Pith. sign in

REVIEW 3 minor 3 cited by

Perfect spin hydrodynamics is proven nonlinearly causal and stable in all four formulations.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-03 20:31 UTC pith:5MV4H7PQ

load-bearing objection Clean proof that three new spin-hydro formulations are divergence-type and nonlinearly causal/stable; limited only by a stated convergence domain.

arxiv 2511.19295 v1 pith:5MV4H7PQ submitted 2025-11-24 hep-ph nucl-th

Nonlinear causality and stability of perfect spin hydrodynamics and its nonperturbative character

Samapan Bhadury , Zbigniew Drogosz , Wojciech Florkowski , Sudip Kumar Kar , Valeriya Mykhaylova This is my paper
classification hep-ph nucl-th
keywords spin hydrodynamicsdivergence-type theorynonlinear causalitynonlinear stabilitygenerating functionFermi-Dirac statisticsBoltzmann statisticsnonperturbative
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 proves that all four standard formulations of perfect spin hydrodynamics for spin-1/2 particles — classical or quantum spin, Boltzmann or Fermi–Dirac statistics — belong to the class of divergence-type theories, and that their conservation laws are nonlinearly causal and stable. The proof works by constructing a generating function for each formulation and showing that a certain matrix-vector product M^λ = M^λ_AB Z^A Z^B is always a future-directed timelike four-vector, because it equals an integral over phase space of p^λ times a nonnegative weight. That property puts the hydrodynamic equations into symmetric hyperbolic form, which guarantees well-posed initial-value problems and rules out superluminal propagation and exponential growth. Crucially, the positivity holds only when the exact distribution functions are used, not their expansions in the spin-polarization tensor; the paper calls this the nonperturbative character of the approach. A sympathetic reader should care because this makes spin hydrodynamics a reliable computational framework for quark-gluon plasma, where spin polarization has been observed and large initial spin values can cause numerical blow-ups in truncated schemes.

Core claim

The central claim is that perfect spin hydrodynamics for spin-1/2 particles, in all four combinations of spin treatment (classical four-vector s vs. quantum spin projection) and statistics (Boltzmann vs. Fermi–Dirac), satisfies the divergence-type theory requirements and is nonlinearly causal and stable. For each case, the authors define a scalar generating function χ(ξ, β, ω) from which the particle current N^λ and the conserved currents (baryon current, energy-momentum tensor, spin tensor) follow by differentiation. The key object is M^λ_AB = -∂³χ/(∂ζ_B ∂ζ_A ∂β_λ), symmetric in A,B; causality and stability follow if for any nonvanishing real Z^A the vector M^λ = M^λ_AB Z^A Z^B is future-di

What carries the argument

The generating function χ and the derived symmetric matrix M^λ_AB = -∂³χ/(∂ζ_B ∂ζ_A ∂β_λ). The four-vector M^λ = M^λ_AB Z^A Z^B, built from arbitrary real Z^A, is shown to equal ∫ dP dS p^λ w with w ≥ 0, which makes the conservation laws (11) symmetric hyperbolic. The positivity w ≥ 0 is what carries the argument; it is established separately for each of the four cases using the exact distribution functions.

Load-bearing premise

The proof only works when the integrals defining the generating functions converge, which restricts the admissible strength of the spin-polarization tensor relative to mass and temperature; outside that domain the exact distribution functions are not integrable and the positivity argument for M^λ has nothing to rely on.

What would settle it

Evaluate the generating function (26) for a Fermi–Dirac spin gas with a spin-polarization tensor exceeding the convergence bound cited in the paper (roughly |ω| bounded by m/T and flow); the integral diverges, so M^λ is undefined and the symmetric-hyperbolic form cannot be established, showing the theorem's domain boundary. Within the domain, a direct numerical check of Eq. (43) for random Z^A, momenta, and parameter values — verifying w ≥ 0 — would confirm or refute the positivity claim.

