The EOS of neutron matter, and the effect of Λ hyperons to neutron star structure

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1 The EOS of neutron matter, and the effect of Λ hyperons to neutron star structure Stefano Gandolfi Los Alamos National Laboratory (LANL) Nuclear Structure and Reactions: Weak, Strange and Exotic International Workshop XLIII on Gross Properties of Nuclei and Nuclear Excitations Hirschegg, Kleinwalsertal, Austria, January 11-17, Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 1 / 35

2 Neutron stars Neutron star is a wonderful natural laboratory D. Page Atmosphere: atomic and plasma physics Crust: physics of superfluids (neutrons, vortex), solid state physics (nuclei) Inner crust: deformed nuclei, pasta phase Outer core: nuclear matter Inner core: hyperons? quark matter? π or K condensates? Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 2 / 35

3 Nuclei and hypernuclei Several thousands of binding energies for normal nuclei. Only few, 5, for hypernuclei. Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 3 / 35

4 Homogeneous neutron matter Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 4 / 35

5 Outline The model and the method Equation of state of neutron matter, role of three-neutron force Symmetry energy Neutron star structure (I) Λ-hypernuclei Λ-neutron matter Neutron star structure (II) Conclusions Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 5 / 35

6 Nuclear Hamiltonian Model: non-relativistic nucleons interacting with an effective nucleon-nucleon force (NN) and three-nucleon interaction (TNI). H = 2 2m A i=1 2 i + i<j v ij + i<j<k v ij NN fitted on scattering data. Sum of operators: V ijk v ij = O p=1,8 ij v p (r ij ), O p ij = (1, σ i σ j, S ij, L ij S ij ) (1, τ i τ j ) NN interaction - Argonne AV8. Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 6 / 35

7 Phase shifts, AV S Argonne V8 SAID S1 Argonne V8 SAID 1 P1 Argonne V8 SAID δ (deg) 2 δ (deg) 6 4 δ (deg) E lab (MeV) 2 3 P E lab (MeV) -1 3 P1 Argonne V8 SAID E lab (MeV) 3 2 Argonne V8 SAID 3 P2 δ (deg) -2 Argonne V8 SAID δ (deg) δ (deg) E lab (MeV) -1 3 D1 Argonne V8 SAID E lab (MeV) Argonne V8 SAID 3 D E lab (MeV) 8 6 ε 1 δ (deg) -2 δ (deg) 3 δ (deg) Argonne V8 SAID E lab (MeV) E lab (MeV) E lab (MeV) Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 7 / 35

8 Scattering data and neutron matter Two neutrons have k E lab m/2, k F that correspond to k F ρ (E lab m/2) 3/2 /2π 2. E lab =15 MeV corresponds to about.12 fm 3. E lab =35 MeV to.44 fm 3. Argonne potentials useful to study dense matter above ρ =.16 fm 3 Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 8 / 35

9 Three-body forces Urbana Illinois V ijk models processes like π π π π π π π π π π + short-range correlations (spin/isospin independent). Urbana UIX: Fujita-Miyazawa plus short-range. Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 9 / 35

10 Quantum Monte Carlo H ψ( r 1... r N ) = E ψ( r 1... r N ) ψ(t) = e (H E T )t ψ() Ground-state extracted in the limit of t. Propagation performed by ψ(r, t) = R ψ(t) = dr G(R, R, t)ψ(r, ) Importance sampling: G(R, R, t) G(R, R, t) Ψ I (R )/Ψ I (R) Constrained-path approximation to control the sign problem. Unconstrained calculation possible in several cases (exact). Ground state obtained in a non-perturbative way. Systematic uncertainties within 1-2 %. Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 1 / 35

11 Overview Recall: propagation in imaginary-time e (T +V ) τ ψ e T τ e V τ ψ Kinetic energy is sampled as a diffusion of particles: e 2 τ ψ(r) = e (R R ) 2 /2 τ ψ(r) = ψ(r ) The (scalar) potential gives the weight of the configuration: Algorithm for each time-step: do the diffusion: R = R + ξ compute the weight w e V (R) τ ψ(r) = wψ(r) compute observables using the configuration R weighted using w over a trial wave function ψ T. For spin-dependent potentials things are much worse! Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 11 / 35

12 GFMC and AFDMC Because the Hamiltonian is state dependent, all spin/isospin states of nucleons must be included in the wave-function. Example: spin for 3 neutrons (radial parts also needed in real life): GFMC wave-function: a a a a ψ = a a a a A correlation like 1 + f (r)σ 1 σ 2 can be used, and the variational wave function can be very good. Any operator accurately computed. AFDMC wave-function: [ ( ) ( ) ( )] a1 a2 a3 ψ = A ξ s1 ξ b s2 ξ 1 b s3 2 b 3 We must change the propagator by using the Hubbard-Stratonovich transformation: e 1 2 to2 = 1 dxe x2 2 +x to 2π Auxiliary fields x must also be sampled. The wave-function is pretty bad, but we can simulate larger systems (up to A 1). Operators (except the energy) are very hard to be computed, but in some case there is some trick! Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 12 / 35

