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) 54th International Winter Meeting on Nuclear Physics Bormio, Italy, January 25-29, Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 1 / 27

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) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 2 / 27

3 Nuclei and hypernuclei Few thousands of binding energies for normal nuclei are known. Only few tens for hypernuclei. Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 3 / 27

4 Homogeneous neutron matter Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 4 / 27

5 Outline The model and the method Equation of state of neutron matter Neutron star structure (I) - radius Λ-hypernuclei and Λ-neutron matter Neutron star structure (II) - maximum mass Conclusions Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 5 / 27

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 (Argonne AV8 ) 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 ) Urbana Illinois V ijk models processes like π π π π π π π π π π + short-range correlations (spin/isospin independent). Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 6 / 27

7 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) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 7 / 27

8 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 7/2 + 5/2 + 7/2 7/2 3/2 3/2 + 5/2 + 1/2 5/2 1/2 + 3/2 1 Be , B C Carlson, Gandolfi, et al., Rev. Mod. Phys. 87, 167 (215) Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 8 / 27

9 Neutron matter equation of state Neutron matter is an exotic system. Why do we care? 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. Determines properties of neutron stars. Theory Esym, L Neutron stars Experiments Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 9 / 27

10 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) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 1 / 27

11 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) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 11 / 27

12 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) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 12 / 27

13 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. Knowing E sym or L useful to constrain 3N! (within this model...) Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 13 / 27

14 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) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 14 / 27

15 Neutron star matter Neutron star radii sensitive to EOS around ρ (1 2)ρ Maximum mass depends to higher densities Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 15 / 27

16 Neutron star structure 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). Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 16 / 27

17 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 constrain the EOS, E sym and L. (Systematic uncertainties still under debate...) Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 17 / 27

18 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) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 18 / 27

19 High density neutron matter If chemical potential large enough (ρ 2 3ρ ), nucleons produce Λ, Σ,... Non-relativistic BHF calculations suggest that available hyperon-nucleon Hamiltonians support an EOS with M > 2M : Schulze and Rijken PRC (211). Vidana, Logoteta, Providencia, Polls, Bombaci EPL (211). Note: (Some) other relativistic model support 2M neutron stars. Hyperon puzzle Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 19 / 27

20 Λ-systems, outline Λ-hypernuclei and hypermatter H = H N + 2 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) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 2 / 27

21 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 Unfortunately NN data, 3 of ΛN data. Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 21 / 27

22 ΛN and ΛNN interactions ΛNN has the same range of ΛN N N N N N N Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 22 / 27

23 Λ hypernuclei B [MeV] NN (I) N A 1. NN (II) A -2/3 Lonardoni, Gandolfi, Pederiva, PRC (213) and PRC (214). V ΛNN (II) is a new form where the parameters have been readjusted. ΛNN crucial for saturation. see Lonardoni Thursday afternoon Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 23 / 27

24 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 Quantum Monte Carlo results. Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 24 / 27

25 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, Gandolfi, Pederiva, PRL (215) No hyperons up to ρ =.5 fm 3 using ΛNN (II)!!! Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 25 / 27

26 Λ-neutron matter PNM 2. PSR J M [M ] N + NN (II) N + NN (I) PSR J N R [km] Lonardoni, Lovato, Gandolfi, Pederiva, PRL (215) Drastic role played by ΛNN. Calculations can be compatible with neutron star observations. Note: no v ΛΛ, no protons, and no other hyperons included yet... Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 26 / 27

27 Summary EOS of pure neutron matter qualitatively well understood. Λ-nucleon data very limited, but ΛNN seems very important. Role of Λ in neutron stars far to be understood. We cannot conclude anything for neutron stars with present models... We cannot solve the puzzle, too many pieces are missing! My wishes: Accurate and precise measurement of E sym and L. More ΛN experimental data needed. Input from Lattice QCD? Light and medium Λ-nuclei measurements needed, especially N Z. Acknowledgments J. Carlson, D. Lonardoni (LANL) A. Lovato (ANL) F. Pederiva (Trento) S. Reddy (INT) A. Steiner (UT/ORNL) Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 27 / 27

28 Extra slides Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 1 / 21

29 Nuclear Hamiltonian Chiral interactions permit to understand the evolution of theoretical uncertainties with the increasing of A. Slide by Joel Lynn, Scidac NUCLEI meeting 214. Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 2 / 21

30 Neutron matter Equation of state of neutron matter using NN chiral forces: 15 AFDMC LO AFDMC NLO AFDMC N 2 LO R =1. fm E/N [MeV] 1 5 R =1.2 fm n [fm -3 ] Gezerlis, Tews, et al., PRL (213), PRC (214) Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 3 / 21

31 Chiral three-body forces For a finite cutoff, there are additional V D and V E diagrams that contribute in pure neutron matter. They have been often neglected in existing neutron matter calculations! All the above terms have been written in coordinate space, and included into AFDMC. Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 4 / 21

