Armen Sedrakian. ITP, Goethe-Universität, Frankfurt am Main. Southampton, 13 September, Microphysics for neutron star oscillations.

Transcription

1 1 / 26 Armen Sedrakian ITP, Goethe-Universität, Frankfurt am Main Southampton, 13 September, 2016

2 2 / 26 : static properties Equation of state Composition Pairing properties (superfluidity and superconductivity) : dynamic properties Viscosities (first, second), thermal conductivity Electrical conductivity in conducting plasma Mutual friction for superfluid/superconducting matter This talk Electrical conductivity of a warm crust of a neutron star in magnetic field, following Harutyunyan and Sedrakian, PRC 94, (2016). Complete transport for 2SC color superconducting matter, following Alford, Nishimura, Sedrakian, PRC 90, (2014).

3 3 / 26 log 10 T [MeV] T C T F T m T p Boltzmann gas Liquid -2 Quantum solid 8 Solid Ni Ni Ni Kr Ge 79-3 Cu Fe Ni Se Zn Ni log 10 ρ [g cm -3 ] log 10 T [K] Plasma parameter Γ = Z2 e 2 ( ) 3 1/3, a i = Ta i 4πn i Coulomb and melting temperatures T c = Z2 e 2 a i, T m = Z2 e 2 Γ ma i Plasma temperature ( 4πZ 2 e 2 ) 1/2 n i T p = M if Γ 1 - weakly interacting Boltzmann gas if Γ 1 - strongly coupled plasma forγ > Γ m 160 lattice of nuclei - phonons and impurity scattering. forγ < Γ m liquid state - electron ion scattering. for T T p quantization of lattice becomes important

4 4 / 26 The kinetics of electrons is described by the Boltzmann eq. f t + v f r e(e+[v H]) f p = I[f], The collision integral for electron-ion scattering has the form I = (2π) For small perturbation M δ (4) (p+p 2 p 3 p 4 )[f(1 f 3 )g 2 f 3 (1 f)g 4 ], f = f 0 +δf, δf = φ f 0 ( ) 2π 3/2 ε, g(p) = n i e βε. MT The most general form of the perturbation is given by eτ ] φ = 1+(ω cτ) 2 v i [δ ij ω cτε ijk h k +(ω cτ) 2 h i h j E j where the relaxation time dωdq τ 1 (ε) = (2π) 5 dp 2 M q p p 2 δ(ε ε 3 ω)δ(ε 2 ε 4 +ω)g 2 1 f f 0.

5 5 / 26 Defintion of the electrical conduction j i = 2 If B-field is in the direction of z axis dp (2π) 3 ev iφ f 0 ε = σ ije j. ˆσ = σ 0 σ 1 0 σ 1 σ σ The longitudinal conductivity does not depend on the field σ = e2 3π 2 0 dpp 2 v 2 τ f 0 ε The transversal (σ 0 ) and Hall (σ 1 ) conductivities depend on the B-field σ 0 = e2 3π 2 dpp 2 v 2 0 τ f 0 1+(ω cτ) 2 ε, σ 1 = e2 3π 2 dpp 2 v 2 0 τ 2 ω c f 0 1+(ω cτ) 2 ε.

6 6 / 26 For two limiting cases of strongly degenerate and non-degenerate electrons the conductivities are given ly. Low temperature limit corresponds to the Drude formula σ = nee2 τ F σ, σ 0 = ε F 1+(ω cf τ F ) 2, σ 1 = (ω cf τ F )σ 0, (1) High temperature limit T T F similar formula is good to 20% with ε 3T σ = nee2 τ σ, σ 0 = ε 1+( ω c τ) 2, σ 1 = ( ω c τ)σ 0, The anisotropy clear depends on the parameter ω cτ if ω cτ 1 (weak magnetic fields), σ 0 σ,σ 1 ω cτσ σ, therefore the tensor is approximately isotropic σ ik δ ik σ. if ω cτ 1 (strong magnetic fields), σ 0 σ(ω cτ) 2 σ, σ 1 σ(ω cτ) 1 σ, i.e., the transverse conductivities are strongly suppressed.

7 7 / 26 The relaxation time is then found as M = J 0J 0 q 2 +Π + L J µ J µ = e ū s 3(p 3 )γ µ u s (p), J t J t q 2 ω 2 +Π T = Ze v µ = Ze (1, p /M), τ 1 (ε) = πz2 e 4 n ε m i εp 3 dωe ω/2t f 0 (ε ω) f 0 (ε) q+ dq(q 2 ω 2 + 2εω)S(q)F 2 1 (q) e ω2 /2q 2 θ 2 e q2 /8MT q 2πθ { (2ε ω) 2 q 2 q 2 +Π +θ 2(q2 ω 2 )[(2ε ω) 2 + q 2 ] 4m 2 q 2 } L 2 q 2 q 2 ω 2 +Π T 2. The polarization tensors are evaluated in the Hard-Thermal-Loop approximation to QED plasma.

