Hydrodynamics of the superfluid CFL phase and r-mode instabilities Cristina Manuel Instituto de Ciencias del Espacio (IEEC-CSIC) Barcelona Hirschegg 2009
Outline Introduction Superfluid hydrodynamics Hydrodynamics in the CFL phase r-modes in CFL matter with M. Mannarelli and B. Sa d arxiv:0802.0321;0807.3264 A. Dobado and F.Llanes-Estrada hep-ph/0406058;arxiv:0705.3909
Motivation Understanding the hydrodynamical behavior of dense quark matter needed to provide signatures in astrophysical scenarios at high density, weakly coupled QCD is in a color superconducting phase (consequence of Cooper s theorem). CFL is the energetically favored phase deriving the hydrodynamical behavior of the CFL phase is theoretically challenging, but needed!
Color-Flavor-Locked phase Alford, Wilczek, Rajagopal,98 q ia L qjb L = ( δ a i δb j δa j δb i ) m q 0 (m s < 2 µ) Symmetry breaking pattern SU(3) C SU(3) L SU(3) R U(1) B SU(3) C+L+R All gluons, quarks are gapped and heavy Goldstone bosons associated to SSB of flavor χ-symmetries (π 0, π ±,K 0, K 0,K ±, η, η ) One true, massless, Goldstone mode: the superfluid phonon
Superfluidity in condensed-matter systems Quantum phenomenon associated to the appearance of a condensate (BEC) ψ = ψ 0 e iφ Hydrodynamics complicated: two-fluid model superfluid: v s = φ no dissipation 2m normal component: dissipative processes are possible ρ = ρ n + ρ s j = ρ n v n + ρ s v s One can define more transport coefficients than in a standard fluid (κ, η, ζ 1, ζ 2, ζ 3 ) Superfluids in rotation: appearance of quantized vortices
Relativistic superfluid hydrodynamics - Non-dissipative Poisson bracket approach, Son 01 n σ = n 0 u σ V 2 σ ϕ T ρσ =(ρ + P ) u ρ u σ P η ρσ + V 2 ρ ϕ σ ϕ ρ T ρσ =0, µ n µ =0 u µ µ ϕ + µ 0 =0 ρ = s 0 T 0 + n 0 µ 0 P dp = s 0 dt 0 + n 0 dµ 0 + 1 2 V 2 d( µ ϕ) 2
Relativistic superfluid hydrodynamics - dissipative case in a comoving frame n µ =(n, V 2 ϕ), u µ = (1, 0, 0, 0) T 00 = ɛ Gusakov, 07 ω µ = ( µ ϕ + µu µ ) dissipative additions to energy-momentum tensor T 0i d = κ i T T ij d = η ( i u k + k u i 23 ) δ ik j u j δ ik (ζ 1 div(ρ s w)+ζ 2 divu) another modification u µ µ ϕ = µ 0 χ χ = ζ 3 div(v 2 w) ζ 4 divu ζ 1 = ζ 4, ζ 2 1 ζ 2 ζ 3
Low energy effective theory for the CFL phonon Son,02 At high density, g(µ) 1 integrate out heavy modes from QCD L eff = 3 [ ( 0 4π 2 ϕ µ) 2 ( i ϕ) 2] 2 Classical equations can be viewed as hydrodynamical laws n µ = δl eff δ( µ ϕ) = n 0 v µ µ n µ = µ (n 0 v µ ) = 0 T µν =(ɛ + P )v µ v ν g µν P µ T µν =0 v ρ D ρ ϕ = ρ ϕ (µ, 0) Superfluid velocity
Phonons on the superfluid background superfluid velocity S[φ] = 1 2 ϕ(x) = ϕ(x)+φ(x) phonon Expand Son s action about the classical solution S[ϕ] =S[ ϕ]+s[φ]+ d 4 x GG µν µ φ ν φ Relativistic acoustic metric ( ) G µν = η µν + 1 c 2 1 v µ v ν s Phonons can be viewed as bosons moving in a curved space-time Gravity anologue model Unruh 81, Volovik 00
Transport equation for the phonons From 0=G µν dx µ dx ν one derives for the f(x,p) L[f] p α f x α Γ α βγ pβ p γ f p α = C[f] Christoffel symbol T αβ ph = p α p β f(x, p)dp Phonon energy-momentum tensor is not conserved! They can exchange energy with the superfluid background! ( GT ) µα α ph + GT να ph Γµ να =0
