Phenomenological formulation of relativistic spin hydrodynamics Hidetoshi TAYA (Fudan University) based on arXiv: 1901.06615 [hep-th] in collaboration with K. Hattori (YITP), M. Hongo (RIKEN), X.-G. Huang (Fudan), M. Matsuo (UCAS) @YITP 2019/03/27
Ultra-relativistic heavy ion collisions Aim: study quark-gluon plasma (QGP) Found: QGP behaves like a perfect liquid and hydrodynamics works so well
Another interesting physics: Largest ω and B ω , , B Question: What happens under huge ω and/or B ? Specifically, any changes to QGP properties?
Naïve expectation: QGP is polarized Magnetic field B effect Zeeman splitting (Landau quantization) 𝐹 → 𝐹 − 𝑡 ⋅ 𝒓𝐶 ➡ charge dependent spin polarization Rotation ω effect Bernett effect 𝐹 → 𝐹 − 𝑡 ⋅ 𝜕 ➡ charge in in dependent spin polarization
Experimental fact STAR (2018) See also talk by T.Niida
How about theory? Hydrodynamics for spin polarized QGP? ➡ Far from complete See also talk by X.-G. Huang
Hydrodynamics for spin polarized QGP “Hydro simulations” exist, but… usual hydro (i.e., hydro w/o /o spin) is solved (1) Compute velocity gradient at freezeout and define thermal 𝜕 𝜈𝜉 ≡ 𝜖 𝜈 𝑣 𝜉 /𝑈 − 𝜖 𝜉 𝑣 𝜈 /𝑈 vorticity Becattini, Chandra, Del Zanna, Grossi (2013) (2) Use Cooper-Frye formula with spin 𝑔 𝐹 → 𝑔(𝐹 − 𝑡 ⋅ 𝜕) , Becattini, Florkowski, where 𝜕 is spin chemical potential ( ≠ 𝜕 in general) Speranza (2018) (3) Assume 𝜕 = 𝜕 (true only for global equilibrium) (4) Get spin-dependent hadron yield Formulation of relativistic spin hydrodynamics is still developing
Current status of formulation of spin hydro Non-relativistic case e.g. Eringen (1998); Lukaszewicz (1999) Already established (e.g. micropolar fluid) e.g. spintronics: - applied to pheno. and is successful Takahashi et al. (2015) - sp spin in must t be dis issipati tive because of mutual conversion between spin and orbital angular momentum Relativistic case Some trials exist, but - only for “ideal” fluid (no dissipative corrections) - some claim sp spin sh should be co conse served
Purpose of this talk Formulate relativistic spin hydrodynamics with 1 st order dissipative corrections for the first time Clarify spin should be dissipative Outline 1. Introduction 2. Formulation based on an entropy-current analysis 3. Linear mode analysis 4. Summary
Outline 1. Introduction 2. Formulation based on an entropy-current analysis 3. Linear mode analysis 4. Summary
