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arxiv: 2507.20302 · v2 · submitted 2025-07-27 · ✦ hep-ph · nucl-th

Analytic structure of stress-energy response functions and new Kubo formulae

Sangyong Jeon , Alina Czajka , Juhee Hong This is my paper

Pith reviewed 2026-05-19 02:23 UTC · model grok-4.3

classification ✦ hep-ph nucl-th
keywords stress-energy tensorKubo formulaeviscosityrelativistic hydrodynamicscorrelation functionstransport coefficientsanalytic structurequark-gluon plasma
0
0 comments X p. Extension

The pith

Energy conservation fixes the analytic structure of stress-energy correlation functions at low frequency and wavenumber, yielding new Kubo formulae.

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

The paper determines the low-frequency and low-wavenumber analytic structures of all stress-energy correlation functions using energy-conservation laws and gravity-hydrodynamics analysis in the medium rest frame. This structure leads to new Kubo formulae when the zero-frequency limit is taken first. Comparing the results with diffusion and sound spectra from second- and third-order relativistic hydrodynamics shows that the interpretation of Kubo formulae for relaxation times changes with added higher-order terms. The work also resolves subtleties in taking simultaneous zero-frequency and zero-wavenumber limits inside skeleton diagrams.

Core claim

Using the energy-conservation laws and the results from the gravity-hydrodynamics analysis, the low-frequency and low-wavenumber analytic structures of all stress-energy correlation functions in the rest frame of the medium are determined. Various new Kubo formulae are derived in the limit where the zero-frequency limit is taken first. The meaning of the Kubo formulae for relaxation times can change when higher-order terms are added to hydrodynamics, and a subtle issue of taking the zero frequency and zero wavenumber limits when using skeleton diagrams is addressed as well.

What carries the argument

Energy-conservation laws combined with gravity-hydrodynamics analysis that fix the pole positions, residues, and branch cuts of the stress-energy correlation functions.

If this is right

  • New Kubo formulae become available for shear viscosity, bulk viscosity, and other transport coefficients.
  • The derived analytic structures are consistent with the diffusion and sound-mode spectra of both second-order and third-order relativistic hydrodynamics.
  • Kubo formulae that extract relaxation times acquire different interpretations once higher-order hydrodynamic terms are included.
  • The order of limits (zero frequency before zero wavenumber) must be respected when evaluating correlation functions via skeleton diagrams.

Where Pith is reading between the lines

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

  • These fixed structures could reduce the computational cost of lattice evaluations of transport coefficients by supplying exact relations that must hold at small momenta.
  • Analogous conservation-law arguments might be applied to other conserved currents such as baryon number or electric charge in relativistic fluids.
  • The results suggest that hydrodynamic simulations of heavy-ion collisions could incorporate more precise constraints on the form of the stress-energy response at long wavelengths.

Load-bearing premise

The gravity-hydrodynamics results and the hydrodynamic spectra from second- and third-order theories accurately capture the analytic structure of the stress-energy correlators in the low-frequency, low-wavenumber regime of the medium rest frame.

What would settle it

A direct computation of any stress-energy correlation function at small nonzero frequency and wavenumber whose expansion deviates from the predicted poles or residues fixed by energy conservation would falsify the structures and the resulting new Kubo formulae.

read the original abstract

Determining the transport properties of Quark-Gluon Plasma is one of the most important aspects of relativistic heavy ion collision studies. Field-theoretical calculations of the transport coefficients such as the shear and bulk viscosities require Kubo formulae which in turn require real-time correlation functions of stress-energy tensors. Consequently, knowing the analytic structure of these correlation functions is essential in any such studies. Using the energy-conservation laws and the results from the gravity-hydrodynamics analysis, we determine the low-frequency and low-wavenumber analytic structures of all stress-energy correlation functions in the rest frame of the medium. By comparing with the diffusion and sound spectra from the second-order and the third-order relativistic hydrodynamics, various new Kubo formulae are derived in the limit where the zero-frequency limit is taken first. We also show that the meaning of the Kubo formulae for relaxation times can change when higher-order terms are added to hydrodynamics. A subtle issue of taking the zero frequency and zero wavenumber limits when using skeleton diagrams is addressed as well.

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

2 major / 2 minor

Summary. The manuscript claims that energy conservation laws combined with analytic structures obtained from gravity-hydrodynamics determine the low-frequency and low-wavenumber poles and residues of all stress-energy tensor correlation functions in the medium rest frame. Comparing these structures to the diffusion and sound-mode spectra of second- and third-order relativistic hydrodynamics produces new Kubo formulae when the zero-frequency limit is taken first; the paper also shows that the interpretation of relaxation-time Kubo formulae changes upon inclusion of higher-order hydrodynamic terms and addresses subtleties in taking simultaneous zero-frequency and zero-wavenumber limits within skeleton diagrams.

