Extra Dimensions - A Note

Extra Dimensions A Note Johar M. Ashfaque Abstract Extra dimensions may form the basis of some fundamental theories in physics. You may wonder why we would believe in extra dimensions, how they are described in physics and how we search for them. Space-time coordinates form the basis of Einstein’s theory of General Relativity, which describe the geometry of the universe, the behaviour of gravity and therefore the physics at large scales. However, for physics at the very small scales we resolve to the Standard Model (SM). The SM describes electromagnetism, the weak and the strong nuclear forces and it is build upon the framework of quantum mechanics. The quest now is to find one unifying theory that includes both GR and quantum mechanics in one formally coherent mathematical structure. Unfortunately so far it has been extremely difficult, if not impossible, to reconcile GR with quantum mechanics and in particular to unify gravity with the other forces of nature. The quest for unification may be considered as the holy grail of modern physics and it turns out that extra dimensions provide a very useful tool in addressing the problems. In fact, the most promising unifying theories, including String theory, are only consistently written in a universe that consists of 10 or 11 space-time dimensions. This implies the existence of an extra 6 or 7 spatial dimensions.

In models with extra dimensions the usual (3+1)-dimensional space-time xµ = (x0 , x1 , x2 , x3 ) is extended to include additional spatial dimensions parametrized by coordinates x4 , x5 , ..., x3+N where N is the number of extra dimensions. String theory arguments would suggest that N can be as large as 6 or 7. Depending on the type of metric in the bulk, ED models fall into one of the following two categories: • Flat, also known as universal ED models (UED). • Warped ED models.

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Flat Extra Dimensions

The metric on the extra dimensions is chosen to be flat. For simplicity N = 1 or N = 2 UEDs are considered. In the simplest case of N = 1 a compact extra dimension x4 would have the topology of a circle S 1 . However, to implement the chiral fermions of the SM in UED models one must use an orbifold S 1 /Z2 . The size of the extra dimension is simply parametrized by the radius of the circle R.

In the case of N = 2, there are are few possibilities for compactification. One of the many is known as the chiral square corresponding to T 2 /Z4 . The two extra dimensions have equal size and the boundary conditions are such that adjacent sides of the chiral square are identified. The notion of KK parity is fundamental to UED models, which springs from the geometrical symmetries of the compactification. In N = 1 case, KK parity corresponds to the reflection symmetry with respect to the centre of the line segment. Similarly, for the N = 2 case, KK parity is due to the symmetry with respect to the centre of the chiral square. In general, UED models respect KK parity and has important consequences for their collider and astrophysical phenomenology.

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The Mass Spectrum

Since the extra dimensions are compact the extra-dimensional components of the momentum of any SM particle are quantized in units of 1/R. From the four dimensional point of view these momentum components are interpreted as masses. Therefore in the UED models, each SM particle is accompanied by

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Figure 1: (a) Compactification of N = 1 extra dimension on a circle with opposite points identified. (b) Compactification of N = 2 extra dimensions on the chiral square with adjacent sides identified. In each case, the arrows indicate the corresponding identification The dots represent fixed boundary points. See [1] an infinite tower of heavy KK particles with masses n/R where integer n counts the number of quantum units of extra-dimensional momentum. All KK particles at a given n are said to belong to the n-th KK level and at leading order appear to be exactly degenerate. In the case of N = 1 minimal UED, the SM particle content is simply duplicated at the n = 1 level.

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Warped Extra Dimensions

The UED models are very peculiar: the extra dimension is an interval with flat background geometry and KK parity is realized as a geometric reflection about the midpoint of the extra dimension. In fact, KK parity has a larger parent symmetry, the KK number conservation, which is broken only by the interactions living on the orbifold boundary points.

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The Randall-Sundrum Solution

The Randall Sundrum solution is based on a slice of AdS5 bounded by two parallel (3 + 1)-branes, called the Planck-brane and the weak-brane. Our world is constrained to live on the weak brane, but the forces are unified on the Planck Brane. The background space-time metric of AdS5 is ds2 = e−2ky ηµν dxµ dν − dy 2 where k is the AdS curvature of the order of the Planck scale fixed by the bulk cosmological constant. The y dependence in the metric is known as the warp factor. The RS take into account the mass of the branes which leads to a deformation of the space-time in between the branes. This deformation is called warping and it affects the strength of gravity as we measure it at different points in space. RS state that all forces have equal strength at the Planck-brane, but due to the warping of space-time, gravity seems much weaker on the weak brane than at the Planck brane. Since AdS5 is dual to a 4D strongly coupled conformal field theory. It is conjectured that the RandallSundrum solution is dual to the composite Higgs models. 2

References [1] K. Kong, K. Matchev, G. Servant Extra Dimensions at the LHC, Particle Dark Matter: Observations, Models and Searches, Cambridge University Press, November 2013.

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