Transport in the Outer Core of Neutron Stars

Transcription

1 Stephan Stetina Institute for Theoretical Physics Vienna UT Transport in the Outer Core of Neutron Stars SEWM 018, Barcelona Ermal Rrapaj (University of Guelph), Sanjay Reddy (INT Seattle) [S. Stetina, E. Rrapaj, S. Reddy, Phys.Rev. C97 (018) no.4, ] [S. Stetina, in preparation]

2 The outer core of neutron stars homogeneous plasma of electrons, muons, protons, and neutrons stable homogeneous nuclear matter degenerate QED plasma (e, μ, p + ) p, n form strongly interacting Fermi liquid ß equilibrium and charge neutrality μ n μ p = μ e = μ μ, n e + n μ = n p critical densities stability of hom. phase (spinodal point) n c 0.6 n 0 onset of muons (μ e = m μ ) [Weber, J. Phys. G7, 465 (001)] n μ 0.75 n n 0 electrons under NS conditions are always relativistic, degenerate, weakly interacting important contribution to transport

3 Neutron star phenomenology transport is determined by electromagnetic response: - screening & damping - collective modes transport is relevant for neutron stars (T μ) - damping of hydro/r modes - spin evolution - thermal relaxation correlations of strong & EM int. [S. Stetina, E. Rrapaj, S. Reddy, Phys. Rev. C 97, ] fermion dispersions: - screening & damping - collective modes supernovae [S. Stetina, in preparation]

4 Strongly interacting Fermi Liquid (I) Landau energy functional: [N. Chamel, P. Haensel, PRC.73, 04580] E n, j = T=0,1 δ T0 ħ m τ T + C T n n b n T + C T τ n T τ T + C T j jt Functional dependence on nucleon density n, kinetic energy density τ, and current density j V ab = δ δn a δn b E n, j, V ab ij = δ δj a,i δj b,j E[n, j] Current-Current interaction are related to effective masses (L=1 Landau parameter) C τ = C j ħ m a = δe δτ a = ħ m a + C 0 τ C 1 τ n + C 1 τ n a n,τ,j C 0,1 related to Skyrme parameters. Here: NRAPR, SKRA, SQMC700, LNS, KDE0v1 [M. Dutra, O. Lourenço, J. S. Sá Martins, A. Delfino, J. R. Stone, P. D. Stevenson, PRC 85,03501] matching to relativistic field theory variables required

5 Strongly interacting Fermi Liquid (II)

6 RPA photon propagator Relativistic one-loop resummation ( Random Phase Approximation, RPA) D μν (q) D μν (q) = Π μν q Π μν q photon polarization tensor Dressed photon propagator in Coulomb Gauge: D μν q 0, q = q q q Π L 1 P L μν + q Π 1 P μν hard region: q = q 0, q k f soft region (medium effects): q e k f Weak screening approximation: 1 D L q m, D D q i 1 q 0 q q f Extensively used in the calculation of transport in NS:, m D = 4α f π μk f, q f = α f k f [E. Flowers and N. Itoh, Astrophys. J.06, 18 (1976)] [P.S. Shternin, D.G. Yakovlev Phys.Rev.D78 (008), ] [P.S. Shternin, D.G. Yakovlev Phys.Rev.D75 (007), ] Hard dense loop (HDL) approximation q k f (requires m k f )

7 Photon polarization (soft region) full 1-loop vs. hard dense loops (HDL): q 0 μ, q k f Electrons (full 1-loop) Electrons (HDL) real parts protons imaginary parts (Landau damping) at n = n 0 : μ e 10 MeV μ p 600 MeV m p 575 MeV q 0. μ

8 Photon spectrum: multi-component QED Generalize to internal flavour space (i e, μ, p + ) Π diag Π e, Π μ, Π p, γ μ c i γ μ, c i = 1, 1, 1 dressed photon propagator: D μν q 0, q = q q q Tr [Π L ] 1 P L μν + q Tr Π 1 P μν RPA resummation in multi-component plasma is well established: [C. Horowitz, K. Wehrberger, Nucl. Phys. A 531, 665 (1991) ] [S. Reddy, M. Prakash, J.M. Lattimer, J.A. Pons, PRC 59, 888 (1999)] Excitations in degenerate e, μ, p + matter: Protons are quasiparticles (strongly interacting Fermi liquid) with effective masses m p Collective modes (oscillation in the densities of quasiparticles): Plasmon (longitudinal), Photon (transverse)

9 Collective modes: multi-component QED compare to: [M. Baldo, C. Ducoin, PRC 79, (009)] Spectrum (electrons, muons, protons) one gapped (real) plasmon mode (e) ω L two transverse modes ω Two Bohm-Staver sound modes (m, p) [D. Bohm, T. Staver, Phys. Rev. 84, 836 (1950)] Light particles dynamically screen heavier ones. Three overdamped solutions (e, m, p) (roughly located below v f,i q ) plamon decay in dense matter [E. Braaten, D. Segel, Phys.Rev. D48 (1993) ]

