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arxiv: 2504.12044 · v2 · pith:5RKCDJKFnew · submitted 2025-04-16 · ✦ hep-th

Anatomy of the simplest renormalon

Marcos Marino This is my paper

Pith reviewed 2026-05-25 08:35 UTC · model grok-4.3

classification ✦ hep-th
keywords IR renormalonO(N) scalar theorylarge N expansiontrans-seriesBorel summabilitytwo-dimensional QFTnon-perturbative correctionspole mass
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The pith

The IR renormalon in the two-dimensional O(N) scalar theory matches the exact large-N solution and determines the full trans-series of corrections.

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

The paper studies the ground state energy of a superrenormalizable O(N) scalar field theory in two dimensions with a quartic interaction and negative mass squared. It establishes that the infrared renormalon previously identified in perturbation theory at next-to-leading order in the 1/N expansion reproduces the correct asymptotic expansion of the exact solution obtained in the large-N limit. The analysis also constructs the complete trans-series that incorporates all non-perturbative corrections to this perturbative result. The same framework is applied to the O(N)-invariant two-point function, which remains infrared finite order by order in perturbation theory yet develops an infrared renormalon and is not Borel summable, with its pole mass arising entirely from non-perturbative effects.

Core claim

In the scalar O(N) theory in two dimensions, the IR renormalon located at next-to-leading order in the 1/N expansion supplies the correct asymptotic series for the ground-state energy computed from the exact large-N solution. The full trans-series of non-perturbative corrections is determined explicitly. The two-point function, although infrared finite in perturbation theory, is likewise afflicted by an IR renormalon, is not Borel summable, and has a pole mass that is purely non-perturbative; its trans-series is also obtained at the same order.

What carries the argument

The infrared renormalon singularity appearing at next-to-leading order in the 1/N perturbative expansion, validated by direct comparison to the exact large-N saddle-point solution of the model.

If this is right

  • The trans-series gives a complete non-perturbative description of the ground-state energy at this order.
  • The two-point function acquires an infrared renormalon despite being infrared finite in ordinary perturbation theory.
  • The pole mass of the scalar field receives only non-perturbative contributions.
  • This model supplies an explicit benchmark where renormalon effects can be checked against an exact result.

Where Pith is reading between the lines

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

  • Similar renormalon structures may be testable in other superrenormalizable theories that admit exact large-N solutions.
  • The explicit trans-series obtained here could serve as input for resurgence-based resummation methods in related models.
  • The appearance of renormalons in the two-point function suggests that correlation functions may require non-perturbative completions even when perturbative infrared divergences are absent.

Load-bearing premise

The large-N limit supplies an independent exact solution that can be compared directly to the next-to-leading-order perturbative result.

What would settle it

A numerical mismatch between the asymptotic coefficients predicted by the renormalon and the series expansion extracted from the exact large-N solution would falsify the identification.

read the original abstract

Perhaps the simplest IR renormalon occurs in the ground state energy of a superrenormalizable model, the scalar $O(N)$ theory in two dimensions with a quartic potential and negative squared mass. We show that this renormalon, found previously in perturbation theory at next-to-leading order (NLO) in the $1/N$ expansion, gives indeed the correct asymptotic expansion of the exact large $N$ solution of the model, and we determine explicitly the complete trans-series of non-perturbative corrections to the perturbative result. We also use this framework to study the $O(N)$-invariant two-point function of the scalar field. As expected, it is IR finite in perturbation theory, but it is afflicted as well with an IR renormalon singularity and is not Borel summable. The pole mass is purely non-perturbative and its trans-series can be also fully determined at NLO in $1/N$

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 / 1 minor

Summary. The manuscript analyzes the IR renormalon in the ground-state energy of the two-dimensional O(N) scalar theory with quartic interaction and negative mass squared. It asserts that the renormalon previously identified at NLO in the 1/N expansion of perturbation theory reproduces the correct asymptotic expansion of the exact large-N solution, determines the full trans-series of non-perturbative corrections, and extends the same framework to the O(N)-invariant two-point function (which is IR-finite in PT but afflicted by an IR renormalon and not Borel summable), with the pole mass being purely non-perturbative.

