Gia first argued for the following result. Suppose there exists N particle species whose mass is of order M. Further suppose that these species transform under exact gauged discrete symmetries. Then there is a lower bound on the Planck scale:

$M_p > N^{1/2} M$

The proof goes via black holes. As argued in the old paper by Krauss and Wilczek, gauged discrete symmetries should be respected by quantum gravity. Therefore, if we make a black hole out of particles charged under a gauged discrete symmetry, the total charge will be conserved. For example, take a very large number N of particles, each carrying a separate Z2 charge. Form a black hole using an odd number of particles from each species, so that the black hole carries a Z2^N charge. Then wait and see what happens. According to Hawking, the black hole should evaporate. But it cannot emit the charged particles and reduce its charge before its temperature becomes of order M. The relation between the black hole temperature and mass goes like $T \sim M_p^2/M_{BH}$. Thus, by the time the charge starts to be emitted, the black hole mass is reduced to $M_{BH} = M_p^2/M$. To get rid of all its charge the black hole must emit at least N particles of mass M, so its mass at this point must satisfy $M_{BH} > N M$. From this you easily obtain Gia's bound.The bound has several interesting consequences. One is that it can be used to drown the hierarchy problem in the multitude of new particles. Just assume there exists something like 10^32 new charged particle species at the TeV scale. If that is the case, the Planck scale cannot help being 16 orders of magnitude higher than the TeV scale. For consistency, gravity must somehow become strongly interacting at the TeV scale, much as in the ADD or RS model, so that the perturbative contributions to the Higgs mass are cut off at the TeV scale. Thus, in Gia's scenario the LHC should also observe the signatures of strongly interacting gravity.

You might say this sounds crazy...and certainly it does. But, in fact, the idea is not more crazy than the large extra dimensions of the ADD model. The latter is also an example of many-species solution to the hierarchy problem. In that case there are also 10^32 degrees of freedom - the Kaluza-Klein modes of the graviton, which make gravity strongly interacting at TeV. The difference is that most of the new particles is much lighter than TeV, which creates all sorts of cosmological and astrophysical problems. In the present case these problems can be more readily circumvented.

Transparencies available on the workshop page.

## 5 comments:

> Suppose there exists N particle species

stupid question. could it be possible that one has N copies of the standard model, at different (random) energies, which do not interact with each other, except via gravity? would this explain the hierarchy problem and dark matter?

In fact, the example of N copies of the standard model appeared in Gia's talk. There are some cosmological issues one would have to face (e.g. reheating after inflation should not populate all these standard models with the same abundance), but one can certainly do model building along these lines.

Hi Jester,

interesting post. So let me take this further a bit: Imagine there are N distinct SM families, all degenerate in mass, and distinguished by some as-of-yet unfathomed quantum number, say serendipity S. This would also solve some of the issues, but we would only be faced with finding a clue about S, not about N families.

Would this be a scenario ?

Cheers,

T.

Hi Jester -- thanks for the post and summary. A quick question: Is it fair, to call this scenario a `solution' to the Hierarchy problem? Doesn't it just shift the mystery of 10^30 orders of magnitude from a mass hierarchy into a `hierarchy' in the number of gauged symmetries? Or is a huge number of discrete symmetries considered less `problematic' than a mass hierarchy? (If so, why?)

Cheers,

Flip

Sorry guys for a late replay.

FlipTomato: Of course, just postulating a huge number of new species is no solution. You need to find a good reason. One possible reason is large extra dimensions. Gia showed that ANY mechanism that produces a huge number of new states at TeV automatically addresses the hierarchy problem. And this generalization is what i find interesting and inspiring.

Tommaso: i'm not sure i understand your question. But if i understood correctly, the same answer as above applies ;-)

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