Compact Stars within a SU(3) chiral Quark Meson Model Andreas Zacchi - - PowerPoint PPT Presentation

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

Andreas Zacchi1 Matthias Hanauske1,3 Laura Tolos2 J¨ urgen Schaffner-Bielich1

1Institute for Theoretical Physics

Goethe University Frankfurt

2Institut de Ciencies de l Espai

Bellaterra, Spain

3FIAS

Frankfurt Institute for Advanced Studies

Astro coffee, January 17 2017

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Abstract

1

Compact Objects and Ultradense Matter

2

Micro,- Macrophysics and constraints

3

The SU(3) Quark Meson Model

4

Compact Stars

5

Summary and Outlook

(2 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Supernova remnants: Compact stars

Compact stars are either: Neutron stars (consisting mainly

• f neutrons)

Hybrid stars (consisting of a hadronic shell and a quark-matter core) Quark stars (consisting of quark matter only) Size: R ≈ 10 − 15km Mass: M ≈ 1.5M⊙ Density: ρ0 ≈ 2.5 · 1014 g

cm3 ≈ 145 MeV fm3

Credit: Chandra X-ray observatory (NASA) (3 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Different types of compact stars

Characteristic mass radius relations of

1 Neutron Stars

(M ∝ R−3)

2 Quark Stars (M ∝ R3) 3 Hybrid Stars (Can

masquerade as Neutron star)

4 Twin Stars (Two stable

branches)

1 2 5 10 15 M/Msun

Neutron Star sequence

5 10 15

Quark Star sequence

1 2 5 10 15 M/Msun R[km]

Hybrid Star sequences

1 2 5 10 15 M/Msun R[km]

Hybrid Star sequences

1 2 5 10 15 M/Msun R[km]

Hybrid Star sequences

5 10 15 20 R[km]

Twin Star sequences

5 10 15 20 R[km]

Twin Star sequences

(4 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

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? → Equation of state (EoS)

Credit: Fridolin Weber (5 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Constraints on different solutions

1

Schwarzschild Radius → R > 2GM/c2

2

P < ∞ → R > 9GM/4c2

3

Causality → R > 2.9GM/c2

4

Star can not spin faster than 716Hz → 1

ν ≃ (0.96 ± 0.03)

• M⊙

Mnr

1/2 Rnr

10km

3/2

5

Solutions need to fulfill the 2M⊙ constraint → next slide

Credit: Lattimer and Prakash (6 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

New pulsar mass measurements

Recent measurements

1 Demorest et. al; 2010

PSR J1614-2230 with M = 1.97 ± 0.04M⊙

2 Fonseca et. al; 2016

PSR J1614-2230 with M = 1.928 ± 0.017M⊙

3 Antoniadis et. al; 2013

PSR J0348+0432 with M = 2.01 ± 0.04M⊙ ...also a constraint for quark matter based stars?

Credit: Science Magazine (7 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Skalar and vector nonet

Skalar and vector nonet as a base to constuct Lagrangian density ϕ = 1 √ 2    

σn+a0 √ 2

a+ K +

s

a−

σn−a0 √ 2

K 0

s

K −

s

¯ K 0

s

σs     V µ = 1 √ 2   

ωµ

n +ρµ0

√ 2

ρµ+ K ∗µ+ ρµ−

ωµ

n −ρµ0

√ 2

K ∗µ0 K ∗µ− ¯ K ∗µ0 φµ   

(8 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

The SU(3) Quark Meson Model

SU(3) Lagrangian L = LFn,s + Lϕ + LV LFn,s = ¯ Ψn,s

• i✓

∂ − gωγ0ω − gρ τγ0ρ − gφγ0φ − gn,sσn,s

• Ψn,s

Lϕ = tr(∂µϕ)†(∂µϕ) − λ1[tr(ϕ†ϕ)]2 − λ2tr(ϕ†ϕ)2 − m2

0tr(ϕ†ϕ) − tr[ ˆ

H(ϕ + ϕ†)] + c

• det(ϕ†) + det(ϕ)
• LV

= tr(∂µV )†(∂µV ) − m2

vtr(V †V )

couples quarks to mesons via Yukawa type coupling effective quark masses generated by the σn and σs-fields ω, ρ and φ: vector mesons as repulsive mediators

