  In the present paper, we investigate regularization of the one-loop quantum
corrections with infinite Kaluza-Klein (KK) states and evaluate Casimir vacuum
energy from extra dimensions. The extra dimensional models always involve the
infinite massless or massive Kaluza-Klein states, and therefore, the
regularization of the infinite KK corrections is highly problematic. In order
to avoid the ambiguity, we adopt the proper time integrals and the Riemann zeta
function regularization in evaluating the summations of infinite KK states. In
the calculation, we utilize the KK regularization method with exchanging the
infinite summations and the infinite loop integrals. At the same time, we also
evaluate the correction by the the dimensional regularization method without
exchanging the summations and the loop integrals. Then, we clearly show that
the regularized Casimir corrections from the KK states have the form of
$\propto 1/R^2$ for the Higgs mass and $\propto 1/R^4$ for the cosmological
constant, where $R$ is the compactification radius. We also evaluate the
Casimir energy in supersymmetric extra-dimensional models. The contributions
from bulk fermions and bulk bosons are not offset because we choose SUSY
breaking boundary conditions. The non-zero supersymmetric Casimir corrections
from extra dimensions undoubtedly contribute to the Higgs mass and the
cosmological constant. We conclude the coefficients of such corrections are
enhanced compared to the case without bulk supersymmetry.
