhat different approach is used by the NPLQCD collaboration [124, 125, 126, 127]. 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 χ EFT results obtained from hadronic studies. Performing calculations specifically to match LQCD results to low-energy effective field theories will provide a means for first predictions at the physical quark mass limit. This allows also to predict quantities beyond those calculated with LQCD. In [127], for example, the NPLQCD collaboration report the results of calculations of nucleon-nucleon interactions in the ^{3}S_{1} −^{3}D_{1} coupled channels and the ^{1}S_{0} channel at a pion mass m_{ π } = 450 MeV. For that pion mass, the n-p system is overbound and even the di-neutron becomes a bound states. However, extrapolations indicate that at the physical pion mass the observed properties of the two-nucleon systems will be approached.
5.4.6 Infinite Hypermatter
The saturation properties of symmetric pure (p, n)-matter are very satisfactorily described: The saturation point is located within the experimentally allowed region at ρ_{ sat } = 0.166 fm^{−3} and ε(ρ_{ sat }) = −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 until a new minimum for 10% Λ-content is reached at ρ_{ min } = 0.21 fm^{−3} with a binding energy of ε(ρ_{ min }) = −18 MeV. Increasing either ξ_{ Λ } and/or the density, the binding energy approaches eventually zero, as marked by the red line In Fig. 5.20. 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 finally p, n, Λ matter becomes unbound.
5.5 SU(3) Constraints on In-Medium Baryon Interactions
The SU(3) relations among the coupling constants of the octet baryons and the 0^{−}, 0^{+}, 1^{−} meson nonets are conventionally used as constraints at the tree-level interactions. In this section we take a different point of view. First of all, the mixing of singlet and octet mesons, to be discussed below, is considered. An interesting observation, closely connected to the mixing, is that in each interaction channel the three fundamental SU(3) constants g_{ D }, g_{ F }, g_{ S } are already fixed by the NN vertices with the isoscalar and isovector octet mesons and the isoscalar singlet meson, under the provision that the octet-singlet mixing angles are known. As shown below, the mixing angles depend only on meson masses. Since the Brueckner-approach retains the meson masses, the relations fixing the mixing angles are conserved by the solutions of the Bethe-Salpeter or Lippmann-Schwinger equation, respectively. Moreover, SU(3) symmetry in general will be conserved, as far as interactions are concerned. The only substantial source of symmetry breaking is due to the use of physical masses from which one might expect SU(3) violating effects of the order of 10%. Although in low-energy baryon interactions the exchange particles are far off their mass shell, the on-shell mixing relations will persist because the BB′ T-matrices are symmetry conserving.
5.5.1 Meson Octet-Singlet Mixing
Octet-singlet meson mixing used to determine the in-medium vertices
Channel | f | f′ | a | θ(^{∘}) |
---|---|---|---|---|
Pseudo-scalar | η | η′ | π | -25.65 |
Vector | ω | ϕ | ρ | +36.0 |
Scalar | σ | σ′ | a _{0} | -50.73 |
SU(3) relations for the ω, ρ and ϕ baryon coupling constants, relevant for the mean-field sector of the theory
Vertex | Coupling constant |
---|---|
NNω | \(g_{NN\omega }=g_S \cos {}(\theta _v) +\frac {1}{\sqrt {6}} \left (3 g_F -g_D\right ) \sin {}(\theta _v)\) |
NNϕ | \(g_{NN\phi }=g_S \sin {}(\theta _v) -\frac {1}{\sqrt {6}} \left (3 g_F -g_D\right ) \cos {}(\theta _v)\) |
