Pith. sign in

REVIEW 1 major objections 5 minor 7 cited by

An introduction to relativistic spin hydrodynamics

T0 review · 1 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper derives relativistic spin hydrodynamics from angular momentum conservation and the local second law, fixing the antisymmetric stress so that spin relaxes toward thermal vorticity with two new transport coefficients.

desk verdict Useful pedagogical review, but a sign error in the central constitutive relation makes it violate the second law it claims to derive. read the letter →

arxiv 2411.11753 v2 pith:HCK3J25W submitted 2024-11-18 nucl-th hep-phnucl-exphysics.flu-dyn

Xu-Guang Huang This is my paper · ORCID
classification nucl-thhep-phnucl-exphysics.flu-dyn PACS 47.75.+f25.75.-q05.70.Ln
keywords relativisticspinhydrodynamicsangularmomentumconservationtensorthermalvorticityrotationalviscositypseudo-gaugeambiguitypolarizationheavy-ioncollisions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Relativistic spin hydrodynamics can be built from two inputs rather than guessed: conservation of angular momentum and the local second law of thermodynamics. The central result is a constitutive relation for the antisymmetric part of the energy-momentum tensor which says that spin is generated by the mismatch between the fluid's acceleration, temperature gradient, and spin potential, and that spin relaxes toward the thermal vorticity. The derivation introduces two new transport coefficients, the boost heat conductivity and the rotational viscosity, both forced to be non-negative by entropy production. If the construction is right, spin polarization in heavy-ion collisions becomes an output of hydrodynamics, computable from the same flow fields that generate the observed momentum anisotropies.

What carries the argument

The load-bearing identity is angular momentum conservation in the form $\partial_\mu\Sigma^{\mu\nu\rho}=\Theta^{\rho\nu}-\Theta^{\nu\rho}$: it makes the antisymmetric part of the energy-momentum tensor the source that converts orbital angular momentum into spin. Around this, the paper constructs the covariant entropy current $s^\mu=P\beta^\mu+\Theta^{\mu\nu}\beta_\nu-\tfrac12\alpha_{\rho\sigma}\Sigma^{\mu\rho\sigma}$ with $\alpha_{\rho\sigma}=\mu_{\rho\sigma}/T$, and demands $\partial_\mu s^\mu\ge0$ at first order in gradients. That requirement pins down $\Theta^{\mu\nu}_a$ exactly as in Eqs. (39)--(41) and forces the boost heat conductivity $\lambda$ and rotational viscosity $\eta_s$ to be non-negative. The spin density is a quasi-hydrodynamic mode, not a conserved charge: it relaxes to the local equilibrium set by the thermal vorticity, which is why the framework is a quasi-hydrodynamics rather than a strict hydrodynamic theory.

What would settle it

A decisive check is a microscopic calculation of the antisymmetric part of the energy-momentum tensor to first order in gradients in a weakly coupled spin-$\tfrac12$ plasma with small spin density. If quantum kinetic theory reproduces exactly $\phi^{\mu\nu}=\eta_s\Delta^{\mu\rho}\Delta^{\nu\sigma}(\mu_{\rho\sigma}-T\varpi_{\rho\sigma})$ and $q^\mu$ from Eq. (40) with a single positive $\eta_s$, the construction is supported; any independent tensor structure or a negative extracted $\eta_s$ in a regime where the second law should hold would falsify the constitutive ansatz.

Watch

Extended reading notes

Core claim

The article establishes that a closed first-order theory of relativistic spin hydrodynamics follows from angular momentum conservation and covariant local thermodynamics. The spin density $S^{\rho\sigma}=u_\mu\Sigma^{\mu\rho\sigma}$ is treated as a quasi-hydrodynamic variable of order $\mathcal{O}(\partial)$ relative to energy density and flow, and the first law is extended to $T\,ds+\tfrac12\mu_{\mu\nu}\,dS^{\mu\nu}=d\varepsilon$. Requiring $\partial_\mu s^\mu\ge0$ for the covariant entropy current then fixes the antisymmetric part of the energy-momentum tensor to $\Theta^{\mu\nu}_a=q^\mu u^\nu-q^\nu u^\mu+\phi^{\mu\nu}$, with $q^\mu=\lambda[\beta\nabla^\mu T+Du^\mu-2\mu^{\mu\nu}u_\nu]$ and $\phi^{\mu\nu}=\eta_s\Delta^{\mu\rho}\Delta^{\nu\sigma}(\mu_{\rho\sigma}-T\varpi_{\rho\sigma})$, where $\varpi^{\mu\nu}=\tfrac12(\partial^\nu\beta^\mu-\partial^\mu\beta^\nu)$ is the thermal vorticity. Positivity of entropy production forces $\lambda\ge0$ and $\eta_s\ge0$, identifying these as the transport coefficients that govern spin--orbit conversion. The same construction is then shown to reorganize under pseudo-gauge changes, under large vorticity where the theory becomes gyrohydrodynamics, and into a freeze-out formula that maps the hydrodynamic fields onto the measured spin vector in momentum space.

Load-bearing premise

The construction assumes the spin density is a small, first-order-in-gradients correction to the energy density and flow, an assumption motivated by the few-percent hyperon polarization seen in heavy-ion collisions; once the spin density or the vorticity becomes large, the gradient expansion behind Eqs. (39)--(41) breaks down.

