Phenomenological formulation ofrelativistic spin hydrodynamics

Hidetoshi TAYA (Fudan; h_taya@fudan.edu.cn)

with K. Hattori (YITP), M. Hongo (Illinois), X.-G. Huang (Fudan), M. Matsuo (UCAS)
Ref: Hattori, Hongo, Huang, Matsuo, HT, Phys. Lett. B795, 100 (2019) [arXiv:1901.06615]

1. Introduction
1-1. Spin-polarization in HIC

1-2. Theoretical status of spin hydro✅ huge vorticity 𝝎 magnetic field 𝑩 ✅ spin-polarization by 𝝎, 𝑩
in non-central collisions
e.g. Barnett effect, Zeeman splittingB✅ experimentally confirmed already !

ωHydrodynamics is a powerful framework to describe QGP.
However, it is far from complete for spin-polarized QGP✅ “hydro simulations” exist, but…
- ordinary hydro (i.e., hydro w/o spin) is solved- thermal vorticity 𝜔෥ 𝜇𝜈 ≡ 2𝜕 [𝜇 𝛽𝑢𝜈] is converted into spin via Cooper-Frye✅ relativistic spin hydro is still under construction- established for non-relativistic case, but not for relativistic case 😡

QGPe.g. micropolar fluid: Eringen (1998); Lukaszewicz (1999)- some trials exist, but only at the “ideal” level (no dissipative corr.) 😡
- controversy exists 😡 on the conservation of spin
⇒ some relativistic people claim spin is conserved, while it is establishedin non-relativistic spin hydro that spin must NOT be conserved[STAR, PRC 98, 014910 (2018)]

Aim(1) Formulate relativistic spin hydro including 1st order dissipative corrections for the first time(2) Clarify that spin must be a dissipative quantity

2. Phenomenological formulation based on entropy-current analysis
2-1. hydro without spin (review)

2-2. hydro with spin

step 1 Write down the conservation law

step 1 Write down the conservation laws

Conservation of energy-momentum 0 = 𝜕𝜇 𝑇
✅ 4 equations

Conservation of EM 0 = 𝜕𝜇 𝑇𝜇𝜈 and total angular mom. 0 = 𝜕𝜇 𝑀𝜇,𝛼𝛽✅ 4 + 6 equations✅ Define spin tensor by Σ𝜇,𝛼𝛽 ≡ 𝑀𝜇,𝛼𝛽 − 𝐽𝜇,𝛼𝛽 ≡ 𝑀𝜇,𝛼𝛽 − (𝑥 𝛼 𝑇𝜇𝛽 − 𝑥 𝛽 𝑇𝜇𝛼 )𝛼𝛽𝜇,𝛼𝛽𝜇,𝛼𝛽𝛼𝛽𝛽𝛼⇒ 0 = 𝜕𝑢 𝑀
⇔ 𝜕𝑢 Σ= 𝑇 − 𝑇 ≡ 2𝑇 a(canonical) orbital
angular momentum⇒ spin is not conserved if 𝑇 𝜇𝜈
≠0a𝜇𝜈
✅ There is no reason 𝑇 must be symmetric⇒ spin must not be a hydro mode in a strict sense𝜇𝜈⇒ nevertheless, it behaves like a hydro mode if 𝑇 a ≪ 1,for which inclusion of dissipation is crucially important𝜇𝜈

step 2 Express 𝑻𝝁𝝂 i.t.o hydro variables (constitutive relation)[2-1] Define hydro variables
Slow modes that (you expect to) survive in IR limit: 𝛽, 𝑢𝜇 w/ 𝑢2 = −1
✅ 1 + 3 = 4 DoGs = # of equations[2-2] 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 )𝝁𝝂𝝁,𝜶𝜷Express𝑻,𝜮
i.t.o hydro variables (constitutive relations)step 2[2-1] Define hydro variables

