相対論的流体力学の理論的進展(傾向と対策)本郷 優 (新潟大学)

2022/04/30, 名古屋大学 & Zoom (ハイブリッド)
第39回Heavy Ion Cafe & 第35回Heavy Ion Pub 合同研究会「ポストQM2022」注意 (言い訳)New developments in relativistic hydrodynamics
Nora Weickgenannt

Quark Matter 22, Krakow, Poland | April 9, 2022

What is relativistic hydrodynamics really ?

CRC-TR←ここに焦点Hydrodynamics successful even in regimes where not expected to be applicable,in particular far from local equilibrium
Nora Weickgenannt

Relativistic hydrodynamics

3 / 35

流体モデルを拡張する
1. 新しい自由度を導入する

臨界モード，パイオン，スピン

2. 新しい項を見つける
Thermal shear(3. 上の2つを現象に適用する)
スピン偏極

流体モデルを拡張する
1. 新しい自由度を導入する

臨界モード，パイオン，スピン

2. 新しい項を見つける
Thermal shear(3. 上の2つを現象に適用する)
スピン偏極

臨界ダイナミクス その1
Talk by Eduardo Grossi

So8 pions near the QCD chiral
cri=cal point: transport anddynamics
Eduardo Grossi
IPhT Saclay, Ecole Polytechnique
E.G., A.Soloviev, D. Teaney, F. Yan PRD (2020)
E.G., A. Soloviev, D. Teaney, F. Yan PRD (2021)
A. Florio, E.G., A. Soloviev, D, Teaney PRD (2022)

QM 2022 Krakow 5-4-20221

Mo=va=on

We are neglec=ng any hydro-dynamics of the chiral condensate !
2

Setup: the O(4) phase transi=on
The (approximated) conserved quan==es of 2 ﬂavour QCD are
µ
JV

µ
JA

Iso-vector (isospin)µV
0q̄γ tI q

Iso-axialµA
0q̄γ γ5 tI q

µν

TStressµ(T, u )
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The approximate ﬂavour symmetry SU (2)L × SU (2)R ∼ O(4)<latexit sha1_base64="ZkUHOyG6jyHszI2CT9Cl8VjyjwM=">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</latexit>

The order parameter is the chiral condensate
hq̄qi ⇠ φα = (σ, ϕα ) = (sigma, pions)

We need the hydrodynamic theory of the charge and the orderparameter
4

Rajagopal Wilczek (93)
Son and Stephanov (02)Equa=on of mo=on (Model G)
Rajagopal Wilczek (93)

Chiral condensate ϕa + Axial and Vector charge nab = χ0 μab∂t φa + g0 µab φb = Γ0 r2 φaΓ0 (m20 + λφ2 )φa + Γ0 Ha + θa ,2i∂t nab + g0 r · (rφ[a φb] ) + H[a φb] = D0 r nab + ∂i Ξab .

Ideal part

Dissipa=ve partGaussian Noise•

The ideal part is charge conserva=on and Josephson constraint•

Two dissipa=ve coeﬃcient Γ0 and D0 and noise•

The simula=on of the stochas=c process is done with an
ideal step and metropolis update.
Diﬀusion at high temperature, pion propaga=on at low
temperature as the vev develops
5High temperature
1
Gσσ (t, k) ⌘ hσ(t, k)σ(0, k)ic ,V
We focus on the sta=s=cal
1 X
Gππ (t, k) ⌘hπs (t, k)πs (0, k)ic ,3V scorrelator at k = 0
1 X s
hnA (t, k)nsA (0, k)ic ,GAA (t, k) ⌘3V s
50

GAA/100
GG 

G() / 4030

z=3.8720
100-0.4 -0.3 -0.2 -0.1

0


0.1

0.2

0.3

0.4

The axial charge is almost conserved the O(4) ﬁeld are simply
dissipate with a broad width7

Quark diﬀusion T>>Tc

450
1

zpc
0.7zpcz=0
z=-2.2

0.8
0.6

400
350
GAA() / 0

GAA(t) / 0

Propaga=on of axial charge across thetransi=on

0.4
0.2
0

300
250
200
150

-0.2

100

-0.4

50

-0.6

0
-0.03 -0.02 -0.01

0200 400 600 800 1000 1200 1400
t

0


0.01

0.02

0.03

Quasipar=cle Pions T<<Tc
Around Tpc the axial charge start changing form a diﬀusive form
to a quasipar=cle one
8Dynamical scaling on the cri=cal line
On the Cri=cal line z=0 we should have scaling with a dynamical
cri=cal exponent ζ1