(9 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Symmetries of QCD realized in the SU(3) Quark Meson Model

L = LFn,s + Lϕ + LV respects symmetries of QCD Apart from color- and flavour symmetry, L exhibits chiral symmetry

spontaneously broken due to chiral condensate < ¯ Ψn,sΨn,s > explicitly broken due to non vanishing quark masses mq = 0

Restoration of chiral symmetry

Chiral symmetry restored at high densities, i.e. chemical potential µq Quarks (nearly) massless Right handed and left handed quarks are (nearly) indistinguishable

(10 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

The EoS from the SU(3) Quark Meson model at T = 0

Once the pressure is known... p = − λ1 4 (σ2

n + σ2 s )2 − λ2

4 (σ4

n + σ4 s ) − m2

2 (σ2

n + σ2 s ) + σ2 nσs

2 √ 2 c +hnσn + hsσs − B + 1 2

• m2

ωω2 + m2 ρρ2 + m2 φφ2

− 3 π2

• f =u,d,s

kf

F

dk · k2

• k2 + ˜

m2

f − ˜

µf

• ... the energy density follows from the Gibbs-Duhem relation,

Ω = ǫ +

f µf nf , where nf = (kf F)3/π2 is the density associated

to each quark flavour.

(11 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Important Parameters of the model

Repulsive vector meson coupling gω = gρ = √ 2gφ Models the repulsive interaction (0 ≤ gω ≤ 10) The Bag constant B1/4 B1/4 acts as an additive vacuum pressure (0 ≤ B1/4 ≤ 150 MeV)

(Depending on the used model)

Sigma meson mass mσ experimentally not well determined Variation: 400 ≤ mσ ≤ 1000 MeV

(12 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Hybrid stars: Maxwell construction

SU(3)-EoS for the ultradense core and DD2-EoS for the stars outer layer

100 200 300 400 500 600 700 800 500 1000 1500 2000 p [MeV] ε [MeV] B=80MeV B=100MeV B=120MeV B=140MeV B=160MeV B=180MeV DD2 1 2 100 200 0.5 1 1.5 2 2.5 5 10 15 20 25 M/Msun R(km) B=80MeV B=100MeV B=120MeV B=140MeV B=160MeV B=180MeV DD2

The other papameters: mq = 300 MeV, mσ = 600 MeV and gω = 0 (13 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Hybrid Stars → Twin Stars

100 200 300 400 500 600 700 800 500 1000 1500 2000 p [MeV/fm3] ε [MeV/fm3] Set A Set B Set C DD2 0.8 1 1.2 1.4 1.6 1.8 7 8 9 10 11 12 13 14 M/Msun R(km) Set A unstable Set A stable Set B unstable Set B stable Set C unstable Set C stable

Parametrizied quark matter EoS yields Twin Stars p(ǫ) = c2

s (ǫ − ǫ∗) ,

with: ǫ∗ := ǫt + ∆ǫ − 1 c2

s

pt , Transition pressure pt and jump in energy density ∆ǫ under direct influence (c2

s = 1 3)

(14 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Twin stars hard to find in microscopic modelling

SU(3)-EoS for the ultadense core and DD2-EoS for the stars crust do not yield (reasonable) Twin Star solutions

0.5 1 1.5 2 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 1.0 2.0 3.0 4.5 6 ∆ε/εt pt/εt ε/ε0

gω=0 Bag=140MeV M1 < M2 M1 > M2 0 < gω < 2 ∆ gω= 0.5 M1 < M2 M1 > M2 Set A Set B Set C 80 < Bag < 190 ∆ Bag= 4MeV constraint

0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 2.0 3.0 4.5 7.0 10.012.5 15 ∆ε/εt pt/εt ε/ε0

gω=0 gω=1 gω=2 gω=3

Twin Stars 80 < Bag < 190 ∆ Bag= 4MeV 60 < Bag < 180 ∆ Bag= 4MeV 40 < Bag < 200 ∆ Bag= 8MeV 50 < Bag < 200 ∆ Bag= 25MeV constraint

constraint

Blue shaded area: Twin Star region

(15 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Twin stars hard to find in microscopic modelling

SU(3)-EoS for the ultadense core and DD2-EoS for the stars crust do not yield (reasonable) Twin Star solutions

