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REVIEW 4 major objections 3 minor 29 references

This paper verifies that all eight order-reversed Kubo formulas yield consistent shear and bulk viscosity coefficients when evaluated with the massive Anderson-Witting kinetic-theory response functions.

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2026-08-01 21:11 UTC pith:ZOLBU5HJ

load-bearing objection A useful but not fully shown consistency check of the authors' own order-reversed Kubo formulas in the massive Anderson-Witting model; the claimed verification rests on an unreported symbolic computation and an unresolved bulk-viscosity discrepancy. the 4 major comments →

arxiv 2607.16148 v1 pith:ZOLBU5HJ submitted 2026-07-17 hep-ph

Order-reversed Kubo formulas in relativistic kinetic theory

Sangyong Jeon , Juhee Hong , Alina Czajka This is my paper
classification hep-ph
keywords Kubo formulasshear viscositybulk viscosityAnderson-Witting modelrelaxation time approximationkinetic theorystress-energy response functionsanalytic structure
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.

The paper sets out to prove that a recently derived family of 'order-reversed' Kubo formulas—transport-coefficient extraction rules in which the small-frequency limit is taken before the long-wavelength limit—are self-consistent and correct in a nontrivial setting. Using the retarded stress-energy tensor response functions of the Anderson-Witting relaxation-time model for a massive Boltzmann gas, and expanding in the small ratio ξ = m/T, the authors show that all eight formulas return the same shear viscosity η/h0 = τ_R(1/5 − ξ²/60 + ξ⁴/96) and bulk viscosity ζ/h0 = (5/432)τ_R ξ⁴. For a massless gas the formulas collapse to η/h0 = τ_R/5 and ζ = 0. A sympathetic reader should care because the new formulas exploit subtle analytic properties of correlators and are hard to evaluate; this kinetic-theory check is a necessary intermediate step before applying them in full field theory. The paper also shows the correlators satisfy the small-frequency and small-wavenumber analytic conditions required by the formulas.

Core claim

The central claim is that the five independent retarded response functions G_L, G_LT, G_T, G_1, G_2 constructed from the linearized massive Anderson-Witting equation have the analytic structure required by the new Kubo formulas: they obey the small-ω forms with the correct ω^{-2} leading behavior, the small-k relations such as 2G1 - GL + GLT = O(k^2), and the minimum-form structure with coefficients DR, ZR, ZI, and related parameters. Inserted into the eight order-reversed Kubo formulas, they all produce the same transport coefficients, η/h0 = τ_R(1/5 − ξ²/60 + ξ⁴/96) + O(ξ⁵) and ζ/h0 = (5/432)τ_R ξ⁴ + O(ξ⁵), with speed of sound v_s² = 1/3 − ξ²/36 + 5ξ⁴/864. The massless limit yields the kno

What carries the argument

The load-bearing object is the set of five retarded stress-energy response functions, built from angle and energy integrals Ω of the linearized Anderson-Witting collision term. These correlators contain hydrodynamic poles in the small-frequency, small-wavenumber regime—a property that makes them suitable for the order-reversed limits. The calculation relies on a small-m/T expansion of the angle integrals using modified Bessel and Bickley-Naylor functions, plus symbolic manipulation, to carry out the limits explicitly.

Load-bearing premise

The verification assumes the algebraic correctness of the five massive response functions (Eqs. 27–32) and of the small-m/T expansion of the angle integrals through O(m⁴); if either contains an error—or if the unresolved disagreement with one earlier bulk-viscosity calculation points to one—the claimed consistency of the eight formulas collapses.

What would settle it

Compute the angle integrals in Appendix A order by order without the small-m/T expansion (or to one higher order, O(ξ⁵)) and insert them into the eight Kubo formulas; if the eight results for η/h0 and ζ/h0 no longer coincide, or if they differ from τ_R(1/5 − ξ²/60 + ξ⁴/96) and (5/432)τ_R ξ⁴ at the expected orders, the central claim is falsified.

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

If this is right

  • All eight order-reversed Kubo formulas produce identical η/h0 and ζ/h0 from kinetic-theory correlators, establishing mutual consistency of the new formulas in a nontrivial model.
  • The massive relaxation-time correlators satisfy the analytic small-ω and small-k consistency conditions, so they are a valid arena for extracting transport coefficients via the new formulas.
  • The bulk viscosity from these formulas is of order τ_R (m/T)^4, confirming the expected mass scaling and providing a definite prediction for the massive relaxation-time model.
  • The minimum-form coefficients imply v_s² R_R = 6 τ_R, and the paper speculates that n_R = v_s²/(τ_R R_R) may count the total number of relaxing modes in the response functions.

