Phenomenological formulation ofrelativistic spin hydrodynamics
Hidetoshi TAYA (Fudan University)
based on arXiv: 1901.06615 [hep-th]
in collaboration withK. 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 liquidand 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 effectZeeman splitting (Landau quantization)
𝐸 → 𝐸 − 𝑠 ⋅ 𝒒𝐵
➡ charge dependent spin polarization
 Rotation ω effectBernett effect
𝐸 →𝐸−𝑠⋅𝜔➡ charge 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 spin) is solved(1) Compute velocity gradient at freezeout and define thermal
Becattini, Chandra, Del Zanna,vorticity 𝜔෥ 𝜇𝜈 ≡ 𝜕𝜇 𝑢𝜈 /𝑇 − 𝜕 𝜈 𝑢𝜇 /𝑇Grossi (2013)(2) Use Cooper-Frye formula with spin 𝑓 𝐸 → 𝑓(𝐸 − 𝑠 ⋅ 𝜔),Florkowski,
where 𝜔 is spin chemical potential (≠ 𝜔෥ in general) Becattini,
Speranza (2018)(3) Assume 𝜔 = 𝜔
෥ (true only for global equilibrium)(4) Get spin-dependent hadron yield Formulation of relativistic spin hydrodynamics isstill developingCurrent status of formulation of spin hydro Non-relativistic casee.g. Eringen (1998); Lukaszewicz (1999)

Already established (e.g. micropolar fluid)- applied to pheno. and is successful

e.g. spintronics:
Takahashi et al. (2015)- spin must be dissipative because of mutual conversion
between spin and orbital angular momentum

 Relativistic caseSome trials exist, but- only for “ideal” fluid (no dissipative corrections)- some claim spin should be conservedPurpose of this talk
 Formulate relativistic spin hydrodynamics with
1st order dissipative corrections for the first time Clarify spin should be dissipative
Outline1. 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: Write down the conservation law: 0 = 𝜕𝜇 𝑇𝜇𝜈

4 eqs

Step 2: Express 𝑻𝝁𝝂 i.t.o hydro variables (constitutive relation)
1 + (4-1) = 4 DoGs
- define hydro variables: {𝛽, 𝑢𝜇 } (𝑢2 = −1) chemical potential for 𝑃𝜇- write down all the possible tensor structures of 𝑇𝜇𝜈𝑇𝜇𝜈 = 𝑓1 𝛽 𝑔𝜇𝜈 + 𝑓2 𝛽 𝑢𝜇 𝑢𝜈
+𝑓3 𝛽 𝜖 𝜇𝜈𝜌𝜎 𝜕𝜌 𝑢𝜎 + 𝑓4 𝛽 𝜕𝜇 𝑢𝜈 + 𝑓5 𝛽 𝜕 𝜈 𝑢𝜇+𝑓6 𝛽 𝑔𝜇𝜈 𝜕𝜌 𝑢𝜌 + 𝑓7 𝛽 𝑢𝜇 𝑢𝜈 𝜕𝜌 𝑢𝜌 + 𝑓8 𝛽 𝑢𝜇 𝜕𝜇 𝑢𝜈 + ⋯ + 𝑂(𝜕 2 )- 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 𝑇𝜇𝜈 i.t.o derivatives𝜇𝜈𝜇𝜈𝑇𝜇𝜈 = 𝑇(0) + 𝑇(1) + 𝑂(𝜕 2 )In static equilibrium 𝑇𝜇𝜈𝜇𝜈→ 𝑇(0) = (𝑒, 𝑝, 𝑝, 𝑝), so that𝜇𝜈

𝑇(0) = 𝑒𝑢𝜇 𝑢𝜈 + 𝑝(𝑔𝜇𝜈 + 𝑢𝜇 𝑢𝜈 )1st law of thermodynamics says
𝑑𝑠 = 𝛽𝑑𝑒, 𝑠 = 𝛽(𝑒 + 𝑝)
By using EoM 0 = 𝜕𝜇 𝑇𝜇𝜈 , div. of entropy current 𝑆𝜇 = 𝑠𝑢𝜇 + 𝑂(𝜕) can be evaluated as𝜇𝜈𝜕𝜇 𝑆𝜇 = −𝑇 1 𝜕𝜇 𝛽𝑢𝜈 + 𝑂(𝜕 3 )2st law of thermodynamics requires 𝜕𝜇 𝑆𝜇 ≥ 0, which is guaranteed if RHS isexpressed as a semi-positive bilinear as𝜇𝜈𝜇𝜈−𝑇 1 𝜕𝜇 𝛽𝑢𝜈 = σ𝑋 ∈ 𝑇 𝜆𝑖 𝑋𝑖 𝑋𝑖 𝜈𝜇 ≥ 0 with 𝜆𝑖 ≥ 0𝑖1𝜇ex) heat current: 2ℎ(𝜇 𝑢𝜈) ≡ ℎ𝜇 𝑢𝜈 + ℎ𝜈 𝑢𝜇 ∈ 𝑇 𝜇𝜈
1 (𝑢𝜇 ℎ = 0)𝜇𝜈⇒ 𝑇 1 𝜕𝜇 𝛽𝑢𝜈 = −𝛽ℎ𝜇 𝛽𝜕⊥𝜇 𝛽 −1 + 𝑢𝜈 𝜕𝜈 𝑢𝜇 ≥ 0⇒ ℎ𝜇 = −𝜅 𝛽𝜕⊥𝜇 𝛽−1 + 𝑢𝜈 𝜕𝜈 𝑢𝜇 with 𝜅 ≥ 0

