A perturbative Liouville prescription for the celestial three-gluon amplitude

TL;DR

This paper develops a perturbative approach to celestial three-gluon amplitudes in a dilaton background using Liouville theory, providing a controlled expansion that reproduces known amplitudes and includes computable quantum corrections.

Contribution

It introduces a unique resolution to the Liouville-Mellin variable ambiguity, enabling finite-$b$ corrections and explicit quantum loop calculations in celestial amplitudes.

Findings

Leading term matches tree-level Yang-Mills amplitude

One-loop correction expressed with Bessel functions

Soft limit shows geometric and logarithmic separation

Abstract

We study the celestial three-gluon amplitude in a dilaton background through the Mellin-Liouville formulation proposed by Stieberger, Taylor and Zhu (STZ). The original map contains an ambiguity in the identification of Liouville and Mellin variables; we resolve it by requiring global conformal covariance and compatibility with the semiclassical expansion of Liouville theory. This uniquely fixes the operator normalization and the parameter dictionary, and leads to a controlled expansion in the Liouville coupling $b$. Starting from the full Liouville DOZZ three-point function, we derive the leading and first subleading terms in the $b^2$ expansion. The leading term reproduces the tree-level Yang-Mills amplitude in the small total momentum limit, as anticipated in the STZ proposal. The one-loop correction can be written in closed form using modified Bessel functions, and its soft limit exhibits a clear separation into geometric and logarithmic contributions. The resulting framework extends the STZ proposal to finite-$b$ corrections in a consistent and computable way.