Hyperons and Resonances in Nuclear Matter

  • Horst Lenske
  • Madhumita Dhar
Part of the Lecture Notes in Physics book series (LNP, volume 948)


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].

The group-theoretical approach introduced independently by Murray Gell-Mann and Yuval Ne’eman in the beginning of the sixties of the last century was the long awaited for breakthrough towards 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 1970s [9], Quantum Chromo Dynamics (QCD) has evolved into the nowadays accepted standard model of strong interaction physics. Since long, QCD theory has become part of the solid foundations of modern science. As indispensable part of the scientific narrative QCD gauge theory has become a central topic in particle and nuclear physics text books, for instance the one by Cheng and Li [8]. The \(\frac {1}{2}^+\) baryon octet and the \(\frac {3}{2}^+\) baryon decuplet together with their valence quark structures are displayed in Fig. 5.1.
Fig. 5.1

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 \(I_3=Q- \frac {1}{2}Y\) 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. [8]

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 Δ33 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 P11(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 Δ33(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

In Fig. 5.2 the SU(3) multiplets are shown which are considered in the following. The lowest \(J^P=\frac {1}{2}^+\) baryon octet, representing the baryonic ground state multiplet, is taken into account together with three meson multiplets, namely the pseudo-scalar nonet (\(\mathcal {P}\)) with J P  = 0, the scalar nonet (\(\mathcal {S}\)) with J P  = 0+, and the vector nonet (\(\mathcal {V}\)) with J P  = 1, which, in fact, consists of four subsets, \(\mathcal {V}=\{\mathcal {V}^\mu \}_{|\mu =0\ldots 3}\), 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. 5.2. Masses and lifetimes of baryons are displayed in Table 5.1.
Fig. 5.2

The SU(3) multiplets considered in this section: the \( \frac {1}{2}^+\) 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 5.1

Mass, lifetime, and valence quark configuration of the \(J^P= \frac {1}{2}^+\) octet baryons


Mass (MeV)

Lifetime (s)




> 6.62 × 10+36




880.2 ± 1




2.63 × 10−10


Σ +


0.80 × 10−10


Σ 0


7.4 × 10−20




1.48 × 10−10


Ξ 0


2.90 × 10−10




1.64 × 10−10


The data are taken from Ref. [65]

The mesons of each multiplet couple to the baryons by vertices of a typical, generic operator structure characterized the Lagrangian densities
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathcal{L}^{BB'}_{\mathcal{M}}(0^-)&\displaystyle =&\displaystyle -\quad g_{BB'\mathcal{M}}\bar \psi_{B'}\frac{1}{m_\pi}\gamma_5\gamma_\mu\psi_B\partial^\mu\varPhi_{\mathcal{M}} {} \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathcal{L}^{BB'}_{\mathcal{M}}(1^-)&\displaystyle =&\displaystyle -\quad g_{BB'\mathcal{M}}\bar \psi_{B'}\gamma_\mu\psi_B V^\mu_{\mathcal{M}} +\frac{f_{BB'\mathcal{M}}}{2(M_B+M_{B'})}\bar{\psi}_{B'}\sigma_{\mu\nu}\psi_B F^{\mu\nu}_{\mathcal{M}}{} \end{array} \end{aligned} $$
given by Lorentz-covariant bilinears of baryon field operators ψ B and the Dirac-conjugated field operators \(\bar {\psi }_B=\gamma _0\psi ^\dag _B\) and Dirac γ-matrices, to which meson fields (\(\mathcal {V,S}\)) or their derivatives (\(\mathcal {P}\)) are attached, such that in total a Lorentz-scalar is obtained. Vector mesons may also couple to the baryons through their field strength tensor \(F^{\mu \nu }_{\mathcal {M}}\) (see Eq. (5.11)) and the relativistic rank-2 tensor operator \(\sigma _{\mu \nu }=\frac {i}{2}[\gamma _\mu ,\gamma _\nu ]\), given by the commutator of γ matrices. The tensor coupling involves a separate tensor coupling constant \(f_{BB'\mathcal {M}}\). The mass factors in the \(\mathcal {P}\) and the tensor part of the \(\mathcal {V}\) vertices are serving to compensate the energy-momentum scales introduced by derivative operators for the sake of dimensionless coupling constants. The Bjorken-convention [66] is used for spinors and γ-matrices.
We introduce the flavor spinor \(\varPsi _{\mathcal {B}}\)
$$\displaystyle \begin{aligned} \begin{array}{rcl} \varPsi_{\mathcal{B}} = \left( N, \varLambda, \varSigma, \varXi \right)^{\text{T}} {} \end{array} \end{aligned} $$
being composed of the isospin multiplets
$$\displaystyle \begin{aligned} \begin{array}{rcl} {} N = \left( \begin{array}{c} p \\ n \end{array} \right),\quad \varSigma = \left( \begin{array}{c} \varSigma^+ \\ \varSigma^0 \\ \varSigma^- \end{array} \right),\quad \varXi = \left( \begin{array}{c} \varXi^0 \\ \varXi^- \end{array} \right) \end{array} \end{aligned} $$
and the iso-singlet Λ. Each baryon entry is given by a Dirac spinor. For later use, we also introduce the mesonic isospin doublets
$$\displaystyle \begin{aligned} K=\left(\begin{array}{c} K^{+} \\ K^{0} \end{array} \right), \ \ \ K_{c}=\left(\begin{array}{c} \overline{K^{0}} \\ -K^{-} \end{array} \right) \quad , {} \end{aligned} $$
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 also serving to fix phases [67].
The full Lagrangian is given by
$$\displaystyle \begin{aligned} \mathcal{L} = \mathcal{L}_{\mathcal{B}} + \mathcal{L}_{\mathcal{M}} + \mathcal{L}_{int} \end{aligned} $$
accounting for the free motion of baryons with mass matrix \(\hat {M}\),
$$\displaystyle \begin{aligned} \mathcal{L}_{\mathcal{B}} = \overline{\varPsi}_{\mathcal{B}} \left[ i\gamma_\mu\partial^\mu - \hat{M} \right] \varPsi_{\mathcal{B}} \quad , \end{aligned} $$
and the Lagrangian density of massive mesons,
$$\displaystyle \begin{aligned} \mathcal{L}_{\mathcal{M}} =\frac{1}{2} \sum_{i \in \{ \mathcal{P},\mathcal{S},\mathcal{V} \}} \left(\partial_\mu\varPhi_i\partial^\mu\varPhi_i - m_{\varPhi_i}^2\varPhi_i^2\right) - \frac{1}{2} \sum_{\lambda \in \mathcal{V}} \left( \frac{1}{2} F^{{}^2}_\lambda - m^2_\lambda V^2_{\lambda} \right) \end{aligned} $$
where \(\mathcal {P},\mathcal {S},\mathcal {V}\) denote summations over the lowest nonet pseudo-scalar, scalar, and vector mesons. The field strength tensor of the vector meson fields \(V^\mu _\lambda \), λ ∈{ω, ρ, K, ϕ, γ} is defined by
$$\displaystyle \begin{aligned} \begin{array}{rcl} F^{\mu\nu}_{\lambda} = \partial^\mu V^\nu_{\lambda} - \partial^\nu V^\mu_{\lambda} \quad . {} \end{array} \end{aligned} $$
For the scattering of charged particles and in finite nuclei, the electromagnetic vector field \(V^\mu _\gamma \) 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 pseudo-scalar and the vector mesons are well identified as stable 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 \(\kappa =K^*_0(800)\) as found in the compilations of the Particle Data Group (PDG) [65]. 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 m κ  = 682 ± 29 MeV is recommended [65]. We use m κ  = 700 MeV. The octet baryons are listed in Table 5.1, the meson parameters are summarized in Table 5.2.
Table 5.2

Intrinsic quantum numbers and the measured masses [65] of the mesons used in the calculations



Mass (MeV)

Cut-off (MeV/c)










K 0, +



























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 [65]

Last but not least we consider the interaction Lagrangian \(\mathcal {L}_{int}\). 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, thus defining a guideline and reducing the number of free parameters considerably. The eight \(J^P={\textstyle \frac {1}{2}}^+\) baryons are collected into a traceless matrix \(\mathcal {B}\), which is given by a superposition of the eight Gell-Mann matrices λ i combined with the eight baryons B i  ∈{N, Λ, Σ, Ξ}, leading to the familiar form
$$\displaystyle \begin{aligned} \mathcal{B} =\sum_{i=1\cdots 8}{\lambda_i B_i}= \left( \begin{array}{ccc} {\displaystyle\frac{\varSigma^{0}}{\sqrt{2}}+\frac{\varLambda}{\sqrt{6}}} & \varSigma^{+} & p \\ {} \varSigma^{-} & {\displaystyle-\frac{\varSigma^{0}}{\sqrt{2}} +\frac{\varLambda}{\sqrt{6}}} & n \\ {} -\varXi^{-} & \varXi^{0} & {\displaystyle-\frac{2\varLambda}{\sqrt{6}}} \end{array} \right), \end{aligned} $$
which is invariant under SU(3) transformations. The pseudo-scalar (\(\mathcal {P}\)), vector (\(\mathcal {V}\)), and the scalar (\(\mathcal {S}\)) meson octet matrices are constructed correspondingly by replacing the baryons B i by the appropriate mesons M i . Taking the J P  = 0 pseudo-scalar mesons as an example we obtain the (traceless) octet matrix
$$\displaystyle \begin{aligned} \mathcal{P}_{8} = \left( \begin{array}{ccc} {\displaystyle\frac{\pi^{0}}{\sqrt{2}}+\frac{\eta_{8}}{\sqrt{6}}} & \pi^{+} & K^{+} \\ {} \pi^{-} & {\displaystyle-\frac{\pi^{0}}{\sqrt{2}} +\frac{\eta_{8}}{\sqrt{6}}} & K^{0} \\ {} K^{-} & \overline{{K}^{0}} & {\displaystyle-\frac{2\eta_{8}}{\sqrt{6}}} \end{array} \right). \end{aligned} $$
which, for the full nonet, has to be completed by the singlet matrix \(\mathcal {P}_1\), given by the 3 × 3 unit matrix multiplied by \(\eta _1/\sqrt {3}\). Thus, the full pseudo-scalar nonet is described by \(\mathcal {P}=\mathcal {P}_8+\mathcal {P}_1\). We define the SU(3)-invariant baryon-baryon-meson vertex combinations
$$\displaystyle \begin{aligned} \begin{array}{rcl} \left[\overline{\mathcal{B}}\mathcal{B}\mathcal{P}\right]_{D}&\displaystyle =&\displaystyle \mathrm{Tr}\left(\left\{\overline{\mathcal{B}},\mathcal{B}\right\}\mathcal{P}_{8}\right) \quad , \quad \left[\overline{\mathcal{B}}\mathcal{BP}\right]_{F} = \mathrm{Tr}\left(\left[\overline{\mathcal{B}},\mathcal{B}\right]\mathcal{P}_{8}\right) \quad , \\ \left[\overline{\mathcal{B}}\mathcal{BP}\right]_{S} &\displaystyle =&\displaystyle \mathrm{Tr}(\overline{\mathcal{B}}\mathcal{B})\mathrm{Tr}(\mathcal{P}_{1}) \end{array} \end{aligned} $$
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 pseudo-scalar interaction Lagrangian
$$\displaystyle \begin{aligned} \mathcal{L}^{\mathcal{P}}_{int} = - \sqrt{2}\left\{ g_D\left[\overline{\mathcal{B}}\mathcal{BP}_8\right]_{D} + g_F\left[\overline{\mathcal{B}}\mathcal{BP}_8\right]_{F} \right\}\, - \, g_S\frac{1}{\sqrt{3}}\left[\overline{\mathcal{B}}\mathcal{BP}_1\right]_{S}, {} \end{aligned} $$
with the generic SU(3) coupling constants {g D , g F , g S }. The de Swart convention [67], underlying the Nijmegen and the Jülich approaches, is given by using g8 ≡ g D  + g F , α ≡ g F /(g D  + g F ), and g1 ≡ g S leading to the equivalent representation
$$\displaystyle \begin{aligned} \mathcal{L}^{\mathcal{P}}_{int} = -g_{8}\sqrt{2}\left\{ \alpha\left[\overline{\mathcal{B}}\mathcal{BP}_8\right]_{F}+ (1-\alpha)\left[\overline{B}\mathcal{BP}_8\right]_{D}\right\}\, - \, g_{1}{\textstyle\frac{1}{\sqrt{3}}} \left[\overline{\mathcal{B}}\mathcal{BP}_1\right]_{S}, {} \end{aligned} $$
In order to evaluate the couplings we define the pseudo-vector derivative vertex operator \(m_\pi \varGamma _{\mathcal {P}}=\gamma _{5}\gamma _{\mu }\partial ^{\mu }\), following Eq. (5.1). From the F- and D-type couplings, Eq. (5.15), we obtain the pseudo-scalar octet-meson interaction Lagrangian in an obvious, condensed short-hand notation, going back to de Swart [67], The—in total 16—pseudo-scalar BB-meson vertices are completely fixed by the three nonet coupling constants (g D , g F , g S ) or, likewise, by (g8, g1, α).
Corresponding relations exist also for interactions induced by the vector and the scalar meson nonets. As in the pseudo-scalar case, they are given in terms of octet (\(\mathcal {V}^\mu _{8},\mathcal {S}_{8}\)) and singlet multiplets (\(\mathcal {V}^\mu _{1},\mathcal {S}_{1}\)), resulting in \(\mathcal {V}^\mu =\mathcal {V}^\mu _8+\mathcal {V}^\mu _1\) and \(\mathcal {S}=\mathcal {S}_8+\mathcal {S}_1\), respectively, and having their own sets of respective coupling constants \((g_{D},g_{F},g_{S})_{\mathcal {S,V}}\). The \(\mathcal {BBV}\) coupling constants are obtained in analogy to Eq. (5.17) by the mapping {K, π, η8, η1}→{K, ρ, ω8, ϕ1}. Correspondingly, the scalar couplings \(\mathcal {BBS}\) are obtained from Eq. (5.17) 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 \(\mathcal {M}\in \{\mathcal {P},\mathcal {V},\mathcal {S}\}\) are of a common structure which allows to express the \(\mathcal {BBM}\) coupling constants in generic manner. For that purpose, we denote the isoscalar octet meson by f ∈{η8, ω8, σ8}, the isovector octet meson by a ∈{π, ρ, δ}, and the iso-doublet mesons by K ∈{K0, +, K∗0, +} and \(\kappa =\{K^{*,0}_0,K^{*,+}_0 \}\). Irrespective of the particular interaction channel, the coupling constants are then given by the relations [67] where g NNa  = g8 and, depending on the case, {g D , g F , g S } denote either the pseudo-scalar, vector, or scalar set of basic 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 where again the proper g S  ≡ g1 coupling constant for the \(\{\mathcal {P},\mathcal {V},\mathcal {S}\}\) multiplet under consideration has to be inserted. The complete interaction Lagrangian is given by the sum over the partial interaction components
$$\displaystyle \begin{aligned} \mathcal{L}_{int}=\sum_{\mathcal{M}\in \{\mathcal{P},\mathcal{S},\mathcal{V} \}}{\mathcal{L}^{\mathcal{M}}_{int}} \quad . \end{aligned} $$
The coupling constants above define the tree-level interactions entering into calculations of T-matrices. The corresponding diagrams contributing to the and interactions in the S = −1 sector are shown in Fig. 5.3.
Fig. 5.3