H=0.002
H=0.003
H=0.004
H=0.006
H=0.01

0.6

0.8

z=0

0.6
 
GAA(t) / 0

0.8

GAA(t) / 0

1

0.4
0.2

0.4

H=0.002
H=0.003
H=0.004
H=0.006
H=0.01

0.2

0

0

-0.2

-0.2

Href = 0.002

-0.4

-0.4
0

500

1000
t

1500

2000
0.8

0

300

600
900
t(Href/ H) 1.47  c

1200

Mesure: ζ = 1.47 ± 0.01
Expected: ζ = d/2 = 1.5

GAA (t, H)= YAc H ζνc t ,χ09Rajagopal Wilczek (93)

Eﬀec=ve Boltzmann equa=on
E.G., A.Soloviev, D. Teaney, F. Yan PRD (2020)

From the linear propagator we can deﬁne (using the Wignertransform) an eﬀec=ve kine=c descrip=on of the so8 pions
distribu=on func=on∂Ep ∂fπ∂Ep ∂fπ−= interaction terms∂ t fπ +
∂q ∂x
∂x ∂q

Well below the phase transi=on the pions propagate likequasipar=lce with a modiﬁed energy dispersion from the medium222

Ep = v (p + m )<latexit sha1_base64="7kqIgQfW621jxKZYQZAtQARsnmI=">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</latexit>

Depends on σ̄
10

So8 pion enhancement
E.G., A. Soloviev, D. Teaney, F. Yan PRD (2021)

The dispersion curve get modiﬁed form the phase transi=on
1n(Ep ) = E /T
Ep = v 2 (p)(p2 + m2 )e p −1<latexit sha1_base64="3M5GDnVn1wdk9jsz2Dvdo7jAexw=">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</latexit><latexit sha1_base64="q+p8i9HXqz1QWUkMKMZU9KU0Y1w=">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</latexit>

2

1

Vacum

critical

1.8

0.8

Nmodified /Nvac

ratio

v2(p)

1.6

0.6
0.4

1.4
Λ/πT=0.8

Cri=calΛ/πT=1.2

1.2

0.2

Guessed dispersion curve

100

vacuum

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
p (GeV)0

0.2

Pion enhanced p < 0.5 GeV
11

0.4

0.6

p (GeV)

0.8

1

臨界ダイナミクス その2
Talk by Xin An

Evolution of Non-Gaussian Hydrodynamic Fluctuations
Xin AnÌ

based on PRL 127, 072301 and work in progresswith G. Başar, M. Stephanov and H.-U. Yee

XXIXth Quark Matter, Kraków, Poland
Apr 5, 2022

Ì

Non-Gaussian & non-equilibrium fluctuations• Gaussian distribution : normality manifested by central limit theorem .
GaussianP(x ) =  12  e-( x -) 2

2 2

-



+

• Fluctuations in general deviate from normal distribution (i.e., non-Gaussian)and/or evolve towards equilibration (i.e., non-equilibrium) .

Xin An (UNC)

Evolution of Non-Gaussian Hydrodynamic Fluctuations

3/16

Stochastic description• Langevin equation for a set of variables ψi :(Newton’s 2nd law)

∂ t ψ i = Fi + ξ i ,hξi (x1 )ξj (x2 )i = 2Qij δ (4) (x1x2 ) ,(FDT)

where Fi ⌘ drift force ; ξi ⌘ random force (noise) ; Qij ⌘ Onsager matrix .

10

10

5

500

5

5

0

0

0

0

−5

−500

−5

−5

10 −1000

−10
−10

−10
−10

−5

0
x (fm)

5ξ

1,1

y (fm)

3ξ

1,1

y (fm)

3

1000(Gev/fm )10(Gev/fm )• It suffers from the problem of infinite noise and multiplicative noise .

