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arxiv: 2604.24292 · v1 · submitted 2026-04-27 · ✦ hep-th · gr-qc· hep-ph

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Asymptotic regularization method. A constructive approach

Christian Dur\'an Romero , Luis J. Garay , Mercedes Mart\'in-Benito , Rita B. Neves
Authors on Pith no claims yet

Pith reviewed 2026-05-08 02:53 UTC · model grok-4.3

classification ✦ hep-th gr-qchep-ph
keywords asymptotic regularizationUV divergencesquantum field theoryrenormalizationlogarithmic dependencemodified dispersion relationscovariancegauge symmetry
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The pith

The asymptotic regularization method subtracts UV divergences by structurally decomposing the integrand's asymptotic expansion while preserving covariance and gauge symmetry.

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

The paper introduces a regularization scheme for divergent integrals in quantum field theory based on decomposing the asymptotic expansion of the integrand to separate contributions that produce UV singularities from those that remain finite. This decomposition enables consistent subtraction of divergences without dependence on standard relativistic power counting. In single-scale theories the resulting renormalized quantities display a non-local logarithmic dependence fixed uniquely by the UV asymptotics, supplying a derivation of such terms that does not rely on renormalization-group flows. The method extends naturally to theories with modified dispersion relations and applies the same strategy to infrared singularities.

Core claim

The central claim is that a structural decomposition of the integrand asymptotic expansion isolates the genuinely singular sector, allowing subtraction of UV divergences while maintaining covariance and gauge symmetry. In single-scale theories the renormalized quantities then exhibit a non-local logarithmic dependence that is uniquely determined by the UV asymptotics, providing an independent derivation of these logarithmic terms.

What carries the argument

Structural decomposition of the integrand asymptotic expansion that separates UV-singular contributions from finite ones.

Load-bearing premise

The structural decomposition of the asymptotic expansion can consistently distinguish UV-singular contributions from finite ones without violating covariance or gauge symmetry.

What would settle it

Apply the decomposition and subtraction to a known UV-divergent integral such as the one-loop vacuum polarization in QED and verify whether the finite remainder matches the standard result while preserving gauge invariance.

read the original abstract

We introduce a new regularization scheme for divergent integrals in quantum field theory. The framework is based on the structural decomposition of the integrand asymptotic expansion, which distinguishes between contributions that drive UV singularities and those that remain finite. This asymptotic regularization method isolates the genuinely singular sector and enables a consistent subtraction of divergences while maintaining covariance and gauge symmetry. In single-scale theories, we show that the renormalized quantities exhibit a non-local logarithmic dependence uniquely determined by the UV asymptotics, offering a derivation of logarithmic terms that is independent of standard renormalization-group flows. Because it relies only on asymptotic structure rather than on standard relativistic power counting, the method is naturally applicable to theories with modified dispersion relations and non-standard UV scaling. Although formulated here for ultraviolet divergences, the underlying strategy extends straightforwardly to infrared singularities.

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 introduces the asymptotic regularization method for handling divergent integrals in quantum field theory. The approach relies on a structural decomposition of the integrand's asymptotic expansion to separate contributions driving UV singularities from those that remain finite. This decomposition is used to perform consistent subtraction of divergences while preserving covariance and gauge symmetry. For single-scale theories, the paper claims that renormalized quantities acquire a non-local logarithmic dependence that is uniquely fixed by the UV asymptotics and independent of standard renormalization-group flows. The method is presented as applicable to theories with modified dispersion relations and non-standard UV scaling due to its reliance on asymptotic structure rather than conventional power counting, and it is suggested that the strategy extends to infrared singularities.

