rved value of the “hierarchy”, $\lambda_{obs}$, and the observed value of the effective four-dimensional cosmological constant, which we take to be zero. Thus, we take as our renormalization conditions $$V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}(\lambda_{obs})
={dV_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}\over d\lambda}(\lambda_{obs})=0. 
\label{renc}$$ If there are other bulk fields, such as the graviton, which give additional classical or quantum mechanical contributions to the radion potential, then those should be included in $V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}$. From the renormalization conditions (\[renc\]) the unknown coefficients $\alpha$ and $\beta$ can be found, and then the mass of the radion is calculable. In Fig. \[fig1\] we plot (\[confveff\]) for a fermionic field and a chosen value of $\lambda_{obs}$.

=10 cm\

From (\[renc\]), we have $$\beta = - A (1-\lambda_{obs})^{-5},\quad \alpha= -\beta 
\lambda_{obs}^5.
\label{consts}$$ These values correspond to changes $\delta \sigma_{\pm}$ on the positive and negative brane tensions which are related by the equation $$\delta\sigma_+ =  -\lambda^5_{obs}\ \delta\sigma_-.
\label{reltensions}$$ As we shall see below, Eq. (\[reltensions\]) is just what is needed in order to have a static solution according to the five dimensional equations of motion, once the casimir energy is included.

We can now calculate the mass of the radion field $m_\phi^{(-)}$ from the point of view of the negative tension brane. For $\lambda_{obs}\ll 1$ we have: $$m^{2\ (-)}_\phi = 
\lambda_{obs}^{-2}\ m^{2\ (+)}_\phi = \lambda_{obs}^{-2}\ 
   {d^2 V_{\hbox{\footnotesize\it \hspace{-6pt} eff\,}}\over
  d\phi^2}\approx \mp
\lambda_{obs} \left({5\pi^3 \zeta'_R(-4)\over 6 M^3 l^5}\right).
\label{massconf}$$ The contribution to the radion mass squared is negative for bosons and positive for fermions. Thus, depending on the matter content of the bulk, it is clear that the radion may be stabilized due to this effect.

Note, however, that if the “observed” i