−50
x (fm)

5

10 −10regularize the infinite noise in stochastic sampling Singh et al., 2019Landau et al., 1959; Kapusta et al., 2012; Young et al., 2015; Sakaida et al., 2017; Nahrgang et al., 2018; etc. See also Posters by De and Pihan
Xin An (UNC)

Evolution of Non-Gaussian Hydrodynamic Fluctuations

5/16

Deterministic description• The Langevin equation is complementary to Fokker-Planck equation forprobability distribution P [t; ψi ] :probability flux

∂t P =z⇣}|⌘{
Fi P + (Qij P ),j,i,(Ito’s prescription)

where (. . .),i = δ(. . .)/δψi andentropy

S = log Peq

Poisson/Onsager matrix

Mij = Ωij + Qij

drift force

Fi = Mij S,j + Mij,j• Evolution equation for n-th cumulant (n-point connected function)
Gcn ⌘ hψ1 . . . ψn ic : see An et al., PRL, 2021 for details∂t P
Xin An (UNC)=)∂t Gcn = F [{Gcn }] .

Evolution of Non-Gaussian Hydrodynamic Fluctuations

6/16

Small parameters• Small parameters in, e.g., hydrodynamics : Akamatsu et al., 2017; An et al., 2019parameter expression"

meaningrole`mic /`

Knudsen number

controls gradient expansion(`mic /`)3

inverse of uncorrelated DOF

controls loop expansionb~-1hydro-cellmrℓmic ~ T -1

micro-scaleinℓ ~ q -1system scale

fluctuation scale

L ~ k -1ℓmic ≪ b < ℓ ≪ L
T ≫ >q≫ k
Xin An (UNC)

Evolution of Non-Gaussian Hydrodynamic Fluctuations

7/16

Truncated equations• Effective power counting :Mij ∼ εδ 2 ,S ∼ ε−1 ,Gcn ∼ εn−1 .
o

o
or

E
E

e

e

E

E

E

E

•

.

G2 ~

E

E

G3 ~

2

G4 ~

c

2

G4 ~

3

• Perturbed in small parameters , the evolution equations for cumulants can besystematically truncated and iteratively solved (e.g., in hydrodynamics) :∂t Gcn = Ftree [{Gcn }] + higher orders
É .
{z
}
|
,

,

,

,

,

O(εn−1 δ 2 )EoXin An (UNC)

E

E

E

E

E

o

Evolution of Non-Gaussian Hydrodynamic Fluctuations

E

8/16

Multi-point Wigner function• We introduced the novel n-point Wigner function (cumulant in q-space)Wnc (x, q1 , . . . , qn ) =

Otto

x=

Z "

n
Y

3

d yi e−iqi ·yii=1x1 + x2 + + xn
n

x1

#

δ

(3)

n
X

1
yi
n i=1

!

Gcn (x + y1 , . . . , x + yn ),q1 + q2 + + qn = 0

Otto

xn ≡ x + yn

q1

qn

x
x-spaceOtto

q-spacex2

q2

being traditional Wigner function when n = 2 :W2 (x, q, −q) =Z

d3 y e−iq·y G2

⇣

y⌘
y
.
x + ,x −2
2• Evolution equation for Wnc :∂t Wnc = F [{Wnc }] .
Xin An (UNC)

Evolution of Non-Gaussian Hydrodynamic Fluctuations

10/16

Cumulant evolution equations for diffusive charge• Evolution equations for Wn=2,3,... of diffusive charge :∂t W2 (q1 , q2 ) =⇥γq12 W2 (q1 , q2 ) + λq1 · q2 Perm. ,⇤(Hydro+ equation)



1 2
1 0 2
∂t W3 (q1 , q2 , q3 ) =γq1 W3 (q1 , q2 , q3 ) + γ q1 W2 ( q2 , q2 )W2 ( q3 , q3 )22i
+ λ 0 q 1 · q 2 W 2 ( q 3 , q3 )(non-Gaussian Hydro+),Perm.

· · · where γ = λα0 ⌘ diffusion coefficient .
• Equilibrium solutions match thermodynamics :1eq
W2 = 0 ,α• In critical regime (Tλ ∼ ξ,03
2

λ ∼ξ ,

W3eq =
1

α00
,
03
α

W4c,eq =

α000
3α002
+ 05 ,04αα···⌧ ξ ⌧ q 1 ) : Hohenberg et al., 1977; Stephanov, 2009

γ∼ξ

1

,

0

γ ∼ξ

1
2

,

5n−6
c
Wn ∼ ξ 2 ,c
2 5n−8∂ t Wn ∼ q ξ 2 .