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

If this is right

  • All four formulations of perfect spin hydrodynamics are divergence-type theories, so the standard machinery of symmetric hyperbolicity applies to each.
  • The hydrodynamic equations are nonlinearly causal and stable, guaranteeing well-posed initial-value problems for numerical simulations.
  • Truncated expansions of the distribution functions in the spin-polarization tensor can break the positivity of w, explaining numerical instabilities observed for large spin polarization; exact distributions are required for the theorem to hold.
  • The existence of generating functions χ in all four cases means a full thermodynamic/Hamiltonian structure underlies spin hydrodynamics.
  • The framework provides a consistent starting point for simulating spin dynamics in quark-gluon plasma produced in relativistic heavy-ion collisions.

Where Pith is reading between the lines

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

  • The paper's distinction between exact and expanded distribution functions suggests a practical diagnostic: simulations that linearize in ω may become acausal even while the exact theory remains causal; comparing the two could locate the safe range of initial spin polarization.
  • The convergence bounds on the generating-function integrals set a physical maximum for the spin-polarization tensor (relative to m/T), beyond which a hydrodynamic description built on these kinetic definitions does not exist; this could serve as a criterion for whether a spin-hydrodynamics initial state is valid.
  • The positivity argument likely extends to other kinetic-theory-based hydrodynamic formulations (e.g., magnetized fluids or higher-spin particles) as long as exact distribution functions are kept, since the structure relies only on f(1-f) ≥ 0 and the spin-projection identities.
  • An immediate testable consequence is that numerical codes using exact Fermi–Dirac distributions for spin should remain stable for initial polarizations up to the convergence bound, whereas codes using linearized distributions should diverge in the same regime.

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

0 major / 3 minor

Summary. The paper analyzes four formulations of perfect spin hydrodynamics for spin-1/2 particles, distinguished by classical vs. quantum treatment of spin and Boltzmann vs. Fermi–Dirac statistics. For each formulation it constructs a scalar generating function from which the conserved currents follow, casts the conservation laws in divergence-type form with a symmetric matrix M^λ_AB, and proves nonlinear causality and stability by showing that M^λ = M^λ_AB Z^A Z^B is a positive-weighted integral of the future-timelike on-shell four-momentum p^λ, hence itself future-timelike. The argument uses exact distribution functions and is therefore nonperturbative in the spin-polarization tensor; convergence restrictions on the generating-function integrals are acknowledged in Sec. IV.

Significance. If the result holds, it is a significant unification: it extends the earlier divergence-type causality/stability proof of Abboud et al. (Ref. [36]) from the classical-spin Boltzmann case to the Fermi–Dirac and quantum-spin formulations actually used in heavy-ion spin-hydrodynamics simulations. The proof is clean, parameter-free, and has no circular fitting: the positivity of the weights follows from f(1-f)>0 for Fermi–Dirac and from explicit sums for the quantum-spin cases. The nonperturbative character—using exact distributions rather than expansions in ω—is conceptually important and explains why truncated expansions can fail. The main limitation is that the theorem is conditional on the convergence domain of the integral representations (16), (26), (33), (44), which the authors explicitly state.

minor comments (3)
  1. [Sec. III.C and III.D, Eqs. (42) and (50)] The claim that w(x,p)≥0 in Eq. (43) is not immediate from the displayed expressions. Equations (42) and (50) contain an explicit factor j outside the square bracket, so the nonnegativity of the bracket (proved via α²−(n·α)²) alone does not imply nonnegativity of the total contribution. The missing step is the summation over j: for Boltzmann statistics, Σ_j j g^{ij}_eq is proportional to e^s−e^{-s} (positive for s>0), and for Fermi–Dirac statistics it is f(y_+)−f(y_-) (also positive). The conclusion is correct, but this step should be written out explicitly.
  2. [Sec. IV / Theorem statements in Sec. III] The causality/stability result is conditional on convergence of the generating-function integrals (16), (26), (33), (44). The authors do state this in Sec. IV, but the main claims in Secs. III.A–III.D are phrased as unconditional. The hypotheses should be included in the theorem statements (or a prominent remark should state the domain restriction) so that the conditional nature of the result is clear at the point of use.
  3. [Sec. II, Eqs. (4), (6), (9), (21)] The symbol N^λ is used both for the conserved particle current (e.g., Eq. (21)) and for the thermodynamic potential current in Eqs. (4), (6), and (9). These are different objects, and the notation is confusing. Consider using a different symbol or explicitly disambiguating them in the text.