NNρ | \(g_{NN\rho }=\sqrt {2}(g_F+g_D)\) |
ΛΛω | \(g_{\varLambda \varLambda \omega }=g_S \cos {}(\theta _v) -\sqrt {\frac {2}{3}} g_D \sin {}(\theta _v)\) |
ΛΛϕ | \(g_{\varLambda \varLambda \phi }=g_S \sin {}(\theta _v) +\sqrt {\frac {2}{3}} g_D \cos {}(\theta _v)\) |
ΣΣω | \(g_{\varSigma \varSigma \omega }=g_S \cos {}(\theta _v) +\sqrt {\frac {2}{3}} g_D \sin {}(\theta _v)\) |
ΣΣϕ | \(g_{\varSigma \varSigma \phi }= g_S \sin {}(\theta _v) -\sqrt {\frac {2}{3}} g_D \cos {}(\theta _v)\) |
ΣΣρ | \(g_{\varSigma \varSigma \rho }= \sqrt {2}g_F\) |
ΛΣρ | \(g^\rho _{\varLambda \varSigma }=\sqrt {\frac {2}{3}}g_D\) |
ΞΞω | \(g_{\varXi \varXi \omega }=g_S \cos {}(\theta _v) -\frac {1}{\sqrt {6}} \left (3 g_F +g_D\right ) \sin {}(\theta _v)\) |
ΞΞϕ | \(g_{\varXi \varXi \phi }=g_S \sin {}(\theta _v) +\frac {1}{\sqrt {6}} \left (3 g_F +g_D\right ) \cos {}(\theta _v)\) |
ΞΞρ | \(g_{\varXi \varXi \rho }=\sqrt {2}(g_F-g_D)\) |
5.5.2 SU(3) In-medium Vertices
The isovector interactions, however, do not follow the naïve quark-model scaling hypothesis. There, one finds scaling constants of the order of unity. In hypermatter with more than a single hyperon, sizable condensed isoscalar-singlet fields will evolve to which the Λ, Σ and Ξ baryons will couple. The Ξ-interactions, for example, are dominated by the isoscalar-singlet and the isovector-octet channels which might shed new light on the dynamics of S = −2 hypernuclei.
5.5.3 Mean-Field Self-Energies of Octet Baryons in Infinite Nuclear Matter
With increasing proton-neutron asymmetry the isovector potentials gain strength and are inducing a splitting of the potentials within the iso-multiplets. The effect is most pronounced in the limiting, albeit hypothetical case of pure neutron matter. The Lambda, however, is not affected because of its isoscalar nature. In Fig. 5.25 the mean-field potentials for the three iso-multiplets are displayed. The proton and neutron potentials are showing the known behavior of a deepening of the proton and a reduction of the neutron potential depth, respectively. The three Sigma hyperons have obtained now mean-fields of a quite different depth: While the Σ^{−} feels a strongly repulsive interaction the Σ^{+} potential has become attractive. The latter may indicate that there might be bound exotic Σ^{+} nuclei. With increasing asymmetry the Cascade potentials remain throughout repulsive, indicating that it is unlikely to find exotic bound Ξ − A systems or even double-Lambda hypernuclei with S = −2.
5.5.4 SU(3) Symmetry Breaking for Lambda Hyperons
5.5.5 ΛΣ^{0} Mixing in Asymmetric Nuclear Matter
5.6 Theory of Baryon Resonances in Nuclear Matter
5.6.1 Decuplet Baryons as Dynamically Generated, Composite States
Mass, width, lifetime, and valence quark configuration of the \(J^P= \frac {3}{2}^+\) decuplet baryons (taken from Ref. [65])
State | Mass (MeV) | Width (MeV) | Lifetime (s) | Configuration |
---|---|---|---|---|
Δ ^{++} | 1230 | 120 | 10^{−23} | [uuu] |
Δ ^{+} | 1232 | 120 | 10^{−23} | [uud] |
Δ ^{0} | 1234 | 120 | 10^{−23} | [udd] |
Δ ^{−} | 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 | 0.8 × 10^{−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 [129] and functional methods, e.g. [130], will be able to treating 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 [131, 132] 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 partial waves of L = 1 and resonant states to determine the weight of the meson-baryon component in the Δ(1232) resonance and the other members of the \(J^P = \frac {3}{2}^+\) baryon decuplet.
