THE EMERGENCE OF BARNETT EFFECT

WITHIN RELATIVISTIC MAGNETOHYDRODYNAMICS WITH SPIN

Radoslaw Ryblewski
The H. Niewodniczański Institute of Nuclear Physics
Polish Academy of Sciences
In collaboration with:
Samapan Bhadury, Wojciech Florkowski (IFT UJ, Kraków),
Amaresh Jaiswal (NISER Bhubaneswar),
Avdhesh Kumar (IOP Academia Sinica Taipei)
based on:
Phys. Lett. B 814 136096 (2021); Phys. Rev. D 103, 014030 (2021); Phys. Rev. Lett 129, 192301 (2022)

XVI POLISH WORKSHOP ON RELATIVISTIC HEAVY-ION COLLISIONS

DECEMBER 2-3, 2023, KIELCE, PL

SONATA BIS 8 Grant No. 2018/30/E/ST2/00432

RELATIVISTIC HEAVY-ION COLLISIONS PROBE QCD PHASE DIAGRAM

Shen, C., Yan, L. NUCL SCI TECH 31, 122 (2020)

2
EXPERIMENTAL DATA SUGGESTS THAT QGP IS THE MOST PERFECT FLUID

J. Bernhard, J. Moreland, S. Bass, Nat. Phys. 15, 1113–1117 (2019)

hydrodynamics is applicable

inclusion of dissipative effects is required

3
 SPIN-ORBIT TRANSFER OF ANGULAR MOMENTUM IN HIC

non-central heavy-ion collisions create reballs polarized emitted hadrons

with large global orbital angular momenta
F. Becattini, F. Piccinini, J. Rizzo, PRC 77 (2008) 024906

part of the angular momentum can be

transferred from the orbital to the spin part

emitted particles are expected to be globally

polarized along the system’s angular momentum
Z.-T. Liang and X.-N. Wang, Phys. Rev. Lett. 94, 102301 (2005); 96, 039901(E) (2006)
Z.-T. Liang and X.-N. Wang, Phys. Lett. B 629, 20 (2005)
S. A. Voloshin, arXiv:nucl-th/0410089.
B. Betz, M. Gyulassy, and G. Torrieri, Phys. Rev. C 76, 044901 (2007).
F. Becattini and F. Piccinini, Ann. Phys. (Amsterdam) 323, 2452 (2008).

gure: R. Ryblewski

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 MEASUREMENT OF Λ AND Λ̄ SPIN POLARIZATION

Self-analysing parity-violating hyperon weak

decay allows to measure polarization of Λ
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 MEASUREMENT OF Λ AND Λ̄ SPIN POLARIZATION

gure: adapted from T. Niida

Self-analysing parity-violating hyperon weak

decay allows to measure polarization of Λ
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PARTICLES EMERGING FROM HIC ARE POLARIZED

L. Adamczyk et al. (STAR) (2017), Nature 548 (2017) 62-65

7
 SPIN POLARIZATION IN EQUILIBRATED QGP

In thermodynamic equilibrium one can

establish a link between spin and vorticity
Becattini F, Chandra V, Del Zanna L, Grossi E. AP 338:32 (2013)
F. Becattini, L. Csernai, and D. J. Wang, PRC 88, 034905 (2013)
Fang R, Pang L,Wang Q,Wang X. PRC 94:024904 (2016)
F. Becattini, I. Karpenko, M. Lisa, I. Upsal, and S. Voloshin PRC 95, 054902
(2017)

1 ∫ dΣ λ p λ
nF (1 − nF) ϖρσ
S μ(p) = − ϵ μρστpτ
8m ∫ dΣλ p λnF

μ
1 u
ϖμν = − (∂μ βv − ∂ν βμ) μ
β =
2 T

Spin is enslaved to thermal vorticity!

gure: D.D. Chinellato

8
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 SPIN POLARIZATION IN EQUILIBRATED QGP

