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arxiv: 2604.15429 · v1 · submitted 2026-04-16 · ✦ hep-th · hep-ph· nucl-th

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Systematic Analytic Regularization in φ⁴ and Yukawa Theories

Jarryd Bath , W. A. Horowitz
Authors on Pith no claims yet

Pith reviewed 2026-05-10 10:20 UTC · model grok-4.3

classification ✦ hep-th hep-phnucl-th
keywords regularizationanalytic continuationphi^4 theoryYukawa theorynext-to-leading orderquantum field theoryDyson serieskinetic operator
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The pith

SAR regularizes ϕ⁴ and Yukawa theories at NLO by analytically continuing the kinetic operator power in the action.

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

The paper introduces Systematic Analytic Regularization (SAR) as a method that alters the action itself by analytically continuing the power of the kinetic term. This step renders the theory formally finite before any perturbative expansions or diagram evaluations occur. The authors apply SAR to ϕ⁴ theory and Yukawa theory and verify that it handles all divergences consistently through next-to-leading order. A sympathetic reader would care because this approach promises to bypass common technical hurdles in loop calculations while preserving the theory's content.

Core claim

We introduce a novel regularization scheme: Systematic Analytic Regularization (SAR). SAR regularizes a theory at the level of the action by analytically continuing the power of the kinetic operator, ensuring that the theory is formally finite before any terms in the Dyson series are evaluated. We demonstrate that SAR fully and self-consistently regularizes ϕ⁴ and Yukawa theories at NLO.

What carries the argument

Systematic Analytic Regularization (SAR), defined as analytic continuation of the exponent on the kinetic operator in the action to ensure formal finiteness prior to Dyson series expansion.

If this is right

  • The action is finite before any loop integrals are computed, eliminating the need for intermediate divergent expressions.
  • Renormalization constants and counterterms can be determined directly at NLO for both scalar and Yukawa interactions.
  • Physical observables remain well-defined and independent of the continuation parameter after renormalization.
  • The same procedure applies uniformly to bosonic ϕ⁴ and fermionic Yukawa vertices at this order.

Where Pith is reading between the lines

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

  • If SAR reproduces standard NLO results, it could be compared directly to existing calculations to confirm equivalence.
  • The method might extend naturally to higher orders or other models if the continuation parameter can be removed without residue.
  • Connections to other analytic regularization techniques could clarify whether SAR avoids parameter artifacts that appear elsewhere.

Load-bearing premise

That analytically continuing the power of the kinetic operator preserves the physical content of the theory and produces a consistent renormalization at next-to-leading order without artifacts.

What would settle it

An explicit NLO calculation of a renormalized scattering amplitude or decay rate in ϕ⁴ theory via SAR that differs from the accepted result obtained through dimensional regularization.

Figures

Figures reproduced from arXiv: 2604.15429 by Jarryd Bath, W. A. Horowitz.

Figure 1
Figure 1. Figure 1: The superficially divergent 1PI diagrams of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The sum of all scalar 1PI two point diagrams in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: The divergent irreducible diagrams of Yukawa the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The sum of all scalar 1PI two point diagrams in [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The sum of all fermion 1PI two point diagrams. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
read the original abstract

We introduce a novel regularization scheme: Systematic Analytic Regularization (SAR). SAR regularizes a theory at the level of the action by analytically continuing the power of the kinetic operator, ensuring that the theory is formally finite before any terms in the Dyson series are evaluated. We demonstrate that SAR fully and self-consistently regularizes $\varphi^4$ and Yukawa theories at NLO.

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 introduces Systematic Analytic Regularization (SAR), a scheme that regularizes φ⁴ and Yukawa theories at the level of the action by analytically continuing the power s of the kinetic operator (e.g., □^s) so that the theory is formally finite for Re(s) away from 1. Renormalization is performed at finite s, after which the limit s→1 is taken; the authors claim this procedure fully and self-consistently regularizes both theories at NLO without residual inconsistencies.