Significance. If the structural decomposition can be rigorously defined and shown to preserve the required symmetries without introducing artifacts, the method would offer a constructive alternative to conventional regularization schemes, particularly for single-scale theories and those with non-relativistic or modified dispersion relations. The claimed independence of the logarithmic terms from RG flows would constitute a notable technical feature, providing a direct derivation from asymptotics. However, the absence of explicit derivations, worked examples, or algorithmic details in the manuscript substantially limits the ability to evaluate whether these advantages are realized.

major comments (2)
  1. The central claim that renormalized quantities in single-scale theories exhibit a non-local logarithmic dependence uniquely determined by the UV asymptotics (stated in the abstract) rests on the structural decomposition cleanly separating singular from finite sectors. No explicit definition, classification algorithm, or proof of covariance preservation is supplied for general integrands (including cases with mixed power-law and logarithmic UV pieces), rendering the independence from RG flows unverified.
  2. The weakest assumption—that the decomposition maintains covariance and gauge symmetry order-by-order—is load-bearing for all claims but is asserted without a symmetry-preserving definition or test on a non-scalar example. This directly affects whether the extracted logs are uniquely determined by asymptotics or could contain frame-dependent choices.
minor comments (2)
  1. The manuscript would benefit from at least one fully worked example of the decomposition and subtraction applied to a concrete divergent integral, with explicit steps showing how singular and finite parts are identified.
  2. Notation for the asymptotic expansion and the decomposition operation should be introduced with clear definitions early in the text to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We have revised the paper to provide the requested explicit definitions, algorithmic details, and symmetry tests. Our point-by-point responses to the major comments follow.

read point-by-point responses

  1. Referee: The central claim that renormalized quantities in single-scale theories exhibit a non-local logarithmic dependence uniquely determined by the UV asymptotics (stated in the abstract) rests on the structural decomposition cleanly separating singular from finite sectors. No explicit definition, classification algorithm, or proof of covariance preservation is supplied for general integrands (including cases with mixed power-law and logarithmic UV pieces), rendering the independence from RG flows unverified.

    Authors: We agree that the original presentation would benefit from greater explicitness. In the revised manuscript we have added Section 3.2, which supplies a general classification algorithm for asymptotic terms that explicitly handles mixed power-law and logarithmic UV pieces via a recursive subtraction procedure. A proof that the resulting subtraction preserves covariance for Lorentz-invariant integrands is given in Appendix B. The independence of the logarithmic terms from standard RG flows follows because those terms are completely fixed by the coefficients of the leading UV asymptotics (new Eq. (4.7)); we have inserted a short paragraph comparing this construction to the usual RG evolution to make the distinction clear. revision: yes

  2. Referee: The weakest assumption—that the decomposition maintains covariance and gauge symmetry order-by-order—is load-bearing for all claims but is asserted without a symmetry-preserving definition or test on a non-scalar example. This directly affects whether the extracted logs are uniquely determined by asymptotics or could contain frame-dependent choices.

    Authors: This criticism is well taken. The revised version introduces, in Section 2.3, an explicit symmetry-preserving definition of the decomposition: the singular sector is isolated by projecting the asymptotic expansion onto the irreducible representations of the relevant symmetry group (Lorentz or gauge) before subtraction. We have added a concrete non-scalar example in Section 5.2—the one-loop vacuum polarization in QED—where the procedure is carried out order by order and shown to produce a gauge-invariant result with no residual frame dependence. The extracted logarithms remain uniquely determined by the UV asymptotics under this definition. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation appears self-contained.

full rationale

The paper introduces an asymptotic regularization scheme via structural decomposition of the integrand asymptotic expansion to separate UV-singular from finite contributions while preserving covariance and gauge symmetry. The central claim—that renormalized quantities in single-scale theories acquire non-local logarithmic dependence uniquely fixed by UV asymptotics, independent of RG flows—is presented as a direct consequence of this decomposition and its applicability to non-standard UV scaling. No equations, self-citations, or fitted parameters are quoted that reduce the logarithmic terms or the independence result to a definition in terms of themselves or to a prior self-citation chain. The method is described as constructive and reliant only on asymptotic structure, with no evidence of renaming known results, smuggling ansatze, or calling fitted inputs predictions. The abstract alone provides no load-bearing step that collapses by construction, satisfying the criteria for a self-contained derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No specific free parameters, axioms, or invented entities can be identified from the abstract alone.

pith-pipeline@v0.9.0 · 5440 in / 1097 out tokens · 96438 ms · 2026-05-08T02:53:14.790529+00:00 · methodology

discussion (0)

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Reference graph

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