Editorial extensions

If this is right

  • Spin--orbit conversion is a dissipative process: the spin density decays toward the thermal vorticity with rate $\Gamma_s=\eta_s/\chi_s$, so spin is not conserved separately from orbital angular momentum.
  • The second law requires two new non-negative transport coefficients, $\lambda$ and $\eta_s$, that must be supplied by microscopic calculation or by data; they control how fast spin equilibrates with flow.
  • The pseudo-gauge ambiguity means the split of angular momentum into spin and orbital parts is not unique; physical predictions must be invariant under the transformations in Eqs. (52)--(53), and any reported spin potential must specify the pseudo-gauge used.
  • For strongly vortical fluids, treating the thermal vorticity as order one leads to an anisotropic, magnetohydrodynamics-like theory (gyrohydrodynamics) with different pressures parallel and transverse to the vorticity and with additional odd viscosity coefficients not constrained by the second law.
  • A freeze-out formula converts the spin potential, temperature, flow velocity, and thermal shear on the decoupling surface into the measured momentum-space spin vector, so hyperon polarization can be computed from the same hydrodynamic fields that determine the momentum spectra.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If these constitutive relations survive comparison with kinetic theory, measurements of the time and momentum dependence of hyperon polarization could indirectly pin down $\eta_s$ and $\lambda$, turning spin polarization into a probe of dissipative spin--orbit coupling in the quark-gluon plasma.
  • The pseudo-gauge dependence of the freeze-out formula suggests that claimed contributions such as thermal-shear polarization are not universal: a measurement that confirms one pseudo-gauge's prediction may simply be selecting the pseudo-gauge that matches how hadronization projects spin.
  • The same entropy-current machinery could be extended to chiral spin magnetohydrodynamics by promoting magnetic flux to an order-one variable alongside vorticity, offering a common framework for vorticity, magnetic fields, and chirality in heavy-ion collisions; the paper only lists this as a future direction.
  • A sharp testable extension is the spin alignment of vector mesons: the paper notes a similar freeze-out formula exists for spin-one particles, and comparing that formula's predictions with the measured $\rho_{00}$ matrix element would check the same constitutive relations through an independent observable.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. This paper is a pedagogical review of relativistic spin hydrodynamics. It first reviews first-order relativistic hydrodynamics as an effective theory, then constructs spin hydrodynamics from energy-momentum and angular momentum conservation, taking the spin density S^{ρσ} to be O(∂) and using a covariant entropy current to fix the antisymmetric part of the energy-momentum tensor. The paper then discusses pseudo-gauge ambiguity, strong-vorticity or gyrohydrodynamics, and a spin Cooper-Frye formula for Dirac fermions, and closes with outlooks on spin magnetohydrodynamics, transport coefficients, and numerical simulations.

Significance. If corrected, this would be a useful reference for the heavy-ion and relativistic-fluid community: it collects the pseudo-gauge, power-counting, and freeze-out issues of spin hydrodynamics into one pedagogical narrative, gives explicit constitutive relations with the two new transport coefficients λ and η_s, and provides a phase-space spin formula suitable for phenomenological applications. The organization is clear, the references are extensive, and the paper explicitly identifies the quasi-hydrodynamic nature of spin density. However, the central derivation currently contains a sign inconsistency that makes the displayed q-sector violate the second law that the derivation is based on; because this is the key construction of the paper, the manuscript needs correction before it can serve as a reliable introduction to the subject.

major comments (1)
  1. [III, Eqs. (38)-(41)] The sign of the q-sector contribution to the entropy-production rate is inconsistent with the stated constitutive relation. With βν = βuν and the thermal vorticity defined in Eq. (42), one has uνTϖ^{μν} = (1/2)(Du^μ + β∇^μT) up to terms that vanish under u·q = 0. Therefore the q-part of Θ^a in Eq. (39) contributes to Eq. (38) as 2q_μ u_ν (μ^{μν} − Tϖ^{μν}) = −q_μ(β∇^μT + Du^μ − 2μ^{μν}u_ν) = −|q|^2/λ. For λ ≥ 0 this is strictly negative whenever q ≠ 0, contradicting the claimed semi-positive entropy production. The limiting case μ^{μν}=0, u^μ=(1,0), and a static temperature gradient gives T∂_μs^μ = −λ|β∇T|^2 < 0 and describes heat flowing up the temperature gradient. The sign in Eq. (40), or equivalently the relative sign between q^μ in Eq. (39) and the entropy current in Eq. (37), must be corrected; Eqs. (43)-(44) inherit this correction.
minor comments (5)
  1. [IV C, Eqs. (96)-(99)] The transition from the first-order Wigner function in Eq. (96) to the phase-space spin vector in Eq. (98) is summarized as "after some calculations"; for a pedagogical review, the Dirac traces and the use of Eq. (97) should be displayed or the intermediate steps should be given explicitly, since Eq. (99) is one of the main outputs of the subsection.
  2. [IV B, Eq. (73)] The derivation of the entropy-production rate in Eq. (73) is stated without intermediate steps, and the power counting of n5 is described only in passing. The reader is told that n5 is O(∂^3) by Eq. (65), but the counting of the terms in Eq. (65) is not transparent; a short explanation would improve the pedagogical value.
  3. [IV B, Eqs. (78)-(80)] The notation with symmetrization brackets such as Ξ^{μ(ρ}b^{σ)} and b^{μ(ρ}b^{σ)} would be clearer if the convention for (anti)symmetrization were stated explicitly, since b^{μν} was defined earlier as a cross projector.
  4. [Throughout] There are several typographical slips: "ultilize" in Section III, "tenor" for "tensor" after Eq. (76), "tracelss" in the discussion of Eq. (105), and inconsistent hyphenation of "pseudo-gauge". A careful proofreading pass is recommended.
  5. [Section II, Eq. (27)] The sign conventions for ζ and η in Eq. (27) are standard, but the reader may benefit from an explicit sentence noting that the signs of the dissipative terms depend on the metric convention η^{μν} = diag(1,−1,−1,−1) adopted in the paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is a pedagogical review that reproduces its derivations explicitly; self-citations are attributions, not load-bearing premises.