Hyperon-nucleon OBE interactions in the S = −1 sector. Interactions without and with strangeness-exchange are displayed in the upper and lower row, respectively. The interactions from S = 0 scalar (σ, δ) and vector mesons (ω, ρ) mesons will contribute to the hypernuclear mean-field self-energies

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 θP,V,S, one for each meson multiplet, which relate the physical, dressed isoscalar mesons to their bare octet and singlet counterparts.

SU(3) symmetry, however, is broken at several levels and SU(3) relations will not be satisfied exactly. An obvious one is the non-degeneracy of the physical baryon and meson masses within the multiplets, also reflecting the non-degeneracy of (u, d, s) quark masses. As discussed below, this splitting leads to additional complex structure in the set of coupled equations for the scattering amplitudes because the various baryon-baryon (BB) channels open at different threshold energies \(\sqrt {s_{BB'}}=m_B+m_{B'}\). At energies \(s<s_{BB'}\) a given BB-channel does not contain asymptotic flux but contributes as a virtual state. Thus, scattering, for example, will be modified at any energy by virtual or real admixtures of channels. For pratical purposes, a scheme taking into account the physical hadron masses is of clear advantage, if not a necessity. Irrespective of the mass symmetry breaking, total strangeness and charge are always conserved. That is assured by using the particle basis, which at the same time allows to use physical hadron masses. The corresponding charge-strangeness conserving multiplets are listed in Table 5.3.
Table 5.3

Baryon-baryon channels for fixed strangeness S and total charge Q


Q = −2

Q = −1

Q = 0

Q = 1

Q = 2

S = 0





S = −1


Σ n

Λn Σ 0 n Σ p

Λp Σ + n Σ 0 p

Σ + p

S = −2


Ξ n Σ Λ Σ Σ 0

ΛΛ Ξ0n Ξp Σ+Λ Σ+Σ0

Ξ0p Σ+Λ Σ+Σ0

Σ + Σ +

S = −3


Ξ Λ Ξ 0 Σ Ξ Σ 0

Ξ 0 Λ Ξ 0 Σ 0 Ξ Σ +

Ξ 0 Σ +


S = −4


Ξ Ξ 0

Ξ 0 Ξ 0


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 [8] and for a practical application in [24]. However, as seen from Table 5.1, SU(3) symmetry is obviously broken on the mass scale: The average octet mass is \({{\bar m}_8} = {\mathrm {1166}}{\mathrm {.42}}\) MeV, the octet mass splitting is found as \(\varDelta {m_8} = {{\bar m}_\varXi } - {{\bar m}_N} = {\mathrm {379}}{\mathrm {.19}}\) 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. 5.4 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:
$$\displaystyle \begin{aligned} q^2(s)=\frac{1}{4s}\left((s-(m_1-m_2)^2)(s-(m_1+m_2)^2) \right), \end{aligned} $$
Fig. 5.4

The channel momentum q(s) is shown as a function of the Mandelstam variable s for the nucleon-nucleon and the S = −1 nucleon-hyperon two-body channels. The threshold is defined by q(s) = 0

where at the channel threshold s = s thres  = (m1 + m2)2 and q2(s thres ) = 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 data and numerical results. An example is found in Fig. 5.5. The spike showing up in the elastic Λp cross section at about p lab  ∼ 640 MeV/c is due to the coupling to the Σ0p channel which crosses the kinematical threshold at the \(\sqrt {s}=M_{\varSigma ^0}+M_p\) reached that channel momentum, namely at p lab  ≃ 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.
Fig. 5.5

Total cross sections as a function of p lab 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 [25] (dashed), and the Nijmegen NSC97f model [19] (solid curve) (from Ref. [32])

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.

In the /-system, for example, the tree-level interactions are given by
$$\displaystyle \begin{aligned} \begin{array}{rcl} {V_{N\varLambda ,N\varLambda }} &\displaystyle =&\displaystyle V_{N\varLambda ,N\varLambda }^{(\eta )} + V_{N\varLambda ,N\varLambda }^{(\sigma )} + V_{N\varLambda ,N\varLambda }^{(\omega )} \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} V_{N\varSigma ,N\varSigma }^{(I)} &\displaystyle =&\displaystyle V_{N\varSigma ,N\varSigma }^{(\eta )} + V_{N\varSigma ,N\varSigma }^{(\sigma )} + V_{N\varSigma ,N\varSigma }^{(\omega )} \\ &\displaystyle &\displaystyle + \left( {V_{N\varSigma ,N\varSigma }^{(\pi )} + V_{N\varSigma ,N\varSigma }^{(\delta )} + V_{N\varSigma ,N\varSigma }^{(\rho )}} \right)\left\langle {I||{\tau _N} \cdot {\tau _\varSigma }||I} \right\rangle \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} V_{N\varLambda ,N\varSigma }^{(I)} &\displaystyle =&\displaystyle \left( {V_{N\varLambda ,N\varSigma }^{(\pi )} + V_{N\varLambda ,N\varSigma }^{(\delta )} + V_{N\varLambda ,N\varSigma }^{(\rho )}} \right)\left\langle {I = \frac{1}{2}||{\tau _N} \cdot {\tau _\varSigma}||I = \frac{1}{2}} \right\rangle , \end{array} \end{aligned} $$
where τN,Σ denote the isospin operators acting on the nucleon and the Σ hyperon, respectively. The strengths of the interactions inhibits 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, \(\mathcal {V}\) indicates the Born terms, \(\mathcal {G}\) denotes the diagonal matrix of channel Green’s functions, and the resulting T-matrix is defined by
$$\displaystyle \begin{aligned} \mathcal{T}(q',q|\,P)=\mathcal{V}(q',q|\,P)+\int{\frac{d^4k}{(2\pi)^4}\mathcal{V}(q',k|\,P) }\mathcal{G}(k,P)\mathcal{T}(k,q|\,P) \end{aligned} $$
describing the transition of the system for the (off-shell) four-momenta q′→ q and fixed center-of-mass four-momentum P with s = P2. 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. [68, 69]. A widely used scheme is the so-called Blankenbecler-Sugar (BbS) reduction [70] 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:
$$\displaystyle \begin{aligned} \mathcal{T}(\mathbf{q}',\mathbf{q}|s)=\mathcal{V}(\mathbf{q}',\mathbf{q}|s)+ \int{\frac{d^3k}{(2\pi)^3}}\mathcal{V}(\mathbf{q}',\mathbf{k}|s)g_{BbS}(k,s)\mathcal{T}(\mathbf{k},\mathbf{q}|s) \end{aligned} $$
where the propagator is now replaced by the Blankenbecler-Sugar propagator with (diagonal) elements
$$\displaystyle \begin{aligned} g_{BbS}(k,s)=\frac{1}{s-(E_1(k)+E_2(k))^2+i\eta} \end{aligned} $$
where \(E_{1,2}(k)=\sqrt {m^2_{1,2}+k^2}\) is the relativistic energy of the particles in that given channel. The T-matrix may be expressed by the K-matrix:
$$\displaystyle \begin{aligned} \mathcal{T}=\left(1-i\mathcal{K} \right )^{-1}\mathcal{K} \end{aligned} $$
and very often the scattering problem is solved in terms of the (real-valued) K-matrix [68],
$$\displaystyle \begin{aligned} \mathcal{K}(\mathbf{q}',\mathbf{q}|s)=\mathcal{V}(\mathbf{q}',\mathbf{q}|s)+ \int{\frac{d^3k}{(2\pi)^3}}\mathcal{V}(\mathbf{q}',\mathbf{k}|s)\frac{P}{s-(E_1(k)+E_2(k))^2}\mathcal{T}(\mathbf{k},\mathbf{q}|s) \end{aligned} $$
given by Cauchy principal value integral. In practice, moreover, a decomposition into invariants and partial wave matrix elements is performed which reduces the problem 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. [68].
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. 5.6 where the s-wave matrix elements for 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.
Fig. 5.6

The np s-wave interactions in the (S = 0, I = 1) singlet-even channel in Born approximation, for which U(q, q) denotes the s-wave component of \(\mathcal {V}(q,q)\) (upper curve, red), and for the full K-matrix result, for which U(q, q) corresponds to the s-wave component of \(\mathcal {K}(q,q)\), (lower curve, blue). The difference between the Born-term and the fully summed K-matrix solution of the Lippmann-Schwinger is indicated by the shaded (blue coloured) area. The K-matrix results reproduce the measured scattering phase shift

Representative results for nucleon-hyperon scattering are shown in Fig. 5.5, comparing total cross sections obtained by EFT and the Jülich and Nijmegen OBE approaches, respectively. The latest χ EFT account for interactions up to next-to-leading-order (NLO) [71]. The resulting phase shifts in the spin-singlet channel are shown in Fig. 5.7 for a few baryon-baryon channels: the pp (S = 0), + (S = −1), and Σ+Σ+ (S = −2). From the pp results it is seen that the NLO calculations are trustable up to about p lab  ∼ 300 MeV/c.
Fig. 5.7

The pp, Σ+p, and Σ+Σ+ phase shifts in the1S0 partial wave. The filled bands represent the results at NLO [71]. The pp phase shifts of the GWU SAID-analysis [72] (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 (from Ref. [71])

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 Q F (p1, p2) = 1 − P F (p1, p2), projecting to the momentum region outside of the respective Fermi-spheres:
$$\displaystyle \begin{aligned} Q_F(\mathbf{p}_1,\mathbf{p}_2)=\varTheta(p^2_1-k^2_{F1})\varTheta(p^2_2-k^2_{F2}). \end{aligned} $$
The particle momenta p1,2 are given in the nuclear matter rest frame. Together with the time-like energy variables p0 they define the four-momenta \(p_{1,2}=(p^0_{1,2},\mathbf {p}_{1,2})^T\) which are related the total two-particle four-momentum P and the relative momentum k by the Lorentz-invariant transformation
$$\displaystyle \begin{aligned} p_{1,2}=\pm k+x_{1,2}P \end{aligned} $$
$$\displaystyle \begin{aligned} x_1=\frac{s-m^2_2+m^2_1}{2s}\quad ;\quad x_2=\frac{s-m^2_1+m^2_2}{2s} \end{aligned} $$
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
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} &\displaystyle &\displaystyle {K_{{B_1}{B_2},{B_3}{B_4}}}(q,q') = {{\tilde V}_{{B_1}{B_2},{B_3}{B_4}}}(q,q') \\ &\displaystyle &\displaystyle \quad + \sum_{{B_5}{B_6}} {P\int {\frac{{{d^3}k}}{{{{(2\pi )}^3}}}} } {V_{{B_1}{B_2},{B_5}{B_6}}}(q,k){G_{{B_5}{B_6}}}(k,{q_s}){Q_F}({p^2_5},{p^2_6}){K_{{B_5}{B_6},{B_3}{B_4}}}(k,q')\qquad \quad \end{array} \end{aligned} $$
where the integration has to be performed as a Cauchy Principal Value integral and the propagator \(G_{{B_5}{B_6}}(k,{q_s})\) 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. The coupled equations have to be 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 [19, 73].
The calculation of in-medium interactions is in fact a very involved self-consistency problem, see e.g. [45, 74]. 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. 5.8 where the nested structure is recognized. However, as 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 [29].
Fig. 5.8

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 Q F , Eq. (5.30), while P F is the projector onto the single particle Fermi sphere

