Nuclear Thermodynamics from Chiral Low-Momentum Interactions

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1 Nuclear Thermodynamics from Chiral Low-Momentum Interactions arxiv: (214) Corbinian Wellenhofer 1, Jeremy W. Holt 2, Norbert Kaiser 1, Wolfram Weise 1,3 1 Technische Universität München 2 University of Washington, Seattle 3 ECT, Villazzano (Trento) ECT Workshop Three-Body orces: from Matter to Nuclei May 7, 214 Work supported in part by DG and NSC (CRC 11)

2 Motivation Nuclear Equation of State for Astrophysics Neutron star structure and evolutions, stellar collisions ( gravitational waves), core-collapse supernovae Need realistic EoS of neutron matter with small admixture of protons Constraints from Nuclear Thermodynamics (Symmetric Matter) Bulk properties at zero temperature (saturation point, compressibility) Liquid-gas phase transition at nite temperature Critical point from multifragmentation, ssion and compound nuclear decay experiments c = 17.9 ±.4 MeV Elliott et al.; Phys. Rev. C 87, (213) T

3 Nuclear orces in Chiral Eective ield Theory Hierarchy of nuclear forces controlled by chiral expansion parameter Q Λχ Q m π 14 MeV, Λ χ 4πf π 1.2 GeV Two types of interactions: pion-exchange (long range) & contact (short range) Nuclear forces parametrized by Low-Energy Constants (LECs) LO (Q/Λ χ) NNorce 3N orce 4N orce NLO (Q/Λ χ) 2 NNLO (Q/Λ χ) c E c 1, c 3, c 4 c D N 3 LO (Q/Λ χ) not considered here not considered here Regularization of nuclear forces in order to restrict to low-momentum region: Q < Λ < Λ χ Nuclear potentials VNN = V NN (Λ; c i ) and V3N = V 3N (Λ; c i, c E, c D ) c i (Λ) from NN-scattering phase shifts, c D (Λ), c E (Λ) from 3N & 4N observables

4 Chiral Low-Momentum Interactions V NN (Λ, n): NN potential with smooth regulator, LECs retted for dierent Λ, n [ ( ) f Λ,n (k k 2n ( k ) 2n ], k) = exp Λ V low-k (Λ): RG-evolution of partial-wave matrix-elements (universality for Λ 2.1 fm 1 ) c i 's: Nijmegen LECs, c E (Λ) & c D (Λ) tted to few-body observables Λ R Bogner, urnstahl, Schwenk; Prog. Part. Nucl. Phys. 65, 94 (21) Summary: sets of low-momentum NN and 3N potentials used in this work identier Λ n c E c D c1 [GeV 1 ] c3 [GeV 1 ] c4 [GeV 1 ] n3lo5 2.5 fm n3lo fm fm fm fm

5 Phase Shifts with n3lo NN Potentials (red), n3lo45 (blue), n3lo5 (black) L. Coraggio, J. W. Holt, N. Itaco, R. Machleidt, and. Sammarruca, Phys. Rev. C 87, (213)

6 Many-Body Perturbation Theory for Nuclear Matter Imaginary-Time ormalism + Kohn-Luttinger-Ward Method ree energy density (as function of non-interacting chemical potential/density): [ (µ, T ) = (µ, T ) + 1(µ, T ) + 2,normal (µ, T ) + 2,anomalous (µ, T ) 1 ( 1/ µ) 2 ] 2 2 / µ 2 ρ(µ, T ) = (µ, T ) µ [... ] T k 2 k 1 k 1 k 3 k 1 k 3 k 4 k 2 (1,NN) (1,3N) (2,normal) k 3 k 1 k 2 (2,anomalous)

7 Approximate Treatment of Three-Body orces at Second Order Temperature- and density-dependent eective NN potential ṼDDNN(ρ, T ) Constructed from genuine 3N forces by integrating out (i.e. closing) one nucleon-line, DDNN regulator consistent with NN regulator our additional diagrams at second order k3 k3 k1 k3 k4 k1 k1 k3 k4 k1 k T= MeV 3N (Λ 3N =2.1 fm-1 ) 3N (Λ 3N = ) NN [DDNN] (a) T= MeV 3N (Λ 3N = ) NN [DDNN] (b) n3lo5

8 Approximate Treatment of Three-Body orces at Second Order Temperature- and density-dependent eective NN potential ṼDDNN(ρ, T ) Constructed from genuine 3N forces by integrating out (i.e. closing) one nucleon-line, DDNN regulator consistent with NN regulator our additional diagrams at second order k3 k3 k1 k3 k4 k1 k1 k3 k4 k1 k ,normal [NN] ,normal [mixed] 2,normal [DDNN] 2,normal [DDNNtotal] 2,normal [total] T= MeV (a) n3lo ,normal [NN] 2,normal [mixed] T= MeV 2,normal [total] ,normal ,normal (b)

9 Approximate Treatment of Three-Body orces at Second Order Temperature- and density-dependent eective NN potential ṼDDNN(ρ, T ) Constructed from genuine 3N forces by integrating out (i.e. closing) one nucleon-line, DDNN regulator consistent with NN regulator our additional diagrams at second order k3 k3 k1 k3 k4 k1 k1 k3 k4 k1 k T=5 MeV T=15 MeV 2,anomalous [NN] 2,anomalous [mixed] 2,anomalous [DDNN] totanom [DDNN] T=5 MeV T=1 MeV T=15 MeV totanom [mixed] totanom [NN]

