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Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions Simple Neutron Star Model To give an estimate of the density inside a neutron star we want to use first a simple model. We assume that It consists of neutrons only. The temperature is zero. The neutrons behave like an ideal Fermi gas. The star is spherically symmetric. Force balance inside the star: Fermi pressure ←→ Gravity. Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions Force Balance Inside the Neutron Star We derive the force balance by considering chemical and thermal equilibrium. Chemical Potential µ = F (r ) + Φgr (r ) = constant. Here F (r ) is the Fermi energy and Φgr (r ) the gravitational potential. We use relativistic dispersion for the neutrons with kF (r ) the local Fermi momentum q F (r ) = c 2 ~2 kF2 (r ) + m2 c 4 , Z kF3 (r ) d3 k n(r ) = 2 θ(k (r ) − k) = . F (2π)3 3π 2 Energy µ R1 Radius 0 0 1 0 1 0 1 0 1 0 1 1111111 0000000 0 00000001 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 Φ Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions Differential Equation for the Density Newtonian Gravitational Potential Since the neutron star is not homogeneous in the radial direction, the gravitational potential complicated, but from Gauss’s law we know the force Z Gm2 r 0 0 dr 4π(r 0 )2 n(r 0 ) . (1) −Φgr (r ) = − 2 r 0 Differential Equation Taking the derivative of the chemical potential F (r ) + Φgr (r ) = µ with respect to the radius gives d F (r ) + Φ0gr (r ) = 0 . dr Notice that −dF (r )/dr is thus the force due to the Fermi pressure. (2) Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions Numerical Solution Maximum Mass Density Profile 14 Star Radius HkmL 1.75 Density HkgΜm3 L 1.5 1.25 1 0.75 0.5 12 10 8 6 4 2 0.25 0 0 0 0 5 10 Radius HkmL 15 20 We solved the equation with the boundary conditions n(0) ≡ nc 0 and n (0) = 0 . The total star mass 1.44 M 1 4 2 3 Star Mass HSolar MassL 5 With different nc we notice that The maximum mass is 5.7M . Above 2.8 M the radius is smaller than the Schwarzschild radius. Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions General Relativity This shows that general relativity is required. General Spherical Symmetric Line Element ds2 = −e 2φ(r ) (cdt)2 + e 2λ(r ) dr2 + r 2 dΩ2 . (3) Outside the Neutron Star The stress-energy tensor is zero, the solution to the Einstein equations is the Schwarzschild solution. Inside the Neutron Star For the stress-energy tensor we take a perfect fluid, this is a general fluid without viscosity or heat conduction. With pressure P and energy density ρ uµ uν uµ uν . (4) Tµν = ρ 2 + P gµν + 2 c c Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions Interior Solution: Tolman-Oppenheimer-Volkoff Equation In 1939 Tolman, Oppenheimer and Volkoff solved the Einstein equation for a perfect fluid. The solution gives a differential equation for the pressure and energy density as a function of radius. Tolman-Oppenheimer-Volkoff (TOV) Equation dP(r ) = −(P(r ) + ρ(r )) dr with m(r ) = 1 c2 Z r Gm(r ) + 4πGr 3 P(r )/c 2 r (c 2 r − 2Gm(r )) ρ(r 0 ) 4π(r 0 )2 dr 0 . , (5) (6) 0 To solve this equation we need the equation of state (EOS) P(ρ). Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions Interior Solution Maximum Mass with TOV Star Radius HkmL The EOS for relativistic neutrons can be derived from homogeneous matter. We thus have Z d3 k θ(kF − k)(k) ρ=2 (2π)3 20 15 10 5 0 and the thermal relation ∂E = nµ−ρ . ∂V We can express P(r ) and ρ(r ) in terms of the particle density n(r ). P=− 0 0.2 0.4 0.6 0.8 Star Mass HSolar MassL The maximum mass is 0.7 M . The central density in this star is about 1.5 fm−3 . 1 Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions Baryonic Matter with Interactions through Mesons The density is high enough to produce more massive baryons than neutrons. The interaction between the baryons is relevant for the phase transition. Effective Interaction Model for Baryons Model the interaction through meson exchange and use a mean-field approximation. Use three types of mesons (scalar, vector and isovector). Determine the couplings by comparison with data and extrapolation. Use 13 baryon types (n0 , p+ , Λ, Σ+,0,− , ∆++,+,0,− , Ξ0,− , Ω− ). Use two types of leptons, the electron and muon. Tauon is too heavy. Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions Deconfined Quarks in Neutron Star Core The density in a neutron star can be much greater than the nuclear density and, therefore, there can possible be deconfined quarks inside. The MIT Bag Model Description of hadrons in QCD is very complex. Can be modeled by giving the quark vacuum extra energy BV , with B the Bag constant. The energy density ρ = ρf + B , with ρf the energy density of free quarks, and the pressure ∂E P=− = Pf − B . ∂V Perturbative Vacuum Real Vacuum The value of B is not known exactly, but about 200 MeV/fm3 . Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions Phase Transition with Two Conserved Quantities Consider two phases, the nuclear phase and the quark phase. Nuclear phase: many particles like neutron, proton, electron, . . . Quark phase: three relevant flavors: up, down and strange Conservation of Electric Charge and Baryon Number Two conserved charges: two chemical potentials Electric charge and → µQ and µB Baryon number The particle chemical potentials (µn , µp , µe , µup , . . . ) are determined from µQ and µB . The EOS can be computed for both phases individually. The Coulomb force can be included by demanding the matter to be locally neutral. Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions The Mixed Phase Pressure of the Mixed Phase The phase with lowest grand potential Ω is the one the system is in. For the grand potential we have: Ω = −PV 35 3·10 35 2·10 P 35 1·10 1 2000 1750 There does not always exist a single phase with neutral configuration and lowest Ω or highest P. This leads to a neutral mixed phase. 0 1500 ΜB -600 -400 1250 -200 1000 0 ΜQ The blue lines are the charge neutral phases. The yellow line is the neutral mixed phase. Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions Solution of Mixed Baryon Quark Matter Maximum Mass Star Mixed Phase: Particle Fractions 1. -225 Particle Fraction p+ ds u L0 0.1 -150 S- eΜ- 0.01 0 0.5 S0 -75 S+ 1 1.5 2 Baryon Density Hfm-3 L 2.5 3 Chemical potential HMeVL n0 Μq Particle Density Hfm-3 L 1.4 s 1.2 d 1 0.8 u S S0 L0 0.6 p+ 0.4 0.2 n0 0 2 4 6 Radius HkmL 8 0 Quark bag constant B is 230 MeV/fm3 . The maximum star mass is 1.5 M in this model. No pure quark phase in a neutron star. Possibility with correct geometrical structure. In center µB = 1500 MeV. Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions The Strong Interaction Quarks interact with the strong interaction described by QCD. The quarks can have three colors (red, green, blue). QCD based on nonabelian SU(3) gauge transformation, with gauge bosons called gluons (Aaµ ). The QCD Lagrangian LQCD = ψ̄(∂/ + m)ψ + 12 Aaµ ∂µ ∂ν − ∂ 2 δµν Aaν −ig ψ̄γµ t a ψAaµ + 12 g ∂µ Aaν − ∂ν Aaµ f abc Abµ Acν + 14 g 2 f abc f ade Abµ Acµ Adν Aeν Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions Effective Quark-Quark Interaction We can integrate out the gluon fields to get an effective interaction. Effective Action Z Z 1 αβγδ 4 Seff = d x d4 y ψ̄fαi (x)ψfβj (x)Vijkl (x − y )ψ̄gγk (y )ψgδl (y ) 2 (7) with the potential αβγδ Vijkl (x − y ) = g 2 γµαβ γνγδ tija tkla Dµν (x − y ) The effective interaction contains the SU(3) group generators tija , the gamma matrices for the spinor structure, and the gluon propagator. Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions Attractive Quark-Quark Interaction Effective Interaction αβγδ Vijkl (x − y ) = g 2 γµαβ γνγδ tija tkla Dµν (x − y ) (8) The interaction is similar to the effective electron-electron interaction in QED. The difference are the group generators tija tkla . The sign of this part determines whether the interaction is attractive or repulsive. Group structure 1 1 tija tkla = − (δij δkl − δil δkj ) + (δij δkl + δil δkj ) 3 6 The red part is the antisymmetric part in j and l and is attractive! Simple Model General Relativity Nuclear matter Deconfinement QCD Conclusions Conclusions Neutron Star A neutron star is dense enough to have a deconfined quark phase. This deconfined phase is most likely not pure, but is mixed with the baryonic phase. The baryon chemical potential is above 1500 MeV in the center of the neutron star. Superconductivity Quarks have a fundamental attractive interaction and because a neutron star is relatively cold, there can exist a superconducting phase. This is presumably a 2SC phase as in ultracold atomic gases.