General equilibrium second-order hydrodynamic coefficients for stress-energy tensor and axial current

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1 General equilibrium second-order hydrodynamic coefficients for stress-energy tensor and axial current Matteo Buzzegoli Dipartimento di Fisica e Astronomia & INFN, Firenze March Based on a work in preparation with Eduardo Grossi and Francesco Becattini Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

2 Motivations Relativistic hydrodynamic Astrophysic Cosmology Heavy ions collisions General Equilibrium Quantum-relativistic formulation Vorticity effects Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

3 Introduction Quantum theory T µν = L ( µ ψ a ν ψ a g µν L Statistical operator ρ Corrispondence principle [ T µν = tr ρ T µν] = T µν [ j µ = tr ρ ĵ µ] = ĵ µ Hydro equations µ T µν = µ j µ = Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

4 Global Generalized Equilibrium General covariant form of the local thermodynamic equilib. density operator ρ = 1 ( [ ] Z exp dσ µ T µν (xβ ν (x ĵ µ [Zubarev et al 1979; (xζ(x Van Weert 1982] Σ Global equilibrium conditions: µ β ν + ν β µ = and µ ζ = Killing equation Solution in Minkowski space: β µ = b µ + ϖ µν x ν ζ, b µ, ϖ µν are const. ϖ µν = ϖ νµ = ν β µ Thermal Vorticity Then, the global generalized equilib. density operator in Minkowski space is ρ GE = 1 [ exp b µ P µ + 1 Z GE 2 ϖ µνĵµν + ζ Q ] Lorentz Generators Ĵ µν = dσ λ (x µ T λν x ν T λµ Σ Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

5 Global Generalized Equilibrium, Special cases β µ = b µ + ϖ µνx ν Homogeneous equilibrium b µ = const., ϖ = ρ β = 1 ( exp β Z P + ξ Q β [Vilenkin 198] T µν ideal = (ρ + puµ u ν p g µν Equilibrium with rotation b µ = (1/T,, ϖ µν = ω/t (δµδ 1 ν 2 δµδ 2 ν 1 ρ ω = 1 ( exp Ĥ/T + ωĵz/t + µ Q/T Z ω Equilibrium with acceleration b µ = (1/T,,,, ϖ µν = (a/t (g µg 3ν g 3µg ν ρ a = 1 ( exp Z Ĥ/T + a K z/t a Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

6 Stress energy tensor mean value ρ GE = 1 bµ P e µ +ζ Z GE ϖµν Q+ 2 Ĵ µν = 1 [ exp Z GE Ĵ µν x Ĵ µν x µ P ν + x ν P µ β µ (x P µ + ζ Q ϖ µνĵµν x How does thermal vorticity affects hydrodynamic equations? T [ ] µν (x = tr ρ GE T µν (x = T µν µ T µν (x = ] T µν = (ρ + pu µ u ν p g µν +δt µν ( β u µ = βµ β 2 Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

7 Cumulant expansion ρ GE = 1 [ exp Z GE β µ (x P µ + ζ Q ϖ µνĵµν x ] Expansion for ϖ 1 (it is adimensional ] [ ρ Ô(x =tr GE Ô(x = Ô( β(x + ϖ µν 2 β + ϖ µνϖ ρσ 8 β 2 dτ T τ (Ĵ µν iτuô( β(x,c dτ 1 dτ 2 T τ (Ĵ µν iτ1uĵ ρσ iτ2uô( β(x,c + O(ϖ 3 with Ô 1 [ ] β(x [ ]tr e βµ(x P µ +ζ Q Ô tr e βµ(x P µ +ζ Q Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

8 Acceleration and vorticity decomposition Tetrad: {u, α, w, γ} ν β µ = ϖ µν = ɛ µνρσ w ρ u σ + α µ u ν α ν u µ thermal acceleration: α µ = β 2 u ν ν u µ = 1 T aµ thermal angular velocity: w µ = 1 2 ɛµνρσ β 2 u σ νu ρ = 1 T ωµ transverse vector: γ µ = ɛ µνρσ w να ρu σ = (α ϖ λ λµ At local rest frame α = ħ a c k B T, ħ ω w = k B T Numerical estimate in heavy ions collisions T = 3 MeV, a c ω.5 c 2 /fm ϖ 1 2 Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

