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:


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


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?

I have a real conflict about this model, the point of solving the hierarchy of scales.

  • 1
    $\begingroup$ I haven't looked at the papers you linked to, but normally the point of the extra dimensional scenarios is that the "true" value of the Planck scale in the Universe considering the extra dimensions is close to the TeV scale, and in our world we observe an effective larger Planck scale because of some geometric property of the extra dimensions -- for example maybe the volume if large, or maybe there is some warping in the extra dimension. $\endgroup$
    – Andrew
    Aug 27 at 13:35
  • $\begingroup$ 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 $\endgroup$
    – Dr. phy
    Aug 27 at 14:42
  • 1
    $\begingroup$ 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. $\endgroup$
    – Andrew
    Aug 27 at 15:14
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    $\begingroup$ There is a very respectable position that the "hierarchy problem" isn't really a problem because Nature can choose any laws of physics and parameters it wants and it only looks like a problem from the perspective of a man-made theoretical prospect that prefers but doesn't rule out the parameters that we actually see. $\endgroup$
    – ohwilleke
    Aug 27 at 16:04

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