Search for Extra Dimensions using Missing Energy at CDF


From the 5th dimension of the 1920's to the nth 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 4-dimensional 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 5-dimensional 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 5-dimensional gravity unified 4-dimensional gravity with 4-dimensional 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.

Figure 1

A couple of years ago Nima Arkani-Hamed, 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 extra-dimensions-induced modifications of gravity competing with Van-der-Waals and Casimir forces at the sub-millimeter 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 multi-dimensional space, as shown in fig. 2. This "space oasis" is described in words as follows:

  • The world we are able to see and measure today is confined on a 4 dimensional wall.
  • Gravity exists in a 4+n "bulk" where n is the number of extra dimensions and "bulk" is the spatially enriched volume.
  • Gravity feels weak to us on the wall because of the enormity of the bulk volume.
  • Gravitons, the gravity mediators, live in the "bulk" Their bulk momentum is interpreted by us as mass. Thus we see a tower of massive graviton states that we call Kaluza-Klein states. The masses take discrete values because the extra dimensions have finite size.
  • Each Kaluza Klein graviton state couples to our wall with 4- dimensional Planck suppressed strength.
  • The number of Kaluza Klein states is huge; for a process involving energy E the number goes as ~(ER)n where R is the compactification length and n is the number of extra dimensions.
  • The sum over all Kaluza Klein states is suppressed by the 4+n dimensional Planck scale, not the 4 dimensional Planck scale. If this scale is TeV then we do have sizable graviton reactions to measure..

Kaluza-Klein states? Cannot picture it ([3-5])

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.

From our lower dimensions point of view these momentum states that live in the higher dimension look like massive particles with mass given by expression (2) where m0 is the 0th mass-less state, which lives on the 4-dimensioanl wall.

m2=m02+(k/R)2    (2)

Figure 4. Kaluza-Klein tower of states

When the compactification radius becomes very large then these momentum states form an almost continuum tower called the Kaluza-Klein 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 table-top 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 Arkani-Hamed, 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 3-space momentum carried away by the Kaluza-Klein graviton

It is possible that we live in a world that has more than the usual 1 time + 3 space dimensions of everyday experience. Indeed, one way to explain the peculiar properties of gravity is that the universe extends in 4+n dimensional space (the bulk) while we are trapped to live in the familiar 4 dimensional world (the brane).


The movie shown here illustrates a process that might occur in such a scenario. The bulk here is shown as the big black void containing our brane, which is represented by the box grid populated by galaxies, stars and familiar elementary particles.



Two such particles, for example a proton and an antiproton, shown as generic blue balls, can collide on the brane, and if the energy is high enough, produce a graviton (the particle that mediates the force of gravity), along with jets of familiar elementary particles (all shown as generic blue balls). The graviton flies out of the brane (the big blue ball) carrying away energy and momentum. Because the graviton moves off into the extra dimensions, we call it a Kaluza-Klein particle.

An observer on the brane witnessing the outcome of the collision would see the usual particles produced in such experiments, with a large imbalance of energy and momentum. Instead of detecting the Kaluza-Klein graviton directly, we observe its "missing energy" signature. This is similar to the way that the existence of neutrinos is inferred in collider experiments.

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 non-hermeticity 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:

In the figure above we see in colors what we expect from the standard model and in points what from a graviton signal. The overlay here is for the sake of shape comparison only and shows that the extra dimensions signal we look for, extends at high missing energies. 4. We optimize (essentially maximize the signal over background ratio) the search to be more sensitive to find the signal. 5. After all is calculated and the optimization is done, then we compare how many events we expect and how many we actually observe in the data. If there is a statistically significant excess we would have a clue for some new physics (we have to work harder to say whether that might have been extra dimensions or what).

In this analysis we came even within uncertainties so we can set a 95% C.L. bound on the number of signal events above which we would have seen the signal if it was there (upper limit on the number of signal events). The number is 62. If a new physics model (with some input parameters) after the analysis is applied, expects more than 62 events passing the analysis requirements then for some region of the input parameters the model is not viable. So we turn our result into lower limits on the effective Planck scale for n=2,4,6 extra dimensions. For a particular compactification method this can be expressed into an upper limit on the size of the extra dimension (R):

This analysis is published in the arXive hep-ex/0309051

For more info mail Kevin Burkett and Maria Spiropulu at CDF
Movie created by Liubo Borissov. Find a short documentory about detectors here.


[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 Arkani-Hamed, Savas Dimopoulos, Gia Dvali, Phys.Lett. B429 (1998) 263-272

[3] Quantum gravity and extra dimensions at high-energy colliders, G.F.Giudice, R.Rattazzi and J.D.Wells, Nucl.\ Phys.\ B {\bf 544}, 3 (1999)

[4] On Kaluza-Klein 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: 397-424 (2002)

[6] Laboratory Tests of Gravitational Physics

[7] The Physics of Extra Dimensions Lecture Slides/Videos from AAAS 2003

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