Quantum vacuum energy and geometry of extra dimension

TL;DR

This work analyzes the UV sensitivity of the 5D energy-momentum tensor for a theory on $S^1/Z_2$, comparing bosonic and fermionic contributions across general extra-dimensional geometries. By computing the full 5D propagators for scalars and fermions, it reveals how the $y$-dependent profiles of different spins affect UV cancellation; the leading ${\cal O}(\Lambda_{\rm cut}^5)$ and ${\cal O}(\Lambda_{\rm cut}^4)$ terms cancel under Pauli-like sum rules, but in generic geometries the remaining divergences do not cancel due to mismatched $y$-dependence. In flat space, all positive-power UV terms cancel when mass relations satisfy the constraints, while in AdS slices cancellations depend on warp factors and mass tuning; in non-AdS geometries the mismatch persists, implying a possible dynamical preference for flat or AdS geometries. The finite parts reduce to Casimir-like energies that contribute to radion stabilization and to the effective 5D cosmological constant, highlighting a link between quantum vacuum structure and extra-dimensional geometry.

Abstract

We discuss the cancellation of the ultraviolet cutoff scale $Λ_{\rm cut}$ in the calculation of the expectation value of the five-dimensional (5D) energy-momentum tensor $\langle T_{MN}\rangle$ ($M,N=0,1,\cdots,4$). Since 5D fields feel the background geometry differently depending on their spins, the bosonic and the fermionic contributions to the $Λ_{\rm cut}$-dependent part $\langle T_{MN}\rangle^{\rm UV}$ may have different profiles in the extra dimension. In that case, there is no chance for them to be cancelled with each other. We consider arbitrary numbers of scalar and spinor fields with arbitrary bulk masses, calculate $\langle T_{MN}\rangle$ using the 5D propagators, and clarify the dependence of $\langle T_{MN}\rangle^{\rm UV}$ on the extra-dimensional coordinate $y$ for a general background geometry of the extra dimension. We find that if the geometry is not flat nor (a slice of) anti-de Sitter (AdS) space, it is impossible to cancel $\langle T_{MN}\rangle^{\rm UV}$ between the bosonic and the fermionic contributions. This may suggest that the flat (or AdS) space is energetically favored over the other geometries, and thus the dynamics forces the compact space to be flat (or AdS).