Circularity Check

0 steps flagged

No significant circularity: the causal/stability proof is a direct parameter-free consequence of the stated generating functions; the only substantive caveat is integral convergence, which the paper itself flags.

full rationale

The paper's central claim is that, for each of the four formulations, M^λ = M^λ_AB Z^A Z^B is future-directed and timelike, placing the conservation laws in symmetric hyperbolic form. This is established by direct calculation, not by assuming the conclusion. For classical spin/Boltzmann, Eq. (22) is a sum of nonnegative square terms times positive Boltzmann factors, so M^λ is a positive-weighted integral of timelike p^λ. For classical spin/Fermi-Dirac, Eq. (32) contains f^±(1-f^±)>0 factors, again giving the required positivity. For quantum spin in both statistics, the nontrivial combination of Eqs. (40)+(42) and (49)+(50) is shown to be nonnegative via α²-(n·α)² and the sum over j; the paper states w≥0 in Eq. (43). These are genuine algebraic consequences of the chosen generating functions, not restatements of the desired causality property. No parameter is fitted to target data and no 'prediction' is statistically forced. The use of earlier same-group papers [45], [46] is to adopt generating functions and convergence ranges, not to import the causality conclusion; the causality check is performed explicitly here. The paper also explicitly limits its theorem to the domain where the generating-function integrals (16), (26), (33), (44) converge, and notes this was studied in [49] and [46]. That is an honest conditional restriction, not a circular step. Thus the only mild caveat is reliance on same-group constructions for the generating-function forms, which is not load-bearing for the mathematical derivation itself. Score 2 reflects that minor non-independent input; no actual circular reduction was found.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central claim rests on no fitted numbers; it depends on the chosen kinetic definitions of equilibrium distributions, the divergence-type equivalence theorem from [36–41], and a convergence bound on the spin polarization taken from prior work by the same group.

axioms (6)
  • domain assumption Equilibrium distribution functions have the exact Boltzmann/Fermi–Dirac forms given in Eqs. (17), (23), (37), (47).
    The positivity used to prove M^λ timelike is a property of these exact forms; linearized/truncated versions fail (Sec. IV).
  • standard math The divergence-type criterion: future-directed timelike M^λ implies symmetric hyperbolicity, nonlinear causality and stability (Geroch-Lindblom [37,41], Gavassino [40], Abboud [36]).
    Invoked in Sec. II around Eqs. (11)–(13); not re-derived in this paper.
  • domain assumption Classical spin phase space uses the Mathisson spin vector with measure dS = (m/(πℓ)) d⁴s δ(s²+ℓ²) δ(p·s), normalized to spin degeneracy 2.
    Eqs. (18)–(19); defines the classical-spin integration used in Secs. III.A and III.B.
  • domain assumption Quantum spin is encoded through the axial-vector a^μ constructed from ω and p (Eq. 34), with generating functions cosh√(−a²) for Boltzmann and F(y^{ij}) for Fermi–Dirac.
    This is the quantum spin-1/2 model taken from the authors' earlier work [45,46]; it is an input to the proof, not proved here.
  • domain assumption Generating-function integrals converge only for bounded spin polarization, roughly |ω| ≲ m/T, as stated in Sec. IV and proved in [49]/[46].
    If this fails, the exact distribution integrals diverge and the M^λ positivity proof has no finite objects to apply to.
  • standard math Standard special-function identities: d ln(1−f)/dy = f, dilogarithm representation, and dual tensor identities.
    Used in Eqs. (27)–(31) and (35).

