Strangeness in Neutron Stars

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1 Strangeness in Neutron Stars Jürgen Schaffner-Bielich Institut für Theoretische Physik Laboratori Nazionali di Frascati, Italy, March 28, 2013 HGS-HIRe Helmholtz Graduate School for Hadron and Ion Research p.1

2 Outline Introduction: measuring neutron star masses Hyperons in Neutron Stars: incompatible with massive pulsars? Rescue the hyperons I: change hyperon potentials Rescue the hyperons II: going from SU(6) to SU(3) symmetry Rescue the hyperons III: add quark matter What is the maximum possible neutron star mass? Constraints from causality Constraints from subthreshold production of kaons in heavy-ion collisions Summary p.2

3 Neutron Stars produced in core collapse supernova explosions compact, massive objects: radius 10 km, mass 1 2M extreme densities, several times nuclear density: n n 0 = g/cm 3 in the middle of the crab nebula: a pulsar, a rotating neutron star! p.3

The Pulsar Diagram the diagram for pulsars: period versus period change (P-Ṗ) dipole model for pulsars: characteristic age: τ = P/(2P) and magnetic field B = 2 10 19 (P P) 1/2 Gauss anomalous x-ray

4 The Pulsar Diagram the diagram for pulsars: period versus period change (P-Ṗ) dipole model for pulsars: characteristic age: τ = P/(2P) and magnetic field B = (P P) 1/2 Gauss anomalous x-ray pulsars: AXP, soft-gamma ray repeaters: SGR, young pulsars in supernova remnants: SNR (ATNF pulsar catalog) rapidly rotating pulsars (millisecond pulsars): mostly in binary systems (old recycled pulsars!) p.4

5 Masses of Pulsars (Stairs 2006) nearly 2000 pulsars known with 140 binary pulsars best determined mass: M = (1.4414±0.0002)M Hulse-Taylor pulsar (Weisberg and Taylor, 2004) mass of PSR J corr. from M = (2.1±0.2)M to M = ( )M (Nice et al. 2008) extremely rapid rotations: up to 716 Hz (1.397 ms) (PSR J ad) p.5

6 Mass measurement of pulsar PSR J (Freire 2009) s. ω r r ω. s measure post-keplerian parameters from pulsar timing Shapiro delay parameters r and s alone constrain M = (1.67±0.11)M combined with periastron advance ω: M = (1.67±0.01)M rotation of the companion star could influence ω (follow-up observation with Hubble planned) p.6

Mass measurement of pulsar PSR J1614-2230 (Demorest et al. 2010) 30 20 10 Inclination Angle (deg) Timing residual (µs) 89.24 89.22 89.2 89.18 89.16 89.14 89.

7 Mass measurement of pulsar PSR J (Demorest et al. 2010) Inclination Angle (deg) Timing residual (µs) Orbital Phase (turns) Probability Density extremely strong signal for Shapiro delay Shapiro delay parameters r and s alone give M = (1.97±0.04)M - new record! by far the highest precisely measured pulsar mass! considerable constraints on neutron star matter properties! Companion Mass (solar) Pulsar Mass (solar) p.7

8 Mass (solar) GR P < rotation Constraints on the Mass Radius Relation (Lattimer and Prakash 2004) J causality AP3 ENG AP4 SQM3 J FSU SQM1 J PAL6 GM3 GS1 Double NS Systems MPA1 MS1 PAL1 MS2 MS Nucleons Nucleons+Exotic Strange Quark Matter Radius (km) spin rate from PSR B of 641 Hz: R < 15.5 km for M = 1.4M Schwarzschild limit (GR): R > 2GM = R s causality limit for EoS: R > 3GM mass limit from PSR J (red band): M = (1.97±0.04)M p.8

9 Hyperons in Neutron Stars p.9

10 Neutron Star Matter for a Free Gas (Ambartsumyan and Saakyan, 1960) Hadron p,n Σ Λ others appears at: n 0 4n 0 8n 0 > 20n 0 but the corresponding equation of state results in a maximum mass of only M max 0.7M < 1.44M (Oppenheimer and Volkoff, 1939) = effects from strong interactions are essential to describe neutron stars! p.10

