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REVIEW 2 major objections 3 minor 68 references

Curved freeze-out hypersurfaces spin-align vector mesons at leading order in gradients.

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2026-08-04 15:13 UTC pith:ZZ7J2BJF

load-bearing objection New curvature-induced tensor polarization term, but the central formula rests on an unproved off-shell δ' cancellation. the 2 major comments →

arxiv 2509.20200 v2 pith:ZZ7J2BJF submitted 2025-09-24 hep-ph nucl-th

Tensor spin polarization induced by curved freeze-out hypersurface

Zhong-Hua Zhang , Xu-Guang Huang This is my paper
classification hep-ph nucl-th
keywords tensor spin polarizationspin alignmentvector mesonsfreeze-out hypersurfacecurvature tensorWigner functionrelativistic heavy-ion collisionssmall collision systems
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper argues that the geometric shape of the freeze-out hypersurface, the surface on which particles decouple in a heavy-ion collision, contributes to the tensor spin polarization (spin alignment) of massive vector mesons such as phi mesons. Working from the Proca Lagrangian through the Wigner function formalism, the authors derive a formula in which the surface curvature tensor B_mu nu multiplies first-order hydrodynamic quantities (thermal shear and net spin potential) to generate the polarization tensor T^mu nu. This contribution is one order lower in gradients than the purely hydrodynamic tensor polarization found in earlier work. In Bjorken and Gubser flows, the curvature anisotropy yields delta Theta_yy of roughly -10^-4 to -10^-3 for phi mesons, and the effect grows as the system shrinks, reaching order -10^-2 in central O-O collisions. If correct, spin-alignment measurements in small systems could become a clean probe of the collision's freeze-out geometry.

Core claim

The central result is Eq. (52): the on-shell tensor spin polarization of a massive neutral vector boson is T^mu nu(x,k) = (1+n_B)/(6 n.k) B_alpha beta [ -xi^rho sigma (eta_rho sigma + k_rho k_sigma/(2 m^2)) (Xi^alpha mu Xi^beta nu - (1/3) Delta^mu nu(k)(eta^alpha beta + khat^alpha khat^beta)) + xi^rho sigma (Xi^alpha rho Xi^beta (mu Delta^nu) sigma(k) - (1/3) Delta^mu nu(k) Xi^alpha rho Xi^beta sigma) - delta Omega^rho sigma Xi^alpha rho Xi^beta (mu Delta^nu) sigma(k) ] + O(delta, kappa^2). Here B_mu nu is the curvature tensor of the freeze-out hypersurface, xi_mu nu the thermal shear, delta Omega_mu nu the net spin potential, and Xi^mu nu = eta^mu nu - khat^mu n^nu. The curvature thus enter

What carries the argument

The load-bearing object is the curvature tensor B_mu nu(x) of the freeze-out hypersurface Sigma, defined by B_mu nu = -Delta^alpha_(n) mu partial_alpha n_nu, i.e., the tangential gradient of the surface normal; for an analytic surface F(x)=0 it equals -(1/v) Delta^rho_(n)mu Delta^sigma_(n)nu partial^2 F / partial x^rho partial x^sigma. The calculation proceeds through the covariant Wigner function of the Proca field, a local-equilibrium density operator defined on Sigma, and a gradient expansion in which surface curvature is treated as an independent small parameter. The derivative operator D_alpha arising from integration by parts over the tangent plane acts on the particle propagators; kee

Load-bearing premise

The calculation drops the off-shell term delta'(k0 - E_k) produced by the derivative operator, assuming it vanishes after averaging over the full freeze-out hypersurface; the cancellation is asserted rather than proven, and if it fails, extra same-order terms would change the predicted delta Theta_yy.