For nuclear structure work, the low-energy behaviour of scattering amplitudes is of particular importance. The low momentum behaviour of the s-wave K-matrix is described by the effective range expansion for vanishing momentum q,
$$\displaystyle \begin{aligned} q\cot{\delta}_{|q\to 0}= -\frac{1}{a}+\frac{1}{2}q^2 r + \mathcal{O}(q^4) \end{aligned} $$
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. [75]. In Table 5.4 the low energy parameters are given for several OBE approaches and compared to leading order (LO) χ EFT results.
Table 5.4

Low energy parameters for the indicated YN system


a (fm)

r (fm)


ΣN (\(I=\frac {1}{2}\))



Jülich 04

ΣN (\(I=\frac {3}{2}\))



Jülich 04




NSC97f 04









ΛN (\(I=\frac {1}{2}\))



Jülich 04













Results from the OBE-oriented Jülich 04 [25], NSC97f [19], the recent Giessen Boson Exchange (GiBE) [29] models are compared to the leading order chiral EFT results [76]

In Fig. 5.9 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 density, i.e. k F  = 209 MeV/c the cross sections are reduced by a factor of \(\frac {1}{3}\) (NN) and a factor of \(\frac {1}{2}\) (Σ+p), respectively. At higher densities the reduction factors converge to an almost constant value close to the one in Fig. 5.9 seen for nuclear saturation density and k F  = 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. (5.33). 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.
Fig. 5.9

Free-space and in-medium total cross sections for pp (up) and Σ+p (down) elastic scattering. As indicated, the cross sections are evaluated in free space (ρ = 0), at half (ρ = ρ sat /2) and at full nuclear matter density ρ = ρ sat  = 0.16/fm3, respectively (from Ref. [29])

The elastic scattering amplitudes for B1B2 → B1B2 we may separate into effective coupling constants \(\tilde {g}_{B_1B_1}\tilde {g}_{B_2B_2}\) and a matrix element \(\bar {M}_{B_1B_2}\), amputated by all coupling constants. Thus, the cross section are given as
$$\displaystyle \begin{aligned} \sigma_{B_1B_2}\sim \tilde{g}^2_{B_1B_1}\tilde{g}^2_{B_2B_2}\bar{\sigma}_{B_1B_2} \end{aligned} $$
$$\displaystyle \begin{aligned} \bar{\sigma}_{B_1B_2}=\frac{4\pi}{q^2}|\bar{M}_{B_1B_2}|{}^2 \end{aligned} $$
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 \(\bar {\sigma }\) is an universal quantity within a given flavor multiplet, we find
$$\displaystyle \begin{aligned} R^2_{B_2B_3}=\frac{\sigma_{B_1B_2}}{\sigma_{B_1B_2}}=\frac{\tilde{g}^2_{B_2B_2}}{\tilde{g}^2_{B_3B_3}} \end{aligned} $$
From Fig. 5.9 we then obtain \(R_{p\varSigma ^+}\sim 0.62\ldots 0.77\) for ρ = 0 to ρ = ρ sat and similar results are found also for other channels. These values are surprisingly close to assumptions of the naïve quark model, namely that hyperon interactions scale essentially with the number of u and d valence quarks, discussed e.g. in [56, 77] and in Sect. 5.4.
By evaluating the low-energy parameters as function of the background density we gain further insight into the density dependence of interactions. In Fig. 5.10 the scattering lengths of the coupled and 0 channels are shown as a functions of the Fermi momentum of symmetric nuclear matter, k F  = (3π2ρ/2)1/3. For k F  → 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.
Fig. 5.10

The in-medium scattering lengths a s for the coupled and 0 channels are shown as a function of the Fermi momentum \(k_{F_N}\) of symmetric nuclear matter (from Ref. [29])

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.

In the following, the expansion of a fully resummed interaction in terms of dressed vertices and boson propagators is sketched. For the reaction amplitude of the process \(\mathcal {B}=(B_1B_2)\to \mathcal {B}'=(B_3B_4)\) we use a the ansatz
$$\displaystyle \begin{aligned} T_{\mathcal{BB}'}(\mathbf{q},\mathbf{q}'|q_sk_F)=\sum_{a}{\varGamma^{(a)\dag}_{B_1B_3}(q_s,k_F)U^{(a)}_{\mathcal{BB}'}(\mathbf{q},\mathbf{q}')\varGamma^ {(a)}_{B_2B_4}(q_s,k_F)} \end{aligned} $$
$$\displaystyle \begin{aligned} U^{(a)}_{\mathcal{BB}'}(\mathbf{q},\mathbf{q}')=\mathcal{M}^{(a)}_{\mathcal{BB}'} D_a(\mathbf{q},\mathbf{q}') \end{aligned} $$
is given by the invariant matrix element
$$\displaystyle \begin{aligned} \mathcal{M}^{(a)}_{\mathcal{BB}'}=\langle B_4B_3|\mathcal{O}^\dag_a(1)\cdot \mathcal{O}_a(2)|B_1B_2\rangle= \langle B_4|\mathcal{O}^\dag_a|B_2\rangle \cdot \langle B_3|\mathcal{O}_a|B_1\rangle \end{aligned} $$
containing the vertex operators \(\mathcal {O}_a\) belonging to the interaction of type a ∈{A, P, S, T, V } defined above, but without coupling constants. The scalar product of the two operators is indicated by the dot-product. Non-relativistically, the operator set is given by \(\mathcal {O}_a=\{1_\sigma ,\boldsymbol {\sigma } \cdot \mathbf {q},\boldsymbol {\sigma },\boldsymbol {\sigma }\times \mathbf {q}\}\otimes \{1_\tau ,\boldsymbol {\tau }\}\) for scalar, pseudo-scalar, vector, and pseudo-vector interactions of isoscalar and isovector character, respectively. The propagator D a (q, q) describes the exchange of the boson a, representing the specific interaction channel. Since the vertices are attached to the matrix element \(\mathcal {M}^{(a)}_{\mathcal {BB}'}\), the propagators are in fact of simple Yukawa-form,
$$\displaystyle \begin{aligned} D_a(\mathbf{q},\mathbf{q}')=\frac{1}{(\mathbf{q}-\mathbf{q}')^2+m^2_a}, \end{aligned} $$
thus depending only on the momentum transfer if meson self-energies are neglected, as above. Accordingly, the Born-terms are given by
$$\displaystyle \begin{aligned} V_{\mathcal{BB}'a}=U^{(a)}_{\mathcal{BB}'}g^2_{\mathcal{BB}'a} \end{aligned} $$
where \(g^2_{\mathcal {BB}'a}\) denotes the bare coupling constants in free space. Leaving out for simplicity the baryon indices and momentum arguments, we find the important relations
$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\delta V}{\delta U^{(a)}}&\displaystyle =&\displaystyle g^2_a {} \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\delta T}{\delta U^{(a)}}&\displaystyle =&\displaystyle \varGamma^{(a)\dag}\varGamma^{(a)} {}. \end{array} \end{aligned} $$
Formally, the Lippmann-Schwinger equation is solved by
$$\displaystyle \begin{aligned} T=\left(1-\int{dk V\mathcal{G}^*Q_F} \right)^{-1}V \end{aligned} $$
and applying Eqs. (5.43) and (5.44), we obtain
$$\displaystyle \begin{aligned} \varGamma^{(a)\dag}\varGamma^{(a)}=\left(1-\int{dk V\mathcal{G}^*Q_F}\right)^{-2}g^2_a+\ldots \end{aligned} $$
where the dots indicate terms given by the variational derivatives of baryon (and meson) self-energies which, in general, are contained in the intermediate baryon-baryon propagator \(\mathcal {G}^*\). Thus, the coupling constants of the dressed vertices are determined in leading order by
$$\displaystyle \begin{aligned} \varGamma^{(a)}\simeq\left(1-\int{dk V\mathcal{G}^*Q_F}\right)^{-1}g_a. \end{aligned} $$
Hence, we have recovered the well known integral equation for dressed vertices, generalizing Migdal’s theorem for dressed vertices in interacting many-body systems [91, 92] to interactions in infinite nuclear matter. Restoring the full index structure, we find
$$\displaystyle \begin{aligned} \varGamma_{\mathcal{BB}'a}(q_s,k_F)\simeq \frac{1}{1-\int{dq' V_{a}\mathcal{G}^*Q_F}}_{|{}_{\mathcal{BB}'}}g_{\mathcal{BB}'a}. \end{aligned} $$
The diagrammatic structure is depicted in Fig. 5.11. Inspecting Eq. (5.48), one finds that dressed and bare vertices are related by a susceptibility matrix
$$\displaystyle \begin{aligned} \chi_{\mathcal{BB}'a}(q_s,k_F)=\left(1-\int{dq' V_{a}\mathcal{G}^*Q_F}\right)^{-1}_{|{}_{\mathcal{BB}'}} \end{aligned} $$
depending on the center-of-mass energy \(\sqrt {s}\) through q s and the set of Fermi momenta \(k_F=\{ k_{F_B}\}_{|B=n,p\ldots }\).
Fig. 5.11

Diagrammatic structure of the dressed vertex functionals \(\varGamma _{BB'a}\) (filled square) in terms of the bare coupling constant \(g_{BB'a}\) (filled circle) and the interaction \(U_{\mathcal {BB}'a}\), 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)

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 .

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 Σ aB (k F ) due to the exchange of the boson a felt by the baryon B is given by
$$\displaystyle \begin{aligned} \varSigma_{aB}(k_F)=\varGamma_{BBa}(k_F)\frac{1}{m^2_a}\sum_{B'}{\varGamma_{B'B'a}(k_F)\rho^{(a)}_{B'}} \end{aligned} $$
where on the right side we have inserted the above decomposition. Thus, a set of quadratic forms is obtained, bilinear in the vertex functions Γ BBa (k F ). \(\rho ^{(a)}_{B}\) 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
$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\varGamma_{B_1B_1a}(k_F)}{\varGamma_{B_2B_2a}(k_F)}&\displaystyle =&\displaystyle \frac{\varSigma_{aB_1}(k_F)}{\varSigma_{aB_2}(k_F)} \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} \varGamma^2_{aB}(k_F)&\displaystyle =&\displaystyle \frac{\varSigma^2_{aB}(k_F)}{\sum_{B'}{\rho_{aB'}\varSigma_{aB'}(k_F)}}. \end{array} \end{aligned} $$
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 [74], those dependencies are cancelling in the above expressions to a large extent. As discussed in [87] 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 [74] are shown in Fig. 5.12 and applications to stable and exotic nuclei and neutrons stars are found e.g. in [41, 42, 43, 46, 87]. Hypernuclear results are discussed below.
Fig. 5.12

In-medium NNα isoscalar (α = σ, ω) and isovector (α = δ, ρ) vertices determined from DBHF self-energies in asymmetric nuclear matter as explained in [74]

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 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 [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.

The essential difference of the Lagrangian discussed here to the one of Eq. (5.8) 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 constraints is to postulate a dependence on the invariant density operator \(\hat {\rho }=j_{\mu }j^\mu \) of the baryon 4-current operator \(j^\mu =\bar {\varPsi }_B\gamma ^\mu \varPsi _{\mathcal {B}}\). Thus, we use \(g^*_{BB'\mathcal {M}}(\varPsi _{\mathcal {B}})=g^*_{BB'\mathcal {M}}(\hat {\rho })\). Applying the SU(3) relations, this implies immediately a corresponding structure for the fundamental interaction vertices, \(\{g_D,g_F,g_S\} \to \{g^*_D(\hat {\rho }),g^*_F(\hat {\rho }),g^*_S(\hat {\rho })\}\). Thus, the interactions with density functional (DF) vertices are described by
$$\displaystyle \begin{aligned} \mathcal{L}^{DF}_{int} = - \sqrt{2}\sum_{\mathcal{M}\in \{\mathcal{P,S,V}\}}{\left\{ g^{*(\mathcal{M})}_D\left[\overline{\mathcal{B}}\mathcal{BP}_8\right]_{D} + g^{*(\mathcal{M})}_F\left[\overline{\mathcal{B}}\mathcal{BP}_8\right]_{F} - g^{*(\mathcal{M})}_S\frac{1}{\sqrt{6}}\left[\overline{\mathcal{B}}\mathcal{BP}_1\right]_{S}\right\}}. {} \end{aligned} $$
By the same relations as in Eq. (5.17) we obtain baryon-meson vertices \(g^*_{BB'M}(\hat {\rho })\) but with the complexity of intrinsic functional structures.
With the standard methods of field theory we obtain the meson field equations which are of the form
$$\displaystyle \begin{aligned} \begin{array}{rcl} \left(\partial_\mu\partial^\mu+m^2_{\mathcal{M}} \right)\varPhi^s_{\mathcal{M}}&\displaystyle =&\displaystyle \sum_{BB'}{g^*_{BB'\mathcal{M}}(\hat{\rho})\rho^{BB's}} \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} \left(\partial_\mu\partial^\mu+m^2_{\mathcal{M}} \right)V^\lambda_{\mathcal{M}}&\displaystyle =&\displaystyle \sum_{BB'}{g^*_{BB'\mathcal{M}}(\hat{\rho})\rho^{BB'\lambda}} \end{array} \end{aligned} $$
where Lorentz-scalar and Lorentz-vector fields are described by the first and second equation, respectively. The baryons are obeying the field equations
$$\displaystyle \begin{aligned} \left(\gamma_\mu\left(p^\mu-\varSigma^\mu_{\mathcal{B}}(\hat{\rho})\right) -M_{\mathcal{F}}+\varSigma^{(s)}_{\mathcal{B}}(\hat{\rho}) \right)\varPsi_{\mathcal{B}}=0 \end{aligned} $$
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 [41, 42]. As discussed above, the density dependence of the in-medium meson-baryon vertices is chosen by a functional structures given by \(\hat {\rho }^2=j_{\mu }j^\mu _{B}\) 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.
The rearrangement self-energies account for the static polarization of the surrounding medium due to the presence of the baryon. As was proven in [42], the rearrangement self-energies ensure the thermodynamical consistency of the theory. Their neglection would violate the Hugenholtz-van Hove theorem. Similar conclusion have been drawn before by Negele for non-relativistic BHF theory [97]. In mean-field approximation, the self-energies will become Hartree type mean-fields. The diagrammatic structure of the mean-field self-energies is depicted in Fig. 5.13. The structure of the rearrangement self-energies is discussed in detail in [45, 97].
Fig. 5.13