11 Experimental Status of Hypernuclear Systems NΛ: attractive Λ-hypernuclei for A = U Λ = 30 MeV at n = n 0 NΣ: 4 Σ He hypernucleus bound by isospin forces Σ atoms: potential is repulsive NΞ: attractive 7 Ξ hypernuclear events U Ξ = 28 MeV at n = n 0 quasi-free production of Ξ: U Ξ = 18 MeV ΛΛ: attractive 3 ΛΛ hypernuclei measured YY: Y= Λ,Σ,Ξ, unknown! hypernuclear programs: JLab, J-PARC, and PANDA, CBM@FAIR: access to strange dibaryons, multi-hypernuclei! p.11

12 Onset of Hyperons in Neutron Star Matter Hyperons appear at n 2n 0! (based on hypernuclear data!) relativistic mean field (RMF) models (Glendenning 1985; Knorren, Prakash, Ellis 1996; JS and Mishustin 1996) nonrelativistic potential model (Balberg and Gal 1997) quark-meson coupling model (Pal et al. 1999) relativistic Hartree Fock (Huber, Weber, Weigel, Schaab 1998) Brueckner Hartree Fock (Baldo, Burgio, Schulze 1998, 2000; Vidana et al. 2000; Schulze, Polls, Ramos, Vidana 2006) chiral effective Lagrangian using SU(3) symmetry (Hanauske et al. 2000; Schramm and Zschiesche 2003; Dexheimer and Schramm 2008) density-dependent hadron field theory (Hofmann, Keil, Lenske 2001) G-matrix calculation (Nishizaki, Takatsuka, Yamamoto 2002) RG approach with V low k (Djapo, Schäfer, Wambach 2008) neutron stars are giant hypernuclei!!! p.12

13 Maximum mass and modern many-body approaches modern many-body calculations (using Nijmegen soft-core YN potential) Vidana et al. (2000): M max = 1.47M (NN and YN interactions), M max = 1.34M (NN, NY, YY interactions) Baldo et al. (2000): M max = 1.26M (including three-body nucleon interaction) Schulze et al. (2006): M max < 1.4M Djapo et al. (2008): M max < 1.4M (V low k ) Schulze and Rijken (2011): M max < 1.4M (ESC08) EoS too soft, too low masses! p.13

14 Maximum masses of neutron stars with hyperons Brueckner-Hartree-Fock calculation with most recent soft core Nijmegen potential ESC08 includes repulsive three-body forces (TBF, UIX ) overall findings: M < 1.4M when hyperons are included their conclusion: need a stiff quark matter core! (Schulze and Rijken 2011) is it really necessary? And does it really work? p.14

15 Rescue the hyperons I: change hyperon potentials p.15

16 Varying the Σ potential for neutron stars: EoS use RMF model with set GM1 EoS for different U Σ = 40,...40 MeV p [MeV fm -3 ] n B / n 0 lower curves: model 1 (without φ meson) upper curves: model 1 + φ meson set U Ξ = 28 MeV Σ baryons do not soften the EoS for repulsive potentials (Weissenborn, Chatterjee, JSB 2011) p.16

17 Varying the Σ potential for neutron stars: masses mass-radius relation for different U Σ = 40,...40 MeV M / M solar lower curves: model 1 (without φ meson) upper curves: model 1 + φ meson Σ baryons do not change the mass-radius relation for repulsive potentials R [km] (Weissenborn, Chatterjee, JSB 2011) moderate increase in maximum mass for repulsive Σ potential p.17

18 Varying the Ξ potential for neutron stars: EoS p [MeV fm -3 ] use RMF model with set GM1 mass-radius relation for different U Ξ = 40,...40 MeV lower curves: model 1 (without φ meson) n B / n 0 (Weissenborn, Chatterjee, JSB 2011) upper curves: model 1 + φ meson repulsive Ξ potential stiffens the EoS at high densities set U Σ = +30 MeV p.18

19 Varying the Ξ potential for neutron stars: masses use RMF model with set GM1 M / M solar mass-radius relation for different U Ξ = 40,...40 MeV lower curves: model 1 (without φ meson) upper curves: model 1 + φ meson R [km] repulsive Ξ potential helps to increase maximum mass (Weissenborn, Chatterjee, JSB 2011) p.19

20 Varying hyperon potentials for neutron stars (N) U Σ [MeV] (N) U Ξ [MeV] contour plots for different Σ and Ξ potentials upper plot: model 1 (without φ meson) lower plot: model 1 + φ (N) U Ξ [MeV] meson new mass limit only compatible with (repulsive) φ meson (N) U Σ [MeV] (Weissenborn, Chatterjee, JSB 2011) p.20