full rationale

I checked the derivation chain in Sections II and III. The spin-hydrodynamic constitutive relations (39)-(41) are not obtained by fitting data or by defining an output in terms of an input; they follow from the displayed local first law (33), the covariant entropy current (37), and the entropy-production expression (38), with the cited prior work [68] serving as attribution rather than as an unverified premise that the paper itself relies on. The same holds for the completely antisymmetric spin-tensor version in Section IV.A, which is derived from the same entropy-current ansatz with the relation (64) and cited to [62]. The review does not present a new prediction whose value is fixed by a self-citation chain, nor does it invoke a uniqueness theorem from the authors to forbid alternatives. Although there are many self-citations (e.g., Refs. [62,65,68,72,130,131]), these are used in the standard review sense: they point to where the displayed results were first obtained, while the present paper restates the derivational steps. The skeptical note about a possible sign inconsistency in Eq. (40) is a correctness concern about whether the displayed relation satisfies the second law, not a circularity concern; even if the sign were wrong, the argument would be internally inconsistent rather than self-referential. Under the hard rules, I therefore find no circular step and assign score 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No new particles, fields, or forces are introduced. The free parameters listed are phenomenological transport coefficients that appear in the constitutive relations; they are not fitted to data in the paper but are part of the general framework.

free parameters (2)
  • λ (boost heat conductivity)
    Introduced in Eq. (40) as a coefficient in the constitutive relation for the heat current qμ; not determined within the paper.
  • η_s (rotational viscosity)
    Introduced in Eq. (41) controlling relaxation of spin density to thermal vorticity; not determined within the paper.
assumptions (4)
  • domain assumption Local thermodynamic first law holds with spin potential μ^{μν} conjugate to spin density S^{μν} (Eq. (33)).
    Assumed without derivation; an extension of standard thermodynamics to spin degrees of freedom.
  • domain assumption The spin density S^{ρσ} is O(∂), i.e. parametrically small compared to energy density and velocity (Eq. (32)).
    Motivated by the few-percent hyperon polarization observed in heavy-ion collisions; justifies gradient expansion but may fail for strongly polarized systems.
  • domain assumption The second law of local thermodynamics (semi-positive entropy production) constrains the constitutive relations.
    Used to fix signs and forms of dissipative coefficients; standard in hydrodynamics.
  • domain assumption The Wigner function calculation for the spin Cooper-Frye formula assumes free Dirac fermions and a flat freeze-out hypersurface.
    Explicitly stated in Section IV.C; simplifies the derivation but limits generality.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An introduction to relativistic spin hydrodynamics." pith.science (2026). https://pith.science/paper/HCK3J25W

@misc{pith2026241111753,
  author       = {Pith},
  title        = {Pith review of: An introduction to relativistic spin hydrodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HCK3J25W}},
  note         = {Machine review of arXiv:2411.11753}
}
read the original abstract

Spin polarization and spin transport are common phenomena in many quantum systems. Relativistic spin hydrodynamics provides an effective low-energy framework to describe these processes in quantum many-body systems. The fundamental symmetry underlying relativistic spin hydrodynamics is angular momentum conservation, which naturally leads to inter-conversion between spin and orbital angular momenta. This inter-conversion is a key feature of relativistic spin hydrodynamics, closely related to entropy production and introducing ambiguity in the construction of constitutive relations. In this article, we present a pedagogical introduction to relativistic spin hydrodynamics. We demonstrate how to derive the constitutive relations by applying local thermodynamic laws and explore several distinctive aspects of spin hydrodynamics. These include the pseudo-gauge ambiguity, the behavior of the system in the presence of strong vorticity, and the challenges of modeling the freeze-out of spin in heavy-ion collisions. We also outline some future prospects for spin hydrodynamics.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 7 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Theory of Spinful Relativistic Superfluids

    nucl-th 2026-07 accept novelty 7.5 of 10

    A single quantized topological term in the dual effective action of a spinful relativistic superfluid produces the relativistic Mermin-Ho relation, anomalous Ettingshausen/Hall transport, and anomalous Hall viscosity.

  2. Local Spin Polarization in Anisotropic Gubser Flow: Suppression Mechanism and Formulation Dependence

    nucl-th 2026-08 conditional novelty 6.0 of 10

    In an anisotropic Gubser flow, the longitudinal spin polarization's sign depends on which shear formulation is used, and two popular formulations show exact or near-exact cancellation between vorticity and shear contr...

  3. Fluid Acceleration in Heavy-Ion Collisions

    nucl-th 2026-03 conditional novelty 6.0 of 10

    In AMPT and UrQMD simulations, fluid acceleration in heavy-ion collisions reaches a few hundred MeV, peaks at fireball boundaries, and evolves from stopping-dominated deceleration at low energies to boundary/tilt puls...

  4. Tensor spin polarization induced by curved freeze-out hypersurface

    hep-ph 2025-09 conditional novelty 6.0 of 10

    The curvature of the freeze-out hypersurface induces a tensor spin polarization of vector mesons at leading gradient order, with predicted phi-meson spin alignment around -10^-4 to -10^-3.

  5. Vector and Tensor Spin Polarization for Vector Bosons at Local Equilibrium

    hep-ph 2024-12 conditional novelty 6.0 of 10

    Vector meson spin alignment at local equilibrium is shown to arise only at second order in thermodynamic gradients, with explicit analytic formulas for the contributing terms.

  6. Quasiparticle second-order dissipative hydrodynamics at finite chemical potential

    hep-ph 2024-12 conditional novelty 5.0 of 10

    A quasiparticle kinetic theory with a bag term yields second-order equations of relativistic dissipative hydrodynamics with baryon diffusion and chemical-potential-dependent transport coefficients.

  7. Spin hydrodynamics

    hep-ph 2024-11 conditional novelty 2.0 of 10

    The paper proposes a hybrid perfect and dissipative spin hydrodynamics built on generalized tensor thermodynamic relations, but it contains no new derivation beyond the cited prior works.

