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arxiv: 2606.28167 · v1 · pith:LGTTTIASnew · submitted 2026-06-26 · ✦ hep-lat · hep-ph· hep-th

Spectral densities from Euclidean correlators via integral transforms: theoretical framework

Leonardo Giusti , Matteo Saccardi , Diego Toniolo This is my paper

Pith reviewed 2026-06-29 01:31 UTC · model grok-4.3

classification ✦ hep-lat hep-phhep-th
keywords spectral densityinverse Laplace transformEuclidean correlatorlattice field theorysmearing functionintegral transformfinite temporal extent
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0 comments X

The pith

Analytic formulae enable extraction of spectral densities from Euclidean correlators via inverse Laplace transforms.

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

The paper develops analytic expressions to invert the Laplace transform and recover spectral densities from the Euclidean time dependence of correlation functions. These expressions apply to both continuum data and discrete lattice samplings, and they extend to regulated or smeared versions of the densities. When the lattice correlator is O(a)-improved, the reconstructed density approaches its continuum counterpart with only O(a^2) discretization effects. For lattices of finite temporal extent the method introduces incomplete transforms whose smearing functions allow rigorous upper bounds on the unknown spectral regions provided the transform decays rapidly enough at large conjugate variable.

Core claim

Analytic formulae exist that perform the inverse Laplace transform on either continuous or discrete Euclidean correlators, yielding spectral densities that include regulated and smeared cases; the lattice version converges to the continuum result up to O(a^2) when the input correlator is O(a)-improved, while incomplete transforms on finite lattices permit controlled bounds on unknown contributions whenever the smearing function’s integral transform decays sufficiently fast.

What carries the argument

Analytic formulae for the inverse Laplace transform, including incomplete integral transforms paired with smearing functions whose transforms decay rapidly enough to bound unknown spectral contributions.

If this is right

  • Lattice computations can extract spectral densities with discretization errors under explicit control.
  • Smearing functions can be selected so that contributions from unknown spectral regions remain rigorously bounded.
  • Lattice temporal extent and spacing can be chosen in advance to meet a target precision on the reconstructed density.
  • The same formulae apply without modification to Euclidean correlators arising in any other field of research.

Where Pith is reading between the lines

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

  • The controlled bounds may allow systematic improvement of precision in non-perturbative calculations of hadronic spectral functions.
  • Analogous incomplete-transform techniques could be tested on exactly solvable models to verify convergence rates and bound tightness.
  • The method supplies a concrete route for propagating statistical and systematic uncertainties from correlator data into the final spectral density.

Load-bearing premise

The integral transform of the chosen smearing function decays fast enough with the conjugate variable that the contribution of unknown regions to the smeared density can be rigorously bounded.

What would settle it

A numerical check in which an O(a)-improved lattice correlator produces a reconstructed density that deviates from the continuum result by more than O(a^2), or in which the bound on the smeared density is violated despite rapid decay of the smearing transform.

read the original abstract

Spectral densities link experimental measurements to dynamical properties of a quantum field theory which, in turn, can be resolved non-perturbatively from the Euclidean time-dependence of correlation functions. By making extensive use of integral transforms, we present analytic formulae to carry out the inverse Laplace transform so as to extract spectral densities from either the continuum or the discrete sampling of correlation functions in the Euclidean time. Formulae extend to regulated and/or smeared spectral densities as well. We explicitly show that the proposed lattice solution tends to its continuum counterpart up to $O(a^2)$ effects in the lattice spacing $a$ if the lattice correlator is $O(a)$-improved. In practical computations, lattices have necessarily a finite Euclidean temporal extent, a lack of knowledge which suggests to introduce incomplete integral transforms and the corresponding incomplete smeared spectral densities. The contribution from the unknowns to a smeared spectral density can then be rigorously bound and kept under control if the integral transform of the smearing function decays fast enough with the conjugate variable. Conversely, the bound can be used to plan lattices so as to achieve a given target precision on the reconstructed spectral density of interest. The formulae presented here in the context of lattice field theory can be easily applied or extended to other areas of research.

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

0 major / 2 minor

Summary. The manuscript develops analytic formulae based on integral transforms to perform the inverse Laplace transform and extract spectral densities from Euclidean correlators, covering both continuum and discrete (lattice) samplings. The formulae are extended to regulated and smeared spectral densities. The work shows that the lattice solution converges to the continuum limit up to O(a²) effects when the correlator is O(a)-improved, and derives rigorous bounds on the unknown contribution from finite temporal extent to a smeared spectral density, provided the integral transform of the smearing function decays sufficiently rapidly with the conjugate variable. The framework is presented in the context of lattice field theory but noted as extensible.

Significance. If the derivations hold, the paper supplies a parameter-free analytic approach to spectral density reconstruction with explicit control over systematic effects from discretization, smearing, and finite extent. The O(a²) consistency result and the bound on incomplete transforms are particularly valuable for guiding lattice computations toward target precision, strengthening the link between Euclidean lattice data and physical spectral densities.

minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a short explicit statement of the key assumptions (e.g., analyticity properties of the correlator) before presenting the formulae.
  2. Notation for the incomplete integral transforms and the corresponding incomplete smeared densities is introduced without a dedicated comparison table to the complete case; adding one would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

Derivation self-contained via integral transform identities

full rationale

The paper constructs analytic formulae for the inverse Laplace transform and associated bounds directly from the definitions and properties of integral transforms applied to Euclidean correlators (both continuum and discrete). The O(a²) continuum limit for O(a)-improved data follows as a mathematical expansion of the lattice kernel, and the rigorous bound on unknown contributions is obtained by requiring the smearing kernel's transform to decay sufficiently fast; neither step reduces to a fitted input renamed as a prediction nor to any self-citation chain. The framework is therefore independent of external fitted quantities and contains no load-bearing self-referential definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, ad-hoc axioms, or new postulated entities; the framework rests on standard properties of Laplace transforms and lattice improvement.

axioms (2)
  • standard math Standard mathematical properties of the Laplace transform and its inverse hold for the correlation functions under consideration.
    Invoked implicitly when claiming analytic inversion formulae exist.
  • domain assumption The lattice correlator can be made O(a)-improved.
    Required for the O(a^2) convergence statement.

pith-pipeline@v0.9.1-grok · 5755 in / 1402 out tokens · 49407 ms · 2026-06-29T01:31:35.293335+00:00 · methodology

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