21 Rescue the hyperons II: going from SU(6) to SU(3) symmetry p.21

22 SU(3) symmetry consider Yukawa coupling of baryons with mesons in SU(3): coupling of two octets: 8 8 = 1 8 S 8 A general coupling scheme for baryon octet with mesons: L SU(3) = g 8 αtr([ B,B]M)+g 8 (1 α)tr({ B,B}M) g 1 Tr( B,B)Tr(M) parameters: octet coupling constant g 8, α = F/(F + D) ratio (F: antisymmetric, D: symmetric coupling), singlet coupling constant g 1, meson nonet mixing angle θ SU(6) model: SU(3) flavour plus SU(2) spin symmetry spin independence of Λ and Σ vector coupling: α V = 1 for vector nonet: ideal mixing tanθ V = 1/ 2 (φ meson: pure s s state) nucleon does not couple to φ mesons: g 1 = 6g 8 everything fixed in terms of one (nucleon) coupling constant in SU(6) p.22

23 Varying hyperon coupling z: coupling constants g Λω / g Nω g Σω / g Nω g Ξω / g Nω g Νφ / g Nω g Λφ / g Nω g Σφ / g Nω g Ξφ / g Nω SU(6) z = g 8 / g 1 (Weissenborn, Chatterjee, JSB 2011) baryon coupling constants for different z = g 8 /g 1 = 0,...2/ 6 SU(6) value: z = 1/ , equal coupling for z = 0 always repulsive vector interactions except for g Nφ Λ and Σ coupling constants are always equal p.23

24 Varying hyperon coupling z: EoS z = 0.0 z = 0.1 z = 0.2 z = 0.3 SU(6) z = 0.5 z = 0.6 z = 0.7 z = 0.8 varying z = 0,...2/ 6 use set GM1 with model 1 + φ p [MeV fm -3 ] adjust scalar couplings to U Σ = +30 MeV, U Ξ = 28 MeV SU(6) value: z = 1/ always repulsive vector interactions except for g Nφ ε [MeV fm -3 ] (Weissenborn, Chatterjee, JSB 2011) p.24

25 Varying hyperon coupling z for neutron stars z = 0.0 z = 0.1 z = 0.2 z = 0.3 SU(6) z = 0.5 z = 0.6 z = 0.7 z = 0.8 allow z = g 8 /g 1 to vary 2 use set GM1 with model 1 + φ Mass [M solar ] mass-radius relation for different z = 0,...2/ 6 (repulsive vector interactions except for g Nφ ) SU(6) value: z = 1/ drastic difference in maximum mass Radius [km] (Weissenborn, Chatterjee, JSB 2011) p.25

26 Maximum mass with hyperons for different z NL3, nuc NL3, hyp TM1, nuc TM1, hyp GM1, nuc GM1, hyp 2.6 M max / M solar z = g 8 / g 1 (Weissenborn, Chatterjee, JSB 2011) maximum mass with varying hyperon couplings z = g 8 /g 1 different RMF parameter sets continous drop in maximum mass with increasing z values of M > 2M possible p.26

27 Maximum mass with hyperons M max / M solar PL-Z NL-SH PL-40 NL1 NL3 NL-Z PSR J GM1 nucleons z = 0.0 z = 0.1 z = 0.2 z = 0.3 SU(6) z = 0.5 z = 0.6 z = 0.7 z = 0.8 TM1 GM m N * / mn (Weissenborn, Chatterjee, JSB 2011) maximum mass for different effective masses of RMF parameter sets crosses: parameter sets with standard SU(6) baryon couplings values of M > 2M possible with hyperons for stiff nuclear EoS p.27

28 Neutron Stars with Hyperons: Composition hyperons within RMF model: couples to scalar and vector fields scalar coupling: fixed by hyperon potential depths (hypernuclei) vector coupling: controlled by SU(3) symmetry, vary singlet to octet strength: z = g 8 /g 1 (Weissenborn, Chatterjee, JSB 2012) p.28

29 Neutron Star Masses and Strangeness Fraction m N * / mn = 0.55 α V free, z = α V = 1.0, z free 2.4 NL3 M max / M solar GM1 PSR J m N * / mn = f s (Weissenborn, Chatterjee, JSB 2012) RMF model for different nuclear parameter sets and hyperon coupling neutron star maximum mass versus the strangeness fraction f s unique relation: M max (f s ) = M max (0) 0.6M (f s /0.1) p.29