Assume spin is slow and introduce spin chemical potential 𝜔𝜇𝜈
as a new hydro variable: 𝛽, 𝑢𝜇 , 𝝎𝝁𝝂 w/ 𝜔𝜇𝜈 = −𝜔𝜈𝜇
✅ 4 + 6 = 10 DoGs = # of equations✅ in general, 𝜔𝜇𝜈 ≠ 2𝜕[𝜇 (𝛽𝑢𝜈] ) = (thermal vorticity)[2-2] Write down all the possible tensor structures of 𝑻𝝁𝝂 , 𝜮𝝁,𝜶𝜷[2-3] Simplify the tensor structures with some assumptions(1) symmetry(2) power counting (gradient expansion)(3) thermodynamics (entropy-current analysis)𝑇𝜇𝜈 = too many terms , Σ 𝜇,𝛼𝛽 = (too many terms)[2-3] Simplify the tensor structures with some assumptions

Define energy 𝑒 & pressure 𝑝 by (matching condition)𝜇𝜈𝜇𝜈𝜇𝜈
𝜇𝜈
2𝜇 𝜈𝜇𝜈𝜇 𝜈𝑇 ≡ 𝑇 0 + 𝑇 1 + 𝑂 𝜕 ≡ 𝑒𝑢 𝑢 + 𝑝(𝑔 + 𝑢 𝑢 ) + 𝑇 1 + 𝑂 𝜕 2(1) symmetry(2) power counting (gradient expansion)(3) thermodynamics (entropy-current analysis)

1st law of thermodynamics: d𝑠 = 𝛽d𝑒, 𝑠 = 𝛽(𝑒 + 𝑝)𝜇𝜈𝜇nd
2 law of thermodynamics: 0 ≤ 𝜕𝜇 𝑠 = −𝑇 1 𝜕𝜇 𝛽𝑢𝜈 + 𝑂(𝜕 3 )Define spin density 𝜎 𝜇𝜈 by= σ𝑋 ∈ 𝑇𝑖

1𝜇𝜈
𝜆𝑖 𝑋𝑖 𝑋𝑖 𝜈𝜇≥ 0 with 𝜆𝑖 ≥ 0✅ give strong constraints on 𝑇𝜇𝜈𝜇𝜈ex) heat current: 2ℎ(𝜇 𝑢𝜈) ≡ ℎ𝜇 𝑢𝜈 + ℎ𝜈 𝑢𝜇 ∈ 𝑇 1✅ RHS is a semi-positive bilinear ⇒ constraints on 𝑇𝜇𝜈 , 𝛴𝜇,𝛼𝛽✅ in global equilibrium, 𝜔𝜇𝜈 = 2𝜕[𝜇 (𝛽𝑢𝜈] ) = (thermal vorticity) holds(𝑢𝜇 ℎ𝜇 = 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

𝜮𝝁,𝜶𝜷 = 𝒖𝝁 𝝈𝜶𝜷𝑻𝝁𝝂 = 𝒆𝒖𝝁 𝒖𝝂 + 𝒑 𝒈𝝁𝝂 + 𝒖𝝁 𝒖𝝂
(𝝁 −𝟏<𝝁 𝝂>(𝝁𝝂)−𝟐𝜿 𝑫𝒖 + 𝜷𝝏⊥ 𝜷𝒖 − 𝟐𝜼𝝏⊥ 𝒖 − 𝜻 𝝏𝝁 𝒖𝝁 𝜟𝝁𝝂𝑻𝝁𝝂 = 𝒆𝒖𝝁 𝒖𝝂 + 𝒑 𝒈𝝁𝝂 + 𝒖𝝁 𝒖𝝂
(𝝁 −𝟏<𝝁 𝝂>(𝝁𝝂)−𝟐𝜿 𝑫𝒖 + 𝜷𝝏⊥ 𝜷𝒖 − 𝟐𝜼𝝏⊥ 𝒖 − 𝜻 𝝏𝝁 𝒖𝝁 𝜟𝝁𝝂heat current

