Vacuum polarization in an anti-de Sitter space as an origin for a cosmological constant in a brane world

We show that the vacuum polarization of quantum fields in an anti-de Sitter space can naturally give rise to a small but nonzero cosmological constant in a brane world living in it. To explain the extremely small ratio of mass density in the cosmological constant to the Planck mass density in our universe ( approximate to 10(-123)) as suggested by cosmological observations, all we need is a four-dimensional brane world (our universe) living in a five-dimensional anti-de Sitter space with a curvature radius r(0) similar to 10(-3) cm and a fundamental Planck energy M-p - 10(9) GeV, and a scalar field with a mass m similar to r(0)(-1) similar to 10(-2) eV. Probing gravity down to a scale - 10(-3) cm, which is attainable in the near future, will provide a test of the model.