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 graph 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. 5.8

With our choice of density dependence, only the vector self-energies are modified by rearrangement effects. Thus, the scalar self-energy \(\varSigma ^s_{\mathcal {F}}\) is obtained as a diagonal matrix with the elements of pure Hartree-structure
$$\displaystyle \begin{aligned} \varSigma^{(s)}_{B}(\hat{\rho})=\sum_{\mathcal{M}\in S}{\varPhi_M(\hat{\rho})g^*_{BB\mathcal{M}}(\hat{\rho})}. \end{aligned} $$
The vector self-energies, contained in the matrix \(\varSigma ^\mu _{\mathcal {F}}\), are consisting of the direct meson field contributions
$$\displaystyle \begin{aligned} \varSigma^{(d)\mu}_{B}(\hat{\rho})=\sum_{\mathcal{M}\in V}{V^\mu_{\mathcal{M}}(\hat{\rho})g^*_{BB\mathcal{M}}(\hat{\rho})} \end{aligned} $$
and the rearrangement self-energies
$$\displaystyle \begin{aligned} \varSigma^{(r)\mu}_{B}(\hat{\rho})=\sum_{B'B''\mathcal{M}}{\frac{\partial g^*_{B'B''M}(\hat{\rho})}{\partial \rho^{BB}_\mu}\frac{\delta}{\delta g^*_{B'B'f'M} }\mathcal{L}^{DF}_{int} } \quad , \end{aligned} $$
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 [41, 42, 46].
A substantial simplification is obtained in mean-field approximation. As discussed in [46], the vertex functionals are replaced by ordinary functions of the expectation value of their argument. The meson fields are replaced by static classical 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
$$\displaystyle \begin{aligned} \begin{array}{rcl} \varSigma^{(s)}_{B}(\rho_{\mathcal{B}})=\sum_{B',\mathcal{M}\in \mathcal{S}}{\frac{1}{m^2_{\mathcal{M}}}g^*_{BB\mathcal{M}}(\rho_{\mathcal{B}})g^*_{B'B'\mathcal{M}}(\rho_{\mathcal{B}})\rho^{(s)}_{B'}} \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} \varSigma^{(d)}_{B}(\rho_{\mathcal{B}})=\sum_{B',\mathcal{M}\in \mathcal{V}}{\frac{1}{m^2_{\mathcal{M}}}g^*_{BB\mathcal{M}}(\rho_{\mathcal{B}})g^*_{B'B'\mathcal{M}}(\rho_{\mathcal{B}})\rho_{B'}} \end{array} \end{aligned} $$
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, \(\rho _{\mathcal {B}}=\rho _N+\rho _Y\),
$$\displaystyle \begin{aligned} \varSigma^{(r)}_{B}(\rho_{\mathcal{B}})=\sum_{B',\mathcal{M}}\frac{\partial g^*_{B'B'\mathcal{M}}(\rho_{\mathcal{B}})}{\partial \rho_B}\frac{\partial \mathcal{L}_{int}}{\partial g^*_{B'B'\mathcal{M}}}, \end{aligned} $$
where the usual number and scalar densities are denoted by ρ B and \(\rho ^{(s)}_B\), respectively. For later use we define the total time-like baryon vector self-energies
$$\displaystyle \begin{aligned} \varSigma^{(v)}(\rho_{\mathcal{B}})=\varSigma^{(d)}_{B}(\rho_{\mathcal{B}})+\varSigma^{(r)}_{B}(\rho_{\mathcal{B}}). \end{aligned} $$

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 Rσ,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 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
$$\displaystyle \begin{aligned} \left(\boldsymbol{\alpha\cdot \hat{p}}+\varSigma^{(v)}_B(r)+\gamma^0M^*_B(r)-\varepsilon_{n\ell j}\right)\psi_{n\ell jm}=0 \end{aligned} $$
with \(\mathbf {\hat {p}}=-i\boldsymbol {\nabla }\) and α = γ0γ. We assume spherical symmetry. The Dirac spinors ψ nℓjm with radial quantum number n and total angular momentum j and projection m are eigenfunctions to the eigenenergy ε nℓj . The orbital angular momentum of the upper component is indicated by . The direct vector self-energies are
$$\displaystyle \begin{aligned} \varSigma^{(d)}_B(r)=g^*_{BB\omega}(\rho_{\mathcal{B}})V^0_\omega(r)+\langle\tau_3\rangle_B g^*_{BB\rho}(\rho_{\mathcal{B}})V^0_\rho(r)+q_BV^0_\gamma(r). \end{aligned} $$
\(\rho _{\mathcal {B}}(r)=\rho _p(r)+\rho _n(r)+\rho _\varLambda (r)\) is the radial dependent total baryon density with a volume integral normalized to the total baryon number A B  = A n  + A p  + A λ where in a single Λ nucleus A Λ  = 1. q B 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 \(M^*_B\), are given by
$$\displaystyle \begin{aligned} \varSigma^{(s)}_B(r)=g^*_{BB_\sigma}(\rho_{\mathcal{B}})\varPhi_\sigma(r)+\langle\tau_3\rangle_B g^*_{BB_\delta}(\rho_{\mathcal{B}})\varPhi_\delta(r) . \end{aligned} $$
The condensed meson fields are determined by the classical field equations
$$\displaystyle \begin{aligned} \begin{array}{rcl} \left(-\nabla^2+m^2_\omega \right)V^0_\omega&\displaystyle =&\displaystyle g^*_{NN\omega}(\rho_{\mathcal{B}})\left(\rho_p+\rho_n \right)+g^*_{\varLambda\varLambda\omega}(\rho_{\mathcal{B}})\rho_\varLambda \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} \left(-\nabla^2+m^2_\rho \right)V^0_\rho&\displaystyle =&\displaystyle g^*_{NN\omega}(\rho_{\mathcal{B}})\left(\rho_p-\rho_n \right) \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} -\nabla^2V^0_\gamma&\displaystyle =&\displaystyle e_p\rho^{(c)}_p \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} \left(-\nabla^2+m^2_\sigma \right)\varPhi_\sigma&\displaystyle =&\displaystyle g^*_{NN\sigma}(\rho_{\mathcal{B}})\left(\rho^{(s)}_p+\rho^{(s)}_n \right)+g^*_{\varLambda\varLambda\sigma}(\rho_{\mathcal{B}})\rho^{(s)}_\varLambda \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} \left(-\nabla^2+m^2_\delta \right)\varPhi_\delta&\displaystyle =&\displaystyle g^*_{NN\delta}(\rho_{\mathcal{B}})\left(\rho^{(s)}_p-\rho^{(s)}_n \right), \end{array} \end{aligned} $$
where e p is the proton charge and \(\rho ^{(c)}_p\) denotes the proton charge density, including the proton charge form factor. The vector and scalar densities ρ B and \(\rho ^{(s)}_B\) 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 [42, 43, 45]. From the field-theoretical energy-momentum tensor the energy density is obtained as the ground state expectation value \(\mathcal {E}_A(\mathbf {r};Z,N,Y)=\langle T^{00} \rangle \) for a nucleus with A p protons, A n neutrons and A Λ hyperons [40, 42]. Of particular interest are the nuclear binding energies defined by
$$\displaystyle \begin{aligned} B_A=\frac{1}{A}\left[ \int{d^3r\mathcal{E}_A(\mathbf{r})-A_p M_p-A_n M_n-A_\varLambda M_\varLambda} \right], \end{aligned} $$
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 [40]. The lightest nuclei were left out of the fitting procedure because they are not well suited for a mean-field description. From a χ2 procedure, we find that the two scaling constants are correlated linearly, leading to a χ2 distribution with a steep, but long stretched valley. Two solutions of comparable quality are \(\left ( R_{1\sigma },R_{1\omega }\right )=\left (0.506,0.518\right )\) and \(\left ( R_{2\sigma },R_{2\omega }\right )=\left (0.490,0.553\right )\). On a 20% level both sets are compatible with the quark model hypothesis and with the SU(3)-symmetry based results [98]. Taking into account also the light nuclei, the values show a considerably large spread. In Fig. 5.14 relativistic mean-field 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.
Fig. 5.14

Separation energies of the known S = −1 single Λ hypernuclei as a function of the mass number A to power \(\gamma =- \frac {2}{3}\). Results of two sets of scaling parameter sets are compared to measured separation energies. For the  > 0 levels the spin-orbit splitting is indicated. The theoretical uncertainties are marked by colored/shaded bands. For A → the limiting value \(S^{\infty }_\varLambda \simeq 28\) 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. The data are from refs. [99, 100, 101, 102, 103]

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.

In Fig. 5.15 results of the Nijmegen group are shown for comparison. The ESC08 interaction with and without pomeron-exchange (see [23]) 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.
Fig. 5.15

Energy spectra of single-Λ hypernuclei (\(^{13}_{\varLambda }C\), \(^{28}_{\varLambda }Si\), \(^{51}_{\varLambda }V\), \(^{139}_{\varLambda }La\), \(^{208}_{\varLambda }Pb\)) derived by the multi-pomeron exchange interaction MPa (solid lines) and the bare ESC08 interaction (dotted lines). Details of the Nijmegen approach are found in [23]. The theoretical results are compared to experimental values, marked by open circles (from [23])

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.

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 core-particle spin-spin energy to the Λ eigenenergies
$$\displaystyle \begin{aligned} e_{(jJ_C)J\varLambda}=\varepsilon^{RMF}_{j\varLambda}+E_{jJ_c}\langle (jJ_C)J|\boldsymbol{j}_\lambda\cdot\boldsymbol{J}_C|(jJ_C)J\rangle \quad . \end{aligned} $$
giving rise to a multiplet of states. 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 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 [104], modifying the effective Λ-spin-orbit potential to
$$\displaystyle \begin{aligned} U^{(so)}_\varLambda=\frac{1}{r}\mathbf{r}\cdot\boldsymbol{\nabla}\left[\left(2\frac{M^*_B}{M_B} \frac{f_{\varLambda\varLambda\omega}}{g^*_{\varLambda\varLambda\omega}}+1\right)\varSigma^\varLambda_\omega+\varSigma^\varLambda_\sigma \right]. \end{aligned} $$
Here, the tensor strength f appears as an additional parameter. For the NN case, f NNω 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 cancelling the conventional spin-orbit potential, given by the sum of vector and scalar self-energies. This should happen for f/g ∼−1 as seen by considering that \(U^{so}_\varLambda \) 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, f Λλω /g Λλω  = 0, thus agreeing with the NN-case. Our results for the Λ single particle spectra in the two nuclei are found in Table 5.5. The averaged spin-orbit splitting is about 223 and 283 keV and the spin-spin interaction amounts to \(E_{jJ_c}=106\) keV and \(E_{jJ_c}=61.3\) keV in Vanadium and Yttrium, respectively. The experimentally obtained and the theoretical spectra of \(^{89}_\varLambda Y\) are compared in Fig. 5.16.
Fig. 5.16

Single Λ binding energies in \(^{89}_\varLambda Y\). The DDRH results (dotted, dashed, and full lines) are compared to the data of Hotchi et al. [99], 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

Table 5.5

DDRH results for Λ single particle energies


\(^{89}_{\varLambda }Y\) (MeV)

\(^{41}_{\varLambda }V\) (MeV)


− 22.94 ± 0.64

− 19.8 ± 1.4


− 17.02 ± 0.07

− 11.8 ± 1.3


− 16.68 ± 0.07

− 11.4 ± 1.3


− 10.26 ± 0.07

− 2.7 ± 1.2


− 9.71 ± 0.07

− 1.9 ± 1.2


− 3.04 ± 0.11


− 3.04 ± 0.11

The experimentally unresolved fine structure due to the residual core-particle spin-spin interactions inhibits a precise determination of the genuine, reduced Λ single particle energies, defined by subtracting the core-particle spin-spin interaction energies. The shown errors are taking into account the resulting uncertainties in the reduced DDRH single-particle energies

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 12C 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 4He 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 ΛΛ1S0 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 | < |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 channel. In the ESC08c model, however, the apparently required ΛΛ attraction is obtained without giving rise to unphysical bound states.

