= Goldberger–Wise stabilization mechanism
{c}
{title2=$kL\simeq\epsilon^{-1}\log(v_0/v_L)$}
The <Goldberger–Wise stabilization mechanism> generates a <radion> potential in the <Randall–Sundrum model> using a weakly backreacting bulk <scalar field>. On $0\leq y\leq L$, its static profile obeys $\chi''-4k\chi'-m_\chi^2\chi=0$, hence $\chi=Ae^{(4+\epsilon)ky}+Be^{-\epsilon ky}$, where $\epsilon=\sqrt{4+m_\chi^2/k^2}-2$. Boundary potentials approximately fix $\chi(0)=v_0$, $\chi(L)=v_L$. For $\epsilon\ll1$ and $kL\gg1$, the leading radius-dependent potential is proportional to $e^{-4kL}(v_L-v_0e^{-\epsilon kL})^2$. Its leading minimum satisfies $kL\simeq\epsilon^{-1}\log(v_0/v_L)$. 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.
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