For confluentized equations see An et al., to appear; for the EFT approach, see Poster by Sogabe
Xin An (UNC)

Evolution of Non-Gaussian Hydrodynamic Fluctuations

12/16

Evolution near critical point• System evolution in the crossover region near CP :Tevolution trajectory
T(ti )

mapping Ising critical region
crossoverξQCD = ξIsing (r(µ, T ), h(µ, T ))CP
1st-orderT(t f )
critical region

For more realistic EoS,
see e.g., Contributions by Kapusta and Rattiμ• Fluctuations are closer to equilibrium on smaller scale (larger q) , andretain longer memory on larger scale (smaller q) :
W2

W4c

W3
1.0

1.0

0.8

equilibrium
0.0

q=0.3

0.5

q=0.1
0.6
0.0

-0.5

0.4
-0.5
0.2

-1.0
-1.0

0.0
-1.0

-0.5

0.0

t
Xin An (UNC)

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

-1.0

t
Evolution of Non-Gaussian Hydrodynamic Fluctuations

-0.5

0.0

0.5

t
13/16

1.0

Freeze-out• W4c (∼ kurtosis) in and out of equilibrium along the freeze-out curve :• Expected non-monotonic behavior : Stephanov, 2011; STAR, 2021; Pradeep et al., 2022 (also Poster)
3

2

σdata = σ2stat+σ2sys

1.0

equilibrium

0.8

Poisson baseline
poly4 (χ2/ndf = 1.3)

2

q=0.3
0.6

q=0.1

W4c

0.4

0.2

0.0

E

D

0.5

-0.4

0
-0.35

-c / c

-0.30

-0.25

2
(F)κσoooo√ sNN (GeV)

0

-0.2

-0.40

Xin An (UNC)

1

-0.20

-0.15

10

20

30

40

50

60

Non-monotonic: 3.1 σ

−0.5

10

Evolution of Non-Gaussian Hydrodynamic Fluctuations

20

30

40

50

60

15/16

流体モデルを拡張する
1. 新しい自由度を導入する

臨界モード，パイオン，スピン

2. 新しい項を見つける
Thermal shear(3. 上の2つを現象に適用する)
スピン偏極

シアーからのスピン偏極1
Talk by Baochi Fu

Shear-Induced Polarization&
Spin Hall Effects in heavy-ion collisions
Baochi Fu
Peking University

with S. Liu, L.-G. Pang, H. Song and Y. Yin
Shear-Induced Polarization: Phys.Rev.Lett. 127 14, 142301(2021)
Spin Hall Effects: arXiv: 2201.12970
1

Global polarization
STAR, Phys.Rev.C 98 (2018) 014910

 Spin-orbital coupling in non-central heavy ion collisions
 Signals observed at STAR BES energy:STAR Collaboration, Nature 548, 62 (2017)

 Data described be the statistic calculation

Hydrodynamics:ܵఓ ‫ ݌‬՚ ߸ఔఘ ሺ‫ݔ‬ሻ

I. Karpenko, F. Becattini, Eur.Phys.J.C 77 (2017) 4, 213
BF, K. Xu, X-G, Huang, H. Song, Phys.Rev.C 103 (2021) 2, 024903

Transport model:
H. Li, L. Pang, Q. Wang, X. Xia, Phys.Rev. C96 (2017) 054908
D. Wei, W. Deng, X. Huang, Phys.Rev. C99 (2019) 014905

2

ůŽĐĂůƉŽůĂƌŝǌĂƚŝŽŶ͗͚^ŝŐŶƉƵǌǌůĞ͛See also:
Karpenko, Becattini, EPJC 77 (2017) 4, 213

 Different trend/sign in ܲ௬ ሺ߶ሻ and ܲ௭ ሺ߶ሻ results

 Long exist in hydrodynamic and transport calculationsܲ ௬ ሺ߶ሻ

D. Wei, et al., PRC 99 (2019) 014905
X. Xia, et al., PRC 98 (2018) 024905
Becattini, Karpenko, PRL 120 (2018) 012302

ܲ ௭ ሺ߶ሻ

3

BF, K. Xu, X-G, Huang, H. Song, Phys.Rev.C 103 (2021) 2, 024903

Shear Induced Polarization (SIP)

BF, S. Liu, L. -G. Pang, H. Song, Y. Yin,
Phys.Rev.Lett. 127 14, 142301(2021)

Axial Wigner function from CKT (Chen, Son, Stephanov, PRL 115 (2015) 2, 021601)

Expand ࣛఓ to 1st order gradient of the fields:
Vorticity

T gradient(spin Nernst effect) ߪ ఓఔ ǣ shear stress tensor (symmetric)