Thermal equilibrium with the Gravity analogue model for the phonons The GM model allows us to treat phonons in a fully covariant way T µν ph =(ρ ph + P ph /c 2 s)u µ u ν P ph η µν P ph ( 1 c 2 s 1 ) v µ v ν ρ ph = P ph c 2 s ( 4c 2 s G µν u µ u ν G µν u µ u ν ) P ph = G T 4 π 2 90 1 (G µν u µ u ν ) 2 P ph = T 4 π 2 90 1 c 3 (1 w2 s c 2 s ) 2 good non-relativistic limit! u µ (1, u) v µ (1, v) w = u v landau, khalatnikov
Phonon interactions and collisions Similar features than in a non-relativistic BEC L CFL = 3 [ eff ( 0 4π 2 ϕ µ) 2 ( ϕ) 2] 2 + ( ) L BEC eff = c 1 0 ϕ ( ϕ)2 + c 2 ( 0 ϕ ( ϕ)2 2m 2 2m ) 2 + At low p Damping rate of the CFL phonon ~ same sort of laws that of the phonon of He4 γ(t = 0) p 5, γ T 4 p Mean free path (m.f.p.) ϕ ϕϕ l µ 4 /T 5 ϕϕ ϕϕ l µ 8 /T 9
Transport coefficients due to phonons Solve the transport equation, in the easiest frame using a variational method Technical difficulties: collinear singularities due to massless exchange in C; near zero-modes Introduce corrections (non-conformal) to compute bulk viscosity η =1.3 10 4 µ8 T 5 ζ 2 =0.011 m4 s T ζ 1, ζ 3 on the way... m.f.p. > the radius of a compact star ~ 10 km at T < 0.01 Mev!
Other contributions to viscosities for T ~ 0.1 MeV electrons contribute to shear viscosity η T σ T, σ T = 8 3 π ( αe m e Some uncertainty in the kaon masses, but they can be very light; contribution to ζ due to electroweak interactions ) 2 Alford, Braby, Reddy, Schafer, 07 Ζ g cm s 10 30 10 24 10 18 10 12 10 6 1 unpaired 0.01 2SC CFL K 0 m 0.5 MeV 0.05 CFL m 0.5 MeV Φ Φ Φ 0.10 0.50 1.00 5.00 10.00 T MeV K 0 ϕ, ϕ ϕϕ K 0 ϕϕ δm m K 0 µ
R-modes in a CFL quark star r-modes: non-radial pulsations modes that grow via G.W. emission, unless counteracted by dissipation Andersson 98 r-modes in CFL quark matter Madsen, 00; Sa d 08; Jaikumar, Rupak, Steiner 08 analysis complicated: 2-fluid hydrodynamics and vortex dynamics; we will work in the non-relativistic limit! consider two different regimes T < 0.01 MeV ~ m.f.p. > R no thermal phonon fluid! T > 0.01 MeV ~ m.f.p. < R one has a thermal fluid!
CFL Superfluid in rotation Vortices appear, with quantized circulation κ = 2π µ Area density of vortices N v = 2Ω κ Iida, Baym,02 Superfluid flow produces a r 0 Magnus force high velocity, low pressure lift force F M = κρ s (v s v L ) ẑ low velocity, high pressure
Scattering of phonon-vortex incident wave! scattered wave It can be studied with different techniques (Fetter 64, etc) Phonon-vortex scattering, using the gravity analogue model use a vortex configuration in G v = κ µν eg, 2πr e θ σ = ( 1 c 2 ) 2 κ 2 s 8c 2 s k, σ = ( 1 c 2 ) κ s The scattering of quasiparticles produces a force on the vortex c s F N = D(v n v L )+D ẑ (v n v L ) D = σ c s ρ n, D = σ c s ρ n
Vortices are mediators At equilibrium, balance of forces on a vortex F N + F M =0 v L = v s + α (v n v s )+αẑ (v n v s ) α = d d 2 + (1 d ) 2, 1 α = 1 d d 2 + (1 d ) 2, d = D κρ s, d = D κρ s. Force of the vortex on the superfluid component mutual friction force F SN = N v κρ s α(v s v n )+N v κρ s α [ẑ (v s v n )]
Mutual friction in the CFL and r-modes r-modes in compact stars: growth time scale 1 τ GR = 1 3.26 ( Ω 2 πg N ρ ) 3 Dissipation time-scale due to mutual friction at T < 0.01 Mev 1 τ MF 2 18.1 ( T µ ) 5 Ω Stability only when ν ν c 2 10 5 ( T µ ) Hz typical values in the regime ~ 1Hz
Outlook Still to look in the T > 0.01 MeV Hybrid stars (nuclear + CFL quark matter) Study other quark phases: CFLK0, LOFF, etc