13 Light nuclei spectrum computed with GFMC Energy (MeV) He 6 He Li Li 7/2 5/2 5/2 7/2 1/2 3/ He Argonne v 18 with Illinois-7 GFMC Calculations 8 Li AV18 8 Be AV18 +IL7 9 Li 5/2 1/2 3/2 Expt. 9 Be Carlson, et al., arxiv: /2 + 5/2 + 7/2 7/2 3/2 3/2 + 5/2 + 1/2 5/2 1/2 + 3/2 1 Be , B C Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 13 / 35

14 Charge form factor of 12 C F (q) = ψ ρ q ψ ρ q = i ρ q (i) + i<j ρ q (ij) F(q) ρ ch (r) r (fm) 1-3 exp ρ 1b ρ 1b+2b q (fm -1 ) Lovato, Gandolfi, Butler, Carlson, Lusk, Pieper, Schiavilla, PRL (213) Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 14 / 35

15 Neutron matter equation of state Why to study neutron matter at nuclear densities? EOS of neutron matter gives the symmetry energy and its slope. The three-neutron force (T = 3/2) very weak in light nuclei, while T = 1/2 is the dominant part. No direct T = 3/2 experiments available. Why to study symmetry energy? Theory Esym, L Neutron stars Experiments Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 15 / 35

16 What is the Symmetry energy? symmetric nuclear matter pure neutron matter Symmetry energy E = -16 MeV ρ =.16 fm -3 Nuclear saturation Assumption from experiments: E SNM (ρ ) = 16MeV, ρ =.16fm 3, E sym = E PNM (ρ ) + 16 At ρ we access E sym by studying PNM. Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 16 / 35

17 Neutron matter We consider different forms of three-neutron interaction by only requiring a particular value of E sym at saturation. 12 Energy per Neutron (MeV) E sym = 33.7 MeV different 3N different 3N: V 2π + αv R V 2π + αv µ R (several µ) V 2π + αṽr V 3π + αv R Neutron Density (fm -3 ) Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 17 / 35

18 Neutron matter Equation of state of neutron matter using Argonne forces: Energy per Neutron (MeV) E sym = 35.1 MeV (AV8 +UIX) E sym = 33.7 MeV E sym = 32 MeV E sym = 3.5 MeV (AV8 ) Neutron Density (fm -3 ) Gandolfi, Carlson, Reddy, PRC (212) Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 18 / 35

19 Neutron matter and symmetry energy From the EOS, we can fit the symmetry energy around ρ using E sym (ρ) = E sym + L 3 ρ AV8 +UIX L (MeV) E sym =32. MeV E sym =33.7 MeV 35 AV E sym (MeV) Gandolfi et al., EPJ (214) Tsang et al., PRC (212) Very weak dependence to the model of 3N force for a given E sym. Chiral Hamiltonians give compatible results. Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 19 / 35

20 Neutron matter and neutron star structure TOV equations: dp dr = G[m(r) + 4πr 3 P/c 2 ][ɛ + P/c 2 ] r[r 2Gm(r)/c 2, ] dm(r) dr = 4πɛr 2, J. Lattimer Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 2 / 35

21 Neutron star structure EOS used to solve the TOV equations. M (M O ) Causality: R>2.9 (GM/c 2 ) 32 error associated with E sym ρ central =3ρ E sym = 3.5 MeV (NN) ρ central =2ρ (4) M O 1.4 M O R (km) Gandolfi, Carlson, Reddy, PRC (212). Accurate measurement of E sym put a constraint to the radius of neutron stars, OR observation of M and R would constrain E sym! Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 21 / 35

22 Neutron stars Observations of the mass-radius relation are becoming available: 2.5 4U EXO U M (M ) M (M ) M (M ) R (km) R (km) R (km) M (M ) M M (M ) ω Cen M (M ) X R (km) R (km) R (km) Steiner, Lattimer, Brown, ApJ (21) Neutron star observations can be used to measure the EOS and constrain E sym and L. Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 22 / 35

23 Neutron star matter Neutron star matter model: ( ) α ( ) β ρ ρ E NSM = a + b, ρ < ρ t ρ ρ (form suggested by QMC simulations), and a high density model for ρ > ρ t i) two polytropes ii) polytrope+quark matter model Neutron star radius sensitive to the EOS at nuclear densities! Direct way to extract E sym and L from neutron stars observations: E sym = a + b + 16, L = 3(aα + bβ) Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 23 / 35