Introduction to hydrodynamics w/o spin (3/3)
 Constitutive relation up to 1st order w/o spin𝜇𝜈𝑇(0) = 𝑒𝑢𝜇 𝑢𝜈 + 𝑝(𝑔𝜇𝜈 + 𝑢𝜇 𝑢𝜈 )(𝜇𝜇𝜈<𝜇𝑇(1) = −2𝜅 𝐷𝑢(𝜇 + 𝛽𝜕⊥ 𝛽−1 𝑢𝜈) − 2𝜂𝜕⊥ 𝑢𝜈> − 𝜁 𝜕𝜇 𝑢𝜇 Δ𝜇𝜈heat current

shear viscous effectbulk viscous effect Hydrodynamic equation w/o spin
Hydrodynamic eq. = conservation law + constitutive relation

Euler eq.

0 = 𝜕𝜇 𝑇𝜇𝜈𝑇𝜇𝜈 = 𝑇(0)Navier-Stokes eq.

0 = 𝜕𝜇 𝑇𝜇𝜈𝑇𝜇𝜈 = 𝑇(0) + 𝑇(1)⋮⋮𝜇𝜈𝜇𝜈

⋮𝜇𝜈Formulation of hydrodynamics with spin (1/4)
 Strategy is the same
Phenomenological formulation
Step 1: Write down the conservation law
Step 2: Construct a constitutive relation- 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) energy conservation(2) total angular momentum conservation

0 = 𝜕𝜇 𝑀𝜇,𝛼𝛽0 = 𝜕𝜇 𝑇𝜇𝜈(canonical)

𝜓(𝑥) → 𝑆 Λ 𝜓(Λ−1 𝑥)= 𝜕𝜇 (𝐿𝜇,𝛼𝛽 +Σ 𝜇,𝛼𝛽 )= 𝜕𝜇 (𝑥 𝛼 𝑇𝜇𝛽 − 𝑥 𝛽 𝑇𝜇𝛼 + Σ 𝜇,𝛼𝛽 )∴ 𝜕𝜇 Σ𝜇,𝛼𝛽 = 𝑇 𝛼𝛽 − 𝑇𝛽𝛼𝜇𝜈

 Spin is not conserved if (canonical) 𝑇𝜇𝜈 has anti-symmetric part 𝑇(a) There’s no a priori reason (canonical) 𝑇𝜇𝜈 must be symmetricConsequence(1) Spin must not be a hydro mode in a strict sense(2) Nevertheless, it behaves like a hydro mode if 𝑇(a) ≪ 1𝜇𝜈

➡ inclusion of dissipative nature is importantFormulation of hydrodynamics with spin (3/4)
Step 2: Construct a constitutive relation for 𝑇𝜇𝜈 , Σ 𝜇,𝛼𝛽(1) define hydro variables

4 + 6 = 10 DoGs = # of EoMs

Introduce spin chemical potential 𝛽, 𝑢𝜇 , 𝜔 𝜇𝜈with 𝜔 𝜇𝜈 = −𝜔 𝜈𝜇𝛽, 𝑢𝜇 , 𝜔 𝜇𝜈 are independent w/ each other at this stage (𝜔 𝜇𝜈 ≠thermal vorticity)(2) simplify the tensor structure by thermodynamics
Expand 𝑇𝜇𝜈 , Σ 𝜇,𝛼𝛽 , i.t.o derivatives𝜇𝜈𝑇𝜇𝜈 = 𝑒𝑢𝜇 𝑢𝜈 + 𝑝 𝑔𝜇𝜈 + 𝑢𝜇 𝑢𝜈 + 𝑇 1 + 𝑂 𝜕 2 ,Σ 𝜇,𝛼𝛽 = 𝑢𝜇 𝜎 𝛼𝛽 + 𝑂(𝜕1 )where I defined spin density 𝝈𝜶𝜷Generalizing 1st law of thermodynamics with spin as
𝑑𝑠 = 𝛽(𝑑𝑒 − 𝜔𝜇𝜈 𝑑𝜎𝜇𝜈 ), 𝑠 = 𝛽(𝑒 + 𝑝 − 𝜔𝜇𝜈 𝜎 𝜇𝜈 )With EoMs, div. of entropy current 𝑆𝜇 = 𝑠𝑢𝜇 + 𝑂(𝜕) can be evaluated as𝜇𝜈 𝜕𝜇 𝛽𝑢𝜈 + 𝜕𝜈 𝛽𝑢𝜇𝜇𝜈 𝜕𝜇 𝛽𝑢𝜈 − 𝜕𝜈 𝛽𝑢𝜇
𝜕𝜇 𝑆𝜇 = −𝑇 1s− 𝑇 1a− 2𝛽𝜔𝜇𝜈 + 𝑂(𝜕 3 )22 In global equilibrium 𝜕𝜇 𝑆𝜇 = 0, so that 𝜔 = thermal vorticity.
𝜇𝜈

