⚛️ high-energy theory

Celestial Regge theory

This paper establishes an alternative definition of the celestial Mellin transform to derive a celestial dispersion relation and defines the celestial Regge limit, thereby connecting celestial CFT data to the partial amplitudes of bulk Regge scattering.

Original authors: Eduardo Casali, Riccardo Giordana Pozzi

Published 2026-02-04

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CC BY 4.0

Original authors: Eduardo Casali, Riccardo Giordana Pozzi

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 movie screen. In physics, we usually try to understand what's happening on the screen by looking at the "bulk"—the deep, 3D space where particles zoom around and crash into each other. But there's a fascinating idea called Holography, which suggests that all the information about this 3D movie is actually encoded on a 2D screen at the very edge of the universe (like the surface of a sphere).

This paper, titled "Celestial Regge Theory," is about figuring out how to translate the chaotic, high-speed collisions happening in the 3D bulk into a calm, organized pattern on that 2D screen.

Here is the breakdown of their discovery using simple analogies:

1. The Two Languages: Bulk vs. Sky

The Bulk (The 3D Movie): Physicists usually describe particle collisions using energy and momentum. It's like describing a car crash by measuring the speed of the cars and the force of the impact.
The Celestial Sphere (The 2D Screen): The authors are translating this into a "Celestial" language. Instead of speed, they describe the crash using angles and positions on a sphere (like the sky). This is similar to how a 2D shadow can tell you about a 3D object.

2. The Problem: The "Regge" Limit

In physics, there is a specific scenario called the Regge limit. Imagine two cars crashing at incredibly high speeds, but they just barely miss each other, grazing past at a very shallow angle.

In the 3D world, this is a messy, high-energy situation that is hard to calculate.
The authors wanted to know: What does this high-speed, grazing collision look like on the 2D sky screen?

3. The Magic Trick: The "Contour Trick"

To translate the 3D crash to the 2D screen, the authors use a mathematical tool called a Mellin transform. Think of this as a special translator or a prism that turns a white light beam (the messy 3D data) into a rainbow (the organized 2D pattern).

Usually, this translation is very hard to do because the 3D data is complicated. The authors' main innovation is a "Contour Trick."

The Analogy: Imagine you are trying to count the number of people in a crowded room by looking at a single door. It's hard. But if you could magically draw a loop around the whole room and count everyone who crosses that loop, it becomes easier.
The Trick: The authors found a way to redraw the mathematical "loop" (contour) around the problem. Instead of trying to calculate the messy details directly, they moved the loop to pick up only the most important "poles" (the main actors in the play) and "discontinuities" (the sudden jumps in the story).
The Result: This gave them a new, simpler formula. It's like realizing that instead of tracking every single car in a traffic jam, you only need to look at the traffic lights and the main intersections to understand the flow.

4. The Discovery: A "Dispersion Relation"

Using their trick, they derived a Celestial Dispersion Relation.

The Metaphor: Think of a drum. If you hit it, the sound you hear (the vibration) is determined by the tension of the drum skin and the shape of the drum.
The Finding: The authors showed that the pattern on the 2D sky screen (the "sound") is directly determined by the specific "poles" and "jumps" in the 3D collision data (the "tension"). You don't need to know every tiny detail of the collision; just knowing the main "resonances" (the poles) is enough to reconstruct the whole picture on the sky.

5. Connecting the Dots: The "Regge Limit" on the Sky

Finally, they compared their new 2D sky pattern to what is already known about 2D patterns in standard physics (Conformal Field Theory).

They found that the high-speed grazing collisions in the 3D bulk look exactly like a specific type of pattern on the 2D sky, governed by a set of rules called "Regge theory."
They successfully matched the "ingredients" of the 3D crash (the partial amplitudes) with the "ingredients" of the 2D pattern (the conformal data).

Summary

In short, the authors built a mathematical bridge. They took a very difficult, high-energy physics problem (particles crashing at the edge of the universe) and used a clever "loop-drawing" trick to translate it into a clean, understandable pattern on a 2D sphere. They proved that the chaotic 3D world and the organized 2D sky are two sides of the same coin, and they gave us the exact dictionary to translate between them.

Technical Summary: Celestial Regge Theory

Problem Statement
The paper addresses the challenge of defining and understanding the holographic dual of flat space scattering amplitudes, specifically within the framework of Celestial Conformal Field Theory (CCFT). While the AdS/CFT correspondence is well-established, flat space holography lacks a general construction from string theory. The authors aim to bridge the gap between bulk scattering amplitudes in Minkowski space and celestial correlators on the sphere at null infinity. A central difficulty lies in translating the standard Regge limit (high-energy, fixed momentum transfer) of bulk amplitudes into the language of celestial CFT, particularly given the distributional nature of celestial correlators and the specific kinematic constraints imposed by momentum conservation.

Methodology
The authors employ a bottom-up approach, leveraging the analytic properties of scattering amplitudes and the $SL(2, \mathbb{C})$ symmetry of the celestial sphere. The core methodological innovation is the introduction of an alternative, yet equivalent, definition of the Mellin transform used to obtain celestial correlation functions.

