Hybrid stars within a SU(3) chiral Quark Meson Model

Transcription

1 Hybrid stars within a SU(3) chiral Quark Meson Model Andreas Zacchi 1 Matthias Hanauske 1,2 Jürgen Schaffner-Bielich 1 1 Institute for Theoretical Physics Goethe University Frankfurt 2 FIAS Frankfurt Institute for Advanced Sciences Astro coffee, May

2 Abstract 1 Ultradense Matter in Compact Objects (2 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

3 Death of a star If the nuclear fuel is exhausted any stars life will end differently Stars with M 8M reject outer layers in a planetary nebula White dwarf Stars with M 8M end in a Supernova explosion Compact star Stars with M 2M end in a Black hole (3 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

4 Supernova remnants: Compact stars Compact stars are either: neutron stars (consisting of mainly neutrons) hybrid stars (consisting of a hadronic shell and a quark-matter core) quark stars (consisting of quark matter) Size: R 1 15km Mass: M 1.5M Density: ρ g 145 MeV cm 3 fm 3 Credit: Chandra X-ray observatory (NASA) (4 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

5 New pulsar mass measurements Recent measurements 1 Demorest et. al; 21 PSR J with M = 1.97 ±.4M 2 Antoniadis et. al; 213 PSR J with M = 2.1 ±.4M set new constraints on thermodynamic quantities. Until 21 the Hulse Taylor Pulsar with M = ±.35M was the heaviest. Credit: Science Magazine (5 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

6 The stars interior What do certain models predict for the star to consist of? 1...just neutrons? 2...hyperons? 3...kaon condensates? 4...quark matter (QM)? 5...color superconducting QM? Credit: Fridolin Weber (6 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

7 Micro: Calculation of an equation of state (EoS) 1 Polytropic EoS: Fermi Gas - no interactions considered p(ɛ) = Kɛ γ where K = const. and e.g. γ = 4 3 ; see: Compact stars for undergraduates, Sagert et. al 25 2 Nuclear matter EoS: respects n-n interactions formation of clusters density dependent see: Composition and thermodynamics of nuclear matter with light clusters, Typel et. al 21 3 Quark matter EoS Incorporation of QCD properties Exchange of scalar- and vector mesons as mediators of strong interaction Modelling confinement via vacuum energy density term B see: Compact stars in a SU(3) Quark Meson Model, Zacchi et. al 215 (7 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

8 Macro: How to compute compact stars The Tolman-Oppenheimer-Volkoff equations (TOV) are: general relativistic equations to determine the mass-radius relations of compact stars Input is an Equation of State (EoS): p(ɛ) where ɛ(r) dm dr dp dr = 4πr 2 ɛ(r) = Gɛ(r)m(r) r 2 ( 1 + p(r) ) ( 1 + 4πr 3 ) ( p(r) 1 2Gm(r) ) 1 ɛ(r) m(r) r (8 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

9 TOV equations: Boundary conditions 1 Start integration of the TOV equations at the center with boundary conditions m(r start = ) = 2 End integration of the TOV equations at the stars surface where the pressure vanishes m(r end ) = M star (9 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

10 8 From an EoS TOV - equations Mass-Radius Relations pressure p [MeV/fm 3 ] polytrope DD2 SU(3) M/M s un polytrope DD2 SU(3) energydensity ε [MeV/fm 3 ] R(km) Each EoS predicts a specific mass vs. radius line Quark stars: Selfbounded objects Neutron stars: Bounded by gravity (1 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

11 SU(3) Quark Meson Model Setting up a realistic Quark model Ultradense matter might be a phase of deconfined quarks Computation Compute the EoS and solve the TOV equations Combination Combine the QM EoS with a hadronic EoS (DD2) Hybrid- and maybe even Twin stars (11 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

12 The SU(3) Lagrangian Properties of hybrid stars depend on Quark Matter EoS derived by a Lagrangian density L = L Fn,s + L φ + L V L Fn,s = Ψ n ( i g ω γ ω g ρ τγ ρ g n σ n ) Ψn + Ψ s ( i g s σ s g φ γ φ ) Ψ s L φ = tr( µ φ) ( µ φ) λ 1 [tr(φ φ)] 2 λ 2 tr(φ φ) 2 ( ) mtr(φ 2 φ) tr[ĥ(φ + φ )] + c det(φ ) + det(φ) L V = tr( µ V ) ( µ V ) m 2 v tr(v V ) (12 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

13 The Equation of State: Solve the Lagrangian With Z = Dσ a Dπ a a ( β D ΨDΨe dτ ) V d3 r(l+ Ψγ µψ) and p = lnz β = Ω ɛ = p + f =u,d,s µ f n f + Ts we have a relation for the necessary values. (13 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

