Celestial String Integrands & their Expansions

We transform the one-loop four-point type $\mathrm{I}$ open superstring gluon amplitude to correlation functions on the celestial sphere including both the (non-)orientable planar and non-planar sector. This requires a Mellin transform with respect to the energies of the scattered strings, as well as to integrate over the open-string worldsheet moduli space. After accomplishing the former we obtain celestial string integrands with remaining worldsheet integrals $\Psi\left(\beta\right)$, where $\beta$ is related to the conformal scaling dimensions of the conformal primary operators under consideration. Employing an alternative approach of performing an $\alpha'$-expansion of the open superstring amplitude first and Mellin transforming afterwards, we obtain a fully integrated expression, capturing the pole structure in the $\beta$-plane. The same analysis is performed at tree-level yielding similar results. We conclude by solving $\Psi\left(\beta\right)$ for specific values of $\beta$, consistently reproducing the results of the $\alpha'$-expansion ansatz. In all approaches we find that the dependence on $\alpha'$ reduces to that of a simple overall factor of $\left(\alpha'\right)^{\beta-3}$ at loop and $\left(\alpha'\right)^{\beta}$ at tree level, consistent with previous literature.