30 Rescue the hyperons III: add quark matter? p.30

31 Hybrid Stars with a stiff nuclear EoS a 4 =0.4 Maximum Mass [M solar ] M solar 1.44 M solar a 4 =0.5 a 4 =1.0 PSR J Maxwell hybrid stars Maxwell, quark core Gibbs, quark core Gibbs, mixed core n c ~ 0.1 1/fm 3 n c ~ 0.2 1/fm 3 n c ~ 0.3 1/fm 3 n c ~ 0.4 1/fm 3 n c ~ 0.5 1/fm B eff 1/4 [MeV] (Weissenborn, Sagert, Pagliara, Hempel, JSB 2011) nuclear phase: relativistic mean field model with parameter set NL3 (fitted to properties of nuclei) match with Gibbs (lines) or Maxwell construction (shaded area) solid lines: pure quark matter cores, dashed lines: mixed phase cores p.31

32 Hybrid Stars with a soft nuclear EoS a 4 =0.4 Maximum Mass [M solar ] a 4 =0.5 a 4 = M solar Maxwell, hybrid stars Maxwell, quark core 1.6 Gibbs, quark core Gibbs, mixed core n 1.5 c ~ 0.1 1/fm M n solar c ~ 0.2 1/fm 3 n c ~ 0.3 1/fm n c ~ 0.4 1/fm 3 n c ~ 0.5 1/fm 3 n c ~ 0.6 1/fm B eff 1/4 [MeV] PSR J (Weissenborn, Sagert, Pagliara, Hempel, JSB 2011) nuclear phase: relativistic mean field model with parameter set TM1 (fitted to properties of nuclei) match with Gibbs (lines) or Maxwell construction (shaded area) solid lines: pure quark matter cores, dashed lines: mixed phase cores no pure quark cores compatible with data for a soft nuclear EoS p.32

33 Hybrid Stars with a NJL model (Bonanno and Sedrakian 2011) uses Nambu-Jona-Lasinio model for quark matter matches to nuclear EoS with hyperons (RMF with set NL3) 2SC quark matter below green line δ = R CFL /R: amount of CFL quark matter p.33

34 What is the maximum possible neutron star mass? p.34

35 Maximum central density of a compact stars (Lattimer and Prakash 2011) maximally compact EoS: p = s(ǫ ǫ c ) with s = 1 stiffest possible EoS (Zeldovich 1961) gives upper limit on compact star mass: M max = 4.2M (ǫ sat. /ǫ f ) 1/2 (Rhoades and Ruffini 1974, Hartle 1978, Kalogera and Baym 1996, Akmal, Pandharipande, Ravenhall 1998) p.35

36 Input to transport models: the nucleon potential E/A [MeV] DBHF Bonn A BHF AV 18 BHF AV BF var AV 18 +δv var AV 18 +δv+3-bf Skyrme (K=380) Skyrme (K=200) Skyrme BHF DBHF var ρ/ρ 0 (Fuchs 2006) crucial input to control the amount of compression: the nucleon potential study two extreme cases using the Skyrme model hard EoS: stiffness parameter K = 380 MeV, soft EoS: K = 200 MeV p.36

37 Heavy-ion collisions: density range probed with kaons (Hartnack, Oeschler, Leifels, Bratkovskaya, Aichelin 2012) kaon production by associated production: NN NΛK, NN NNKK produced in a baryon-rich medium at densities of 2n 0 up to 3n 0 long mean-free path of kaons: escape from the high-density region p.37

38 Kaon production in heavy-ion collisions (M K+ /A) Au+Au / (M K+ /A) C+C E/A [MeV] soft EOS hard EOS KaoS ρ/ρ E lab [GeV] nuclear matter is compressed up to 2 3n 0! long mean-free path of kaons: kaons can escape high density matter clear trend indicating high compression a soft EoS Sturm et al. (KaoS collaboration), PRL 2001 Fuchs, Faessler, Zabrodin, Zheng, PRL 2001 p.38