pith-pipeline@v1.3.0-alltime-deepseek · 8264 in / 18886 out tokens · 172836 ms · 2026-08-03T20:31:27.494919+00:00 · methodology

0 comments
read the original abstract

Four formulations of perfect spin hydrodynamics for spin-1/2 particles, distinguished by their treatment of spin (classical vs. quantum) and by the underlying particle statistics (Boltzmann vs. Fermi-Dirac), are analyzed and shown to satisfy the requirements of a divergence-type theory. Moreover, for all the formulations, we define the generating functions associated with the relevant thermodynamic currents and demonstrate that the constructed hydrodynamic theory is nonlinearly causal and stable. The latter is achieved by employing the exact expressions for the distribution functions, indicating a nonperturbative character of our approach.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 3 Pith papers

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

  1. Modeling $\Lambda$ polarization in Au$+$Au collisions at $\sqrt{s_{\rm NN}}=200$ GeV using relativistic spin hydrodynamics

    nucl-th 2026-05 unverdicted novelty 7.0

    A (1+1+2)D relativistic spin hydrodynamics model with transverse expansion and longitudinal spin acceleration reproduces the observed quadrupole pattern in longitudinal Lambda polarization for Au+Au collisions at 200 GeV.

  2. Modeling $\Lambda$ polarization in Au$+$Au collisions at $\sqrt{s_{\rm NN}}=200$ GeV using relativistic spin hydrodynamics

    nucl-th 2026-05 unverdicted novelty 6.0

    A (1+1+2)D spin hydrodynamics model with longitudinal spin acceleration and transverse expansion reproduces the quadrupole pattern in longitudinal Lambda polarization and matches Au+Au data at 200 GeV while predicting...

  3. Boost-invariant perfect Fermi-Dirac spin hydrodynamics

    hep-ph 2026-04 unverdicted novelty 4.0

    Fermi-Dirac statistics in boost-invariant perfect spin hydrodynamics produce evolution differences about one order of magnitude smaller than spin-feedback corrections, with special functions conveniently parametrized.

Reference graph

Works this paper leans on

49 extracted references · 34 linked inside Pith · cited by 2 Pith papers

  1. [1]

    Adamczyket al.(STAR), Nature548, 62 (2017), arXiv:1701.06657 [nucl-ex]

    L. Adamczyket al.(STAR), Nature548, 62 (2017), arXiv:1701.06657 [nucl-ex]

    Pith/arXiv arXiv 2017

  2. [2]

    Adamet al.(STAR), Phys

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

    Pith/arXiv arXiv 2018

  3. [3]

    Adamet al.(STAR), Phys

    J. Adamet al.(STAR), Phys. Rev. Lett.123, 132301 (2019), arXiv:1905.11917 [nucl-ex]

    arXiv 2019

  4. [4]

    Acharyaet al.(ALICE), Phys

    S. Acharyaet al.(ALICE), Phys. Rev. Lett.125, 012301 (2020), arXiv:1910.14408 [nucl-ex]

    Pith/arXiv arXiv 2020

  5. [5]

    B. E. Aboonaet al.(STAR), Phys. Lett. B870, 139891 (2025), arXiv:2505.05046 [nucl-ex]

    Pith/arXiv arXiv 2025

  6. [6]

    M. S. Abdallahet al.(STAR), Nature614, 244 (2023), arXiv:2204.02302 [hep-ph]

    arXiv 2023

  7. [7]

    Li (STAR), EPJ Web Conf.316, 05005 (2025)

    D. Li (STAR), EPJ Web Conf.316, 05005 (2025)

    2025

  8. [8]

    STAR, (2025), arXiv:2509.17487 [hep-ex]

    Pith/arXiv arXiv 2025

  9. [9]

    Shen and Q

    D. Shen and Q. Yang (STAR), PoSSPIN2023, 236 (2024)

    2024

  10. [10]