 No free parameter Identical form by linear response theory
with arbitrary mass (S. Liu and Y. Yin, JHEP 07 (2021) 188)

Shear-Induced Polarizationఓܳఓఔ ൌ െ ‫݌‬ୄ ‫݌‬ୄఔ Ȁ‫݌‬ୄଶ ൅ ȟఓఔ Ȁ͵
ͳ ఓ ఔ
ͳ ఓఔ
ఔ ఓ
ఓఔ
ߪ
ൌ
߲ୄ ‫ ݑ‬൅ ߲ୄ ‫ ݑ‬െ ȟ ߲ୄ ‫ݑ ڄ‬ʹ͵5Shear Induced Polarization (SIP)

BF, S. Liu, L. -G. Pang, H. Song, Y. Yin,
Phys.Rev.Lett. 127 14, 142301(2021)

Axial Wigner function from CKT (Chen, Son, Stephanov, PRL 115 (2015) 2, 021601)
To one-loop order (in charge neutral fluid)
Thermal vorticity
Expand ࣛఓ to 1st order gradient of the fields:
Vorticityͳ߸ఓఔ ൌ ሺ߲ఔ ߚ‫ݑ‬ఓ െ ߲ఓ ߚ‫ݑ‬ఔ ሻʹT gradient(spin Nernst effect)

Shear-Induced Polarization

 No free parameter

Total ܲఓ = [Vorticity] + [T gradient] + [Shear] Different sensitivity of each term
ͳ ఓ ఔ
ఓఔ

 Identical form by linear response theory
with arbitrary mass (S. Liu and Y. Yin, arXiv: 2103.09200)

ߪ

ൌ

ʹ

߲ୄ ‫ݑ‬

൅ ߲ୄఔ ‫ݑ‬ఓͳ ఓఔെ ȟ ߲ୄ ‫ݑ ڄ‬͵7

Shear Induced Polarization (SIP)

BF, S. Liu, L. -G. Pang, H. Song, Y. Yin,
Phys.Rev.Lett. 127 14, 142301(2021)

Axial Wigner function from CKT (Chen, Son, Stephanov, PRL 115 (2015) 2, 021601)
To one-loop order (in charge neutral fluid)
Thermal vorticity
Expand ࣛఓ to 1st order gradient of the fields:
Vorticityͳ߸ఓఔ ൌ ሺ߲ఔ ߚ‫ݑ‬ఓ െ ߲ఓ ߚ‫ݑ‬ఔ ሻʹT gradient(spin Nernst effect)

Shear-Induced Polarization

 No free parameter

Total ܲఓ = [Vorticity] + [T gradient] + [Shear] Different sensitivity of each term
ͳ ఓ ఔ
ͳ ఓఔ
ఔ ఓ
ఓఔ
ߪ
ൌ
߲ ‫ ݑ‬൅ ߲ୄ ‫ ݑ‬െ ȟ ߲ୄ ‫ݑ ڄ‬
Total ܲఓ = [Thermal vorticity] + [Shear] ʹ ୄ͵ Identical form by linear response theory
mass(S. Liu and Y. Yin, arXiv: 2103.09200)The only new effectSeewithalso: arbitrary
Kumar@Tues.
& Buzzegoli@Tues.
8

S. Liu, L. -G. Pang, H. Song, Y. Yin,͚Ȧ ĞƋƵŝůŝďƌŝƵŵ͛ǀƐ͘͚^-ƋƵĂƌŬŵĞŵŽƌǇ͛ BF,
Phys.Rev.Lett. 127 14, 142301(2021)

Spin Cooper-Frye:͚Ȧ ĞƋƵŝůŝďƌŝƵŵ͛ఓܲஃ߬ୱ୮୧୬ǡ ஃ ՜ ͲPolarization of Ȧ-hyperonఓܲஃ ሺ‫݌‬ሻȦ equilibrium

F. Becattini (2013)and later hydrodynamic(transport) calculations

ఓ

ܲୱ
QGP

Hadron resonance gas
hadronization

Hydro evolution

͚S-quark memory͛S-quark memory߬ୱ୮୧୬ǡ ஃ ՜ λ߬Polarization of S-quarkఓఓܲஃ ‫ ݌‬ൌ ܲୱ ‫݌‬Z.-T. Liang, X.-N. Wang, PRL 94 (2005) 102301
9

Competition of ܲ௭ : Grad T vs. SIP

BF, S. Liu, L. -G. Pang, H. Song, Y. Yin, PRL 127 14, 142301(2021)

Total ܲఓ = [vorticity] + [T grad] + [SIP][SIP][SIP]ј

[vort][T grad]݉ஃ ൌ ͳǤͳͳ͸ ݉௦ ൌ ͲǤ͵ Competition between:

 [SIP]: ̶ ൅ ሺʹ߶ሻ̶ structure for ܲ௭ (same as exp.)

