Modeling the EOS. Christian Fuchs 1 & Hermann Wolter 2. 1 University of Tübingen/Germany. 2 University of München/Germany

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1 Modeling the EOS Christian Fuchs 1 & Hermann Wolter 2 1 University of Tübingen/Germany 2 University of München/Germany Christian Fuchs - Uni Tübingen p.1/20

2 Outline Christian Fuchs - Uni Tübingen p.2/20

3 Outline Overview models Christian Fuchs - Uni Tübingen p.2/20

4 Outline Overview models Symmetric nm Christian Fuchs - Uni Tübingen p.2/20

5 Outline Overview models Symmetric nm Isospin dependence Christian Fuchs - Uni Tübingen p.2/20

6 Outline Overview models Symmetric nm Isospin dependence Constraints from HICs Christian Fuchs - Uni Tübingen p.2/20

7 Overview models Ab inito approaches Brueckner: BHF (Catania,..), DBHF (T übingen,..), variational appr. (Urbana) realistic NN-interaction, no parameters Effective field theory Density functionals (Furnstahl, Serot,...), ChPT (Weise) peturbativ, scale arguments (m π /M, k F /M), few parameters (< 2) Empirical density functionals Skyrme, Relativistic Mean Field many parameters (6-10), high precison fits to finite nuclei Christian Fuchs - Uni Tübingen p.3/20

8 Saturation of Nuclear Matter DBHF: realistic NN force, no parameter E B [MeV] V K=? ρ/ρ 0 correlated uncorrelated wave r E/A [MeV] Tuebingen (Bonn) BM (Bonn) Bonn A, ps Reid CD Bonn Bonn AV k F [fm 1 ] see e.g. nucl-th/ Coester line = relativistic! Christian Fuchs - Uni Tübingen p.4/20

9 Hadronic many-body theory Relativistic Brueckner: N+OBEP ( V = σ, ω, π, ρ, η, δ) = 2-N correlations in hole-line expansion = self-consistent sum of ladder diagrams Dyson-Equation: G = G0 + G0ΣG = + Σ Bethe-Salpeter-Equation: T = V + i V GGQT T T = + Self Energy (Hartree-Fock): Σ(ρ, k) = qɛf < q T (q, k) q >= ΣS γ0σ0 + γ kσv = T T Σ Christian Fuchs - Uni Tübingen p.5/20

10 Saturation mechanism Non-relativistic: tensor force essential 2nd order 1 π exchange: large and attractive Pauli-blocking = saturation Relativistic: tensor force quenched Banerjee & Tjon NPA 708 (2002) 303 cancellation of large scalar and vector fields difference of vector and scalar density = saturation principally similar to RMF theory C.F., Lect. Notes Phys. 641 (2004) 119 Christian Fuchs - Uni Tübingen p.6/20

11 BHF versus DBHF BHF: 3-body forces necessary (Zuo et al., NPA 706 (2002) 418) E/A [MeV] variational Tuebingen (Bonn) BM (Bonn) Reid CD-Bonn Bonn AV 18 AV BF AV 18 +δv AV 18 +δv+3-bf E/A [MeV] DBHF Bonn A DBHF Bonn B BHF AV 18 BHF AV BF Skyrme hard (K=380) Skyrme soft (K=200) ChPT var AV 18 +δv var AV 18 +δv+3-bf -25 1,2 1,4 1,6 1,8 2,0 k F [fm -1 ] ρ/ρ 0 All microscopic EOS are soft! Christian Fuchs - Uni Tübingen p.7/20

12 Example for EFT: ChPT ChPT: pion dynamics + cut-off = expansion in k F : = soft EOS fine tunig to finite nuclei: = hard EOS E/A (MeV) chiral pion exchange case 2 case ρ (fm 3 ) ChPT: Finelli et al. NPA 735 (2004) 449 Σ v [MeV] Σ s [MeV] DB CHPT + point-couplings ρ NM k F [fm -1 ] DBHF: Gross-Boelting, C.F., Faessler, NPA 648 (1999) 105 Christian Fuchs - Uni Tübingen p.8/20

13 Neutron matter EOS 100 E b [MeV] 50 DBHF var AV BF NL3 DDRH Typel ChPT neutron matter 0 0 0,1 0,2 0,3 0,4 n B [ fm -3 ] nuclear matter DBHF EOS is soft (K=230 MeV); but asy-stiff Christian Fuchs - Uni Tübingen p.9/20

14 Symmetry energy from Skyrme Baran, Di Toro et al. nucl-th/ Christian Fuchs - Uni Tübingen p.10/20

15 Symmetry energy [ ] 2 E b (n B, β) E sym (n B ) = 1 2 β 2 β=0 E b (n B, β = 1) E b (n B, β = 0) β = Y n Y p 100 model E sym [MeV] Skyrme <30 Skyrme (SkLy) 32 RMF DBHF (Bonn A) 34.4 Lenske (Bonn C) 28 E sym [MeV] DBHF Pandaripande AV BF a 4 =30 a 4 =32 a 4 =34 a 4 =36 a 4 =38 ChPT p 0 =3.5 MeV/fm 3 (ρ=0.166) ChPT (Finelli et al.) ,1 0,2 0,3 0,4 0,5 n B [ fm -3 ] RMF: Vretenar et al., PRC 68 ( 03) Christian Fuchs - Uni Tübingen p.11/20

