⚛️ high-energy theory

Analytic thermal bootstrap in momentum space: From thermal OPE to QNMs

This paper establishes an analytic thermal bootstrap framework that connects ultraviolet thermal OPE data to infrared observables like quasinormal modes by constructing meromorphic completions of thermal blocks, deriving inversion formulae and sum rules, and validating these results across various theories including free models, CFTs, and $\mathcal{N}=4$ SYM.

Original authors: Julien Barrat, Deniz N. Bozkurt, Enrico Marchetto, Alessio Miscioscia, Elli Pomoni

Published 2026-07-29

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CC BY 4.0

Original authors: Julien Barrat, Deniz N. Bozkurt, Enrico Marchetto, Alessio Miscioscia, Elli Pomoni

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible orchestra playing a symphony of particles. In the world of quantum physics, scientists try to understand this music by listening to how particles interact. Usually, they study these interactions in a quiet, cold room where the temperature is absolute zero. But in the real world, everything is warm. When you heat up a system, the particles start dancing wildly, creating a "thermal" noise that changes the music entirely. This is the realm of Quantum Field Theory at finite temperature.

To make sense of this chaotic thermal dance, physicists use two main tools. The first is the Operator Product Expansion (OPE). Think of this as a recipe book. If you look at two particles very close together, the recipe tells you exactly what other particles they are "made of" or transforming into at that tiny distance. It's a precise, short-distance rule. The second tool is the Retarded Correlator, which is like a microphone recording how the system reacts to a shout. It tells us how a disturbance travels through the hot soup of particles over time and space. The big mystery has been: How do we connect the tiny, precise recipe (the OPE) to the big, messy sound of the reaction (the correlator)? For a long time, these two tools lived in separate worlds, and bridging them was like trying to translate a microscopic blueprint into a macroscopic city map without losing any details.

This paper, titled "Analytic thermal bootstrap in momentum space," is a bold attempt to build that bridge. The authors, a team of physicists from DESY and Stony Brook, have initiated a "bootstrap" program. In physics, "bootstrapping" means pulling yourself up by your own bootstraps—using a few known rules to deduce everything else without needing to know every single detail of the universe. Here, they take the short-distance "recipe" data (the thermal OPE) and use it to predict the long-distance, low-frequency behavior of the system, specifically looking for Quasinormal Modes (QNMs). You can think of QNMs as the "ringing tones" or natural frequencies of a black hole or a hot plasma, much like the specific note a bell makes when you strike it.

The paper's main finding is that they successfully constructed a mathematical "inversion formula." This is a special equation that acts like a decoder ring. If you feed it the ringing tones (the QNMs) of a thermal system, it spits out the recipe ingredients (the OPE coefficients). Conversely, if you know the ingredients, it can predict the ringing tones. They demonstrated that this works by testing it on several known systems, including free theories, the O(N) model (a famous model for magnetism), and the 3D Ising model (which describes how magnets flip). In the case of the 3D Ising model, they compared their predictions against massive computer simulations (Monte Carlo data) and found that their "truncated" recipe matched the simulation results very well, especially in the region where the particles are close together.

However, the authors are careful to note that this is a work in progress. They explicitly rule out the idea that the "arc contribution"—a mathematical term for the messy, non-repeating parts of the signal that don't fit the neat recipe—can be ignored or easily calculated from the recipe alone in all cases. In fact, they show that for some systems, this "arc" part is uniquely fixed by the rules of the game, but in others, it remains a separate piece of the puzzle. They also clarify that while their method works beautifully for systems with a finite number of ringing tones (like the large-N O(N) model), most real-world systems have an infinite, complex tower of tones, making the math much harder.

The paper suggests that this new method opens a door to understanding the deep structure of hot quantum systems. By treating the thermal correlator as a sum of "Thermal Polyakov blocks" (which are like the individual musical notes that make up the thermal song), they can reorganize the chaos into something predictable. They tested their ideas on the 3D Ising model and found good agreement, but they admit that for more complex, strongly interacting systems, the full picture is still being assembled. They also point out that while their method works great for "UV" (ultraviolet, or high-energy/short-distance) data, using it to predict "IR" (infrared, or low-energy/long-distance) behavior is tricky and requires careful handling of the "deep IR" regime, which they explored using toy models.

In essence, this paper doesn't claim to have solved the entire mystery of hot quantum matter. Instead, it provides a powerful new set of tools and a clear roadmap. It shows that the short-distance rules and the long-distance ringing tones are deeply connected, and that by understanding one, we can start to reconstruct the other. It's a significant step forward in the "thermal bootstrap," turning a vague hope into a concrete mathematical strategy that other physicists can now use to explore the hot, noisy universe.

Technical Summary: Analytic Thermal Bootstrap in Momentum Space

Problem Statement
Thermal correlation functions in quantum field theories (QFTs) are constrained by the interplay between their analytic structure, the local Operator Product Expansion (OPE), and the Kubo–Martin–Schwinger (KMS) condition. While recent progress in the thermal conformal bootstrap has successfully utilized OPE data to constrain thermal observables, these results have largely been confined to the short-distance (UV) regime where the OPE converges. A significant gap remains in understanding the relation between this ultraviolet data and infrared (IR) observables, specifically the low-frequency behavior and quasinormal modes (QNMs) of the retarded correlator. Furthermore, standard momentum-space OPEs are typically understood only as asymptotic expansions at large Euclidean momenta, lacking a globally defined decomposition valid at arbitrary spatial momentum $k$ . This paper initiates a bootstrap program to bridge this gap, aiming to relate UV thermal OPE data to IR spectral data (QNMs) and to construct a momentum-space decomposition of thermal correlators.

