Equations of State of Relativistic Mean-Field Models with Different Parametrisations of Density Dependent Couplings

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1 Equations of State of Relativistic Mean-Field Models with Different Parametrisations of Density Dependent Couplings Stefan Typel 100 7th International Symposium on Nuclear Symmetry Energy GANIL, Caen, France September 4 7, 2017 energy per baryon E (MeV) baryon density n B (fm -3 ) September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 1

2 Outline Motivation Relativistic Mean-Field Models Medium Dependence of Effective Interaction Parametrisation of Couplings Determination of Parameters Choice of Functionals Results Coupling Functions Equations of State (Symmetric and Neutron Matter) Density Dependence of Symmetry Energy Nuclear Matter Parameters Effects of Rearrangement Contributions Summary and Outlook September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 2

3 Motivation symmetry energy of nuclear matter density dependence densities below saturation: convergence of different theoretical approach, consistency with experimental constraints densities above satuation: large uncertainties characteristic parameters at saturation symmetry energy at saturation J rather well constrained slope parameter L still has large uncertainty theoretical description choice of energy density functional? effects on extrapolation? probability dp/dj [MeV -1 ] probability dp/dl [MeV -1 ] J = (31.7 +/- 3.2) MeV symmetry energy at saturation J [MeV] symmetry energy slope coefficient L [MeV] L = (58.7 +/- 28.1) MeV (M. Oertel et al., Rev. Mod. Phys. 89 (2017) ) September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 3

4 Relativistic Mean-Field Models field theoretical approach energy density functional derived from Lagrangian density phenomenological description various versions interaction: exchange of scalar and vector mesons (σ,ω,ρ,... ) minimal coupling of mesons to nucleons with nonlinear self-interactions with density dependent couplings without explicit meson fields point-coupling models applications description of finite nuclei and excitations nuclear matter and equation of state many parametrizations for different purposes (M. Dutra et al., Phys. Rev. C 90 (2014) ) September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 4

5 Medium Dependence of Effective Interaction interaction contribution to Lagrangian nonlinear (NL) RMF models with meson self-interactions L int = ψg σσψ A 3 σ3 B 4 σ4 ψg ωω µγ µ ψ + C 4 with constants g σ, g ω, g ρ, A, B, C,... ( ωµω µ)2 ψg ρ ρ µ τγ µ ψ (ususally scalar and vector contributions not coupled, cross terms added later) density dependent (DD) RMF models L int = ψγ σσψ ψγ ωω µγ µ ψ ψγ ρ ρ µ τγ µ ψ with functionals Γ σ, Γ ω, Γ ρ,... depending on Lorentz scalars constructed from ψ, ψ (more flexible than NL models) September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 5

6 Parametrisation of Couplings dependence of Γ i on vector density (v) = j µ j µ with current j µ = ψγ µ ψ standard choice scalar density (s) = ψψ not really explored so far... form of dependence (introduced in S. Typel and H.H. Wolter, Nucl. Phys. A 565 (1999) 331) general parametrisation: Γ i( ) = Γ i( ref)f i(x) with x = / ref and reference density ref rational function: f i(x) = a i 1 + b i(x + d i) c i(x + d i) 2 with additional constraints on f i exponential: f i(x) = exp[ a i(x 1)] other September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 6

7 Determination of Parameters fit to properties of nuclei minimisation of functionχ 2 ({p k}) = N data n=1 with parameters {p k} set of 12 nuclei: [ O (exp) n ] O n (model) 2 ({p k }) O n 16 O, 24 O, 40 Ca, 48 Ca, 56 Ni, 68 Ni, 90 Zr, 100 Sn, 114 Sn, 132 Sn, 140 Ce, 208 Pb observables O n with assumed errors O n: binding energy (0.1 MeV, 12 data) charge radius (0.01 fm, 8 data) diffraction radius (0.01 fm, 5 data) surface thickness (0.005 fm, 5 data) spin-orbit splitting (0.1 MeV, 14 data) 44 data in total no fit to nuclear matter parameters (derived quantities) September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 7

8 Choice of Functionals density dependence V : dependence of Γ ω, Γ σ, Γ ρ on vector density (v) S : dependence of Γ ω, Γ σ, Γ ρ on scalar density (s) M : dependence of Γ ω, Γ ρ (Γ σ) on vector (scalar) density (v) ( (s) ) functional form for ω and σ mesons 1+b rational function f i(x) = a i (x+d i ) 2 i with conditions f 1+c i (x+d i ) i(1) = 1 and 2 P : f i (0) = 0, d i > 0 (positive) Z : f i (0) = 0, d i = 0 (zero) N : f i (0) = 0, d i < 0 (negative) functional form for ρ meson E : exponential function f i(x) = exp[ a i(x 1)] 1+b R : rational function f i(x) = a i (x+d i ) 2 i with conditions 1+c i (x+d i ) 2 f i(1) = 1, f i (0) = 0, d i = 0, f i (1)/f i(1) = f i (1)/f i (1) 18 models with 9 parameters (including ref and m σ ), similar quality in describing nuclei September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 8

