Are Extra Dimensions Real or Are They Just Science Fiction?

The recent proposal of theories with compactified large extra dimensions is reviewed. We pay especial attention to brane world models with low tension where the only relevant degrees of freedom at low energies are the Standard Model (SM) particles and the branons, which are transversal brane oscillations. By using an effective Lagrangian, we study some phenomenological consequences of these scenarios in a model independent way. ARE EXTRA DIMENSIONS REAL OR ARE THEY JUST SCIENCE FICTION? Most of the motivations for considering extra dimension have a theoretical origin. In the last thirty years most of the new developments in theoretical high energy physics required the introduction of extra dimensions. Well known examples are modern KaluzaKlein theories (KK), where the isometries of the extra dimensions appear as gauge symmetries, Supersymmetry (SUSY), which in the superfield formulation can be understood as a symmetry involving extra Grassmann dimensions, Supergravity (SUGRA), Superstrings defined consistently only in 10 dimensions and M-theory, which is supposed to be the eleven dimensional theory underlying the five known superstring theories and eleven dimensional SUGRA. One important exception to this rule are the Grand Unified Theories (GUT) but still the most interesting examples from the phenomenological point of view are supersymmetric since they give rise to gauge coupling unification. The first attempts to extend general relativity to include electromagnetism date back to Theodor Kaluza and Oscar Klein [1]. The discovery of 11D SUGRA produced a revival of the KK ideas in the early 80’s. The first string revolution, triggered by the realization that superstrings can provide a theory of gravitation instead of a theory of strong interactions, the cancelation of the anomalies and the discovery of the heterotic string which opened the door for phenomenology, traslated the interest to 10 dimensions with six dimensional compactified spaces (Calabi-Yau, orbifolds...). The second string revolution of the 90’s introduced new ideas such as non-perturbative strings, dualities, branes and the unification of the five known superstring theories in the conjectured Mtheory (these two periods of development of string theory can be reviewed in [2] and [3] respectively). 1 Talk given by A. Dobado in the X Mexican School of Particles and Fields, Playa del Carmen, México, 2002 The main phenomenological problem of the old string theories is that they could not be tested since stringy effects were expected to appear at the Plank scale MP which is of the order of 1019GeV. However the new ideas coming from M-theory have inspired new scenarios that could be testable. These scenarios were developed in principle to address the hierarchy problem by putting it in a different setting. The first one was proposed by Arkani-Hamed, Dimopoulos and Dvali (ADD) [4]. The main idea is that our universe is a 3-brane living in a higher D = 4 +N dimensional space (the bulk space) being the extra dimensions compactified to some small volume (Brane World scenario). In this picture the SM particles are confined to the 3-brane but gravitons can propagate along the whole bulk space. Now the fundamental scale of gravity is not the Planck scale any more but another scale MD which is supposed to be of the order of the electroweak scale in order to solve the hierarchy problem. Then it is possible to find the relation M2 P = VNM 2+N D , where VN is the volume of the compatified extra dimension manifold B. Thus the hierarchy between the Planck and the electroweak (TeV) scale is generated by the large volume of the extra dimensions. The typical size R of the extra dimensions ranges from a fraction of mm for N = 2 to about 10 fm for N = 6 (the case N = 1 is already ruled out by the observations in our solar system). The most interesting property of the ADD scenario (in which we will concentrate in the rest of this work) is that it is compatible with the present experimental data, but at the same time it gives rise to many new phenomena that could be tested in the near future. There are also scenarios where the scale of the extra dimensions R is of the order of 1 TeV−1 [5]. All or some of the SM particles are allowed to propagate along the bulk. This set up is quite appropriate for model building and to deal with gauge coupling unification, SUSY breaking, neutrino spectrum, fermion masses and many other things even if it does not solve the hierarchy problem. In addition there are also models where the hierarchy is generated by the curvature of the extra dimensions as it is the case of the Randall-Sundrum model where the geometry of the space-time is AdS5 and thus it cannot be factorized [6]. IS OUR UNIVERSE A 3-BRANE? The most obvious consequence of the the Brane World scenario is the modification of the Newton’s Law at short distances i.e. distances of the order of R. Some recent experiments are trying to test the Newton Law’s at the submillimeter scale to explore this possibility (see for instance [7]). From the point of view of particle physics the main new effects in the ADD scenario are related with the KK mode expansion of the bulk gravitational field gμν(x,~y) = ∑ ~k g ~k μν(x)e i~k.~y/R (1) where a toroidal compactification has been assumed for simplicity, ~y represents the N extra dimension coordinates and ~k is a N dimensional vector with components ka = 0,1,2, .... Therefore a bulk graviton can be understood as a KK tower of four dimensional massive gravitons with masses of the order of k/R with k being any natural number (for the N = 1 case) so that the distance in the mass spectrum between two consecutive KK gravitons is of the order of 1/R. This means in particular that the KK graviton spectrum can be considered as approximately continuous for large extra dimensions. In principle we expect two kinds of effects from the KK graviton tower, namely graviton production and virtual effects on other particle production or observables (see for instance [8] and references therein). The rates for the different processes can be computed by linearizing the bulk gravitational field and by coupling the graviton field to the SM energy momentum tensor T μν SM . Then expanding the gravitational field in terms of the KK modes one finds the corresponding Feynman rules. As an example one could consider the process e+e− → γ + G where G stands for any KK graviton. To compute the total cross section one must sum (integrate for large extra dimensions) over all the KK gravitons. The result is