39 Confirmed KaoS data analysis: the nuclear EoS is soft! (M Au A -1 Au )/(M C A-1 C ) E beam = 0.8 AGeV IQMD with KN pot. IQMD w/o KN pot. E beam = 1.0 AGeV IQMD with KN pot. IQMD w/o KN pot. data range data range K N [MeV] Forster et al. (KaoS collaboration) 2007 Hartnack, Oeschler, Aichelin, 2006 kaon production (K + ) far below threshold double ratio: multiplicity per mass number for C+C collisions and Au+Au collisions at 0.8 AGeV and 1.0 AGeV only calculations with a compression modulus of K N 200 MeV and smaller can describe the data = the nuclear equation of state is SOFT! p.39

40 Constraint from heavy-ion data: nucleon potential at 2 3n 0 (Sagert, Tolos, Chatterjee, JSB, Sturm 2012) input to transport simulations: nucleon potential kaon production is sensitive to densities of n = 2 3n 0 (n 0 = 0.17 fm 3 ) constraint: nucleon potential must be below the curve for the Skyrme model with K = 200 MeV at a fiducial density of n f = 2...3n 0 p.40

41 Constraint for neutron stars: causality plus heavy-ion (Sagert, Sturm, Chatterjee, Tolos, JSB 2011) maximum mass for neutron stars versus the fiducial density same trend for the various nuclear models main difference: the density dependence of the asymmetry energy upper limit of a maximum mass of M max = 3M for n f = 2n 0 p.41

42 Summary: hyperons can substantially soften the EoS hyperons can substantially reduce the maximum mass of neutron stars too low masses in modern many-body models rescue the hyperons in neutron stars: make hyperon potentials repulsive: does not really work add φ mesons and break SU(6) symmetry: works = limits on the amount of hyperons in neutron stars add quark matter in the core: works = both nuclear and quark matter must be stiff constraints from causality and heavy-ion data: new firm upper mass limit for neutron stars of 3M! p.42

43 Thanks to: Simon Weissenborn (Heidelberg) Debarati Chatterjee (Heidelberg) Giuseppe Pagliara (Ferrara) Matthias Hempel (Basel) Irina Sagert (Michigan State) Christian Sturm (GSI Darmstadt) Laura Tolos (Barcelona) p.43

44 Composition of Neutron Star Matter Baryon fraction 10 0 n GM1 p e µ Λ Ξ Ξ 0 Σ 0 U Σ = 30 MeV U Ξ = 28 MeV 10 4 Σ Density (fm 3 ) Σ + attractive potential for Σs and Ξs Σ appear shortly before Λs around n = 2n 0 Λs present in matter at n = 2.5n 0, Ξ before n = 3n 0 p.44

45 Composition of Neutron Star Matter Baryon fraction 10 0 n GM1 p e µ Λ Ξ 0 Ξ U Σ =+30 MeV U Ξ = 18 MeV Density (fm 3 ) Λs are present close to n = 2n 0 repulsive potential for Σs: Σ hyperons do not appear at all! population is highly sensitive to the in-medium potential! p.45

46 Spin orbit splitting of hypernuclei: theory Simple Quark Model (SU(6) symmetry): in quark picture: Λ = (uds), where the u- and d-quark couple to spin zero nucleon couples only to u- and d-quarks: no spin-orbit term! for Σ 0 = (uds) u- and d-quarks couple to spin one big spin-orbit splitting! for Ξ = (dss) or Ξ 0 = (uss) nucleon couples to one light quark only same spin-orbit strength as for nucleons summary: spin-orbit strength goes as N : Λ : Σ : Ξ = 1 : 0 : 2 : 1 Relativistic potentials and SU(6) symmetry: cancellation effect due to large tensor term for Λs (Jennings 1990) p.46

47 Maximum mass without hyperons (nucleons only) PL-Z PL-40 NL1 NL-Z NL-SH NL3 K = 300 MeV K = 275 MeV K = 250 MeV K = 225 MeV K = 200 MeV 2.6 M max / M solar TM1 GM1 GM3 2 PSR J m * N / m N (Weissenborn, Chatterjee, JSB 2011) maximum mass for different effective masses m /m and compressibility K change in maximum mass for different compressibilities: at most 0.1M change in maximum mass for different m /m: up to 1M! values of M > 2M possible for reasonable values of m /m p.47

48 At which density do new particles appear? (Page and Reddy (2006)) Baryon chemical potential (MeV) µ B 0 = µ n µ B - = µ n +µ e µ B + = µ n -µ e Σ Σ + Λ n B /n 0 hyperons appear, when its in-medium energy equals its chemical potential: µ(y) = ω(y) = m Y +U Y (n) with µ(y) = B(Y) µ n Q(Y) µ e thin lines: free gas, thick lines: with mean-field NN potential note: horizontal lines are hyperon vacuum masses, onset of hyperons will change due to YN potentials p.48