9 Results (Preliminary) Coupling Functions coupling Γ ω ω meson vector or scalar density n (v) or n (s) (fm -3 ) coupling Γ σ σ meson vector or scalar density n (v) or n (s) (fm -3 ) coupling Γ ρ ρ meson vector or scalar density n (v) or n (s) (fm -3 ) VPE VZE VNE VPR VZR VNR SPE SZE SNE SPR SZR SNR MPE MZE MNE MPR MZR NNR similar smooth functions for P and Z parametrisations minimum in functions for N parametrisations (ω and σ mesons) only small differences between E and R parametrisations (ρ meson) September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 9

10 Results (Preliminary) Equations of State VPE VZE energy per baryon E (MeV) symmetric nuclear matter energy per baryon E (MeV) neutron matter VNE VPR VZR VNR SPE SZE SNE SPR SZR SNR MPE MZE MNE MPR MZR NNR baryon density n B (fm -3 ) baryon density n B (fm -3 ) very similar below saturation density divergence above saturation density strong stiffening for N parametrisations September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 10

11 Results (Preliminary) Equations of State VPE VZE energy per baryon E (MeV) symmetric nuclear matter energy per baryon E (MeV) neutron matter VNE VPR VZR VNR SPE SZE SNE SPR SZR SNR MPE MZE MNE MPR MZR NNR baryon density n B (fm -3 ) baryon density n B (fm -3 ) very similar below saturation density divergence above saturation density strong stiffening for N parametrisations September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 11

12 Results (Preliminary) Density Dependence of Symmetry Energy VPE symmetry energy E sym (MeV) symmetry energy E sym (MeV) VZE VNE VPR VZR VNR SPE SZE SNE SPR SZR SNR MPE MZE MNE MPR MZR NNR baryon density n B (fm -3 ) baryon density n B (fm -3 ) very similar below saturation density divergence above saturation density, stiffest for S paremetrisations convergence problems for some parametrisations September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 12

13 Nuclear Matter Parameters I energy per nucleon E(n B,δ) = E 0 (n B ) + E sym (n B )δ 2 +O(δ 4 ) with baryon density n B = n n + n p and isospin asymmetry δ = (n n n p )/n B energy per nucleon in symmetric nuclear matter E 0 (n B ) = m nuc B sat Kx Qx with x = n B n sat 3n sat symmetry energy E sym (n B ) = J + Lx K symx parameters n sat, B sat, K, Q, J, L, K sym,... September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 13

14 Results (Preliminary) Nuclear Matter Parameters II binding energy per nucleon B (MeV) VPE VZE VNE VPR VZR VNR SPE SZE SNE SPR SZR SNR MPE MZE MNE MPR MZR MNR saturation density n sat (fm -3 ) Dirac effective nucleon mass m eff /m nuc VPE VZE VNE VPR VZR VNR SPE SZE SNE SPR SZR SNR MPE MZE MNE MPR MZR MNR mass of σ meson m σ (MeV/c 2 ) small variation in n sat and B, some spread in m eff, a few outliers for m σ September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 14

15 Results (Preliminary) Nuclear Matter Parameters III skewness Q (MeV) VPE VZE VNE VPR VZR VNR SPE SZE SNE SPR SZR SNR MPE MZE MNE MPR MZR MNR slope parameter L (MeV) VPE VZE VNE VPR VZR VNR SPE SZE SNE SPR SZR SNR MPE MZE MNE MPR MZR MNR incompressibility K (MeV) symmetry energy J (MeV) large spread in K, Q, and L systematics of Q with K and correlation of L with J September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 15

16 Effects of Rearrangement Contributions I essential for thermodynamic consistency of model general form of potentials (in symmetric nuclear matter): S = Γ σ σ + S (R) V = Γ ω ω 0 + V (R) with S (R) = Γ σ (s) n σσ Γ ω (s) n ωω 0 V (R) = Γ ω (v) n ωω 0 Γ σ (v) n σσ with source densities n σ and n ω dependence of couplings on scalar density (s) rearrangement contribution S (R) to scalar potential S dependence of couplings on vector density (v) rearrangement contribution V (R) to scalar potential V (as in standard DD-RMF models) September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 16

17 Effects of Rearrangement Contributions II dependence of Γ ω on scalar density (s) with Γω < 0, zero temperature (s) Dirac effective mass m nuc (eff) = m nuc S can approach zero scalar density (s) = n σ 0 σ 0 scalar potential S Γω Γ ω (s) m 2 ω n 2 ω limiting maximum density n max = ( Γω (s) ) 1 mnucm 2 ω Γ ω dependence of Γ σ on vector density (v) with Γσ 0 (v) at finite temperature T, baryon density n B = 0 n ω = 0, ω 0 = 0, n σ > 0, σ > 0 (antiparticles!) vector potential V = V (R) = Γσ Γ σ n 2 (v) mσ 2 σ baryon chemical potential µ B = V 0 model should work in whole hadronic part of phase diagram constraints on density dependent couplings exclusion of some parametrisations September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 17

18 Summary and Outlook extension of RMF models with density dependent couplings different dependencies on vector and scalar densities model parameters from fit to properties of finite nuclei similar quality in description of all 18 parametrisations equations of state, symmetry energy and nuclear matter parameters only small variations below saturation density divergence above saturation density exceptional behavior of N parametrisations some parameters (K, Q, L) not well constrained and model dependent constraints from rearrangement contributions to potentials next steps only parametrisations VZR, SZR, and Mxx viable, others problematic include δ meson include pairing effects detailed analysis of uncertainties and correlations September 4, 2017 NuSYM17, GANIL, Caen, France S. Typel 18