14 EoS - Grandcanonical Potential Ω V = 1 2 Having performed the T approximation the resulting grandcanonical potential is Ω = V + 3 π 2 f =u,d,s k f F dk k 2 ( k 2 n,s + m 2 µ f ) where ( m 2 ω ω 2 + m 2 ρρ 2 + m 2 φ φ2) + λ 1 4 (σ2 n + σ 2 s ) 2 + λ 2 4 (σ4 n + σ 4 s ) + m2 2 (σ2 n + σ 2 s ) 2σ2 nσ s 2 c h n σ n h s σ s + B (14 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

15 The energy density and the pressure are then determined to ɛ = ɛ e + λ 1 4 (σ2 n + σ 2 s ) 2 + λ 2 4 (σ4 n + σ 4 s ) + m2 2 (σ2 n + σ 2 s ) and 2σ2 nσ s c h n σ n h s σ s + B k f ( F ) π 2 dk k 2 kn,s 2 + m 2 p = 1 2 f =u,d,s ( m 2 ω ω 2 + m 2 ρρ 2 + m 2 φ φ2) ( m 2 ω ω 2 + m 2 ρρ 2 + m 2 φ φ2) + λ 1 4 (σ2 n + σ 2 s ) 2 + λ 2 4 (σ4 n + σ 4 s ) + m2 2 (σ2 n + σs 2 ) 2σ2 nσ s c h n σ n h s σ s + B k f ( F ) π 2 dk k 2 kn,s 2 + m 2 µ f f =u,d,s (15 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

16 Lagrangian parameters The vacuum expectation values σ n = f π = 92.4MeV σ s = 2f K f π 2 = 94.47MeV Scalar and vector couplings g n = m q f π with m q = 3 MeV g s = 2g n g ω = g ρ = g φ 2 (16 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

17 Lagrangian parameters Spontaneous breaking controlled by λ 1 via m 2, m2 V and m σ λ 2 from meson masses (m π,m K,...) and decay constants (PDG) Such as c, which describes axial anomaly (m η ) Explicit symmetry breakers h n = f π m 2 π h s = 2f K m 2 K h n 2 (17 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

18 Parameter space Constituent quark mass m q g n = mq f π and g s = g n 2, where gs is adopted from SU(3) symmetry considerations. Vector coupling g ω g n The φ-meson coupling is also fixed by SU(3) symmetry Mass of the σ-meson m σ covers a range from 4 MeV m σ 8 MeV The Bag constant B B models the confinement; 6 MeV B 2 MeV. (18 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

19 Pure SU(3) quark stars within a tiny parameter space Contour plot of vacuum pressure B vs. Sigma meson mass 1 Above the 2-flavour-line: Iron, i.e. hadronic matter more stable 2 Below the 3-flavour-line: Pure quark matter more stable Pure quark-stars with M 2 possible within a narrow area! B 1/4 12 [MeV] ,748 2,3 2,313 2,595 2-flavor-line 1,183 1,465 1, flavor-line 2, ,878 2, m [MeV] The other papameters: m q = 3 and g ω = 2.,9 1,465 2,3 2,595 3,16 (19 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

20 Hybrid stars: Maxwell construction Combining a nuclear matter EoS and a Quark matter EoS: Pressure p has to be dominant vs. chem. Potential µ 3 pressure p [MeV/fm 3 ] HM QM Hybrid chemical potential µ [MeV] (2 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

21 Results for g ω 3 From the intersecting point in the p µ plane the EoS changes from the HM EoS to the QM EoS p [MeV/fm 3 ] g w = g w =1 g w =2 g w =3 DD µ [MeV] ε [MeV/fm 3 ] p [MeV/fm 3 ] g w = g w =1 g w =2 g w =3 DD2 m q = 3 MeV, m σ = 6 MeV, B = 1 MeV (21 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

22 The corresponding Speed of Sound (SoS) p [MeV/fm 3 ] g w = g w =1 g w =2 g w =3 DD2 The density within the star changes abruptly ε [MeV/fm 3 ] m q = 3 MeV, m σ = 6 MeV, B = 1 MeV c s 2 =dp/dε 1 g w = g w =1 g.8 w =2 g w =3 DD g ω = g ω =1 g ω =2 g ω = ε [MeV/fm 3 ] p trans : transition pressure ɛ trans : transition energy density ɛ : Difference in energy density between the two EoS (22 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

23 Results for g ω 3 At a certain central pressure the star configurations get unstable R(km) g w = g w =1 g w =2 g w =3 DD2 Star Star M/M sun M/M sun g w = g w =1 g w =2 g w =3 DD2 Star Star p c (MeV/fm 3 ) R(km) m q = 3 MeV, m σ = 6 MeV, B = 1 MeV (23 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

