  In this paper a framework is introduced to remove the huge discrepancy
between the empirical value of the cosmological constant and the contribution
to the cosmological constant predicted from the vacuum energy of quantum
fields. An extra dimensional space with metric reversal symmetry and $R^2$
gravity is considered to this end. The resulting 4-dimensional energy-momentum
tensor (obtained after integration over extra dimensions) consists of terms
that contain off-diagonally coupled pair of Kaluza-Klein modes. This, in turn,
generically results in vanishing of the vacuum expectation value of the
energy-momentum tensor for quantum fields, and offers a way to solve the
problem of huge contribution of quantum fields to the vacuum energy density.
