# How Randall-Sundrum model has solved the hierarchy problem?

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!

$$M$$ 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
$$$$m = e^{-k r_c \pi}m_0$$$$ where $$m_0 \sim M$$ and $$e^{-k r_c \pi}$$ is a warping factor that accounts for the warped geometry of the extra dimension.
In slightly more detail, the metric is (Eq 12) $$$$ds^2 = e^{-2 k r_c |\phi|} \eta_{\mu\nu} dx^\mu dx^\nu + r_c^2 d\phi^2$$$$ where $$-\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.