5.4.4 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. 5.5 and Ref. [32]. Already rather early the Munich and the Jülich groups have applied χ EFT also to hypernuclei. The early applications as in [116] were based on the leading order (LO) and next-to-leading order (NLO) diagrams shown in Fig. 5.17.
Fig. 5.17

Leading order (LO) and next-to-leading order (NLO) diagrams used in the χ EFT 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

The results obtained by Finelli et al. in [116] 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. 5.18 the single-Λ separation energies are shown. The LO and NLO nucleon-hyperon interactions were derived by fits to the spectra of \(^{13}_{\varLambda }C\), \(^{16}_{\varLambda }O\), \(^{40}_{\varLambda }Ca\), \(^{89}_{\varLambda }Y\), \(^{139}_{\varLambda }La\), and \(^{208}_{\varLambda }Pb\). 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.
Fig. 5.18

χ EFT 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 (from Ref. [116])

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.

A different aspect of hypernuclear physics is considered by the Darmstadt group of Roth and collaborators. In [119] 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 shell 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. 5.19. 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.
Fig. 5.19

Absolute and excitation energies of \(^{13}_\varLambda C\). The convergence properties of the calculated energies on the number of harmonic oscillator basis states is displayed, denoted by the number of principal oscillator shells N max . (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). Energies are determined with respect to the corresponding ground states (from Ref. [119])

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 3S1 −3D1 coupled channels and the 1S0 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

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
$$\displaystyle \begin{aligned} \rho_{\mathcal{B}}=\sum_B{tr_s\int{\frac{d^3k}{(2\pi)^3} n_{sB}(k,k_{F_B})} }\quad . \end{aligned} $$
where the trace is to be evaluated with respect to spin s. In cold spin saturated matter the occupation numbers n sB are independent of s and are given by \(n_B=\varTheta (k^2_{F_B}-k^2)\) resulting in
$$\displaystyle \begin{aligned} \rho_B=\frac{N_s}{3\pi^2}k^3_{F_B} \end{aligned} $$
and N s  = 2 is the spin multiplicity for a spin-1/2 particle. Frequently we use \(\rho _B=\xi _B\rho _{\mathcal {B}}\) where the fractional baryon numbers \(\xi _B=\rho _B/\rho _{\mathcal {B}}\) add up to unity. The scalar densities are defined by
$$\displaystyle \begin{aligned} \rho^{(s)}_B=N_s\int{\frac{d^3k}{(2\pi)^3}\frac{M^*_B}{E^*_B(k)}} \end{aligned} $$
where \(E^*_B(k)=\sqrt {k^2+M^{*2}_B}\). The integral is easily evaluated in closed form and one finds \(\rho ^{(s)}_B=\rho _Bf_s(z_B)\) where
$$\displaystyle \begin{aligned} f_s(z)=\frac{3}{2z^3}\left(z\sqrt{1+z^2}-\log{(z+\sqrt{1+z^2})}\right) \quad . \end{aligned} $$
is a positive transcendental function with f s  ≤ 1 depending on \(z_B=k_{F_B}/M^*_B\). Since \(M^*_B\) depends on \(\rho ^{(s)}_B\) 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
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathcal{E}(\rho_{\mathcal{B}}) &\displaystyle =&\displaystyle \langle T^{00} \rangle = \sum_{B} \frac{1}{4} \left[ 3E_{F_B}\rho_B + m_B^*\rho^{(s)}_B \right] \\ &\displaystyle &\displaystyle + \frac{1}{2} \left[ m_{\sigma}^2\varPhi_{\sigma}^2 + m_{\delta}^2\varPhi_{\delta}^2 + m_{\sigma'}^2\varPhi_{\sigma'}^2 + m_{\omega}^2{V^0_{\omega}}^2 + m_{\rho}^2{V^0_{\rho}}^2 + m_{\phi}^2{V^0_{\phi}}^2 \right]\qquad \end{array} \end{aligned} $$
where the sum runs over all baryons and their energies are weighted by the partial vector and scalar densities ρ B and \(\rho ^{(s)_B}\) 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
$$\displaystyle \begin{aligned} \varepsilon(\rho_{\mathcal{B}})=\mathcal{E}(\rho_{\mathcal{B}})/\rho_{\mathcal{B}}-\sum_B{\xi_B M_B} \quad . \end{aligned} $$
In Fig. 5.20 results of DDRH calculation in the scaling approximation for \(\epsilon (\rho _{\mathcal {B}})\) 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 ξ p  = ξ n and ξ Λ  = 1 − 2ξ p .
Fig. 5.20

Binding energy per baryon of (n, p, Λ) matter. The Λ fraction is defined as ξ Λ  = ρ Λ /ρ and the background medium is chosen as symmetric (p, n) matter, ξ n  = ξ p . The absolute minimum is marked by a filled circle. The line ε = 0 is indicated by a red line

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

In the quark model, the isoscalar mesons have flavor wave functions
$$\displaystyle \begin{aligned} \begin{array}{rcl} |\,f_1\rangle&\displaystyle =&\displaystyle \frac{1}{\sqrt{3}}\left( |u\bar u\rangle + |d\bar d\rangle + |s\bar s\rangle \right) \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} |\,f_8\rangle&\displaystyle =&\displaystyle \frac{1}{\sqrt{6}}\left( |u\bar u \rangle + |d\bar d\rangle -2|s\bar s\rangle \right) \end{array} \end{aligned} $$
Since they are degenerate in their spin-flavor quantum numbers the physical states f8 → f and f1 → f′ will be superpositions of the bare states. Taking this into account, the physical mesons are written as
$$\displaystyle \begin{aligned} \begin{array}{rcl} |\,f_m \rangle&\displaystyle =&\displaystyle \cos{\theta_m}|\,f_0\rangle+\sin{\theta_m}|\,f_8\rangle \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} |\,f^{\prime}_m\rangle&\displaystyle =&\displaystyle \sin{\theta_m}|\,f_0\rangle-\cos{\theta_m}|\,f_8\rangle \end{array} \end{aligned} $$
where f m  ∈{η, ω, σ} and \(f^{\prime }_m\in \{\eta ',\phi ,\sigma _s \}\). Ideal mixing is defined if f m does not contain a \(s\bar s\) component and \(f^{\prime }_m\) is given as a pure \(s\bar s\) configuration, requiring \(\theta _{ideal}=\pi /2-\arcsin (\sqrt {(}2/3))\simeq 35.3^{\circ }\). Physical mixing within a meson nonet, however, is reflected by mass relations of the Gell-Mann Okubo-type. The widely used linear mass relation [65]
$$\displaystyle \begin{aligned} \tan{\theta_m}=\frac{4m_K-m_{a}-3m_{f'}}{2\sqrt{2}(m_{a}-m_K)} \end{aligned} $$
leads to (η, η′) mixing with θ P  = −24.6 and (ω, ϕ)-mixing with θ V  = +36.0, respectively. For the scalar nonet, the situation is less well understood, mainly because of the unclear structure of those mesons. The 0++ multiplets are typically strongly mixed with two- and multi-meson configurations or corresponding multi-\(q\bar q\)-configurations, leading to broad spectral distributions. For the purpose of low-energy baryon-baryon and nuclear structure physics we follow the successful strategy and identify the lowest scalar resonances as the relevant degrees of freedom. Thus, we choose the scalar octet consisting of the isoscalar σ = f0(500), the isovector a0(980) and the isodoublet \(\kappa =K^*_0(800)\) mesons, respectively. While these mesons are observed at least as broad resonances [65], the isoscalar-singlet partner σ′ of the σ-meson is essentially unknown. From the SU(3) mixing relation, however, one easily derives the instructive mass relation [65],
$$\displaystyle \begin{aligned} (m_f+m_{f'})(4m_K-m_a)-3m_fm_{f'}-8m^2_K+8m_Km_a-3m^2_a=0 \quad , \end{aligned} $$
serving as a constraint among the physical masses. Solving this equation for the scalar singlet meson we find the mass \(m_{f'}=m_{\sigma '}=936^{+406}_{-88}\) MeV. The large uncertainty range indicates the uncertainties of the choice of the σ- and the \(\kappa =K^*_0\)-masses, mentioned before. We use the mean mass values m σ  = 475 MeV and m κ  = 740 MeV, leading the scalar-singlet mass \(m_{\sigma '}=936\) MeV, which lies close to mass of the f0(980) state, in good compliance with general expectations [65]. Then, the corresponding scalar mixing angle is θ S  = −50.73. In Table 5.6 the mixing results are collected.
Table 5.6

Octet-singlet meson mixing used to determine the in-medium vertices



















a 0


Meson-mixing affects directly the baryon interactions. The transformed BB-vector nonet coupling constants are displayed in Table 5.7 and corresponding relations hold for the pseudo-scalar {ω, ρ, K, ϕ}→{η, π, K, η′} and the scalar nonets, {ω, ρ, K, ϕ}→{σ, a0, κ, σ′}.
Table 5.7

SU(3) relations for the ω, ρ and ϕ baryon coupling constants, relevant for the mean-field sector of the theory


Coupling constant


\(g_{NN\omega }=g_S \cos {}(\theta _v) +\frac {1}{\sqrt {6}} \left (3 g_F -g_D\right ) \sin {}(\theta _v)\)


\(g_{NN\phi }=g_S \sin {}(\theta _v) -\frac {1}{\sqrt {6}} \left (3 g_F -g_D\right ) \cos {}(\theta _v)\)


\(g_{NN\rho }=\sqrt {2}(g_F+g_D)\)


\(g_{\varLambda \varLambda \omega }=g_S \cos {}(\theta _v) -\sqrt {\frac {2}{3}} g_D \sin {}(\theta _v)\)


\(g_{\varLambda \varLambda \phi }=g_S \sin {}(\theta _v) +\sqrt {\frac {2}{3}} g_D \cos {}(\theta _v)\)


\(g_{\varSigma \varSigma \omega }=g_S \cos {}(\theta _v) +\sqrt {\frac {2}{3}} g_D \sin {}(\theta _v)\)


\(g_{\varSigma \varSigma \phi }= g_S \sin {}(\theta _v) -\sqrt {\frac {2}{3}} g_D \cos {}(\theta _v)\)


\(g_{\varSigma \varSigma \rho }= \sqrt {2}g_F\)


\(g^\rho _{\varLambda \varSigma }=\sqrt {\frac {2}{3}}g_D\)


\(g_{\varXi \varXi \omega }=g_S \cos {}(\theta _v) -\frac {1}{\sqrt {6}} \left (3 g_F +g_D\right ) \sin {}(\theta _v)\)


\(g_{\varXi \varXi \phi }=g_S \sin {}(\theta _v) +\frac {1}{\sqrt {6}} \left (3 g_F +g_D\right ) \cos {}(\theta _v)\)


\(g_{\varXi \varXi \rho }=\sqrt {2}(g_F-g_D)\)

5.5.2 SU(3) In-medium Vertices

Since we are primarily interested in mean-field dynamics we consider in this section interactions in the vector and the scalar channels only. From DBHF theory we have available in-medium isoscalar and isovector NN-vector and NN-scalar vertices as density dependent functionals Γ α (ρ), see Fig. 5.12. Thus, the SU(3) relation, Table 5.7, lead to the set of equations for the vector sector
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \varGamma_\omega(\rho)&\displaystyle =&\displaystyle g^{(v)}_{S} \cos{}(\theta_v) +\frac{1}{\sqrt{6}} \left( 3 g^{(v)}_{F} -g^{(v)}_{D} \right )\sin{}(\theta_v) \\ \varGamma_\phi(\rho) &\displaystyle =&\displaystyle g^{(v)}_{S} \sin{}(\theta_v) -\frac{1}{\sqrt{6}} \left( 3 g^{(v)}_{F} -g^{(v)}_{D} \right )\cos{}(\theta_v) \\ \varGamma_\rho(\rho) &\displaystyle =&\displaystyle \sqrt{2}(g^{(v)}_{D}+g^{(v)}_{F}) \end{array} \end{aligned} $$
and accordingly for the scalar sector,
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \varGamma_{\sigma }(\rho) &\displaystyle =&\displaystyle g^{(s)}_{S} \cos{}(\theta_s) +\frac{1}{\sqrt{6}} \left( 3 g^{(s)}_{F} -g^{(s)}_{D} \right )\sin{}(\theta_s) \\ \varGamma_{\sigma'}(\rho) &\displaystyle =&\displaystyle g^{(s)}_{S} \sin{}(\theta_s) -\frac{1}{\sqrt{6}} \left( 3 g^{(s)}_{F} -g^{(s)}_{D} \right )\cos{}(\theta_s) \\ \varGamma_{\delta}(\rho) &\displaystyle =&\displaystyle \sqrt{2}(g^{(s)}_{D}+g^{(s)}_{F}) \end{array} \end{aligned} $$
For the present discussion we assume that the NNf′-singlet vertices \(\varGamma _{\phi ,\sigma '}\) vanish as it would be the case in the quark-model under ideal mixing conditions. Obviously, this constraint is easily relaxed and generalized scenarios with non-vanishing NNf′ coupling can be investigated.
The resulting SU(3) vertices are displayed in Figs. 5.21 and 5.22 for the vector and the scalar nonets, respectively. In both the vector and the scalar channel, g D is found to be negative and with a modulus smaller than g F , g S by a factor 5–10 which is in surprisingly good agreement with the general conclusion that g D should be small. However, here we derive this result from an input of coupling constants which describe perfectly well infinite nuclear matter and nuclear properties. The vertex functionals depend only weakly on the density with variations on the 10% level over the shown density range. The resulting BB vector and scalar vertices are shown in Figs. 5.23 and 5.24. The results may be taken as a confirmation of the widely used scaling hypothesis at least for the Λ-hyperon. Up to saturation density indeed almost constant values of about Rω,σ ∼ 0.5…0.6 are found which are also surprisingly close to the quark model estimate. Roughly the same situation is found for the isoscalar-octet vertices in the Σ- and Ξ-channels: The isoscalar-octet Σ-vertex scaling factors agree to a good approximation with the values found for the Λ. The isoscalar-octet Ξ scaling factors are ranging close to 0.25⋯0.3 which again is surprisingly close to the quark-model expectation of \(\frac {1}{3}\). However, the ΞΞσ vertex involves also a change of sign which would never be obtained by a scaling hypothesis.
Fig. 5.21

In-medium fundamental SU(3) vector vertices g vA (as indicated) versus the baryon density ρ

Fig. 5.22

In-medium fundamental SU(3) scalar vertices g sA (as indicated) versus the baryon density ρ

Fig. 5.23

In-medium SU(3) vector vertices. The NN and ΛΛ vertices are found in the left column, the ΣΣ and ΞΞ vertices are displayed in the right column. Note that the NNϕ and the ΛΛρ coupling constants vanish identically

Fig. 5.24

In-medium SU(3) scalar vertices. The NN and ΛΛ vertices are found in the left column, the ΣΣ and ΞΞ vertices are displayed in the right column. Note that the NNσ′ and the ΛΛδ coupling constants vanish identically.

