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During the last few days I have been interested in the gravitational hierarchy problem and the different explanations for it/solutions to it.

Among the most "concrete" (insofar as anything this complex can be concrete) explanations are several realisations of the so-called "brane cosmology".
In these, whether they be the original ADD model or some string theory model, the standard model particles/modes are confined to a brane (either due to them being open strings or some other effect) while gravity/gravitons can freely propagate into the so-called "bulk", therefore allowing gravity to "leak" away from "our" spacetime thereby explaining why it's so weak.
Common to these theories is that while these dimensions can be taken to be either compact or non-compact (but often are assumed to be compact) they are "large" extra dimensions.

Now in your "garden-variety/lift pitch"-string theory the necessary extra dimensions are assumed to be very, very small and compact. My questions now revolve around the parallels of these Kaluza-Klein compactified theories and the brane cosmology ones in regard to the hierarchy problem:

  1. Can standard model particles/modes access the extra dimensions? I seem to recall that this should only be possible at "high enough" energies but I am unsure.
  2. Can gravity/the graviton access the extra dimensions even at low energies? If so, why? What is the mechanism/explanation for this?
  3. If neither of the above yields a similar solution to the hierarchy problem as in the brane cosmology scenario, how is the hierarchy problem solved in these "classical ", small-extra-dimensions Kaluza-Klein types of theories?
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There are two issues here. One is the existence of the hierarchy: Why are the electroweak scale and the gravitational scale (often called Fermi scale and Planck scale respectively) so far apart?

But the other issue is the stability of the hierarchy: Why aren't electroweak phenomena disrupted by virtual effects coming from heavier scales of physics? Not just gravitational phenomena like micro black holes, but any heavy physics from beyond the standard model?

This latter question is considered far more serious and has received far more attention.

The models with large extra dimensions explain the existence of the hierarchy by saying it is somewhat illusory, and the true Planck scale is much closer to the Fermi scale than we think. But I suppose this might also explain the stability of the hierarchy: in this scenario, the heavier scales of physics just aren't much heavier than the Fermi scale, so their effects need not be disruptive.

For other models of physics, such as those with small extra dimensions, the hierarchy is still large, so the stability problem has to be resolved some other way. Until 2012, the standard expectation was that new physics just above the Fermi scale, such as supersymmetry or compositeness, neutralized the heavy virtual effects, and thereby stabilized the hierarchy.

So I want to emphasize again: Explaining the existence of the hierarchy is about explaining the relative magnitude of two out of the many fundamental quantities in physics. Explaining the stability of the hierarchy is about finding a mechanism that protects the physics of the lower scales from disruption by the physics of the higher scales.

When it comes to explaining the existence of the hierarchy, the only framework I can think of, that even came close, would be string theory; because the values of fundamental constants do have explanations in string theory, at least in principle. The problem is, I'm not aware of a calculation in string theory that actually derives the Fermi scale.

String theory can tell you that the gravitational force and the gauge forces have comparable strengths at the Planck scale, because they both derive from the string coupling; and then it can tell you how strong the gauge forces are at lower scales, such as the Fermi scale. But the thing that makes the Fermi scale actually significant, is that this is the scale of the Higgs condensate; and as far as I know, calculating that from first principles, remains out of reach in all theoretical frameworks, not just string theory.

What a string theorist could tell you is that in a particular model, the size of the Higgs vev corresponds, for example, to the distance between two particular brane stacks. But those geometric quantities are dynamic, and actually calculating their ground-state values remains too hard, even for string theorists.

Summing up what I've said so far: In string theory models with small extra dimensions, the Fermi scale is assumed to be set by some dynamical process that we don't know how to calculate; the strength of the weak force at the Fermi scale is calculated by applying standard methods (the running of coupling constants) to the strength of gauge forces at the Planck scale, which can be calculated; and then the hierarchy is stabilized by new physics, usually supersymmetry.

Earlier, I said this was how the mainstream thought until 2012. That was when the Higgs boson turned up, but without any sign of additional new physics. Particle physicists are still looking for, and hoping for, new physics; but the idea that new particles stabilize the hierarchy, has really taken a beating.

The old alternative to that mechanism, which was not considered to actually be an option, was that all the destabilizing effects due to heavy physics just happen to cancel out by sheer coincidence. But now the anthropic principle has provided a possible reason for the coincidence to occur: what if life (or even just complex molecules) couldn't exist unless the Fermi scale has this kind of value? Logic would force conscious observers, like us, to be in one of the otherwise unlikely worlds where the heavy effects happen to cancel out.

The theoretical situation, regarding the stability of the hierarchy, is therefore a muddle of people still hoping for sensible new physics, people embracing the idea of anthropic finetuning, and various "none of the above" ideas. In the latter category, the one that interests me the most dates back to 2001: "Solving the Hierarchy Problem without Supersymmetry or Extra Dimensions: An Alternative Approach" by Keith Dienes.

This is an argument that supersymmetry-like cancellations can still occur in non-supersymmetric string theory, if you take into account all the states of the string; due to a symmetry of the string called "modular invariance". Twenty years later, this idea is starting to yield results (see recent papers by Dienes and Steven Abel).

In his 2001 paper, Dienes also wrote

A third open issue concerns the origin of the scale of electroweak symmetry breaking. Clearly, our approach does not shed light on this critical issue. This scale is presumably set by some dynamics connected with the stability of the underlying non-supersymmetric string. However, regardless of how this scale is set, our point is that it is then guaranteed to be insensitive to other heavy scales such as the string scale.

Hopefully you can recognize in these words, what I was already saying. The actual hierarchy is to be produced by some kind of stringy geometric dynamics (the details vary greatly according to the specific string model), which ends up making electroweak physics happen at distances of 10^-18 meters - the length scale of the weak force, 10^15 times bigger than a fundamental string; and then the modular invariance of the fundamental string will protect electroweak physics from disruption by gravitational or other processes.

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