The hierarchy problem

Black holes in the laboratory

LXD - Large Extra Dimensions

Ther are problems with hierarchy not only here in our institute and in global politics but also in physics. We all know the experiment with the two electrically charged spheres that repel each other from school physics. Was that that the story with the resin and the cat skin? Today presumably rather plastics and a kerchief, which causes less amok runs by animal-rights activists. Anyway, the two spheres marvellously repel each other. As we all know, the reason is that both carry the same charge and therefore the Coulomb force is repellent.

But as we know also, both spheres have a mass and thus attract each other. But this attraction carries not much weight (haha), because gravity is a lot weaker than the electrical force. Precisely, by the factor "a lot" $= 10^{34}$ . And this is really a great deal. We are only such familiar with gravity because it is (according to today's knowledge) the only of the four interactions which cannot be neutralised. Electrical charge appears no longer on great distances, even atoms are neutral, the same holds for the other forces. What remains is gravity, which therefore dominates physics on large scales.

Newton's law for two masses und reads:

$\begin{displaymath} F_g \propto \frac{1}{m_p^2} \frac{1}{r^2} m_1 m_2 \;\;\; \mbox{wobei $\hbar = c =1$} \end{displaymath}$

But this results in quite a few problems: A unification of gravity and the other forces of the standard model is still to be done. An experimental verification of quantum gravity, whose existence we assume, would be desirable. Unfortunately, the typical energy scale on which this would occur is about

$\begin{displaymath} m_p \approx 10^{19} \mbox{GeV} , \end{displaymath}$

which lies beyond our today's capabilities. And if one takes in consideration that the financial and the spatial size as well as the construction period of particle accelerators increases extremly with the achievable energy, this is not only sad but also socially not reasonable.

Furthermore, this unexplainable maverick role of gravity is unsatisfactory for theorists.

The idea [1,2, 3] is now to postulate the existence of spatial extra dimensions which are compactificated on a radius of the size of . The gravitational law in dimensions reads then

$\begin{displaymath} F \propto \frac{1}{M_f^{2+d}} \frac{1}{r^{2+d}} m_1 m_2 \end{displaymath}$

where is now the new fundmental scale in - dimensional spacetime. Since for large both laws have to coincide because anyway we are observing Newton's law, we get:

$\begin{displaymath} M_f^{2+d}R^d = m_p^2 \end{displaymath}$

Now one can calculate easily that for the radius would be too large to be unnoticed until now. For one gets $R \approx 1/10$ mm, which would be already possible. For $d \geq 2$ the radius shrinks further. The fundamental scale here comes into the TeV-range.

In this way the hierarchy problem can be solved by presence of more than 2 extra dimensions.

Literatur

1

hep-ph/9803315 Arkani-Hamed, Dimopoulos & Dvali "The Hierarchy Problem and new dimensions at a Millimeter"

2

Phys. Rev. D 59 1999 Arkani-Hamed, Dimopoulos & Dvali "Phenomenology, Astrophysics and Cosmology of Theories with Sub-Millimeter Dimensions and TeV Scale Quantum Gravity"

3

hep-ph/0007211 Dvali, Gabadadze & Porrati "On Sub-Millimeter Forces From Extra Dimensions"

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