Small-$b$ expansion of the DOZZ formula for light operators
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Ferrari, Franco Piatek, Marcin R. Pietrykowski, Artur R. |
| author_facet | Ferrari, Franco Piatek, Marcin R. Pietrykowski, Artur R. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_21182 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Small-$b$ expansion of the DOZZ formula for light operators Ferrari, Franco Piatek, Marcin R. Pietrykowski, Artur R. High Energy Physics - Theory Mathematical Physics 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. |
| title | Small-$b$ expansion of the DOZZ formula for light operators |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2509.21182 |