Methodology
The authors develop a formalism based on the "thermal Polyakov blocks," which are KMS-symmetric completions of individual thermal OPE blocks. The methodology proceeds through several key steps:

Thermal Polyakov Blocks in Position Space: Starting from the Euclidean two-point function, the authors utilize a generalized method of images to construct thermal Polyakov blocks. These blocks are defined as sums of images of standard thermal OPE blocks, ensuring they satisfy the KMS condition and the clustering property. This decomposition reorganizes the short-distance expansion into building blocks compatible with the global structure of the thermal correlator.
Fourier Transform to Momentum Space: The authors explicitly compute the Fourier transform of these thermal Polyakov blocks. This yields a representation of the retarded thermal correlator $g_R(\omega, k)$ as a sum over operators involving Gegenbauer polynomials (or Chebyshev polynomials in 2d) and a specific kinematic factor dependent on $\omega$ and $k$ . Crucially, this decomposition is valid for any spatial momentum $k$ , though the sum over operators is generally an asymptotic series.
Inversion Formulae: By exploiting the analytic structure of the retarded correlator in the complex frequency plane, the authors derive inversion formulae. These formulae allow for the extraction of thermal OPE coefficients directly from the retarded correlator.
- Zero Momentum ( $k=0$ ): The inversion formula relates OPE coefficients to the spectrum of QNMs ( $\omega_n$ ) and their residues ( $r_n$ ) via a Mellin-Barnes representation. This leads to a function $Z(u) = \sum r_n \omega_n^{-u}$ , whose analytic properties encode the thermal OPE data.
- Non-Zero Momentum ( $k \neq 0$ ): The inversion formula is generalized to arbitrary $k$ by projecting onto Gegenbauer polynomials and performing a Mellin transform with respect to the radial momentum variable. This resolves the spin dependence of the exchanged operators.
Analytic Assumptions: The derivation assumes the retarded correlator is meromorphic (possessing only isolated poles corresponding to QNMs) in the complex frequency plane. The authors also discuss extensions to non-meromorphic cases involving branch cuts.

Key Contributions and Results

Thermal Polyakov Blocks in Momentum Space: The paper provides the first explicit construction of thermal Polyakov blocks in momentum space, valid for arbitrary spatial momentum. This allows for a decomposition of the retarded correlator that extends the regime of validity of the OPE beyond the strict large-momentum limit.
Universal Asymptotic Formulae for Heavy Operators: As a byproduct of the position-space analysis, the authors derive universal asymptotic formulae for thermal OPE coefficients of heavy operators (double-twist operators). These formulae resolve the dependence on spin and scaling dimension, refining previous Tauberian approximations.
UV-IR Connection via Inversion Formulae:
- Sum Rules and Constraints: Under the meromorphicity assumption, the inversion formulae establish a direct link between UV OPE coefficients and the IR QNM spectrum. Specifically, the function $Z(u)$ must satisfy vanishing conditions at negative integers (unless associated with conformal anomalies) and possess poles at locations determined by operator dimensions. This generates an infinite set of sum rules constraining the QNM frequencies and residues.
- Asymptotic Distribution: For theories with an infinite tower of QNMs, the inversion formulae constrain the asymptotic distribution of QNM frequencies and residues at large mode numbers, relating them to the spectrum of non-integer dimension operators in the thermal OPE.
Applications and Benchmarks: The authors test their formalism in several explicit examples:
- Free Theories and Large- $N$ $O(N)$ Model: The inversion formulae correctly reproduce known OPE coefficients and the finite number of QNMs (or specific towers) in these solvable limits.
- 3d Ising CFT: Using numerical bootstrap data for light operators, the authors construct truncated thermal Polyakov block expansions. The resulting correlators show excellent agreement (<5% discrepancy) with Monte Carlo simulation data within the OPE regime, validating the approach for non-trivial interacting theories.
- $\mathcal{N}=4$ SYM and Holography: The formalism is applied to the R-current correlator in strongly coupled $\mathcal{N}=4$ SYM, successfully reproducing OPE coefficients (including those related to conformal anomalies) from the exact holographic QNM spectrum.
- Deep IR Analysis: The authors explore inverting deep IR (hydrodynamic-like) toy models. While exact UV OPE coefficients are not recovered, the inversion yields results with physically reasonable quantum numbers, suggesting the formulae can probe the full correlator structure beyond the UV limit.

Significance
The paper claims to initiate a new phase in the thermal conformal bootstrap by establishing a rigorous analytic link between ultraviolet data (thermal OPE) and infrared observables (QNMs and low-frequency behavior) in momentum space. By constructing momentum-space thermal Polyakov blocks, the authors provide a tool to reorganize thermal correlators that respects global KMS symmetry and analyticity. The derived inversion formulae offer a new strategy to constrain the QNM spectrum of thermal theories using OPE data, and conversely, to predict the asymptotic behavior of QNMs from the operator spectrum. This work extends the reach of bootstrap methods beyond the short-distance regime and provides a framework for analyzing thermal correlators in both holographic and non-holographic contexts, including critical statistical mechanics models like the 3d Ising model. The authors note that while the current analysis focuses on scalar correlators and meromorphic structures, the framework is designed to be generalized to include branch cuts, line defects, and hydrodynamic modes in future work.

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