What would settle it

Evaluate the omitted delta'(k0 - E_k) term in Eq. (41) explicitly for a concrete curved surface, e.g., the constant-temperature Gubser freeze-out surface, and compute its Cooper-Frye average. If the average is nonzero, Eq. (52) is incomplete at its claimed order; alternatively, a precision measurement of phi-meson delta Theta_yy in central O-O collisions at RHIC would test the predicted magnitude and sign (-10^-2) directly.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For phi mesons in Bjorken flow with freeze-out temperature 150 MeV and proper time 5 fm, the curvature-induced spin alignment is delta Theta_yy ~ -10^-4, with delta Theta_xx negative and delta Theta_zz positive, so longitudinal spin tends to align with the direction of greater curvature.
  • In Gubser flow the estimate is delta Theta_yy ~ -3x10^-3 near the paraxial freeze-out region, and the effect scales with system size: since q ~ 1/L and T0 ~ L, smaller systems show larger curvature contributions.
  • A rough estimate for central O-O collisions gives delta Theta_yy of order -10^-2, making spin-alignment measurements in small systems a possible clean probe of this geometric effect.
  • The curvature contribution vanishes in global equilibrium (zero thermal shear and zero net spin potential), because total angular momentum is hypersurface-independent.
  • The result applies to the space-like part of the freeze-out hypersurface; the time-like part requires future work.
  • The spin alignment along the y direction is negative, meaning the spin-0 state is suppressed relative to spin +/-1 in that direction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Inference: The same geometric mechanism should contribute to the spin alignment of J/psi and D*+ mesons measured at the LHC, since the derivation is generic for massive vector bosons; comparing the magnitude across meson masses could help isolate the curvature term from other spin-alignment mechanisms.
  • Inference: If the omitted off-shell delta'(k0 - E_k) contribution does not vanish after integration, it would act at the same gradient order as Eq. (52) and could shift or even flip the predicted sign of delta Theta_yy; this is the most direct target for an independent check.
  • Inference: A similar curvature-induced contribution should appear in the next-to-leading-order spin polarization of fermions (e.g., Lambda hyperons), where previous analyses have so far neglected surface geometry; the vector-boson result provides a template for that computation.
  • Inference: Because the effect grows as the system shrinks, a transverse-size scan from Au-Au to O-O or Ar-Ar collisions at fixed beam energy would directly test the predicted 1/L scaling of delta Theta_yy.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper derives a covariant formula for the tensor spin polarization of massive vector bosons at local equilibrium, including the effect of a curved freeze-out hypersurface. Starting from the Proca Lagrangian and the Zubarev-type local-equilibrium density operator, the authors perform a first-order gradient expansion of the Wigner function. The central result, Eq. (52), expresses T^{mu nu}(x,k) in terms of the hypersurface curvature tensor B_{alpha beta} multiplied by the thermal shear and net spin potential. Analytic estimates for Bjorken and Gubser flows predict delta Theta_yy of order -10^-4 to -10^-3 for phi mesons, with larger magnitude in smaller systems.

Significance. If the derivation is correct, Eq. (52) identifies a novel geometric contribution to vector-meson spin alignment that is absent in earlier hydrodynamic treatments. The paper is careful to note its limitations (space-like hypersurfaces only, pseudo-gauge dependence, time-like edges left for future work). The analytic Bjorken/Gubser estimates give concrete, falsifiable predictions. The main risk is the unproved cancellation of off-shell delta' terms, which is load-bearing for the central formula; a second, more minor risk is the omission of scalar-distribution curvature corrections that enter the Cooper-Frye normalization.

major comments (2)
  1. [Section IV, after Eq. (41)] The derivation of Eq. (41) omits the off-shell term delta'(k0 - E_k) generated by D_alpha acting on the energy delta, with the statement that it is 'natural that it disappears after we average it over the full hypersurface.' This cancellation is asserted, not proved. The observable Theta_{rs}(k) in Eq. (17) is evaluated at fixed on-shell k through a Cooper-Frye projection; without an explicit projection prescription, a delta' term is not defined before the k0 integral, so it cannot simply be dropped. Because the same derivative acts in the curvature terms of Eq. (42) and the spin-tensor terms of Eq. (46), a surviving delta' contribution would enter the symmetric traceless channel at the same order as Eq. (52). The paper itself cites Ref. [63] for a case where such off-shell terms do contribute to spin alignment. This issue is load-bearing for the central formula and must be resolved, eit
  2. [Section IV, Eq. (50) and Section V/VI, Eqs. (57) and (63)] The spin alignment is defined in Eq. (23) as a ratio involving the scalar distribution f in the denominator. In Eq. (50) the terms proportional to Delta^{mu nu}(k) in the first-order curvature correction are explicitly omitted, with the note that they contribute only to the scalar distribution. However, those terms also contribute to the denominator of Eq. (23). If they do not vanish after the hypersurface average, the numerical estimates of delta Theta_yy in Eqs. (57) and (63) are incomplete at the same order as the numerator. The authors should either display these scalar terms and show their contribution to the ratio is subleading, or include them in the estimates.
minor comments (3)
  1. [Abstract / Section IV (after Eq. 52)] The abstract and introduction state that the curvature contribution is 'one order lower' in gradients than the purely hydrodynamic terms. The later discussion after Eq. (52) correctly notes that for a constant-temperature freeze-out surface the curvature is itself O(delta), making the tensor polarization O(delta^2). The phrasing should be reconciled to avoid overstating the gradient-order reduction.
  2. [Section IV, Eqs. (41)-(50)] The step from Eqs. (41)-(46) to the compact results (49)-(50) is not shown; the authors state they 'present the result' without exhibiting the lengthy algebra or any reproducible scripts. Given the complexity, an appendix or ancillary file would help readers verify the derivation.
  3. [Various] Typos and minor wording: 'govened' (page 2), 'spound' should be 'sound' (page 12), and 'the speed of spound' (Section V). The notation hat{k}^alpha is used without explicit definition near Eq. (49), though it is inferable.

