Dense Baryonic Matter and Strangeness in Neutron Stars

Recent developments of chiral effective field theory (ChEFT) applications to nuclear and neutron matter are summarized, with special emphasis on a (non-perturbative) extension using functional renormalisation group methods. Topics include: nuclear thermodynamics, extrapolations to dense baryonic matter and constraints from neutron star observables. Hyperon-nuclear interactions derived from SU(3) will be discussed with reference to the"hyperon puzzle"in neutron star matter.


I. INTRODUCTION
Understanding nuclei and dense baryonic matter at the interface with QCD is one of the pending challenges in nuclear many-body theory. The present paper reports on recent developments using effective field theory methods applied to dense baryonic matter. Of special interest in this context is the equation-of-state (EoS) of cold and dense matter subject to constraints from neutron star observations (the existence of two-solar-mass stars [1][2][3] and information deduced from gravitational wave signals of two merging neutron stars [4,5] ).
Whatever the detailed composition of matter in the core of neutron stars may be (see also the presentation by G. Baym at QNP2018), its EoS must produce sufficiently high pressure in order to meet the empirical benchmark constraints from astrophysics together with those from nuclear physics at lower densities. A further persisting question in this context is about the possible presence of strange quarks or hyperons in neutron stars. This issue, the so-called "hyperon puzzle", will be addressed in a later section of this presentation.

II. NUCLEONS AND PIONS IN LOW-ENERGY QCD
Global symmetries of QCD provide guidance for constructing effective field theories that represent low-energy QCD and thus establish a conceptual frame for nuclear physics and extrapolations to matter at higher densities. In its sector with the two lightest (u and d) quarks, chiral symmetry is a key to understanding low-energy energy hadron structure as well as nuclear forces. Given the small quark masses (m u 2 MeV and m d 5 MeV at a renormalization scale µ 2 GeV), a useful starting point is QCD with a massless isospin doublet (u, d) of quarks. In this limit, chiral invariance is an exact symmetry of QCD. Nonperturbative quark-gluon dynamics implies that this symmetry is spontaneously broken at low energy. The Nambu-Goldstone (NG) boson of spontaneously broken chiral symmetry is identified as the isospin triplet of pions.
Starting from three massless quarks forming a color-singlet, the massive nucleon is generated by quark-gluon dynamics. The three valence quarks, accompanied by a strong gluonic field and a sea of quark-antiquark pairs, are localized in a small volume with a radius of less than a Fermi. Localization (confinement) of the valence quarks in a finite volume implies breaking of chiral symmetry. The following low-energy QCD based picture of the nucleon, relevant to nuclear physics, thus emerges. The underlying strong-interaction processes which generate the nucleon mass M N induce at the same time the spontaneous breaking of chiral symmetry. The three valence quarks, carrying one unit of baryon number, form a compact core of the nucleon. Strong vacuum polarization effects surround this baryonic core with multiple quark-antiquark pairs. The low-energy physics associated with this mesonic (qq) cloud is governed by pions as NG bosons of spontaneously broken chiral symmetry, in such a way that the total axial vector current of the "core + cloud" system is conserved apart from small mass corrections (PCAC).
Interactions between two nucleons at long and intermediate distances are generated as an inward-bound hierarchy of one-, two-, multi-pion exchange processes. The framework for systematically organizing this hierarchy is chiral effective field theory (ChEFT). It is based on the separation of scales delineating low energies and momenta characteristic of nuclear physics from the chiral symmetry breaking "gap" scale, Λ χ = 4πf π ∼ 1 GeV ∼ M N . In this context the pion decay constant, f π 0.09 GeV, figures as an order parameter for spontaneously broken chiral symmetry.

