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arxiv: 2606.01429 · v1 · pith:HI23GKRMnew · submitted 2026-05-31 · ✦ hep-th · nucl-th

Pseudo-Gauge Stabilizers and Fibration Structure of the Cooper--Frye Map at Freeze-Out

Jiahua Tian This is my paper

Pith reviewed 2026-06-28 16:26 UTC · model grok-4.3

classification ✦ hep-th nucl-th
keywords pseudo-gauge transformationsCooper-Frye mapfreeze-outspin hydrodynamicsfibration structurepolarization observablesheavy-ion collisions
0
0 comments X

The pith

The Cooper-Frye map at freeze-out factors through a quotient by a universal pseudo-gauge stabilizer.

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

The paper establishes that the Cooper-Frye map, which converts freeze-out hypersurface data into particle observables in relativistic spin hydrodynamics, factors through the quotient by a universal stabilizer group of pseudo-gauge transformations. This produces a stratified fibration over the space of thermodynamic Lagrange multipliers. Observables are thereby classified as base (invariant) or fiber (sensitive) types. A sympathetic reader would care because this bounds the number of independent pseudo-gauge-sensitive observables and enforces consistency relations among them, while also providing a structural explanation for observed tensions in heavy-ion polarization measurements.

Core claim

The Cooper--Frye map is shown to factor through the quotient of freeze-out data by a universal stabilizer, yielding a stratified fibration over the space of thermodynamic Lagrange multipliers. This classifies observables into base and fiber types, bounds the number of independent PGT-sensitive observables by the family-restricted fiber dimension, and implies cross-observable consistency relations. Applied to heavy-ion polarization data, the fibration structure provides a structural interpretation of the tension between Lambda polarization and phi-meson spin alignment as evidence that the vorticity-dominated response sector may need to be enlarged with local field-correlation data. Weyl-anoma

What carries the argument

The universal stabilizer of the pseudo-gauge transformation freedom at freeze-out, through which the Cooper-Frye map factors to produce the fibration over thermodynamic Lagrange multipliers.

If this is right

  • Observables are classified into base and fiber types.
  • The number of independent PGT-sensitive observables is bounded by the family-restricted fiber dimension.
  • Cross-observable consistency relations are implied among polarization and alignment measurements.
  • Weyl-anomaly-induced currents are base observables.
  • The Belinfante--canonical obstruction is recovered from the stabilizer condition.

Where Pith is reading between the lines

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

  • The fibration may suggest adding local field-correlation data to the vorticity response sector to resolve specific polarization tensions.
  • The consistency relations could be checked directly by comparing polarization observables across multiple particle species in the same collision events.
  • The same quotient construction might apply to other hydrodynamic settings that retain pseudo-gauge freedom.

Load-bearing premise

A universal stabilizer exists for the pseudo-gauge transformation freedom at freeze-out and the Cooper-Frye map factors cleanly through the resulting quotient.

What would settle it

A measurement in heavy-ion data showing either more independent PGT-sensitive observables than permitted by the family-restricted fiber dimension or a violation of the predicted cross-observable consistency relations.

read the original abstract

We study the pseudo-gauge transformation (PGT) freedom at freeze-out in relativistic spin hydrodynamics. The Cooper--Frye map is shown to factor through the quotient of freeze-out data by a universal stabilizer, yielding a stratified fibration over the space of thermodynamic Lagrange multipliers. This classifies observables into base and fiber types, bounds the number of independent PGT-sensitive observables by the family-restricted fiber dimension, and implies cross-observable consistency relations. Applied to heavy-ion polarization data, the fibration structure provides a structural interpretation of the tension between $\Lambda$ polarization and $\phi$-meson spin alignment as evidence that the vorticity-dominated response sector may need to be enlarged with local field-correlation data. We show that Weyl-anomaly-induced currents studied recently are classified as base observables and recover the known Belinfante--canonical obstruction $\Omega_{ab}\neq\varpi_{ab}$ from the stabilizer condition.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims that the Cooper--Frye map in relativistic spin hydrodynamics factors through the quotient of freeze-out data by a universal stabilizer arising from pseudo-gauge transformation (PGT) freedom. This yields a stratified fibration over the space of thermodynamic Lagrange multipliers, which classifies observables into base and fiber types, bounds the number of independent PGT-sensitive observables by the family-restricted fiber dimension, and implies cross-observable consistency relations. Applied to heavy-ion polarization data, the structure interprets the tension between Λ polarization and φ-meson spin alignment as evidence that the vorticity-dominated response sector may need enlargement with local field-correlation data. Weyl-anomaly-induced currents are classified as base observables, and the known Belinfante--canonical obstruction Ω_ab ≠ ϖ_ab is recovered from the stabilizer condition.

