Does the Cosmological Constant really indicate the existence of a Dark Dimension?

Abstract: According to the "dark dimension" (DD) scenario, we might live in a universe with a single compact extra dimension, whose mesoscopic size is dictated by the measured value of the cosmological constant. This scenario is based on swampland conjectures, that lead to the relation $\rho_{\rm swamp}\sim m_{{\rm KK}}^4$ between the vacuum energy $\rho{\rm swamp}$ and the size of the extra dimension $m_{{\rm KK}}^{-1}$ ($m{{\rm KK}}$ is the mass scale of a Kaluza-Klein tower), and on the corresponding result $\rho{{\rm EFT}}$ from the EFT limit. We show that $\rho{{\rm EFT}}$ contains previously missed UV-sensitive terms, whose presence invalidates the widely spread belief (based on existing literature) that the calculation gives automatically the finite result $\rho{{\rm EFT}}\sim m{{\rm KK}}^4$ (with no need for fine-tuning). This renders the matching between $\rho{\rm swamp}$ and $\rho_{_{\rm EFT}}$ a non-trivial issue. We then comment on the necessity to find a mechanism that implements the suppression of the aforementioned UV-sensitive terms. This should finally allow to frame the DD scenario in a self-consistent framework, also in view of its several phenomenological applications based on EFT calculations.