The calculations predict an appreciable πN fraction in the Δ(1232) wave function, as large as 60%. At first sight this is a surprising result which, however, 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 \(u\bar {u} N\) and \(d\bar {d} N\) components on a level of up to 30% [133, 134]. 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. Thus, 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 Δ(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 \(\bar {K} N\) channel but that threshold is much higher than the pion-nucleon one. The S = −2 baryons would couple preferentially to \(\bar {K} \varLambda \) or \(\bar {K} \varSigma \) channels with even higher thresholds and so on. The Aceti-Oset approach was further extended by investigating the formation of resonances by interactions of \(\frac {3}{2}^+\) decuplet baryons with pseudo-scalar mesons from the lowest 0^{−} octet [135] and vector mesons from the lowest 1^{−} octet [136], 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 afore mentioned Λ(1405) state [137] but also the Λ(1520) [138] resonances. A compelling insight from those and similar studies is that the baryons above the lowest \(\frac {1}{2}^+\) 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) J^{ P } = 1^{−} vector meson which is known to be a pronounced ππ p-wave resonance [65]. 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 [139] 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 [140] 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^{∗}Δ and Ξ^{∗}Ω systems were calculated. The so restricted study led to rather strongly attractive decuplet interaction, producing deeply bound ΣΔ 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 [141]. Here, we are less ambitious and consider mainly interactions of the Δ baryon and few other resonances in nuclear matter.
5.6.2 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 amounts 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 \(V_{NN^*}\). The appropriate theoretical frame work for that process is given by the polarization propagator formalism [92], also underlying, for example, the approaches in [142, 143, 144].
In the non-relativistic limit of cold infinite matter with nucleons filling up the Fermi sea up to the Fermi momentum k_{ F }, we have \(n(\textbf {p})=\theta (k^2_F-\mathbf {p}^2)\). The nucleon propagator Eq. (5.112) consists of the vacuum part and the in-medium part (∝ n(p)). The Δ propagator Eq. (5.113) includes the vacuum part only, since we have neglected the presence of Δ excitations in nuclear matter. Both propagators take into account effective mass corrections if present.
In the literature the values of the Landau-Migdal parameters are not fixed unambiguously by theory but must be constrained on phenomenological grounds. Within a simple universality assumption \(g_{NN}^\prime = g_{N\varDelta }^\prime = g_{\varDelta \varDelta }^\prime \equiv g_{BW}^\prime \), which is the so-called Bäckmann-Weise choice (see Ref. [157] and Refs. therein), one gets \(g_{BW}^\prime = 0.7 \pm 0.1\) from the best description of the unnatural parity isovector states in ^{4}He, ^{16}O and ^{40}Ca. However, the same calculations within the Migdal model [158] assumption \(g_ {N\varDelta }^\prime = g_{\varDelta \varDelta }^\prime = 0\) produce \(g_{NN}^\prime = 0.9\ldots 1\). The description of the quenching of the Gamow-Teller matrix elements requires, on the other hand, \(g_{\varDelta \varDelta }^\prime = 0.6\ldots 0.7\) (assuming \(g_{N\varDelta }^\prime = g_{\varDelta \varDelta }^\prime \)) [159]. The real part of the pion optical potential in π-atoms implies \(g_{N\varDelta }^\prime = 0.2\) and \(g_{\varDelta \varDelta }^\prime = 0.8\) [158]. The pion induced two-proton emission is the best described with \(g_{N\varDelta }^\prime = 0.25\ldots 0.35\).