32 Neutron matter with chiral forces Preliminary! Contribution of additional V D and V E terms: Energy per neutron π-exchange 2π-exchange + V D + V E NN ρ=.16 fm -3 NN R =1. fm NN R =1.2 fm R (fm) of three-body force Note: Contribution of FM (2π exchange) about.9 MeV with AV8 Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 5 / 21

33 Neutron matter with chiral forces Preliminary! Equation of state of neutron matter at N 2 LO. energy per neutron (MeV) NN + V3 (R 3N =1.2 fm) NN + V3 (R 3N =1. fm) NN + V3 (R 3N =.8 fm) NN NN (R=1. fm) NN (R=1.2 fm) ρ (fm -3 ) Note: the real V D and V E terms are not included yet. Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 6 / 21

34 Neutron matter with chiral forces Preliminary! Equation of state of neutron matter, a comparison. energy per neutron (MeV) NN + V3 (R 3N =1.2 fm) NN + V3 (R 3N =1. fm) NN + V3 (R 3N =.8 fm) NN AV8 AV8 +UIX NN (R=1. fm) NN (R=1.2 fm) ρ (fm -3 ) Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 7 / 21

35 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 and AV6. Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 8 / 21

36 Variational wave function E E = ψ H ψ ψ ψ = dr1... dr N ψ (r 1... r N )Hψ (r 1... r N ) dr1... dr N ψ (r 1... r N )ψ (r 1... r N ) Monte Carlo integration. Variational wave function: Ψ T = f c (r ij ) f c (r ijk ) 1 + u ijk f p (r ij )O p ij Φ i<j i<j<k i<j,p k where O p are spin/isospin operators, f c, u ijk and f p are obtained by minimizing the energy. About 3 parameters to optimize. Φ is a mean-field component, usually HF. Sum of many Slater determinants needed for open-shell configurations. BCS correlations can be included using a Pfaffian. Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 9 / 21

37 Quantum Monte Carlo Projection in imaginary-time t: 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, ) dr dr 1 dr 2..., G(R, R, t) G(R 1, R 2, δt) G(R 2, R 3, δt)... Importance sampling: G(R, R, δt) G(R, R, δt) 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) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 1 / 21

38 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) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 11 / 21

39 Branching The configuration weight w is efficiently sampled using the branching technique: Configurations are replicated or destroyed with probability int[w + ξ] (1) Note: the re-balancing is the bottleneck limiting the parallel efficiency. Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 12 / 21

40 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) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 13 / 21

41 Propagator We first rewrite the potential as: V = i<j [v σ (r ij ) σ i σ j + v t (r ij )(3 σ i ˆr ij σ j ˆr ij σ i σ j )] = 3N = i,j σ iα A iα;jβ σ jβ = 1 2 n=1 O 2 nλ n where the new operators are O n = jβ σ jβ ψ n,jβ Now we can use the HS transformation to do the propagation: 1 n λo2 n ψ = dxe x2 2 + λ τxo n ψ 2π e τ 1 2 Computational cost (3N) 3. n Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 14 / 21

42 Three-body forces Three-body forces, Urbana, Illinois, and local chiral N 2 LO can be exactly included in the case of neutrons. For example: O 2π = cyc [ {X ij, X jk } {τ i τ j, τ j τ k } + 1 ] 4 [X ij, X jk ] [τ i τ j, τ j τ k ] = 2 cyc {X ij, X jk } = σ i σ k f (r i, r j, r k ) The above form can be included in the AFDMC propagator. Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 15 / 21

43 The Sign problem in one slide Evolution in imaginary-time: ψ I (R )Ψ(R, t + dt) = dr G(R, R, dt) ψ I (R ) ψ I (R) ψ I (R)Ψ(R, t) note: Ψ(R, t) must be positive to be Monte Carlo meaningful. Fixed-node approximation: solve the problem in a restricted space where Ψ > (Bosonic problem) upperbound. If Ψ is complex: ψ I (R ) Ψ(R, t + dt) = dr G(R, R, dt) ψ I (R ) ψ I (R) ψ I (R) Ψ(R, t) Constrained-path approximation: project the wave-function to the real axis. Multiply the weight term by cos θ (phase of Ψ(R ) Ψ(R) ), Re{Ψ} > not necessarily an upperbound. Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 16 / 21

44 Unconstrained-path After some equilibration within constrained-path, release the constraint: 16 O, AV6 +Coulomb 16 O, AV7 +Coulomb E (MeV) E (MeV) τ (MeV -1 ) τ (MeV -1 ) The difference between CP and UP results is mainly due to the presence of LS terms in the Hamiltonian. Same for heavier systems. Work in progress to improve Ψ and to fully include three-body forces. Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 17 / 21

45 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) Difference AV8 -AV18 less than.2 MeV per nucleon up to A=12. Stefano Gandolfi (LANL) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 18 / 21

46 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) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 19 / 21

47 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) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 2 / 21

48 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) - stefano@lanl.gov Effect of Λ in neutron matter and the neutron star structure 21 / 21