8 8 / 26 S(q) Γ = 0.1 Γ = 1 Γ = 10 Γ = 40 Γ = a i q F(q) qr c Left: ion structure factor suppresses the scattering with small q; for a i q 1 we find S(q) 1. Right: Nuclear formfactor where r c = 1.15A 1/3 is the nuclear charge. The structure factor depends on the value of the plasma parameter Γ and is derived in the case fo one sort of ions. It originates from the s of Monte Carlo s of Galam and Hansen forγ > 2 and expressions by Tamashiro forγ < 2. The nuclear structure factor assumes spherical nuclei, which is a good approximation in the warm and dilute crust (below neutron drip).

9 9 / (a) σ (d) σ log 10 σ [s -1 ] (b) σ (c) σ Fe 18 0,01 0, T [MeV] (e) σ 0 (f) σ 1 12 C 0, The temperature dependence shows a minimum at approximately T/T F = 0.3 (transition from degenerate to the non-degenerate regime). Dots show the degenerate regime.

10 10 / 26 log 10 σ [s -1 ] (a) σ 0 At high densities or small magnetic fields ω cτ 1 (isotropic region) and σ 0 σ, σ 1 σω cτ B n ee σ2 (2) At low densities or strong magnetic fields ω cτ 1 (anisotropic region) and B = 0 σ 0 56 Fe 12 C (b) σ 1 σ ( nee ) 2σ (ω cτ) 2 1, σ 1 σ B ω = nee cτ B. (3) (c) σ 0 (d) σ 1 log 10 σ [s -1 ] (a) σ 0 56 Fe 12 C (c) σ log 10 ρ [g cm -3 ] B 12 = 1 B 12 = 10 B 12 = (b) σ 1 (d) σ log 10 B [G]

11 11 / 26 Fit formulae are obtained for components of conductivity tensor σ fit = CZ 1 T a F ( T T F The other components of the tensor are given by ) b ( ) T b+c + d, T F σ0 fit = σ ( ) g 1+δ 2 σ 2, σ = σ fit TF, ε F ( σ1 fit = δσ 2 1+δ 2 σ 2, σ = σ fit 1+ T ) h, T F where δ = B(n eec) 1 in c g s. The relative error in σ is γ 11% for 12 C andγ 13% for 56 Fe andβ-equilibrium composition. The relative error in σ 0 andσ 1 is γ 12% for 12 C andγ 15% for 56 Fe andβ-equilibrium composition at temperatures T > 0.15 MeV. Text only tables are available in the arxiv source (free access) and as supplemental material /PhysRevC

12 General form of order parameter 0 ψ a ασ ψb βτ 0 - Antisymmetry in spin σ,τ for the BCS mechanism to work - Antisymmetry in color a, b for attraction - Antisymmetry in flavor α,β to avoid Pauli blocking At low densities 2SC (Bailin and Love 84) (2SC) ǫ ab3 ǫ αβ Important variations on 2SC (crystalline-color-superconductor) (CSC) ǫ ab3 ǫ αβ, δµ 0, m s 0. At high densities we expect 3 flavors of u, d, s massless quarks. The ground state is the color-flavor-locked (CFL) 0 ψ a αl ψb βl 0 = 0 ψa αr ψb βr 0 = ǫabc ǫ αβc Phase diagram in NJL see Buballa-Shovkovy-Rischke, Sandin-Blaschke,... (2006+) 12 / 26

13 13 / 26 Boltzmann equation for fermions: ( t + v 1 x ) f 1 = (2π) 4 ν j M ij 2 j 234 [f 1 f 2 (1 f 3 )(1 f 4 ) f 3 f 4 (1 f 1 )(1 f 2 )]δ 4 (p in p out) f - fermion distribution function, M ij scattering matrix element. ν j - the degeneracy factors (spin, flavor, color) Fermions in the basis: Ψ i = {Ψ bu,ψ bd,ψ e} = {blue up quark (bu), blue down quark (bd), electron (e)}. the indices i and j specify the species of the ungapped fermions in this basis. Further assumptions: - Red and green colors are gapped and do not contribute to the transport - No strangeness (number of s-quarks too small) - High-density, low-temperature regime T, m µ q - Light flavor (isospin) asymmetry typical s µ u µ d (β-equilibrium)