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

An important global test of the SU(3)-constrained approach is the application to nuclear dynamics. For that purpose we consider the mean-field potentials predicted by the present approach. The leading order relativistic baryon potentials in symmetric nuclear matter are shown in Fig. 5.25. Since the isovector self-energy components are vanishing in symmetric matter the members of the nucleon, Sigma and Cascade isospin multiplets are having their respective common mean-field potentials. It is worthwhile to emphasize that without any attempt to fit the value the Λ potential acquires at saturation density a depth of about − 32 MeV. That value is just perfectly in agreement with the afore cited s-wave Lambda separation energy of about S Λ  ≃ 28 MeV. The Σ experiences a slightly attractive potential which, however, is much too weak for the formation of a bound state.
Fig. 5.25

Mean-field potentials of the octet baryons in symmetric nuclear matter (upper right) with proton fraction \(\xi = \frac {Z}{A}= \frac {1}{2}\) and in pure neutron matter where ξ = 0

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

There is an appreciable electro-weak mixing between the ideal isospin-pure Λ and Σ0 states. Exact SU(3) symmetry of strong interactions predicts \(g_{\varLambda \varLambda \pi ^0}=0\). This effect was investigated, in fact, by Dalitz and von Hippel [128] already in the early days of strangeness physics. They derived the effective coupling constant
$$\displaystyle \begin{aligned} g_{\varLambda\varLambda\pi}=c_b g_{\varLambda\varSigma\pi}, \end{aligned} $$
with the symmetry breaking coefficient
$$\displaystyle \begin{aligned} c_b=-2\frac{\langle\varSigma^0|\delta M|\varLambda\rangle}{M_{\varSigma^0}-M_{\varLambda}}, \end{aligned} $$
given by the Σ0-Λ mass difference in the energy denominator and the ΣΛ element of the octet mass matrix,
$$\displaystyle \begin{aligned} \langle\varSigma^0|\delta M|\varLambda\rangle= \left[M_{\varSigma^0}-M_{\varSigma^+}+M_p-M_n\right]/\sqrt{3}. \end{aligned} $$
Substituting the physical baryon masses [65], we obtain c b  = −0.02699…. From the nucleon-nucleon-pion part of the interaction Lagrangian, Eq. (5.17), we find the isospin matrix element which shows that the neutral pion couples with opposite sign to neutrons and protons. This implies that the non-zero induced \(g_{\varLambda \varLambda \pi ^0}\) coupling produces considerable deviations from charge symmetry in Λp and Λn interactions.
This kind of induced SU(3) symmetry breaking is not specific for the pseudo-scalar meson sector. Rather, it is generic for all kinds of isovector mesons. Thus, the same type of mechanism is obtained with the (neutral) ρ vector meson and the δ/a0(980) meson:
$$\displaystyle \begin{aligned} g_{\varLambda\varLambda\rho}=c_b\,g_{\varLambda\varSigma\rho} \quad ,\quad g_{\varLambda\varLambda\delta}=c_b\,g_{\varLambda\varSigma\delta} \end{aligned} $$
and generalizing the above pion-nucleon relation for arbitrary isovector mesons a = ρ, δ confirming the well known fact that isovector mesons couple to the baryonic isovector currents. The symmetry breaking potentials are found as given in terms of the condensed isovector meson fields ϕδ,ρ
$$\displaystyle \begin{aligned} \begin{array}{rcl} U^{(s)}_{\varLambda SB}&\displaystyle =&\displaystyle g_{\varLambda\varLambda \delta}\phi_\delta {} \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} U^{(v)}_{\varLambda SB}&\displaystyle =&\displaystyle g_{\varLambda\varLambda \rho}\phi_\rho {} \end{array} \end{aligned} $$
and SU(3) symmetry breaking resides fully in the coupling constants. In Hartree-approximation, the isovector fields are directly proportional to the differences of vector and scalar proton and neutron densities:
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \phi_\rho&\displaystyle =&\displaystyle g_{NN\rho}\frac{1}{m^2_\rho}\left(\rho^{(v)}_p-\rho^{(v)}_n \right) \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} \phi_\delta&\displaystyle =&\displaystyle g_{NN\delta}\frac{1}{m^2_\delta}\left(\rho^{(s)}_p-\rho^{(s)}_n \right), \end{array} \end{aligned} $$
as anticipated by the isovector vertex structure, Eq. (5.94). The field ϕ ρ is the static time-like component of the full rho-meson vector field.
The mean-fields of Eq. (5.97) are also defining the SU(3) symmetry conserving ΛΣ mixing potentials
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} U^{(s)}_{\varLambda\varSigma}(\rho)&\displaystyle =&\displaystyle g_{\varLambda\varSigma \delta}\phi_\delta{} \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} U^{(v)}_{\varLambda\varSigma}(\rho)&\displaystyle =&\displaystyle g_{\varLambda\varSigma \rho}\phi_\rho {}, \end{array} \end{aligned} $$
which allow to express Eqs. (5.95) and (5.96) in a rather intriguing form
$$\displaystyle \begin{aligned} \begin{array}{rcl} U^{(s)}_{\varLambda SB}(\rho)&\displaystyle =&\displaystyle c_b U^{(s)}_{\varLambda\varSigma}(\rho) {} \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} U^{(v)}_{\varLambda SB}(\rho)&\displaystyle =&\displaystyle c_b U^{(V)}_{\varLambda\varSigma}(\rho) {} \end{array} \end{aligned} $$
showing explicitly the intimate relation between the electro-weak SU(3) symmetry violation—contained in the symmetry breaking coefficient c b —and the SU(3) symmetry conserving ΛΣ mixing potentials.
In Fig. 5.26 the in-medium g ΛΣρ and g ΛΣδ vertices are shown together with the resulting ΛΣ0 mixing potentials. At low density, both potentials are of similar strength but the vector potential starts to dominate at higher densities. The SU(3)-symmetry breaking effective scalar and vector potentials are displayed in Fig. 5.27. As an example, results are displayed for asymmetric nuclear matter with Z/A = 0.4, as e.g. found in \(^{208}_{\varLambda }Pb\) or also \(^6_{\varLambda }He\). The symmetry breaking potentials amount, in fact, to contributions of a few hundred keV only. Even in pure neutron matter the magnitudes range well below 1 MeV at saturation density and also at higher densities never exceed a few MeV. However, for high precision spectroscopic investigations the effect must be taken into account.
Fig. 5.26

In-medium ΛΣ0 mixing and SU(3) symmetry breaking. The (symmetry conserving) ΛΣ0 interaction vertices resulting from the isovector-vector and isovector-scalar interactions are shown in left panel. The ΛΣ0 mixing scalar (Eq. (5.99), lower curves) and vector (Eq. (5.100), upper corves) potentials are shown in the right panel for asymmetric nuclear matter with ξ = Z/A = 0.4 and ξ = Z/A = 0.1, respectively

Fig. 5.27

In-medium SU(3) symmetry breaking scalar, Eq. (5.101), and vector potentials, Eq. (5.102), of the Lambda hyperon are shown for asymmetric nuclear matter with ξ = Z/A = 0.4

5.5.5 ΛΣ0 Mixing in Asymmetric Nuclear Matter

As discussed in the previous section, SU(3) symmetry breaking and SU(3) mixing are intimately connected as stated by Eqs. (5.101) and (5.102), respectively. The SU(3) symmetry conserving ΛΣ0 mixing by isovector mesons is included, of course, on the level of two-body NY interactions when solving the Bethe-Goldstone equations. This leads to the coupling of and 0 channels. A new phenomenon, however, is encountered in asymmetric nuclear matter. The non-vanishing isovector rho- and delta-meson mean-fields are inducing a superposition of Λ and Σ0 states with respect to the nucleonic core, such that the two hyperon single particle states are becoming mixtures of the free-space hyperon states, i.e.
$$\displaystyle \begin{aligned} \begin{array}{rcl} |\tilde{\varLambda} \rangle &\displaystyle =&\displaystyle \cos{\alpha} |\varLambda \rangle + \sin{\alpha} |\varSigma^0 \rangle \\ |\tilde{\varSigma}^0 \rangle &\displaystyle =&\displaystyle -\sin{\alpha} |\varLambda \rangle + \cos{\alpha} |\varSigma^0 \rangle . \end{array} \end{aligned} $$
The total isospin of the hyperon-nuclear compound system is of course conserved by virtue of the background medium. The coupling is determined by the effective mixing self-energy
$$\displaystyle \begin{aligned} U_m(\rho)=\gamma^0 U^{(s)}_{\varLambda SB}(\rho)+U^{(v)}_{\varLambda SB}(\rho) \end{aligned} $$
and a two-by-two coupled system of equations has to be solved. Without going into the details, here we limit the discussion to two aspects. First, we note that the (density dependent) mixing angle is given in leading order by
$$\displaystyle \begin{aligned} \tan{\alpha(\rho)} = \frac{\langle \varLambda|U_m(\rho)|\varSigma^0\rangle}{M_{\varSigma^0-M_{\varLambda}}} \end{aligned} $$
where we neglected certain higher order terms from differences of the diagonal Λ and Σ0 mean-field Hamiltonians.
Secondly, we remark that the diagonal Λ and Σ0 Hamiltonians should include also weak interaction effects leading to the decay of the octet hyperons. Usually, they are neglected because of their smallness against the strong interaction self-energies. In a mixing situation, however, they should be included. The dispersive interactions may be accounted for by a purely anti-hermitian weak interaction self-energies
$$\displaystyle \begin{aligned} 2\varSigma^{(w)}_{\varLambda,\varSigma}=-i\varGamma_{\varLambda,\varSigma}=-i\frac{1}{\tau_{\varLambda,\varSigma}}, \end{aligned} $$
where we have denoted the free space lifetimes by τΛ,Σ and the corresponding spectral widths by ΓΛ,Σ.
A particular interesting aspect is that the mixing will modify the lifetime of the Λ hyperon. The diagonalization leads to the effective in-medium Λ width
$$\displaystyle \begin{aligned} \varGamma^*_\varLambda(\rho)=\varGamma_\varLambda \left(1+\frac{\varGamma_\varSigma}{\varGamma_\varLambda}\tan^2{\alpha(\rho)} \right) \end{aligned} $$
which is density dependent because of the in-medium mixing angle α(ρ). This width is giving rise to a reduced in-medium lifetime of the Lambda hyperon
$$\displaystyle \begin{aligned} \tau^*_\varLambda\simeq \tau_\varLambda\frac{1}{1+\frac{\tau_\varLambda}{\tau_\varSigma}\tan^2{\alpha} }. \end{aligned} $$
In Fig. 5.28 the lifetime \(\tau ^*_\varLambda \) is shown together with the mixing angle α. Already small admixtures are changing the Λ lifetime drastically. Assuming that this mechanism is the source for the lifetime reduction observed for the hyper-triton, admixtures of α ≃ 4 × 10−6 rad will be sufficient to produce a lifetime of \(\tau ^*_\varLambda ({ }^3H)\simeq 180\) ps.
Fig. 5.28

The ΛΣ0 mixing angle α, Eq. (5.105), and the effective in-medium lifetime \(\tau ^*_\varLambda \) of the mixed in-medium Λ-like state are shown as functions of the nuclear matter density. Note that the lifetime is given in picoseconds

5.6 Theory of Baryon Resonances in Nuclear Matter

5.6.1 Decuplet Baryons as Dynamically Generated, Composite States

Almost all of the decuplet baryons, Fig. 5.1 and Table 5.8, are decaying by strong interactions to octet states [65], thus giving them lifetimes of the order of \(t_{\frac {1}{2}}\sim 10^{-23}\) s. An exception is the S = −3 Ω state with its seminal [sss] valence quark structure. The Ω baryon decays by weak interaction with a probability of ∼ 68% into the ΛK channel with \(t_{\frac {1}{2}}\sim 0.8\times 10^{-10}\) s. In Table 5.8 masses, lifetimes, and valence quark configurations of the decuplet baryons are listed.
Table 5.8

Mass, width, lifetime, and valence quark configuration of the \(J^P= \frac {3}{2}^+\) decuplet baryons (taken from Ref. [65])


Mass (MeV)

Width (MeV)