16 Neutron-proton mass splitting Comparison of different approaches = careful! Many different defnitions of effective masses are used! Non-relativistic mass: [ m NR = M + 1 k d dk U s.p. ] 1 Dirac mass: m D = M + Σ S Relativistic: U s.p. m D E Σ S + Σ 0 Christian Fuchs - Uni Tübingen p.12/20

17 Neutron-proton mass splitting BHF: m NR,n > m NR,p RMF: m D,n < m D,p ; m NR,n < m NR,p (ρ + δ) Baran, Di Toro et al. nucl-th/ DBHF with Σ extracted by fit method: m D,n > m D,p Alonso & Sammarunca, nucl-th/ DBHF with projection method: m D,n < m D,p de Jong & Lenske, PRC 58 ( 98) 890, van Dalen, C.F., Faessler, NPA 744 ( 04) 227 non-rel. mass in DBHF: m NR,n > m NR,p Christian Fuchs - Uni Tübingen p.13/20

18 Neutron-proton mass splitting 1100 nonrelativistic mass Dirac mass Neutron effective mass [MeV c -2 ] Neutron Fermi momentum Momentum k [fm -1 ] β = 0.0 β = 0.4 β = 0.6 β = 1.0 n B = fm Momentum k [fm -1 ] DBHF: van Dalen, C.F., Faessler in preparation Christian Fuchs - Uni Tübingen p.14/20

19 Constraints from HICs: Kaons (M K+ /A) Au+Au / (M K+ /A) C+C E/A [MeV] hard soft ρ/ρ 0 E thr Symmetric part of EOS: Subthreshold K + production Far subthreshold: highly sensitive to collective effects E lab [GeV/c] soft EOS, with pot hard EOS, with pot Kaos, Sturm et al., PRL 86 (2001) Christian Fuchs - Uni Tübingen p.15/20

20 Constraints from HICs: Kaons (M K+ /A) Au+Au / (M K+ /A) C+C E/A [MeV] E lab [GeV/c] soft EOS, with pot hard EOS, with pot Kaos, Sturm et al., PRL 86 (2001) 0 hard soft ρ/ρ 0 E thr Symmetric part of EOS: Subthreshold K + production Far subthreshold: highly sensitive to collective effects KaoS data = soft EOS! C.F. et al., PRL 86 (2001) 1974 Christian Fuchs - Uni Tübingen p.15/20

21 Constraints from HICs: Flow K=210 MeV (m /m = 0.7/0.65) and K=380 MeV (ruled out) Danielewicz, NPA 673 (2000) 375 Christian Fuchs - Uni Tübingen p.16/20

22 Constraints from HICs: Flow K=210 MeV (m /m = 0.7/0.65) and K=380 MeV (ruled out) Danielewicz, NPA 673 (2000) 375 Christian Fuchs - Uni Tübingen p.16/20

23 Constraints from HICs: Flow directed flow = momentum dependence Christian Fuchs - Uni Tübingen p.17/20

24 Constraints from HICs: Flow directed flow = momentum dependence elliptic flow = density dependence Danielewicz, NPA 673 (2000) 375 Christian Fuchs - Uni Tübingen p.17/20

25 Constraints from HICs: Flow directed flow = momentum dependence elliptic flow = density dependence 0,05 elliptic flow 0,00-0,05-0,10 EOS/E895 FOPI DB-CNM DB-LDA E beam [AGeV] C.F., Gaitanos, NPA 714 (2003) 643 Danielewicz, NPA 673 (2000) 375 Gaitanos, C.F., Wolter, EPJA 12 (2001) 421; NPA 650 Christian (1999) Fuchs 97- Uni Tübingen p.17/20

26 Constraints from HICs Isospin dependence: HICs with equal mass isotopes = isospin diffusion: GSI (FOPI), MSU data BUU: Chen, Ko, B-A Li, nucl-th/ , E sym = 31.6(ρ/ρ 0 ) γ Christian Fuchs - Uni Tübingen p.18/20

27 Preliminary Summary Christian Fuchs - Uni Tübingen p.19/20

28 Preliminary Summary EOS from ab inito calculations are soft for symmetric nm Christian Fuchs - Uni Tübingen p.19/20

29 Preliminary Summary EOS from ab inito calculations are soft for symmetric nm EOS from ab inito calculations are asi-stiff Christian Fuchs - Uni Tübingen p.19/20

30 Preliminary Summary EOS from ab inito calculations are soft for symmetric nm EOS from ab inito calculations are asi-stiff Consistent with information from hics Christian Fuchs - Uni Tübingen p.19/20