Circularity Check

0 steps flagged

No significant circularity: Eq. (52) is derived from the Proca Lagrangian and the standard local-equilibrium density operator via an explicit Wigner-function expansion; the only flagged gap is an asserted off-shell cancellation, which is a correctness risk, not a circular reduction.

full rationale

The paper's central formula, Eq. (52), is obtained by expanding the Wigner function under the local-equilibrium density operator (24), with the freeze-out hypersurface curvature B_{\mu\nu} entering through the explicit Taylor expansion (32)-(33) and the surface-element substitution leading to Eqs. (41), (42), (45), and (46). No parameter is fitted to the target spin alignment; the Bjorken and Gubser estimates use external inputs (T_f=150 MeV, tau_f=5 fm, q^{-1}=4.3 fm, Ttilde0=2.99) and known analytic flow solutions. The paper does rely on Ref. [27] for a standard Wigner-function integral technique, and that reference includes the present authors, but the cited technique is not the claimed curvature result and is independently documented; the new contribution is derived, not imported. No uniqueness theorem or forced-choice argument is borrowed from the authors' prior work. The one genuinely under-supported step is the statement after Eq. (41) that the off-shell term delta'(k0-E_k) 'disappears after we average it over the full hypersurface'; this is asserted, not proven, and the paper itself cites Ref. [63] for cases where off-shell terms contribute to spin alignment. That is a load-bearing assumption and a potential correctness risk, but it is not circular: it is an extra hypothesis about the support of the Wigner function after the Cooper-Frye average, not a built-in identity between the claimed prediction and an input. Therefore, no circular step is established, and the circularity score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The derivation is a first-principles calculation within a well-established formalism. It introduces no new free parameters fitted to the target result; the numerical estimates use standard external inputs (T_f=150 MeV, tau_f=5 fm, m_phi=1.019 GeV, c_s^2, q=1/4.3 fm^-1, Ttilde0=2.99) from prior literature. The main burden is the modeling assumptions listed above, especially the unproved off-shell cancellation and the quadratic approximation of the hypersurface.

axioms (6)
  • domain assumption The local equilibrium density operator has the Zubarev form with canonical spin tensor (Eq 24).
    Section III, Eq (24). The entire calculation uses this density operator. The paper notes results depend on the pseudo-gauge choice, so this is a modeling assumption.
  • domain assumption Delta beta and Omega are O(delta) and the gradient expansion converges; the Taylor expansion of beta(y) around x is valid within the correlated region.
    Section III, Eqs (27)-(30). Standard in spin hydrodynamics; the paper states the convergence condition |beta|/|delta beta| >> l.
  • domain assumption The freeze-out hypersurface is analytic, space-like, and can be replaced by its quadratic approximation within the correlation length (|n.(y'-y)| << l).
    Section III, Eq (31)-(33) and Section IV. This is the basis for defining the curvature tensor B.
  • ad hoc to paper The off-shell delta'(k0 - E_k) terms vanish after the Cooper-Frye hypersurface average.
    Section IV, after Eq (41). The paper asserts this without proof. It is the weakest assumption.
  • domain assumption For the numerical estimates, the produced particle is at rest in the fluid cell (k^mu = m u^mu).
    Sections V and VI. This simplifies the formula; real phi mesons have finite p_T.
  • domain assumption The hydrodynamic backgrounds (Bjorken and Gubser flows with the stated parameters) are accurate for the systems considered.
    Sections V and VI, using Ref [30] and [67]. The estimates inherit errors from these backgrounds.

