Tiny black holes could offer a richer view of physics than their better known, moremassive relations, as Aurélien Barrau and Julien Grain explain.

Unification is the dream of high-energy physicists. Attaining this conceptual – and actual – convergence undoubtedly constitutes the fundamental challenge in the search for elementary objects and interactions. There have been many suCCEesses in this area. If, however, it sometimes looks as if the main pieces of the puzzle have yet to be put together, it is because the opportunities nature gives us to attain an all encompassing and global view of the most disparate properties – described by the general theory of relativity and quantum mechanics – are extremely rare.

The early universe is often considered to be the only instance of a situation in which quantum and gravitational processes are equally important. However, black holes with small masses could also provide insight into these unusual conditions and would certainly constitute, if they exist, the only objects in today’s cosmos where such extreme physical processes can oCCEur. Even if it remains speculative, the search for small black holes is certainly warranted, especially since many models predict their existence in fundamentally different frameworks and contexts.

From the astrophysics point of view, it is thought that only massive black holes – with masses several times that of the Sun – are possible, as they are the only ones able to form in the final stages of stellar evolution. Although they have many fascinating properties, these large black holes are not as rich as their smaller cousins could be. The lighter the black hole the greater its surface gravity – and the more interesting the associated physical effects. This is simply due to the fact that Newton’s gravitational force is linearly dependent on mass but quadratically dependent on the inverse distance (which is itself proportional to the Schwarzschild radius of the black hole). In particular, the phenomenon of Hawking evaporation is significant only in the case of small black holes: the tidal effect becomes so great near the surface that the particle pairs produced by quantum vacuum fluctuations may be broken, one particle falling into the black hole and the other being projected outwards. This process has yet to be observed, precisely because astrophysical black holes are too massive and therefore too cold, but it is certainly one of the most important predictions of quantum field theory in curved space-time. Contrary to the usual ideas of general relativity, black holes are capable of emitting particles. They can even be very hot and very bright if their mass is sufficiently small. Indeed, the principles of thermodynamics apply to black holes, the essential variables being temperature, entropy and internal energy, as opposed to surface gravity, area and mass in the case of general relativity.

### Where can we find mini black holes?

The absence of notable excesses of particles in cosmic radiation – especially in the form of antiprotons or gamma rays – compared with the fluxes expected in a “standard” astrophysics context allows strict constraints to be placed on the density of black holes evaporating in today’s universe. In particular, it can be deduced that their contribution to the total mass of the universe is today no higher than one ten millionth. As these small black holes are likely to have been produced in the early cosmos thanks to the fluctuations in density present at that time – and with masses that were arbitrarily low – it is possible to obtain vital information about the universe’s degree of inhomogeneity shortly after the period of inflation. This route of investigation is all the more remarkable in that the relevant scales for the black holes of the early universe are completely beyond the usual observables of cosmology, namely the 3K background radiation and large-scale structure. There is therefore a genuine complementarity between these approaches. Many cosmological scenarios – involving phase transitions, the breaking of scale invariance, blue power spectra, positive running of the spectral index of scale fluctuations, phases of double inflation, topological defects, collisions of bubbles of “real” vacuum in a background of “false” vacuum and softening of the equation of state – may be excluded or severely constrained by the study of small black holes.

In addition to these astrophysical and cosmological aspects, there is another route of investigation that is particularly promising for microscopic black holes, namely at particle accelerators. In response to the persistent problem of hierarchy – why is the Planck scale 16 orders of magnitude higher than the electroweak scale? – a hypothesis put forward a few years ago offers a neat and efficient lead: the existence of large extra dimensions. The novelty of this idea lies in the fact that it is no longer necessary to assume that these dimensions are of sizes close to the Planck length (~10^{-33}cm). Rather, they can be as large as around a millimetre if we suppose that the fields of matter live in the 3+1 dimensional hypersurface of our 3-brane and that only gravity can benefit from new dimensions. The constraints (~10^{-16}cm ) usually derived via the interactions of gauge bosons in extra dimensions can therefore be ignored and only experiments involving the direct measurement of Newtonian gravity put limits on the size of extra dimensions to a value of less than a few tenths of a millimetre. Using such an approach, the traditional Planck energy, E_{PI}~10^{19}GeV, is no more than an effective scale and the real D-dimensional fundamental Planck scale is given by E_{D} = (E_{PI}^{2}/V_{D-4})^{1/(D-2)}, where V_{D-4} is the volume associated with the D-4 extra dimensions. For D=10 and radii associated with the extra dimensions of the Fermi scale, we obtain E_{D}~TeV. If this model has any meaning, it is effectively a natural choice (and not an arbitrary one based on phenomenological motivations) because it essentially resolves the problem of hierarchy. This approach uses the geometrical properties of space to link completely different energy scales.

A spectacular consequence of such a model is the possibility of being able to produce black holes with the next generation of particle colliders. If the centre-of-mass energy of two elementary particles is indeed higher than the Planck scale E_{D}, and their impact parameter b is lower than the Schwarzschild radius R_{H}, a black hole must be produced. If the Planck scale is thus in the TeV range, the 14 TeV centre-of-mass energy of the Large Hadron Collider (LHC) could allow it to become a black-hole factory with a production rate as high as about one per second. Many studies are underway to make a precise evaluation of the cross-section for the creation of black holes via parton collisions, but it appears that the naive geometric approximation σ~πR^{2}_{H} is quite reasonable for setting the orders of magnitude.

