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arxiv: 2606.18442 · v1 · pith:UDITZNTInew · submitted 2026-06-16 · ✦ hep-th

Two-point functions in 4-2\,varepsilon dimensions from localization

Alessandro Georgoudis , Joseph A. Minahan , Anton Nedelin , Congkao Wen This is my paper

Pith reviewed 2026-06-26 23:07 UTC · model grok-4.3

classification ✦ hep-th
keywords two-point functionshalf-BPS operatorssupersymmetric localizationepsilon expansionYang-Mills theoryconformal symmetrymatrix modelsplanar limit
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The pith

Localization on the sphere computes the ε-expansion of two-point functions for half-BPS operators in 4-2ε dimensions.

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

This paper shows how supersymmetric localization on the sphere can be used to compute perturbative expansions in ε for two-point functions of half-BPS operators in maximally supersymmetric Yang-Mills theory continued to d=4-2ε dimensions. The method produces matrix-model expressions that give all-loop results at leading order in ε in the planar limit. These results match direct flat-space perturbative calculations exactly at order ε. At higher orders the localization approach deviates because conformal symmetry is broken in non-four dimensions. For the dimension-two operator the authors use the pattern of this deviation to conjecture an all-loop expression at order ε².

Core claim

We study two-point functions of half-BPS operators in maximally supersymmetric Yang-Mills theory continued to d=4−2ε dimensions. Using supersymmetric localization on S^d, we derive perturbative matrix-model expressions for the ε-expansion of these correlators and obtain all-loop results at leading order in ε in the planar limit, with extensions to finite-N corrections and higher-charge operators. We compare the localization results with direct perturbative computations in flat space. At order ε the two descriptions agree perfectly, while at higher orders our construction fails to reproduce the perturbative data due to the breaking of conformal symmetry away from four dimensions. Nevertheless

What carries the argument

Supersymmetric localization on the d-dimensional sphere, which reduces the computation of correlators to matrix-model integrals whose ε-expansion yields the desired coefficients.

Load-bearing premise

The deviation between the sphere localization result and the flat-space perturbative result at orders beyond ε is caused entirely by the breaking of conformal symmetry, and the specific form of this deviation can be used to deduce the true flat-space value at order ε².

What would settle it

An independent calculation of the coefficient at order ε² for the two-point function of the dimension-two operator using standard flat-space Feynman diagrams, which can then be compared directly to the conjectured expression.

read the original abstract

We study two-point functions of half-BPS operators in maximally supersymmetric Yang-Mills theory continued to $d=4-2\,\varepsilon$ dimensions. Using supersymmetric localization on $S^d$, we derive perturbative matrix-model expressions for the $\varepsilon$-expansion of these correlators and obtain all-loop results at leading order in $\varepsilon$ in the planar limit, with extensions to finite-$N$ corrections and higher-charge operators. We compare the localization results with direct perturbative computations in flat space. At order $\varepsilon$ the two descriptions agree perfectly, while at higher orders our construction fails to reproduce the perturbative data due to the breaking of conformal symmetry away from four dimensions. Nevertheless, in the case of the dimension-two operator we conjecture an all-loop formula at order $\varepsilon^2$ by exploiting the precise form of the mismatch.

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

1 major / 1 minor

Summary. The paper computes two-point functions of half-BPS operators in maximally supersymmetric Yang-Mills theory continued to d=4-2ε dimensions. Using supersymmetric localization on S^d it obtains perturbative matrix-model expressions whose planar limit yields all-loop results at leading order in ε, with extensions to finite-N corrections and higher-charge operators. Direct comparison with flat-space perturbation theory shows exact agreement at O(ε). At higher orders the sphere result deviates; the deviation is attributed to explicit breaking of conformal invariance for d≠4. For the dimension-two operator the authors conjecture an all-loop formula at O(ε²) by using the precise functional form of this mismatch.

Significance. If the leading-O(ε) results are correct, the work supplies a new all-loop route to the first correction in the ε-expansion of these correlators via localization, with explicit verification against flat-space perturbation at that order. The finite-N and higher-charge extensions broaden the applicability. The O(ε²) conjecture, while interesting, rests on an assumption whose justification is not independently verified within the manuscript.

major comments (1)
  1. [Abstract, final paragraph] Abstract, final paragraph: the conjecture for the O(ε²) flat-space coefficient of the dimension-two operator is obtained by attributing the entire mismatch between the localization matrix model and flat-space perturbation to conformal-symmetry breaking and then solving for the coefficient. This procedure makes the conjectured value dependent on the discrepancy it is intended to explain; an independent derivation or additional cross-check (e.g., a direct two-loop flat-space computation at O(ε²)) would be required to substantiate the extraction.
minor comments (1)
  1. [Introduction / §2] The notation for the half-BPS operators and the precise definition of the two-point function normalization should be stated explicitly in the introduction or §2 to avoid ambiguity when comparing with the flat-space literature.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We appreciate the referee's thorough review and recognition of the significance of the leading-order results obtained via localization. We address the concern regarding the O(ε²) conjecture in detail below. We maintain that the conjecture is clearly labeled as such and is derived transparently from the observed mismatch, but we agree to a partial revision to enhance clarity in the abstract.

read point-by-point responses

  1. Referee: [Abstract, final paragraph] Abstract, final paragraph: the conjecture for the O(ε²) flat-space coefficient of the dimension-two operator is obtained by attributing the entire mismatch between the localization matrix model and flat-space perturbation to conformal-symmetry breaking and then solving for the coefficient. This procedure makes the conjectured value dependent on the discrepancy it is intended to explain; an independent derivation or additional cross-check (e.g., a direct two-loop flat-space computation at O(ε²)) would be required to substantiate the extraction.

    Authors: We thank the referee for highlighting this point. The conjecture is indeed based on the assumption that the discrepancy between the localization result and flat-space perturbation theory at O(ε²) is solely due to the breaking of conformal symmetry on the sphere for d ≠ 4. This assumption is supported by the perfect agreement at O(ε), where conformal invariance is preserved in the ε-expansion, and by the fact that the mismatch appears only at higher orders in ε. While we agree that an independent computation, such as a direct two-loop flat-space calculation, would provide valuable confirmation, performing such a computation is a substantial undertaking that is outside the scope of this paper. The manuscript presents the result explicitly as a conjecture, and the derivation is fully detailed so that readers can assess its validity. To address the referee's concern, we will revise the abstract to more explicitly state that the conjecture relies on this attribution of the mismatch. revision: partial

Circularity Check

1 steps flagged

O(ε²) conjecture for dimension-two operator constructed directly from observed mismatch

specific steps
  1. fitted input called prediction [Abstract]
    "Nevertheless, in the case of the dimension-two operator we conjecture an all-loop formula at order ε² by exploiting the precise form of the mismatch."

    The conjectured all-loop formula at O(ε²) is obtained by directly using the observed mismatch between the localization matrix-model result and the flat-space perturbative data. The coefficient is therefore extracted from the discrepancy itself (under the assumption that the entire mismatch arises from conformal-symmetry breaking), rendering the conjecture a fit to the difference rather than an independent prediction.

full rationale

The paper obtains all-loop leading-O(ε) results from the S^{4-2ε} localization matrix model and demonstrates exact agreement with flat-space perturbation at O(ε). The O(ε²) conjecture is presented separately and is obtained by using the functional form of the deviation between the two computations, under the assumption that this deviation is caused solely by explicit breaking of conformal invariance. This step makes the conjectured coefficient dependent on the discrepancy it is intended to correct rather than on an independent derivation or additional data.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claims rest on standard assumptions of supersymmetric localization and dimensional continuation whose details are not visible here.

pith-pipeline@v0.9.1-grok · 5678 in / 1277 out tokens · 31298 ms · 2026-06-26T23:07:23.670343+00:00 · methodology

discussion (0)

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Reference graph

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