9 Stress-energy tensor corrections Thermodynamic coefficient definitions Expansion up to second order on ϖ µν = ν β µ T µν (x (ρ α 2 U α w 2 U w u µ u ν (p α 2 D α w 2 D w µν + A α µ α ν + W w µ w ν + G 1 u µ γ ν + G 2 u ν γ µ U α= D α= D w= A = dτ1dτ2 2 β 2 dτ1dτ2 2 β 2 dτ1dτ2 β 2 G1 = G2 = ( Tτ K3 iτ1 ( Tτ K3 iτ1 dτ 1dτ 2 2 β 2 Tτ (Ĵ 3 ( T τ K1 iτ1 iτ1 Ĵ 3 iτ2 K 3 T dτ1dτ2 ( iτ2 T,c U w = 2 β Tτ 3 (Ĵ 2 K 3 T 11 ( iτ2 T,c T 11 ( T,c K 2 T (12 ( iτ2 T,c W = ( dτ1dτ2 2 β Tτ { K 1 2 iτ 1, Ĵ 2 iτ 2 } T 3 ( T,c dτ1dτ2 2 β 2 Tτ ( { K 1 iτ 1, Ĵ 2 iτ 2 } T 3 ( T,c dτ1dτ2 3 β 2 ( Tτ K1 iτ1 dτ 1dτ 2 3 β Tτ 1 (Ĵ 2 iτ1 Ĵ 2 iτ2 iτ1 Ĵ 3 iτ2 T ( T,c K 2 T (12 ( iτ2 T,c T (12 ( T,c dτ1dτ2 1 T τ (Ĵ β Ĵ 2 T (12 ( 2 iτ1 iτ2 T,c Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

10 Stress-energy tensor corrections Features T µν (x (ρ α 2 U α w 2 U w u µ u ν (p α 2 D α w 2 D w µν + A α µ α ν + W w µ w ν + G 1 u µ γ ν + G 2 u ν γ µ Because are evaluated at global equilibrium they are non-dissipative µ T µν = implies relations between coefficients ( U α = β Dα + A ( D α + A β ( U w = β Dw + W D w + 2A 3W β G 1 + G 2 = 2 ( D α + D w + A + β β W + 3W Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

11 Currents corrections ĵ µ (x = n u µ + (α 2 N α + w 2 N ω u µ + G V γ µ N α = N w = G V = 1 6 β β β 2 ( dτ 1 dτ 2 T τ K3 iτ1 K3 iτ2 ĵ ( T,c dτ 1 dτ 2 T τ (Ĵ 3 iτ1 Ĵ 3 iτ2 ĵ ( T,c dτ 1 dτ 2 T τ ({ K iτ 1 1, Ĵ iτ 2 2 } ĵ 3 ( T,c. ĵ µ A (x = wµ W A W A = 1 β 3 dτ T τ (Ĵ iτ ĵ 3,A ( T,c Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

12 Comparison with previous determinations Landau frame T µν = ρ(t, µ u µ L uν L ( p(t, µ α 2 D α w 2 D w µν L +Aαµ α ν + W w µ w ν p D α T = D α U α ρ T p D w T = D w U w ρ T n p n µ µ T n ρ n µ µ T n p n µ µ T n ρ n µ µ T p T N α ρ T p T N w ρ T ρ p ρ µ µ T n ρ n µ µ T ρ p ρ µ µ T n ρ n µ µ T [Baier et al 28; Romatschke 21; Moore and Sohrabi 211] T µν =ρu µ u ν ( p + ξ 3 Ω αβ Ω αβ ξ 4 λρ λ ln s ρ ln s µν Ω µν = 1 2 µν νβ ( α u β β u α Transformation equalities A T 2 = 9λ 4; λ 3 Ω µ λ Ων λ + λ 4 µ ln s ν ln s W T 2 = λ 3; D w ( T 2 = 2ξ 3 λ 3 ; 3 D α T 2 = (9ξ 4 3λ 4 Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

13 Free fields Charged free scalar field T µν = µ φ ν φ + ν φ µ φ gµν ( φ φ m 2 φ φ 2ξ( µ ν φ φ Free Dirac field T µν [ ψγ µ ( ν ψ γ µ ( ν ψ ψ] can = i 2 T sym µν = i [ ψγ µ ( ν ψ + ψγ ν ( µ ψ ( ν µ ψγ ψ ( ν µ ψγ ψ] 4 Example coefficient dτ β 1 U α = dτ 2 2 d 3 x d 3 y sym sym T (X T (Y T ( T,c x 3 y 3 Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