Introduction to hydrodynamics w/o spin (1/3) Hydrodynamics is a low energy effective theory that describes spacetime evolution of IR modes (hydro modes) textbook by Landau, Lifshitz Phenomenological formulation (EFT construction) Step 1: Writ rite do down the the conservation law: 0 = 𝜖 𝜈 𝑈 𝜈𝜉 4 eqs Express 𝑼 𝝂𝝃 i.t. Step 2: Expr i.t.o hy hydro varia riables ( constitutive relation ) 1 + (4-1) = 4 DoGs - define hydro variables: {𝛾, 𝑣 𝜈 } (𝑣 2 = −1) chemical potential for 𝑄 𝜈 - write down all the possible tensor structures of 𝑈 𝜈𝜉 𝑈 𝜈𝜉 = 𝑔 1 𝛾 𝜈𝜉 + 𝑔 2 𝛾 𝑣 𝜈 𝑣 𝜉 4 𝛾 𝜖 𝜈 𝑣 𝜉 + 𝑔 3 𝛾 𝜗 𝜈𝜉𝜍𝜏 𝜖 𝜍 𝑣 𝜏 + 𝑔 5 𝛾 𝜖 𝜉 𝑣 𝜈 +𝑔 8 𝛾 𝑣 𝜈 𝜖 𝜈 𝑣 𝜉 + ⋯ + 𝑃(𝜖 2 ) 6 𝛾 𝜈𝜉 𝜖 𝜍 𝑣 𝜍 + 𝑔 7 𝛾 𝑣 𝜈 𝑣 𝜉 𝜖 𝜍 𝑣 𝜍 + 𝑔 +𝑔 - simplify the tensor structures by (assumptions in hydro) (1) symmetry (2) power counting ➡ gradient expansion (3) other physical requirements ➡ thermodynamics (see next slide) Hydrodynamic eq. = conservation law + constitutive relation
Introduction to hydrodynamics w/o spin (2/3) Constraints by thermodynamics Expand 𝑈 𝜈𝜉 i.t.o derivatives 𝜈𝜉 + 𝑈 𝜈𝜉 + 𝑃(𝜖 2 ) 𝑈 𝜈𝜉 = 𝑈 (0) (1) In static equilibrium 𝑈 𝜈𝜉 → 𝑈 𝜈𝜉 = (𝑓, 𝑞, 𝑞, 𝑞) , so that (0) 𝜈𝜉 = 𝑓𝑣 𝜈 𝑣 𝜉 + 𝑞( 𝜈𝜉 + 𝑣 𝜈 𝑣 𝜉 ) 𝑈 (0) 1 st law of thermodynamics says 𝑒𝑡 = 𝛾𝑒𝑓 , 𝑡 = 𝛾(𝑓 + 𝑞) By using EoM 0 = 𝜖 𝜈 𝑈 𝜈𝜉 , div. of entropy current 𝑇 𝜈 = 𝑡𝑣 𝜈 + 𝑃(𝜖) can be evaluated as 𝜖 𝜈 𝑇 𝜈 = −𝑈 1 𝜈𝜉 𝜖 𝜈 𝛾𝑣 𝜉 + 𝑃(𝜖 3 ) 2 st law of thermodynamics requires 𝜖 𝜈 𝑇 𝜈 ≥ 0 , which is guaranteed if RHS is expressed as a semi-positive bilinear as 𝜈𝜉 𝜖 𝜈 𝛾𝑣 𝜉 = σ 𝑌 𝑗 ∈ 𝑈 1 𝜇 𝑗 𝑌 𝑗 𝜈𝜉 𝑌 𝑗 𝜉𝜈 ≥ 0 with 𝜇 𝑗 ≥ 0 −𝑈 1 ex) heat current: 2ℎ (𝜈 𝑣 𝜉) ≡ ℎ 𝜈 𝑣 𝜉 + ℎ 𝜉 𝑣 𝜈 ∈ 𝑈 1 𝜈𝜉 (𝑣 𝜈 ℎ 𝜈 = 0) 𝜈𝜉 𝜖 𝜈 𝛾𝑣 𝜉 = −𝛾ℎ 𝜈 𝛾𝜖 ⊥𝜈 𝛾 −1 + 𝑣 𝜉 𝜖 𝜉 𝑣 𝜈 ≥ 0 ⇒ 𝑈 1 ⇒ ℎ 𝜈 = −𝜆 𝛾𝜖 ⊥𝜈 𝛾 −1 + 𝑣 𝜉 𝜖 𝜉 𝑣 𝜈 with 𝜆 ≥ 0
Introduction to hydrodynamics w/o spin (3/3) Constitutive relation up to 1 st order w/o spin 𝜈𝜉 = 𝑓𝑣 𝜈 𝑣 𝜉 + 𝑞( 𝜈𝜉 + 𝑣 𝜈 𝑣 𝜉 ) 𝑈 (0) 𝜈𝜉 = −2𝜆 𝐸𝑣 (𝜈 + 𝛾𝜖 ⊥ (𝜈 𝛾 −1 𝑣 𝜉) − 2𝜃𝜖 ⊥ <𝜈 𝑣 𝜉> − 𝜂 𝜖 𝜈 𝑣 𝜈 Δ 𝜈𝜉 𝑈 (1) shear sh ear visc scous eff effect bulk visc scous eff effect hea eat t curr rren ent Hydrodynamic equation w/o spin Hydrodynamic eq. = conservation law + constitutive relation 𝑈 𝜈𝜉 = 𝑈 Euler eq . 𝜈𝜉 0 = 𝜖 𝜈 𝑈 𝜈𝜉 (0) Navier-Stokes eq. 𝜈𝜉 + 𝑈 𝑈 𝜈𝜉 = 𝑈 𝜈𝜉 0 = 𝜖 𝜈 𝑈 𝜈𝜉 (0) (1) ⋮ ⋮ ⋮
Formulation of hydrodynamics with spin (1/4) Strategy is the same Phenomenological formulation Step 1: Writ rite do down the the co cons nservatio ion la law Step 2: Co Construct a co constitutiv ive rela latio ion - define hydro variables - write down all the possible tensor structures - simplify the tensor structures (1) symmetry (2) gradient expansion (3) thermodynamics