14 14 / 26 Gauge bosons: write the covariant derivative as ( D µψ = µ i a A a µq a) Ψ (4) Two basis for gauge bosons - standard (T 8, Q)and rotated (X, Q) related by rotations via mixing angleϕ A µ = A T 8 µ T 8 + A Q µq = A X µx + A Q µ Q. (5) A X µ = cosϕa T 8 µ + sinϕa Q µ (6) A Q µ = sinϕa T 8 3g µ + cosϕa Q µ cosϕ = (7) e 2 + 3g2. -In the rotated basis the Q charge is massless, i.e., Q color magnetic field penetrates the 2SC -In the rotated basis the X charge is massive, i.e., there is a Meissner effect (more precisely color magnetic flux tubes)

15 15 / 26 The charges Q a are defined to be the product of the coupling constant and the charge matrix for the ungapped fermions: Q T 8 Q Q in the standard (T 8, Q) basis and Q X Q Q ( = g cosϕ diag ( = g diag 1, 1 ), = e diag (+ 23, 13 ) (8), tan2 ϕ 3 = e cosϕ diag(1, 0, 1), 1+tan2 ϕ, ) 3 tan 2 ϕ 3 (9) in the rotated (X, Q) basis. - The longitudinal part of the screening is evaluated in the standard basis - The transverse part of the screening is evaluated in the rotated basis

16 Computing the matrix element for scattering: p 1i + p 2j p 3i + p 4j (flavor i, j) Standard Feynman rules give: ( ) M ij = J µ a,i D ab µν Jb,j ν (10) J µ a,i = Q a i ū(p 3)γ µ u(p 1 )/2p 1 J ν b,j = Qb j ū(p 4)γ ν u(p 2 )/2p 2 (11) where the most general form of the propagator is given by ( ) 1 ( D ab µν = gµν ω 2 q 2) δ ab +Π ab µν (12) Screening in a plasma is taken into account via self-energies Π µν Decomposition all the quantities (matrix elements, gauge propagators) into longitudinal and transverse parts: Π aa l Π aa t = i = i M ij = Ja,i 0 J0 a,j q 2 +Π aa J t a,i Jt a,j a={t 8,Q} l q 2 ω 2 +Π aa (13) t a={x, Q} ( ) q a 2χl 2χl D,i + 4( qd,c) a in the(t 8, Q) basis ( ) q a 2χt 2χt ( 2χsc D,i + 4( qd,c) a + 4 qd,c) a in the(x, Q) basis 16 / 26

17 17 / 26 The screening functions, χ l andχ t in the static limit (Hard Thermal Loop approximation) χ l = 1, χ t = i π 4 ω q, χsc = 1 3. (15) (better done by Rischke and co-workers). To leading order inω/q, we thus have Π T 8T 8 l Π QQ l Π XX t = i = i (Q T 8 i ) 2 µ2 i π 2 + 4(QT 8 C )2µ2 C π 2 (16) (Q Q i )2 µ2 i π 2 + 4(QQ C )2µ2 C π 2 (17) = 4 3 (QX C )2µ2 C π 2 (18) Π Q Q t = i ω q Λ2 where Λ 2 i (Q Q i )2 µ2 i 4π (19) The Q s can be found in the paper.

18 18 / 26 The squared matrix element summed over the final spins and averaged over the initial spins is M ij 2 Q = L l a 2 i Qa j q 2 +Π aa + L t a={t 8,Q} l a={x, Q} 2L lt R Q a i Qa j q 2 +Π aa a={t 8,Q} l where L l = L lt = L t = Q a i Qa j q 2 ω 2 +Π aa t Q a i Qa j +δ ijγ q 2 ω 2 +Π aa int t a={x, Q} (20) ( )( ) 1 q2 4p 2 1 q2 1 4p 2 2 ( ) 1/2 ( ) 1/2 1 q2 4p 2 1 q2 1 4p 2 cosθ (21) ( )( ) 2 1 q2 4p q2 4p 2 2 cos 2 θ + q2 4p 2 + q2 1 4p The interferenceγ int term is small and is neglected.