24 Results for g ω 3 Particle composition of two individual stars: (hybrid 1.8M ) and (hadronic 1.4M ) Star Star.8 n n Star Star (continuous) (dotted) ε [MeV/fm 3 ] 6 4 particle fraction.6.4 d u 2.2 s p+e p+e r [km] r [km] m q = 3 MeV, m σ = 6 MeV, B = 1 MeV (24 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

25 Criterion for stable hybrid stars ɛ crit ɛ = p trans 2 ɛ trans ε/ε t ε/ε Twins Star Area 1 constraint constraint No QM core: Neutron Star Area.5 Hybrid Star Area p t /ε t Effect of QM core on the star is determined by ɛ Small ɛ: Quark matter has similar energydensity than nuclear matter Large ɛ: Star becomes unstable, since QM-core is unable to counteract gravitational attraction (25 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

26 Twin stars: Same mass - different radii Considering a perturbation causing the star to collapse: Possible scenarii: 1 Star collapses into a black hole 2 Star stabilizes again One EoS may yield two stable branches. (26 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

27 Twin stars: Hard to find ε/ε t ε/ε Twin Stars 1 constraint 8 < Bag < 19 Bag= 4MeV 6 < Bag < 18 Bag= 4MeV 4 < Bag < 2 Bag= 8MeV 5 < Bag < 2 Bag= 25MeV constraint ε/ε t ε/ε < g ω = < 3 g ω =.5 < g Bag=6MeV ω = < 3 g ω =.5 < g Bag=18MeV ω = < 3 g ω =.5 < g ω = < 3 Twin Stars g ω =.5 constraint Bag=14MeV constraint.5 g ω = g ω =1 g ω =2 g ω =3.5 Bag=1MeV p t /ε t p t /ε t m q = 3 MeV and m σ = 6 MeV (27 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

28 The influence of the speed of sound p [MeV/fm 3 ] Set A Set B Set C DD2 M/M sun Set A unstable Set A stable Set B unstable Set B stable Set C unstable Set C stable ε [MeV/fm 3 ] R(km) Twin Star Area scanned with SoS dependent EoS p(ɛ) = cs 2 (ɛ ɛ ), with: ɛ := ɛ t + ɛ 1 cs 2 p t, p t /ɛ t and ɛ/ɛ t under direct influence, c 2 s = 1 3 (28 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

29 The influence of the speed of sound p [MeV/fm 3 ] Set A Set B Set C DD2 M/M sun Set A unstable Set A stable Set B unstable Set B stable Set C unstable Set C stable ε [MeV/fm 3 ] R(km) Twin Star Area scanned with SoS dependent EoS p(ɛ) = cs 2 (ɛ ɛ ), with: ɛ := ɛ t + ɛ 1 cs 2 p t, p t /ɛ t and ɛ/ɛ t under direct influence, c 2 s = 1 3 (28 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

30 Twin stars with 2M hard to model ε/ε < g ω < 2 M < M M > M g ω =.5 M 1 < M 2 ε/ε t M 1 > M 2 Set A Set B Set C 8 < Bag < 19 Bag= 4MeV constraint Bag=14MeV.5 g ω = p t /ε t Star sequence p t /ɛ t ɛ/ɛ t M 1 R 1 M 2 R 2 Set A Set B Set C (29 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

31 Conclusions on Hybrids Hybrid stars 1 m q and m σ show little effect. 2 Large g ω and B: 2M but tiny QM core, star configuration soon unstable 3 Small g ω and B: less then 2M with acceptable QM core, hybrid star configuration remarkably stable 4 No direct influence on p t and ɛ t (3 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

32 Conclusions on Twins Twin stars 1 Twins within our model hard to achieve, swerve on speed of sound (SoS) independent EoS 2 Parametrization via SoS of the EoS leads to Twin stars: Large cs 2 advantageous for Twins 3 Chances are best for low transition pressure p t and large jump in energydensity ɛ The appearing QM core does not destabilize the star 4 Twin stars explain the two component structure of Short Radio Bursts (31 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

33 Neutron star merger, both stars with 1.4 M Animation by Filippo Galeazzi, Goethe University Frankfurt (32 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich

34 Summary 1 Combination of Hadronic and Quark Matter EoS solve the TOV equations 2 Maxwell construction to study hybrid stars 3 Do the Mass Radius relations exhibit 2M Twin Star solutions? 4 Generally: Twin Star solutions 2M hard to find Outlook 1 More properties - more sophisticated QM-EoS, e.g. inclusion of self enery loops? 2 Gibbs construction 3 Finite T calculation (Supernova EoS) 4 Cooling process via neutrino emission 5 Compact star merger 6 Gravitational wave emission (33 / 33) Hybrid stars within a SU(3) Quark Meson model A. Zacchi, M. Hanauske, J. Schaffner-Bielich