III. FROM NUCLEONS TO COMPRESSED BARYONIC MATTER
Extrapolations from normal nuclear systems to the high-density regime require a detailed assessment of size scales of the nucleon itself. As mentioned the "chiral" nucleon is viewed as a compact valence quark core surrounded by a multi-pion cloud. Early descriptions of the nucleon as a topological soliton derived from a non-linear chiral meson Lagrangian [6] suggested an r.m.s. radius of about 0.5 fm for the baryonic core, together with a significantly larger charge radius determined primarily by the charged meson cloud surrounding the core.
The ratio of the corresponding volumes of baryon no. and isoscalar charge distributions, , indicates yet another separation of scales relevant to the window of applicability for ChEFT.
Empirical tests delineating the size scale of the nucleon core begin to be accessible in deeply-virtual Compton scattering experiments at J-Lab and their theoretical analysis [7][8][9]]. An interesting development in this respect is the measurement of form factors of the nucleon's energy momentum tensor (see also the presentation by B. Pasquini at QNP2018).
First explorations of the distribution of pressure inside the proton [10] appear to be consistent with previous chiral quark-soliton model predictions [11,12] and the picture of a compact baryonic core surrounded by a pionic cloud.
Assuming a typical 1/2-fm radius of the baryon core, let us consider compressed baryonic matter and examine up to which baryon densities, n = B/V , one can still expect nucleons (rather than free-floating quarks) to be the relevant baryonic degrees of freedom.
A schematic picture is drawn in Fig. 1 Pions couple to these baryonic sources and act in the space between these cores. The pion field incorporates multiple exchanges of pions between nucleons, and those mechanisms are properly dealt with in chiral EFT. While the longer-range pionic field configurations will undergo drastic changes in highly compressed baryonic matter, the sizes of the compact baryonic cores themselves are expected to remain stable as long as they continue to be separated. As illustrated in Fig. 1, even at n = 5 n 0 , corresponding to an average distance d N N 1.1 fm between nucleons, the individual baryon distributions are indeed still well identifiable, with just small overlaps at their touching surfaces. At such high densities the (non-linear) pion field between baryonic sources is accumulating much strength. Non-perturbative chiral field theory methods must be employed to treat these strong-field configurations.
In this picture the nucleons start loosing their identities once the density reaches n 8 n 0 and the baryon distributions begin to merge (percolate). Quark matter with strong pairing (diquarks) is supposed to take over [13]. However, such extreme densities are already beyond the baryon densities typically encountered in the central regions of neutron stars if their radii are larger than about ten kilometers. Furthermore the strong short-distance repulsion between nucleons makes a "soft" merging scenario for nucleons energetically quite expensive. The repulsive hard core of the N N interaction with its typical range of about half a Fermi has a long phenomenological history in nuclear physics. More recently this hard core has been established by deducing an equivalent local N N potential from lattice QCD computations [14,15] .

IV. NUCLEAR CHIRAL EFFECTIVE FIELD THEORY
Chiral effective field theory starts out as a non-linear theory of pions and their interactions, with symmetry-breaking mass terms added. Nucleons are introduced as "heavy" sources of the NG bosons. A systematically organized low-energy expansion in powers of derivatives of the pion field (power-counting) provides a remarkably successful quantitative description (chiral perturbation theory) of pion-pion scattering and of pion-nucleon interactions at momenta and energies small compared to the chiral symmetry breaking scale, ChEFT with inclusion of nucleons is the basis for a successful theory of the nucleon-nucleon interaction. In recent years this theory has also become the widely accepted framework for the treatment of nuclear many-body systems (see e.g. [16,17] and references therein for recent overviews).
Nuclear forces in ChEFT are constructed in terms of a systematically organized hierarchy of explicit one-, two-and multi-pion exchange processes, constrained by chiral symmetry, plus a complete set of contact terms encoding unresolved short-distance dynamics [18][19][20].
Three-body forces enter at next-to-next-to-leading order (N 2 LO) in this hierachy and turn out to be important in reproducing the equilibrium properties of nuclear matter.
Chiral EFT is basically a perturbative framework. In a nuclear medium, the new "small" scale that enters in addition to three-momenta and pion mass is the Fermi momentum, p F , of the nucleons. Its value at the equilibrium density of N = Z nuclear matter, n 0 = The perturbative chiral EFT approach to nuclear and neutron matter relies on a convergent expansion of the energy-density in powers of p F /Λ, the ratio of Fermi momentum and a suitably chosen momentum space cutoff. Typical ChEFT cutoffs are Λ ∼ 400 − 500 MeV, about half of the chiral symmetry breaking scale Λ χ = 4πf π . At densities n ∼ 5n 0 the Fermi momenta are comparable to Λ and therefore a perturbative expansion cannot be expected to work. A non-perturbative method needs to be developed. In particular, multi-nucleon correlations grow rapidly with increasing density and must be treated through proper resummations.
Here our method of choice is the functional renormalization group (FRG) [26] applied to nuclear many-body systems. We give a brief outline and summary of developments and results reported in [23][24][25]. The underlying logic is the following: in the domain of spontaneously broken chiral symmetry, the active degrees of freedom are pions and nucleons. This non-perturbative FRG framework based on chiral nucleon-meson field theory yields results for symmetric and asymmetric nuclear matter as well as pure neutron matter that are consistent with those of perturbative chiral EFT at moderate temperatures and densities. In particular, nuclear thermodynamics including the liquid-gas phase transition is well reproduced as discussed in more detail in ref. [23].

