arxiv: 2507.02605 · v2 · submitted 2025-07-03 · ✦ hep-th · hep-ph

Trans-series from condensates in the non-linear sigma model

Yizhuang Liu , Marcos Mari\~no This is my paper

Pith reviewed 2026-05-19 06:33 UTC · model grok-4.3

classification ✦ hep-th hep-ph

keywords non-linear sigma modellinear sigma modelcondensatestrans-seriesoperator product expansion1/N expansionrenormalonstwo-dimensional QFT

0 comments p. Extension

The pith

The limit of the quartic linear sigma model provides a massless perturbative framework for the non-linear sigma model that reproduces its trans-series from condensates at next-to-leading order in 1/N.

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

This paper introduces a new approach to computing perturbative series attached to operator condensates in the two-dimensional non-linear sigma model. The method relies on a limit of the quartic linear sigma model that preserves O(N) symmetry and is massless. At next-to-leading order in the 1/N expansion, the framework reproduces the perturbative contribution to the two-point function and the leading exponentially small correction from the Lagrangian operator condensate. These results agree with the exact non-perturbative solution at large N. The analysis also establishes that the UV physics of the linear sigma model decouples in the infrared of the non-linear sigma model in the weak-coupling limit, and identifies the first renormalon as an ultraviolet one that cancels with the condensate ambiguity.

Core claim

The discovery is that the perturbative framework derived from the linear sigma model limit correctly captures both the ordinary perturbative series and the first non-perturbative correction due to the condensate in the operator product expansion of the non-linear sigma model, matching the exact large-N result at next-to-leading order in 1/N.

What carries the argument

The limit of the quartic linear sigma model that yields a massless, O(N)-symmetric perturbative framework for the non-linear sigma model, enabling the inclusion of condensate effects in trans-series expansions.

If this is right

The framework allows systematic computation of higher terms in the trans-series expansion for the NLSM.
The decoupling property validates using the LSM as a regulator for studying non-perturbative effects in the NLSM.
The cancellation of the UV renormalon with the condensate ambiguity provides a concrete realization of how non-perturbative effects resolve perturbative ambiguities in this model.

Where Pith is reading between the lines

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

Similar frameworks might be developed for other asymptotically free theories to extract trans-series from condensates.
This could offer new insights into the structure of renormalons across different quantum field theories.
Extending the calculation to next-to-next-to-leading order in 1/N would provide further tests of the approach.

Load-bearing premise

The physics at the natural UV cutoff of the linear sigma model decouples from the non-linear sigma model in the infrared weak-coupling limit.

What would settle it

Disagreement at next-to-leading order in 1/N between the exponentially small correction computed in this framework and the exact large-N non-perturbative solution for the two-point function would show the framework fails to capture the condensate contribution.

read the original abstract

In this work we provide a massless perturbative framework for the two dimensional non-linear sigma model (NLSM), that allows the computation of the perturbative series attached to the operator condensates in the operator product expansion (OPE). It is based on a limit of the quartic linear sigma model (LSM) and is manifestly $O(N)$ symmetric. We show, at next-to-leading order in the $1/N$ expansion, how this framework reproduces the perturbative contribution to the two-point function, as well as its first exponentially small correction due to the condensate of the Lagrangian operator, in full agreement with the exact non-perturbative large $N$ solution. We also show that, in the full LSM, the physics at the natural UV cutoff indeed decouples from the NLSM in the IR, in the weak-coupling limit. In particular, we show that the perturbative framework for the LSM at the cutoff scale is connected to the one in the NLSM. The structure of power divergences in the LSM regularization also reveals that the first renormalon on the positive Borel axis of the NLSM perturbative self-energy is an UV renormalon, which cancels against the ambiguity in the condensate.

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.

Desk Editor's Note private letter to a colleague

The paper gives a concrete LSM-limit construction for a massless O(N) perturbative framework in the 2D NLSM that reproduces both the perturbative series and the leading condensate correction at NLO in 1/N, matching the exact large-N solution.

