The idea that spacetime could have more than four dimensions was first proposed by Theodor Kaluza and Oskar Klein at the beginning of the twentieth century. Higher-dimensional gauge theories are interesting for several reasons. In string theory, extra dimensions are required in order to cancel quantum anomalies. But even in an ordinary higher-dimensional quantum field theory context one can address several of the shortcomings of the Standard Model, for example, the gauge hierarchy problem, the smallness of neutrino masses, or the dark matter problem.
In order to have avoided detection so far, any extra dimensions have to be compact and small. Naively, the limits on the size of extra dimensions are small enough to rule out an observation of them in the forseeable future. However, in recent years, different models have been proposed that avoid these stringent constraints. These models could potentially be tested in high-energy experiments, such as the Large Hadron Collider (LHC) at CERN, Geneva, Switzerland, which has been operating since 2009.
Universal extra dimensions
In the universal extra dimensions (UED) scenario, all the fields of the Standard Model (SM) are allowed to probe the extra dimensions. In this model, there is a conserved quantum number known as KK parity, under which all the odd-numbered KK modes are charged. KK parity is analogous to R-parity in supersymmetric models. The conservation of KK parity has the consequence that KK particles can only be produced in pairs, and hence, experimental constraints are avoided. Like the ADD model (see below), the UED model would give rise to observable effects at the LHC. Another important implication of KK parity is that it ensures that the lightest KK particle (LKP) is stable, and hence, it could be a dark matter candidate. This kind of dark matter is known as Kaluza-Klein dark matter.
Figure: The person on the cable can only move backward and forward, not left and right, nor up and down, and experiences only one dimension. However, things that live on a smaller scale, such as ants, can move in an extra dimension - circularly around the cable.
Large extra dimensions
Large extra dimensions were first proposed by Arkani-Hamed, Dvali, and Dimopoulos (ADD) in 1998, their model being known as the ADD model. The novel feature of this model is the assumption that the SM fields are confined to a so-called brane, which is a four-dimensional manifold residing in the full bulk spacetime. This brane is to be identified with ordinary four-dimensional spacetime. Since the SM fields are not allowed to probe the extra dimensions, experimental constraints on their size are avoided to a large extent. Gravity, on the other hand, carries no SM charges and is allowed to probe the extra dimensions. In principle, the assumption that gravity lives in a higher-dimensional spacetime leads to deviations from Newton's inverse-square law at short distances. However, because of its weakness relative to the SM forces, gravity has only been tested down to distances of the order of micrometers, and hence, the experimental constraints are still quite weak.
In the ADD model, the internal space is assumed to be flat and compactified on an n-dimensional torus. One possible solution to the problem of the hierarchy between the electroweak scale and the radius of the internal space is to drop this assumption and consider different geometries. In particular, it has been argued that a compact hyperbolic internal space is a better alternative, and we have calculated the LHC signals for such a model.