0.5 1 1.5 2 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 1.0 2.0 3.0 4.5 6 ∆ε/εt pt/εt ε/ε0

gω=0 Bag=140MeV M1 < M2 M1 > M2 0 < gω < 2 ∆ gω= 0.5 M1 < M2 M1 > M2 Set A Set B Set C 80 < Bag < 190 ∆ Bag= 4MeV constraint

0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1.0 2.0 3.0 4.5 7.5 10.0 15 ∆ε/εt pt/εt ε/ε0 Bag=100MeV Bag=140MeV Bag=60MeV Bag=180MeV Twin Stars 0 < gω=0 < 3 ∆ gω= 0.5 0 < gω=0 < 3 ∆ gω= 0.5 0 < gω=0 < 3 ∆ gω= 0.5 0 < gω=0 < 3 ∆ gω= 0.5 constraint

constraint

Blue shaded area: Twin Star region

(16 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Quark Stars: Stable?

Contour plot of vacuum pressure B1/4 vs. repulsive coupling gω Bodmer Witten hypothesis

1 Above the 2-flavour-line:

Iron, i.e. hadronic matter more stable

E A

• u,d ≥ 311 MeV

2 Below the 3-flavour-line:

Pure quark matter more stable E

A

• u,d ≤ 311 MeV

The other papameters: mq = 300 MeV and mσ = 600 MeV (17 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Nontrivialities in the SU(3)-EoS → Twin Stars?

Different values of the vacuum energy term B1/4...

1 ...shifts the EoS in positive

pressure range

2 ...crossover chiral phase

transition

Small B: Nontriviality in positive pressure range!

20 40 100 200 300 400 p [MeV/fm3] B1/4=37.5 MeV B1/4=52.5 MeV B1/4=65.0 MeV B1/4=100 MeV 20 40 60 80 100 200 300 400 scalar fields [MeV] ε [MeV/fm3]

σn σs

The other papameters: mσ = 600 MeV and gω = 4.0 (18 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

SU(3) Twin stars and constraints

Shaded areas are excluded

20 40 100 200 300 400 p [MeV/fm3] B1/4=37.5 MeV B1/4=52.5 MeV B1/4=65.0 MeV B1/4=100 MeV 20 40 60 80 100 200 300 400 scalar fields [MeV] ε [MeV/fm3] σn σs 1 2 3 10 20 30 M/Msun R[km] B1/4=37.5 MeV B1/4=52.5 MeV B1/4=65.0 MeV B1/4=100 MeV

small B → large and heavy stars (but excluded by rotation)

(mσ = 600 MeV and gω = 4)

(19 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Baryon number conservation

Perturbation might cause collapse from one to the other branch

(2M⊙ constraint)

1 2 3 10 20 30 M/Msun R[km] B1/4=37.5 MeV B1/4=52.5 MeV B1/4=65.0 MeV B1/4=100 MeV 57.25 57.3 57.35 57.4 57.45 57.5 57.55 57.6 57.65 57.7 200 400 600 800 1000 1200 1400 1600 1800 log NB ε [MeV/fm3] B1/4=37.5 MeV B1/4=52.5 MeV B1/4=65.0 MeV B1/4=100 MeV

B1/4 = 52.5 MeV has a sibling at same baryon number

(mσ = 600 MeV and gω = 4)

(20 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Contour lines of Twin stars

Shaded areas are allowed

mσ = 600 MeV gω = 4

10 20 30 40 50 60 70 80 1 2 3 4 5 6

B1/4 [MeV] gω crossover phase transition 1st order phase transition

gω=gn M1 < M2 M1 = M2 M1 > M2 2-flavour line 3-flavour line gω=gn=mq/fπ 10 20 30 40 50 60 70 80 1 2 3 4 5 6

B1/4 [MeV] gω

> 2 Msun M1 < M2 M1 = M2 M1 > M2 2-flavour line 2 Msun 716Hz line

Relatively large parameter space where all constraints are fulfilled

(21 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Varying mσ and gω at B1/4 = 37.5 MeV