19 coefficients - definitions of electrical and thermal conductivities and shear viscosity j α = σ αu = h α = κ αt = σ αβ = ηv αβ = d 3 p evαδ f (22) 3 (2π) d 3 p (ǫ µ) vαδ f (23) 3 (2π) d 3 p (2π) 3 pαv β δ f (24) where V αβ is the traceless part of the spatial derivative of fluid velocity V, V αβ = αv β + β V α 2 3 δ αβ V. (25) Comparing the left-hand-sides we obtain a universal relation ξy = i ν i d 3 p (2π) 3φ iδ f i (26) where ν i is a spin factor for a particle flavor i, ξ stands σ, κ, orη, - Y stands αu, αt, or V αβ 19 / 26

20 20 / 26 Linearization of the Boltzmann equation is given by Relaxation time approximation f i = f 0 i + δ f i = ξ i = 3τ iν i γ 1 e (ǫ µ i)/t + 1 f i 0 ǫ Φ i (27) Φ i = 3τ i ψ i Y (28) d 3 p (2π) 3 (φ i ψ i ) f i 0 ǫ γ = δα α = 3 for the electrical ) and thermal conductivities γ = (δ α αδβ β +δα α 2δα α /3 /2 = 5 for the shear viscosity. From Eq. (26), we can now define transport coefficient of each componentξ i as (29) ξ = i ξ i = ξ bu +ξ bd +ξ e (30)

21 Linearization of collision integral ψ i Y f 0 1 ǫ 1 = (2π)4 T ν j M ij 2 j 234 Using the same procedure as for the drift term ξ i = 9τ i γ In the limit ω, T µ q f 0 1 f 0 2 (1 f 0 3 )(1 f 0 4 )δ4 (p in p out)(φ 1 +Φ 2 Φ 3 Φ 4 ). (31) (2π) 4 ν i ν j M ij 2 T j 1234 f 0 1 f 0 2 (1 f 0 3 )(1 f 0 4 )δ4 (p in p out)φ 1 [τ i (ψ 1 ψ 3 )+τ j (ψ 2 ψ 4 )].(32) ξ i = τ i γ j ν i ν j 36Tµ 2 i µ2 j (2π) 5 0 ( ) ω/2t 2 qm 2π dθ dω dq sinh(ω/2t) 0 0 2π M ij 2 φ 1 [τ i (ψ 1 ψ 3 )+τ j (ψ 2 ψ 4 )] (33) q M = min[2p 1, 2p 2 ] = min[2µ i, 2µ j ] is the maximum momentum transfer, andθ is again the angle between p 1 + p 3 and p 2 + p 4. In the limit T/µ q 1 p 1, p 2 µ i,µ j. Comparing Eqs. (29) and (33) we obtain relaxation times τ i for the three gapless fermion species. 21 / 26

22 22 / 26 Qualitative understanding - in the 2SC occurs via the ungapped fermions: the blue up quark, the blue down quark, and the electron. - is dominated by the fermion that feels the least influence from surrounding particles (i.e. long relaxation time or mean-free-path) Relevant interactions longitudinal strong interaction (T 8 ) - Debye screened (short range) longitudinal electromagnetic interaction (Q), - Debye screened (short range) transverse rotated strong interaction (X) - Meissner screening (short ranged) transverse rotated electromagnetic interaction ( Q) (not screened, only Landau damped - long ranged at low T) At low-t the bu quark and electron carry Q charge, bd does not. is dominated by bd quarks (!) At high T the Landau damping of the Q is more significant. Relaxation times are dominated by the X and T 8 interactions. Electron, which has no T 8 charge and only a very small X charge, dominates transport. A transition from the regime dominated by the bd quark to a regime dominated by electrons as the temperature is rised.

23 η bu µ q = (T/µ q) 5/ (T/µ q) 2, η e µ q = (T/µ q) 5/ (T/µ q) 2 (34) Numerical of shear viscosity as a function of temperature, takingα s = 1. In this temperature range we see electron and quark contributing equally at high temperature and electron domination at low temperature. 23 / 26

24 24 / 26 κ bu µ q = (T/µ q), κ e µ q = (T/µ q) 2/3 (35) Numerically calculated thermal conductivity in units of quark chemical potential µ q in the 2SC with α s = 1. In this temperature range we see the crossover from electron domination at high temperature to blue down quark domination at low temperature.

25 25 / 26 σ bu µ q = (T/µ q) 5/ (T/µ q) 2, σ e µ q = 1.46 (T/µ q) 5/ (T/µ q) 2 (36) Numerically calculated electrical ( Q) conductivity as a function of temperature, both expressed in units of the quark chemical potential µ q, taking strong interaction coupling α s = 1. The electrons dominate because the bu relaxation time is shortened by its strong interaction with the bd quarks.

26 26 / 26 Highly accurate results for conductivity of dilute and warn compact star matter including dynamical screening effects. Other transport coefficients will follow. Text-only tables and fit formulas are available for application in MHD computations of compact stars Complete set of transport coefficients (thermal and electrical conductivity and shear viscosity) for 2SC ; second viscosity is also in the literature Fit formulas for all transport 2SC and flux-fermion (quasi) mutual friction are available. Future: continue systematically computing accurate transport coefficients in regimes relevant for compact stars.