The possible presence of extra dimensions would be doubly beneficial for the production of black holes. The key point is that it allows the Planck scale to be reduced to accessible values, but it also allows the Schwarzschild radius to be significantly increased, thus making the condition b<R_{H} distinctly easier to satisfy. It is important to note that the resulting mini black holes have radii that are much smaller (of the order of 10^{-4}fm in the case of those that can be expected from the LHC) than the size of extra dimensions, and that they can therefore be considered as totally immersed in a D-dimensional space, which has, to a good approximation, a time dimension and D-1 non-compact space dimensions. The black hole thus acts like a quasi-selective source of S waves and sees our brane in the same way as the “bulk” associated with the extra dimensions. As the particles residing in the brane greatly outnumber those living in the bulk (essentially gravitons), the black hole evaporates into particles of the Standard Model. Its lifetime is very short (of the order of 10^{-26}s) and its temperature (typically about 100 GeV here) is much lower than it would be with the same mass in a four-dimensional space. The black hole nevertheless retains its characteristic spectrum in the form of a quasi-thermal law peaked around its temperature. From the point of view of detection, it is not too difficult to find a signature for such events: they have a high multiplicity, a large transverse energy, a “democratic” coupling to all particles and a rapid increase in the production cross-section with energy.

### Particle physics and mini black holes

At first glance the production of black holes in colliders could be bad news. It could mean the end of particle physics since the presence of a horizon would obscure all the microphysics processes that could occur behind it. However, it would in fact open up very good opportunities.

First of all the reconstruction of temperature (determined by the energy spectrum of the particles emitted when the black hole evaporates) as a function of mass (determined by the total energy deposited) allows information to be gained about the dimensionality of space- time. In the case of Planck scales close to the TeV mark, the number of extra dimensions could thus be revealed quite easily by the characteristics of the emitted particles. However, one can go further. In particular, quantum gravity effects could be revealed, as behaviour during evaporation in the Planck region is sensitive to the details of the gravitational theory used.

Approaches of the Gauss-Bonnet type, which include quadratic terms in scalar curvature in the Lagrangian, are good candidates for a description beyond general relativity as they can be supported both by theoretical arguments (heterotic strings in particular) and by phenomenological arguments (Taylor expansion in curvature). In such a case, the coupling constant of the Gauss-Bonnet term, namely the quantum character of the gravitational theory used (and the link with the underlying string theory) can also be reconstructed and the LHC would become a very valuable tool for studying speculative gravitation models.

Other promising avenues are also being investigated for new physics. Firstly, the black holes formed may be excellent intermediate states for highlighting new particles. When the collision energy is higher than the Planck scale E_{D}, the cross-section for the creation of black holes is quite large (~500 pbarn) and has no suppression factor. Moreover, when the temperature of the black hole is higher than the mass of a particle, the particle must be emitted during evaporation in proportion to its number of internal degrees of freedom. There is thus a definite potential for the search for the Higgs or for supersymmetric particles in the evaporation products of black holes, possibly with cross-sections much greater than for the direct processes. Finally, taking account of a D-dimensional cosmological constant also modifies the evaporation law. If the constant is sufficiently high – which is possible without contradicting the low value measured in our brane – the temperature and the coupling coefficients with the entities emitted could be the signature of this particular structure of space-time. It would be quite neat and certainly surprising that a measurement of the cosmological constant in the bulk should come from the LHC!

Microscopic black holes are thus a paradigm for convergence. At the intersection of astrophysics and particle physics, cosmology and field theory, quantum mechanics and general relativity, they open up new fields of investigation and could constitute an invaluable pathway towards the joint study of gravitation and high-energy physics. Their possible absence already provides much information about the early universe; their detection would constitute a major advance. The potential existence of extra dimensions opens up new avenues for the production of black holes in colliders, which would become, de facto, even more fascinating tools for penetrating the mysteries of the fundamental structure of nature.

### Postscript

It should be stated, in conclusion, that these black holes are not dangerous and do not threaten to swallow up our already much-abused planet. The theoretical arguments and the obvious harmlessness of any black holes that, according to these models, would have to be formed from the interaction of cosmic rays with celestial bodies, mean that we can regard them with perfect equanimity.

### Further reading

N Arkani-Hamed, S Dimopoulos and G R Dvali 1998 *Phys. Lett. B***429** 257.

A Barrau *et al.* 2002 Astronom. *Astrophys.***388** 676.

A Barrau *et al.* 2004 *Phys. Lett. B***584** 114.

S Dimopoulos and G Landsberg 2001 *Phys. Rev. Lett.***87** 161602.

J L Feng and A D Shapere 2002 *Phys. Rev. Lett.***88** 021303.

S B Giddings and S Thomas 2002 *Phys. Rev. D***65** 056010.

P Kanti 2004 *Int. J. Mod. Phys.* (in press) www.arxiv.org/hep-ph/0402168. G Landsberg 2002 *Phys. Rev. Lett.* **88** 181801.