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read the original abstract

We investigate how the curvature of the freeze-out hypersurface polarizes massive vector bosons in relativistic heavy-ion collisions. Starting from the Proca Lagrangian and using the Wigner function formalism, we perform a systematic gradient expansion to obtain a covariant spin-polarization tensor expressed in terms of hydrodynamic fields and the curvature tensor of the freeze-out hypersurface. Analytic results for Bjorken and Gubser flows show that curvature anisotropy generates a nonzero tensor polarization. For $\phi$ meson, we estimate the curvature contribution to its spin alignment as $ \delta\Theta_{yy} \sim -10^{-4} $ to $ -10^{-3} $. We also find that the curvature contribution grows as the system size decreases. A rough estimate for central O-O collisions gives a spin alignment of order $-10^{-2}$, suggesting that spin-alignment measurements in such small systems may provide a clean probe of this geometric effect.

Figures

Figures reproduced from arXiv: 2509.20200 by Xu-Guang Huang, Zhong-hua Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1. We draw a sketch of Σ and Σ [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

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Reference graph

Works this paper leans on

68 extracted references · 55 linked inside Pith

  1. [1]

    Huang, Vorticity and Spin Polarization — A Theoretical Perspective, Nucl

    X.-G. Huang, Vorticity and Spin Polarization — A Theoretical Perspective, Nucl. Phys. A1005, 121752 (2021), arXiv:2002.07549 [nucl-th]. 19

    Pith/arXiv arXiv 2021

  2. [2]

    Liu and X.-G

    Y.-C. Liu and X.-G. Huang, Anomalous chiral transports and spin polarization in heavy-ion collisions, Nucl. Sci. Tech.31, 56 (2020), arXiv:2003.12482 [nucl-th]

    Pith/arXiv arXiv 2020

  3. [3]

    Huang, J

    X.-G. Huang, J. Liao, Q. Wang, and X.-L. Xia, Vorticity and Spin Polarization in Heavy Ion Collisions: Transport Models, Lect. Notes Phys.987, 281 (2021), arXiv:2010.08937 [nucl-th]

    Pith/arXiv arXiv 2021

  4. [4]

    Becattini and M

    F. Becattini and M. A. Lisa, Polarization and Vorticity in the Quark–Gluon Plasma, Ann. Rev. Nucl. Part. Sci.70, 395 (2020), arXiv:2003.03640 [nucl-ex]

    Pith/arXiv arXiv 2020

  5. [5]

    Becattini, Spin and polarization: a new direction in relativistic heavy ion physics, Rept

    F. Becattini, Spin and polarization: a new direction in relativistic heavy ion physics, Rept. Prog. Phys. 85, 122301 (2022), arXiv:2204.01144 [nucl-th]

    Pith/arXiv arXiv 2022

  6. [6]

    Becattini, M

    F. Becattini, M. Buzzegoli, T. Niida, S. Pu, A.-H. Tang, and Q. Wang, Spin polarization in relativistic heavy-ion collisions, Int. J. Mod. Phys. E33, 2430006 (2024), arXiv:2402.04540 [nucl-th]

    Pith/arXiv arXiv 2024

  7. [7]

    Adamczyket al.(STAR), Global Λ hyperon polarization in nuclear collisions: evidence for the most vortical fluid, Nature548, 62 (2017), arXiv:1701.06657 [nucl-ex]

    L. Adamczyket al.(STAR), Global Λ hyperon polarization in nuclear collisions: evidence for the most vortical fluid, Nature548, 62 (2017), arXiv:1701.06657 [nucl-ex]

    Pith/arXiv arXiv 2017

  8. [8]

    Adamet al.(STAR), Global polarization of Λ hyperons in Au+Au collisions at √sN N = 200 GeV, Phys

    J. Adamet al.(STAR), Global polarization of Λ hyperons in Au+Au collisions at √sN N = 200 GeV, Phys. Rev. C98, 014910 (2018), arXiv:1805.04400 [nucl-ex]

    Pith/arXiv arXiv 2018

  9. [9]

    M. S. Abdallahet al.(STAR), Global Λ-hyperon polarization in Au+Au collisions at √sN N=3 GeV, Phys. Rev. C104, L061901 (2021), arXiv:2108.00044 [nucl-ex]

    arXiv 2021

  10. [10]

    Adamet al.(STAR), Global Polarization of Ξ and Ω Hyperons in Au+Au Collisions at √sN N = 200 GeV, Phys

    J. Adamet al.(STAR), Global Polarization of Ξ and Ω Hyperons in Au+Au Collisions at √sN N = 200 GeV, Phys. Rev. Lett.126, 162301 (2021), [Erratum: Phys.Rev.Lett. 131, 089901 (2023)], arXiv:2012.13601 [nucl-ex]

    arXiv 2021

  11. [11]