Significance. If the factorization and fibration structure hold, the work supplies a geometric classification of observables under PGT freedom at freeze-out and a structural explanation for observed tensions in heavy-ion spin data. The recovery of the Belinfante--canonical obstruction from the stabilizer condition is a concrete consistency check that strengthens the formal framework. The bound on independent PGT-sensitive observables and the implied consistency relations could guide model-building in spin hydrodynamics.

major comments (1)
  1. [Abstract, paragraph 2] Abstract, paragraph 2: The central claim that a universal stabilizer exists for the PGT freedom at freeze-out and that the Cooper--Frye map factors cleanly through the resulting quotient (yielding a stratified fibration) is load-bearing for all subsequent results on classification, bounds, and data interpretation. The manuscript states the factorization but does not supply the explicit definition of the stabilizer, the construction of the quotient map, or the verification steps showing that the fibration is stratified as claimed; without these the soundness of the fibration structure cannot be assessed.
minor comments (2)
  1. The abstract is dense; expanding the statement of the stabilizer condition with a brief equation or reference to its defining property would improve readability.
  2. The notation Ω_ab ≠ ϖ_ab for the Belinfante--canonical obstruction would benefit from a short reminder of the definitions of these tensors in the text or a reference to the original literature.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and for highlighting the importance of explicit constructions in establishing the fibration structure. We address the single major comment below and will incorporate the requested clarifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract, paragraph 2] Abstract, paragraph 2: The central claim that a universal stabilizer exists for the PGT freedom at freeze-out and that the Cooper--Frye map factors cleanly through the resulting quotient (yielding a stratified fibration) is load-bearing for all subsequent results on classification, bounds, and data interpretation. The manuscript states the factorization but does not supply the explicit definition of the stabilizer, the construction of the quotient map, or the verification steps showing that the fibration is stratified as claimed; without these the soundness of the fibration structure cannot be assessed.

    Authors: We agree that the current presentation would benefit from a more self-contained and explicit treatment of the stabilizer, quotient, and stratification to permit direct verification. In the revised manuscript we will add a dedicated subsection that (i) defines the universal stabilizer explicitly as the subgroup of pseudo-gauge transformations preserving the on-shell freeze-out data up to redefinition of the thermodynamic multipliers, (ii) constructs the quotient map via the orbit-stabilizer correspondence in the PGT group action, and (iii) verifies the stratified fibration property by exhibiting local trivializations over open sets in the base space of Lagrange multipliers together with the explicit fiber dimension bound. These additions will be placed immediately after the statement of the main theorem so that the subsequent classification, bounds, and phenomenological interpretation rest on fully spelled-out steps. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the formal derivation

full rationale

The paper's central claim is a mathematical factorization of the Cooper-Frye map through the quotient by a stabilizer derived directly from the pseudo-gauge transformation freedom at freeze-out, producing a stratified fibration over thermodynamic Lagrange multipliers. This is presented as a direct consequence of the stabilizer condition without any reduction to fitted parameters, self-citations that carry the load of the result, or ansatzes smuggled in from prior work. The construction classifies observables and recovers the known Belinfante-canonical obstruction as an external consistency check. No load-bearing step equates a prediction or uniqueness claim to its own inputs by definition. The derivation is therefore self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities beyond the universal stabilizer itself. The stabilizer is introduced to quotient the PGT freedom; no independent evidence for it is stated.