 2nd law of thermodynamics 𝜕𝜇 𝑆𝜇 ≥ 0 gives strong constraint on 𝑇 1

Formulation of hydrodynamics with spin (4/4)
 Constitutive relation for 𝑇𝜇𝜈

up to 1st order with spin𝜇𝜈

𝑇(0) = 𝑒𝑢𝜇 𝑢𝜈 + 𝑝(𝑔𝜇𝜈 + 𝑢𝜇 𝑢𝜈 )heat current

shear viscous effect bulk viscous effect(𝜇𝜇𝜈<𝜇𝑇(1) = −2𝜅 𝐷𝑢(𝜇 + 𝛽𝜕⊥ 𝛽−1 𝑢𝜈) − 2𝜂𝜕⊥ 𝑢𝜈> − 𝜁 𝜕𝜇 𝑢𝜇 Δ𝜇𝜈[𝜇[𝜇“boost heat current”“rotational (spinning) viscous effect”

𝜇−2𝜆 −𝐷𝑢[𝜇 + 𝛽𝜕⊥ 𝛽−1 + 4𝑢𝜌 𝜔 𝜌[𝜇 𝑢𝜈] − 2𝛾 𝜕⊥ 𝑢𝜈] − 2Δ𝜌 Δ𝜆𝜈 𝜔 𝜌𝜆NEW !

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 1st order with spin𝜇𝜈𝜇𝜈

0 = 𝜕𝜇 (𝑇 0 + 𝑇 1 + 𝑂(𝜕 2 ))𝛼𝛽𝛽𝛼

𝜕𝜇 (𝑢𝜇 𝜎 𝛼𝛽 ) = 𝑇(1) − 𝑇 1 + 𝑂(𝜕 2 )Outline

1. Introduction

2. Formulation based on an entropy-current analysis
3. Linear mode analysis
4. SummaryLinear mode analysis (1/2)
Setup: small perturbations on top of static equilibrium
propagate if Im 𝜔 = 0𝑖𝜕𝑡𝛿𝛽𝛿𝛽𝜇= 𝑀 𝛿𝑢𝜇 + 𝑂(𝛿 2 )
𝛿𝑢
𝛿𝜔 𝜇𝜈
𝛿𝜔 𝜇𝜈
hydro evolution𝛽 = 𝛽0𝛽 = 𝛽0 + 𝛿𝛽

𝑢𝜇 = 1, 𝟎𝑢𝜇 = 1, 𝟎 + 𝛿𝑢𝜇𝜔 𝜇𝜈 = 0𝜔 𝜇𝜈 = 0 + 𝛿𝜔 𝜇𝜈

dissipate if Im 𝜔 < 0unstable if Im 𝜔 > 0

Linear mode analysis (2/2)
 Hydro w/o spin {𝛽, 𝑢𝜇 }

 Hydro with spin {𝛽, 𝑢𝜇 , 𝜔 𝜇𝜈 }4 modes

4 modes2 sound modes 𝜔 = ±𝑐s 𝑘 + 𝑂(𝑘 2 )𝜂𝑘 2
2 shear modes 𝜔 = −𝑖+ 𝑂(𝑘 4 )𝑒+𝑝where 𝑐s2 ≡ 𝜕𝑝/𝜕𝑒

2 sound modes 𝜔 = ±𝑐s 𝑘 + 𝑂(𝑘 2 )𝜂𝑘 22 shear modes 𝜔 = −𝑖+ 𝑂(𝑘 4 )𝑒+𝑝+ 6 dissipative modes

3 “boost” modes 𝜔 = −2𝑖𝜏b−1 + 𝑂(𝑘 2 )3 “spin” modes 𝜔 = −2𝑖𝜏s−1 + 𝑂(𝑘 2 )𝜕𝜎𝑖𝑗 Τ𝜕𝜔𝑖𝑗
𝜕𝜎𝑖0 Τ𝜕𝜔𝑖0
where 𝜏s ≡, 𝜏b ≡4𝛾
4𝜆 We explicitly confirmed that spin is dissipative The time-scale of the dissipation is controlled by the new viscous constants 𝛾, 𝜆Outline

1. Introduction

2. Formulation based on an entropy-current analysis
3. Linear mode analysis
4. Summary

Summary Spin polarization of QGP is one of the hottest topics inHIC. But, its theory, in particular hydrodynamicframework, has not been developed well
 Relativistic spin hydrodynamics with 1st order dissipative
corrections is formulated based on the phenomenologicalentropy-current analysis

 Spin must be dissipative because of the mutual conversion
between the orbital angular momentum and spin Linear mode analysis of the spin hydrodynamic equation
also suggests that spin must be dissipative
Outlook: extension to 2nd order, Kubo formula,application to cond-mat, numerical simulations,and start something new with you!