Contour Trick: Instead of the standard Mellin transform integral over positive real frequencies ( $\omega$ ), the authors utilize Cauchy's theorem to rewrite the function in terms of a contour integral in the complex plane. By deforming the contour to encircle the positive real axis (or negative real axis depending on the kinematic channel), they derive a representation where the Mellin transform is effectively performed, leaving an integral over the complex Mandelstam variable (or a related spectral parameter) that encodes the analytic structure (poles and discontinuities) of the bulk amplitude.
Regge Limit Analysis: The authors analyze the bulk amplitude in the Regge limit ( $s \to \infty$ at fixed $t$ ) using the Sommerfeld-Watson transform and Froissart-Gribov expansion. They separate contributions by signature (even and odd spins) and identify the leading behavior governed by Regge poles $j(t)$ .
Mapping to Celestial Sphere: Using the contour trick, they apply the Mellin transform to the Regge-limit amplitude. This allows them to express the celestial correlator directly in terms of the Regge pole residues and the Regge trajectory, bypassing the need for explicit partial wave coefficients in the intermediate steps.
Comparison with CFT: The derived celestial expression is compared to the known Regge limit of standard Euclidean CFTs. By analytically continuing to Lorentzian signature and matching the functional forms, the authors establish a dictionary between bulk Regge data (residues $\rho_\pm$ ) and celestial CFT data (OPE coefficients $\sigma_\pm$ ).

Key Contributions and Results

Alternative Mellin Definition: The paper provides a new integral representation for the celestial correlator $g(\beta, z)$ :
$g(\beta, z) = \frac{2^{-3-\beta}\pi}{\sin(\pi\beta/2)z^2} \left( \sum_{\mu_i} \text{Res}[(\dots)] + \sum_{C_i} \int_{C_i} \dots \text{Disc}_\mu[T(z, \mu)] \right)$
This formulation explicitly relates the reduced celestial correlator to the poles and discontinuities of the bulk amplitude, establishing a "celestial dispersion relation."
Celestial Regge Limit: The authors derive the leading behavior of the celestial four-point function in the Regge limit. For the cross-ratio $z = e^{i\theta}\zeta$ (with $\zeta \to 0$ ), the correlator behaves as:
$f^\pm_{\Delta_i}(z, \bar{z}) \approx -2\pi\delta(\theta) \int_{-\infty}^{+\infty} d\nu \, e^{i\pi j(\nu)} \alpha^\pm(\nu) \zeta^{1-j(\nu)}$
where the coefficients $\alpha^\pm(\nu)$ are determined by the Regge pole residues $\rho_\pm$ and the trajectory $j(\nu)$ .
Bulk-Boundary Dictionary: By comparing the celestial Regge result with the standard CFT Regge limit (which involves conformal blocks and OPE coefficients $\sigma_\pm$ ), the authors identify a precise relationship between bulk partial wave coefficients and boundary conformal data:
$\sigma_\pm(\nu) = i\pi e^{2\pi i j(\nu)} 2^{2j(\nu)-\beta-5} \Delta(\nu)^{\beta/2-1} \frac{e^{\pi i \frac{\Delta_{12}-\Delta_{34}}{2}}}{\sin(\pi\beta/2)} K_{\Delta(\nu), j(\nu)} \rho_\pm(\nu)$
Here, $\Delta(\nu) = 1 + i\nu$ , and $K$ represents a normalization factor related to the conformal blocks.
Froissart-Gribov on the Celestial Sphere: The paper extends the Froissart-Gribov expansion to the celestial sphere, showing that the celestial correlator can be expanded in terms of contour-integrated partial amplitudes, providing a bulk origin for celestial data.

Significance and Claims
The paper claims that this work provides a "celestial dispersion relation" that connects the celestial correlator directly to the analytic structure of the bulk amplitude without requiring explicit knowledge of the full amplitude's functional form, only its singularities. The primary significance lies in:

Defining the Celestial Regge Limit: It establishes a rigorous definition of the Regge limit for celestial CFTs, analogous to the standard CFT case but adapted to the specific kinematics of flat space scattering (where the cross-ratio is constrained to the real line).
Unifying Bulk and Boundary Data: It explicitly identifies the celestial CFT data (OPE coefficients) in terms of the bulk Regge data (partial amplitudes and residues), offering a concrete mechanism for how bulk scattering dynamics manifest in the dual celestial theory.
Methodological Utility: The "contour trick" for the Mellin transform is presented as a useful tool that may be applicable in other contexts, allowing for the manipulation of celestial amplitudes while keeping dynamical information implicit in the complex plane.

The authors remain modest, noting that while the results are derived for massless scalars, the extension to particles with spin is qualitatively similar, differing mainly by tensor structures. They do not claim to solve the general problem of flat space holography but rather provide a specific, calculable link between Regge physics in the bulk and conformal data on the boundary.