Where Pith is reading between the lines

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

  • If the analytic-structure consistency is generic, the order-reversed formulas could be applied to correlators that already contain hydrodynamic poles (for example, from resummed ladder diagrams), offering a route to transport coefficients where the standard thermodynamic limit is ill-defined.
  • The disagreement with one prior bulk-viscosity calculation suggests a sharp test: evaluating the massive response functions numerically without the small-m/T expansion, then checking whether all eight formulas still agree at larger ξ.
  • The hinted relation between R_R and the number of relaxing modes, if confirmed, would connect the new Kubo formulas to the number of non-hydrodynamic modes in second-order hydrodynamics, giving a physical interpretation to that count.

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

4 major / 3 minor

Summary. The paper computes stress-energy tensor retarded response functions in the massive Anderson-Witting (relaxation time) kinetic theory, working to small m/T. Five independent response functions are written down in Eqs. (27)-(32), and it is claimed that they have the small-ω and small-k analytic structure required by the authors' earlier order-reversed Kubo formulas [1]. The main new results are the transport coefficients extracted from all eight Kubo formulas: η/h0 = τ_R(1/5 - ξ²/60 + ξ⁴/96)+O(ξ⁵) and ζ/h0 = (5/432)τ_R ξ⁴+O(ξ⁵).

Significance. If the verification is correct, this is a nontrivial consistency check of the order-reversed Kubo formulas in a concrete relativistic kinetic model, and it also determines the small-mass corrections to η/h0 and ζ/h0 in the massive Anderson-Witting model. The explicit expressions for the response functions and the small-ξ transport coefficients are useful. However, the central demonstration is not actually presented in the manuscript: the paper repeatedly states that the identities were verified with symbolic manipulation programs, but provides no code, no worksheets, and very little intermediate algebra. The unexplained disagreement with Ref. [18] for ζ is an additional unresolved point.

major comments (4)
  1. [Sec. V, Eqs. (75)-(76); Appendix A] The central claim is that all eight Kubo formulas reduce to Eqs. (75)-(76) in the small m/T limit. The only verification offered is "with the help of symbolic manipulation programs" (Sec. V, twice). No Mathematica worksheet, no pseudo-code, and no intermediate simplified expressions are provided. Without these, the equalities in Eqs. (75)-(76) are not independently checkable from the manuscript. This is a load-bearing omission: if even one of the eight limits gives a different coefficient, the consistency claim fails. Please include the verification as an ancillary file or as a detailed appendix that shows, for each of Eqs. (39)-(48), how the response functions of Eqs. (27)-(32) together with the Appendix A expansions lead to (75)-(76).
  2. [Sec. V, paragraph after Eq. (76)] The paper states that the bulk viscosity result disagrees with Ref. [18], but does not resolve the discrepancy. The ζ/h0 coefficient in Eq. (76) is central to the consistency check, so a disagreement with an existing kinetic-theory calculation is not a minor remark. The authors should identify whether the difference is due to a different definition of ζ, a different enthalpy normalization, a different ordering of limits, or an error in one of the calculations. Without this the quoted O(ξ⁴) coefficient is not established.
  3. [Sec. IV, Eqs. (64)-(74)] The section 'Minimum Requirements' asserts that the kinetic theory response functions satisfy the minimal forms (64)-(68), but no explicit matching of Eqs. (27)-(32) to Eqs. (64)-(74) is shown. In particular, the identification DR = η/h0, ZR = v_s², ZI = ..., and the values of F_LT R, F_T1R, F_T2R are stated without derivation. This is load-bearing because the reversed-limit Kubo formulas use the exact ω,k dependence of these response functions, not just the leading-order terms in Eqs. (33)-(38). Please display the matching explicitly or at least give the algebra that fixes the minimal-form coefficients.
  4. [Sec. III, paragraph after Eq. (48) and Sec. IV, after Eq. (63)] The abstract and the summary describe the response functions as 'fully consistent' with the Kubo formulas, but the text itself notes that the thermodynamic coefficients κ and κ* cannot be obtained (Sec. III). Moreover, the analytic conditions in Eqs. (33)-(38) are checked only to leading order in ω and k, not as exact statements. Since the new Kubo formulas require taking frequency derivatives before the k→0 limit, subleading terms can in principle contribute. The claim should be sharpened to 'satisfy the required analytic conditions at the order needed for the limits considered' or, alternatively, exact checks should be provided.
minor comments (3)
  1. [Sec. II, after Eq. (16)] There is a typo: 'N i1···in n' should be 'N i1···in' or 'N i1···il' as in the surrounding text.
  2. [Sec. V, Eq. (77)] Eq. (77) is cited as giving h0 and v_s², but the equation is numbered within the same paragraph as (75)-(76). It would be clearer to place these thermodynamic inputs before the transport coefficients, since they are used in the verification.
  3. [Appendix A, Eq. (A3)] The notation A = (1 - iτ_R ω)/(i k_z τ_R) is introduced, but the sign conventions in Eqs. (27)-(32) would be easier to follow if the small-ω expansion of the denominator were written out once, including the leading terms used in the main text.