Lifetime (s)


Δ ++





Δ +





Δ 0










Σ ∗+





Σ ∗0





Σ ∗−





Ξ ∗0





Ξ ∗−








0.8 × 10−10


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 -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 ΣΔ and ΞΩ 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 NN−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 NN−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 brief, the Delta-hole approach consists of calculating simultaneously the pion self-energies and effective vertices by the coupling to the ΔN−1 excitations of the nuclear medium. These requirements are illustrated in Fig. 5.29 where the diagrams representing the approach are shown. As seen in that figure, the Dyson equations for the propagation of pions and Δ’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.
Fig. 5.29

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

An elegant and transparent formulation of the ΔN−1 problem is obtained by the polarization propagator method [92]. We consider first the Green’s function of the interacting system. Here, we limit the investigations to the coupling of NN−1 and ΔN−1 modes. For the non-interacting system the propagator is given by a diagonal matrix of block structure
$$\displaystyle \begin{aligned} \mathcal{G}^{(0)}(\omega ,\textbf{q}) = \left( {\begin{array}{*{20}{c}} {G_{N{N^{ - 1}}}^{(0)}(\omega ,\textbf{q})}&0\\ 0&{G_{\varDelta {N^{ - 1}}}^{(0)}(\omega ,\textbf{q})} \end{array}}\right) \end{aligned} $$
where \(G^{(0)}_{NN^{ - 1}}\) and \(G^{(0)}_{\varDelta N^{ - 1}}\) describe the unperturbed propagation of two-quasiparticle NN−1 and ΔN−1 states through the nuclear medium, including, however, their self-energies \(\varSigma _{N,N^*}\) of Fig. 5.30.
Fig. 5.30

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)

These propagators are given by the Lindhard functions [92]
$$\displaystyle \begin{aligned} \begin{array}{rcl} \phi_N(k) &\displaystyle = &\displaystyle i \int {d^4 p \over (2\pi)^4} G_N(p) G_N(p+k)~, {} \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} \phi_\varDelta(\pm k) &\displaystyle = &\displaystyle i \int {d^4 p \over (2\pi)^4} G_N(p) G_\varDelta(p \pm k)~. {} \end{array} \end{aligned} $$
Here, p denotes the 4-momentum and G N (p) and G Δ (p) are the nucleon and Δ propagators :
$$\displaystyle \begin{aligned} \begin{array}{rcl} G_N(p) &\displaystyle = &\displaystyle \frac{1}{ p^0 - \varepsilon(\textbf{p})-\varSigma_N(p) + i0} + 2 \pi i\, n(\textbf{p}) \delta(p^0 - \varepsilon(\textbf{p})-\varSigma_N(p^2) )~, {} \\ \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} G^{\mu\nu}_\varDelta(p) &\displaystyle = &\displaystyle \frac{ 1}{ p^0 - \varepsilon_\varDelta(\textbf{p}) -\varSigma_\varDelta(p^2) }\delta^{\mu\nu}~, {} \end{array} \end{aligned} $$
with the single particle (reduced kinetic) energies εN,Δ, and n(p) is the nucleon occupation number. We follow the general practice and approximate the Delta-propagator by the leading order term resembling a spin-\(\frac {1}{2}\) Green’s function, thus leaving away the complexities of a Rarita-Schwinger propagator [145]. This is an acceptable approximation in the low-energy limit used below where the neglected terms will be suppressed, anyway. Since Σ Δ is generically of non-hermitian character, we can omit the infinitesimal shift into the complex plane.

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.

After the contour integration over p0 (c.f. [92]) the nucleon-hole Lindhard function, Eq. (5.110), takes the following form :
$$\displaystyle \begin{aligned} \phi_N(k) = - \int {d^3 p \over (2\pi)^3} \left( { n(\textbf{p}+\textbf{k})(1-n(\textbf{p})) \over \varepsilon^*(\textbf{p}+\textbf{k}) - k^0 - \varepsilon^*(\textbf{p}) + i0 } + { n(\textbf{p})(1-n(\textbf{p}+\textbf{k})) \over \varepsilon^*(\textbf{p}) + k^0 - \varepsilon^*(\textbf{p}+\textbf{k}) + i0 } \right)~. {} \end{aligned} $$
where for simplicity, we have introduced the quasiparticle energies \(\varepsilon ^*_{N}(\textbf {p})=\varepsilon _N(\textbf {p})+\varSigma _N(p)\). Corresponding expressions are found for the Delta-hole Lindhard function, Eq. (5.111):
$$\displaystyle \begin{aligned} \phi_\varDelta(\pm k) = - \int {\frac{d^3 p}{ (2\pi)^3} { n(\textbf{p}) \over \varepsilon^*_N(\textbf{p}) - \varepsilon^*_\varDelta(\textbf{p} \pm \textbf{k}) \pm k^0 }}~. {} \end{aligned} $$
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. [92].
Including the residual NN−1 and ΔN−1 interactions by
$$\displaystyle \begin{aligned} \mathcal{V} = \left( {\begin{array}{*{20}{c}} {{V_{NN}}}&{{V_{N\varDelta }}}\\ {{V_{\varDelta N}}}&{{V_{\varDelta \varDelta }}} \end{array}} \right) \end{aligned} $$
the Green function of the interacting system is given by the Dyson equation for the 4-point function
$$\displaystyle \begin{aligned} \mathcal{G}(\omega,\mathbf{q})=\mathcal{G}^{(0)}(\omega,\mathbf{q})+\mathcal{G}^{(0)}(\omega,\mathbf{q})\mathcal{VG}(w,\mathbf{q}) \end{aligned} $$
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 operator \(\mathcal {O}_a(\mathbf {q})\sim e^{i\mathbf {q}\cdot \mathbf {r}} \boldsymbol {\sigma }^{S_a}\boldsymbol {\tau }^{T_a}\) where a = (S, T) denotes spin (S a  = 0, 1), isospin (T A  = 0, 1) and momentum (∼ eiqr) transfer, is described by the polarization propagators of the non-interacting system
$$\displaystyle \begin{aligned} \varPi^{(0)}_{ab}(\omega,\mathbf{q})=\langle A|\mathcal{O}^\dag_b \mathcal{G}{(0)} \mathcal{O}_a |A\rangle \end{aligned} $$
and the interacting system
$$\displaystyle \begin{aligned} \varPi_{ab}(\omega,\mathbf{q})=\langle A|\mathcal{O}^\dag_b \mathcal{G} \mathcal{O}_a |A\rangle \end{aligned} $$
which—by definition—has the same functional structure as the propagator \(\mathcal {G}\). For a single resonance the polarization tensor is given by a 2-by-2 tensorial structure
$$\displaystyle \begin{aligned} \left( {\begin{array}{*{20}{c}} {\varPi _{NN}^{}}&{\varPi _{N\varDelta }^{}}\\ {\varPi _{\varDelta N}^{}}&{\varPi _{\varDelta \varDelta }^{}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\varPi _{NN}^{(0)}}&{\varPi _{N\varDelta }^{(0)}}\\ {\varPi _{\varDelta N}^{(0)}}&{\varPi _{\varDelta \varDelta }^{(0)}} \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {\varPi _{NN}^{(0)}}&{\varPi _{N\varDelta }^{(0)}}\\ {\varPi _{\varDelta N}^{(0)}}&{\varPi _{\varDelta \varDelta }^{(0)}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{V_{NN}}}&{{V_{N\varDelta }}}\\ {{V_{\varDelta N}}}&{{V_{\varDelta \varDelta }}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\varPi _{NN}^{}}&{\varPi _{N\varDelta }^{}}\\ {\varPi _{\varDelta N}^{}}&{\varPi _{\varDelta \varDelta }^{}} \end{array}} \right) \end{aligned} $$
where the reference to the transition operators \(\mathcal {O}_{a,b}\) is implicit. The diagrammatic structure is shown in Fig. 5.31. The mixing of the NN−1 and the ΔN−1 configurations by the residual interactions is resulting in the mixed polarization tensors Π and Π ΔN , respectively. The polarization tensor contains the full multipole structure as supported by the nuclear system, the interaction \(\mathcal {V}\), and the external transition operators \(\mathcal {O}_{a,b}\). Thus, a decomposition into irreducible tensor 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. [146, 147, 148].
Fig. 5.31

The RPA polarization propagator. The N−1N → N−1N (left), the mixed N−1N → N−1Δ and the N−1Δ → N−1Δ 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

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
$$\displaystyle \begin{aligned} R_{ab}(\omega,\mathbf{q})=-\frac{1}{\pi}Im\left [ \varPi_{ab}(\omega,q) \right] \end{aligned} $$
The response function techniques are frequently used in lepton induced reactions like (e, e′p) or neutrino-induced pion production (ν, μπ). In [149, 150] 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 [151, 152].
A widely used choice for \(\mathcal {V}\) is a combination of pion-exchange and contact interactions of Landau-Migdal type corresponding to the afore mentioned OPEM approach, see e.g. [142, 143, 144]. The diagrammatic structure of the QPEM interactions is shown in Fig. 5.32. Inclusion of other mesons, e.g. the ρ-meson, is easily obtained. The pion exchange part takes care of the long-range interaction component. In non-relativistic reduction (c.f. [153]), but using relativistic kinematics, the πNN and πNΔ interactions, the Lagrangians are:
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathcal{L}_{\pi N N} &\displaystyle = &\displaystyle {f \over m_\pi} \psi^\dag \boldsymbol{\sigma} \boldsymbol{\tau} \psi \cdot \nabla \boldsymbol{\pi}~, {} \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathcal{L}_{\pi N \varDelta} &\displaystyle = &\displaystyle {f_\varDelta \over m_\pi} \psi^\dag_\varDelta \mathbf{S} \textbf{T} \psi \cdot\nabla \boldsymbol{\pi} + h.c.~, {} \end{array} \end{aligned} $$
where ψ, ψ Δ and π are the nucleon, Δ 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 Δ  = 2.202, see e.g. [144, 154]. 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. [155]. Pion-exchange is seen to be of spin-longitudinal structure. Occasionally, also ρ-meson exchange is treated explicitly, e.g. [146, 150] introducing an explicit spin-transversal interaction component into \(\mathcal {V}\) which, however, can be decomposed into spin-spin and spin-longitudinal terms [156].
Fig. 5.32

Direct (a) and exchange (b) diagrams of the OPEM approach for the process N1N2 → N3Δ4. The wavy line denotes either π (and ρ) exchange or the contact interaction \(\propto g_{N\varDelta }^\prime \)

The short range pieces are subsumed into the contact interactions, defined by the following Lagrangian:
$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathcal{L}_{SRC} &\displaystyle =&\displaystyle -{f^2 \over 2 m_\pi^2} g_{NN}^\prime (\psi^\dag \boldsymbol{\sigma} \boldsymbol{\tau} \psi)\cdot (\psi^\dag \boldsymbol{\sigma} \boldsymbol{\tau} \psi) \\ &\displaystyle &\displaystyle - \left[ {f_\varDelta \over m_\pi^2} g_{N\varDelta}^\prime (\psi^\dag \boldsymbol{\sigma} \boldsymbol{\tau} \psi)\cdot (\psi^\dag_\varDelta \mathbf{S} \textbf{T} \psi) + \mbox{h.c.} \right] \\ &\displaystyle &\displaystyle - {f_\varDelta^2 \over m_\pi^2} g_{\varDelta\varDelta}^\prime (\psi^\dag_\varDelta \mathbf{S} \textbf{T} \psi)\cdot (\psi^\dag \mathbf{S}^\dag \textbf{T}^\dag \psi_\varDelta) \\ &\displaystyle &\displaystyle - \left[ {f_\varDelta^2 \over 2 m_\pi^2} g_{\varDelta\varDelta}^\prime (\psi^\dag_\varDelta \mathbf{S} \textbf{T} \psi)\cdot (\psi^\dag_\varDelta \mathbf{S} \textbf{T} \psi) + \mbox{h.c.} \right] {} \end{array} \end{aligned} $$
\(g_{NN}^\prime \), \(g_{N\varDelta }^\prime \) and \(g_{\varDelta \varDelta }^\prime \) 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 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 4He, 16O and 40Ca. 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 \(g_{N\varDelta }^\prime = 0.25\ldots 0.35\).

The Delta self-energies are a heavily studied subject because they are of particular importance for pion-nucleus interactions. c.f. [156, 160]. 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 [161, 162]
$$\displaystyle \begin{aligned} \varGamma_\varDelta(\omega,\rho ) = - 2 Im \varSigma_\varDelta (\omega,\rho )\sim\varGamma _{free}(\omega) + \varGamma _{Pauli}(\omega,\rho ) + \varGamma _{abs}(\omega,\rho ) \end{aligned} $$
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
$$\displaystyle \begin{aligned} \varGamma_\varDelta(\omega) = \varGamma_{abs}(\omega) {\rho \over \rho_{sat}} + \varGamma_\varDelta^0\left({q(\omega,m_N^*,m_\pi) \over q(m_\varDelta,m_N,m_\pi)}\right)^3 {m_\varDelta^* \over \omega} {\beta_0^2+q^2(m_\varDelta^*,m_N^*,m_\pi) \over \beta_0^2+q^2(\omega,m_N^*,m_\pi)}~, {} \end{aligned} $$
where β0 is an adjustable parameter and ρ sat  = 0.16 fm−3 is the nuclear saturation density. \(\omega =\sqrt {s_\varDelta }\) 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 Γ abs and a simple scaling law is used for the density dependence. The Lorentz-invariant center-of-mass momentum is defined as usual,
$$\displaystyle \begin{aligned} q^2(\omega,m_1,m_2)=(\omega^2-(m_1+m_2)^2)(\omega^2-(m_1-m_2)^2)/4\omega^2. \end{aligned} $$
The free and effective nucleon and Delta in-medium masses are denoted by mN,Δ and \(m^*_{N,\varDelta }\), respectively. Using a (subtracted) dispersion relation the real part can be reconstructed by a Cauchy Principal Value integral
$$\displaystyle \begin{aligned} Re(\varSigma_\varDelta(\omega))= -\frac{\omega-m_\varDelta}{\pi}P\int{d\omega'\frac{\varGamma_{\varDelta}(\omega')}{(\omega'-m_\varDelta)(\omega'-\omega)}}, \end{aligned} $$
by which the dispersive self-energy is completely determined.

