  We discuss Casimir effect of a massless, minimally coupled scalar field in a
6D warped flux compactification model and its implications for the hierarchy
and cosmological constant problems, which are longstanding puzzles in
phenomenology and cosmology. Due to the scale invariance of the background
theory, the 4D effective theory contains a volume modulus. To stabilize the
modulus, we consider one-loop corrections to the effective potential by the
Casimir effect. The one-loop effective potential for the volume modulus has a
form which is very similar to Coleman-Weinberg potential. We evaluate
coefficients appearing in the effective potential by employing zeta function
regularization and heat kernel analysis. The volume modulus is stabilized for
smaller degrees of warping, below a critical value, which depends on deficit
angle of the reference brane. After stabilizing the modulus, it is possible to
obtain observed values of the mass ratio between the fundamental energy scales
and a tiny effective cosmological constant (though its sign is negative). The
degree of warping should be tuned to be close to the critical value, not as
severely as the original fine-tuning.