Circularity Check

0 steps flagged

No significant circularity: the massive Anderson-Witting correlators are derived independently and the recovered transport coefficients are benchmarked against external RTA results.

full rationale

The paper's central step is to derive the stress-energy retarded correlators from the linearized Anderson-Witting kinetic equation (Eq. (1)) and then to evaluate the Kubo formulas of Ref. [1] on those correlators. The correlators in Sec. II (Eqs. (27)-(32)) are obtained by solving the kinetic equation, with the angle/energy integrals of Appendix A; they are not built to satisfy the Kubo formulas. The transport coefficients (75)-(76) are the outputs of the Kubo formulas, not fitted inputs: no parameter is adjusted to make the formulas agree. The paper also verifies the massless limit and compares with the independent massive-RTA results of Refs. [20] and [22], which provides an external benchmark. The 'minimum requirements' of Sec. IV are consistency conditions imported from Ref. [1], but the paper checks the kinetic-theory correlators against them rather than using them to construct the correlators; the coefficients D_R, Z_I, etc. are identified only after the correlation functions are already computed. The admitted limitations (inability to extract kappa and kappa*; unresolved disagreement with Ref. [18] in bulk viscosity; reliance on 'symbolic manipulation programs' without displayed algebra) are correctness/robustness issues, not circular reductions. The self-citation to Ref. [1] is central to the paper's purpose, but it is the hypothesis being tested, not the evidence for itself. No equation in the paper is equivalent to its own input by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The check rests on model assumptions (relaxation-time approximation, Landau matching), on formulas imported from the authors' prior Ref. [1], and on correlation functions from the kinetic theory literature. No new particles, forces, or conserved quantities are introduced. τ_R and ξ are model inputs, not fitted constants.

free parameters (2)
  • Relaxation time τ_R = not fitted (model input)
    Anderson-Witting relaxation-time approximation collision scale; appears as the overall prefactor in η/h0 and ζ/h0. No value is fit; the paper's claim is about the functional form and consistency.
  • Mass-to-temperature ratio ξ = m/T = expansion parameter, not fitted
    Used as the small expansion parameter; results are truncated at O(ξ⁴) or O(ξ⁵). It is an input of the model, not a fitted number.
axioms (5)
  • domain assumption Anderson-Witting relaxation-time approximation with a single relaxation time τ_R (Eq. (1)) correctly describes the response of the gas.
    The entire calculation is performed in this model. The authors themselves note in the Introduction that the full collision integral is not used and the model is not fully equivalent to ladder diagram resummation.
  • domain assumption Landau matching conditions and the first-order relations δT/T0 = v_s² δT00/h0 and u_i = δT0i/h0 (Eqs. (13)-(14)).
    Used to close the linearized kinetic equation and to express the stress-energy correlators in terms of the response functions.
  • ad hoc to paper The eight order-reversed Kubo formulas from Ref. [1] are valid candidate identities (Eqs. (39)-(48)).
    The paper imports these formulas from the authors' own prior work and uses them as the benchmark to be tested; they are not rederived in this paper.
  • domain assumption The massive response functions from Refs. [18,20] and from Sec. II are correct, and the Appendix A m/T expansion through O(m⁴) captures the leading viscosity contributions.
    The final verification depends on these expressions and expansions, but the paper does not show the full symbolic check, and the bulk-viscosity result disagrees with Ref. [18].
  • domain assumption Ladder diagram resummation can be reorganized to recover linearized kinetic theory (Refs. [8,11-15]).
    Used in the Introduction to motivate kinetic theory as an important intermediate step toward full field-theory calculations.