49 Varying hyperon coupling α v : coupling constants g Λω / g Nω g Σω / g Nω g Ξω / g Nω g Nφ / g Nω g Λφ / g Nω g Σφ / g Nω g Ξφ / g Nω F/(F+D) ratio α V (Weissenborn, Chatterjee, JSB 2011) baryon coupling constants for different α v = 0,...1 SU(6) value: α v = 1 always repulsive vector interactions Λω coupling constant does not change p.49

50 Varying hyperon coupling α v : EoS α V = 0.0 α V = 0.1 α V = 0.2 α V = 0.3 α V = 0.4 α V = 0.5 α V = 0.6 α V = 0.7 α V = 0.8 α V = 0.9 α V = 1.0 RMF set GM1, model 1 + φ p [MeV fm -3 ] adjust scalar couplings to U Σ = +30 MeV, U Ξ = 28 MeV varying α v = 0,...1 use set GM1 with model 1 + φ SU(6) value: α v = 1 softest EoS for SU(6) case ε [MeV fm -3 ] (Weissenborn, Chatterjee, JSB 2011) p.50

51 Varying hyperon coupling α v for neutron stars α V = 0.0 α V = 0.1 α V = 0.2 α V = 0.3 α V = 0.4 α V = 0.5 α V = 0.6 α V = 0.7 α V = 0.8 α V = 0.9 α V = 1.0 varying α v = 0, use set GM1 with model 1 + φ M max / M solar adjust scalar couplings to U Σ = +30 MeV, U Ξ = 28 MeV mass-radius relation for different α v, SU(6) value: α v = substantially increasing mass with decreasing α v R [km] (Weissenborn, Chatterjee, JSB 2011) p.51

52 Maximum Masses of Neutron Stars Causality Skyrme Bsk8 n crit =2.5 n 0 n crit =2 n 0 n crit =3 n M [M solar ] R [km] (Sagert, Sturm, Chatterjee, Tolos, JSB 2011) Skyrme parameter set BSK8: fitted to masses of all known nuclei above a fiducial density (determined from the analysis of the KaoS heavy-ion data) transition to stiffest possible EoS causality argument: p = ǫ ǫ c above the fiducial density ǫ f Rhoades, Ruffini (1974), Kalogera, Baym (1996): M max = 4.2M (ǫ 0 /ǫ f ) 1/2 = new upper mass limit of about 2.8M from heavy-ion data! p.52

53 Maximum Masses of Neutron Stars Causality Skyrme SLy4 n crit =2.5 n 0 n crit =2 n 0 n crit =3 n M [M solar ] R [km] (Sagert, Sturm, Chatterjee, Tolos, JSB 2011) Skyrme parameter set Sly4: fitted to properties of spherical nuclei above a fiducial density (determined from the analysis of the KaoS heavy-ion data) transition to stiffest possible EoS causality argument: p = ǫ ǫ c above the fiducial density ǫ f Rhoades, Ruffini (1974), Kalogera, Baym (1996): M max = 4.2M (ǫ 0 /ǫ f ) 1/2 = new upper mass limit of about 2.8M from heavy-ion data! p.52

54 Maximum Masses of Neutron Stars Causality RMF TM1 n crit =2.5 n 0 n crit =2 n 0 n crit =3 n M [M solar ] R [km] (Sagert, Sturm, Chatterjee, Tolos, JSB 2011) RMF parameter set TM1: fitted to properties of spherical nuclei above a fiducial density (determined from the analysis of the KaoS heavy-ion data) transition to stiffest possible EoS causality argument: p = ǫ ǫ c above the fiducial density ǫ f Rhoades, Ruffini (1974), Kalogera, Baym (1996): M max = 4.2M (ǫ 0 /ǫ f ) 1/2 = new upper mass limit of about 2.8M from heavy-ion data! p.52

Constraints on Neutron Star Properties from KaoS Heavy-Ion Data

Constraints on Neutron Star Properties from KaoS Heavy-Ion Data Jürgen Schaffner-Bielich Institute for Theoretical Physics and Heidelberg Graduate School for Fundamental Physics and ExtreMe Matter Institute