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 \(\alpha_i=b\sigma_i\) as \(b\to0\). In this limit, the exact DOZZ function factorizes into a prefactor \({\cal P(b;\sigma_1,\sigma_2,\sigma_3)\) and a power series in \(b^2\): \[ C(b\sigma_1,b\sigma_2,b\sigma_3)={\cal P}(b;\sigma_i)\Bigg[1+\sum_{n\ge1}b^{2n}\,\Omega_n(\sigma_1,\sigma_2,\sigma_3)\Bigg]. \] Using Thorn's asymptotic expansion of the \(\Upsilon_b\)-function we derive closed-form expressions for the leading coefficients \(\Omega_n(\sigma_i)\) and show that each \(\Omega_n\) 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.