Quantum vacuum energy and geometry of extra dimension

We discuss the cancellation of the ultraviolet cutoff scale 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 cut-off dependent part 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 their cut-off dependence 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 the divergent part of \(\langle T_{MN}\rangle\) 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).