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arxiv: 2605.04020 · v1 · submitted 2026-05-05 · ✦ hep-th · cond-mat.stat-mech· hep-ph· nucl-th

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Late-Time Relaxation from Landau Singularities

Dong-Lin Wang , Shi Pu
Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:57 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mechhep-phnucl-th
keywords late-time relaxationLandau singularitiesnonlinear hydrodynamicsSchwinger-Keldysh effective field theorypower-law decayrelaxation modestwo-point functionsloop integrals
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The pith

Landau singularity analysis extracts nonlinear relaxation modes from effective field theory loop integrals without explicit computation.

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

The paper develops a method using Landau singularity analysis to identify the singularities in loop corrections to two-point functions in Schwinger-Keldysh effective field theories. These singularities determine the late-time relaxation behavior of fluctuations under nonlinear interactions. When gapless modes are involved, the relaxation shifts from exponential to power-law decay. This approach avoids the complexity of performing loop integrations directly, offering a systematic way to understand nonlinear effects in a wide range of macroscopic theories. A sympathetic reader would care because it provides insight into how nonlinear hydrodynamics alters the long-time dynamics of systems like fluids or plasmas.

Core claim

Nonlinear hydrodynamic interactions can change the relaxation of fluctuations from exponential to power-law decay at late times. By applying Landau singularity analysis to two-point functions in effective field theories, the singularities induced by nonlinear interactions are determined without explicit loop integrations. From these frequency-space singularities, nonlinear relaxation modes are extracted that control the late-time behavior, leading to power-law decay when gapless modes are present.

What carries the argument

Landau singularity analysis applied to two-point functions in Schwinger-Keldysh effective field theories to identify singularities from nonlinear interactions.

Load-bearing premise

The assumption that Landau singularity analysis can be directly applied to the loop integrals arising from nonlinear terms in Schwinger-Keldysh EFTs to extract the relevant relaxation modes without explicit integration or additional approximations.

What would settle it

An explicit one-loop computation of a two-point function in a simple nonlinear model with a cubic interaction, followed by checking whether the resulting singularities and late-time exponents match those predicted by the Landau analysis.

Figures

Figures reproduced from arXiv: 2605.04020 by Dong-Lin Wang, Shi Pu.

Figure 1
Figure 1. Figure 1: Illustration of a banana diagram. Wavy and solid view at source ↗
read the original abstract

Nonlinear hydrodynamic interactions can change the relaxation of fluctuations from exponential to power-law decay at late times. Schwinger-Keldysh effective field theory provides a standard framework for describing such fluctuation effects, where the nonlinear late-time behavior is encoded in loop corrections. Extracting this behavior requires identifying the singularities of loop integrals, whose structure becomes increasingly intricate beyond simple models. We apply Landau singularity analysis to two-point functions in effective field theories and determine the singularities induced by nonlinear interactions without performing the loop integrations explicitly. From these frequency-space singularities, we extract nonlinear relaxation modes that control the late-time behavior. When gapless modes are present, these modes produce power-law decay at late times. Our results give a systematic singularity-based description of nonlinear late-time relaxation in a broad class of macroscopic effective theories.

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 paper claims that Landau singularity analysis can be applied directly to the loop integrals generated by nonlinear vertices in Schwinger-Keldysh effective field theories to identify the frequency-space singularities induced by those interactions, without performing explicit integrations. These singularities are then mapped to nonlinear relaxation modes that control late-time behavior, yielding power-law decay (rather than exponential) when gapless modes are present. The approach is presented as a systematic tool for a broad class of macroscopic EFTs.

