⚛️ high-energy theory

Regge in Wonderland

This paper investigates the asymptotic behavior of large-spin operators in conformal field theories by mapping their three-point coefficients to thermal one-point functions on a pp-wave background, deriving predictions for high- and low-temperature regimes via thermal effective field theory and the analytic bootstrap, and validating these results in free scalar theories and the large- $N$ $\mathrm{O}(N)$ model.

Original authors: Alessio Miscioscia, Dmitrii Pavshinkin, Fedor K. Popov

Published 2026-09-29

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CC BY 4.0

Original authors: Alessio Miscioscia, Dmitrii Pavshinkin, Fedor K. Popov

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

In the vast landscape of modern physics, there exists a special class of theories known as conformal field theories. These are not descriptions of specific materials like water or steel, but rather mathematical frameworks that describe how nature behaves at the very edge of change, where systems lose their sense of scale and look the same whether you zoom in or out. Physicists rely on these theories to understand the fundamental rules governing phase transitions, such as when a magnet loses its magnetism or when a fluid becomes a superfluid. To make sense of these theories, scientists catalog a set of data: the list of all possible building blocks, called operators, and the rules for how they interact. While we have learned a great deal about the behavior of simple, light building blocks, the heavy, complex ones have remained stubbornly difficult to study. These heavy states, often carrying enormous amounts of spin or rotation, are thought to hold the key to understanding the most chaotic and strongly interacting parts of the universe, yet their internal dynamics have been largely a mystery.

A team of researchers has now found a new way to look at these heavy, spinning states by placing the theory into a very specific, curved environment. Instead of trying to calculate the interactions of these massive objects directly in flat space, they imagined the theory living on a background that resembles a ripple in spacetime known as a pp-wave. In this setting, the heavy spinning objects behave like a gas of particles sitting in a thermal bath, a state of matter defined by a specific temperature. This shift in perspective allowed the scientists to translate a difficult problem about heavy particles into a much simpler problem about how a gas of particles responds to heat. They discovered that by studying how these particles absorb and emit energy in this curved space, they could predict exactly how the original heavy operators in the flat universe would interact with lighter ones.

The researchers focused on two distinct temperature regimes to map out the behavior of these systems. In the high-temperature limit, where the thermal energy is intense, the system behaves in a way that is surprisingly simple and universal. The heavy particles act as if they are in a flat, infinite space, and their interactions follow a predictable pattern dictated by the local temperature. The team derived a precise formula for this behavior, showing that the strength of the interaction depends on the temperature and the size of the particles in a specific, calculable way. This result suggests that even in the most complex, high-energy states, there is a layer of order that can be described using standard tools of thermal physics.

In the low-temperature regime, the story becomes more intricate. Here, the behavior is dominated by the lightest possible excitations in the theory, the smallest ripples above the vacuum state. The researchers found that the interactions in this cold limit are controlled by the properties of these lightest particles, specifically their "twist," a measure of their complexity. They derived a new mathematical expression that describes how the heavy spinning states interact with light probes in this cold environment. This expression depends on the distance between the particles and the specific quantum numbers of the lightest states, providing a detailed map of the system's structure when it is not swamped by thermal noise.

To ensure their findings were not just theoretical guesses, the team tested their predictions against several known models. They applied their methods to free theories, where particles do not interact with each other, and found that their formulas perfectly matched the known results. They also tackled a more complex, interacting model known as the large-N O(N) model. Since this model is too difficult to solve exactly by hand, the researchers used powerful numerical simulations to solve the equations on the pp-wave background. The results of these simulations aligned beautifully with their theoretical predictions for both the high- and low-temperature limits. This agreement confirmed that their method of using the pp-wave background is a robust and reliable way to probe the hidden dynamics of heavy operators.

