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 ${\alpha }_i=b{\sigma }_i$ as $b\to 0$. 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 $b^2$: $$C(b{\sigma }_1,b{\sigma }_2,b{\sigma }_3)=\mathcal{P}(b;{\sigma }_i)[1+\sum _{n\ge 1}{b}^{2n}{\mathrm{\Omega }}_n({\sigma }_1,{\sigma }_2,{\sigma }_3)]\mathrm{.}$$ Using Thorn’s asymptotic expansion of the ${\mathrm{{\rm Y}}}_b$-function we derive closed-form expressions for the leading coefficients ${\mathrm{\Omega }}_n({\sigma }_i)$ and show that each ${\mathrm{\Omega }}_n$ is a symmetric polynomial in the variables ${\sigma }_i$. 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.