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read the original abstract

Using the stress-energy tensor response functions obtained in the Anderson-Witting model of kinetic theory with non-zero mass, we verify that these response functions satisfy the required analytic conditions and are fully consistent with the recently derived Kubo formulas in Ref.[1].

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Reference graph

Works this paper leans on

29 extracted references · 23 linked inside Pith

  1. [1]

    S. Jeon, A. Czajka, and J. Hong, Analytic structure of stress-energy response functions and new Kubo formulas, Phys. Rev. C112, 064905 (2025), arXiv:2507.20302 [hep- ph]

    Pith/arXiv arXiv 2025

  2. [2]

    Jeon, Hydrodynamic transport coefficients in relativis- tic scalar field theory, Phys

    S. Jeon, Hydrodynamic transport coefficients in relativis- tic scalar field theory, Phys. Rev. D52, 3591 (1995), arXiv:hep-ph/9409250

    Pith/arXiv arXiv 1995

  3. [3]

    Jeon and L

    S. Jeon and L. G. Yaffe, From quantum field theory to hydrodynamics: Transport coefficients and effective ki- netic theory, Phys. Rev. D53, 5799 (1996), arXiv:hep- ph/9512263

    arXiv 1996

  4. [4]

    M. E. Carrington, D.-f. Hou, and R. Kobes, Shear vis- cosity in phi**4 theory from an extended ladder resum- mation, Phys. Rev. D62, 025010 (2000), arXiv:hep- ph/9910344

    arXiv 2000

  5. [5]

    Wang and U

    E. Wang and U. W. Heinz, Nonperturbative calculation of the shear viscosity in hot phi**4 theory in real time, Phys. Lett. B471, 208 (1999), arXiv:hep-ph/9910367. 6

    Pith/arXiv arXiv 1999

  6. [6]

    Wang and U

    E. Wang and U. W. Heinz, Shear viscosity of hot scalar field theory in the real time formalism, Phys. Rev. D67, 025022 (2003), arXiv:hep-th/0201116

    Pith/arXiv arXiv 2003

  7. [7]

    P. B. Arnold, G. D. Moore, and L. G. Yaffe, Transport coefficients in high temperature gauge theories. 1. Lead- ing log results, JHEP11, 001, arXiv:hep-ph/0010177

    Pith/arXiv arXiv

  8. [8]

    P. B. Arnold, G. D. Moore, and L. G. Yaffe, Effective kinetic theory for high temperature gauge theories, JHEP 01, 030, arXiv:hep-ph/0209353

    Pith/arXiv arXiv

  9. [9]

    P. B. Arnold, G. D. Moore, and L. G. Yaffe, Transport coefficients in high temperature gauge theories. 2. Beyond leading log, JHEP05, 051, arXiv:hep-ph/0302165

    Pith/arXiv arXiv

  10. [10]

    P. B. Arnold, C. Dogan, and G. D. Moore, The Bulk Viscosity of High-Temperature QCD, Phys. Rev. D74, 085021 (2006), arXiv:hep-ph/0608012

    Pith/arXiv arXiv 2006

  11. [11]

    Jakovac, Time evolution in linear response: Boltz- mann equations and beyond, Phys

    A. Jakovac, Time evolution in linear response: Boltz- mann equations and beyond, Phys. Rev. D65, 085029 (2002), arXiv:hep-ph/0112188

    Pith/arXiv arXiv 2002

  12. [12]

    Jakovac, Generalized Boltzmann equations for on- shell particle production in a hot plasma, Phys

    A. Jakovac, Generalized Boltzmann equations for on- shell particle production in a hot plasma, Phys. Rev. D 66, 125003 (2002), arXiv:hep-ph/0207085

    Pith/arXiv arXiv 2002

  13. [13]

    P. B. Arnold, G. D. Moore, and L. G. Yaffe, Photon and gluon emission in relativistic plasmas, JHEP06, 030, arXiv:hep-ph/0204343

    Pith/arXiv arXiv

  14. [14]

    Fillion-Gourdeau, J.-S

    F. Fillion-Gourdeau, J.-S. Gagnon, and S. Jeon, All or- ders Boltzmann collision term from the multiple scatter- ing expansion of the self-energy, Phys. Rev. D74, 025010 (2006), arXiv:hep-ph/0603212

    Pith/arXiv arXiv 2006

  15. [15]

