Celestial String Integrands & their Expansions

Abstract

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 ${\alpha }^{\prime }$-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 ${\alpha }^{\prime }$-expansion ansatz. In all approaches we find that the dependence on ${\alpha }^{\prime }$ reduces to that of a simple overall factor of ${\left({\alpha }^{\prime }\right)}^{\beta -3}$ at loop and ${\left({\alpha }^{\prime }\right)}^{\beta }$ at tree level, consistent with previous literature.

37 pages, 6 figures, 1 table