Quantum effects, brane tension, and large hierarchy in the brane world

The semiclassical Einstein equation in five dimensions with a negative cosmological constant and conformally invariant bulk matter fields is examined in the brane world scenario with $S^1{/Z}_2$ compactification. When numbers $N_{b,f}$ of bosonic and fermionic fields satisfy $I<\mathrm{}{\kappa }^2{l}^{-3}{(N}_b-N_f)<\mathrm{}32I,$ we obtain an exact semiclassical solution which has two static branes with positive tension and for which the warp factor can be arbitrarily large. Here, ${\kappa }^2$ is the five-dimensional gravitational constant, l is a length scale determined by the negative five-dimensional cosmological constant, and I is a dimensionless positive constant of order unity. However, in order to obtain a large warp factor, fine-tuning of brane tensions is required. Hence, in order to solve the hierarchy problem, we need to justify the fine-tuning by another new mechanism.