How Randall-Sundrum model has solved the hierarchy problem?

Question

I'm trying to understand how the RS model solved the hierarchy problem from this mass relation

$$ M^2_p = \frac{M^3}{k} \Large[1- e^{-2k\pi r} \Large],$$

Equ. 16 in their paper:

https://arxiv.org/abs/hep-ph/9905221

With k as large as the Planck scale, the exponential will be so small and almost has no effect, which leads to (is this correct? ), as they say in page 6, $M \approx M_p$!

So the conflict rises here, cause $M_p$ is the four dimensional effective Planck scale $\sim 10^{18}$ GeV, while $M$ is the higher 5-dimensional Planck scale assumed to be at TeV scale, so what does $M \approx M_p$ mean?

Any help is appreciated!

See also the discussion in this thread: The hierarchy problem

Answer 1

$M$ is indeed approximately equal to the 4-d Planck scale $M_{\rm pl}$. The hierarchy problem is why the Higgs mass $m_H$ is so much smaller than the Planck scale. In the original version of the Randall-Sundrum model, this happens because the standard model fields live on a brane inside a warped throat. The warping factor leads to an exponential suppression of all particle masses relative to the Planck scale. See Eq. 21 of https://arxiv.org/abs/hep-ph/9905221

\begin{equation} m = e^{-k r_c \pi}m_0 \end{equation} where $m_0 \sim M$ and $e^{-k r_c \pi}$ is a warping factor that accounts for the warped geometry of the extra dimension.

In slightly more detail, the metric is (Eq 12) \begin{equation} ds^2 = e^{-2 k r_c |\phi|} \eta_{\mu\nu} dx^\mu dx^\nu + r_c^2 d\phi^2 \end{equation} where $-\pi \leq \phi \leq \pi$. The standard model lives on a brane at $\phi=\pi$ (see Eq 3), where leading to coefficient of the Minkowski metric above to be $e^{-2 k r_c \pi}$, which leads to the exponential suppression of the particle masses after computing the effective four dimensional theory on the brane.