The phase diagram of neutral quark matter

Transcription

1 The phase diagram of neutral quark matter Verena Werth 1 Stefan B. Rüster 2 Michael Buballa 1 Igor A. Shovkovy 3 Dirk H. Rischke 2 1 Institut für Kernphysik, Technische Universität Darmstadt 2 Institut für Theoretische Physik, J. W. Goethe-Universität Frankfurt a.m. 3 Frankfurt Institute for Advanced Studies, J. W. Goethe-Universität Frankfurt a.m. Virtual Institute, Workshop II, October 6th to 8th 2005

2 Outline 1 Motivation 2 Model 3 Neutral quark matter 4 The effect of neutrino trapping 5 Summary Phys. Rev. D 72, hep-ph/

3 QCD phase diagram T early universe CSC ψψ 0 ψψ = 0 ψψ 0 neutron stars µ

4 Nambu Jona-Lasinio model Lagrangian L eff = ψ(i / ˆm)ψ + L qq + L qq

5 Nambu Jona-Lasinio model Lagrangian L eff = ψ(i / ˆm)ψ + L qq + L qq quark-antiquark interaction: 8 [ ( L qq = G ψτa ψ ) 2 ( + ψiγ5 τ a ψ ) ] 2 a=0 ( K [det f ψ(1 + γ5 )ψ ) ( + det f ψ(1 γ5 )ψ )]

6 Nambu Jona-Lasinio model Lagrangian L eff = ψ(i / ˆm)ψ + L qq + L qq quark-antiquark interaction: 8 [ ( L qq = G ψτa ψ ) 2 ( + ψiγ5 τ a ψ ) ] 2 a=0 ( K [det f ψ(1 + γ5 )ψ ) ( + det f ψ(1 γ5 )ψ )] quark-quark interaction: L qq = H ( ψiγ5 τ A λ A C ψ T) ( ψciγ 5 τ A λ A ψ T) A=2,5,7 A =2,5,7

7 Mean-field approximation 6 condensates 3 quark-antiquark condensates: ūu, dd, ss φ a = āa a = u, d, s 3 diquark condensates: ud, us, ds ud = 2 H ψ T Cγ 5 τ 2 λ 2 ψ us = 2 H ψ T Cγ 5 τ 5 λ 5 ψ ds = 2 H ψ T Cγ 5 τ 7 λ 7 ψ

8 Mean-field approximation 6 condensates 3 quark-antiquark condensates: ūu, dd, ss φ a = āa a = u, d, s 3 diquark condensates: ud, us, ds ud = 2 H ψ T Cγ 5 τ 2 λ 2 ψ us = 2 H ψ T Cγ 5 τ 5 λ 5 ψ ds = 2 H ψ T Cγ 5 τ 7 λ 7 ψ dynamically generated quark masses M u = m u 4Gφ u + 2Kφ d φ s M d = m d 4Gφ d + 2Kφ u φ s M s = m s 4Gφ s + 2Kφ u φ d

9 Thermodynamic potential Thermodynamic potential in mean-field approximation Ω(T, µ) = T n d 3 ( ) p 1 1 (2π) 3 Tr ln 2 T S 1 (iω n, p) + V + Ω lept V = 4Kφ u φ d φ s + 2G(φ 2 u + φ 2 d + φ 2 s ) + 1 4H ( 2 ud + 2 us + 2 ds)

10 Thermodynamic potential Thermodynamic potential in mean-field approximation Ω(T, µ) = T n d 3 ( ) p 1 1 (2π) 3 Tr ln 2 T S 1 (iω n, p) + V + Ω lept V = 4Kφ u φ d φ s + 2G(φ 2 u + φ 2 d + φ 2 s ) + 1 4H ( 2 ud + 2 us + 2 ds) Stable solutions minima of the thermodynamic potential: Ω φ f = 0, Ω i = 0 gap equations solve selfconsistently!

11 Chemical potentials without neutrino trapping

12 Chemical potentials without neutrino trapping conserved charges: n, n 3, n 8, n Q

13 Chemical potentials without neutrino trapping conserved charges: n, n 3, n 8, n Q chemical potentials: µ, µ 3, µ 8, µ Q

14 Chemical potentials without neutrino trapping conserved charges: n, n 3, n 8, n Q chemical potentials: µ, µ 3, µ 8, µ Q quark chemical potentials µ f,c = µ + a µ Q + b µ 3 + c µ 8

15 Chemical potentials without neutrino trapping conserved charges: n, n 3, n 8, n Q chemical potentials: µ, µ 3, µ 8, µ Q quark chemical potentials µ f,c = µ + a µ Q + b µ 3 + c µ 8 lepton chemical potentials µ e = µ Q µ µ = µ Q

