Radius stabilization ($V\prime (R_0)=0,m_R^2>0$)

Radius stabilization fixes the size of a compactification dynamically rather than choosing it as an undetermined parameter. If the four-dimensional low-energy effective action contains a radion $R$ with positive kinetic coefficient $Z(R)$ and scalar potential $V(R)$, a stable radius $R_0$ requires $V^{\prime }(R_0)=0$ and $m_R^2=V^{\prime \prime }(R_0)/Z(R_0)>0$. This matters for the hierarchy problem: the volume or warp factor explains a hierarchy only if a consistent dynamics selects and maintains the required geometry.

The Goldberger–Wise stabilization mechanism generates a radion potential in the Randall–Sundrum model using a weakly backreacting bulk scalar field. On $0\le y\le L$, its static profile obeys ${\chi }^{\prime \prime }-4k{\chi }^{\prime }-m_{\chi }^2\chi =0$, hence $\chi =A{e}^{(4+ϵ)ky}+B{e}^{-ϵky}$, where $ϵ=\sqrt{4+m_{\chi }^2/k^2}-2$. Boundary potentials approximately fix $\chi (0)=v_0$, $\chi (L)=v_L$. For $ϵ\ll 1$ and $kL\gg 1$, the leading radius-dependent potential is proportional to $e^{-4kL}(v_L-v_0{e}^{-ϵkL}{)}^2$. Its leading minimum satisfies $kL\simeq {ϵ}^{-1}\log (v_0/v_L)$. Thus a modest ratio of boundary values can generate a large warp exponent. Corrections shift this minimum; the approximation requires small scalar field backreaction, and the four-dimensional cosmological constant still needs adjustment.