    Becattini, V

    F. Becattini, V. Chandra, L. Del Zanna, and E. Grossi, Relativistic distribution function for particles with spin at local thermodynamical equilibrium, Annals Phys.338, 32 (2013), arXiv:1303.3431 [nucl- th]

    Pith/arXiv arXiv 2013

  12. [12]

    Fang, L.-g

    R.-h. Fang, L.-g. Pang, Q. Wang, and X.-n. Wang, Polarization of massive fermions in a vortical fluid, Phys. Rev. C94, 024904 (2016), arXiv:1604.04036 [nucl-th]

    Pith/arXiv arXiv 2016

  13. [13]

    Y.-C. Liu, K. Mameda, and X.-G. Huang, Covariant Spin Kinetic Theory I: Collisionless Limit, Chin. Phys. C44, 094101 (2020), [Erratum: Chin.Phys.C 45, 089001 (2021)], arXiv:2002.03753 [hep-ph]

    Pith/arXiv arXiv 2020

  14. [14]

    Adamet al.(STAR), Polarization of Λ ( ¯Λ) hyperons along the beam direction in Au+Au collisions at √sN N = 200 GeV, Phys

    J. Adamet al.(STAR), Polarization of Λ ( ¯Λ) hyperons along the beam direction in Au+Au collisions at √sN N = 200 GeV, Phys. Rev. Lett.123, 132301 (2019), arXiv:1905.11917 [nucl-ex]

    arXiv 2019

  15. [15]

    Acharyaet al.(ALICE), Polarization of Λ and ¯Λ Hyperons along the Beam Direction in Pb-Pb Collisions at √sN N=5.02 TeV, Phys

    S. Acharyaet al.(ALICE), Polarization of Λ and ¯Λ Hyperons along the Beam Direction in Pb-Pb Collisions at √sN N=5.02 TeV, Phys. Rev. Lett.128, 172005 (2022), arXiv:2107.11183 [nucl-ex]

    Pith/arXiv arXiv 2022

  16. [16]

    Becattini, M

    F. Becattini, M. Buzzegoli, and A. Palermo, Spin-thermal shear coupling in a relativistic fluid, Phys. Lett. B820, 136519 (2021), arXiv:2103.10917 [nucl-th]

    Pith/arXiv arXiv 2021

  17. [17]

    Becattini, M

    F. Becattini, M. Buzzegoli, G. Inghirami, I. Karpenko, and A. Palermo, Local Polarization and Isother- mal Local Equilibrium in Relativistic Heavy Ion Collisions, Phys. Rev. Lett.127, 272302 (2021), arXiv:2103.14621 [nucl-th]

    Pith/arXiv arXiv 2021

  18. [18]

    S. Y. F. Liu and Y. Yin, Spin polarization induced by the hydrodynamic gradients, JHEP07, 188, arXiv:2103.09200 [hep-ph]

    Pith/arXiv arXiv

  19. [19]

    B. Fu, S. Y. F. Liu, L. Pang, H. Song, and Y. Yin, Shear-Induced Spin Polarization in Heavy-Ion Collisions, Phys. Rev. Lett.127, 142301 (2021), arXiv:2103.10403 [hep-ph]

    Pith/arXiv arXiv 2021

  20. [20]

    C. Yi, S. Pu, and D.-L. Yang, Reexamination of local spin polarization beyond global equilibrium in relativistic heavy ion collisions, Phys. Rev. C104, 064901 (2021), arXiv:2106.00238 [hep-ph]

    Pith/arXiv arXiv 2021

  21. [21]

    Wu, L.-G

    H.-Z. Wu, L.-G. Pang, X.-G. Huang, and Q. Wang, Local spin polarization in high energy heavy ion collisions, Phys. Rev. Research.1, 033058 (2019), arXiv:1906.09385 [nucl-th]

    Pith/arXiv arXiv 2019

  22. [22]

    Liu and X.-G

    Y.-C. Liu and X.-G. Huang, Spin polarization formula for Dirac fermions at local equilibrium, Sci. China Phys. Mech. Astron.65, 272011 (2022), arXiv:2109.15301 [nucl-th]

    Pith/arXiv arXiv 2022

  23. [23]

    Buzzegoli, Pseudogauge dependence of the spin polarization and of the axial vortical effect, Phys

    M. Buzzegoli, Pseudogauge dependence of the spin polarization and of the axial vortical effect, Phys. Rev. C105, 044907 (2022), arXiv:2109.12084 [nucl-th]

    Pith/arXiv arXiv 2022

  24. [24]