14 Results T µν (x (ρ α 2 U α w 2 U w u µ u ν (p α 2 D α w 2 D w µν A generic coefficient can be written as + A α µ α ν + W w µ w ν + G 1 u µ γ ν + G 2 u ν γ µ 1 dp ( π 2 β 2 n B/F E (E p µ + n B/F (E p + µ (Ap 4 + Bm 2 p 2 + Cm 4 p Bose-Einstein/Fermi-Dirac distribution function n B/F (x = 1 exp( β x η Bose Field G sym 1 = G sym 2 Coeff. depends on ξ W = [Moore and Sohrabi 211] Dirac Field A = W = G sym 1 = G sym 2 G can 1 = 2G can 2 W [Moore and Sohrabi 211] A and W =[Megias and Valle 214] Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

15 Results T µν (x (ρ α 2 U α w 2 U w u µ u ν (p α 2 D α w 2 D w µν A generic coefficient can be written as + A α µ α ν + W w µ w ν + G 1 u µ γ ν + G 2 u ν γ µ 1 dp ( π 2 β 2 n B/F E (E p µ + n B/F (E p + µ (Ap 4 + Bm 2 p 2 + Cm 4 p Bose-Einstein/Fermi-Dirac distribution function n B/F (x = 1 exp( β x η Bose Field G sym 1 = G sym 2 Coeff. depends on ξ W = [Moore and Sohrabi 211] Dirac Field A = W = G sym 1 = G sym 2 G can 1 = 2G can 2 W [Moore and Sohrabi 211] A and W =[Megias and Valle 214] Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

16 Massless fields U α D α A U w D w W G Bose µ = 1 6ξ 6 T ξ 1 9 T ξ Dirac (1 + 3β2 µ 2 (1 + 3β2 µ 2 6 T 4 (1 + 3β2 µ 2 1 4ξ 6 T ξ 3 T ξ 1 Dirac case: (1 + 3β2 µ 2 6 T 4 (1 + 3β2 µ 2 1+ξ 18 T π T 4 2 π T 4 2 π T 4 2 π T 4 2 π T ρ eff T µν u µ u ν = ρ α 2 U α w 2 U w, p eff 1 3 T µν µν = p α 2 D α w 2 D w [ ρ eff = ρ 1 5 ] 14π 2 (α2 + 3w 2 r(ζ = 3p eff Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

17 Dirac Field W=A= U α/ρ U w/ρ D α/p D w/p G/p Scalar Field ξ = 1/ 6 U α /ρ U w /ρ -D α /p D w /p G/p.1.2 A/p -W/p.5 T T m m Figure: The coefficients divided by the energy ρ or the pressure p with m >,µ = Quark gluon plasma Values: k B T = 3 MeV, a = 1 3 m s 2 α2 U α ρ 1 3 Cold atoms of 6 3Li: Values: T = 1 7 K, n = 1 2 m 3, a = 9.8 m s 2, ω = 1 Hz Specific heat: c eff ρ eff T = c c α c w ; c α c 1 16, c w c 1 6 Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

18 Dirac Field W=A= U α/ρ U w/ρ D α/p D w/p G/p Scalar Field ξ = 1/ 6 U α /ρ U w /ρ -D α /p D w /p G/p.1.2 A/p -W/p.5 T T m m Figure: The coefficients divided by the energy ρ or the pressure p with m >,µ = Quark gluon plasma Values: k B T = 3 MeV, a = 1 3 m s 2 α2 U α ρ 1 3 Cold atoms of 6 3Li: Values: T = 1 7 K, n = 1 2 m 3, a = 9.8 m s 2, ω = 1 Hz Specific heat: c eff ρ eff T = c c α c w ; c α c 1 16, c w c 1 6 Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

19 Non relativistic limit T µν (x (ρ α 2 U α w 2 U w u µ u ν (p α 2 D α w 2 D w µν Every coefficient 1 π 2 β 2 + A α µ α ν + W w µ w ν + G 1 u µ γ ν + G 2 u ν γ µ dp ( n B/F E (E p µ + n B/F (E p + µ (Ap 4 + Bm 2 p 2 + Cm 4 p in the limit m/t 1 becomes m dn dx 3 f(m/t where f remains finite and dn/dx 3 is the classical expression of particle density in Boltzmann limit α = ħ a c k B T, ħ ω w = k B T Quantum effects Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