VI. CHIRAL SYMMETRY RESTORATION AND ORDER PARAMETERS
Whereas the applicability of perturbative chiral EFT is limited to baryon densities n 2 n 0 , the non-perturbative chiral FRG approach can in principle be extended to compressed baryonic matter at higher densities. A necessary condition for this to work is that matter remains in the hadronic phase characterized by the spontaneously broken Nambu-Goldstone realisation of chiral symmetry.
Lattice QCD computations [27,28] at µ = 0 demonstrate the existence of a crossover transition towards restoration of chiral symmetry in its Wigner-Weyl realisation at temperatures T > T c 0.15 GeV. Chiral symmetry is presumably also restored at large baryon chemical potentials µ and low temperature. However, the critical value of µ at which this transition might take place is unknown.
Several calculations of isospin-symmetric matter using Nambu & Jona-Lasinio or chiral quark-meson models have predicted a first-order chiral phase transition at vanishing temperature for quark chemical potentials, µ q , around 300 MeV (see, e.g., Refs. [29][30][31][32][33]). Translated into baryonic chemical potentials, µ 3µ q , chiral symmetry would then be restored not far from the equilibrium point of normal nuclear matter, µ 0 = 923 MeV. Nuclear physics with its well-established empirical phenomenology teaches us that this can obviously not be the case. However, these calculations -apart from the fact that they operate with (quark) degrees of freedom that are not appropriate for dealing with the hadronic phase of QCD -work mostly within the mean-field approximation. It is therefore of great importance to examine how fluctuations beyond mean-field can change this scenario. Using chiral nucleon-meson field theory in combinaton with FRG, we observe indeed that the mean-field approximation cannot be trusted: it is likely that fluctuations, such as repulsive multi-nucleon correlations in the presence of the Pauli principle, shift the chiral transition to extremely high baryon densities beyond six times n 0 .
In chiral nucleon-meson (ChNM) field theory with its chiral field (π, σ), the expectation value of the scalar field, σ (µ, T ) = f * π (µ, T ), or equvalently, the in-medium pion decay constant, acts as order parameter for the spontaneous breaking of chiral symmetry. It is instructive to examine this order parameter in the T -µ phase diagram of symmetric nuclear matter around the liquid-gas phase transition. Figure 3 [34,35] . Chiral nuclear forces treated up to four-body interactions [35] at N 3 LO were shown to work moderately against the leading linearly decreasing chiral condensate with increasing density around and beyond n n 0 .
The non-perturbative FRG approach permits an extrapolation to higher densities. Results are presented in Fig. 4. In mean-field approximation the order parameter σ /f π shows a first-order chiral phase transition at a density of about 3 n 0 . However, the situation changes shows a steep rise towards pressures exceeding 100 MeV/fm 3 in the region relevant for the core of heavy neutron stars. Such an amount of pressure is indeed capable of supporting a two-solar-mass neutron star. Its radius is predicted to be R 11.5 km. Notably the baryon density in the center of such an object does not exceed n ∼ 5 n 0 . Following the previous discussions it then appears justified to work with nucleons and pion fields as relevant degrees of freedom even at such extreme but not hyperdense conditions.
A further interesting property of compressed baryonic matter is its velocity of sound. For a non-interacting relativistic Fermi gas the squared sound velocity has a canonical value, , which is supposed to be reached at asymptotically high densities. The inset of Fig. 5 shows that the squared sound velocity of the FRG -ChNM equation-of-state exceeds c 2 s = 1/3 at a baryon density around n ∼ 4 n 0 and continues to grow as it approaches neutron star core densities. This behaviour can be traced to the continuously rising strength of repulsive many-body correlations driven in part by the Pauli principle as the density increases. At much higher densities, once nucleon clusters dissolve and quark matter takes over, c 2 s is expected to decrease again and ultimately approach its asymptotic value of 1/3 from below at ultrahigh densities [36] .
An important effort presently pursued and steadily being improved is to constrain the neutron star equation-of-state systematically from observational data, together with an interpolation between theoretical limits provided by nuclear physics at low densities and perturbative QCD at asymptotically high densities. Fig. 6 shows a recent example. Nuclear constraints as represented by ChEFT calculations [16,37] set the low-density limit of P (E) at energy densities E 200 MeV/fm 3 . Sophisticated perturbative QCD calculations [38] determine the pressure at extreme energy densities, E > 10 GeV/fm 3 . Constraints in the region between these extremes are introduced by studying large sets of parametrized equationsof-state subject to the condition that they all produce a maximum neutron star mass of at least 2 M and at the same time fulfill the tidal deformability limit deduced from the updated LIGO & Virgo gravitational wave analysis. The shaded area in Fig. 6 defines the region of acceptable neutron star equations-of-state under such conditions [39,40]. Remarkably, the EoS computed using the FRG-ChNM approach and shown in Fig. 5 satisfies these constraints up to the densities relevant for the core of massive neutron stars. Of course, one order of magnitude in the pressure P still separates this neutron star domain from the perturbative QCD limit. It is in the range of densities n > 5 n 0 where one can expect a (probably continuous) hadrons-to-quarks transition [13] to take place.