Reference graph

Works this paper leans on

143 extracted references · 3 canonical work pages · cited by 7 Pith papers

  1. [1]

    Wang, ed., Quark-Gluon Plasma 5 (World Scientific, New Jersey, 2016)

    X.-N. Wang, ed., Quark-Gluon Plasma 5 (World Scientific, New Jersey, 2016)

    2016

  2. [3]

    Zhang, D

    Y . Zhang, D. Zhang, and X. Luo, Nucl. Tech.46 (2023), 10.11889/j.0253-3219.2023.hjs.46.040001

    arXiv 2023

  3. [4]

    Shou et al., Nucl

    Q.-Y . Shou et al., Nucl. Sci. Tech. 35, 219 (2024), arXiv:2409.17964 [nucl-ex]

    arXiv 2024

  4. [5]

    Chen et al., Nucl

    J. Chen et al., Nucl. Sci. Tech. 35, 214 (2024), arXiv:2407.02935 [nucl-ex]

    arXiv 2024

  5. [6]

    Ollitrault, Phys

    J.-Y . Ollitrault, Phys. Rev. D46, 229 (1992)

    1992

  6. [7]

    Adamczyk et al

    L. Adamczyk et al. (STAR), Nature 548, 62 (2017), arXiv:1701.06657 [nucl-ex]

    arXiv 2017

  7. [8]

    Liang and X.-N

    Z.-T. Liang and X.-N. Wang, Phys. Rev. Lett. 94, 102301 (2005), [Erratum: Phys.Rev.Lett. 96, 039901 (2006)], arXiv:nucl-th/0410079

    arXiv 2005

  8. [9]

    S. A. V oloshin, (2004), arXiv:nucl-th/0410089

    arXiv 2004

Show all 143 references
  1. [10]

    Liang and X.-N

    Z.-T. Liang and X.-N. Wang, Phys. Lett. B 629, 20 (2005), arXiv:nucl-th/0411101

    2005 arXiv

  2. [11]

    Adam et al

    J. Adam et al. (STAR), Phys. Rev. C98, 014910 (2018), arXiv:1805.04400 [nucl-ex]

    2018 arXiv

  3. [12]

    M. S. Abdallah et al. (STAR), Phys. Rev. C104, L061901 (2021), arXiv:2108.00044 [nucl-ex]

    2021

  4. [13]

    Abou Yassine et al

    R. Abou Yassine et al. (HADES), Phys. Lett. B 835, 137506 (2022), arXiv:2207.05160 [nucl-ex]

    2022 arXiv

  5. [14]

    M. I. Abdulhamid et al. (STAR), Phys. Rev. C108, 014910 (2023), arXiv:2305.08705 [nucl-ex]

    2023 arXiv

  6. [15]

    M. S. Abdallah et al. (STAR), Nature 614, 244 (2023), arXiv:2204.02302 [hep-ph]

    2023

  7. [16]

    Acharya et al

    S. Acharya et al. (ALICE), Phys. Rev. Lett. 131, 042303 (2023), arXiv:2204.10171 [nucl-ex]. 29

    2023 arXiv

  8. [17]

    Deng and X.-G

    W.-T. Deng and X.-G. Huang, Phys. Rev. C 93, 064907 (2016), arXiv:1603.06117 [nucl-th]

    2016 arXiv

  9. [18]

    Jiang, Z.-W

    Y . Jiang, Z.-W. Lin, and J. Liao, Phys. Rev. C 94, 044910 (2016), [Erratum: Phys.Rev.C 95, 049904 (2017)], arXiv:1602.06580 [hep-ph]

    2016 arXiv

  10. [19]

    Deng, X.-G

    X.-G. Deng, X.-G. Huang, Y .-G. Ma, and S. Zhang, Phys. Rev. C 101, 064908 (2020), arXiv:2001.01371 [nucl-th]

    2020 arXiv

  11. [20]

    Becattini, V

    F. Becattini, V . Chandra, L. Del Zanna, and E. Grossi, Annals Phys.338, 32 (2013), arXiv:1303.3431 [nucl-th]

    2013 arXiv

  12. [21]

    Fang, L.-G

    R.-H. Fang, L.-G. Pang, Q. Wang, and X.-N. Wang, Phys. Rev. C 94, 024904 (2016), arXiv:1604.04036 [nucl-th]

    2016 arXiv

  13. [22]

    Y .-C. Liu, K. Mameda, and X.-G. Huang, Chin. Phys. C 44, 094101 (2020), [Erratum: Chin.Phys.C 45, 089001 (2021)], arXiv:2002.03753 [hep-ph]

    2020 arXiv

  14. [23]

    Karpenko and F

    I. Karpenko and F. Becattini, Eur. Phys. J. C 77, 213 (2017), arXiv:1610.04717 [nucl-th]

    2017 arXiv

  15. [24]

    Y . Xie, D. Wang, and L. P. Csernai, Phys. Rev. C 95, 031901 (2017), arXiv:1703.03770 [nucl-th]

    2017 arXiv

  16. [25]

    Li, L.-G

    H. Li, L.-G. Pang, Q. Wang, and X.-L. Xia, Phys. Rev. C 96, 054908 (2017), arXiv:1704.01507 [nucl-th]

    2017 arXiv

  17. [26]

    S. Shi, K. Li, and J. Liao, Phys. Lett. B 788, 409 (2019), arXiv:1712.00878 [nucl-th]

    2019 arXiv

  18. [27]

    X.-L. Xia, H. Li, Z.-B. Tang, and Q. Wang, Phys. Rev. C 98, 024905 (2018), arXiv:1803.00867 [nucl-th]

    2018 arXiv

  19. [28]

    Wei, W.-T

    D.-X. Wei, W.-T. Deng, and X.-G. Huang, Phys. Rev. C99, 014905 (2019), arXiv:1810.00151 [nucl- th]

    2019 arXiv

  20. [29]

    Vitiuk, L

    O. Vitiuk, L. V . Bravina, and E. E. Zabrodin, Phys. Lett. B 803, 135298 (2020), arXiv:1910.06292 [hep-ph]

    2020 arXiv

  21. [30]

    Y . B. Ivanov, V . D. Toneev, and A. A. Soldatov, Phys. Rev. C100, 014908 (2019), arXiv:1903.05455 [nucl-th]

    2019 arXiv

  22. [31]