Significance. If the mapping from Landau conditions to realized singularities and dominant late-time modes holds, the method would provide a computationally efficient, integration-free route to nonlinear relaxation in hydrodynamic and related fluctuation problems. It would strengthen the singularity-based understanding of late-time tails in EFTs and could be extended to higher-point functions or other effective theories.

major comments (2)
  1. [Application to two-point functions (around the discussion of nonlinear interactions)] The central claim that Landau equations suffice to determine the singularities realized in SK loop integrals (without explicit integration or contour checks) is not verified by any benchmark calculation. No section compares the predicted singularities against a fully integrated simple model (e.g., a cubic vertex with known retarded/advanced/Keldysh propagators) to confirm that the pinch actually produces a non-analyticity after integration over the measure and distribution functions.
  2. [Mapping singularities to relaxation modes] The extraction of relaxation modes from the frequency-space singularities assumes that these singularities dominate the late-time asymptotics and are not canceled or subdominant due to the specific structure of SK propagators or higher-order terms. This assumption is load-bearing for the power-law decay claim but lacks an explicit argument or contour-deformation analysis showing why other contributions cannot overtake them at late times.
minor comments (2)
  1. [Preliminaries] Notation for the Keldysh, retarded, and advanced components is introduced without a compact summary table; a single table listing the propagator definitions and their analytic properties would improve readability.
  2. [Introduction] The abstract and introduction refer to 'a broad class of macroscopic effective theories' but the explicit examples are limited to two-point functions; clarifying the scope (e.g., whether the method extends immediately to n-point functions) would help readers assess generality.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the two major comments point by point below, providing clarifications based on the manuscript's content and indicating the revisions we will make.

read point-by-point responses

  1. Referee: [Application to two-point functions (around the discussion of nonlinear interactions)] The central claim that Landau equations suffice to determine the singularities realized in SK loop integrals (without explicit integration or contour checks) is not verified by any benchmark calculation. No section compares the predicted singularities against a fully integrated simple model (e.g., a cubic vertex with known retarded/advanced/Keldysh propagators) to confirm that the pinch actually produces a non-analyticity after integration over the measure and distribution functions.

    Authors: The manuscript applies the standard Landau equations to the analytic structure of SK propagators in two-point loop integrals, relying on the fact that these equations locate the pinch singularities without requiring explicit evaluation of the integrals. While no explicit benchmark comparison with a fully integrated cubic model is presented, the approach follows directly from the established properties of Landau singularities in QFT loop integrals, adapted to the SK contour and distribution functions. To strengthen the presentation, we will add a benchmark example in a revised section or appendix, explicitly comparing Landau-predicted singularities for a simple cubic vertex against the result of direct integration. revision: partial

  2. Referee: [Mapping singularities to relaxation modes] The extraction of relaxation modes from the frequency-space singularities assumes that these singularities dominate the late-time asymptotics and are not canceled or subdominant due to the specific structure of SK propagators or higher-order terms. This assumption is load-bearing for the power-law decay claim but lacks an explicit argument or contour-deformation analysis showing why other contributions cannot overtake them at late times.

    Authors: The mapping follows from the Fourier transform relation between frequency-space singularities and time-domain asymptotics, where the leading non-analyticity determines the dominant late-time power-law decay. In the SK formalism, the combination of retarded, advanced, and Keldysh components, together with the presence of gapless modes, ensures that the nonlinear Landau singularities are not canceled by linear terms or higher-order contributions. We will add an explicit discussion, including a sketch of the relevant contour deformation, to demonstrate why these singularities control the asymptotics. revision: partial

Circularity Check

0 steps flagged

No circularity: Landau analysis applied as external tool to EFT loop structure

full rationale

The derivation applies the standard Landau singularity conditions to the frequency-space structure of loop integrals generated by nonlinear vertices in Schwinger-Keldysh EFTs. Singularities are located from the pinch conditions on the propagators without explicit integration or fitting to the late-time decay; relaxation modes are then read off from those singularities. No step reduces by definition or by self-citation to the target power-law result, no parameters are tuned to match observed decay, and the method is presented as a direct, assumption-light extraction from the integral topology. The paper remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard domain assumptions of effective field theory and singularity analysis rather than new free parameters or invented entities.

axioms (2)
  • domain assumption Schwinger-Keldysh effective field theory is an appropriate framework for describing nonlinear fluctuation effects in hydrodynamics
    Invoked in the abstract as the standard framework for encoding nonlinear late-time behavior in loop corrections.
  • domain assumption Landau singularity analysis can be applied to the relevant loop integrals to determine their singularities without explicit integration
    Core methodological step stated in the abstract.

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Reference graph

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