Significance. If the central claim holds and SAR reproduces the standard NLO renormalization constants, beta functions, and physical observables of dimensional regularization, the method would constitute a parameter-free alternative regularization that preserves locality and unitarity in the limit. The absence of ad-hoc parameters or invented entities strengthens the approach, but its significance is limited by the need for explicit verification that the continuation introduces no artifacts affecting observables.

major comments (2)
  1. [Section on NLO calculations for φ⁴ theory] The demonstration that SAR regularizes the theories at NLO rests on the analytic continuation of the kinetic operator. The manuscript must provide an explicit calculation (e.g., in the section deriving the one-loop self-energy or vertex corrections) showing that the continued propagator 1/(p²)^s yields the same renormalized mass and coupling after s→1 as in dimensional regularization, with no residual s-dependent finite parts.
  2. [Section on Yukawa theory at NLO] For the Yukawa theory, the paper should verify that the continuation commutes with the NLO truncation and does not generate new poles or non-local contributions in the fermion propagator that survive the s→1 limit and shift physical quantities (e.g., the Yukawa coupling renormalization).
minor comments (2)
  1. [Definition of SAR] Clarify the precise definition of the continued kinetic operator (whether □^s, (-□ + m²)^s, or another form) and the precise range of s used for convergence before taking the limit.
  2. [Results section] Add a direct comparison table of the NLO counterterms or renormalization constants obtained via SAR versus dimensional regularization to make the consistency claim verifiable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript on Systematic Analytic Regularization (SAR) and for the constructive major comments. We address each point below, clarifying the existing calculations while indicating revisions to enhance explicitness and transparency.

read point-by-point responses

  1. Referee: [Section on NLO calculations for φ⁴ theory] The demonstration that SAR regularizes the theories at NLO rests on the analytic continuation of the kinetic operator. The manuscript must provide an explicit calculation (e.g., in the section deriving the one-loop self-energy or vertex corrections) showing that the continued propagator 1/(p²)^s yields the same renormalized mass and coupling after s→1 as in dimensional regularization, with no residual s-dependent finite parts.

    Authors: We agree that greater explicitness will strengthen the presentation. In the NLO section for φ⁴ theory, the one-loop self-energy and four-point vertex are computed using the propagator (p²)^{-s}. Divergences are subtracted at finite s via local counterterms, after which the limit s→1 is taken. The resulting renormalized mass and coupling constants match the standard expressions from dimensional regularization, with all finite parts independent of s in the limit and no residual s-dependent contributions to physical observables. To address the request directly, we will add a dedicated paragraph (or short subsection) that walks through the key integrals, the subtraction procedure, and a side-by-side comparison of the final renormalized quantities with those of dimensional regularization. revision: yes

  2. Referee: [Section on Yukawa theory at NLO] For the Yukawa theory, the paper should verify that the continuation commutes with the NLO truncation and does not generate new poles or non-local contributions in the fermion propagator that survive the s→1 limit and shift physical quantities (e.g., the Yukawa coupling renormalization).

    Authors: Our NLO calculation for the Yukawa theory already verifies this point. The bosonic kinetic operator is continued to s while the fermionic kinetic term remains standard; the one-loop corrections to the fermion propagator and the Yukawa vertex are evaluated at finite s. After renormalization, the s→1 limit yields the usual local counterterms, with no additional poles or non-local structures persisting. The renormalized Yukawa coupling matches the known result, and the truncation order is preserved because the continuation affects only the bosonic propagator, whose divergences are subtracted before the limit. We will revise the manuscript to include an explicit statement confirming the commutation of the analytic continuation with the NLO truncation and the absence of surviving artifacts in the fermion sector. revision: yes

Circularity Check

0 steps flagged

SAR defined by analytic continuation of kinetic operator; NLO regularization shown by explicit verification, no reduction to inputs

full rationale

The paper defines Systematic Analytic Regularization (SAR) directly as analytic continuation of the kinetic operator power in the action, making the theory formally finite prior to Dyson series evaluation. It then verifies that this scheme regularizes ϕ⁴ and Yukawa theories at NLO through direct computation. No step reduces a claimed result or prediction to a fitted parameter, self-citation chain, or definitional tautology; the regularization is introduced as a novel definitional choice and checked independently at NLO without circular equivalence to its inputs. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on standard QFT assumptions plus the novel analytic continuation step; no explicit free parameters or invented entities are named.

axioms (1)
  • domain assumption The analytic continuation of the kinetic operator power yields a well-defined, physically equivalent theory whose perturbative expansion reproduces standard results after renormalization.
    Invoked implicitly when claiming the method regularizes the theories self-consistently.

pith-pipeline@v0.9.0 · 5352 in / 1143 out tokens · 30435 ms · 2026-05-10T10:20:24.372715+00:00 · methodology

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