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The idea that quantum gravity can be realized at the TeV scale is extremely attractive to theorists and experimentalists alike. This proposal leads to extra spacial dimensions large compared to the Planck scale. Here, we give a very systematic view of the foundations of the theories with large extra dimensions and their physical consequences.

The idea of extra dimensions slipped into the realm of physics in the 1920s when Kaluza et al. [

The first indication of large extra dimensions in string theory came in 1988 from studies of the problem of supersymmetry breaking by Antoniadis et al. [

In 1996, Witten [

In 1998, Arkani-Hamed, Dimopoulos and Dvali (ADD) [

The simplest ADD scenario is characterized by the SM fields localized on a four dimensional submanifold of thickness

In this paper we would like to present most of the features of ADD model. Other recent reviews of the subject can be found in [

Why are not any of the standard model particles or fields, in any experiment so far conducted disappearing into extra dimensions? Answering this question will naturally lead us towards theories with SM matter and fields localized to branes. Then, the new question arises, what is the mechanism by which the standard model fields are localized to the brane? The idea of localizing particles on walls (brane) in a higher dimensional space goes back to Akama [

Rubakov and Shaposhnikov [

A mechanism for gauge field localization within the field theory context was proposed by Dvali and Shifman [

The standard model fields are only localized on the brane of width

Notice that while energy can be lost into the extra dimensions, electric charge (or any other unbroken gauge quantum number) cannot be lost. This is because the massless photon is localized in our Universe and an isolated charge cannot exist in the region where electric field cannot penetrate, so charges cannot freely escape into the bulk, although energy may be lost in the form of neutral particles propagating in the bulk. Similar conclusions can be reached by considering a soft photon emission process in [

The important question that we would like to answer is how large the extra dimensions could possibly be without us having them noticed until now. For this, we need to understand how the effectively four-dimensional world that we observe would be arising from the higher-dimensional theory. Let us call the fundamental (higher dimensional) Planck scale of the theory

The easiest derivation is a trivial application of Gauss’ Law. The

Suppose now that a point mass

For

The two test masses of mass

On the other hand, if the masses are placed at distances

In this method, we first write down the action for the higher dimensional gravitational theory, including the dimensionful constants and then dimensionally reduce it to compare the quantities. Here, we use the mass dimensions of the various quantities for analysis.

The higher dimensional line element is given by

Now, to compare these two actions, we assume that spacetime is flat and that the

Substituting these quantities in (

The factor

Finally, we can understand this result purely from the 4-dimensional point of view as arising from the sum over the Kaluza-Klein excitations of the graviton. From the 4d point of view, a

An important issue in extra dimensional theories is the mechanism by which extra dimensions are hidden, so that the spacetime is effectively four dimensional in so far as known physics is concerned. The most plausible way of achieving this is by assuming that these extra dimensions are finite and are compactified. Then, one would need to be able to probe length scales corresponding to the size of the extra dimensions to be able to detect them. If the size of the extra dimensions is small, then one would need extremely large energies to be able to see the consequences of the extra dimensions. Thus, by making the size of the extra dimensions very small, one can effectively hide these dimensions. So the most important question that one needs to ask is how large could the size of the extra dimensions be without getting into conflict with observations?

The new physics will only appear in the gravitational sector when distances as short as the size of the extra dimension are actually reached. However, it is very hard to test gravity at very short distances. The reason is that gravity is a much weaker interaction than all the other forces. Over large distances, gravity is dominant; however, as one starts going to shorter distances, intermolecular van der Waals forces and eventually bare electromagnetic forces will be dominant, which will completely overwhelm the gravitational forces. This is the reason why the Newton law of gravitational interactions has only been tested down to about a fraction of a millimeter using essentially Cavendish-type experiments [

Let us check how large a radius one would need, if in fact

For

For

In this section, we compactify

The

Clearly, the graviton is a

This brings down the number of degrees of freedom by

Now, let us discuss the different modes

The above KK modes satisfy the following equation of motions:

The different four-dimensional fields are coming from the different blocks in the bulk graviton metric, which is represented aesthetically as follows:

The bulk graviton is given by a

For nonzero modes, the upper left

In this section, we would like to explicitly construct the generic interaction Lagrangians between the matter on the brane and the various graviton modes. For our discussion, we will follow the work of Giudice et al. [

The action with minimal gravitational coupling of the general scalar

and we have used

For the KK modes, we replace

In the following, we present only three-point vertex Feynman rules and the energy-momentum tensor for scalar bosons, gauge bosons, and fermions where we have used the following symbols:

The four-point and five-point vertex Feynman rules and their derivation can be found in [

The conserved energy-momentum tensor for scalar bosons is

The conserved energy-momentum tensor for gauge vector bosons is

The conserved energy-momentum tensor for fermions is

In the following, we will briefly list some of the most interesting constraints on these models. The four principal means of investigating these theories are as follows.

