
From the 5^{th} dimension of the 1920's to the n^{th} dimension of 2000 In the beggining of the 20th century Gunnar Nordstrom, Theodor Kaluza and Oscar Klein independently proposed an extension of Einstein's newly invented theory of general relativity to include an extra dimension of space. As opposed to 4dimensional gravity, which is what we see and describe using Einstein's insight that gravity is the result of spacetime curvature, Nordstrom, Kaluza and Klein wrote down a theory of 5dimensional gravity. To make sense of this radical proposal, Klein suggested that the extra spatial dimension was "compactified". What does this mean? It means that he curled up the 5th dimension on a circle, a circle so microscopically small that it is not directly observed in everyday physics. The remarkable result of Nordstrom, Kaluza and Klein was that their theory of 5dimensional gravity unified 4dimensional gravity with 4dimensional electromagnetism. However our knowledge of the physics laws that describe elementary particles and their interactions was, at that time, neither experimentally nor theoretically advanced enough to be able to make something of this idea.
A couple of years ago Nima ArkaniHamed, Savas Dimopoulos and Gia Dvali [2] worked through the same idea. Namely that there can be more than 3 spatial dimensions and the extra dimensions can be curled up and thus compactified around circles (Figure 1) so that we cannot feel them and the compactification radius size is small enough that we have not measured them. The authors pointed out that we had not measured the gravitational force law at distances less than a millimeter (at that time). Thus if there were extra curled up dimensions that only affected gravity, they could be as large as a millimeter! When we read their paper in Nov 98, a bunch of us experimentalists were stunned by the fact that there might be extradimensionsinduced modifications of gravity competing with VanderWaals and Casimir forces at the submillimeter scale.
The paradigm of extra dimensions does much more that excite our imagination. It solves the so called "hierarchy of scales" problem. Simply stated the problem is that if gravity is unified with the rest of the forces it seems that this can only be done in energies close to the very beginning of the universe. The energy scale of Standard Model physics (called Weak in Figure 2) is many orders of magnitude away from the energy scale where the strength of the forces of nature are comparable (called Planck in Figure 2; this is actually 1 over the square root of Newton's gravitational constant in particle physics units The huge energy gap between these seemingly fundamental scales, the Weak and the Planck, would seem to be an "energy desert" unpopulated by physics. However from the extra dimensions point of view the true higher dimensional Planck scale can be as low as the Weak scale. The "energy desert" goes away. In fact the energy desert is replaced by a kind of "space oasis" in the the multidimensional space, as shown in fig. 2. This "space oasis" is described in words as follows:
KaluzaKlein states? Cannot picture it ([35]) If a spatial dimension is periodic (circle) then the momentum palong his dimension is quantized as shown in Figure 3; It is given by the expression p=k/R where R is the compactification radius of the extra dimensions .Figure 3. Momentum uantization in the extra dimension; Looks like particle in a box.
Figure 4. KaluzaKlein tower of states When the compactification radius becomes very large then these momentum states form an almost continuum tower called the KaluzaKlein tower of states as shown in Figure 3. You got to measure it to believe it : examples So what would we need to measure and how, in order to discover this extra space in which we are embedded? One category of experiments is those measuring Newton's Law at distances less than a millimeter. These are tabletop Cavendish [6] type experiments. They measure the gravitational strength down to 100 microns and with better than 10% precision thus being sensitive to many of the predictions of this new age multidimensional low scale quantum gravity. Small extra dimensions can be detectable at particle collisions. Within the ArkaniHamed, Dimopoulos and Dvali "space oasis" extra dimensions scenario there are two kind of searches that we can do to detect effects from gravity living in the "bulk". One type is the "direct kind" (e.g. our analysis with CDF data presented here): direct means that there is a reaction between Standard Model particles and in the final products of the reaction there is an actual Kaluza Klein graviton emitted into the bulk together with some other standard model stuff which remains in the "wall" or "brane". Take for example the standard model reaction of quark+antiquark to graviton+ gluon: this is the case of direct graviton emission the Kaluza Klein graviton is the missing energy and Figure 5
the jet in the detector is the gluon in this example. In Figure 5 there is a simulation of such an event. The Kaluza Klein graviton cannot be detected as it "escapes" in the bulk, but the gluon of the final state has a large energy and shows in the detector as a cluster of energy. The missing energy in the event arises from the energy and 3space momentum carried away by the KaluzaKlein graviton
The other type of searches is a reaction between standard model particles with final states being standard model particles. So, where is the graviton involved ? There are quantum fluctuations in this reaction that can involve gravitational interaction, so although the graviton is not directly produced is affecting the observables of the standard model final states. These are called "indirect searches" or searches where there is a "virtual graviton exchange". Our analysis looking for the graviton with missing energy and a jet The crucial part of the search is to determine all the standard processes that would account for events that exhibit large missing energy. In addition the way the detector is built, namely the nonhermeticity of the detector  the unavoidable fact that the detector has holes, is causing the presence of events with large missing energy. So not only do we need to know very well our standard model particles but also we need to understand the details of the experiment accurately and simulate them in order to have a description and a calculation of the number of cases where other processes mimic the exotic signal that we are hunting for. In this analysis more than a third of the events that we start with have large missing energy as a result of instrumental conditions.The steps for the analysis are the following: 1. We clean up the data from events that are obviously fake, due to the instrument for example. 2. We have the theoretical model which tells us what is the "signature" of the escaping graviton events  We simulate this, using the detector specifications and we see what we expect to observe from such a theory (in shapes and numbers). We also note down all the processes from the standard theory that have the same signature and we estimate how many we expect and with what shapes. For example: 

This analysis is published in the arXive hepex/0309051 For more info mail Kevin Burkett and Maria Spiropulu at CDFMovie created by Liubo Borissov. Find a short documentory about detectors here. 

References/Links [1]Kaluza Th.Sitzungsber. Preuss. Akad. Wiss. Berlin, Math Phys K 1 (1921) 966, Klein O. Z.Phys. 37 (1926) 895, Nature 118, 516 (1926); G. Nordstrom, Z. Phys. 15, 504 (1914) [2] The Hierarchy Problem and New Dimensions at a Millimeter, Nima ArkaniHamed, Savas Dimopoulos, Gia Dvali, Phys.Lett. B429 (1998) 263272 [3] Quantum gravity and extra dimensions at highenergy colliders, G.F.Giudice, R.Rattazzi and J.D.Wells, Nucl.\ Phys.\ B {\bf 544}, 3 (1999) [4] On KaluzaKlein states from large extra dimensions, T.Han, J.D.Lykken and R.J.Zhang, Phys. Rev. D 59, 105006 (1999) [5] For a review, see JoAnne Hewett and Maria Spiropulu, Ann. Rev. Nucl. Part. Sci., Vol. 52: 397424 (2002) [6] Laboratory Tests of Gravitational Physics [7] The Physics of Extra Dimensions Lecture Slides/Videos from AAAS 2003
