On the perturbative expansion of exact bi-local correlators in JT gravity

We study the perturbative series associated to bi-local correlators in Jackiw-Teitelboim (JT) gravity, for positive weight $\lambda$ of the matter CFT operators. Starting from the known exact expression, derived by CFT and gauge theoretical methods, we reproduce the Schwarzian semiclassical expansion beyond leading order. The computation is done for arbitrary temperature and finite boundary distances, in the case of disk and trumpet topologies. A formula presenting the perturbative result (for $\lambda \in \mathbb{N}/2$) at any given order in terms of generalized Apostol-Bernoulli polynomials is also obtained. The limit of zero temperature is then considered, obtaining a compact expression that allows to discuss the asymptotic behaviour of the perturbative series. Finally we highlight the possibility to express the exact result as particular combinations of Mordell integrals.