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

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.

💡 Research Summary

This paper develops a systematic small‑b expansion of the Liouville DOZZ three‑point structure constant in the light‑operator regime, where the Liouville momenta are scaled as α_i = b σ_i with fixed σ_i while the coupling b → 0. In this limit the exact DOZZ function factorizes into a universal prefactor P(b;σ_1,σ_2,σ_3) and a power series in b²: \