In thermodynamic equilibrium one can

establish a link between spin and vorticity
Becattini F, Chandra V, Del Zanna L, Grossi E. AP 338:32 (2013)
F. Becattini, L. Csernai, and D. J. Wang, PRC 88, 034905 (2013)
Fang R, Pang L,Wang Q,Wang X. PRC 94:024904 (2016)
F. Becattini, I. Karpenko, M. Lisa, I. Upsal, and S. Voloshin PRC 95, 054902
(2017)

1 ∫ dΣ λ p λ
nF (1 − nF) ϖρσ
S μ(p) = − ϵ μρστpτ
8m ∫ dΣλ p λnF

μ
1 u
ϖμν = − (∂μ βv − ∂ν βμ) μ
β =
2 T

Spin is enslaved to thermal vorticity!

gure: D.D. Chinellato

9
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 SPIN POLARIZATION IN EQUILIBRATED QGP

In thermodynamic equilibrium one can

establish a link between spin and vorticity
Becattini F, Chandra V, Del Zanna L, Grossi E. AP 338:32 (2013)
F. Becattini, L. Csernai, and D. J. Wang, PRC 88, 034905 (2013)
Fang R, Pang L,Wang Q,Wang X. PRC 94:024904 (2016)
F. Becattini, I. Karpenko, M. Lisa, I. Upsal, and S. Voloshin PRC 95, 054902
(2017)

1 ∫ dΣ λ p λ
nF (1 − nF) ϖρσ
S μ(p) = − ϵ μρστpτ
8m ∫ dΣλ p λnF

μ
1 u
ϖμν = − (∂μ βv − ∂ν βμ) μ
β =
2 T

Spin is enslaved to thermal vorticity!

gure: D.D. Chinellato

10
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MEASUREMENT VS SPIN-THERMAL APPROACH: GLOBAL POLARIZATION

F. Becattini, J. Liao, M. Lisa Lect.Notes Phys. 987 (2021) 1-14

Global polarization data supports

the spin-thermal approach

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MEASUREMENT VS SPIN-THERMAL APPROACH: GLOBAL POLARIZATION

F. Becattini, J. Liao, M. Lisa Lect.Notes Phys. 987 (2021) 1-14

Global polarization data supports

the spin-thermal approach

Azimuthal modulation is not captured

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 LONGITUDINAL (BEAM-AXIS) POLARIZATION

Flow structures in the

plane transverse to beam
(jet, ebe uctuations etc.)
may generate
longitudinal polarization
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 LONGITUDINAL POLARIZATION — ‘SPIN SIGN’ PUZZLE

Experiment Theory

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 LONGITUDINAL POLARIZATION — ‘SPIN SIGN’ PUZZLE

Experiment Theory

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 FLUID DYNAMICS OF SPIN?!

Spin-thermal approach does not capture properly

phenomena seen in experiment.

If spin polarization is trully hydrodynamic quantity

it should not be enslaved to thermal vorticity.
W. Florkowski, B. Friman, A. Jaiswal, E. Speranza, PRC 97 (4) (2018) 041901

Fluid dynamics with spin

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 CONSERVED CURRENTS IN QUANTUM FIELD THEORY

Noether’s theorem:
for each continuous symmetry of the action there is a corresponding conserved (canonical) current
S. Weinberg, The Quantum Theory Of Fields Vol. 1 Cambridge University Press (1995)

invariance under space-time translation conservation of energy and linear momentum

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 CONSERVED CURRENTS IN QUANTUM FIELD THEORY

Noether’s theorem:
for each continuous symmetry of the action there is a corresponding conserved (canonical) current
S. Weinberg, The Quantum Theory Of Fields Vol. 1 Cambridge University Press (1995)

invariance under space-time translation conservation of energy and linear momentum

invariance under rotations and boosts conservation of total angular momentum

orbital part spin part asymmetric part

spin is sourced by antisymmetric

part of stress-energy tensor
18
 ORBITAL AND SPIN ANGULAR MOMENTUM

orbital part spin part

orbital angular momentum of a point particle

its dual is

relativistic generalization is

for relativistic uid one has

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 CONSERVATION LAWS AND LAGRANGE MULTIPLIERS

conservation laws + local equilibrium hydrodynamics

conservation of charge (baryon number, electric charge, …)