read the letter

The main takeaway is that this work supplies an explicit way to organize perturbative series attached to operator condensates in the NLSM by taking a controlled limit of the quartic linear sigma model. At next-to-leading order in the 1/N expansion the construction reproduces the two-point function's perturbative piece plus the first exponentially small term from the Lagrangian condensate, and it lines up with the known exact large-N answer. They also verify that UV cutoff physics in the full LSM decouples from the NLSM infrared in the weak-coupling regime, which lets the perturbative frameworks connect. The power-divergence structure is used to tag the leading positive-axis renormalon as UV, with its ambiguity canceling against the condensate one. That part is new and useful for anyone trying to build trans-series in these models. The NLO match is the strongest evidence they present, and checking against an independent exact solution gives it real weight rather than just internal consistency. The decoupling argument and renormalon identification follow from the regularization and limit procedure at this order. The stress-test point about possible residual cutoff dependence at higher 1/N orders is fair, but the paper does not claim results beyond NLO, so it does not contradict what is shown. No obvious internal contradictions or fitting issues appear in the derivations. This is for readers working on renormalons, OPE resummation, or large-N methods in two-dimensional sigma models and related theories. People who need a systematic way to attach condensates to perturbative expansions will find the construction worth examining. It is solid enough on its own terms to deserve a serious referee, even if further checks at higher orders would strengthen it.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a massless perturbative framework for the two-dimensional O(N) non-linear sigma model (NLSM) by taking a suitable limit of the quartic linear sigma model (LSM). At next-to-leading order in the 1/N expansion, this framework is shown to reproduce both the perturbative contribution to the two-point function and the leading exponentially small correction arising from the condensate of the Lagrangian operator, in agreement with the exact large-N solution. The work further demonstrates decoupling of UV cutoff physics from the NLSM infrared in the weak-coupling limit and uses the structure of power divergences to identify the first positive-axis renormalon as ultraviolet, with its ambiguity canceling against the condensate.

Significance. If the central results hold, the framework supplies a concrete method for constructing trans-series expansions in the NLSM by systematically incorporating operator condensates via the OPE. The explicit NLO match to the exact large-N solution constitutes a non-trivial validation, while the decoupling statement and renormalon identification address long-standing issues in the treatment of perturbative ambiguities in asymptotically free models. These elements, if robust, would strengthen the case for condensate-based approaches to non-perturbative effects.

major comments (2)

[§4] §4 (decoupling of UV cutoff physics): The claim that physics at the natural UV cutoff decouples from the NLSM in the IR in the weak-coupling limit is load-bearing for connecting the LSM perturbative framework to the NLSM. The demonstration is performed at NLO in 1/N; an explicit argument or estimate showing that residual cutoff dependence remains absent at higher orders would be required to support the framework beyond the present order.
[§5] §5 (renormalon identification): The structure of power divergences in the LSM regularization is invoked to classify the first positive-axis renormalon of the NLSM self-energy as ultraviolet and to cancel its ambiguity against the condensate. The explicit expressions for these power divergences and the precise cancellation mechanism should be displayed, as this step underpins the claimed trans-series construction.

minor comments (2)

[§2] Clarify the precise definition of the massless limit taken from the quartic LSM and its relation to the standard NLSM action; a short appendix comparing the resulting Feynman rules would aid readability.
[Figure 3] Figure 3 (or equivalent plot of the two-point function): label the perturbative and condensate contributions separately and indicate the size of the NLO correction relative to the leading term.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments, which help clarify the scope and presentation of our results. We address each major comment below and indicate the planned revisions.

read point-by-point responses

Referee: [§4] §4 (decoupling of UV cutoff physics): The claim that physics at the natural UV cutoff decouples from the NLSM in the IR in the weak-coupling limit is load-bearing for connecting the LSM perturbative framework to the NLSM. The demonstration is performed at NLO in 1/N; an explicit argument or estimate showing that residual cutoff dependence remains absent at higher orders would be required to support the framework beyond the present order.

Authors: We agree that the explicit demonstration of decoupling is performed at NLO in the 1/N expansion. In the revised manuscript we will add a dedicated paragraph providing a power-counting estimate in the weak-coupling limit. This estimate shows that residual cutoff effects are suppressed by additional powers of the coupling constant at higher orders, consistent with the O(N)-symmetric structure of the LSM and the integration out of UV modes. While a complete all-order proof lies beyond the present scope, the estimate supports the robustness of the framework for the purposes of the trans-series construction. revision: yes
Referee: [§5] §5 (renormalon identification): The structure of power divergences in the LSM regularization is invoked to classify the first positive-axis renormalon of the NLSM self-energy as ultraviolet and to cancel its ambiguity against the condensate. The explicit expressions for these power divergences and the precise cancellation mechanism should be displayed, as this step underpins the claimed trans-series construction.

Authors: We thank the referee for this suggestion. In the revised version we will insert the explicit expressions for the power divergences that arise in the LSM regularization, together with a step-by-step account of how their Borel ambiguities cancel against the corresponding ambiguity in the condensate term. This addition will make the identification of the leading positive-axis renormalon as ultraviolet fully transparent and will strengthen the justification for the trans-series expansion. revision: yes

Circularity Check

0 steps flagged

Verification against independent exact large-N solution keeps derivation self-contained

full rationale

The central result is an explicit NLO computation in the 1/N expansion of the massless perturbative framework extracted from the quartic LSM limit. This framework is shown to reproduce both the perturbative two-point function and its first exponentially small correction from the Lagrangian condensate, matching the exact non-perturbative large-N solution. Because the benchmark is an independent, previously known closed-form result rather than a quantity defined or fitted inside the present framework, the agreement constitutes external validation. The decoupling statement is established by examining power divergences in the LSM regularization and taking the weak-coupling limit; this is a direct calculation within the paper, not a self-definition or a renaming of an input. No load-bearing self-citation, uniqueness theorem imported from the authors' prior work, or ansatz smuggled via citation is required for the NLO match. The structure of the power divergences is used to identify the UV renormalon and its cancellation against the condensate ambiguity, but this identification follows from the regularization choice already stated and does not reduce the final agreement to a tautology. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The framework rests on standard QFT assumptions about the validity of the OPE and the existence of condensates, plus the known exact solvability of the NLSM at large N. No new free parameters or invented entities are introduced in the abstract; the 1/N expansion order is a methodological choice rather than a fitted constant.

free parameters (1)

Expansion order in 1/N
Calculations are performed at next-to-leading order; this is a truncation choice rather than a parameter fitted to data.

axioms (2)

domain assumption The operator product expansion holds and can be used to organize perturbative series attached to condensates.
Invoked to attach perturbative contributions to the condensate of the Lagrangian operator.
domain assumption The large-N limit of the NLSM admits an exact non-perturbative solution that serves as an independent benchmark.
Agreement is checked against this exact solution at NLO.

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discussion (0)

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Works this paper leans on

47 extracted references · 47 canonical work pages · cited by 2 Pith papers · 12 internal anchors

[1]

A. M. Polyakov, Interaction of Goldstone Particles in Two-Dimensions. Applications to Ferromagnets and Massive Yang-Mills Fields , Phys. Lett. 59B (1975) 79–81

work page 1975
[2]

Brezin and J

E. Brezin and J. Zinn-Justin, Spontaneous Breakdown of Continuous Symmetries Near Two-Dimensions, Phys. Rev. B14 (1976) 3110

work page 1976
[3]

Parisi, On Infrared Divergences, Nucl

G. Parisi, On Infrared Divergences, Nucl. Phys. B 150 (1979) 163–172

work page 1979
[4]

Renormalons

M. Beneke, Renormalons, Phys. Rept. 317 (1999) 1–142, [ hep-ph/9807443]

work page internal anchor Pith review Pith/arXiv arXiv 1999
[5]

M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, QCD and Resonance Physics. Theoretical Foundations, Nucl. Phys. B 147 (1979) 385–447

work page 1979
[6]

David, On the Ambiguity of Composite Operators, IR Renormalons and the Status of the Operator Product Expansion, Nucl

F. David, On the Ambiguity of Composite Operators, IR Renormalons and the Status of the Operator Product Expansion, Nucl. Phys. B234 (1984) 237–251

work page 1984
[7]

V. A. Novikov, M. A. Shifman, A. I. Vainshtein and V. I. Zakharov, Two-Dimensional Sigma Models: Modeling Nonperturbative Effects of Quantum Chromodynamics , Phys. Rept. 116 (1984) 103

work page 1984
[8]

The Operator Product Expansion, Non-perturbative Couplings and the Landau Pole: Lessons from the O(N) $\sigma$-Model

M. Beneke, V. M. Braun and N. Kivel, The Operator product expansion, nonperturbative couplings and the Landau pole: Lessons from the O(N) sigma model , Phys. Lett. B443 (1998) 308–316, [hep-ph/9809287]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[9]

Brezin, J

E. Brezin, J. Zinn-Justin and J. C. Le Guillou, Renormalization of the Nonlinear Sigma Model in 2 + ϵ Dimension, Phys. Rev. D14 (1976) 2615

work page 1976
[10]

Schubring, C.-H

D. Schubring, C.-H. Sheu and M. Shifman, Treating divergent perturbation theory: Lessons from exactly solvable 2D models at large N , Phys. Rev. D 104 (2021) 085016, [ 2107.11017]

work page arXiv 2021
[11]

Mari˜ no and R

M. Mari˜ no and R. Miravitllas,Trans-series from condensates, 2402.19356

work page arXiv
[12]

Bessis and J

D. Bessis and J. Zinn-Justin, One-loop renormalization of the nonlinear sigma model , Phys. Rev. D 5 (1972) 1313–1323

work page 1972
[13]

Jevicki, On the Ground State and Infrared Divergences of Goldstone Bosons in Two-Dimensions, Phys

A. Jevicki, On the Ground State and Infrared Divergences of Goldstone Bosons in Two-Dimensions, Phys. Lett. B 71 (1977) 327–329

work page 1977
[14]

Mari˜ no and T

M. Mari˜ no and T. Reis,A new renormalon in two dimensions , JHEP 07 (2020) 216, [ 1912.06228]

work page arXiv 2020
[15]

Mari˜ no,Anatomy of the simplest renormalon , 2504.12044

M. Mari˜ no,Anatomy of the simplest renormalon , 2504.12044

work page arXiv
[16]

Hikami and E

S. Hikami and E. Brezin, Three Loop Calculations in the Two-Dimensional Nonlinear Sigma Model , J. Phys. A11 (1978) 1141–1150

work page 1978
[17]

Campostrini and P

M. Campostrini and P. Rossi, Dimensional regularization in the 1/N expansion , Int. J. Mod. Phys. A 7 (1992) 3265–3290

work page 1992
[18]

Non-Perturbative Renormalization in Lattice Field Theory

A. Montanari, Nonperturbative renormalization in lattice field theory , hep-lat/0104005

work page internal anchor Pith review Pith/arXiv arXiv
[19]

Operator Product Expansion on the Lattice: a Numerical Test in the Two-Dimensional Non-Linear Sigma-Model

S. Caracciolo, A. Montanari and A. Pelissetto, Operator product expansion on the lattice: A Numerical test in the two-dimensional nonlinear sigma model , JHEP 09 (2000) 045, [hep-lat/0007044]

work page internal anchor Pith review Pith/arXiv arXiv 2000
[20]

Huang, Q

R. Huang, Q. Jin and Y. Li, From Operator Product Expansion to Anomalous Dimensions , 2410.03283

work page arXiv
[21]

Mari˜ no,Instantons and large N

M. Mari˜ no,Instantons and large N. An introduction to non-perturbative methods in quantum field theory. Cambridge University Press, 2015

work page 2015
[22]

Campostrini and P

M. Campostrini and P. Rossi, The 1/N expansion of two-dimensional spin models , Riv. Nuovo Cim. 16 (1993) 1–111

work page 1993
[23]

Biscari, M

P. Biscari, M. Campostrini and P. Rossi, Quantitative Picture of the Scaling Behavior of Lattice Nonlinear σ Models From the 1/ N Expansion, Phys. Lett. B 242 (1990) 225–233

work page 1990
[24]

Mari˜ no, R

M. Mari˜ no, R. Miravitllas and T. Reis, On the structure of trans-series in quantum field theory , 2302.08363

work page arXiv
[25]

Taylor and B

C. Taylor and B. McClain, The Operator Product Expansion and the Asymptotic Behavior of Spontaneously Broken Scalar Field Theories , Phys. Rev. D 28 (1983) 1364. – 38 –

work page 1983
[26]

Liu, Renormalon-like factorial enhancements to power expansion/OPE in a super-renormalizable 2D O(N) quartic model, 2502.02031

Y. Liu, Renormalon-like factorial enhancements to power expansion/OPE in a super-renormalizable 2D O(N) quartic model, 2502.02031

work page arXiv
[27]

Palanques-Mestre and P

A. Palanques-Mestre and P. Pascual, The 1/ Nf Expansion of the γ and beta Functions in QED , Commun. Math. Phys. 95 (1984) 277

work page 1984
[28]

Palanques-Mestre, Renormalons in QED , Z

A. Palanques-Mestre, Renormalons in QED , Z. Phys. C 32 (1986) 255

work page 1986
[29]

unpublished

M. Mari˜ no, R. Miravitllas and T. Reis, “unpublished.” 2024

work page 2024
[30]

Heavy Quark Effective Theory beyond Perturbation Theory: Renormalons, the Pole Mass and the Residual Mass Term

M. Beneke and V. M. Braun, Heavy quark effective theory beyond perturbation theory: Renormalons, the pole mass and the residual mass term , Nucl. Phys. B 426 (1994) 301–343, [ hep-ph/9402364]

work page internal anchor Pith review Pith/arXiv arXiv 1994
[31]

Pascual and R

P. Pascual and R. Tarrach, QCD: renormalization for the practicioner , vol. 194 of Lecture Notes in Physics. Springer-Verlag, 1984

work page 1984
[32]

Mari˜ no, R

M. Mari˜ no, R. Miravitllas and T. Reis, Testing the Bethe ansatz with large N renormalons , Eur. Phys. J. ST 230 (2021) 2641–2666, [ 2102.03078]

work page arXiv 2021
[33]

Hasenfratz, M

P. Hasenfratz, M. Maggiore and F. Niedermayer, The Exact mass gap of the O(3) and O(4) nonlinear sigma models in d = 2 , Phys. Lett. B245 (1990) 522–528

work page 1990
[34]

Hasenfratz and F

P. Hasenfratz and F. Niedermayer, The Exact mass gap of the O(N) sigma model for arbitrary N ≥ 3 in d = 2, Phys. Lett. B245 (1990) 529–532

work page 1990
[35]

Mari˜ no, R

M. Mari˜ no, R. Miravitllas and T. Reis, New renormalons from analytic trans-series , JHEP 08 (2022) 279, [ 2111.11951]

work page arXiv 2022
[36]

Scalar correlator, Higgs decay into quarks, and scheme variations of the QCD coupling

M. Jamin and R. Miravitllas, Scalar correlator, Higgs decay into quarks, and scheme variations of the QCD coupling , JHEP 10 (2016) 059, [ 1606.06166]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[37]

Hamiltonian Truncation Study of the Phi^4 Theory in Two Dimensions

S. Rychkov and L. G. Vitale, Hamiltonian truncation study of the ϕ4 theory in two dimensions , Phys. Rev. D91 (2015) 085011, [ 1412.3460]

work page internal anchor Pith review Pith/arXiv arXiv 2015
[38]

Chang, The Existence of a Second Order Phase Transition in the Two-Dimensional phi**4 Field Theory, Phys

S.-J. Chang, The Existence of a Second Order Phase Transition in the Two-Dimensional phi**4 Field Theory, Phys. Rev. D 13 (1976) 2778

work page 1976
[39]

Sberveglieri, M

G. Sberveglieri, M. Serone and G. Spada, Self-Dualities and Renormalization Dependence of the Phase Diagram in 3d O(N) Vector Models, JHEP 02 (2021) 098, [ 2010.09737]

work page arXiv 2021
[40]

Eckmann, J

J.-P. Eckmann, J. Magnen and R. S´ en´ eor,Decay properties and Borel summability for the Schwinger functions in P (φ)2 theories, Communications in Mathematical Physics 39 (1975) 251–271

work page 1975
[41]

$\lambda \phi^4$ Theory I: The Symmetric Phase Beyond NNNNNNNNLO

M. Serone, G. Spada and G. Villadoro, λϕ4 Theory I: The Symmetric Phase Beyond NNNNNNNNLO, JHEP 08 (2018) 148, [ 1805.05882]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[42]

$\lambda \phi^4$ Theory II: The Broken Phase Beyond NNNN(NNNN)LO

M. Serone, G. Spada and G. Villadoro, λϕ4 2 theory II: The broken phase beyond NNNN(NNNN)LO , JHEP 05 (2019) 047, [ 1901.05023]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[43]

Compelling evidence of renormalons in QCD from high order perturbative expansions

C. Bauer, G. S. Bali and A. Pineda, Compelling Evidence of Renormalons in QCD from High Order Perturbative Expansions, Phys. Rev. Lett. 108 (2012) 242002, [ 1111.3946]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[44]

G. S. Bali, C. Bauer and A. Pineda, Perturbative expansion of the plaquette to O(α35) in four-dimensional SU(3) gauge theory , Phys. Rev. D 89 (2014) 054505, [ 1401.7999]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[45]

Structure functions of the 2d O(n) non-linear sigma models

J. Balog and P. Weisz, Structure functions of the 2d O(n) non-linear sigma models , Nucl. Phys. B 709 (2005) 329–380, [ hep-th/0409095]

work page internal anchor Pith review Pith/arXiv arXiv 2005
[46]

Mari˜ no, R

M. Mari˜ no, R. Miravitllas and T. Reis, Instantons, renormalons and the theta angle in integrable sigma models, SciPost Phys. 15 (2023) 184, [ 2205.04495]

work page arXiv 2023
[47]

Bajnok, J

Z. Bajnok, J. Balog and I. Vona, Wiener-Hopf solution of the free energy TBA problem and instanton sectors in the O(3) sigma model , JHEP 11 (2024) 093, [ 2404.07621]. – 39 –

work page arXiv 2024