20 40 60 80 100 200 300 400 p [MeV/fm3] mσ=400MeV, gω=8 mσ=600MeV, gω=4 mσ=800MeV, gω=1 20 40 60 80 100 200 300 400 scalar fields [MeV] ε [MeV/fm3] σn σs 1 2 3 4 5 10 20 30 40 M/Msun R[km] mσ=400MeV, gω=8 mσ=600MeV, gω=4 mσ=800MeV, gω=1

small mσ needs relatively large gω → EoS stiff → large M and R large mσ needs relatively small gω → EoS soft → small M and R

(22 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

The EoS at T = 20 MeV for different repulsive coupling

20 40 60 80 100 200 300 400 500 600 700 800 σs σn σn,s [MeV] µq [MeV] gω=0 gω=1.5 gω=3 gω=6 20 40 60 80 100 0 250 500 750 ω [MeV] µq [MeV]

• 10

10 20 30 250 500 750 ρ [MeV] µq [MeV] 10 20 30 40 250 500 750 φ [MeV] µq [MeV] 100 200 300 400 200 400 600 800 1000 ε [MeV/fm3] p [MeV/fm3] gω=0 gω=1.5 gω=3 gω=6 1 10 100 1000 0.2 0.4 0.6 0.8 1 p [MeV/fm3] nB [1/fm3]

From T = 0 → T = 10 MeV the radius increases ∼ 10 − 20%, the mass is nearly unaffected - work in progress...

(23 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Neutron star merger, both stars with 1.4 M⊙

Animation by Matthias Hanauske

(24 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Compact Objects and Ultradense Matter Micro,- Macrophysics and constraints The SU(3) Quark Meson Model Compact Stars Summary and Outlook

Summary and Outlook

Summary

1

SU(3) Quark Matter EoS → Hybrid Star solutions → ...find Twin Star solutions..

2

Twins hard to find in microscopic modelling

3

Do these stars fulfill the 716Hz constraint?

4

Stability of quark matter?

5

Relatively large parameter space allowed for mσ = 600 MeV

6

Interplay of mσ and gω determine “nonlinearity” in the EoS

7

Twin Stars-from one EoS only - crossover chiral phase transition

Outlook

1

NICER experiment (Radius measurement of compact stars)

2

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)

(25 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

back up slides

(26 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

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 ≥ 20M⊙ end in a → Black hole

www.schoolsobservatory.org.uk (27 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

New pulsar mass measurements

Recent measurements

1 Demorest et. al; 2010

PSR J1614-2230 with M = 1.97 ± 0.04M⊙

2 Antoniadis et. al; 2013

PSR J0348+0432 with M = 2.01 ± 0.04M⊙ set new constraints on thermodynamic quantities. Until 2010 the Hulse Taylor Pulsar with M = 1.4411 ± 0.00035M⊙ was the heaviest.

Credit: Science Magazine (28 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Neutron star merger, both stars with 1.4 M⊙

Animation by Filippo Galeazzi

(29 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

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(ǫ) where ǫ(r) dm dr = 4πr2ǫ(r) dp dr = −Gǫ(r)m(r) r2

• 1 + p(r)

ǫ(r) 1 + 4πr3p(r) m(r) 1 − 2Gm(r) r −1

(30 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

TOV equations: Boundary conditions

1 Start integration of the

TOV equations at the center with boundary conditions m(Rstart = 0) = 0

2 End integration of the

TOV equations at the stars surface where the pressure vanishes m(Rend) = Mstar

(31 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Combining Micro and Macro

From an EoS → TOV - equations → Mass-Radius Relations

100 200 300 400 500 600 700 800 500 1000 1500 2000

pressure p [MeV/fm3] energydensity ε [MeV/fm3] polytrope DD2 SU(3)

0.5 1 1.5 2 2.5 2 4 6 8 10 12 14 16 18

M/Msun R(km) polytrope DD2 SU(3)

Each EoS predicts a specific mass vs. radius line Quark stars: Selfbounded objects Neutron stars: Bounded by gravity

(32 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Hybrid stars: Maxwell construction

Combining a nuclear matter EoS and a Quark matter EoS: Pressure p has to be dominant vs. chem. Potential µ

50 100 150 200 250 300 400 420 440 460 480 500 520 540

pressure p [MeV/fm3] chemical potential µ [MeV] HM QM Hybrid

(33 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Results for 0 ≤ gω ≤ 3

From the intersecting point in the p − µ plane the EoS changes from the HM EoS to the QM EoS

