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Small-$b$ expansion of the DOZZ formula for light operators

Franco Ferrari, Marcin R. Piatek, Artur R. Pietrykowski

TL;DR

The paper develops a systematic small-$b$ expansion of the Liouville DOZZ three-point constant for light operators, revealing a factorized form with a universal prefactor and a power-series of quantum corrections $Ω_n(σ)$. Using Thorn's small-$b$ expansion of $Υ_b$, it derives explicit expressions for the leading coefficients, notably $Ω_0=1$, $Ω_1=-2γ(Σ_i σ_i-1)$, $Ω_2=2γ^2(Σ_i σ_i-1)^2$, and a $Ω_3$ term containing $ amily{ζ}(3)$, with each $Ω_n$ a symmetric polynomial in the $σ_i$. This framework connects the semiclassical Liouville three-point function to a perturbative, loop-like expansion in celestial holography, providing data and bootstrap constraints for celestial amplitudes. It also outlines a concrete program to realize the construction via a Mellin–Liouville map, aiming to produce a controlled celestial three-gluon amplitude expansion and to extend Liouville data to loop-level celestial observables.

Abstract

We present a systematic small-$b$ expansion of the Liouville DOZZ three-point structure constant in the light-operator regime \(α_i=bσ_i\) as \(b\to0\). In this limit, the exact DOZZ function factorizes into a prefactor \({\cal P(b;σ_1,σ_2,σ_3)\) and a power series in \(b^2\): \[ C(bσ_1,bσ_2,bσ_3)={\cal P}(b;σ_i)\Bigg[1+\sum_{n\ge1}b^{2n}\,Ω_n(σ_1,σ_2,σ_3)\Bigg]. \] Using Thorn's asymptotic expansion of the \(Υ_b\)-function we derive closed-form expressions for the leading coefficients \(Ω_n(σ_i)\) and show that each \(Ω_n\) is a symmetric polynomial in the variables \(σ_i\). Our expansion provides explicit perturbative corrections to the semiclassical Liouville three-point function and therefore supplies a practical tool for applications in celestial holography, in particular, for generating loop-level corrections to the tree-level three-gluon scattering amplitude. Finally, we formulate a perturbative Liouville program for celestial amplitudes and outline directions for further development.

Small-$b$ expansion of the DOZZ formula for light operators

TL;DR

The paper develops a systematic small- expansion of the Liouville DOZZ three-point constant for light operators, revealing a factorized form with a universal prefactor and a power-series of quantum corrections . Using Thorn's small- expansion of , it derives explicit expressions for the leading coefficients, notably , , , and a term containing , with each a symmetric polynomial in the . This framework connects the semiclassical Liouville three-point function to a perturbative, loop-like expansion in celestial holography, providing data and bootstrap constraints for celestial amplitudes. It also outlines a concrete program to realize the construction via a Mellin–Liouville map, aiming to produce a controlled celestial three-gluon amplitude expansion and to extend Liouville data to loop-level celestial observables.

Abstract

We present a systematic small- expansion of the Liouville DOZZ three-point structure constant in the light-operator regime as . In this limit, the exact DOZZ function factorizes into a prefactor \({\cal P(b;σ_1,σ_2,σ_3)\) and a power series in : \[ C(bσ_1,bσ_2,bσ_3)={\cal P}(b;σ_i)\Bigg[1+\sum_{n\ge1}b^{2n}\,Ω_n(σ_1,σ_2,σ_3)\Bigg]. \] Using Thorn's asymptotic expansion of the -function we derive closed-form expressions for the leading coefficients \(Ω_n(σ_i)\) and show that each is a symmetric polynomial in the variables . Our expansion provides explicit perturbative corrections to the semiclassical Liouville three-point function and therefore supplies a practical tool for applications in celestial holography, in particular, for generating loop-level corrections to the tree-level three-gluon scattering amplitude. Finally, we formulate a perturbative Liouville program for celestial amplitudes and outline directions for further development.

Paper Structure

This paper contains 7 sections, 1 theorem, 58 equations.

Table of Contents

  1. Introduction
  2. Weak-coupling expansion of the DOZZ formula
  3. Asymptotic expansion of the $\Upsilon_{b}$-function
  4. Semiclassical expansion of the DOZZ formula
  5. Perturbative Liouville for celestial loops
  6. Conclusions
  7. Three- and four-point DOZZ correlators on the Riemann sphere

Key Result

Proposition 2.1

The natural logarithm of the gamma function admits the following power series expansion: which is valid for all $z \in \mathbb{C}$ such that $|z-1| < 1$.See e.g.: [https://proofwiki.org/wiki/Taylor_Series_of_Logarithm_of_Gamma_Function].

Theorems & Definitions (1)

  • Proposition 2.1: Taylor series of $\operatorname{Log}(\Gamma(z))$