Extra Dimensions

Introduction
There has been a revolution in our thinking about extra dimensions. A new understanding of the feasibility oflocalizing four dimensional gauge theories in higher dimensionalspacetimes has led to a variety of phenomenologically viable models,and even to the possibility of localizing gravity. Unlike older theories ofextra dimensions, much of the focus now is on extra dimensions with sizes on the order of one thousandth of a proton width or larger! Thus, there is a potential for discovery at current and soon-to-be-completed colliders,and in some cases table-top experiments. In addition there are tremendous implications for cosmology.
Since Einstein forced physicists to think of time as a fourth dimensionthere have been speculations about a fifth and even higher dimensions. In order to avoid gross violations with experiment and even everyday experienceit was thought that any extra dimensions would have to be compact (like a circle or a sphere) and that their effects, if any, on present day experimentswould be unobservably small. There was the possibility that ordinary particles thatwe know are somehow restricted to a membrane with three spatial dimensions whichis embedded in a higher dimensional spacetime. This possibility was not taken veryseriously since it was not known how to realize this in a quantum field theory in arobust way. However recent developments in string theory show that such membranes arerequired for the consistency of string theory, and so they have acquired a new-foundrespectability and the nickname of branes. What is most exciting about this development is that when realistic models are built that incorporate such branesthere are often experimental signatures that can be seen at current and upcoming experiments.These signatures range from observing a change in the strength of gravity at sub-millimeter distances in table-top experimentsto the production of gravitons with momentum in an extra dimension at colliders. There are also important effects in cosmology which is about to undergo its own revolutiondue to an upcoming abundance of satellite data on the cosmic microwave background.
It is this potential for verification or elimination by experimental data thatis fueling the drive to understand the new possibilities of extra dimensions.
Background

