The influence of the symmetry energy on the structure of hyperon - PowerPoint PPT Presentation

The influence of the symmetry energy on the structure of hyperon stars. I. Bednarek (Katowice) Matter To The Deepest Recent Developments In Physics Of Fundamental Interactions XXXIX International Conference of Theoretical Physics Ustron 2015 1 / 24

Study of nuclear matter at densities above saturation density Motivation: better understanding of the physics of neutron stars examining the possibility of the existence of strange baryons in the very inner part of a neutron star 2 / 24

Modelling neutron star structure and composition Schematic structure of a neutron star atmosphere outer crust - lattice of neutron-rich heavy nuclei, degenerate, relativistic P. Haensel et al. 2007 electrons - correction to radius ∼ 10 percent Outer core - n , p , e , µ matter inner crust - as above plus under β equilibrium degenerate non-relativistic neutrons ε = ε N ( n n , n p ) + ε l ( e , µ ) outer core - homogeneous Equilibrium conditions: nucleonic matter 1 µ n = µ p + µ e inner core - may contain exotic forms of matter 2 µ µ = µ e 3 / 24

Modelling neutron star structure and composition Equilibrium conditions - Threshold chemical potentials contribution of hyperons to β of hyperons equilibrium. 1 µ Ξ − = µ Σ − = µ n + µ e 2 µ Ξ 0 = µ Σ 0 = µ Λ = µ n 3 µ Σ + = µ p = µ n − µ e hyperon onset points - hyperon threshold densities n Y n Y → 0 = ∂ε | eq = µ 0 lim Y ∂ n b P. Haensel et al. 2007 Apppearance of hyperons - at For n b > n th Y hyperon Y become 2 − 3n 0 stable in dense matter. 4 / 24

Modelling neutron star structure and composition - Tolman-Oppenheimer-Volkoff equation M − R relations mc 2 + 4 π r 3 P details about the internal � � ( E + P ) d P − G = structure of a neutron star dr c 4 r ( r − 2Gm / c 2 ) provides data on the dm 4 π r 2 E = impact of a given model on c 2 dr the internal structure of a neutron star Solution of the TOV equations needs supplementation by the equation of state (EoS) of the matter of a neutron star P ( E ( n B )) Lattimer et al. 2013 5 / 24

Measured neutron star masses. There are no precise simultaneous measurements of neutron star mass and radius. Constraints on the mass-radius relation radius - not strong enough mass 1 PSR J1614-2230, NS-WD binary system, M NS = 1 . 97 ± 0 . 4M ⊙ , M WD = 0 . 5M ⊙ P.Demorest et al. 2010 2 PSR J0348+0432, NS-WD binary system, M NS = 2 . 01 ± 0 . 4M ⊙ , M WD = 0 . 172M ⊙ Antoniadis et al. 2013 6 / 24 Lattimer et al. 2013

Hyperon puzzle. M max ≥ M measured ⇒ M max ≥ 2M ⊙ Massive neutron stars - strong constraint on the equation of state - requires stiff equation of state Hyperons soften the equation of state significantly. 200 TM1-npe 180 TM1-weak 160 TM1-ext 140 P (MeV fm -3 ) 120 100 80 60 40 20 0 0 100 200 300 400 500 600 700 ε (MeV fm -3 ) 7 / 24

Equation of state of isospin asymmetric nuclear matter - two component system of N nucleons The energy differences of the states with different composition of protons and neutrons are encoded in the symmetry energy. E sym ( N p , N n ) ≡ E ( N p , N n ) − E ( N p = N / 2 , N n = N / 2 ) δ a = N n − N p = 1 − 2Y p N B E sym ( N , δ a ) ≡ E ( N , δ a ) − E ( N , δ a = 0 ) 1 symmetric nuclear matter (SNM) δ a = 0 ⇒ N n = N p 2 pure neutron matter (PNM) δ a = 1 ⇒ N p = 0 E sym ( n B ) = E ( n B , δ a = 1 ) − E ( n B , δ a = 0 ) E ( n 0 , δ a = 1 ) = E sym ( n 0 ) + E ( n B , δ a = 0 ) 8 / 24

Using the expansion E ( n B , y p ) = E ( n B , y p = 1 / 2 ) + ( 1 − 2y p ) 2 S 2 ( n B ) + . . . S 2 ( n B ) = S v + L n B − n 0 + . . . 3 n 0 S v ≃ 31 MeV, L ≃ 50 MeV 9 / 24

Symmetry energy- connections to neutron star parameters Proton fraction µ p − µ n = ∂ E Tot = 4E sym ( n B )( 1 − 2Y p ) ∂ Y p E Tot = E + E e at saturation n B = n 0 � 3 1 � 4S v Y p ≈ ≈ 0 . 04 3 π 2 n 0 ℏ c Pressure at saturation density � 3 4 − 3S v / L � � p β ( n 0 ) = ≃ L � 4S v 3 n 0 1 − + . . . 3 π 2 n 0 ℏ c 10 / 24

Symmetry energy- connections to neutron star parameters Pressure- radius correlations R = C ( n B , M )( p β / MeVfm − 3 ) 1 / 4 Coefficients C ( n B , 1 . 4M ⊙ ) M ∗ / M ⊙ n 0 1.5n 0 2n 0 1.3 9.30 ± 0.58 6.99 ± 0.30 5.72 ± 0.25 2.0 9.52 ± 0.49 7.06 ± 0.24 5.68 ± 0.14 Coefficients appropriate for n B = n 0 - C ( n 0 , 1 . 4M ⊙ ) Crust-core transition density and pressure Crust thickness 11 / 24

