Quantum vacuum energy and geometry of extra dimension

Abstract

We discuss the cancellation of the ultraviolet cutoff scale $\hat{I}{›}_{\mathrm{cut}}$ in the calculation of the expectation value of the five-dimensional (5D) energy-momentum tensor $⟨T_{MN}⟩$ ($M,N=0,1,\cdots ,4$). Since 5D fields feel the background geometry differently depending on their spins, the bosonic and the fermionic contributions to the $\hat{I}{›}_{\mathrm{cut}}$-dependent part $⟨T_{MN}{⟩}^{\mathrm{UV}}$ 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 $⟨T_{MN}⟩$ using the 5D propagators, and clarify the dependence of $⟨T_{MN}{⟩}^{\mathrm{UV}}$ 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 $⟨T_{MN}{⟩}^{\mathrm{UV}}$ 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).

24 pages plus appendices, no figure