Significance. If the identification holds, the work supplies an explicit, controlled example in which a renormalon singularity extracted from NLO perturbation theory is shown to govern the asymptotics of an exact large-N solution, with the complete trans-series determined. This constitutes a concrete strength for the study of renormalons in a superrenormalizable model.

major comments (2)
  1. [Abstract] Abstract: the central claim equates the renormalon singularity found at NLO in the 1/N expansion with the asymptotic expansion of the exact solution at leading order in 1/N (N=∞). Because the exact large-N solution is obtained from the leading-order gap equation or saddle point, while the renormalon is reported only at NLO, it is necessary to show explicitly that the position and residue of the Borel singularity are already reproduced at leading order; otherwise the identification rests on an unstated assumption that O(1/N) corrections do not shift the singularity.
  2. [Abstract] The abstract states that the NLO renormalon 'gives indeed the correct asymptotic expansion' of the exact large-N result, yet no explicit equation or section is referenced that demonstrates the matching of the Borel singularity between the NLO perturbative series and the leading large-N trans-series. This matching is load-bearing for the claim that the renormalon is not an artifact of the NLO truncation.
minor comments (1)
  1. The abstract mentions determination of the 'complete trans-series' for both the ground-state energy and the pole mass; the manuscript should include a compact summary table or equation block that lists the first few non-perturbative terms with their explicit coefficients.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim equates the renormalon singularity found at NLO in the 1/N expansion of perturbation theory with the asymptotic expansion of the exact solution at leading order in 1/N (N=∞). Because the exact large-N solution is obtained from the leading-order gap equation or saddle point, while the renormalon is reported only at NLO, it is necessary to show explicitly that the position and residue of the Borel singularity are already reproduced at leading order; otherwise the identification rests on an unstated assumption that O(1/N) corrections do not shift the singularity.

    Authors: We thank the referee for highlighting this important subtlety. The exact large-N solution is obtained from the leading-order saddle-point equation. The renormalon singularity is extracted from the NLO term in the 1/N expansion of perturbation theory around this saddle point. In the manuscript we determine the complete trans-series at NLO in 1/N and verify that it reproduces the asymptotic expansion of the exact solution. To make the argument fully explicit, we will add a short paragraph (or subsection) demonstrating that the location and residue of the Borel singularity are fixed already by the leading large-N dynamics and receive no shift from the NLO correction itself. This will be included in the revised version. revision: yes

  2. Referee: [Abstract] The abstract states that the NLO renormalon 'gives indeed the correct asymptotic expansion' of the exact large-N result, yet no explicit equation or section is referenced that demonstrates the matching of the Borel singularity between the NLO perturbative series and the leading large-N trans-series. This matching is load-bearing for the claim that the renormalon is not an artifact of the NLO truncation.

    Authors: We agree that the abstract would benefit from a concrete pointer to the relevant equation or section. In the revised manuscript we will update the abstract to include an explicit reference (e.g., “as shown in Eq. (X.Y) and the trans-series analysis of Section Z”) that directly links the NLO perturbative Borel singularity to the leading large-N trans-series. revision: yes

Circularity Check

0 steps flagged

No significant circularity: exact large-N solution provides independent benchmark for NLO renormalon

full rationale

The derivation compares a renormalon singularity identified at NLO in the 1/N perturbative expansion to the asymptotic behavior of an exact solution obtained in the strict large-N limit. The large-N solution is constructed via standard saddle-point or gap-equation methods that do not invoke the Borel-plane analysis or the NLO perturbative input; the comparison therefore functions as an external check rather than a self-referential identity. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the central chain. The order mismatch between NLO perturbation and leading large-N exact solution is a substantive assumption about 1/N corrections but does not reduce the claimed result to a tautology by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on the existence of an exact solution in the large N limit of the model and the applicability of the 1/N expansion at NLO for comparison.

axioms (1)
  • domain assumption The O(N) scalar theory in 2D admits an exact solution in the large N limit that serves as an independent benchmark
    Invoked to validate that the renormalon gives the correct asymptotic expansion

pith-pipeline@v0.9.0 · 5670 in / 1401 out tokens · 31096 ms · 2026-05-25T08:35:13.910969+00:00 · methodology

discussion (0)

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