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The hierarchy problem revolves around the fact that we don't have a way to explain why gravity is so much weaker than the other forces. It may just be a coincidence, but since physicists tend to understandably not like coincidences very much, we treat it as a "problem" and therefore look for a solution.
One of the proposed solutions is that gravity spreads out in some extra dimensions, and how this would explain the hierarchy problem is clear to me. My question is, would this solve the problem? From what I reckon, this would still lead to gravity to be treated differently from the other forces, and the question would just translate from "why is gravity weaker than the others" to "why does gravity span more dimensions than the others". Is there something I'm missing? Do other proposed solutions also fall into this eventual fallacy?

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  • $\begingroup$ The notion of "weak gravity" is "not even wrong"! It's comparing oranges to apples: it's meaningless to compare gravity with standard model interactions such as the electromagnetic force, if no circumstance (e.g the specific charges and masses in comparison) is provided. See answer to a related question here: physics.stackexchange.com/questions/145518/… $\endgroup$
    – MadMax
    Nov 17 '20 at 18:44
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    $\begingroup$ @MadMax This is a more subtle question than "compare the gravitational and electric force." There is a dimensionful scale associated with the weak interactions, which we is the mass of the Higgs. There is also a dimensionful scale associated with gravity, which is the Planck mass. You can form a dimensionless ratio from these two numbers, and the question is why this dimensionless ratio is so far from 1. The reason it is an issue is because we would generically expect quantum corrections to the Higgs mass from degrees of freedom that are active at or above the Planck scale. $\endgroup$
    – Andrew
    Nov 17 '20 at 19:06
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    $\begingroup$ @Andrew, totally agree with your point. And the weak/Planck gap was actually touched upon in my answer to the related question: physics.stackexchange.com/questions/145518/… $\endgroup$
    – MadMax
    Nov 17 '20 at 19:18
  • $\begingroup$ ah i see. thanks! and, just to be clear, I actually don't think it is clear the hierarchy problem needs to be solved, but I just wanted to clarify that it is a well-defined problem. $\endgroup$
    – Andrew
    Nov 17 '20 at 19:46
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The hierarchy problem asks why the weak scale is so much smaller than the Planck scale. There's a specific, technical reason this is a problem, called "technical naturalness," which is delicate to explain properly. Roughly speaking it boils down to an expectation that unknown physics at the Planck scale should contribute via quantum effects to the weak scale we observe. Just by naive dimensional analysis there is no obvious reason why a sum of different terms, each of which is of order the Planck scale, should cancel amongst each other to produce a much smaller scale (the weak scale we observe).

Anyway, the extra dimensional solution essentially boils down to saying: the world really has 5 dimensions, and the true 5 dimensional Planck scale actually is approximately the same size as the weak scale today. Then there is no hierarchy problem, because there is no hierarchy between the weak scale and the true gravitational scale. The cleverness in the idea is finding a way to warp the extra dimension, and the way our four dimensional world sits inside of it, to produce an effective four-dimensional Planck constant that is much bigger than the weak scale.

If you want to learn more, you should look at the original papers, or subsequent reviews. Here are two classics:

https://arxiv.org/abs/hep-ph/9905221 https://arxiv.org/abs/hep-ph/9803315

It is actually a very elegant idea, but is moving out of fashion as a solution to the hierarchy problem (as are essentially all solutions to the hierarchy problem) as data from the LHC have ruled out significant parts of the interesting parameter space of these models.

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  • $\begingroup$ Ok, but my question is: would the weak force also be "weakened" by this 5th dimension? If not, do we have a guess on why? Wouldn't this create a "dimensional hierarchy" problem? $\endgroup$ Nov 18 '20 at 0:24
  • $\begingroup$ No. For instance in the ADD model, the weak force is confined to the brane and is not affected by the 5th dimension. $\endgroup$
    – Andrew
    Nov 18 '20 at 7:13

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