The work also extended to four-dimensional space, where the theory involves two independent directions of rotation. In this more complex setting, the researchers found that the symmetry of the space forces the thermal response to be uniform across the entire background, simplifying the problem further. They derived predictions for how the interaction strength scales with the size of the spins and the twist of the operators in both the hot and cold limits. By comparing these results with independent methods from the "analytic bootstrap," a technique that uses symmetry to constrain physical theories, they showed that their approach yields consistent and accurate descriptions of the heavy-light interactions.

Ultimately, this research provides a new bridge between the microscopic world of quantum fields and the macroscopic behavior of thermal systems. By mapping the difficult problem of heavy spinning operators onto the more familiar terrain of thermal physics on a curved background, the authors have unlocked a way to calculate previously inaccessible data. Their findings suggest that the chaotic behavior of heavy states is not random but follows strict, universal laws that can be uncovered by looking at the system through the right lens. This approach offers a powerful new tool for physicists to explore the deepest corners of quantum field theory, turning the unknown into the calculable and revealing the hidden order within the most complex states of matter.

Technical Summary: "Regge in Wonderland"

Problem Statement
Conformal Field Theories (CFTs) are characterized by their conformal data: the spectrum of local primary operators (scaling dimensions, Lorentz representations, quantum numbers) and Operator Product Expansion (OPE) coefficients. While significant progress has been made in understanding the CFT spectrum in various limits—such as the numerical bootstrap for low-dimension operators, the Cardy formula for high-energy asymptotics, and effective field theories (EFT) for large global charges—the behavior of operators with large spin remains a rich but partially unexplored territory.

Specifically, the "large-spin" regime involves taking the spin $J$ to infinity. In the conventional light-cone (or analytic bootstrap) regime, the twist $\tau = \Delta - J$ is kept small. However, as the twist increases, the semiclassical picture of weakly interacting constituents breaks down. In $2+1$ dimensions, this occurs when $\tau \sim \sqrt{J_z}$ , and in $3+1$ dimensions (with two large angular momenta $J_1, J_2$ ) when $\tau \sim (J_1 J_2)^{1/3}$ . These limits interpolate between the light-cone regime and a strongly coupled sector of fast-spinning operators. While recent work established that these limits admit a geometric description on an effective pp-wave background, the conformal data beyond the spectrum (specifically, heavy-light-heavy OPE coefficients) in this regime remained largely unknown.

Methodology
The authors utilize the emergent pp-wave geometry to study the asymptotic behavior of large-spin operators. By placing the CFT on a pp-wave background, the problem of computing three-point coefficients involving two large-spin operators and a light operator is mapped to the calculation of thermal one-point functions of the light operator on the pp-wave.

The methodology proceeds in two complementary asymptotic regimes defined by the temperature $T = 1/\beta$ on the pp-wave:

High-Temperature Limit ( $\beta \to 0$ ): Corresponds to the regime where the twist scales with an appropriate power of the spin such that $\tau \gg \sqrt{J}$ (in 2+1d) or $\tau \gg (J_1 J_2)^{1/3}$ (in 3+1d). Here, the thermal wavelength is smaller than the curvature scale of the pp-wave. The authors employ Thermal Effective Field Theory (EFT) to derive the leading behavior, assuming the local thermal state dominates.
Low-Temperature Limit ( $\beta \to \infty$ ): Corresponds to the regime where the twist is parametrically smaller than the spin ( $\tau \ll \sqrt{J}$ or $\tau \ll (J_1 J_2)^{1/3}$ ). In this limit, the thermal partition function is dominated by the lightest excitations above the vacuum. The authors perform a systematic expansion in Boltzmann factors, utilizing the specific spectrum of the pp-wave Hamiltonian and the conformal symmetry of the vacuum to compute the leading contribution.

The results derived on the pp-wave are then translated back to the original CFT on the cylinder ( $\mathbb{R} \times S^{d-1}$ ) via a Weyl rescaling and an inverse Laplace transform (or saddle-point approximation) to extract the asymptotic OPE data.

