Compact Stars within a SU(3) chiral Quark Meson Model

Transcription

1 Compact Stars within a SU(3) chiral Quark Meson Model Andreas Zacchi Matthias Hanauske,3 Laura Tolos Jürgen Schaffner-Bielich Institute for Theoretical Physics Goethe University Frankfurt Institut de Ciencies de l Espai Bellaterra, Spain 3 FIAS Frankfurt Institute for Advanced Studies Astro coffee, January 7 7

2 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Abstract Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints 3 4 Compact Stars 5 ( / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

3 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Supernova remnants: Compact stars Compact stars are either: Neutron stars (consisting mainly of neutrons) Hybrid stars (consisting of a hadronic shell and a quark-matter core) Quark stars (consisting of quark matter only) Size: R 5km Mass: M.5M Density: ρ.5 4 g 45 MeV cm 3 fm 3 Credit: Chandra X-ray observatory (NASA) (3 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

4 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Different types of compact stars Characteristic mass radius relations of Neutron Stars (M R 3 ) Quark Stars (M R 3 ) 3 Hybrid Stars (Can masquerade as Neutron star) 4 Twin Stars (Two stable branches) M/M sun M/M sun Neutron Star sequence Hybrid Star sequences Quark Star sequence Twin Star sequences R[km] R[km] (4 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

5 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars The stars interior What do certain models predict for the star to consist of?...just neutrons?...hyperons? 3...kaon condensates? 4...quark matter (QM)? 5...color superconducting QM? Equation of state (EoS) Credit: Fridolin Weber (5 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

6 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Constraints on different solutions Schwarzschild Radius R > GM/c P < R > 9GM/4c 3 Causality R >.9GM/c 4 Star can not spin faster than 76Hz ν (.96 ±.3) ( M M nr ) / ( Rnr km ) 3/ 5 Solutions need to fulfill the M constraint next slide Credit: Lattimer and Prakash (6 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

7 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars New pulsar mass measurements Recent measurements Demorest et. al; PSR J64-3 with M =.97 ±.4M Fonseca et. al; 6 PSR J64-3 with M =.98 ±.7M 3 Antoniadis et. al; 3 PSR J with M =. ±.4M...also a constraint for quark matter based stars? Credit: Science Magazine (7 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

8 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Skalar and vector nonet Skalar and vector nonet as a base to constuct Lagrangian density σ n+a a + K s + ϕ = a K s σ n a K s Ks σ s V µ = ω µ n +ρ µ ρ µ+ K µ+ ρ µ ωµ n ρ µ K µ K µ K µ φ µ (8 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

9 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars SU(3) Lagrangian L = L Fn,s + L ϕ + L V L Fn,s = Ψ n,s ( i g ω γ ω g ρ τγ ρ g φ γ φ g n,s σ n,s ) Ψn,s L ϕ = tr( µ ϕ) ( µ ϕ) λ [tr(ϕ ϕ)] λ tr(ϕ ϕ) ( ) mtr(ϕ ϕ) tr[ĥ(ϕ + ϕ )] + c det(ϕ ) + det(ϕ) L V = tr( µ V ) ( µ V ) m v tr(v V ) couples quarks to mesons via Yukawa type coupling effective quark masses generated by the σ n and σ s -fields ω, ρ and φ: vector mesons as repulsive mediators (9 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

10 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Symmetries of QCD realized in the SU(3) Quark Meson Model L = L Fn,s + L ϕ + L V respects symmetries of QCD Apart from color- and flavour symmetry, L exhibits chiral symmetry spontaneously broken due to chiral condensate < Ψ n,s Ψ n,s > explicitly broken due to non vanishing quark masses m q Restoration of chiral symmetry Chiral symmetry restored at high densities, i.e. chemical potential µ q Quarks (nearly) massless Right handed and left handed quarks are (nearly) indistinguishable ( / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

11 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars The EoS from the SU(3) Quark Meson model at T = Once the pressure is known... p = λ 4 (σ n + σs ) λ 4 (σ4 n + σs 4 ) m (σ n + σs ) + σ nσ s c +h n σ n + h s σ s B + ( m ω ω + mρρ + mφ φ) 3 k f ) F π dk k ( k + m f µ f f =u,d,s... the energy density follows from the Gibbs-Duhem relation, Ω = ɛ + f µ f n f, where n f = (k f F )3 /π is the density associated to each quark flavour. ( / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

12 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Important Parameters of the model Repulsive vector meson coupling g ω = g ρ = g φ Models the repulsive interaction ( g ω ) The Bag constant B /4 B /4 acts as an additive vacuum pressure ( B /4 5 MeV) (Depending on the used model) Sigma meson mass m σ experimentally not well determined Variation: 4 m σ MeV ( / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