    B. Fu, K. Xu, X.-G. Huang, and H. Song, Phys. Rev. C 103, 024903 (2021), arXiv:2011.03740 [nucl-th]

    2021 arXiv

  23. [32]

    Y . Guo, J. Liao, E. Wang, H. Xing, and H. Zhang, Phys. Rev. C 104, L041902 (2021), arXiv:2105.13481 [nucl-th]

    2021 arXiv

  24. [33]

    Li, X.-L

    H. Li, X.-L. Xia, X.-G. Huang, and H. Z. Huang, Phys. Lett. B 827, 136971 (2022), arXiv:2106.09443 [nucl-th]

    2022 arXiv

  25. [34]

    Deng, X.-G

    X.-G. Deng, X.-G. Huang, and Y .-G. Ma, Phys. Lett. B 835, 137560 (2022), arXiv:2109.09956 30 [nucl-th]

    2022 arXiv

  26. [35]

    X.-Y . Wu, C. Yi, G.-Y . Qin, and S. Pu, Phys. Rev. C105, 064909 (2022), arXiv:2204.02218 [hep-ph]

    2022 arXiv

  27. [36]

    Hidaka, S

    Y . Hidaka, S. Pu, Q. Wang, and D.-L. Yang, Prog. Part. Nucl. Phys. 127, 103989 (2022), arXiv:2201.07644 [hep-ph]

    2022 arXiv

  28. [37]

    Gao, G.-L

    J.-H. Gao, G.-L. Ma, S. Pu, and Q. Wang, Nucl. Sci. Tech. 31, 90 (2020), arXiv:2005.10432 [hep- ph]

    2020 arXiv

  29. [38]

    Becattini and M

    F. Becattini and M. A. Lisa, Ann. Rev. Nucl. Part. Sci. 70, 395 (2020), arXiv:2003.03640 [nucl-ex]

    2020 arXiv

  30. [39]

    Huang, J

    X.-G. Huang, J. Liao, Q. Wang, and X.-L. Xia, Lect. Notes Phys. 987, 281 (2021), arXiv:2010.08937 [nucl-th]

    2021 arXiv

  31. [40]

    Huang, Nucl

    X.-G. Huang, Nucl. Phys. A 1005, 121752 (2021), arXiv:2002.07549 [nucl-th]

    2021 arXiv

  32. [41]

    Liu and X.-G

    Y .-C. Liu and X.-G. Huang, Nucl. Sci. Tech.31, 56 (2020), arXiv:2003.12482 [nucl-th]

    2020 arXiv

  33. [42]

    Becattini, Rept

    F. Becattini, Rept. Prog. Phys. 85, 122301 (2022), arXiv:2204.01144 [nucl-th]

    2022 arXiv

  34. [43]

    Sheng, Z.-T

    X.-L. Sheng, Z.-T. Liang, and Q. Wang, Acta Phys. Sin. 72, 072502 (2023)

    2023

  35. [44]

    Gao, X.-G

    J.-H. Gao, X.-G. Huang, Z.-T. Liang, Q. Wang, and X.-N. Wang, Acta Phys. Sin. 72, 072501 (2023)

    2023

  36. [45]

    Ruan, Z.-B

    L.-J. Ruan, Z.-B. Xu, and C. Yang, Acta Phys. Sin. 72, 112401 (2023)

    2023

  37. [46]

    Gao, X.-L

    J.-H. Gao, X.-L. Sheng, Q. Wang, and P.-F. Zhuang, Acta Phys. Sin. 72, 112501 (2023)

    2023

  38. [47]

    Becattini, M

    F. Becattini, M. Buzzegoli, T. Niida, S. Pu, A.-H. Tang, and Q. Wang, Int. J. Mod. Phys. E 33, 2430006 (2024), arXiv:2402.04540 [nucl-th]

    2024 arXiv

  39. [48]

    Chen, Z.-T

    J.-H. Chen, Z.-T. Liang, Y .-G. Ma, X.-L. Sheng, and Q. Wang, (2024), arXiv:2407.06480 [hep-ph]

    2024 arXiv

  40. [49]

    Sun, C.-S

    X. Sun, C.-S. Zhou, J.-H. Chen, Z.-Y . Chen, Y .-G. Ma, A.-H. Tang, and Q.-H. Xu, Acta Phys. Sin. 72, 072401 (2023)

    2023

  41. [50]

    Ji, X.-Y

    Z. Ji, X.-Y . Zhao, A.-Q. Guo, Q.-H. Xu, and J.-L. Zhang, Nucl. Sci. Tech. 34, 155 (2023), arXiv:2308.15998 [nucl-ex]

    2023 arXiv

  42. [51]

    D. E. Kharzeev, J. Liao, S. A. V oloshin, and G. Wang, Prog. Part. Nucl. Phys. 88, 1 (2016), arXiv:1511.04050 [hep-ph]

    2016 arXiv

  43. [52]

    Huang, Rept

    X.-G. Huang, Rept. Prog. Phys. 79, 076302 (2016), arXiv:1509.04073 [nucl-th]

    2016 arXiv

  44. [53]

    Hattori and X.-G

    K. Hattori and X.-G. Huang, Nucl. Sci. Tech. 28, 26 (2017), arXiv:1609.00747 [nucl-th]

    2017 arXiv

  45. [54]

    D. E. Kharzeev and J. Liao, Nature Rev. Phys. 3, 55 (2021), arXiv:2102.06623 [hep-ph]

    2021 arXiv

  46. [55]

    Zhao, G.-L

    X.-L. Zhao, G.-L. Ma, and Y .-G. Ma, Acta Phys. Sin. 72, 112502 (2023)

    2023

  47. [56]

    D. E. Kharzeev, J. Liao, and P. Tribedy, (2024), arXiv:2405.05427 [nucl-th]

    2024 arXiv

  48. [57]

    Romatschke, Int

    P. Romatschke, Int. J. Mod. Phys. E 19, 1 (2010), arXiv:0902.3663 [hep-ph]. 31

    2010 arXiv

  49. [58]