50 100 150 200 250 300 350 300 350 400 450 500 550 600 p [MeV/fm3] µ [MeV] gw=0 gw=1 gw=2 gw=3 DD2 780 820 860 730 740 750 200 400 600 800 1000 1200 1400 1600 500 1000 1500 2000 2500 3000 3500 4000 p [MeV/fm3] ε [MeV/fm3] gw=0 gw=1 gw=2 gw=3 DD2 50 100 200 400 600

mq = 300 MeV, mσ = 600 MeV, B = 100 MeV

(34 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Results for 0 ≤ gω ≤ 3

At a certain central pressure the star configurations get unstable

5 10 15 20 25 30 1 10 100 1000 10000 0.5 1 1.5 2 2.5 R(km) M/Msun pc (MeV/fm3) gw=0 gw=1 gw=2 gw=3 DD2 Star Star 0.5 1 1.5 2 2.5 5 10 15 20 25 M/Msun R(km) gw=0 gw=1 gw=2 gw=3 DD2 Star Star

mq = 300 MeV, mσ = 600 MeV, B = 100 MeV

(35 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Results for 0 ≤ gω ≤ 3

Particle composition of two individual stars: △(hybrid 1.8M⊙) and (hadronic 1.4M⊙)

200 400 600 800 1000 2 4 6 8 10 12 14 ε [MeV/fm3] r [km] Star Star 0.2 0.4 0.6 0.8 1 2 4 6 8 10 12 14 particle fraction r [km] n p+e d u s n p+e

(continuous) (dotted)

Star Star

mq = 300 MeV, mσ = 600 MeV, B = 100 MeV

(36 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

∆ǫcrit ∆ǫ = 1 2 + 3 2 ptrans ǫtrans

(37 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Twin stars: Hard to find

0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 2.0 3.0 4.5 7.0 10.012.5 15 ∆ε/εt pt/εt ε/ε0

gω=0 gω=1 gω=2 gω=3

Twin Stars 80 < Bag < 190 ∆ Bag= 4MeV 60 < Bag < 180 ∆ Bag= 4MeV 40 < Bag < 200 ∆ Bag= 8MeV 50 < Bag < 200 ∆ Bag= 25MeV constraint

constraint

0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1.0 2.0 3.0 4.5 7.5 10.0 15 ∆ε/εt pt/εt ε/ε0 Bag=100MeV Bag=140MeV Bag=60MeV Bag=180MeV Twin Stars 0 < gω=0 < 3 ∆ gω= 0.5 0 < gω=0 < 3 ∆ gω= 0.5 0 < gω=0 < 3 ∆ gω= 0.5 0 < gω=0 < 3 ∆ gω= 0.5 constraint

constraint

mq = 300 MeV and mσ = 600 MeV

(38 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

The influence of the speed of sound

100 200 300 400 500 600 700 800 500 1000 1500 2000 p [MeV/fm3] ε [MeV/fm3] Set A Set B Set C DD2 0.8 1 1.2 1.4 1.6 1.8 7 8 9 10 11 12 13 14 M/Msun R(km) Set A unstable Set A stable Set B unstable Set B stable Set C unstable Set C stable

Twin Star Area scanned with SoS dependent EoS p(ǫ) = c2

s (ǫ − ǫ∗) ,

with: ǫ∗ := ǫt + ∆ǫ − 1 c2

s

pt , pt/ǫt and ∆ǫ/ǫt under direct influence, c2

s = 1 3

(39 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

The influence of the speed of sound

100 200 300 400 500 600 700 800 500 1000 1500 2000 p [MeV/fm3] ε [MeV/fm3] Set A Set B Set C DD2 0.8 1 1.2 1.4 1.6 1.8 7 8 9 10 11 12 13 14 M/Msun R(km) Set A unstable Set A stable Set B unstable Set B stable Set C unstable Set C stable

Twin Star Area scanned with SoS dependent EoS p(ǫ) = c2

s (ǫ − ǫ∗) ,

with: ǫ∗ := ǫt + ∆ǫ − 1 c2

s

pt , pt/ǫt and ∆ǫ/ǫt under direct influence, c2

s = 1 3

(39 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Looking for Twin star solutions

General analysis and parametrization via transition pressure pt, transition energy density ǫt and jump in the energy density ∆ǫ:

Alford, Han et al. arXiv:1302.4732 and arXiv:1501.07902

0.5 1 1.5 2 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 1.0 2.0 3.0 4.5 6 ∆ε/εt pt/εt ε/ε0

gω=0 Bag=140MeV M1 < M2 M1 > M2 0 < gω < 2 ∆ gω= 0.5 M1 < M2 M1 > M2 Set A Set B Set C 80 < Bag < 190 ∆ Bag= 4MeV constraint

0.8 1 1.2 1.4 1.6 1.8 7 8 9 10 11 12 13 14 M/Msun R(km) Set A unstable Set A stable Set B unstable Set B stable Set C unstable Set C stable

Blue shaded area: Twin Star region

(40 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

First step: Construction of meson matrices

φ = 1 √ 2   

σn √ 2 σn √ 2

σs    V = 1 √ 2   

ωn+ρ0 √ 2

ρ+ ρ−

ωn−ρ0 √ 2

ωs   

1 σn and σs:

scalar mesons

2 ω, ρ and ωs = φ:

vector mesons

(41 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

σn as a function of µq while varying gω

fixed mq = 300MeV and mσ = 600MeV Smaller vector coupling leads to 1.order phase transition µ ∼ 300MeV

(42 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

σs as a function of µq while varying gω

(43 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

The Equation of State: Solve the Lagrangian

With Z =

a

DσaDπa

• D ¯

ΨDΨe

β

0 dτ

• V d3

r(L+¯ Ψγ0µΨ)

• and

p = lnZ β = −Ω ǫ = −p +

• i

µini we have given a relation for the necessary values.

(44 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

EoS - Grancanonical Potential Ω

Having performed the T → 0 approximation the resulting grandcanonical potential is Ω = V + 3 π2

• f =u,d,s

kf

F

dk · k2 k2

n,s + ˜

m2 − ˜ µf

• with

V = −1 2

• m2

ωω2 + m2 ρρ2 + m2 ϕϕ2

+ λ1 4 (σ2

n + σ2 s )2 + λ2

4 (σ4

n + σ4 s )

+ m2 2 (σ2

n + σ2 s ) − 2σ2 nσs

√ 2 · c − hnσn − hsσs + B

(45 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

The energy density and the pressure are then determined to ǫ = ǫe + λ1 4 (σ2

n + σ2 s )2 + λ2

4 (σ4

n + σ4 s ) + m2

2 (σ2

n + σ2 s )

− 2σ2

nσs

√ 2 · c − hnσn − hsσs + B + 1 2

• m2

ωω2 + m2 ρρ2 + m2 φφ2

+ 3 π2

• f =u,d,s

kf

F

dk · k2 k2

n,s + ˜

m2

• and

p = −1 2

• m2

ωω2 + m2 ρρ2 + m2 φφ2

+ λ1 4 (σ2

n + σ2 s )2 + λ2

4 (σ4

n + σ4 s )

+ m2 2 (σ2

n + σ2 s ) − 2σ2 nσs

√ 2 · c − hnσn − hsσs + B + 3 π2

• f =u,d,s

kf

F

dk · k2 k2

n,s + ˜

m2 − ˜ µf

• (46 / 25)

Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Hybrid stars: Maxwell construction

Different EoS and corresponding Mass-Radius Relations

100 200 300 400 500 600 700 800 500 1000 1500 2000 p [MeV] ε [MeV] B=80MeV B=100MeV B=120MeV B=140MeV B=160MeV B=180MeV DD2 1 2 100 200 0.5 1 1.5 2 2.5 5 10 15 20 25 M/Msun R(km) B=80MeV B=100MeV B=120MeV B=140MeV B=160MeV B=180MeV DD2

(47 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich

Hybrid stars: Maxwell construction

The resulting mass radius relations while varying the vacuum pressure B

0.5 1 1.5 2 2.5 5 10 15 20 25 M/Msun R(km) B=80MeV B=100MeV B=120MeV B=140MeV B=160MeV B=180MeV DD2

2M⊙ can be reached The appearence of a QM core destabilizes the star under certain circumstances

(48 / 25) Compact Stars within a SU(3) Quark Meson model Zacchi, Hanauske, Tolos, Schaffner-Bielich