13 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Hybrid stars: Maxwell construction SU(3)-EoS for the ultradense core and DD-EoS for the stars outer layer 8.5 p [MeV] B=8MeV B=MeV B=MeV B=4MeV B=6MeV B=8MeV DD M/M sun.5 B=8MeV B=MeV B=MeV B=4MeV B=6MeV B=8MeV DD ε [MeV] R(km) The other papameters: m q = 3 MeV, m σ = 6 MeV and g ω = (3 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

14 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Hybrid Stars Twin Stars p [MeV/fm 3 ] Set A Set B Set C DD M/M sun Set A unstable Set A stable Set B unstable Set B stable Set C unstable Set C stable 5 5 ε [MeV/fm 3 ] R(km) Parametrizied quark matter EoS yields Twin Stars p(ɛ) = cs (ɛ ɛ ), with: ɛ := ɛ t + ɛ cs p t, Transition pressure p t and jump in energy density ɛ under direct influence (cs = 3 ) (4 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

15 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Twin stars hard to find in microscopic modelling SU(3)-EoS for the ultadense core and DD-EoS for the stars crust do not yield (reasonable) Twin Star solutions ε/ε t.5.5 ε/ε M < M M > M < g ω < g ω =.5 M < M M > M Set A Set B Set C 8 < Bag < 9 Bag= 4MeV constraint g ω = Bag=4MeV p t /ε t ε/ε t ε/ε Twin Stars constraint 8 < Bag < 9 Bag= 4MeV 6 < Bag < 8 Bag= 4MeV 4 < Bag < Bag= 8MeV 5 < Bag < Bag= 5MeV constraint.5 g ω = g ω =3 g ω = g ω = p t /ε t Blue shaded area: Twin Star region (5 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

16 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Twin stars hard to find in microscopic modelling SU(3)-EoS for the ultadense core and DD-EoS for the stars crust do not yield (reasonable) Twin Star solutions ε/ε t.5.5 ε/ε M < M M > M < g ω < g ω =.5 M < M M > M Set A Set B Set C 8 < Bag < 9 Bag= 4MeV constraint Bag=4MeV ε/ε t.5.5 ε/ε < g ω = < 3 g ω =.5 < g Bag=6MeV ω = < 3 g ω =.5 < g Bag=8MeV ω = < 3 g ω =.5 < g ω = < 3 Twin Stars g ω =.5 constraint Bag=4MeV constraint Bag=MeV g ω = p t /ε t p t /ε t Blue shaded area: Twin Star region (6 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

17 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Quark Stars: Stable? Contour plot of vacuum pressure B /4 vs. repulsive coupling g ω Bodmer Witten hypothesis Above the -flavour-line: Iron, i.e. hadronic matter more stable E u,d 3 MeV A Below the 3-flavour-line: Pure quark matter more stable E A u,d 3 MeV B /4 4 [MeV] flavor-line 3-flavor-line,67,35,86 3,49 4,7 4, g,98,35 3,49 4,745 6, The other papameters: m q = 3 MeV and m σ = 6 MeV (7 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

18 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Nontrivialities in the SU(3)-EoS Twin Stars? Different values of the vacuum energy term B / shifts the EoS in positive pressure range...crossover chiral phase transition Small B: Nontriviality in positive pressure range! scalar fields [MeV] p [MeV/fm 3 ] B /4 =37.5 MeV B /4 =5.5 MeV B /4 =65. MeV B /4 = MeV 3 4 σ s σ n 3 4 ε [MeV/fm 3 ] The other papameters: m σ = 6 MeV and g ω = 4. (8 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

19 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars SU(3) Twin stars and constraints Shaded areas are excluded p [MeV/fm 3 ] 4 B /4 =37.5 MeV B /4 =5.5 MeV B /4 =65. MeV B /4 = MeV 3 B /4 =37.5 MeV B /4 =5.5 MeV B /4 =65. MeV B /4 = MeV scalar fields [MeV] σ n M/M sun 3 4 ε [MeV/fm 3 ] 3 R[km] small B large and heavy stars (but excluded by rotation) (m σ = 6 MeV and g ω = 4) (9 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

20 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Baryon number conservation Perturbation might cause collapse from one to the other branch (M constraint) 3 B /4 =37.5 MeV B /4 =5.5 MeV B /4 =65. MeV B /4 = MeV B /4 =37.5 MeV B /4 =5.5 MeV B /4 =65. MeV B /4 = MeV M/M sun log N B R[km] ε [MeV/fm 3 ] B /4 = 5.5 MeV has a sibling at same baryon number (m σ = 6 MeV and g ω = 4) ( / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