    Jeon and U

    S. Jeon and U. Heinz, Int. J. Mod. Phys. E 24, 1530010 (2015), arXiv:1503.03931 [hep-ph]

    2015 arXiv

  50. [59]

    Yan, Chin

    L. Yan, Chin. Phys. C 42, 042001 (2018), arXiv:1712.04580 [nucl-th]

    2018 arXiv

  51. [60]

    Florkowski, M

    W. Florkowski, M. P. Heller, and M. Spalinski, Rept. Prog. Phys. 81, 046001 (2018), arXiv:1707.02282 [hep-ph]

    2018 arXiv

  52. [61]

    G. S. Rocha, D. Wagner, G. S. Denicol, J. Noronha, and D. H. Rischke, Entropy 26, 189 (2024), arXiv:2311.15063 [nucl-th]

    2024 arXiv

  53. [62]

    Hongo, X.-G

    M. Hongo, X.-G. Huang, M. Kaminski, M. Stephanov, and H.-U. Yee, JHEP 11, 150 (2021), arXiv:2107.14231 [hep-th]

    2021 arXiv

  54. [63]

    Grozdanov, A

    S. Grozdanov, A. Lucas, and N. Poovuttikul, Phys. Rev. D 99, 086012 (2019), arXiv:1810.10016 [hep-th]

    2019 arXiv

  55. [64]

    Stephanov and Y

    M. Stephanov and Y . Yin, Phys. Rev. D98, 036006 (2018), arXiv:1712.10305 [nucl-th]

    2018 arXiv

  56. [65]

    Hongo, X.-G

    M. Hongo, X.-G. Huang, M. Kaminski, M. Stephanov, and H.-U. Yee, JHEP 08, 263 (2022), arXiv:2201.12390 [hep-th]

    2022 arXiv

  57. [66]

    Hidaka, M

    Y . Hidaka, M. Hongo, M. A. Stephanov, and H.-U. Yee, Phys. Rev. C 109, 054909 (2024), arXiv:2312.08266 [hep-ph]

    2024 arXiv

  58. [67]

    Li and H.-U

    S. Li and H.-U. Yee, Phys. Rev. D 100, 056022 (2019), arXiv:1905.10463 [hep-ph]

    2019 arXiv

  59. [68]

    Hattori, M

    K. Hattori, M. Hongo, X.-G. Huang, M. Matsuo, and H. Taya, Phys. Lett. B 795, 100 (2019), arXiv:1901.06615 [hep-th]

    2019 arXiv

  60. [69]

    Fukushima and S

    K. Fukushima and S. Pu, Phys. Lett. B 817, 136346 (2021), arXiv:2010.01608 [hep-th]

    2021 arXiv

  61. [70]

    D. She, A. Huang, D. Hou, and J. Liao, Sci. Bull. 67, 2265 (2022), arXiv:2105.04060 [nucl-th]

    2022 arXiv

  62. [71]

    Daher, A

    A. Daher, A. Das, W. Florkowski, and R. Ryblewski, Phys. Rev. C 108, 024902 (2023), arXiv:2202.12609 [nucl-th]

    2023 arXiv

  63. [72]

    Z. Cao, K. Hattori, M. Hongo, X.-G. Huang, and H. Taya, PTEP 2022, 071D01 (2022), arXiv:2205.08051 [hep-th]

    2022 arXiv

  64. [73]

    Hu, Phys

    J. Hu, Phys. Rev. C 107, 024915 (2023), arXiv:2209.10979 [nucl-th]

    2023 arXiv

  65. [74]

    Biswas, A

    R. Biswas, A. Daher, A. Das, W. Florkowski, and R. Ryblewski, Phys. Rev. D 108, 014024 (2023), arXiv:2304.01009 [nucl-th]

    2023 arXiv

  66. [75]

    Becattini, A

    F. Becattini, A. Daher, and X.-L. Sheng, Phys. Lett. B 850, 138533 (2024), arXiv:2309.05789 [nucl- th]

    2024 arXiv

  67. [76]

    Drogosz, W

    Z. Drogosz, W. Florkowski, and M. Hontarenko, (2024), arXiv:2408.03106 [hep-ph]

    2024 arXiv

  68. [77]

    Dey and A

    S. Dey and A. Das, (2024), arXiv:2410.04141 [nucl-th]. 32

    2024 arXiv

  69. [78]

    Yang and L

    L. Yang and L. Yan, (2024), arXiv:2410.07583 [nucl-th]

    2024 arXiv

  70. [79]

    A. D. Gallegos, U. G ¨ursoy, and A. Yarom, SciPost Phys.11, 041 (2021), arXiv:2101.04759 [hep-th]

    2021 arXiv

  71. [80]

    A. D. Gallegos, U. Gursoy, and A. Yarom, JHEP 05, 139 (2023), arXiv:2203.05044 [hep-th]

    2023 arXiv

  72. [81]

    Hu, Phys

    J. Hu, Phys. Rev. D 103, 116015 (2021), arXiv:2101.08440 [hep-ph]

    2021 arXiv

  73. [82]

    Tiwari and B

    A. Tiwari and B. K. Patra, (2024), arXiv:2408.11514 [hep-th]

    2024 arXiv

  74. [83]

    She, Y .-W

    D. She, Y .-W. Qiu, and D. Hou, (2024), arXiv:2410.15142 [nucl-th]

    2024 arXiv

  75. [84]

    Florkowski, B

    W. Florkowski, B. Friman, A. Jaiswal, and E. Speranza, Phys. Rev. C 97, 041901 (2018), arXiv:1705.00587 [nucl-th]

    2018 arXiv

  76. [85]

    Florkowski, A

    W. Florkowski, A. Kumar, and R. Ryblewski, Prog. Part. Nucl. Phys. 108, 103709 (2019), arXiv:1811.04409 [nucl-th]

    2019 arXiv

  77. [86]

    Bhadury, W

    S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, and R. Ryblewski, Phys. Lett. B 814, 136096 (2021), arXiv:2002.03937 [hep-ph]

