# The hierarchy problem

I recently read a couple of scenarios to solve the hierarchy problem: the huge ratio between the electroweak scale $$M_W$$ of order $$100-1000$$ GeV and the Planck scale $$\sim 10^{19}$$ GeV

I think I understand the first study, but I have many conflicts about the second , so any help about that is appreciated!

The first scenarios : Here what I have got from this paper:

https://arxiv.org/abs/hep-ph/9803315

The 4-dimensional Planck scale $$M_p$$ is given in terms of the Planck scale of hidden extra dimensions multiplied by the size of these dimensions $$V$$. By assuming there is no hierarchy in these extra dimensions, and the gravity becomes evident as the other forces at near to the electroweak scale, $$M$$ is set to be of order of TeV scale, which fine-tunes the value of $$V$$.

While in the second scenario

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

They reached equation (16)

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

with k of order the Planck scale, the exponential will has a weak effect and that leads to $$M \sim M_p$$!

Now what I don't get the discussion in page 6: as they say

Until this point, we have viewed $$M \sim M_p$$ as the fundamental scale, and the TeV scale as a derived scale as a consequence of......

How does this solve the hierarchy problem? I mean $$M_p$$ still in the four dimensions $$\sim 10^{18}$$ GeV as the real universe, while they assume $$M$$ of order TeV!

Unless they consider $$M_p$$ is a Planck scale in another 3-brane universe, not ours? Cause the solution from the the beginning is a static solution, equ. (2), while our universe expands?

• But in the second paper how they says $M \sim M_p$, which is direct from the equation. While M is a TeV scale! I just don't get what they mean by that! And what you are saying I understand very much and it's very close to the first paper approach Aug 27 at 14:42
• Ah I see. Yes, in the Randall-Sundrum model, the effective 4-d Planck scale is around $M_{\rm Pl}$, but on the brane where the standard model particles live, the particle masses are exponentially suppressed; see Eq 20 and 21 of arxiv.org/pdf/hep-ph/9905221.pdf. Aug 27 at 15:14