VIII. STRANGENESS IN NEUTRON STARS ?
An issue that still needs to be resolved is the so-called "hyperon puzzle" in neutron stars. At densities around 2-3 times n 0 the neutron Fermi energy reaches a point at which it becomes energetically favourable to replace neutrons by Λ hyperons, as long as only ΛN two-body forces are employed [41,42] . Then, however, the EoS becomes far too soft and misses the 2 M constraint for the neutron star mass.
Ongoing investigations suggest a possible way of resolving this puzzle. The starting breaking effects are incorporated through the physical mass differences within the multiplets.
Hyperon-nucleon interactions have been constructed in this scheme at next-to-leading order (NLO) [43]. Unfortunately, the statistical quality of the existing empirical hyperon-nucleon scattering data is still too limited to warrant more detailed studies beyond NLO. There is an obvious need for a much improved hyperon-nucleon data base.
Such chiral SU (3) calculations indicate strong ΛN → ΣN coupled-channels effects in combination with repulsive short-distance dynamics. Together these effects work to raise the onset condition for the Λ chemical potential, µ Λ = µ n , towards higher densities. Perhaps more significantly, ΛN N three-body forces [44,45] as they emerge from chiral SU (3) EFT introduce additional repulsion [46] that raises µ Λ further with increasing density. Detailed studies are now performed combining these repulsive effects in order to explore whether the condition µ Λ = µ n can still be met in neutron stars. chemical potential is derived from a Brueckner calculation [46] of the Λ single-particle potential in neutron matter. Dashed curve: ΛN two-body interactions only. Solid curve: including ΛN N three-body forces [44,45].
An preliminary impression of how repulsive two-and three-body Λ-nuclear interactions work against the appearance of hyperons in neutron star matter is shown in Fig. 7. This figure is based on a Brueckner calculations of the density-dependent single-particle potential of a Λ in neutron matter [46]. Once repulsive ΛN N three-body forces are turned on, it is likely that the Λ chemical potential does not meet the neutron chemical potential any more at densities relevant to the inner region of neutron stars. Such a mechanism would maintain the stiffness of the EoS and the high pressures needed to fulfill astrophysics constraints.

IX. CONCLUDING REMARKS
The focus in this presentation has been on guiding principles leading from QCD symmetries and symmetry breaking patterns to strongly interacting complex systems such as nuclei and dense baryonic matter. It indeed turns out that chiral symmetry and its spontaneous breakdown in conjunction with the confining and scale-invariance breaking QCD forces provide the basis for constructing effective field theories of low-energy QCD that lead a long way towards the understanding of nuclear forces, nuclear many-body systems and even baryonic matter under more extreme conditions such as they are encountered in the cores of neutron stars.
In this context, a significant outcome from a non-perturbative framework using functional renormalization group methods concerns the appearance of phase transitions in the equationof-state of cold baryonic matter. While the empirically established first-order liquid-gas transition in nuclear matter is well reproduced, strong fluctuations beyond mean-field approximation prevent a first-order chiral phase transition even at densities as high as those encountered in the core of neutron stars. In such a scheme the quest for the emergence of quark-hadron continuity and the transition to freely floating quarks in cold and compressed baryonic matter is passed over to even more extreme density scales.