Goal

shear viscosity[𝝁−𝟐𝝀 −𝑫𝒖

[𝝁 −𝟏+ 𝜷𝝏⊥ 𝜷 + 𝟒𝒖𝝆 𝝎𝝆[𝝁𝝂]𝒖 − 𝟐𝜸[𝝁 𝝂]𝝁 𝝂 𝝆𝝀𝝏⊥ 𝒖 − 𝟐𝜟𝝆 𝜟𝝀 𝝎

rotational (spinning) viscosity
NEW !! boost heat current✅ a generalization of the non-relativistic micropolar fluid

bulk viscosity

Hydro eq. = conservation law + constitutive relation≡ 𝑢𝜇 𝜎 𝛼𝛽 + 𝑂(𝜕1 )1st law w/ spin: d𝑠 = 𝛽(d𝑒 − 𝜔𝜇𝜈 d𝜎 𝜇𝜈 ), 𝑠 = 𝛽(𝑒 + 𝑝 − 𝜔𝜇𝜈 𝜎 𝜇𝜈 )𝜇𝜈𝜇nd2 law w/ spin: 0 ≤ 𝜕𝜇 𝑠 = −𝑇 1s 𝜕𝜇 𝛽𝑢𝜈𝜇𝜈−𝑇 1a 𝜕[𝜇 (𝛽𝑢𝜈] ) − 2𝛽𝜔𝜇𝜈 + 𝑂(𝜕 3 )2nd law is satisfied if RHS is a semi-positive bilinear, i.e.,𝜇𝜈−𝑇 1 𝜕𝜇 𝛽𝑢𝜈

𝜇,𝛼𝛽𝜇,𝛼𝛽Σ≡ Σ 0 + 𝑂 𝜕1

Goal

Hydro eqs. = conservation laws + constitutive relations

3. Linear mode analysis
2-1. setup

2-2. Linearized hydro eq.

2-3. Eigenvalues of 𝑴

Consider perturbations
on top of a global static
thermal equil. config.

Expand hydro eqs. i.t.o. the pert. 𝛿Usual 4 gapless modes + 6 diffusive gapped modes appear

𝛽 = 𝛽0 + 𝛿𝛽𝑢𝜇 = 1, 𝟎 + 𝛿𝑢𝜇𝜔 𝜇𝜈 = 0 + 𝛿𝜔 𝜇𝜈

𝛿𝛽𝛿𝛽𝑖𝜕𝑡 𝛿𝑢𝜇 = 𝑀 𝛿𝑢𝜇 + 𝑂(𝛿 2 )
𝛿𝜔 𝜇𝜈
𝛿𝜔 𝜇𝜈
✅ Eigenvalues 𝜔 of 𝑀 determine how
the perturbations evolve in time:
Im 𝜔 = 0: propagate (𝛿 ∼ const)
Im 𝜔 < 0: diffusive (𝛿 → 0)2)
𝜔=±𝑐𝑘+𝑂(𝑘2 sound modes:s22𝜔=−i𝜂𝑘/(𝑒+𝑝)+𝑂(𝑘)2 shear modes:

usual 4 gapless modes✅ completely the same as hydro w/o spin−12𝜔=−2𝐢𝜏+𝑂(𝑘)3 “spin” modes:
b6 gapped modes associated w/ spin DoG✅ Im 𝜔 < 0 ⇒ spin must be diffusive✅ time-scale is controlled by the newviscous constants 𝛾, 𝜆3 “boost” modes: 𝜔 = −2𝐢𝜏s−1 + 𝑂(𝑘 2 )𝜕𝜎 𝑖𝑗 Τ𝜕𝜔𝑖𝑗
𝜕𝜎 𝑖0 Τ𝜕𝜔𝑖0
where 𝜏s ≡, 𝜏b ≡4𝛾
4𝜆

4. Summary✅ Relativistic spin hydrodynamics with 1st order dissipative corrections is formulated for the first time
based on the phenomenological entropy-current analysis✅ Spin must be dissipative because of the mutual conversion b/w orbital angular momentum and spin✅ Linear mode analysis of the spin hydro equation found 6 new diffusive modes, which means that spin
is dissipative, and time-scale of the dissipation is controlled by the new viscous constants 𝜸, 𝝀✅ Outlook:extension to 2nd order, Kubo formula,
MHD, application to cond-mat,numerical simulations