 Total polarization: a competition between [SIP] and [Grad T]

T-grad: ߳ ఓఔఈఒ ‫ݑ‬ఔ ‫݌‬ఈ ሾߚିଵ ߲ఒ ߚሿ
Shear: ߳ఓఔఈఘ௣ഊ‫ݑ‬ఔ ‫݌‬ఘ߲ ‫ݑ‬
ఢబ 10 ሺఈ ఒሻܲ௭ ሺ߶ሻ with SIP

BF, S. Liu, L. -G. Pang, H. Song, Y. Yin,
Phys.Rev.Lett. 127 14, 142301(2021)

Total ܲఓ = [vorticity] + [T grad] + [SIP]= [thermal vorticity] + [SIP]݉ஃ ൌ ͳǤͳͳ͸ ݉௦ ൌ ͲǤ͵ STAR, PRL 123 (2019) 132301

 /ŶƚŚĞƐĐĞŶĂƌŝŽŽĨ͚S-quark memory͕͛ƚŚĞƚŽƚĂůܲఓ with SIP qualitatively agrees with data11ܲ௬ ሺ߶ሻ with SIP

BF, S. Liu, L. -G. Pang, H. Song, Y. Yin,
Phys.Rev.Lett. 127 14, 142301(2021)

Total ܲఓ = [vorticity] + [T grad] + [SIP]= [thermal vorticity] + [SIP]

STAR, NPA, 982 (2019) 511-514

݉ஃ ൌ ͳǤͳͳ͸ ݉௦ ൌ ͲǤ͵ 

 /ŶƚŚĞƐĐĞŶĂƌŝŽŽĨ͚S-quark memory͕͛ƚŚĞƚŽƚĂůܲఓ with SIP qualitatively agrees with data
12

From RHIC to LHC

Same hydrodynamic model: AMPT + MUSIC(LHC parameter from EPJC 77 (2017) 9, 645) ͞^ƚƌĂŶŐĞDĞŵŽƌǇ͟ƐĐĞŶĂƌŝŽƋƵĂůŝƚĂƚŝǀĞůǇĚĞƐĐƌŝďĞƐƚŚĞĐĞŶƚƌĂůŝƚǇΘ‫ ்݌‬dependence More precise model needed to quantitative description

13

BF & H. Song, in preparation

Summary

Shear-Induced Polarization: Phys.Rev.Lett. 127 14, 142301(2021)
Spin Hall Effects: arXiv: 2201.12970

Total ܲఓ = [vorticity] + [Grad T] + [SIP] + [SHE]Spin Hall Effects

Shear-Induced Polarization

͞Strange memory͟нShear-Induced Polarization
Describes ܲ௭ ሺ߶ሻ and ܲ௬ ሺ߶ሻ qualitatively at top RHIC and LHC͞^ŝŐŶƉƵǌǌůĞ͟

Total: ̶ ൅ ሺʹ߶ሻ ̶

Total: ̶ ൅ ሺʹ߶ሻ ̶ܲേ ‫ ן‬േ ‫݌‬Ԧ ൈ ‫ߤ׏‬஻ Particle ʹAnti-particle
separation

 Relevant for
RHIC-BES and
RHIC/LHC
forward rapidity
 Scenario
independent23

シアーからのスピン偏極2
Talk by Matteo Buzzegoli

Spin-thermal shear
coupling in relativistic
nuclear collisions

Quark Matter 2022 , April 5 2022

Matteo Buzzegoli

and F. Becattini, A. Palermo, G. Inghirami, I. Karpenko

Polarization from Wigner function
F. Becattini, Lect. Notes Phys. 987 (2021) 15-52.

