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

Abstract

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 $I<{\ensuremath{\kappa}}^{2}{l}^{\ensuremath{-}3}{(N}_{b}\ensuremath{-}{N}_{f})<32I,$ 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, ${\ensuremath{\kappa}}^{2}$ 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.