21 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Contour lines of Twin stars Shaded areas are allowed m σ = 6 MeV g ω = st order phase transition crossover phase transition > M sun 7 g ω =g n B /4 [MeV] 5 4 B /4 [MeV] M < M M = M M > M -flavour line 3-flavour line g ω =g n =m q /f π M < M M = M M > M -flavour line M sun 76Hz line g ω g ω Relatively large parameter space where all constraints are fulfilled ( / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

22 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Varying m σ and g ω at B /4 = 37.5 MeV p [MeV/fm 3 ] m σ =4MeV, g ω =8 m σ =6MeV, g ω =4 m σ =8MeV, g ω = 5 4 m σ =4MeV, g ω =8 m σ =6MeV, g ω =4 m σ =8MeV, g ω = 3 scalar fields [MeV] σ s σ n M/M sun ε [MeV/fm 3 ] R[km] small m σ needs relatively large g ω EoS stiff large M and R large m σ needs relatively small g ω EoS soft small M and R ( / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

23 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars The EoS at T = MeV for different repulsive coupling 4 σ n,s [MeV] ω [MeV] g ω = g ω =.5 g ω =3 g ω = µ q [MeV] ρ [MeV] 3 - µ q [MeV] µ q [MeV] φ [MeV] 4 3 σ s σ n µ q [MeV] p [MeV/fm 3 ] p [MeV/fm 3 ] 3 g ω = g ω =.5 g ω =3 g ω =6 ε [MeV/fm 3 ] n B [/fm 3 ] From T = T = MeV the radius increases %, the mass is nearly unaffected - work in progress... (3 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

24 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Neutron star merger, both stars with.4 M Animation by Matthias Hanauske (4 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

25 Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints Compact Stars Summary SU(3) Quark Matter EoS Hybrid Star solutions...find Twin Star solutions.. Twins hard to find in microscopic modelling 3 Do these stars fulfill the 76Hz constraint? 4 Stability of quark matter? 5 Relatively large parameter space allowed for m σ = 6 MeV 6 Interplay of m σ and g ω determine nonlinearity in the EoS 7 Twin Stars-from one EoS only - crossover chiral phase transition Outlook NICER experiment (Radius measurement of compact stars) Simulation of a collaps from first stable branch to second stable branch 3 Finite T calculation (Supernova EoS) - nontriviality already located - work in progress 4 Cooling process via neutrino emission 5 Compact star merger and gravitational wave emission (LISA) (5 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

26 back up slides (6 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

27 Death of a star If the nuclear fuel is exhausted any stars life will end differently Stars with M 8M eject outer layers in a planetary nebula White dwarf Stars with M 8M end in a Supernova explosion Compact star Stars with M M end in a Black hole (7 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

28 New pulsar mass measurements Recent measurements Demorest et. al; PSR J64-3 with M =.97 ±.4M Antoniadis et. al; 3 PSR J with M =. ±.4M set new constraints on thermodynamic quantities. Until the Hulse Taylor Pulsar with M =.44 ±.35M was the heaviest. Credit: Science Magazine (8 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

29 Neutron star merger, both stars with.4 M Animation by Filippo Galeazzi (9 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

30 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 ɛ(r) = Gɛ(r)m(r) r ( + p(r) ) ( + 4πr 3 ) ( p(r) Gm(r) ) ɛ(r) m(r) r (3 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

31 TOV equations: Boundary conditions Start integration of the TOV equations at the center with boundary conditions m(r start = ) = End integration of the TOV equations at the stars surface where the pressure vanishes m(r end ) = M star (3 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

32 Combining Micro and Macro 8 From an EoS TOV - equations Mass-Radius Relations.5 7 pressure p [MeV/fm 3 ] polytrope DD SU(3) M/M s un.5 polytrope DD 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 (3 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

33 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 ] 5 5 HM QM Hybrid chemical potential µ [MeV] (33 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

34 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 = g w = g w =3 DD µ [MeV] ε [MeV/fm 3 ] p [MeV/fm 3 ] g w = g w = g w = g w =3 DD m q = 3 MeV, m σ = 6 MeV, B = MeV (34 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

35 Results for g ω 3 At a certain central pressure the star configurations get unstable R(km) g w = g w = g w = g w =3 DD Star Star.5.5 M/M sun M/M sun.5.5 g w = g w = g w = g w =3 DD Star Star p c (MeV/fm 3 ) R(km) m q = 3 MeV, m σ = 6 MeV, B = MeV (35 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

36 Results for g ω 3 Particle composition of two individual stars: (hybrid.8m ) and (hadronic.4m ) 8 Star Star.8 n n Star Star (continuous) (dotted) ε [MeV/fm 3 ] 6 4 particle fraction.6.4 d u. s p+e p+e r [km] r [km] m q = 3 MeV, m σ = 6 MeV, B = MeV (36 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

37 ɛ crit ɛ = + 3 p trans ɛ trans (37 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