Deviation from Newton’s Law at sub-mm distance.

Virtual graviton exchange colliders.

Real graviton production.

Missing energy in collider experiments.

Missing energy in astrophysical sources, for example, supernovae, sun, Red giants.

Cosmological consequences, for example, dark Energy, dark Matter, inflation, CMBR.

Black hole production at colliders.

We describe only the above-mentioned topics. The interested readers can find other useful references as follows, for the running of couplings and unification in extra dimensions see [

From the relation between the Planck scales of the

Putting

For

Besides the direct production of gravitons, another interesting consequence of large extra dimensions is that the exchange of virtual gravitons can lead to enhancement of certain cross-sections above the SM values. One can also study the effects of the exchange of virtual gravitons in the intermediate state on experimental observables. Virtual graviton exchange may generate numerous higher dimension operators, contributing to the production of SM particles [

Some of the most interesting processes in theories with large extra dimensions involve the production of a single graviton mode at the LHC or NLC. In addition to their traditional role of probing the electroweak scale, they can also look into extra dimensions of space via exotic phenomena such as apparent violations of energy, sharp high-

Since the lifetime of an individual graviton mode of mass

Some of the strongest constraints on the large extra dimensional scenarios come from astrophysics. The gravitons are similar to Goldstone bosons, axions, and neutrinos in at least one respect. They can carry away bulk energy from an astrophysical body and accelerate its cooling dynamics. These processes have been discussed in detail in [

We consider the supernova 1987A. There, the maximum available energy per particle is presumed to be between 20 and 70 MeV. The production of axions in supernovae is proportional to the axion decay constant

For the sun,

Finally, we come to the early universe. The most solid aspect of early cosmology, namely, primordial nucleosynthesis, remains intact in ADD framework. The reason is simple. The energy per particle during nucleosynthesis is at most a few MeV, too small to significantly excite gravitons. Furthermore, the horizon size is much larger than a mm so that the expansion of the universe is given by the usual 4-dimensional Friedmann equations. Issues concerning very early cosmology, such as inflation and baryogenesis, may change. This, however, is not necessary since there may be just enough space to accommodate weak-scale inflation and baryogenesis.

The cosmological models with large extra dimensions offer new ways of understanding the universe [

Some of the strong constraints come from the fact that at large temperatures, emission of gravitons into the bulk would be a very likely process. This would empty our brane from energy density and move all the energy into the bulk in the form of gravitons. To find out at which temperature this would cease to be a problem, one has to compare the cooling rates of the brane energy density via the ordinary Hubble expansions and the cooling via the graviton emission. The two cooling rates are given by

These two are equal at the so-called “normalcy temperature’’

This suggests that after inflation, the reheat temperature of the universe should be such that one ends up below the normalcy temperature, otherwise one would overpopulate the bulk with gravitons and overclose the universe. This is in fact a very stringent constraint on these models, since, for example, for

One of the most amazing predictions of theories with large extra dimensions would be that since the scale of quantum gravity is lowered to the TeV scale, one could actually form black holes from particle collisions at the LHC. Black holes are formed when the mass of an object is within the horizon size corresponding to the mass of the object.

What would be the characteristic size of the horizon in such models? This usually can be read off from the Schwarzschild solution which in four dimensions is given by

The exact solution gives a similar expression except for a numerical prefactor in the above equation. Thus, we know roughly what the horizon size would be, and a black hole will form if the impact parameter in the collision is smaller, than this horizon size. Then, the particles that collided will form a black hole with mass

The cross-section would thus be of order

Even though the extra dimensions look very exotic in the beginning, their inception in modern physics has helped us a great deal in understanding some of the long standing problems in particle physics and cosmology. Over the last twenty years, the hierarchy problem has been one of the central motivations for constructing extensions of the SM, with either new strong dynamics or supersymmetry stabilizing the weak scale. By contrast, ADD have proposed that the problem simply does not exist if the fundamental short-distance cutoff of the theory, where gravity becomes comparable in strength to the gauge interactions, is near the weak scale. This led immediately to the requirement of new sub-mm dimensions and SM fields localized on a brane in the higher-dimensional space. On the other hand, it leads to one of the most exciting possibilities for new accessible physics, since in this scenario the structure of the quantum gravity can be experimentally probed in the near future. In summary, there are many new interesting issues that emerge in ADD framework. Our old ideas about unification, inflation, naturalness, the hierarchy problem, and the need for supersymmetry are abandoned, together with the successful supersymmetric prediction of coupling constant unification [

The authors would like to thank Christos Kokorelis for many suggestions and useful correspondence. They thank Rizwan ul Haq Ansari and Jonathan Perry for going through the first draft of this paper.