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

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.