20 Massive Dirac field at zero temperature ρ eff = ρ α 2 U α w 2 U w, p eff = p α 2 D α w 2 D w At T = we have n F (E µ = θ(µ E Ultra relativistic limit Non relativistic limit (µ NR = µ m α 2 U α ρ w 2 U w ρ α 2 D α p w 2 D w p = 1 ( a 2 µ = 3 ( ω 2 µ = 1 ( a 2 µ = 3 ( ω 2 µ α 2 U α ρ w 2 U w ρ α 2 D α p w 2 D w p = 1 ( a 64 µ NR = 3 32 = 5 64 = ( ω µ NR ( a 2 m µ NR 2 2 m µ NR µ NR 2 ( ω µ NR For white dwarfs and neutron stars these corrections are negligible: Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

21 Axial Vortical Effects ĵ µ A = ψγ µ γ 5 ψ ja = W A T ω, j A and angular momentum J are axial vectors ω W A = 1 dp 2π 2 (n F (E p µ + n F (E p + µ (2p 2 + m 2 β E p Massless field Anomaly? µ ĵ µ A = µ ĵ µ A ( = T 2 j A = 6 + µ2 2π 2 ω Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

22 Axial Vortical Effects Massive field µ ĵ µ A = 2mi ψγ 5 ψ µ ĵ µ A = 2mi ψγ 5 ψ ja = W A T ω At T = W A T = µ2 µ 2 m 2 2π 2 µ 2 For T m [ W A T = m 2 T π 3/2 m 8 ( ( ] 2 T T m + O m For T m W A T = T µ2 2π 2 m2 4π 2 7ζ ( 2 m 4 ( (m 3 8π 2 T 2 + O T Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

23 Summary We have studied quantum relativistic free fields of spin and 1/2 at general thermodynamic equilibrium with non-vanishing acceleration and vorticity. Expressed as correlators of conserved quantities (T µν and Poincaré groups generators These corrections depend on the explicit form of the quantum stress-energy tensor operator (dependence on ξ for the free scalar field. Thus, different tensors are thermodynamically inequivalent Such corrections may be phenomenologically relevant for system with very high acceleration, such as in the early stage of relativistic heavy ion collisions These corrections are pure quantum effects Axial vortical effects without anomaly Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

24 Thank for the attention! Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 22

25 Local Thermal Equilibrium Foliation with 3D spatial hypersurfaces Σ of normal vector n n µ T µν (x = n µ T µν (x n µ ĵ µ (x = n µ j µ (x ρ = 1 ( [ ] Z exp dσ n µ (x T µν (xβ ν (x ĵ µ (xζ(x Σ(τ Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 4

26 Stevin law Consider a non-relativistic fluid with µ = in a gravitational field g = gẑ β µ = 1 T (1 + gz,,, gt 1 T (z = β 2 1 T (1 + gz + O(g 2 The pressure p and the numerical density n are given by ( 3/2 ( m n = m4 T (z 2π 3/2 e m/t (z p = m4 T (z m 2π 3/2 e m/t (z m The coordinate dependence are completely contained on T (z p z m2 n T (z 1 m T z (1 + gz + O(g2 = m n g + O(g 2 5/2 p = p + d g h d mass density h depth Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 4

27 Dirac Bilinears Scalar Pseudoscalar Vector Current ψ ψ(x GE = ψ ψ(x T + α 2 k s + w 2 j s ψγ5 ψ(x GE = α w l PS ĵ α GE = n c u α + ( α 2 U V α + w 2 U V w uα + G V γ α Axial Current Tensor ĵ A α (x GE = w α W A σ αβ (x GE = u α α β A T Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 4

28 Axial Vortical Effects Massless field ω, j A and angular momentum J are axial vectors. ĵ µ wµ A (x = β 3 dτ T τ (Ĵ iτ ĵ 3,A ( T,c ĵ µ A = ψγ µ γ 5 ψ µ ĵ µ A = µ ĵ µ A = Propagator with rotation [Vilenkin 198] ψ(χ, p ψ(χ, q ω = δ(χ χ, p qe ω 1 2 Σ χ iγ (χ + µ iγp (χ + µ 2 + p 2 Σ i = ɛ ijk i 4 [γ j, γ k ] ĵ µ A ω tr [ γ 5 γ µ γ ρ γ σ γ ν] = 4i ɛ µρσν Matteo Buzzegoli (Unifi QCD Chirality Workshop 217 March / 4