5.6.3 Δ Mean-Field Dynamics

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 \(U^\varDelta _0\) and \(U^\varDelta _1\), respectively:
$$\displaystyle \begin{aligned} U_\varDelta ^{(H)} = {U^\varDelta_0} + \frac{2}{A}{U^\varDelta_1}{\boldsymbol{\tau}^\varDelta } \cdot {\boldsymbol{\tau}^N} \end{aligned} $$
where A is the nucleon number and τ Δ  = 2T Δ and τ N denote the Delta and nucleon isospin operators, respectively, with the known properties of isospin
$$\displaystyle \begin{aligned} \tau^N_3|\,p\rangle=+|\,p\rangle \quad ; \quad \tau^N_3|n\rangle=-|n\rangle \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} T^\varDelta_3|\varDelta^{++}\rangle &\displaystyle =&\displaystyle +\frac{3}{2}|\varDelta^{++}\rangle \quad ; T^\varDelta_3|\varDelta^{+}\rangle=+\frac{1}{2}|\varDelta^{+}\rangle \\ T^\varDelta_3|\varDelta^{0}\rangle &\displaystyle =&\displaystyle -\frac{1}{2}|\varDelta^{0}\rangle \quad ; T^\varDelta_3|\varDelta^{-}\rangle=-\frac{3}{2}|\varDelta^{-}\rangle . \end{array} \end{aligned} $$
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
$$\displaystyle \begin{aligned} U_\varDelta ^{(H)}\simeq{U^\varDelta_0} - {U^\varDelta_1}\tau^{\varDelta}_{z} \frac{N - Z}{A}. \end{aligned} $$
Thus, we find
$$\displaystyle \begin{aligned} \begin{array}{rcl} U_{\varDelta^{++}} ^{(H)}&\displaystyle =&\displaystyle {U^\varDelta_0} -3 {U^\varDelta_1} \frac{N - Z}{A} \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} U_{\varDelta^{+}} ^{(H)}&\displaystyle =&\displaystyle {U^\varDelta_0} -{U^\varDelta_1} \frac{N - Z}{A} \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} U_{\varDelta^{0}} ^{(H)}&\displaystyle =&\displaystyle {U^\varDelta_0}+{U^\varDelta_1} \frac{N - Z}{A} \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} U_{\varDelta^{-}} ^{(H)}&\displaystyle =&\displaystyle {U^\varDelta_0} +3 {U^\varDelta_1} \frac{N - Z}{A} \end{array} \end{aligned} $$
and the universality assumption would mean to use \(U^\varDelta _{0,1}=U^N_{0,1}\). At the nuclear center, typical values are \(U^N_0=-40\ldots -60\) MeV and \(U^N_1=+20\ldots +30\) MeV, depending on the chosen form factor.1 These estimates agree quite well with those of the involved many-body calculations in [163]. 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. [59, 60, 61] and also [62, 63] the Delta resonance was included into the RMF treatment. In [60] Δ dynamics in nuclear matter is described by the mean-field Lagrangian density
$$\displaystyle \begin{aligned} {\mathcal L}_\varDelta=\overline{\psi}_{\varDelta\,\nu}\, \left[ i\gamma_\mu \partial^\mu -(m_\varDelta-g_{\sigma\varDelta} \sigma)-g_{\omega\varDelta}\gamma_\mu\omega^\mu -g_{\rho \varDelta} \gamma_\mu I_{3} \rho^{\mu}_3 \right] \psi_{\varDelta}^{\,\nu} \, , \end{aligned} $$
where \(\psi _\varDelta ^\nu \) is the Rarita-Schwinger spinor for the for the full set of Δ(1232)-isobars (Δ++, Δ+, Δ0, Δ) and I3 = diag(3/2, 1/2, −1/2, −3/2) is the matrix containing the isospin charges of the Δs. Assuming SU(6) universality, the same coupling constants as for nucleons may be used. However, contributions form dispersive self-energies will surely spoil SU(6) symmetry and deviations from that rule will occur.

5.6.4 Response Functions in Local Density Approximation

An application of the response function formalism is shown in Fig. 5.33 where the spectra for the Fermi-transition operator \(\mathcal {O}_a=\boldsymbol {\sigma }\tau _+\) for 58, 64, 78Ni are shown. The response functions are normalized to the nucleon numbers A = 58, 64, 78. As discussed above, the polarization tensor, Eq. (5.120), was evaluated in infinite matter at a dense mesh of proton and neutron densities, thus leading to \(\varPi _{BB'}(\rho _p,\rho _n)\) for B, B′ = N, Δ. By mapping the nucleon densities to the radial densities ρp,n(r) of the Ni-isotopes, we obtain in local density approximation \(\varPi _{BB'}(r)\). The response function shown in Fig. 5.33 is obtained finally by integration
$$\displaystyle \begin{aligned} R(\omega,q)=-\frac{1}{\pi}\frac{1}{A}\int{d^3r \rho_A(\mathbf{r})Im\left[\varPi_{NN}(\omega,q,\mathbf{r})+\varPi_{\varDelta\varDelta}(\omega,q,\mathbf{r}) \right]} \end{aligned} $$
where q = |q| and ρ A  = ρ p  + ρ n is the total nuclear density. In the response functions of Fig. 5.33, the quasi-elastic NN−1 and the deep-inelastic ΔN−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. [94]. 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.
Fig. 5.33

RPA response function per nucleon for the operator στ+ in the isotopes 58, 64, 78Ni [164] obtained by an energy density functional as in [94]. Note the apparent shift of the ΔN−1 peak to higher energy which is introduced by mixing with the NN−1 component due to the residual interactions

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 the high energy limit, the electron scattering waves are approximated sufficiently well by plane waves Results of such a calculation [164] are shown in Fig. 5.34 where the double differential cross section for the inclusive 40Ca(e, e′) reaction at the electron incident energy T lab  = 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 N resonance spectral components seems to be slightly too large, indicating that the configuration mixing interaction V was chosen slightly too strong.
Fig. 5.34

Double differential cross section for the inclusive 40Ca(e, e′) reaction at T lab  = 500 MeV [164]. The underlying longitudinal and transversal response functions were obtained by an energy density functional as in [94]. Note the apparent shift of the ΔN−1 peak to higher energy which is introduced by mixing with the quasi-elastic NN−1 component due to the residual interactions

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

Interestingly, neutron stars may be useful systems to study N mean-field dynamics [59, 60, 62] and also in [63]. In [59, 60] nucleons, hyperons and Deltas are described within the same RMF approach, used to investigate the composition of neutron star matter. In Fig. 5.35 particle fractions as a function of the baryon density n B  = ρ B of ρ sat are shown. With the Δ-resonance included the particle mixtures are changed considerably. Furthermore, the onset of the Delta appearance depends on the RMF coupling constant. That effect is illustrated in Fig. 5.35 by varying \(x_{\omega \varDelta }=\frac {g_{\omega \varDelta }}{g_{\omega N}}\).
Fig. 5.35

Particle fractions as functions of the baryon density within the SFHo model: only hyperons (panel (a)), hyperons and Δs (panel (b)) for x σΔ  = x ωΔ  = x ρΔ  = 1. The red line indicates the fraction of the Δ which among the four Δs are the first to appear. The blue and the green vertical lines indicate the onset of the formations of Δ for x ωΔ  = 0.9 and x ωΔ  = 1.1, respectively (from Ref. [60])

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

Exploring the spectral properties of resonances and their dynamics in the nuclear medium is the genuine task of nuclear physics. In Fig. 5.36 interactions are indicated which in heavy ion reactions are leading to excitation of nucleon resonances. In fact, already in the 1970s first experiments were performed at the AGS at Brookhaven with proton-induced reaction on a series of targets between Carbon and Uranium [173]. 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 [174, 175]. About the same time also (n, p) reactions were measured at Los Alamos [176, 177]. A few years later, similar experiments utilizing (p, n) charge exchange were started also at Saclay [178]. 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 [179, 180]. 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 Synchrophasotron 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 [181, 182]. 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 1990s with the DIOGENE detector at SATURNE [183, 184], at KEK using the FANCY detector [185]. However, with respect to the first experiments it took another decade or so before those exclusive data were studied theoretically [186]. A couple of years later, corresponding experiments were done at Dubna, taking advantage of the higher energies of up to p lab  = 4.2 AGeV/c reached at the Synchrophasotron [187, 188, 189, 190]. In [191] the measurements were extended to the detection of up to N→ +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. 5.44.
Fig. 5.36

Resonance production by charge-neutral meson exchange (left) and charged meson exchange (right)

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 , 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 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 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

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 [149, 150, 194, 195]. The first round of experimental data from 1980s and 1990s 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 ΔN1 states. This combination of—at that time—very involved theoretical methods was able to explain the observed puzzling shift of the Delta-peak by ΔM ∼−70 MeV. In Fig. 5.37 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.
Fig. 5.37

Zero-degree triton spectra for the reaction 12C(3He, t) at T lab  = 2 GeV, shown as a function of the excitation energy ω L . The data are taken from Ref. [179]. 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 (from Ref.[150])

Fig. 5.38

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 ± 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 Δ(1232) mass distribution (from Ref. [202])

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 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 3He → 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 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 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.

The best studies case is the Δ33(1232) resonance which, in fact, is a 16-plet formed by four isospin and four spin sub-states. In nucleon-nucleon scattering one observes a centroid mass M Δ  ≃ 1232 MeV and the width Γ Δ  = 115–120 MeV which is in good agreement with the direct observation in pion-nucleon scattering [65]. Modifications of mass and width of the Δ(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 [203] and at GSI by the FOPI collaboration [202]. In [203], 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 Δ33(1232) properties has been interpreted in terms of hadronic density, temperature, and various non-nucleon degrees of freedom in nuclear matter [204, 205, 206]. The invariant mass distributions of correlated nucleon and pion pairs provide, in principle, a direct proof for resonance excitation. As discussed in [202], in the early heavy ion collision experiments a major obstacle for the reconstruction of the resonance spectral distribution was the large background of uncorrelated 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. 5.38.
Fig. 5.39

Double differential cross sections for the DCX process π±O → πX at T lab  = 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 [222], only statistical errors are shown. The GiBUU results are shown as histograms, where the fluctuations indicate the degree of statistical uncertainty (from Ref.[221])

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 →  followed by the decay Δ → . In transport calculations, the pion multiplicity, therefore, depends crucially on the value of the in-medium NN →  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 → (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/ρ0 MeV [163, 220].

The effect of the medium modifications of the NN ↔  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 ↔  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. [221] 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 T lab  = 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 [221]. GiBUU results for the DCX reaction π±O → πX at T lab  = 120, 150 and 180 MeV are shown Fig. 5.39. A good agreement with data was achieved for the total cross section and also for angular distributions and double differential cross sections. Some strength at backward angles and rather low pion energies below T lab  ≈ 30 MeV is still missed. A striking sensitivity on the thickness of neutron skins was found, indicating that such reactions may of potential advantage for studies of nuclear density profiles.
Fig. 5.40

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

5.7.5 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 [12, 152, 223]. 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 NN−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. 5.40.
Fig. 5.41

Results of recent heavy ion charge exchange reactions at the FRS at T lab  = 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 (data taken from Ref. [152])

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 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. 5.41. In the spectra, the quasi-elastic and the resonances regions, discussed in Fig. 5.40, are clearly seen and energetically well resolved.
Fig. 5.42

Exclusive resonance production processes by observation of the decay products

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. 5.42, 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 [183, 184], the Dubna Synchrophasotron [191] and, at slightly lower energies, at the FANCY detector at KEK [185]. In the Dubna experiments single and double pion channels have been measured. The gain in spectroscopic information is already visible in the singles spectra, Fig. 5.43, and even more so in the pππ spectra in Fig. 5.44. In coincidence experiments measuring the decay particles of in-medium resonances are obviously complementary by establishing a connection of meson production on the free nucleon and on nuclei.
Fig. 5.43

Proton-single pion yields measured at the Synchrophasotron in the reaction 12C +12C at p lab  = 4.2 GeV/c per nucleon (from Ref. [182])

Fig. 5.44

Observation of higher resonances in C + C collisions at the Synchrophasotron by N→ +π two-pion decay spectroscopy at the beam momentum p lab  = 4.2 GeV/c. Expected N states are indicated for the two lower structures. The data are from Ref. [191]

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 NN−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.


  1. 1.

    More meaningful values are in fact the volume integrals per nucleon.



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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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikJLU GiessenGießenGermany
  2. 2.Balurghat CollegeBalurghatIndia

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