The covariant Wigner function of the free Dirac field:

where:It allows to calculate the mean spin vector:

4

Spin polarization inducedby thermal shear
Using linear response theory we eventually obtain:

F. Becattini, MB, A. Palermo, Phys. Lett. B 820 (2021) 136519
Same (not precisely the same) formula obtained by Liu and Yin with a different method:
S. Liu, Y. Yin, JHEP 07 (2021) 188
Dependence on a specific vector is not
surprising as this term arises from the correlator

But Q is not a tensor and, unlike J, it
does depend on the hypersurface
8

Application to relativistic
heavy ion collisions
F. Becattini, MB, A. Palermo, G. Inghirami and I. Karpenko, Phys. Rev. Lett. 127 (2021) 27, 272302

Modulus of thermal-vorticity and thermal-shear at the freeze-out hypersurface

9Isothermal local equilibrium
The most appropriate setting for relativistic heavy ion collisions
at very high energy!

At high energy, ΣFO
expected to be T= constant!

Only NOW u can be expanded!

12

Spin mean vector at leading orderwith isothermal local equilibrium (ILE)
Readily found by replacing the gradients of β with those of u

Kinematic vorticity

Kinematic shear

13

Isothermal local equilibrium: result
Apply the new formula (for primary hadrons):·

Sensitive to the decoupling temperature Tdec·

Quantitative agreement with the data for realistic Tdec=150 MeV
14

スピンを含む流体
Talk by Masaru Hongo

Relativistic spin hydrodynamics
with torsion and linear response
theory for spin relaxation

Hydro
≃ +
Spin
?

<latexit sha1_base64="D42sn37g3C7/gnHlmpF31JXk1SQ=">AAACh3ichVFNL0NBFD2ez9ZXsZHYiIZY1TwEsSo2llXaSmia957B8L68N21C0z9gZSdYkViIvT9g4w9Y+AliWYmNhdvXlwiNupOZOXPmnjtncnXXFL5k7KVFaW1r7+jsikS7e3r7+mMDg1nfKXoG
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Masaru Hongo (Niigata University)
2022/04/04, Quark Matter 2022
MH-Huang-Kaminski-Stephanov-Yee, JHEP 2021, 150 (2021) [arXiv:2107.14231]

One-page Summary
Extending hydrodynamics to include spin

Hydro
≃ +
Spin
<latexit sha1_base64="D42sn37g3C7/gnHlmpF31JXk1SQ=">AAACh3ichVFNL0NBFD2ez9ZXsZHYiIZY1TwEsSo2llXaSmia957B8L68N21C0z9gZSdYkViIvT9g4w9Y+AliWYmNhdvXlwiNupOZOXPmnjtncnXXFL5k7KVFaW1r7+jsikS7e3r7+mMDg1nfKXoG
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MH-Huang-Kaminski-Stephanov-Yee, JHEP 2021, 150 (2021) [arXiv:2107.14231]

Three main messages from our paper:(1) Spin hydrodynamic equations in a torsionful geometry(2) Mode mixing between shear and spin modes(3) Green-Kubo formula for a rotational viscosityqfGqtrzC3o0cj2Tf/nWVefVw8On6w6Gy2tdtTRXyf2bzUMScn0nnjLbPNNNqLX/18PwlO58Zr0/QFT1zzktq0D0nNauv2nVaZC4Q5kHJ38fyE2xMJ+SZxHQ6GU8tBSMLYRRjmOS5zCKFVawhx/cKnOAUZ9KAlJQWpMWWVGoLPMP4UtLKB7NokVQ=</latexit>
VRxN5DTLsJxNVXGFoZsi5+meITZtRyhl1RB5tbTcPM9XhePqlrnu1WyxU1b2Tb2oa4rHVJp2o3FKkF+xn0AOQBxBrVnRO2xjDxY0VFCGgAmPsQEFLj8FyCDYzO2gzpzDSPfPBY4QZm+FVYIVCrMl/u7zrhCwJu+bPV3frfEtBr8OO2MYp0e6oRd6oFt6ovdfe9X9Hs1/<latexit
zA==</latexit>
h6JadGZy6K1R1LFHZx2XV9ApEqYqQdXUjE9nnlSXQOX+ZdspiL0hYi8F0RsieimIjSFiwxDj1ZGoyCh1iinU2UV9RrHrE1mb3p8TPfVc0huRkJ6mcrx7Cv/fNKeNndV84X5+detebu2xuYVm4AbchDuY6QGswTPYhBLdMG/gAN5Nf7fmrWXr2iA0M2WYqzA0rOxf90gk
9o19YkfsK/vMfrA/Z+ZKKIeupYdPMWBl0Fh4fX3790TKxWcML0+oMYTAaIp7cbdaHttbDE14SD0p7DEgj+7WHvDd/bdH24+KK8lt9oH9xD7fs0P2BTv1ur/sj1uyeEDZPWRe0a64VKeHOiSkhsL8TcylZ8cdhGgluNY3+34WrePUSFalYuVIVk5kOXRxJTS71KeTU6cV
8dLl+YXFpSs7kd8JbVmyfccPK4JH0lGeLMUqdmQlCCV3hSPLov1Er5e7MoyU7z2Pe4Gsu7zlqaayeYyuSu0pd13eiBqLOZZnNLKnjYIxcmDGpr+UuQU12AUfbOiACxI8iNF2gEOEnyoUgEGAvjok6AvRUrQuoQ9zyHYwSmIER28bf1s4qxqvh3OdMyLaxn9x8BsimYUV
sha1_base64="Mt4fGOo1knlaS4iV9oiX7GdwnCQ=">AAACjHichVE9S8NQFD3Gr1o/WnURXIpFEYdyI0VFEYqCONrWWqWKJPFVQ9MkJGmhFv+AroqDk4KD+BMcXfwDDv0J4qjg4uBtGhAV9YbknXfeOfflcFXb0F2PqNEmtXd0dnWHesK9ff0Dkejg0IZr
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Sketch of our result|(k)| = frequency scale