24 Neutron star matter really matters! Here an astrophysical measurement Probability distribution Quarks, no corr. unc. 8 Quarks No corr. unc. 6 Fiducial Pb (p,p) 4 Masses HIC 2 PDR IAS L (MeV) 1 M (M solar ) E sym = 33.7 MeV E sym =32 MeV Polytropes Quark matter R (km) 32 < E sym < 34 MeV, 43 < L < 52 MeV Steiner, Gandolfi, PRL (212). Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 24 / 35

25 High density neutron matter If chemical potential large enough (ρ 2 3ρ ), heavier particles form, i.e. Λ, Σ,... Non-relativistic BHF calculations suggest that none of the available hyperon-nucleon Hamiltonians support an EOS with M > 2M : Schulze and Rijken PRC (211). (Some) other relativistic model support 2M neutron stars. Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 25 / 35

26 Λ-systems, outline Hypernuclei and hypermatter: 2 H = H N + 2m Λ A i=1 2 i + i<j v ΛN ij + i<j<k V ΛNN ijk Λ-binding energy calculated as the difference between the system with and without Λ. Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 26 / 35

27 Λ-nucleon interaction The Λ-nucleon interaction is constructed similarly to the Argonne potentials (Usmani). Argonne NN: v ij = p v p(r ij )O p ij, O ij = (1, σ i σ j, S ij, L ij S ij ) (1, τ i τ j ) Usmani ΛN: v ij = p v p(r ij )O p ij, O λj = (1, σ λ σ j ) (1, τ z j ) N N K, K N N But NN data, 5 of ΛN data. Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 27 / 35

28 ΛN and ΛNN interactions ΛNN has the same range of ΛN N N N N N N Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 28 / 35

29 Hypernuclei AFDMC for (hyper)nuclei is limited to simple interactions. We found that the Λ-binding energy is quite independent to the details of NN interactions. The inclusion of (simple) three-body forces gives very similar results (unpublished). Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 29 / 35

30 Λ hypernuclei v ΛN and V ΛNN (I) are phenomenological (Usmani). B [MeV] NN (I) N A 1. NN (II) A -2/3 Lonardoni, Pederiva, SG, PRC (213) and PRC (214). V ΛNN (II) is a new form where the parameters have been fine tuned. As expected, the role of ΛNN is crucial for saturation. Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 3 / 35

31 Λ hypernuclei Λ in different states: B Λ [MeV] d f g 4 p 28 s emulsion (K -,π - ) (π +,K + ) (e,e K + ) AFDMC A -2/3 Lonardoni, SG, Pederiva, in preparation. Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 31 / 35

32 Hyper-neutron matter Neutrons and Λ particles: E HNM (ρ, x) = ρ = ρ n + ρ Λ, x = ρ Λ ρ ] ] [E PNM ((1 x)ρ)+m n (1 x)+ [E PΛM (xρ)+m Λ x +f (ρ, x) where E PΛM is the non-interacting energy (no v ΛΛ interaction), and ( ) α ( ) β ρ ρ E PNM (ρ) = a + b ρ ρ f (ρ, x) = c 1 x(1 x)ρ ρ + c 2 x(1 x) 2 ρ 2 ρ 2 All the parameters are fit to AFDMC results. Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 32 / 35

33 Λ-neutron matter EOS obtained by solving for µ Λ (ρ, x) = µ n (ρ, x) E [MeV] n [fm -3 ] particle fraction PNM N + NN (I) N [fm -3 ] Lonardoni, Lovato, Pederiva, SG, arxiv: No hyperons up to ρ =.5 fm 3 using ΛNN (II)!!! Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 33 / 35

34 Λ-neutron matter PNM 2. PSR J M [M ] N + NN (II) N + NN (I) PSR J N R [km] Lonardoni, Lovato, Pederiva, SG, arxiv: Drastic role played by ΛNN. Calculations can be compatible with neutron star observations. Note: no v ΛΛ, no protons, and no other hyperons included Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 34 / 35

35 Summary QMC methods useful to study nuclear systems in a coherent framework: Three-neutron force is the bridge between E sym and neutron star structure. Neutron star observations becoming competitive with experiments. Λ-nucleon data very limited, but ΛNN is very important. Role of Λ in neutron stars far to be understood. More ΛN data needed. Input from Lattice QCD? Conclusion? We cannot conclude anything with present models... Acknowledgments J. Carlson (LANL) D. Lonardoni, A. Lovato (ANL) F. Pederiva (Trento) A. Steiner (UT) Stefano Gandolfi (LANL) Effect of Λ in neutron matter and the neutron star structure 35 / 35