    Gagnon and S

    J.-S. Gagnon and S. Jeon, Leading Order Calculation of Shear Viscosity in Hot Quantum Electrodynamics from Diagrammatic Methods, Phys. Rev. D76, 105019 (2007), arXiv:0708.1631 [hep-ph]

    Pith/arXiv arXiv 2007

  16. [16]

    Romatschke, Retarded correlators in kinetic theory: branch cuts, poles and hydrodynamic onset transitions, Eur

    P. Romatschke, Retarded correlators in kinetic theory: branch cuts, poles and hydrodynamic onset transitions, Eur. Phys. J. C76, 352 (2016), arXiv:1512.02641 [hep- th]

    Pith/arXiv arXiv 2016

  17. [17]

    Hong and D

    J. Hong and D. Teaney, Spectral densities for hot QCD plasmas in a leading log approximation, Phys. Rev. C82, 044908 (2010), arXiv:1003.0699 [nucl-th]

    Pith/arXiv arXiv 2010

  18. [18]

    Hataei, R

    A. Hataei, R. Heydari, and F. Taghinavaz, Kinetic The- ory of Quasiparticles, Retarded Correlators and Hydro- dynamics, (2025), arXiv:2504.14591 [nucl-th]

    Pith/arXiv arXiv 2025

  19. [19]

    Bajec, S

    M. Bajec, S. Grozdanov, and A. Soloviev, Spectra of cor- relators in the relaxation time approximation of kinetic theory, JHEP08, 065, arXiv:2403.17769 [hep-th]

    Pith/arXiv arXiv

  20. [20]

    Bajec and A

    M. Bajec and A. Soloviev, Relaxation for massive parti- cles: Transport and causality, Phys. Rev. D112, 065002 (2025), arXiv:2506.15531 [hep-th]

    Pith/arXiv arXiv 2025

  21. [21]

    J. L. Anderson and H. R. Witting, A relativistic relaxation-time model for the Boltzmann equation, Phys- ica74, 466 (1974)

    1974

  22. [22]

    G. S. Denicol, S. Jeon, and C. Gale, Transport Co- efficients of Bulk Viscous Pressure in the 14-moment approximation, Phys. Rev. C90, 024912 (2014), arXiv:1403.0962 [nucl-th]

    Pith/arXiv arXiv 2014

  23. [23]

    T. S. Biro, P. Levai, and B. Muller, Strangeness produc- tion with ’massive’ gluons, Phys. Rev. D42, 3078 (1990)

    1990

  24. [24]

    M. I. Gorenstein and S.-N. Yang, Gluon plasma with a medium dependent dispersion relation, Phys. Rev. D52, 5206 (1995)

    1995

  25. [25]

    Peshier, B

    A. Peshier, B. Kampfer, O. P. Pavlenko, and G. Soff, A Massive quasiparticle model of the SU(3) gluon plasma, Phys. Rev. D54, 2399 (1996)

    1996

  26. [26]

    Chakraborty and J

    P. Chakraborty and J. I. Kapusta, Quasi-Particle Theory of Shear and Bulk Viscosities of Hadronic Matter, Phys. Rev. C83, 014906 (2011), arXiv:1006.0257 [nucl-th]

    Pith/arXiv arXiv 2011

  27. [27]

    Czajka and S

    A. Czajka and S. Jeon, Kubo formulas for the shear and bulk viscosity relaxation times and the scalar field theory shearτ π calculation, Phys. Rev. C95, 064906 (2017), arXiv:1701.07580 [nucl-th]

    Pith/arXiv arXiv 2017

  28. [28]

    Czajka, S

    A. Czajka, S. Hauksson, C. Shen, S. Jeon, and C. Gale, Bulk viscosity of strongly interacting matter in the re- laxation time approximation, Phys. Rev. C97, 044914 (2018), arXiv:1712.05905 [nucl-th]

    Pith/arXiv arXiv 2018

  29. [29]

    G. S. Rocha, M. N. Ferreira, G. S. Denicol, and J. Noronha, Transport coefficients of quasiparticle mod- els within a new relaxation time approximation of the Boltzmann equation, Phys. Rev. D106, 036022 (2022), arXiv:2203.15571 [nucl-th]. Appendix A: Evaluation ofNi1···il in the smallm/T limit N i1···il = Z dΩv 4π ni1 · · ·nim 1−iτ Rω+iτ Rkzvν (A1) wheren...

    Pith/arXiv arXiv 2022