fast modes<latexit sha1_base64="KI2s+J4/6OozWRQihycoz+CX5pQ=">AAAFZHicjZTPTxNBFMcflVVEFJCYmJgYYsEYD80UjBpPiAcNJyi2xbQNmVmmdez+yu62Ujf9Bzh5M+pJEw/GP8OL/4AH/gET4xETLx548zoEK6Xdadp98+Z9vn1v3uyIwFFRzNj+WObMuHX23MT5yQtTFy9Nz8xeLkV+K7Rl0fYdP9wSPJKO8mQxVrEjt4JQclc4siyaj/R6uS3DSPne07gTyJrLG56qK5vH6CpVH3PX5dszWZZjNOZPGnljZMGMdX82swBV2AEfbGiBCxI8iNF2gEOEnwrkgUGAvhok6AvRUrQuoQuTyLYwSmIER28Tfxs4qxivh3OtGRFt4784+A2RnIdF

Non-hydro regime

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Assumed scale
separation<latexit sha1_base64="dzUdSN+PmzzEpW33nlanv1CvqmU=">AAACknichVE7S8NQGD3Gd33VxyC4FIsiDuVGRKVTtYuDg1prlFokiVcN5nFJ0mIt/gE3J8FOCg7iT3B08Q849CeIo4KLg1/SgKhYv5Dcc88958s9fJowDc9nrN4itba1d3R2dcd6evv6B+KDQ5ue|sound(k)||shear(k)||spin,(k)||spin,||(k)|U3J1ntcd03G3NNXjpmHzvG/4Jt8SLlctzeSKdpQNzpUydz3DsTf8iuBFSz2wjX1DV32ilB1VCNc53o0nWYqFlfgN5AgkEdWqE7/HDvbgQEcJFjhs+IRNqPDoKUAGgyCuiCpxLiEjPOc4RYy8JVJxUqjEHtH3gHaFiLVpH/T0QrdOfzHpdcmZwAR7YrfslT2yO/bMPv7sVQ17BHep0Ko1vFzsDpyN5t7/dVm0+jj8cjVxaKQOddvTBaVpNh/7WAgzGZRRhEyQVm/4yycXr7n0+kR1kl2zF8p5xersgZLa5Tf9Zo2v1xCjQck/x/IbbM6k5LnUzNpsMrMUjawLYxjHFM1lHhksYxX58HbnuERNGpHS0qKUbUillsgzjG8lrXwCmH+UVg==</latexit>≈

Relaxation rate
Spin hydro regime

Pure hydro regimeΓs0Mode mixing!
(Level repulsion)k = wave numberまとめ
1. 新しい自由度を導入する

臨界モード，パイオン，スピン

2. 新しい項を見つける
Thermal shear(3. 上の2つを現象に適用する)
スピン偏極

個人的見解堅実な結果は出ているが“理論的におもしろい(?)”進展はあまり見られなくなった印象(よく言えば分野として成熟した?)
今後の大きな方向性を考える時期?
中性子星も念頭に高密度QCD?
EICを考えてpQCD??

()