    2021 arXiv

  78. [87]

    S. Shi, C. Gale, and S. Jeon, Phys. Rev. C 103, 044906 (2021), arXiv:2008.08618 [nucl-th]

    2021 arXiv

  79. [88]

    Bhadury, W

    S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, and R. Ryblewski, Phys. Rev. D 103, 014030 (2021), arXiv:2008.10976 [nucl-th]

    2021 arXiv

  80. [89]

    Peng, J.-J

    H.-H. Peng, J.-J. Zhang, X.-L. Sheng, and Q. Wang, Chin. Phys. Lett. 38, 116701 (2021), arXiv:2107.00448 [hep-th]

    2021 arXiv

  81. [90]

    Hu, Phys

    J. Hu, Phys. Rev. D 105, 076009 (2022), arXiv:2111.03571 [hep-ph]

    2022 arXiv

  82. [91]

    Weickgenannt, E

    N. Weickgenannt, E. Speranza, X.-l. Sheng, Q. Wang, and D. H. Rischke, Phys. Rev. Lett. 127, 052301 (2021), arXiv:2005.01506 [hep-ph]

    2021 arXiv

  83. [92]

    Weickgenannt, D

    N. Weickgenannt, D. Wagner, E. Speranza, and D. H. Rischke, Phys. Rev. D 106, 096014 (2022), arXiv:2203.04766 [nucl-th]

    2022 arXiv

  84. [93]

    Weickgenannt, D

    N. Weickgenannt, D. Wagner, E. Speranza, and D. H. Rischke, Phys. Rev. D 106, L091901 (2022), arXiv:2208.01955 [nucl-th]

    2022 arXiv

  85. [94]

    Weickgenannt, Phys

    N. Weickgenannt, Phys. Rev. D 108, 076011 (2023), arXiv:2307.13561 [nucl-th]

    2023 arXiv

  86. [95]

    Wagner, (2024), arXiv:2409.07143 [nucl-th]

    D. Wagner, (2024), arXiv:2409.07143 [nucl-th]

    2024

  87. [96]

    Montenegro, L

    D. Montenegro, L. Tinti, and G. Torrieri, Phys. Rev. D 96, 056012 (2017), [Addendum: Phys.Rev.D 96, 079901 (2017)], arXiv:1701.08263 [hep-th]

    2017 arXiv

  88. [97]

    Montenegro and G

    D. Montenegro and G. Torrieri, Phys. Rev. D 100, 056011 (2019), arXiv:1807.02796 [hep-th]

    2019 arXiv

  89. [98]

    Montenegro and G

    D. Montenegro and G. Torrieri, Phys. Rev. D 102, 036007 (2020), arXiv:2004.10195 [hep-th]

    2020 arXiv

  90. [99]

    S. Li, M. A. Stephanov, and H.-U. Yee, Phys. Rev. Lett. 127, 082302 (2021), arXiv:2011.12318 33 [hep-th]

    2021 arXiv

  91. [100]

    D.-L. Wang, S. Fang, and S. Pu, Phys. Rev. D 104, 114043 (2021), arXiv:2107.11726 [nucl-th]

    2021 arXiv

  92. [101]

    Wang, X.-Q

    D.-L. Wang, X.-Q. Xie, S. Fang, and S. Pu, Phys. Rev. D 105, 114050 (2022), arXiv:2112.15535 [hep-ph]

    2022 arXiv

  93. [102]

    Torrieri and D

    G. Torrieri and D. Montenegro, Phys. Rev. D 107, 076010 (2023), arXiv:2207.00537 [hep-th]

    2023 arXiv

  94. [103]

    Hu, Phys

    J. Hu, Phys. Rev. D 106, 036004 (2022), arXiv:2202.07373 [hep-ph]

    2022 arXiv

  95. [104]

    Hu, Phys

    J. Hu, Phys. Rev. D 105, 096021 (2022), arXiv:2204.12946 [hep-ph]

    2022 arXiv

  96. [105]

    Sarwar, M

    G. Sarwar, M. Hasanujjaman, J. R. Bhatt, H. Mishra, and J.-e. Alam, Phys. Rev. D 107, 054031 (2023), arXiv:2209.08652 [nucl-th]

    2023 arXiv

  97. [106]

    Daher, A

    A. Daher, A. Das, and R. Ryblewski, Phys. Rev. D 107, 054043 (2023), arXiv:2209.10460 [nucl-th]

    2023 arXiv

  98. [107]

    Pu and X.-G

    S. Pu and X.-G. Huang, Acta Phys. Sin. 72, 071202 (2023)

    2023

  99. [108]

    Wagner, M

    D. Wagner, M. Shokri, and D. H. Rischke, Phys. Rev. Res. 6, 043103 (2024), arXiv:2405.00533 [nucl-th]

    2024 arXiv

  100. [109]

    X. Ren, C. Yang, D.-L. Wang, and S. Pu, Phys. Rev. D 110, 034010 (2024), arXiv:2405.03105 [nucl-th]

    2024 arXiv

  101. [110]

    D.-L. Wang, L. Yan, and S. Pu, (2024), arXiv:2408.03781 [hep-ph]

    2024 arXiv

  102. [111]

    F. J. Belinfante, Physica 7, 449 (1940)

    1940

  103. [112]

    Rosenfeld, Memoires Acad

    L. Rosenfeld, Memoires Acad. Roy. De Belgique 18, 1 (1940)

    1940

  104. [113]

    F. W. Hehl, P. V on Der Heyde, G. D. Kerlick, and J. M. Nester, Rev. Mod. Phys.48, 393 (1976)

    1976

  105. [114]

    Becattini, W

    F. Becattini, W. Florkowski, and E. Speranza, Phys. Lett. B 789, 419 (2019), arXiv:1807.10994 [hep-th]

    2019 arXiv

  106. [115]

    Speranza and N

    E. Speranza and N. Weickgenannt, Eur. Phys. J. A 57, 155 (2021), arXiv:2007.00138 [nucl-th]

    2021 arXiv

  107. [116]