    Becattini, M

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

    Pith/arXiv arXiv 2021

  11. [11]

    Palermo, E

    A. Palermo, E. Grossi, I. Karpenko, and F. Becattini, Eur. Phys. J. C84, 920 (2024), arXiv:2404.14295 [nucl-th]

    Pith/arXiv arXiv 2024

  12. [12]

    Sheng, F

    X.-L. Sheng, F. Becattini, and D. Roselli, (2025), arXiv:2509.14301 [nucl-th]

    Pith/arXiv arXiv 2025

  13. [13]

    Becattini and R

    F. Becattini and R. Singh, Eur. Phys. J. C85, 1338 (2025), arXiv:2506.20681 [nucl-th]

    Pith/arXiv arXiv 2025

  14. [14]

    Florkowski, B

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

    Pith/arXiv arXiv 2018

  15. [15]

    Florkowski, B

    W. Florkowski, B. Friman, A. Jaiswal, R. Ryblewski, and E. Speranza, Phys. Rev. D97, 116017 (2018), arXiv:1712.07676 [nucl-th]

    Pith/arXiv arXiv 2018

  16. [16]

    Florkowski, A

    W. Florkowski, A. Kumar, and R. Ryblewski, Phys. Rev. C98, 044906 (2018), arXiv:1806.02616 [hep-ph]

    Pith/arXiv arXiv 2018

  17. [17]

    Florkowski, A

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

    Pith/arXiv arXiv 2019

  18. [18]

    Bhadury, W

    S. Bhadury, W. Florkowski, A. Jaiswal, A. Kumar, and R. Ryblewski, Phys. Lett. B814, 136096 (2021), arXiv:2002.03937 [hep-ph]

    Pith/arXiv arXiv 2021

  19. [19]

    Bhadury, W

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

    Pith/arXiv arXiv 2022

  20. [20]

    Weickgenannt, X.-L

    N. Weickgenannt, X.-L. Sheng, E. Speranza, Q. Wang, and D. H. Rischke, Phys. Rev. D100, 056018 (2019), arXiv:1902.06513 [hep-ph]. 12

    Pith/arXiv arXiv 2019

  21. [21]

    Weickgenannt, E

    N. Weickgenannt, E. Speranza, X.-l. Sheng, Q. Wang, and D. H. Rischke, Phys. Rev. D104, 016022 (2021), arXiv:2103.04896 [nucl-th]

    Pith/arXiv arXiv 2021

  22. [22]

    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]

    Pith/arXiv arXiv 2021

  23. [23]

    Weickgenannt, D

    N. Weickgenannt, D. Wagner, E. Speranza, and D. H. Rischke, Phys. Rev. D106, 096014 (2022), arXiv:2203.04766 [nucl-th]

    Pith/arXiv arXiv 2022

  24. [24]

    Weickgenannt and J.-P

    N. Weickgenannt and J.-P. Blaizot, Phys. Rev. D109, 056012 (2024), arXiv:2311.15817 [hep-ph]

    Pith/arXiv arXiv 2024

  25. [25]

    Wagner, M

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

    Pith/arXiv arXiv 2024

  26. [26]

    Hu, Phys

    J. Hu, Phys. Rev. D105, 076009 (2022), arXiv:2111.03571 [hep-ph]

    Pith/arXiv arXiv 2022

  27. [27]

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

    Pith/arXiv arXiv 2021

  28. [28]

    S. Shi, C. Gale, and S. Jeon, Phys. Rev. C103, 044906 (2021), arXiv:2008.08618 [nucl-th]

    Pith/arXiv arXiv 2021

  29. [29]

    Banerjee, S

    S. Banerjee, S. Bhadury, W. Florkowski, A. Jaiswal, and R. Ryblewski, Phys. Rev. C111, 064912 (2025), arXiv:2405.05089 [hep-ph]

    Pith/arXiv arXiv 2025

  30. [30]

    Bhadury, Phys

    S. Bhadury, Phys. Rev. C111, 034909 (2025), arXiv:2408.14462 [hep-ph]

    Pith/arXiv arXiv 2025

  31. [31]