10 Inclusion of Self-Energy Corrections G(k, ω) = G(k, ω) + G(k, ω) Σ (k) G(k, ω) = [ ] 1 ω 2M Σ (k, ω) Σ 1 Σ Eective mass: ε(k; ρ, T ) = ε (k) + Σ (k; ρ, T ) = M ε(k) + U(ρ, M T ) (ρ,t ) 2,normal(µ, T ) M M d 3 k1 (2π) 3 d 3 (2π) 3 d 3 k3 (2π) 3 d 3 k4 (2π) 3 [...] 1 ε(k3)+ε(k4) ε(k1) ε()

11 Inclusion of Self-Energy Corrections G(k, ω) = G(k, ω) + G(k, ω) Σ (k) G(k, ω) = [ ] 1 ω 2M Σ (k, ω) Σ 1 Σ Eective mass: ε(k; ρ, T ) = ε (k) + Σ (k; ρ, T ) = M ε(k) + U(ρ, M T ) (ρ,t ) 2,normal(µ, T ) M M d 3 k1 (2π) 3 d 3 (2π) 3 d 3 k3 (2π) 3 d 3 k4 (2π) 3 [...] 1 ε(k3)+ε(k4) ε(k1) ε() 1.9 T= MeV T=15 MeV 1.9 M*/M.8 n3lo M*/M T= MeV T=15 MeV n3lo µ

12 Results at Dierent Orders in MBPT (no M /MCorrections) n3lo45 n3lo5 T= MeV (a) NN first order, no 3N (b) NN second order, no 3N (c) NN second order, 3N first order (d) NN second order, 3N second order

13 Results at Dierent Orders in MBPT (no M /MCorrections) n3lo45 n3lo5 T= MeV n3lo45 n3lo5 T= MeV (a) NN first order, no 3N (b) NN second order, no 3N (c) NN second order, 3N first order (d) NN second order, 3N second order

14 Results at Dierent Orders in MBPT (no M /MCorrections) n3lo45 n3lo5 T= MeV n3lo45 n3lo5 T= MeV (a) NN first order, no 3N (b) NN second order, no 3N n3lo45 n3lo5 T= MeV (c) NN second order, 3N first order (d) NN second order, 3N second order

15 Results at Dierent Orders in MBPT (no M /MCorrections) n3lo45 n3lo5 T= MeV n3lo45 n3lo5 T= MeV (a) NN first order, no 3N (b) NN second order, no 3N -1 T= MeV -1 T= MeV n3lo45 n3lo n3lo45 n3lo (c) NN second order, 3N first order (d) NN second order, 3N second order

16 Inclusion of M /MCorrections n3lo45 n3lo5 T= MeV (a) NN second order, 3N second order, M /M included

17 Inclusion of M /MCorrections VLK: pressure isotherm crossing!! n3lo: no crossing P [MeV fm -3 ] n3lo45 n3lo5 T= MeV (a) NN second order, 3N second order, M /M included

18 Nuclear Equation of State from n3lo Potential Sets T=: saturation point T=nite: critical point Ē ρ [fm 3 ] K T c ρ c [fm 3 ] P c [MeV/fm 3 ] Empirical ± ±.4.6 ±.2.31 ±.7 n3lo5 (no M ) n3lo45 (M ) (M ) Physical equation of state via Maxwell construction (cf. van der Waals EoS) P [MeV fm -3 ] T= MeV T=1 MeV T=15 MeV T c =17.4 MeV T=2 MeV T gas gas-liquid n3lo5 n3lo45 liquid (a) P (ρ, T ) from (with M /M) (b) T ρ phase diagram

19 Uncertainty Bands from n3lo Potential Sets T= MeV T=1 MeV T=15 MeV n3lo45-5 n3lo5-6 T= MeV T=1 MeV T=15 MeV n=1 n=2 n= n3lo5 vs. n3lo45 vs. Variation of DDNN regulator ()

20 Summary Nuclear thermodynamics from chiral low-momentum interactions using MBPT up to second order (with rst-order self-energy in M /M-approximation) Pure NN results at second order similar for all ve two-body potentials Results from and n3lo45 Well-converged at second order with M /Mcorrections (cf. Coraggio et al.; arxiv: (214)) Good reproduction of saturation point, critical point agrees with empirical constraints, T c = MeV Outlook: isospin-asymmetric matter for astrophysics applications Results from n3lo5 NN potential not well-converged at second order (cf. Coraggio et al.; arxiv: ) Saturation point reproduced only without M /Mcorrections Critical point close to empirical value, T c = 19.1 MeV Outlook: isospin-asymmetric matter for astrophysics applications Results from and T=: Saturation point reproduced only with (second order with M ) T=nite: pressure isotherm crossing due to large second-order DDNN contributions Nijmegen LECs not consistent with nuclear thermodynamics (N2LO three-body force) Particularly problematic: large value of c 3 = 4.78 GeV 1

21 Three-Body orces & Nuclear Thermodynamics Status Largest deviations in results from dierent potential sets due to dierent parametrization of N2LO three-body forces (n3lo vs. Nijmegen LECs) n3lo LECs consistent with nuclear thermodynamics, Nijmegen LECs not! Nuclear thermodynamics can help in constraining uncertainties in LECs/three-body matrix elements! Thermodynamic nuclear equation of state from three dierent sets of N3LO two-body and N2LO three-body potentials (, n3lo45, n3lo5) in agreement with many-body observables! Needs Nuclear thermodynamics with consistently evolved (S)RG many-nucleon forces Inclusion of subleading (N4LO) many-nucleon forces Long term: inclusion of explicit (1232)-isobar degrees of freedom We are happy to collaborate regarding these issues!