Relating Dark Dimension Size with Dark Energy via 5D Quintessence Potential and Casimir Energy
We explore effective field theory (EFT) realizations of the dark dimension (DD) scenario in which the size of the DD is determined by the competition between a 5D quintessence potential and the Casimir energy. When only contributions from massless fields are considered, the Casimir energy does not have a typical energy scale, and the 4D dark energy (DE) scale is set by the height of the 5D quintessence potential. The simple 5D model we first study has an attractive feature that the size of the DD is related to the 4D vacuum energy (VE), as outlined above. However, it turns out that the 4D VE thus obtained cannot work as the 4D DE in the standard cosmological setting. One of the discrepancies between the VE and the DE manifests in the radion dependence of the effective 4D Newton's gravitational constant. This leads us to explore a 5D model in which the 4D Einstein-Hilbert (EH) action is dominantly induced by quantum effects, leaving the radion dependence of the 4D Newton's constant subdominant. This model leads to an attractive cosmological model, realizing the mechanism explained in the beginning. We also explore a 6D model with the total internal volume fixed, to make the 4D Newton's constant independent of the radion. We first study a scenario in which the sizes of the 5th and the 6th dimensions are of the same order. In this scenario, the physical origin of the size of the DD is left to its UV completion. Next, we turn to a scenario in which there is a large hierarchy between the sizes of the 5th and the 6th dimensions. In this scenario, the energy scale of the 5D quintessence potential turns out to be much higher than the energy scale of the 4D DE as inferred from the cosmological observations, so we need to subtract a constant term in 4D to tune the 4D DE. This prevents us from obtaining the scaling relation for the DE expected in the dSDC.