10 Spectral functions: multi-component case (A)-(E): Landau damping in a e, m, p plasma q 0 compare to weak screening approx. : Π L q 0 = 0 = m D, j q v e q j v p q v μ q Π q 0 = 0 = i q 0 q j q f, j

11 QED + strong interactions What's the role of the neutrons within RPA? - Coupling strength to photon tiny in free space V np - BUT: Interactions induced by the polarizability of protons [B. Bertoni, S. Reddy, E. Rrapaj, Phys. Rev. C 91, (015)] - use resummed RPA polarization tensor for protons Π p = Π p (1 V nn Π n ) 1 V nn Π n V pp Π p + V nn V pp V np Π n Π p D μν q 0, q = q q q Π e,l Π μ,l Π p,l 1 PL μν + q Π e, Π μ, Π p, 1 P μν effective interaction L γ n = e V np തn γ μ n A ν (Π L,p P L μν + Π,p P μν ) changes to transverse spectrum are negligible since for protons Π Π L.

12 Induced interactions Nuclear interactions appear nested inside electromagnetic ones V pp V pn - V ab are described by pointlike short-range interactions: e μ p n V μν q = P μν μν L + P q + iε q q V pp V pn 0 0 V np V nn gμ0 gν തV pp തV pn 0 0 തV np തV nn gμi gνj density-density (l=0) current-current (l=1)

13 Induced (strong) screening TD definition: Π L, p q 0 = 0 = μ p μ n n p 1 = m p (1+V nn m n ) 1+V nn m n +V pp m p + V nn V pp V np m n m p homogeneous nuclear matter unstable pure QED (NRAPR), n = 0.65 n 0 induced interactions most pronounced at densities close to the crust-core boundary

14 Spectral functions: QED + strong int. n = 0.65 n 0 n = n 0 n = 1.6 n 0

15 Damping rates of fermions (I) fermion ( p > k f ) and hole ( p < k f ) dispersion relations in degenerate matter soft fermionic excitations include Λ + : particles (or holes), anti-plasmino Λ : anti-particle, plasmino soft fermion spectrum more involved (mode mixing, non-perturbative effects) [J. P. Blaizot, J.Y. Ollitrault, PRD 48, ] [S. Stetina, in preparation] S p = S + Λ +,p + S Λ,p γ 0 S ± = p 0 ε p Σ ± 1 scattering close to the Fermi surface: Λ ± = 1 [1 + γ 0 γ p+m ε p ] fermions p k f are always on-shell, there is no damping at order α f photon is either hard (large angle) or soft (small) angle

16 Damping rates of fermions (II) Imaginary parts from cutting two-loop contributions Γ Im Σ T channel Interference terms U channel (should be small..) Moeller scattering Compton scattering

17 Damping rate of fermions (III) Relevant contribution at T = 0 Γ + = 1 Tr Λ + γ 0 Im Σ R = 1 p 0 Tr γ p + m Im Σ R p 0, p, p 0 = ε p photon spectrum ρ μν = ρ L g μ0 g ν0 + ρ P μν week screening & close to Fermi surface ε p μ m D, u = q 0 / q Γ L e m D 4π v f ε p μ න 0 du u න d q 0 1 m D + q = e 1 3 ε m D v p μ f Γ e 4π m D v f ε p μ න 0 du u න d q 0 q 4 q 16 q 6 + u π m D v f = e 1π v f ε p μ compare to: [C. Manuel, Phys.Rev. D6 (000) ]

18 Longitudinal and transverse damping (I) nonrelativistic: electric interactions dominate, magnetic interactions are down by relativistic: damping due to the exchange of plasmons and photons is equally important [H. Heiselberg, G. Baym, C. J. Pethick, J. Popp, Nuc. Phys. A 544 (199)] electrons at n=n0 v c h e solid: full one-loop dashed: HDL dot-dashed: weak screening (static) HDL approximations work much better in the longitudinal channel!

19 Longitudinal and transverse damping (II) nonrelativistic: electric interactions dominate, magnetic interactions are down by relativistic: damping due to the exchange of plasmons and photons is equally important [H. Heiselberg, G. Baym, C. J. Pethick, J. Popp, Nuc. Phys. A 544 (199)] muons at n=n0 v c q k f hard to fulfill, HDL don t work really well in either channel Γ L overtakes Γ

20 Damping rate of electrons, multiple species energy loss of electrons due to collisions with other electrons, muons, and protons total screening mass: M D = σ a m D,a ρ L = 1 π Tr Im Π 00 Tr ReΠ 00 q + Tr Im Π 00 total scattering rate e close to the crust-core boundary easy to implement in transport calculations (fit as function of ε p μ )

21 Impact of induced interactions Γ L strongly modified near crust-core boundary, barely modified at saturation density electrons: with vs. without induced int. electrons muons caveat: dynamical screening effects should be incorporated in the nuclear sector

22 Outlook Existing description of transport effects in dense nuclear matter should be improved by taking into account dynamical screening effects and induced interactions Where to go from here: Improve the implementation of nuclear interaction potentials. Deeper in the core: protons are superconducting Meissner effect for (transverse) photon induced e n scattering dominates D = 1 q Π, e Π, μ Π,p include magnetic fields

23 Thank you!