Formulation of hydrodynamics with spin (2/4) Step 1: Write down the conservation law (1) (1) ene nergy co cons nservation (2) total ang (2) ngula lar r mom omentum co cons nservatio ion 0 = 𝜖 𝜈 𝑁 𝜈,𝛽𝛾 𝜔(𝑦) → 𝑇 Λ 𝜔(Λ −1 𝑦) = 𝜖 𝜈 (𝑀 𝜈,𝛽𝛾 +Σ 𝜈,𝛽𝛾 ) 0 = 𝜖 𝜈 𝑈 𝜈𝜉 = 𝜖 𝜈 (𝑦 𝛽 𝑈 𝜈𝛾 − 𝑦 𝛾 𝑈 𝜈𝛽 + Σ 𝜈,𝛽𝛾 ) (canonical) ∴ 𝜖 𝜈 Σ 𝜈,𝛽𝛾 = 𝑈 𝛽𝛾 − 𝑈 𝛾𝛽 Spin is not conserved if (canonical) 𝑈 𝜈𝜉 has anti-symmetric part 𝑈 𝜈𝜉 (a) There’s no a priori reason (canonical) 𝑈 𝜈𝜉 must be symmetric Consequence (1) Spin must not be a hydro mode in a strict sense 𝜈𝜉 ≪ 1 (2) Nevertheless, it behaves like a hydro mode if 𝑈 (a) ➡ inclusion of dissipative nature is important
Formulation of hydrodynamics with spin (3/4) Step 2: Construct a constitutive relation for 𝑈 𝜈𝜉 , Σ 𝜈,𝛽𝛾 (1) define hydro variables 4 + 6 = 10 DoGs = # of EoMs Introduce spin chemical potential with 𝜕 𝜈𝜉 = −𝜕 𝜉𝜈 𝛾, 𝑣 𝜈 , 𝜕 𝜈𝜉 𝛾, 𝑣 𝜈 , 𝜕 𝜈𝜉 are independent w/ each other at this stage ( 𝜕 𝜈𝜉 ≠ thermal vorticity) (2) simplify the tensor structure by thermodynamics Expand 𝑈 𝜈𝜉 , Σ 𝜈,𝛽𝛾 , i.t.o derivatives 𝜈𝜉 + 𝑃 𝜖 2 , Σ 𝜈,𝛽𝛾 = 𝑣 𝜈 𝜏 𝛽𝛾 + 𝑃(𝜖 1 ) 𝑈 𝜈𝜉 = 𝑓𝑣 𝜈 𝑣 𝜉 + 𝑞 𝜈𝜉 + 𝑣 𝜈 𝑣 𝜉 + 𝑈 1 where I defined sp spin in den density 𝝉 𝜷𝜸 Generalizing 1 st law of thermodynamics with spin as 𝑒𝑡 = 𝛾(𝑒𝑓 − 𝜕 𝜈𝜉 𝑒𝜏 𝜈𝜉 ) , 𝑡 = 𝛾(𝑓 + 𝑞 − 𝜕 𝜈𝜉 𝜏 𝜈𝜉 ) With EoMs, div. of entropy current 𝑇 𝜈 = 𝑡𝑣 𝜈 + 𝑃(𝜖) can be evaluated as 𝜈𝜉 𝜖 𝜈 𝛾𝑣 𝜉 + 𝜖 𝜉 𝛾𝑣 𝜈 𝜖 𝜈 𝛾𝑣 𝜉 − 𝜖 𝜉 𝛾𝑣 𝜈 𝜖 𝜈 𝑇 𝜈 = −𝑈 1s 𝜈𝜉 − 2𝛾𝜕 𝜈𝜉 + 𝑃(𝜖 3 ) − 𝑈 1a 2 2 In global equilibrium 𝜖 𝜈 𝑇 𝜈 = 0 , so that 𝜕 = thermal vorticity. 2 nd law of thermodynamics 𝜖 𝜈 𝑇 𝜈 ≥ 0 gives strong constraint on 𝑈 1 𝜈𝜉
Formulation of hydrodynamics with spin (4/4) Constitutive relation for 𝑈 𝜈𝜉 up to 1 st order with spin 𝜈𝜉 = 𝑓𝑣 𝜈 𝑣 𝜉 + 𝑞( 𝜈𝜉 + 𝑣 𝜈 𝑣 𝜉 ) 𝑈 (0) hea eat t curr rren ent sh shea ear visc scous eff effect bulk visc scous eff effect 𝜈𝜉 = −2𝜆 𝐸𝑣 (𝜈 + 𝛾𝜖 ⊥ (𝜈 𝛾 −1 𝑣 𝜉) − 2𝜃𝜖 ⊥ <𝜈 𝑣 𝜉> − 𝜂 𝜖 𝜈 𝑣 𝜈 Δ 𝜈𝜉 𝑈 (1) −2𝜇 −𝐸𝑣 [𝜈 + 𝛾𝜖 ⊥ [𝜈 𝛾 −1 + 4𝑣 𝜍 𝜕 𝜍[𝜈 𝑣 𝜉] − 2𝛿 𝜖 ⊥ [𝜈 𝑣 𝜉] − 2Δ 𝜍 𝜈 Δ 𝜇 𝜉 𝜕 𝜍𝜇 “rotational (spinning) viscous effect” “boost heat current” NEW ! e.g. Eringen (1998); Lukaszewicz (1999) Relativistic generalization of a non-relativistic micropolar fluid “boost heat current” is a relativistic effect Hydrodynamics equation up to 1 st order with spin 𝛽𝛾 − 𝑈 1 𝛾𝛽 + 𝑃(𝜖 2 ) 𝜈𝜉 + 𝑈 1 𝜈𝜉 + 𝑃(𝜖 2 )) 𝜖 𝜈 (𝑣 𝜈 𝜏 𝛽𝛾 ) = 𝑈 0 = 𝜖 𝜈 (𝑈 0 (1)
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