5.6.3 Δ Mean-Field Dynamics
5.6.4 Response Functions in Local Density Approximation
There is a large body of data available on inclusive (e, e′) cross sections [165, 166, 167, 168] which are the perfect test case to the response function formalism. Since a detailed discussion of the functional structure of (e, e′) cross sections is beyond the scope of the present work, we refer the reader to the ground-breaking monograph of DeForest and Walecka [169] and the more recent review article of Benhar and Sick [170]. The reactions proceed such that the incoming electron couples via virtual photon emission to the charged nuclear currents, involving electric and magnetic interactions. The cross sections are given by the superposition of response functions for operators of spin-longitudinal (\(\mathcal {O}_L\sim \boldsymbol {\sigma }\cdot \mathbf {q}\)), and spin-transversal (\(\mathcal {O}_T\sim \boldsymbol {\sigma }\times \mathbf {q}\)) structure, where q is the momentum transfer. These operators are defining the corresponding response functions (R_{ LL }), (R_{ TT }), respectively. The cross sections are obtained by weighting the response functions by the proper kinematical factors.
In Xia et al. [171], 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 ΔN^{−1} states, the real 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 Δ-resonance region. Moreover, the interesting result is found that the baryonic vertex form factors obtained for pionic and electromagnetic probes agrees with their interpretation as effective hadronic structure functions.
5.6.5 Resonances in Neutron Stars
Moreover, the investigations in [59, 60] lead to the important conclusion that the onset of Δ-isobars is strictly related to 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 [172], 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 Δ 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.
5.7 Production and Spectroscopy of Baryon Resonances in Nuclear Matter
5.7.1 Resonances as Nuclear Matter Probes
The generic interaction processes shown in Fig. 5.36 are using a meson exchange picture which describes successfully the dynamics of N^{∗} production in ion-ion reactions. The Δ(1232) is produced mainly in NN → ΔN 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. The intermediate population of higher resonances like P_{11}(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 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 [192, 193]. That figure is largely confirmed by the work of Ref. [190] on deuteron induced resonance production at the Synchrophasotron at p_{ lab } = 4.2 AGeV/c, although there a somewhat lower resonance excitation rate of \(15\pm 2^{+4}_{-3}\)% was found which is in the same bulk as the excitation probability of \(16\pm 3^{+4}_{-3}\)% derived from proton induced reactions at the same beam momentum.
5.7.2 Interaction Effects in Spectral Distribution in Peripheral Reactions
In the Delta region the theoretical ^{12}C(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 ΔM_{ DW } ∼−50 MeV [150]. Finite size and a detailed treatment of particle-hole correlations within the ΔN^{−1} configurations contribute the remaining ΔM_{ c } ∼−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. [195]. An interesting results, shown in Fig. 5.37, is that the longitudinal (LO) partial cross section appears to be shifted down to a peak values of ω_{ L } ∼ 240 MeV, while the transversal partial cross section peaks at ω_{ L } ∼ 285 MeV. This is an effect of the ^{3}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 behaviour: 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 + Δ^{++} process is three times larger than that for the p + n → n + Δ^{+} process.
An even more detailed picture emerges from semi-inclusive reactions observing also decay products. For the reactions discussed in [183, 184, 185, 196] at incident energies of about 2 AGeV, the pπ correlations were successfully analyzed and the mass distribution of the Δ(1232) resonance could be reconstructed. In these peripheral reactions on various targets, the resonance mass was found to be shifted by up to ΔM ≃−70 MeV towards lower masses compared to those on protons. In reactions on various nuclei at incident energies around 1 AGeV the mass reduction of the Δ(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 Δ-resonance by the Valencia group [161, 197], 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.
5.7.3 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 collision [192, 193, 198, 199, 200, 201, 202]. 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.