Theoretical predictions for symmetry energy Theoretical considerations predict wide range of symmetry energies for densities below and above saturation density n 0 = 0 . 16fm − 3 . Density dependence of the symmetry energy predicted by various theoretical calculations. (Shetty, 2010) 12 / 24

Nuclear matter with strangeness degrees of freedom - system of nucleons and hyperons Modification of the symmetry energy by the presence of hyperons. E H sym ( n B , δ a , y i ) = E ( n B , δ a , y i ) − E ( n B , δ a = 0 , y i ) In this case: n B = n N + y and y = � i y i - total hperon number density Pure neutron matter − → y = 0 13 / 24

Experimental constraints for symmetry energy parameters. Constraint for the centroid energy of the giant dipole resonance for 208 Pb - S 2 ( 0 . 1 ) ≃ ( 24 . 1 ± 0 . 9 ) MeV Consensus agreement of the six experimental constraints 44 MeV < L < 66 MeV Results of neutron matter studies - direct estimates of S v and L consistent with the results determined from nuclear experiments 14 / 24

Astrophysical considerations Measurements of neutron star radii Estimation of neutron star radii - distant measurement and atmospheric modelling required. Photospheric Radius Expansion Bursts Accreation from the companion (MS star) - overflowing the Roche lobe Unstable burning of the accreated material Spread of the nuclear burning accros stellar surface - sudden increase in X-ray luminosity and temperature X-ray bursts The average neutron star mass and radius implied by these results: ¯ M = 1 . 65 ± 0 . 12M ⊙ , ¯ R = 10 . 77 ± 0 . 65. 15 / 24

Quiescent LMXBs in globular clusters - candidate II QLMXBs Neutron stars in binary system with intermittently accreated matter from evolving companion star. Episodes of accretion separated by long periods of quiescence. Low magnetic field Compression of matter in the crust induces nuclear reactions Sufficient amount of heat is released to warm the star Neutron stars cool via neutrino radiation from their interiors and X-ray from their surfaces The emitted X-ray spectra (for a given composition) depend on: R, T eff , g = GM ( 1 + z ) / R 2 (observed spectra - D and N H ) J.Lattimer, 2014 16 / 24

The model � L = L B + L M + L NL + L L B ψ B ( γ µ iD µ − M ⋆ L B = ¯ B ) ψ B M ⋆ B = M − g B σ σ − g B σ ⋆ σ ⋆ D µ = ∂ µ + ig B ω ω µ + ig B φ φ µ + ig B ρ I B ρ µ L NL = − 1 3g 3 σ 3 − 1 4g 4 σ 4 + � C ijk ω i µ ρ j µ φ k µ i , j , k Constituents of the model Coupling constants baryons: B ∈ vector meson-hyperon - { n , p , Λ , Σ + , Σ 0 , Σ − , Ξ 0 , Ξ − } SU(6) symmetry leptons: L ∈ { e − , µ − } scalar meson-hyperon - mezons: M ∈ { σ, ω µ , ρ a µ } hypernuclear potential in ∪{ σ ∗ , ϕ µ } nuclear matter 17 / 24

The Walecka-type models Very ”stiff” form of the symmetry energy. To provide additional freedom in varying the density dependence of the symmetry energy the model is supplemented by the term: Λ V ( g ω ω ) 2 ( g ρ ρ ) 2 The density dependence of the symmetry energy k 2 k 3 F F E sym ( n B ) = + 12 ( m 2 ρ / g 2 ρ + 2 Λ V ( g ω ω ) 2 ) � ( k 2 F + M 2 6 eff ) for Λ V = 0 the symmetry energy varies linearly with the density. TM1 nonlinear (isovector sector) Λ V 0 0.014 0.015 0.016 0.0165 g ρ 9.264 9.872 9.937 10.003 10.037 L (MeV) 108.58 77.52 75.81 74.16 73.36 18 / 24

Density dependence of symmetry energy 250 Λ v = 0 Λ v = 0.0165 Λ v = 0.03 200 AV14+VII UV14+VII E sym ( MeV ) UV14+TNI 150 100 50 0 0.2 0.4 0.6 0.8 1 1.2 1.4 n b ( fm -3 ) Calculations performed for different values of parameter Λ V and compared with the results obtained for the AV14+VII, UV14+VII and UV14+TNI models.(R.B.Wiringa,1988) The inclusion of ω − ρ coupling softens the symmetry energy. 19 / 24

Modification of the symmetry energy for nuclear matter with hyperons. 36 EXT f S =0.4 34 WEAK f S =0.4 32 EXT f S =0.2 WEAK f S =0.2 30 E sym ( δ a =0.5) [MeV] f S =0 28 26 24 22 20 18 16 0.4 0.5 0.6 0.7 0.8 0.9 n B [fm -3 ] 20 / 24

Modification of neutron star parameters Equations of state Mass-radius relations Results obtained for Neutron star matter with non-strange and and hyperons - the maximum mass strangeness-rich matter for range: different parameterizations. 1 . 86 − 2 . 03M ⊙ 300 TM1 Λ v =0.014 2 TM1 Λ v =0.015 250 TM1 Λ v =0.016 TM1 Λ v =0.0165 200 TM1 Λ v =0.017 1.5 TM1 npe µ 3 ) P (MeV/fm TM1 weak M ( M Θ ) NL3 npe µ 150 TM1 npe µ FSUGold 1 TM1 Λ v =0.014 TM1 Λ v =0.015 100 TM1 Λ v =0.016 TM1 Λ v =0.0165 0.5 TM1 Λ v =0.017 50 FSUGold 0 0 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 10 11 12 13 14 15 3 ) ε (MeV/fm R ( km ) 21 / 24