    Bhadury, J

    S. Bhadury, J. Bhatt, A. Jaiswal, and A. Kumar, Eur. Phys. J. ST 230, 655 (2021), arXiv:2101.11964 [hep-ph]

    2021 arXiv

  108. [117]

    Weickgenannt, D

    N. Weickgenannt, D. Wagner, and E. Speranza, Phys. Rev. D105, 116026 (2022), arXiv:2204.01797 [nucl-th]

    2022 arXiv

  109. [118]

    Singh, (2024), arXiv:2406.02127 [hep-th]

    R. Singh, (2024), arXiv:2406.02127 [hep-th]

    2024

  110. [119]

    Buzzegoli and A

    M. Buzzegoli and A. Palermo, (2024), arXiv:2407.14345 [hep-ph]

    2024 arXiv

  111. [120]

    Hattori, M

    K. Hattori, M. Hongo, and X.-G. Huang, Symmetry 14, 1851 (2022), arXiv:2207.12794 [hep-th]

    2022 arXiv

  112. [121]

    Huang, A

    X.-G. Huang, A. Sedrakian, and D. H. Rischke, Annals Phys. 326, 3075 (2011), arXiv:1108.0602 [astro-ph.HE]. 34

    2011 arXiv

  113. [122]

    Grozdanov, D

    S. Grozdanov, D. M. Hofman, and N. Iqbal, Phys. Rev. D 95, 096003 (2017), arXiv:1610.07392 [hep-th]

    2017 arXiv

  114. [123]

    Hernandez and P

    J. Hernandez and P. Kovtun, JHEP 05, 001 (2017), arXiv:1703.08757 [hep-th]

    2017 arXiv

  115. [124]

    D. N. Zubarev, A. V . Prozorkevich, and S. A. Smolyanskii, Theor. Math. Phys. 40, 821 (1979)

    1979

  116. [125]

    C. G. van Weert, Annals of Physics 140, 133 (1982)

    1982

  117. [126]

    Becattini, L

    F. Becattini, L. Bucciantini, E. Grossi, and L. Tinti, Eur. Phys. J. C 75, 191 (2015), arXiv:1403.6265 [hep-th]

    2015 arXiv

  118. [127]

    Becattini, M

    F. Becattini, M. Buzzegoli, and E. Grossi, Particles 2, 197 (2019), arXiv:1902.01089 [cond-mat.stat- mech]

    2019 arXiv

  119. [128]

    Becattini, M

    F. Becattini, M. Buzzegoli, and A. Palermo, Phys. Lett. B 820, 136519 (2021), arXiv:2103.10917 [nucl-th]

    2021 arXiv

  120. [129]

    Sheng, F

    X.-L. Sheng, F. Becattini, X.-G. Huang, and Z.-H. Zhang, (2024), arXiv:2407.12130 [hep-th]

    2024 arXiv

  121. [130]

    Liu and X.-G

    Y .-C. Liu and X.-G. Huang, Sci. China Phys. Mech. Astron. 65, 272011 (2022), arXiv:2109.15301 [nucl-th]

    2022 arXiv

  122. [131]

    Buzzegoli, Phys

    M. Buzzegoli, Phys. Rev. C 105, 044907 (2022), arXiv:2109.12084 [nucl-th]

    2022 arXiv

  123. [132]

    S. Y . F. Liu and Y . Yin, JHEP07, 188 (2021), arXiv:2103.09200 [hep-ph]

    2021 arXiv

  124. [133]

    Becattini, M

    F. Becattini, M. Buzzegoli, G. Inghirami, I. Karpenko, and A. Palermo, Phys. Rev. Lett. 127, 272302 (2021), arXiv:2103.14621 [nucl-th]

    2021 arXiv

  125. [134]

    B. Fu, S. Y . F. Liu, L. Pang, H. Song, and Y . Yin, Phys. Rev. Lett. 127, 142301 (2021), arXiv:2103.10403 [hep-ph]

    2021 arXiv

  126. [135]

    S. Y . F. Liu and Y . Yin, Phys. Rev. D104, 054043 (2021), arXiv:2006.12421 [nucl-th]

    2021 arXiv

  127. [136]

    Singh, M

    R. Singh, M. Shokri, and S. M. A. T. Mehr, Nucl. Phys. A 1035, 122656 (2023), arXiv:2202.11504 [hep-ph]

    2023 arXiv

  128. [137]

    Bhadury, W

    S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, and R. Ryblewski, Phys. Rev. Lett. 129, 192301 (2022), arXiv:2204.01357 [nucl-th]

    2022 arXiv

  129. [138]

    Kiamari, N

    M. Kiamari, N. Sadooghi, and M. S. Jafari, Phys. Rev. D 109, 036024 (2024), arXiv:2310.01874 [nucl-th]

    2024 arXiv

  130. [139]

    Z. Fang, K. Hattori, and J. Hu, (2024), arXiv:2409.07096 [hep-ph]

    2024

  131. [140]

    Z. Fang, K. Hattori, and J. Hu (2024) arXiv:2410.00721 [hep-ph]

    2024 arXiv

  132. [141]

    Xie, D.-L

    X.-Q. Xie, D.-L. Wang, C. Yang, and S. Pu, Phys. Rev. D 108, 094031 (2023), arXiv:2306.13880 [hep-ph]. 35

    2023 arXiv

  133. [142]

    Daher, W

    A. Daher, W. Florkowski, R. Ryblewski, and F. Taghinavaz, Phys. Rev. D 109, 114001 (2024), arXiv:2403.04711 [hep-ph]

    2024 arXiv

  134. [143]

    S. K. Singh, R. Ryblewski, and W. Florkowski, Phys. Rev. C 111, 024907 (2025), arXiv:2411.08223 [hep-ph]

    2025 arXiv

  135. [144]

    Sapna, S. K. Singh, and D. Wagner, (2025), arXiv:2503.22552 [hep-ph]

    2025

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.