Significance. If the hydrodynamic poles exhaust all singularities that remain finite as ω→0 at fixed k=0, the derived Kubo formulae would supply practical new relations for extracting shear and bulk viscosities as well as relaxation times from real-time correlators, directly relevant to QGP transport studies. The construction is parameter-free and starts from conservation laws plus gravity-hydrodynamics matching rather than fitting, which is a methodological strength. The explicit demonstration that relaxation-time formulae are sensitive to hydrodynamic truncation order provides a useful cautionary result.

major comments (2)
  1. [Abstract and the comparison with hydrodynamic spectra (around the derivation of the new Kubo formulae)] The central derivation assumes that the poles and residues extracted from second- and third-order hydrodynamic spectra (obtained via gravity) capture every singularity that can contribute to the ω→0 limit at fixed k=0. Non-hydrodynamic modes or branch cuts that remain finite in this ordering would alter the residues and therefore the resulting Kubo formulae; the manuscript does not provide an explicit argument or bound showing that such contributions are absent or vanish in the relevant limit.
  2. [Discussion of higher-order hydrodynamics and relaxation-time formulae] The observation that the physical meaning of the relaxation-time Kubo formulae changes when higher-order terms are added to hydrodynamics indicates that the claimed analytic structure is truncation-dependent. This directly affects the load-bearing claim that the zero-frequency-first limit yields unambiguous new formulae; a concrete test or statement of the order at which the structure stabilizes is needed.
minor comments (2)
  1. The term 'skeleton diagrams' is introduced in the abstract when discussing limit ordering; the main text should define the diagrams explicitly and show how the zero-frequency-first prescription is implemented within them.
  2. Notation for the ordering of limits (ω→0 first versus k→0 first) should be introduced with a single consistent symbol or equation early in the manuscript to avoid ambiguity in later sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major comments point by point below, providing our strongest honest defense of the results while indicating where revisions will be made to improve clarity.

read point-by-point responses

  1. Referee: [Abstract and the comparison with hydrodynamic spectra (around the derivation of the new Kubo formulae)] The central derivation assumes that the poles and residues extracted from second- and third-order hydrodynamic spectra (obtained via gravity) capture every singularity that can contribute to the ω→0 limit at fixed k=0. Non-hydrodynamic modes or branch cuts that remain finite in this ordering would alter the residues and therefore the resulting Kubo formulae; the manuscript does not provide an explicit argument or bound showing that such contributions are absent or vanish in the relevant limit.

    Authors: We thank the referee for this important clarification request. The analytic structures we determine follow directly from the energy-momentum conservation equations together with the hydrodynamic spectra obtained via gravity matching. Non-hydrodynamic modes possess a finite imaginary part (gap) that remains nonzero even at k=0; consequently their poles do not reach the origin in the ω→0 limit taken at fixed k=0 and do not contribute to the residues of the hydrodynamic poles. Branch cuts associated with multi-particle continua are likewise assumed to lie at a finite distance from the origin within the hydrodynamic regime, consistent with the effective-theory description underlying our matching. To make this reasoning fully explicit we will insert a short paragraph in the revised manuscript (near the discussion of the hydrodynamic spectra) that recalls the scale separation between gapped non-hydro modes and the hydrodynamic poles, thereby bounding their contribution to zero in the stated limit. revision: yes

  2. Referee: [Discussion of higher-order hydrodynamics and relaxation-time formulae] The observation that the physical meaning of the relaxation-time Kubo formulae changes when higher-order terms are added to hydrodynamics indicates that the claimed analytic structure is truncation-dependent. This directly affects the load-bearing claim that the zero-frequency-first limit yields unambiguous new formulae; a concrete test or statement of the order at which the structure stabilizes is needed.

    Authors: We agree that the physical content of the relaxation-time Kubo formulae is truncation-dependent, and we regard this as a central and useful result of the paper rather than a shortcoming. The manuscript explicitly demonstrates the change between second- and third-order hydrodynamics to illustrate the necessity of consistent truncation. Within any fixed truncation the zero-frequency-first limit produces unambiguous formulae because all transport coefficients up to that order are retained. Stabilization of the functional form occurs once the hydrodynamic expansion includes all coefficients that couple to the stress-tensor correlators at the order under consideration; additional higher-order terms introduce new coefficients but do not retroactively alter the expressions derived at lower orders. We will revise the relevant discussion paragraph to state this explicitly and to note that a model-specific numerical check (for example in a holographic setup) would be a natural extension but lies beyond the present scope. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation uses external conservation laws and hydrodynamics spectra as independent inputs

full rationale

The paper's central derivation begins from energy-conservation laws and the analytic structures obtained from gravity-hydrodynamics analysis (treated as external results), then compares these to diffusion and sound spectra of second- and third-order hydrodynamics to extract new Kubo formulae in the zero-frequency-first limit. No step reduces a claimed prediction or result to a fitted parameter, self-defined quantity, or load-bearing self-citation whose validity is assumed without independent support. The note that relaxation-time interpretations change with higher-order hydrodynamics reflects truncation dependence rather than circular construction. The derivation remains self-contained against external benchmarks and does not rename known results or smuggle ansatze via citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard conservation laws and hydrodynamic mode structures; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Energy-momentum conservation holds for the stress-energy tensor.
    Invoked to constrain the analytic structure of correlation functions.
  • domain assumption Gravity-hydrodynamics correspondence provides the correct low-frequency poles and residues.
    Used as input to determine the structures before matching to hydrodynamics.

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Forward citations

Cited by 1 Pith paper

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  1. Response theory for quantum fields in isolation

    quant-ph 2026-04 unverdicted novelty 2.0

    A review of response theory formalism for isolated quantum fields emphasizing causality, functional techniques, and fluctuation-dissipation relations.

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