Large N Species

One of the talks at the previously mentioned workshop Origin of Time's Arrow was by Gia Dvali. He talked about his recent paper

Black Holes and Large N Species Solution to the Hierarchy Problem
arXiv: 0706.2050v1 [hep-th]

The idea is really cute. First, let me summarize some basics: Numerous results lead us to expect that black holes emit thermal radiation with a temperature proportional to the inverse of the black hole's mass. This means the more mass the hole looses through the radiation, the hotter it becomes. It is unknown whether a collapse into a black hole, and a subsequent complete evaporation really destroys information about the initial state. This process can also violate certain conservation laws like baryon number. But electric charge, as well as energy and other gauge charges are conserved. However, in standard General Relativity black holes have no 'hair', i.e. the asymptotic solution is completely characterized by only their mass, angular momentum, and electromagnetic charges. So their ability to carry additional gauge charges is limited, unless one allows for quantum 'hair' that resides on the horizon [1]. Though this quantum hair does not have long-range fields, its gauge charge is a conserved quantity.

Now consider a black hole with N different such conserved charges, and assume that these charges are (as is the case for the electric charge as well) each bound to massive particles, the lightest of which has a typical mass Λ. Imagine we set up a black hole that carries these charges, one of each, and we let it completely evaporate. During this evaporation, all the N charges need to be re-emitted somehow. But the black hole's temperature has to be high enough - or the mass has to be small enough respectively - before it can start evaporating off the massive particles. The required temperature is T ~ Λ, or the black hole mass is M~m_p²/Λ, where m_p is the Planck mass and roughly ~ 10¹⁶ TeV. To give you a feeling for these numbers: if we were talking about electric charge, the lightest particle is the electron with a mass of roughly .5 MeV, then the black hole can start evaporating off electric charge if its mass has fallen to ~ 10¹⁷ g.

However, there is also an obvious limit to this: the black hole needs to be able to provide the mass of the particles. If the black hole was charged but lighter than an electron it couldn't emit the charge no matter what [2]. If there were many different charges carried by particles with mass scale Λ, one comes to the conclusion that a bound results. The bound arises from the fact that after the black hole started evaporating off the charges, its mass must still have been high enough to provide all the N particles with mass Λ. One thus has N Λ ≤ M, or, if one inserts the above expression for the mass at which the emission of massive particles can start, one finds Λ² ≤ m_p²/N.

The further argument is now the following. We don't know why the gravitational interaction is so much weaker than the other interactions of the standard model (SM). Or, to put it differently, we don't know why the masses of the SM particles are so much smaller than the Planck mass. If we take Λ to be the typical mass of SM particles (Higgs VEV) then there is a gap of roughly sixteen orders of magnitude. Dvali's inequality says if there were very many species particles, then there would necessarily have to be such a hierarchy. Putting in some numbers one finds the 'large' number is indeed very large, and somewhere around N~10³².

Now, as far as I am concerned this doesn't really 'solve' the hierarchy problem, one has just moved it elsewhere (as one also does with the extra dimensional models). Instead of having to explain the gap in the mass-scales one now has to explain where all the other particles are, and why so many of them? However, one can model these as only gravitationally interacting with our beloved standard model which would then only describe a tiny fraction of all there is. The question is of course why there don't seem to exist many particles of this kind around us. But this must stem from some processes in the very early universe, and inflation can easily make small numbers large, and blow up initially only subtle differences. Though it is hard to say at this stage whether it would actually work as desired, I can imagine that such a reformulation of the problem offers the possibility to find a dynamical explanation.

The signatures of such a scenario are in certain regards quite similar to those of extra dimensional models. One has a lot of only very weakly interacting particles whose coupling is given by the Planck mass. But since there are so many of them, their phase space gets really large, cancels the Planck suppression, and the signatures could become observable somewhere around the scale Λ. In contrast to the KK-tower in extra dimensional models however, here the number of species is really finite, so one doesn't have the problem of divergences in the higher dimensional integrals.

I can't say I particularly like the idea of having 10³² particle species, but I like the paper because it is another example for how thought experiments with black holes can lead to sometimes surprising insights. It's a cute idea to play around with that resides somewhere between General Relativity and particle physics, which is - still - a region of large mysteries.

What that has to do with the arrow of time however, I honestly don't know.

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[1] A black hole can e.g. carry quantum hair associated with discrete gauge charges. This can happen when a local continuous gauge symmetry is broken down to a residual discrete subgroup. See ref [1] in Dvali's paper.
[2] However, since the electron mass is so much smaller than the Planck scale, such a black hole would long fall into the quantum gravity regime and no reliable statements can be made anyhow.

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TAGS: PHYSICS, BLACK HOLES