Key Contributions and Results

Universal High-Temperature Behavior:
For a scalar operator $O$ of dimension $\Delta_O$ , the thermal one-point function on the pp-wave in the high-temperature limit is determined by the local Tolman redshift. The result is:
$\langle O(y) \rangle_{pp, \beta} \sim \frac{b_O}{\beta^{\Delta_O} (1+y^2)^{\Delta_O/2}}$
where $b_O$ is the flat-space thermal one-point coefficient. This leads to a prediction for the averaged diagonal matrix elements on the cylinder in the large-twist regime, scaling as $(\tau J)^{\Delta_O/6}$ .
Universal Low-Temperature Behavior:
In the low-temperature limit, the one-point function is dominated by the lightest primary $P$ (with twist $\tau_P$ ) that couples diagonally to the probe $O$ . The authors derive a universal formula involving a hypergeometric function:
$\langle O(y) \rangle_{pp, \beta} \sim e^{-\beta \tau_P} \left(\frac{2}{\beta}\right)^{\Delta_O/2} \frac{C_{PPO}}{C_P} \frac{\Gamma(\tau_P)\Gamma(\Delta_O-1)}{\Gamma(\Delta_O/2)\Gamma(\tau_P+\Delta_O/2-1)} \times {}_2F_1\left(\dots; -\frac{2y^2}{\beta}\right)$
This result is matched to the analytic bootstrap prediction of weakly interacting partons on the cylinder, confirming the equivalence between the pp-wave thermal one-point functions and the heavy-light-heavy (HLH) correlators in the large-spin, low-twist limit.
Two Large Angular Momenta in 3+1 Dimensions:
The authors extend the analysis to CFTs with two large angular momenta $J_1, J_2$ . They identify a distinct pp-wave geometry with enhanced Heisenberg symmetry ( $H_5$ ). Crucially, they find that scalar thermal one-point functions in this background are constant in space due to the unbroken Heisenberg symmetry at finite temperature. This simplifies the high-temperature expansion and leads to specific scaling laws for the HLH correlators depending on the ratio $\tau / (J_1 J_2)^{1/3}$ .
Pseudoscalar Operators:
The paper addresses thermal one-point functions of pseudoscalar operators on the (2+1)d pp-wave. Parity forbids a homogeneous expectation value, but a contribution arises at the first derivative order via the vorticity of the thermal state. The leading high-temperature response is found to be odd in the transverse coordinate $y$ and scales as $\beta^{-(\Delta_O-1)}$ .
Explicit Verification:
The general predictions are tested in several explicit examples:
1. Free Scalar Theories: Exact calculations in 2+1 and 3+1 dimensions confirm the derived asymptotic behaviors, including the specific subleading corrections (e.g., logarithmic terms in 2+1d free scalars).
2. Large-N O(N) Model: The authors solve the large-N gap equation for the Hubbard-Stratonovich field numerically on the pp-wave background. The numerical profiles of the field and the free energy density show excellent agreement with both the high- and low-temperature analytical predictions.
3. Free Fermions: Calculations for free massless Dirac fermions confirm the vorticity-induced response for pseudoscalars.

Significance
The paper establishes a concrete computational framework for determining the asymptotic behavior of heavy-light-heavy OPE coefficients in the large-spin regime of CFTs. By mapping the problem to thermal one-point functions on an emergent pp-wave geometry, the authors provide universal predictions that bridge the gap between the light-cone (analytic bootstrap) regime and the strongly coupled sector.

The work demonstrates that the pp-wave description, previously used primarily for the spectrum, is equally powerful for computing correlation functions. The agreement between the pp-wave thermal EFT results, the analytic bootstrap predictions for weakly interacting partons, and explicit numerical solutions in the large-N O(N) model validates the geometric approach as a robust tool for exploring the "Regge" limit of conformal field theories. The derivation of space-independent one-point functions in the two-spin 3+1d case and the identification of vorticity-induced responses for pseudoscalars represent specific, non-trivial extensions of the thermal EFT paradigm to curved, rotating backgrounds.

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