16 Chemical potentials without and with neutrino trapping conserved charges: n, n 3, n 8, n Q, n Le, n Lµ chemical potentials: µ, µ 3, µ 8, µ Q, µ Le, µ Lµ quark chemical potentials µ f,c = µ + a µ Q + b µ 3 + c µ 8 lepton chemical potentials µ e = µ Q + µ Le µ µ = µ Q + µ Lµ µ νe = µ Le µ νµ = µ Lµ

17 The problem with neutrality Standard BCS-pairing "Cooper-pairs" of quarks condition: p a f p b f p p

18 The problem with neutrality Standard BCS-pairing "Cooper-pairs" of quarks condition: p a f p b f p p Problems M s M u M d charge neutrality different Fermi momenta for different quark species pairing becomes difficult or impossible

19 Phase diagram of neutral quark matter H = 3 4 G 60 first order second order T [MeV] χsb g2sc 10 2SC usc gusc CFL gcfl µ [MeV] normal quark matter: ud = us = ds = 0 2SC phase: ud 0, us = ds = 0 usc phase: ud, us 0, ds = 0 CFL phase: ud, us, ds 0

20 Phase diagram of neutral quark matter H = 3 4 G 60 first order second order T [MeV] χsb g2sc 2SC gusc CFL usc gcfl µ [MeV] interplay between neutrality constraints and thermal smearing normal quark matter: ud = us = ds = 0 2SC phase: ud 0, us = ds = 0 usc phase: ud, us 0, ds = 0 CFL phase: ud, us, ds 0

21 Phase diagram for stronger coupling H = 3 4 G H = G T [MeV] χsb g2sc 10 2SC usc gusc CFL gcfl µ [MeV] T [MeV] gusc 60 g2sc usc gcfl SC χsb CFL µ [MeV]

22 Phase diagram for stronger coupling H = 3 4 G H = G T [MeV] χsb g2sc 10 2SC usc gusc CFL gcfl µ [MeV] T [MeV] gusc 60 g2sc usc gcfl SC χsb CFL µ [MeV] neutrality constraint phase at low temperatures bigger regions of color superconductivity no phase at low temperatures

23 The effect of neutrino trapping additional lepton chemical potentials µ e = µ Q + µ Le µ µ = µ Q + µ Lµ µ νe = µ Le µ νµ = µ Lµ

24 The effect of neutrino trapping additional lepton chemical potentials µ e = µ Q + µ Le µ µ = µ Q + µ Lµ µ νe = µ Le sizable amount of electrons without high electric charge chemical potential µ νµ = µ Lµ

25 The effect of neutrino trapping additional lepton chemical potentials µ e = µ Q + µ Le µ µ = µ Q + µ Lµ µ νe = µ Le µ νµ = µ Lµ sizable amount of electrons without high electric charge chemical potential facilitates pairing of up and down quarks smaller µ Q reduced mismatch of fermi surfaces complicates CFL pairing CFL phase nearly neutral without electrons µ Le must be compensated by positive µ Q

26 Phase diagram with neutrino trapping (H = 3 4 G) µ Le = 0 MeV µ Le = 200 MeV T [MeV] χsb g2sc 10 2SC usc gusc CFL gcfl µ [MeV] T [MeV] χsb g2sc 2SC gusc gcfl usc CFL gcfl µ [MeV]

27 Phase diagram with neutrino trapping (H = 3 4 G) µ Le = 0 MeV µ Le = 200 MeV T [MeV] χsb g2sc 10 2SC usc gusc CFL gcfl µ [MeV] T [MeV] χsb g2sc 2SC gusc usc gcfl CFL gcfl µ [MeV] growing of 2SC phase shrinking of CFL phase

28 More neutrinos... µ Le = 200 MeV µ Le = 400 MeV g2sc g2sc T [MeV] χsb g2sc 2SC gusc usc gcfl T [MeV] χsb g2sc 2SC 10 CFL gcfl µ [MeV] 10 gcfl µ [MeV]

29 More neutrinos... µ Le = 200 MeV µ Le = 400 MeV g2sc g2sc T [MeV] χsb g2sc 2SC gusc usc gcfl T [MeV] χsb g2sc 2SC 10 CFL gcfl µ [MeV] 10 gcfl µ [MeV] even more growing of 2SC phase even more shrinking of CFL phase

30 Lepton fraction 1 Y Le µ [MeV] T = 0 MeV T = 40 MeV solid: µ Le = 200 MeV dashed: µ Le = 400 MeV

31 Lepton fraction 1 Y Le µ [MeV] T = 0 MeV T = 40 MeV solid: µ Le = 200 MeV dashed: µ Le = 400 MeV should be around 0.4 right after the collaps of the progenitor star

32 Summary phase diagrams with self-consistent treatment of quark masses neutral quark matter for two different couplings T = 0 neutron star neutrino trapping ( protoneutron star) lepton chemical potential favors 2SC and disfavors CFL phase CFL phase is unlikely to exist in the first few seconds of stellar evolution