(1 eq / charge)

conservation of energy and linear momentum

(4 eqs)

conservation of angular momentum

(6 eqs)

W. Florkowski, B. Friman, A. Jaiswal, E. Speranza, Phys. Rev. C97 (4) (2018) 041901 spin chemical potential
W. Florkowski, B. Friman, A. Jaiswal, R. R., E. Speranza, Phys. Rev. D97 (2018) 116017
F. Becattini, W. Florkowski, E. Speranza, Phys. Lett. B 789 (2019) 419-425
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 CONSERVED CURRENTS AND CONSTITUTIVE RELATIONS

If the energy-momentum tensor is symmetric the spin tensor is conserved

W. Florkowski, B. Friman, A. Jaiswal, E. Speranza, PRC 97 (4) (2018) 041901
W. Florkowski, B. Friman, A. Jaiswal, R. R., E. Speranza, PRD 97 (2018) 116017
F. Becattini, W. Florkowski, E. Speranza, PLB 789 (2019) 419-425
W. Florkowski, A. Kumar, R. R., PPNP 108 (2019) 103709

What are the constitutive relations which enter equations of motion?

Coarse-graining of underlying microscopic theory is required!

 RELATIVISTIC KINETIC THEORY DERIVATION OF SPIN HYDRODYNAMICS

For dilute systems, the derivation

of uid dynamics can be done
starting from the underlying
kinetic theory

classical RKT method

of moments

22
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 RELATIVISTIC KINETIC THEORY DERIVATION OF SPIN HYDRODYNAMICS

For dilute systems, the derivation

of uid dynamics can be done
starting from the underlying
kinetic theory

classical RKT method

of moments

quantum eld theory

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 RELATIVISTIC KINETIC THEORY WITH SPIN

To include spin in RKT, we start from the Wigner function (WF) that bridges the gap between QFT and RKT

For spin-1/2 particles the WF satis es the quantum kinetic equation

D. Vasak, M. Gyulassy, H.T. Elze, Ann. Phys. 173 (1987) 462–492,

From the LO and NLO of the semi-classical expansion of the WF in powers of Planck's constant,
one obtains two independent kinetic equations

24
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 RELATIVISTIC KINETIC THEORY DERIVATION OF SPIN HYDRODYNAMICS

For dilute systems, the derivation

of uid dynamics can be done
starting from the underlying
kinetic theory

classical RKT method

of moments

semi-classical method
quantum RKT expansion of moments

25
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RELATIVISTIC KINETIC THEORY WITH SPIN

scalar axial vector

26
 CLASSICAL APPROACH TO SPIN HYDRODYNAMICS

In the classical treatments of particles with spin-half one introduces internal angular
momentum tensor of particles
M. Mathisson, APPB 6 (1937) 163-2900

Satis es Frenkel condition

In particle rest frame (PRF)

27
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 RELATIVISTIC KINETIC THEORY WITH SPIN

The distribution function in the extended phase-space is a function of spacetime, momentum,

and internal angular momentum of the particles

The kinetic equation (KE) governing the evolution of the distribution function can be written as
W. G. Dixon, Nuovo Cimento (1955–1965) 34, 317 (1964).
L. Suttorp and S. De Groot, Il Nuovo Cimento A (1965–1970) 65, 245 (1970)
C.G. van Weert, thesis, The University of Amsterdam, 1970.

where

Using the Frenkel condition, one can derive the force (Lorentz and Mathisson) and torque
I. Bailey and W. Israel, Commun. Math. Phys. 42, 65 (1975).

where magnetic dipole moment is

28
 INFINITE CONDUCTIVITY LIMIT

In the limit of in nite conductivity, eld strength tensor is

If the medium is magnetizable, then the Maxwell's equations are given by

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 FROM KT TO SPIN MHD

The particle current, energy-momentum tensor, and spin tensor of the uid can be expressed as
S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, and R. R., Phys. Lett. B 814, 136096 (2021); Phys. Rev. D 103, 014030 (2021).

where we use the notation

The polarization-magnetization tensor is

30

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 FROM KT TO SPIN MHD

Assuming that the microscopic interactions preserve fundamental conservation laws one requires
S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, and R. R., Phys. Lett. B 814, 136096 (2021); Phys. Rev. D 103, 014030 (2021).