5.7.4 The Delta Resonance as Pion Source in Heavy Ion Collisions
Transport calculations are describing accurately most of the particle production channels in heavy-ion collisions already in the early days of transport theory [199, 201, 207, 208]. However, surprisingly the pion yield from heavy-ion collisions at SIS energies (T_{ lab } ∼ 1 AGeV) could not be reproduced properly by the transport-theoretical description. For a long time the pertinent overprediction of the pion multiplicity [209, 210, 211, 212, 213, 214, 215, 216] was a disturbing problem. At the beam energies of a few AGeV the dominating source for pion production is the excitation of the Δ(1232) resonance in a NN collision NN → NΔ followed by the decay Δ → Nπ. In transport calculations, the pion multiplicity, therefore, depends crucially on the value of the in-medium NN → NΔ cross section. A first attempt to solve that issue was undertaken by Bertsch et al. [142]. In the 1990s Helgesson and Randrup [143, 217] took up that issue anew. In their microscopic π + NN^{−1} + ΔN^{−1} model [143] they considered the excitation of ΔN^{−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 Δ(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 Δ 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 [212, 213]. Special emphasis was laid on in-medium pion dispersion relations, the Δ width, pion reabsorption cross sections, the NN → ΔN cross sections and the in-medium Δ 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 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 [218], and by Korpa and Malfliet [219]. 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 afterwards.
Years later, the problem was reconsidered by the Giessen group. Initially, a purely phenomenological quenching prescription was used for fitting the data [216]. The breakthrough was achieved in [144] when the in-medium NN → NΔ(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 Δ resonance were considered. The ΔN^{−1} and the corresponding nuclear NN^{−1} modes, discussed above, were calculated again by RPA theory. In infinite matter the Lindhard functions [92], 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 Δ width and includes the NN^{−1} Lindhard function (see below) in the calculations. The inclusion of the effective mass modifications for the nucleons and Δ’s led to an additional strong reduction of the cross section. Altogether, the total pion multiplicity data [215] measured by the FOPI collaboration on the systems Ca+Ca, Ru+Ru and Au+Au at T_{ lab } = 0.4, 1.0, and 1.5 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 Δ-spreading width for which the prescription of the Valencia group was used: Γ_{ sp } = 80ρ/ρ_{0} MeV [163, 220].
The effect of the medium modifications of the NN ↔ NΔ 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 [144] it was found that including the effective mass modifications in the NN ↔ NΔ 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.
5.7.5 Perspectives of Resonance Studies by Peripheral Heavy Ion Reactions
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.
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 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 encountered 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 [12, 152]. 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.
5.8 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, immersed 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.
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 density (ρ_{ hyp } ∼ 0.21 fm^{−3}) and stronger binding (ε(ρ_{ hyp }) ∼−18 MeV) by adding Λ hyperons. The minimum is reached for a Λ-content of about 10% as shown in Sect. 5.4.6. 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 interactions. Until now, the results rely essentially on a single case, the famous \(^6_{\varLambda \varLambda }He\) 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 \(\overline {\mbox P}\)ANDA 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.
A topic of own interest—which was not discussed here—is the theory of hypernuclear production reactions. For a detailed review we refer to Ref. [18] where the description of hypernuclear production reactions is discussed. In that article a broad range of production scenarios, ranging from proton and antiproton induced reactions to the collision of massive ions, is studied theoretically. 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 hyperon production rates, especially for baryons with strangeness |S| > 1, depend crucially on the accumulation of strangeness in a sequence of reactions by intermediate resonance excitation, involving also Σ^{∗} and Ξ^{∗} states from the \(\frac {3}{2}^+\) decuplet. As a concluding remark on strangeness and hypernuclear production, we emphasize that in-medium reactions induced by heavy-ion beams represent an excellent tool to study in detail the strangeness sector of the hadronic equation of state. The knowledge of the in-medium 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 \(q\bar q\) 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 Δ 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 increasingly so 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^{∗} = Δ. 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 a strong need 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 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 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.
Footnotes
- 1.
More meaningful values are in fact the volume integrals per nucleon.
Notes
Acknowledgements
Many members and guests of the Giessen group have been contributing to the work summarized in this article. Contributions especially by C. Keil and A. Fedoseew, S. Bender, Th. Gaitanos (now at U. Thessaloniki), R. Shyam (Saha Institute, Kolkatta), and V. Shklyar are gratefully acknowledged. Supported by DFG, contract Le439/9 and SFB/TR16, project B7, BMBF, contract 05P12RGFTE, GSI Darmstadt, and Helmholtz International Center for FAIR.
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