axioms (1)
  • domain assumption Existence of a universal stabilizer for pseudo-gauge transformations at freeze-out
    Invoked to define the quotient through which the Cooper-Frye map factors (abstract).
invented entities (1)
  • universal stabilizer no independent evidence
    purpose: Quotient the pseudo-gauge transformation freedom so the Cooper-Frye map factors into a fibration
    Introduced in the abstract as the object that yields the stratified fibration; no falsifiable handle outside the construction is given.

pith-pipeline@v0.9.1-grok · 5685 in / 1374 out tokens · 22144 ms · 2026-06-28T16:26:00.169465+00:00 · methodology

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Reference graph

Works this paper leans on

54 extracted references · 2 canonical work pages

  1. [1]

    Becattini and M.A

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

    arXiv 2020

  2. [2]

    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(2022) 122301 [2204.01144]

    arXiv 2022

  3. [3]

    Huovinen and H

    P. Huovinen and H. Petersen,Particlization in hybrid models,Eur. Phys. J. A48(2012) 171 [1206.3371]

    Pith/arXiv arXiv 2012

  4. [4]

    Cooper and G

    F. Cooper and G. Frye,Comment on the Single Particle Distribution in the Hydrodynamic and Statistical Thermodynamic Models of Multiparticle Production,Phys. Rev. D10(1974) 186

    1974

  5. [5]

    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(2013) 32 [1303.3431]

    Pith/arXiv arXiv 2013

  6. [6]

    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(2022) 272011 [2109.15301]

    arXiv 2022

  7. [7]

    Sheng, F

    X.-L. Sheng, F. Becattini and D. Roselli,An improved formula for Wigner function and spin polarization in a decoupling relativistic fluid at local thermodynamic equilibrium,2509.14301

    arXiv

  8. [8]

    Huang,An introduction to relativistic spin hydrodynamics,Nucl

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

    arXiv 2025

  9. [9]

    Hehl,On the Energy Tensor of Spinning Massive Matter in Classical Field Theory and General Relativity,Rept

    F.W. Hehl,On the Energy Tensor of Spinning Massive Matter in Classical Field Theory and General Relativity,Rept. Math. Phys.9(1976) 55

    1976

  10. [10]

    Becattini and L

    F. Becattini and L. Tinti,Thermodynamical inequivalence of quantum stress-energy and spin tensors,Phys. Rev. D84(2011) 025013 [1101.5251]

    Pith/arXiv arXiv 2011

  11. [11]

    Becattini and L

    F. Becattini and L. Tinti,Nonequilibrium Thermodynamical Inequivalence of Quantum Stress-energy and Spin Tensors,Phys. Rev. D87(2013) 025029 [1209.6212]

    Pith/arXiv arXiv 2013

  12. [12]

    Becattini, W

    F. Becattini, W. Florkowski and E. Speranza,Spin tensor and its role in non-equilibrium thermodynamics,Phys. Lett. B789(2019) 419 [1807.10994]

    Pith/arXiv arXiv 2019

  13. [13]

    Speranza and N

    E. Speranza and N. Weickgenannt,Spin tensor and pseudo-gauges: from nuclear collisions to gravitational physics,Eur. Phys. J. A57(2021) 155 [2007.00138]. – 36 –

    arXiv 2021

  14. [14]

    Drogosz, W

    Z. Drogosz, W. Florkowski, M. Hontarenko and R. Ryblewski,Dynamical constraints on pseudo-gauge transformations,Phys. Lett. B861(2025) 139244 [2411.06249]

    arXiv 2025

  15. [15]

    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(2022) 044907 [2109.12084]

    arXiv 2022

  16. [16]

    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(2023) 042304 [2205.15689]

    arXiv 2023

  17. [17]

    Banerjee, J

    N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla and T. Sharma, Constraints on Fluid Dynamics from Equilibrium Partition Functions,JHEP09(2012) 046 [1203.3544]

    Pith/arXiv arXiv 2012

  18. [18]

    Jensen, M

    K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz and A. Yarom,Towards hydrodynamics without an entropy current,Phys. Rev. Lett.109(2012) 101601 [1203.3556]

    Pith/arXiv arXiv 2012

  19. [19]

    Gallegos, U

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

    arXiv 2021

  20. [20]

    Gallegos, U

    A.D. Gallegos, U. Gursoy and A. Yarom,Hydrodynamics, spin currents and torsion,JHEP 05(2023) 139 [2203.05044]

    arXiv 2023

  21. [21]

    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(2021) 150 [2107.14231]

    arXiv 2021

  22. [22]

    Florkowski, B

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

    Pith/arXiv arXiv 2018

  23. [23]

    Florkowski, A

    W. Florkowski, A. Kumar and R. Ryblewski,Relativistic hydrodynamics for spin-polarized fluids,Prog. Part. Nucl. Phys.108(2019) 103709 [1811.04409]

    arXiv 2019

  24. [24]

    Jaynes,Information Theory and Statistical Mechanics,Phys

    E.T. Jaynes,Information Theory and Statistical Mechanics,Phys. Rev.106(1957) 620

    1957

  25. [25]

    Jaynes,Information Theory and Statistical Mechanics

    E.T. Jaynes,Information Theory and Statistical Mechanics. II,Phys. Rev.108(1957) 171

    1957

  26. [26]

    Zubarev, A.V

    D.N. Zubarev, A.V. Prozorkevich and S.A. Smolyanskii,Derivation of nonlinear generalized equations of quantum relativistic hydrodynamics,Theor. Math. Phys.40(1979) 821

    1979

  27. [27]

    Becattini, L

    F. Becattini, L. Bucciantini, E. Grossi and L. Tinti,Local thermodynamical equilibrium and the beta frame for a quantum relativistic fluid,Eur. Phys. J. C75(2015) 191 [1403.6265]

    Pith/arXiv arXiv 2015

  28. [28]

    Amari and H

    S.-i. Amari and H. Nagaoka,Methods of information geometry, vol. 191, American Mathematical Soc. (2000)

    2000

  29. [29]

    Kubo,Statistical-Mechanical Theory of Irreversible Processes

    R. Kubo,Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems,J. Phys. Soc. Jap.12(1957) 570

    1957

  30. [30]

    Petz and G

    D. Petz and G. Toth,The Bogoliubov inner product in quantum statistics,Lett. Math. Phys. 27(1993) 205

    1993

  31. [31]

    R. Kubo, M. Toda and N. Hashitsume,Statistical Physics II: Nonequilibrium Statistical Mechanics, vol. 31 ofSpringer Series in Solid-State Sciences, Springer, Berlin, Heidelberg, 2 ed. (1991), 10.1007/978-3-642-58244-8

    work page doi:10.1007/978-3-642-58244-8 1991

  32. [32]

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

    arXiv 2020

  33. [33]

    Schilling, P

    K. Schilling, P. Seyboth and G.E. Wolf,On the Analysis of Vector-Meson Production by Polarized Photons,Nucl. Phys. B15(1970) 397. – 37 –

    1970

  34. [34]

    Petz,Quantum Information Theory and Quantum Statistics, Theoretical and Mathematical Physics, Springer, Berlin, Heidelberg (2008), 10.1007/978-3-540-74636-2

    D. Petz,Quantum Information Theory and Quantum Statistics, Theoretical and Mathematical Physics, Springer, Berlin, Heidelberg (2008), 10.1007/978-3-540-74636-2

    work page doi:10.1007/978-3-540-74636-2 2008

  35. [35]

    Belinfante,On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields,Physica7(1940) 449

    F.J. Belinfante,On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields,Physica7(1940) 449

    1940

  36. [36]

    Rosenfeld,Sur le tenseur d’impulsion-´ energie,Mem

    L. Rosenfeld,Sur le tenseur d’impulsion-´ energie,Mem. Acad. Roy. Belg. Cl. Sci.18(1940) 1

    1940

  37. [37]

    Becattini and R

    F. Becattini and R. Singh,On the local thermodynamic relations in relativistic spin hydrodynamics,Eur. Phys. J. C85(2025) 1338 [2506.20681]

    Pith/arXiv arXiv 2025

  38. [38]

    Becattini, M

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

    arXiv 2021

  39. [39]

    de Groot, W.A

    S.R. de Groot, W.A. van Leeuwen and C.G. van Weert,Relativistic Kinetic Theory: Principles and Applications, North-Holland, Amsterdam (1980)

    1980

  40. [40]

    Hilgevoord and S.A

    J. Hilgevoord and S.A. Wouthuysen,On the spin angular momentum of the Dirac particle, Nucl. Phys.40(1963) 1

    1963

  41. [41]

    Foldy and S.A

    L.L. Foldy and S.A. Wouthuysen,On the Dirac theory of spin 1/2 particles and its non-relativistic limit,Phys. Rev.78(1950) 29

    1950

  42. [42]

    Yang, R.-H

    Y.-G. Yang, R.-H. Fang, Q. Wang and X.-N. Wang,Quark coalescence model for polarized vector mesons and baryons,Phys. Rev. C97(2018) 034917 [1711.06008]

    Pith/arXiv arXiv 2018

  43. [43]

    Liang and X.-N

    Z.-T. Liang and X.-N. Wang,Spin alignment of vector mesons in non-central A+A collisions,Phys. Lett. B629(2005) 20 [nucl-th/0411101]

    Pith/arXiv arXiv 2005

  44. [44]

    Karpenko and F

    I. Karpenko and F. Becattini,Study ofΛpolarization in relativistic nuclear collisions at√sNN = 7.7–200 GeV,Eur. Phys. J. C77(2017) 213 [1610.04717]

    Pith/arXiv arXiv 2017

  45. [45]

    Liang and X.-N

    Z.-T. Liang and X.-N. Wang,Globally polarized quark-gluon plasma in non-central A+A collisions,Phys. Rev. Lett.94(2005) 102301 [nucl-th/0410079]

    Pith/arXiv arXiv 2005

  46. [46]

    Sheng, L

    X.-L. Sheng, L. Oliva and Q. Wang,What can we learn from the global spin alignment ofϕ mesons in heavy-ion collisions?,Phys. Rev. D101(2020) 096005 [1910.13684]

    arXiv 2020

  47. [47]

    Sheng, L

    X.-L. Sheng, L. Oliva, Z.-T. Liang, Q. Wang and X.-N. Wang,Relativistic spin dynamics for vector mesons,2206.05868

    arXiv

  48. [48]

    Chen, Z.-T

    J. Chen, Z.-T. Liang, Y.-G. Ma and Q. Wang,Global spin alignment of vector mesons and strong force fields in heavy-ion collisions,Sci. Bull.68(2023) 874 [2305.09114]

    arXiv 2023

  49. [49]

    Kumar, B

    A. Kumar, B. Muller and D.-L. Yang,Spin alignment of vector mesons by glasma fields, Phys. Rev. D108(2023) 016020 [2304.04181]

    arXiv 2023

  50. [50]

    X.-L. Xia, H. Li, X.-G. Huang and H.Z. Huang,Local spin alignment of vector mesons in relativistic heavy-ion collisions,Phys. Lett. B817(2021) 136325 [2010.01474]

    arXiv 2021

  51. [51]

    D. Shen, J. Chen and Z.-W. Lin,The effect of hadronic scatterings on the measurement of vector meson spin alignments in heavy-ion collisions,Chin. Phys. C45(2021) 054002 [2102.05266]

    arXiv 2021

  52. [52]

    Becattini, M

    F. Becattini, M. Buzzegoli, G. Inghirami, I. Karpenko and A. Palermo,Local polarization and isothermal local equilibrium in relativistic heavy ion collisions,Phys. Rev. Lett.127 (2021) 272302 [2103.14621]

    arXiv 2021

  53. [53]

    Yang, J.-H

    S.-Z. Yang, J.-H. Gao, Z.-T. Liang, G.Y. Prokhorov, S. Pu, O.V. Teryaev et al.,Weyl anomaly induced transport in hydrodynamics,2604.23849. – 38 –

    Pith/arXiv arXiv

  54. [54]

    Bjorken,Highly Relativistic Nucleus-Nucleus Collisions: The Central Rapidity Region, Phys

    J.D. Bjorken,Highly Relativistic Nucleus-Nucleus Collisions: The Central Rapidity Region, Phys. Rev. D27(1983) 140. – 39 –

    1983