    Romatschke and U

    P. Romatschke and U. Romatschke, Phys. Rev. Lett.99, 172301 (2007), arXiv:0706.1522 [nucl-th]

    Pith/arXiv arXiv 2007

  32. [32]

    Romatschke, Int

    P. Romatschke, Int. J. Mod. Phys. E19, 1 (2010), arXiv:0902.3663 [hep-ph]

    Pith/arXiv arXiv 2010

  33. [33]

    Jaiswal and V

    A. Jaiswal and V. Roy, Adv. High Energy Phys.2016, 9623034 (2016), arXiv:1605.08694 [nucl-th]

    Pith/arXiv arXiv 2016

  34. [34]

    Florkowski, M

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

    Pith/arXiv arXiv 2018

  35. [35]

    Bhadury, Z

    S. Bhadury, Z. Drogosz, W. Florkowski, and V. Mykhaylova, J. Subatomic Part. Cosmol.4, 100218 (2025), arXiv:2507.01864 [hep-ph]

    arXiv 2025

  36. [36]

    Abboud, L

    N. Abboud, L. Gavassino, R. Singh, and E. Speranza, Phys. Rev. D112, 094043 (2025), arXiv:2506.19786 [nucl-th]

    arXiv 2025

  37. [37]

    R. P. Geroch and L. Lindblom, Phys. Rev. D41, 1855 (1990)

    1990

  38. [38]

    Peralta-Ramos and E

    J. Peralta-Ramos and E. Calzetta, Phys. Rev. D80, 126002 (2009), arXiv:0908.2646 [hep-ph]

    Pith/arXiv arXiv 2009

  39. [39]

    Peralta-Ramos and E

    J. Peralta-Ramos and E. Calzetta, Phys. Rev. C82, 054905 (2010), arXiv:1003.1091 [hep-ph]

    Pith/arXiv arXiv 2010

  40. [40]

    Gavassino, M

    L. Gavassino, M. Antonelli, and B. Haskell, Phys. Rev. D106, 056010 (2022), arXiv:2207.14778 [gr-qc]

    Pith/arXiv arXiv 2022

  41. [41]

    R. P. Geroch and L. Lindblom, Annals of Physics207, 394 (1991)

    1991

  42. [42]

    Mathisson, Acta Phys

    M. Mathisson, Acta Phys. Polon.6, 163 (1937)

    1937

  43. [43]

    Mathisson, General Relativity and Gravitation42, 1011 (2010)

    M. Mathisson, General Relativity and Gravitation42, 1011 (2010)

    2010

  44. [44]

    Drogosz, W

    Z. Drogosz, W. Florkowski, and M. Hontarenko, Phys. Rev. D110, 096018 (2024), arXiv:2408.03106 [hep-ph]

    Pith/arXiv arXiv 2024

  45. [45]

    Bhadury, Z

    S. Bhadury, Z. Drogosz, W. Florkowski, S. K. Kar, and V. Mykhaylova, (2025), arXiv:2505.02657 [hep-ph]

    arXiv 2025

  46. [46]

    S. K. Kar and V. Mykhaylova, (2025), arXiv:2511.09580 [quant-ph]

    arXiv 2025

  47. [47]

    Drogosz, (2025), arXiv:2509.06014 [hep-ph]

    Z. Drogosz, (2025), arXiv:2509.06014 [hep-ph]. 13

    arXiv 2025

  48. [48]

    Drogosz, W

    Z. Drogosz, W. Florkowski, N. Lygan, and R. Ryblewski, Phys. Rev. C111, 024909 (2025), arXiv:2411.06154 [hep-ph]

    arXiv 2025

  49. [49]

    Drogosz, W

    Z. Drogosz, W. Florkowski, and V. Mykhaylova, Phys. Rev. D112, L051901 (2025), arXiv:2506.01537 [hep-ph]

    arXiv 2025