    D. N. Zubarev, A. V. Prozorkevich, and S. A. Smolyanskii, Derivation of nonlinear generalized equations of quantum relativistic hydrodynamics, Theoretical and Mathematical Physics40, 821 (1979)

    1979

  25. [25]

    C. G. van Weert, Maximum entropy principle and relativistic hydrodynamics, Annals of Physics140, 20 133 (1982)

    1982

  26. [26]

    Becattini, M

    F. Becattini, M. Buzzegoli, and E. Grossi, Reworking the Zubarev’s approach to non-equilibrium quan- tum statistical mechanics, Particles2, 197 (2019), arXiv:1902.01089 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2019

  27. [27]

    Zhang, X.-G

    Z.-H. Zhang, X.-G. Huang, F. Becattini, and X.-L. Sheng, Vector and tensor spin polarization for vector bosons at local equilibrium, JHEP07, 224, arXiv:2412.19416 [hep-ph]

    Pith/arXiv arXiv

  28. [28]

    Yang, X.-Q

    S.-Z. Yang, X.-Q. Xie, S. Pu, J.-H. Gao, and Q. Wang, Spin alignment of vector mesons in local equilibrium by Zubarev’s approach, arXiv preprint (2024), arXiv:2412.19400 [hep-ph]

    Pith/arXiv arXiv 2024

  29. [29]

    Sheng, F

    X.-L. Sheng, F. Becattini, X.-G. Huang, and Z.-H. Zhang, Spin polarization of fermions at local equilib- rium: Second-order gradient expansion, Phys. Rev. C110, 064908 (2024), arXiv:2407.12130 [hep-th]

    Pith/arXiv arXiv 2024

  30. [30]

    J. D. Bjorken, Highly Relativistic Nucleus-Nucleus Collisions: The Central Rapidity Region, Phys. Rev. D27, 140 (1983)

    1983

  31. [31]

    Sheng, F

    X.-L. Sheng, F. Becattini, and D. Roselli, An improved formula for Wigner function and spin polar- ization in a decoupling relativistic fluid at local thermodynamic equilibrium, arXiv preprint (2025), arXiv:2509.14301 [nucl-th]

    Pith/arXiv arXiv 2025

  32. [32]

    M. S. Abdallahet al.(STAR), Pattern of global spin alignment ofϕand K ∗0 mesons in heavy-ion collisions, Nature614, 244 (2023), arXiv:2204.02302 [hep-ph]

    arXiv 2023

  33. [33]

    Acharyaet al.(ALICE), First measurement of quarkonium polarization in nuclear collisions at the LHC, Phys

    S. Acharyaet al.(ALICE), First measurement of quarkonium polarization in nuclear collisions at the LHC, Phys. Lett. B815, 136146 (2021), arXiv:2005.11128 [nucl-ex]

    Pith/arXiv arXiv 2021

  34. [34]

    Acharyaet al.(ALICE), Measurement of the J/ψPolarization with Respect to the Event Plane in Pb-Pb Collisions at the LHC, Phys

    S. Acharyaet al.(ALICE), Measurement of the J/ψPolarization with Respect to the Event Plane in Pb-Pb Collisions at the LHC, Phys. Rev. Lett.131, 042303 (2023), arXiv:2204.10171 [nucl-ex]

    Pith/arXiv arXiv 2023

  35. [35]

    Acharyaet al.(ALICE), First measurement of D ∗+ vector spin alignment in Pb-Pb collisions at√sNN =5.02TeV, arXiv preprint (2025), arXiv:2504.00714 [nucl-ex]

    S. Acharyaet al.(ALICE), First measurement of D ∗+ vector spin alignment in Pb-Pb collisions at√sNN =5.02TeV, arXiv preprint (2025), arXiv:2504.00714 [nucl-ex]

    arXiv 2025

  36. [36]

    Sheng, L

    X.-L. Sheng, L. Oliva, and Q. Wang, What can we learn from the global spin alignment ofϕmesons in heavy-ion collisions?, Phys. Rev. D101, 096005 (2020), [Erratum: Phys.Rev.D 105, 099903 (2022)], arXiv:1910.13684 [nucl-th]

    Pith/arXiv arXiv 2020

  37. [37]

    Sheng, L

    X.-L. Sheng, L. Oliva, Z.-T. Liang, Q. Wang, and X.-N. Wang, Spin Alignment of Vector Mesons in Heavy-Ion Collisions, Phys. Rev. Lett.131, 042304 (2023), arXiv:2205.15689 [nucl-th]

    Pith/arXiv arXiv 2023

  38. [38]

    Sheng, S

    X.-L. Sheng, S. Pu, and Q. Wang, Momentum dependence of the spin alignment of theϕmeson, Phys. Rev. C108, 054902 (2023), arXiv:2308.14038 [nucl-th]

    Pith/arXiv arXiv 2023

  39. [39]

    Kumar, B

    A. Kumar, B. M¨ uller, and D.-L. Yang, Spin alignment of vector mesons by glasma fields, Phys. Rev. D108, 016020 (2023), arXiv:2304.04181 [nucl-th]

    Pith/arXiv arXiv 2023

  40. [40]

    Yang, Transverse and longitudinal spin alignment from color fields in heavy ion collisions, Phys

    D.-L. Yang, Transverse and longitudinal spin alignment from color fields in heavy ion collisions, Phys. Rev. D111, 056005 (2025), arXiv:2411.14822 [nucl-th]

    Pith/arXiv arXiv 2025

  41. [41]

    Chen, W.-j

    H.-L. Chen, W.-j. Fu, X.-G. Huang, and G.-L. Ma, Fluctuations and Correlations of Quark Spin in Hot and Dense QCD Matter, Phys. Rev. Lett.135, 032302 (2025), arXiv:2410.20704 [hep-ph]

    arXiv 2025

  42. [42]

    Sheng, Y.-Q

    X.-L. Sheng, Y.-Q. Zhao, S.-W. Li, F. Becattini, and D. Hou, Holographic spin alignment for vector mesons, Phys. Rev. D110, 056047 (2024), arXiv:2403.07522 [hep-ph]

    Pith/arXiv arXiv 2024

  43. [43]

    Xu and M

    K. Xu and M. Huang, Spin alignment of vector mesons induced by local spin density fluctuations, Phys. Rev. D110, 094034 (2024), arXiv:2408.06581 [hep-ph]

    Pith/arXiv arXiv 2024

  44. [44]

    H. A. Ahmed, Y. Chen, and M. Huang, Gluon polarization contribution to the spin alignment of vector mesons from holography, Phys. Rev. D111, 086006 (2025), arXiv:2501.13401 [hep-ph]

    Pith/arXiv arXiv 2025

  45. [45]

    Liang and S

    Y. Liang and S. Lin, Spin alignment of quarkonia in vortical quark-gluon plasma, Chin. Phys. C49, 084105 (2025), arXiv:2502.05866 [hep-ph]

    Pith/arXiv arXiv 2025

  46. [46]

    Zhu, X.-L

    X.-N. Zhu, X.-L. Sheng, and D. Hou, Production of K+K- pairs through the decay ofϕmesons, Phys. Rev. D112, 056011 (2025), arXiv:2503.23919 [hep-ph]

    Pith/arXiv arXiv 2025

  47. [47]

    Sahoo, C

    B. Sahoo, C. R. Singh, and R. Sahoo, Spin alignment of Quarkonia: A Possible Probe of a Deconfined QCD matter in Heavy-ion Collisions at TeV Energies, arXiv preprint (2025), arXiv:2506.09405 [hep- ph]

    Pith/arXiv arXiv 2025

  48. [48]

    Yan and S

    G. Yan and S. Lin, Distorted quarkonia and spin alignment, arXiv preprint (2025), arXiv:2507.22684 [hep-ph]. 21

    arXiv 2025

  49. [49]

    Becattini, Polarization in Relativistic Fluids: A Quantum Field Theoretical Derivation, Lect

    F. Becattini, Polarization in Relativistic Fluids: A Quantum Field Theoretical Derivation, Lect. Notes Phys.987, 15 (2021), arXiv:2004.04050 [hep-th]

    Pith/arXiv arXiv 2021

  50. [50]

    Leader,Spin in Particle Physics, Vol

    E. Leader,Spin in Particle Physics, Vol. 15 (Cambridge University Press, 2001)

    2001

  51. [51]

    Florkowski, B

    W. Florkowski, B. Friman, A. Jaiswal, and E. Speranza, Relativistic fluid dynamics with spin, Phys. Rev. C97, 041901 (2018), arXiv:1705.00587 [nucl-th]

    Pith/arXiv arXiv 2018

  52. [52]

    Hattori, M

    K. Hattori, M. Hongo, X.-G. Huang, M. Matsuo, and H. Taya, Fate of spin polarization in a relativistic fluid: An entropy-current analysis, Phys. Lett. B795, 100 (2019), arXiv:1901.06615 [hep-th]

    Pith/arXiv arXiv 2019

  53. [53]

    Fukushima and S

    K. Fukushima and S. Pu, Spin hydrodynamics and symmetric energy-momentum tensors – A current induced by the spin vorticity –, Phys. Lett. B817, 136346 (2021), arXiv:2010.01608 [hep-th]

    Pith/arXiv arXiv 2021

  54. [54]

    Hongo, X.-G

    M. Hongo, X.-G. Huang, M. Kaminski, M. Stephanov, and H.-U. Yee, Relativistic spin hydrodynamics with torsion and linear response theory for spin relaxation, JHEP11, 150, arXiv:2107.14231 [hep-th]

    Pith/arXiv arXiv

  55. [55]

    Z. Cao, K. Hattori, M. Hongo, X.-G. Huang, and H. Taya, Gyrohydrodynamics: Relativistic spinful fluid with strong vorticity, PTEP2022, 071D01 (2022), arXiv:2205.08051 [hep-th]

    Pith/arXiv arXiv 2022

  56. [56]

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

    Pith/arXiv arXiv 2022

  57. [57]

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

    Pith/arXiv arXiv 2021

  58. [58]

    Huang, An introduction to relativistic spin hydrodynamics, Nucl

    X.-G. Huang, An introduction to relativistic spin hydrodynamics, Nucl. Sci. Tech.36, 208 (2025), arXiv:2411.11753 [nucl-th]

    Pith/arXiv arXiv 2025

  59. [59]

    Becattini, W

    F. Becattini, W. Florkowski, and E. Speranza, Spin tensor and its role in non-equilibrium thermody- namics, Phys. Lett. B789, 419 (2019), arXiv:1807.10994 [hep-th]

    Pith/arXiv arXiv 2019

  60. [60]

    Li and S

    F. Li and S. Y. F. Liu, Tensor polarization and the dissipative damping of vector meson in QCD Medium, arXiv preprint (2022), arXiv:2206.11890 [nucl-th]

    arXiv 2022

  61. [61]

    Buzzegoli, Kubo formulas for spin polarization in dissipative relativistic spin hydrodynamics: a first-order gradient expansion approach, JHEP07, 255, arXiv:2502.15520 [nucl-th]

    M. Buzzegoli, Kubo formulas for spin polarization in dissipative relativistic spin hydrodynamics: a first-order gradient expansion approach, JHEP07, 255, arXiv:2502.15520 [nucl-th]

    Pith/arXiv arXiv

  62. [62]

    Buzzegoli, F

    M. Buzzegoli, F. Becattini, G. Inghirami, I. Karpenko, and A. Palermo, Spin-thermal Shear Coupling in Relativistic Nuclear Collisions, Acta Phys. Polon. Supp.16, 1 (2023), arXiv:2208.04449 [nucl-th]

    Pith/arXiv arXiv 2023

  63. [63]

    Sheng, S.-Y

    X.-L. Sheng, S.-Y. Yang, Y.-L. Zou, and D. Hou, Mass splitting and spin alignment forϕmesons in a magnetic field in NJL model, Eur. Phys. J. C84, 299 (2024), arXiv:2209.01872 [nucl-th]

    Pith/arXiv arXiv 2024

  64. [64]

    Mertig, M

    R. Mertig, M. Bohm, and A. Denner, FEYN CALC: Computer algebraic calculation of Feynman am- plitudes, Comput. Phys. Commun.64, 345 (1991)

    1991

  65. [65]

    Shtabovenko, R

    V. Shtabovenko, R. Mertig, and F. Orellana, New Developments in FeynCalc 9.0, Comput. Phys. Commun.207, 432 (2016), arXiv:1601.01167 [hep-ph]

    Pith/arXiv arXiv 2016

  66. [66]

    Shtabovenko, R

    V. Shtabovenko, R. Mertig, and F. Orellana, FeynCalc 9.3: New features and improvements, Comput. Phys. Commun.256, 107478 (2020), arXiv:2001.04407 [hep-ph]

    Pith/arXiv arXiv 2020

  67. [67]

    S. S. Gubser, Symmetry constraints on generalizations of Bjorken flow, Phys. Rev. D82, 085027 (2010), arXiv:1006.0006 [hep-th]

    Pith/arXiv arXiv 2010

  68. [68]

    S. S. Gubser, S. S. Pufu, and A. Yarom, Entropy production in collisions of gravitational shock waves and of heavy ions, Phys. Rev. D78, 066014 (2008), arXiv:0805.1551 [hep-th]

    Pith/arXiv arXiv 2008