38 Twin stars: Hard to find ε/ε t ε/ε Twin Stars constraint 8 < Bag < 9 Bag= 4MeV 6 < Bag < 8 Bag= 4MeV 4 < Bag < Bag= 8MeV 5 < Bag < Bag= 5MeV constraint ε/ε t.5 ε/ε < g ω = < 3 g ω =.5 < g Bag=6MeV ω = < 3 g ω =.5 < g Bag=8MeV ω = < 3 g ω =.5 < g ω = < 3 Twin Stars g ω =.5 constraint Bag=4MeV constraint.5 g ω = g ω = g ω = g ω =3.5 Bag=MeV p t /ε t p t /ε t m q = 3 MeV and m σ = 6 MeV (38 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

39 The influence of the speed of sound p [MeV/fm 3 ] Set A Set B Set C DD M/M sun Set A unstable Set A stable Set B unstable Set B stable Set C unstable Set C stable 5 5 ε [MeV/fm 3 ] R(km) Twin Star Area scanned with SoS dependent EoS p(ɛ) = cs (ɛ ɛ ), with: ɛ := ɛ t + ɛ cs p t, p t /ɛ t and ɛ/ɛ t under direct influence, c s = 3 (39 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

40 The influence of the speed of sound p [MeV/fm 3 ] Set A Set B Set C DD M/M sun Set A unstable Set A stable Set B unstable Set B stable Set C unstable Set C stable 5 5 ε [MeV/fm 3 ] R(km) Twin Star Area scanned with SoS dependent EoS p(ɛ) = cs (ɛ ɛ ), with: ɛ := ɛ t + ɛ cs p t, p t /ɛ t and ɛ/ɛ t under direct influence, c s = 3 (39 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

41 Looking for Twin star solutions General analysis and parametrization via transition pressure p t, transition energy density ɛ t and jump in the energy density ɛ: Alford, Han et al. arxiv:3.473 and arxiv:5.79 ε/ε t.5 ε/ε M < M M > M < g ω < g ω =.5 M < M M > M Set A Set B Set C 8 < Bag < 9 Bag= 4MeV constraint M/M sun Set A unstable Set A stable Set B unstable Set B stable Set C unstable Set C stable.5 Bag=4MeV g ω = p t /ε t R(km) Blue shaded area: Twin Star region (4 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

42 First step: Construction of meson matrices φ = σ n σ n σ s σ n and σ s : scalar mesons V = ω n+ρ ρ + ρ ωn ρ ω s ω, ρ and ω s = φ: vector mesons (4 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

43 σ n as a function of µ q while varying g ω Smaller vector coupling leads to.order phase transition µ 3MeV fixed m q = 3MeV and m σ = 6MeV (4 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

44 σ s as a function of µ q while varying g ω (43 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

45 The Equation of State: Solve the Lagrangian With Z = Dσ a Dπ a a ( D ΨDΨe β dτ V d3 r(l+ Ψγ ) µψ) and p = lnz β = Ω ɛ = p + i µ i n i we have given a relation for the necessary values. (44 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

46 EoS - Grancanonical Potential Ω V = Having performed the T approximation the resulting grandcanonical potential is Ω = V + 3 π f =u,d,s k f F dk k ( k n,s + m µ f ) with ( m ω ω + m ρρ + m ϕϕ ) + λ 4 (σ n + σ s ) + λ 4 (σ4 n + σ 4 s ) + m (σ n + σ s ) σ nσ s c h n σ n h s σ s + B (45 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

47 The energy density and the pressure are then determined to ɛ = ɛ e + λ 4 (σ n + σ s ) + λ 4 (σ4 n + σ 4 s ) + m (σ n + σ s ) and σ nσ s c h n σ n h s σ s + B k f ( F ) π dk k kn,s + m p = f =u,d,s ( m ω ω + m ρρ + m φ φ) ( m ω ω + m ρρ + m φ φ) + λ 4 (σ n + σ s ) + λ 4 (σ4 n + σ 4 s ) + m (σ n + σs ) σ nσ s c h n σ n h s σ s + B + 3 k f ( F ) π dk k kn,s + m µ f f =u,d,s (46 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

48 Hybrid stars: Maxwell construction Different EoS and corresponding Mass-Radius Relations 8.5 p [MeV] B=8MeV B=MeV B=MeV B=4MeV B=6MeV B=8MeV DD M/M sun.5 B=8MeV B=MeV B=MeV B=4MeV B=6MeV B=8MeV DD ε [MeV] R(km) (47 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

49 Hybrid stars: Maxwell construction The resulting mass radius relations while varying the vacuum pressure B.5 M/M sun.5.5 B=8MeV B=MeV B=MeV B=4MeV B=6MeV B=8MeV DD M can be reached The appearence of a QM core destabilizes the star under certain circumstances R(km) (48 / 5) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich