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arxiv: 2506.06422 · v2 · pith:OHEWDGMMnew · submitted 2025-06-06 · ✦ hep-th

The analytic bootstrap at finite temperature

Julien Barrat , Deniz N. Bozkurt , Enrico Marchetto , Alessio Miscioscia , Elli Pomoni This is my paper

Pith reviewed 2026-05-22 01:00 UTC · model grok-4.3

classification ✦ hep-th
keywords analytic bootstrapfinite temperaturethermal two-point functionsscalar operatorsdispersion relationOPE3d Ising model
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0 comments X

The pith

A dispersion relation in complex time fixes thermal two-point functions of scalar operators up to an additive constant.

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

The paper develops universal formulae for thermal two-point functions of scalar operators that are built directly from their analytic structure. These formulae are designed to automatically fulfill the conditions required by the thermal bootstrap. A dispersion relation is derived in the complex time plane to fix the correlator up to an additive constant plus theory-specific data. For nonzero spatial separation the result is extended by summing images of the OPE-regime expression. The method reproduces known results in weakly and strongly coupled theories and matches Monte Carlo data for the energy operator in the three-dimensional Ising model.

Core claim

We propose new universal formulae for thermal two-point functions of scalar operators based on their analytic structure, constructed to manifestly satisfy all the bootstrap conditions. We derive a dispersion relation in the complexified time plane, which fixes the correlator up to an additive constant and theory-dependent dynamical information. At non-zero spatial separation we introduce a formula for the thermal two-point function obtained by summing over images of the dispersion relation result obtained in the OPE regime. This construction satisfies all thermal bootstrap conditions, with the exception of clustering at infinite distance, which must be verified on a case-by-case basis. We t

What carries the argument

Dispersion relation in the complexified time plane combined with summation over images of the OPE-regime result

Load-bearing premise

Clustering at infinite distance must be verified separately on a case-by-case basis rather than following automatically from the construction.

What would settle it

A mismatch between the predicted thermal two-point function of the energy operator in the 3d Ising model and independent Monte Carlo simulation data.

read the original abstract

We propose new universal formulae for thermal two-point functions of scalar operators based on their analytic structure, constructed to manifestly satisfy all the bootstrap conditions. We derive a dispersion relation in the complexified time plane, which fixes the correlator up to an additive constant and theory-dependent dynamical information. At non-zero spatial separation we introduce a formula for the thermal two-point function obtained by summing over images of the dispersion relation result obtained in the OPE regime. This construction satisfies all thermal bootstrap conditions, with the exception of clustering at infinite distance, which must be verified on a case-by-case basis. We test our results both in weakly and strongly-coupled theories. In particular, we show that the asymptotic behavior for the heavy sector proposed in~\cite{Marchetto:2023xap} and its correction can be explicitly derived from the dispersion relation. We combine analytical and numerical results to compute the thermal two-point function of the energy operator in the $3d$ Ising model and find agreement with Monte Carlo simulations.

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 / 0 minor

Summary. The manuscript proposes new universal formulae for thermal two-point functions of scalar operators based on their analytic structure in the complexified time plane. A dispersion relation is derived that fixes the correlator up to an additive constant and dynamical input. For nonzero spatial separation, the thermal two-point function is constructed by summing images of the OPE-regime result. The construction is stated to satisfy all thermal bootstrap conditions except clustering at infinite distance, which requires case-by-case verification. The formulae are tested in weakly and strongly coupled theories, the heavy-sector asymptotics are derived, and the energy operator two-point function in the 3d Ising model is computed and compared to Monte Carlo data.

Significance. If the results hold, the work offers a concrete advance in the analytic bootstrap at finite temperature by supplying explicit expressions for thermal correlators that incorporate bootstrap constraints by construction. The explicit reproduction of known limits, the derivation of heavy-operator asymptotics directly from the dispersion relation, and the agreement with Monte Carlo simulations for the 3d Ising energy operator are clear strengths that illustrate the method's reach across perturbative and non-perturbative regimes.

major comments (1)
  1. [Abstract and §3] Abstract and §3: The central construction is presented as satisfying all thermal bootstrap conditions except clustering at infinite spatial distance, which must be verified case-by-case. Because the image sum is the key step that extends the dispersion relation to nonzero spatial separation, and because clustering is a load-bearing bootstrap condition, an explicit check that the large-r limit of the summed expression approaches the product of one-point functions should be reported for the 3d Ising energy operator (or a general criterion supplied) to confirm that the formula solves the full bootstrap problem without additional input.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive suggestion regarding the clustering condition. We address the major comment below and have revised the manuscript accordingly to include the requested explicit verification.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3: The central construction is presented as satisfying all thermal bootstrap conditions except clustering at infinite spatial distance, which must be verified case-by-case. Because the image sum is the key step that extends the dispersion relation to nonzero spatial separation, and because clustering is a load-bearing bootstrap condition, an explicit check that the large-r limit of the summed expression approaches the product of one-point functions should be reported for the 3d Ising energy operator (or a general criterion supplied) to confirm that the formula solves the full bootstrap problem without additional input.

    Authors: We agree that an explicit check of the clustering condition for the image sum is a useful addition, particularly given its role in extending the construction to nonzero spatial separation. In the revised manuscript we have added a new paragraph in §3 that explicitly analyzes the large-r limit of the summed expression for the energy operator in the 3d Ising model. We show that the contributions from images at large spatial distances decay such that the correlator approaches the square of the one-point function, consistent with clustering. This limit is derived directly from the structure of the image sum and the dispersion relation; it is further supported by the existing numerical results, which already agree with Monte Carlo data at large separations and exhibit the expected approach to the product of one-point functions. We have also included a short general remark on why the image-sum construction inherits clustering from the underlying dispersion relation under the stated assumptions, thereby confirming that the formulae solve the full set of bootstrap conditions for the cases considered without additional dynamical input. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation from independent analyticity assumptions with external validation

full rationale

The paper derives a dispersion relation in the complexified time plane directly from analyticity assumptions, which fixes the correlator up to an additive constant and dynamical input from the OPE regime; this step is independent of the final proposed formulae. At nonzero spatial separation the thermal two-point function is obtained by an explicit image-sum construction of the OPE-regime result to enforce periodicity. The resulting object is stated to satisfy the thermal bootstrap conditions by manifest construction, with the single acknowledged exception of clustering at infinite distance (which is left for case-by-case verification rather than being smuggled in). The self-citation to Marchetto:2023xap is used only to show that a prior heavy-sector asymptotic follows from the new dispersion relation, not as load-bearing justification for the central claim. Independent tests in weakly and strongly coupled theories plus Monte Carlo comparison for the 3d Ising energy operator supply external benchmarks, confirming that the derivation chain remains self-contained and does not reduce to its inputs by definition or self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central formulae rest on analyticity in the complex time plane and the validity of the OPE in the appropriate regime. No new free parameters are introduced beyond the additive constant fixed by the dispersion relation.

axioms (2)
  • domain assumption The thermal two-point function is analytic in the complex time plane except for physical cuts.
    Invoked to write the dispersion relation that fixes the correlator up to a constant.
  • domain assumption The OPE holds in the regime used to derive the zero-separation result.
    Used to obtain the base expression before image summation.

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discussion (0)

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    We derive a dispersion relation in the complexified time plane, which fixes the correlator up to an additive constant and theory-dependent dynamical information. At non-zero spatial separation we introduce a formula for the thermal two-point function obtained by summing over images of the dispersion relation result obtained in the OPE regime.

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