Zeroth, rst, and ‘spin’ moment of the KE (in absence of the torque) then lead to equations de ning
relativistic magnetohydrodynamics for uid with spin

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 RELATIVISTIC MHD WITH SPIN

Kinetic equation with collision kernel in the relaxation-time approximation (RTA) reads
J. L. Anderson and H. Witting, Physica (Utrecht) 74, 466 (1974)

Using RTA kinetic equation we can write the rst-order gradient correction as

The equilbrium distribution function has the form

32
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 RELATIVISTIC MHD WITH SPIN

The expressions for dissipative currents in terms of the nonequilibrium correction to the
distribution function are

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 EMERGENCE OF BARNETT EFFECT

Equilibrium polarization-magnetization tensor is

In global equilibrium, spin chemical potential

corresponds to rotation of the uid
F. Becattini and F. Piccinini, Ann. Phys. (Amsterdam) 323, 2452 (2008)
F. Becattini, W. Florkowski, and E. Speranza, Phys. Lett. B 789, 419 (2019).

We conclude that rotation of the uid produces magnetization,

which is precisely the physics of Barnett effect.
gure: Journal of the Physical Society of Japan 90, 081003 (2021)
S. J. Barnett, Rev. Mod. Phys. 7, 129 (1935)
A. Einstein and W. de Haas, Deutsch. Phys. Ges., Verh. 17, 152 (1915)

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 CONVERSION BETWEEN VORTICITY AND SPIN

Using the spin matching condition we obtain the evolution equation for the spin polarization
tensor

We observe that the above equation contains information about the connection between
evolution of spin polarization tensor and uid vorticity.

vanishes in absence of electromagnetic eld which leads us to conclusion that the

conversion between spin-polarization and vorticity proceeds via coupling with electromagnetic eld.

35
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 FIRST-ORDER DISSIPATIVE CURRENTS IN SMHD

The expressions for dissipative currents in terms of the nonequilibrium corrections to the DF are

where

These expressions contain gradients of magnetic eld.

Demanding that the divergence of the above entropy current is positive de nite we identify
rst-order dissipative gradient terms

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 CONCLUSIONS

We presented the rst kinetic theory formulation of relativistic dissipative nonresistive MHD
with spin in the limit of small polarization.

We demonstrated that multiple transport coef cients, dissipative as well as non-dissipative, are present.

We showed that our framework naturally leads to the emergence of the relativistic analog of Barnett effect.

We show that the coupling between the magnetic eld and spin polarization appears at gradient order.

Simulation based on our uni ed framework has the potential of explaining the difference of Λ and anti-Λ
polarization.

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THANK YOU FOR YOUR ATTENTION.

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BACKUP SLIDES

39
 PSEUDOGAUGE FREEDOM AND PROBLEM OF LOCALIZATION

Pseudogauge transformation: densities are not uniquely de ned

F.W. Hehl, Rep. Math. Phys. 9 (1976) 55.

The new tensors satisfy conservation equations and preserve Poincare algebra generators

Inequivalence of different pseudogauge pairs was shown.

F. Becattini, L. Tinti, Phys. Rev. D 84 (2011) 025013; F. Becattini, L. Tinti, Phys. Rev. D 87 (2) (2013) 025029

Canonical currents act as sources for Einstein-Cartan theory.

F. W. Hehl, P. von der Heyde, and G. D. Kerlick, Rev. Mod. Phys. 48, 393 (1976)

Belinfante pseudogauge
F. Belinfante, Physica 7 (5) (1940) 449–474

Belinfante energy-momentum tensor gives the Einstein-Hilbert one.

40 gure: Physics World
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