Kaluza-Klein theories
Concrete proposals for the existence of extra dimensions date back to the 1920s, when Kaluzaand Klein attempted to unify gravity and electromagnetism in a five dimensional (5D) theory. If ahypothetical extra dimension were compact (a circle for example) then the propagation ofstandard model particles like the electron through this extra dimension would obviously havephysical consequences. However, if the size of the extra dimension was much smaller than thewavelength of the particles we were observing then the extra dimension could remain hidden.More specifically, with a compact extra dimension the momenta along this direction would bequantized, producing for each field a tower of momentum eigenstates (Kaluza-Klein modes) which wouldlook like massive excitations of the particle at the bottom of the tower which has no momentum in the extra dimension. Since we do not observe any such towers in nature, at least below a few hundred GeV inenergy, we can conclude that extra dimensions in which ordinary matter or gauge bosons can propagatemust have extremely small sizes (i.e. on the order of the inverse weak scale). On the theoreticalside, string theory actually requires extra dimensions, but their natural size, the Planck scale, is even smaller (1 followed by 15 zeroestimes smaller) than the direct experimental limit. Until quite recently it was thought that these arguments required extra dimensions to be irrelevantto long distance physics, however in the mid 1990s our understanding of string theory wasrevolutionized. Through the work of Polchinski and others it became clear that the consistency ofstring theory required the existence of new non-perturbative soliton-like objects called Dirichlet pbranes (Dp branes), which were generalizations of ordinary membranes with p spatial dimensions. Sincestrings could end on Dp branes with Dirichlet boundary conditions (hence the name) itsoon became clear that the low energy physics of N D3 branes was just an SU(N) gauge theory with N = 4 supersymmetry (SUSY)which was constrained to live in the four dimensional spacetime occupied by the D3branes.
Large extra dimensions
Very little time passed before phenomenologists used the idea of matter and gauge fields localized onbranes embedded in extra dimensions. Arkani-Hamed, Dimopoulos, and Dvali proposed thatthe standard model fields live on such a brane embedded in 2 to 6 extra dimensions. Sinceordinary particles did not propagate in the extra dimensions, the usual direct bounds did not apply,but since gravity cannot be restricted to the brane, gravitational effects provided thecrucial constraints. In this type of theory the observed Planck scale (1 followed by 18 zeroes times the proton mass) is only a low energy effectivecoupling that arises from integrating over the extra dimensions. With n extra dimensions the square of the Planck scale isrelated to the underlying gravitational scale (where gravity gets strong) to the 2+n power times the volume of the extra dimensions. At scales smaller than the size of the extradimensions, gravity can become much stronger, allowing for theradical possibility of identifyingthe scale where gravity gets strong with the weak scale (around 1000 times the proton mass). Somewhat surprising, with two extra dimensions the then-current gravitational bounds allowed the size of the extra dimensions to be as large as 1 mm with theunderlying gravitational scale (possibly the string scale!) to be at the weak scale. Refined table-top Cavendish experiments havealready been built and pushed the present direct experimental limit to 0.2 mm.Having three or more extradimensions allowed for a smaller extra-dimensional radius with same gravitational scale. Thus theproblem of explaining the huge hierarchy between Planck scale and the weak scaleis translated into the problem of explaining why the extra dimensions are stabilized at such largeradii relative to the weak scale.
Warped extra dimensions
The discussion so far assumes that the extra dimensions are flat or at least weakly curved, butanother startling possibility was suggested by Randall and Sundrum. They studied an extradimension that was strongly curved (or "warped") by a large negative cosmological constant.(This type of space is known in the literature as an anti-de Sitter (AdS) space, since de Sitter studieda universe with a positive cosmological constant.) They found models where the effective four dimensional (4D) cosmologicalconstant was zero, and where the massless 4D graviton mode was localized on a brane at oneend of the finite (or semi-infinite) extra dimension. Essentiallythe equation for the massless graviton mode is the precise analog of a Schrodinger equation with a binding potential.This provided a new way to make gravitymuch weaker than the weak interactions; if we happen to live on a brane where the graviton isnot localized, then its wavefunction on our brane can be exponentially suppressed. Again theproblem of understanding the hierarchy of Planck and weak scales is translated into understandingthe size of the extra dimension. The Randall-Sundrum (RS) scenario also has some motivation from string theory. A fewyears before Maldacena had proposed a correspondence between supergravity on AdSbackgrounds and conformal field theories (CFT). This AdS/CFT correspondence relatedsupergravity on AdS_5 x S^5 (AdS_5 is a 5 dimensional AdS space and S^5 is a 5 dimensional sphere that acts as a compactinternal space) to a 4D SU(N) gauge theory with N = 4 SUSY inthe large N limit. If one integrates out the KK modes of thesphere and truncates the "radial" direction of the AdS_5 space to a strip, this would yield a particular supersymmetric version of the RSscenario. In fact it is thought thatvariants of the Randall-Sundrum model correspond to some unknown, almost conformal, theoriescoupled to gravity. Subsequent work showed how to make precise calculations with the AdS/CFT correspondence (and its generalizations) which allows for many interesting predictions for4D gauge theories. For example using the correspondence between supergravity onblackhole AdS backgrounds and non-SUSY QCD, my collaborators and I calculated ratios ofglueball masses in three and four dimensions in a strong coupling, large N, limit ofQCD. Although the underlying theory is quite different from real QCD, we found that thesemass ratios are in good agreement with the available lattice data. One of the most remarkableaspects of this correspondence is that a non-perturbative quantum field theorycalculation is reduced to a simpleweakly coupled classical field theory problem. This means that the AdS/CFT correspondence maybe a useful calculational tool in the future for aspects of phenomenology that rely on non-perturbative dynamics (for example dynamical SUSY breaking).

Deconstruction
Recently another new tool for analyzing extra dimensions or reducing extra dimensionalmodels to relatively simple 4D models has been developed whichgoes by the name ``deconstruction". Essentially a 5D gauge theory is latticized in asequence of 4D gauge theories that are connected by link fields. The link fields connecttwo neighboring gauge groups by having gauge interactions under both groups. When the linkfield connecting N gauge groups get vacuum expectation values, the gauge groups break tothe diagonal group, leaving one massless gauge field and a tower of N-1 massivegauge fields. In the large N limit this tower reproduces the KK tower of a 5D gauge field belowthe energy scale associated with the lattice spacing.Since 5D gauge theories are not renormalizable, they must be equipped with a cutoff. Thedeconstruction (latticization) described above is the simplest way to introduce a gauge invariantcutoff and allow for reliable calculations. It also has the potential, by restrictingto a few lattice sites, to lead to new 4D theoriesthat mimic the low-energy behavior of a 5D theory. These 4D theories have more flexibilitythan 5D theories since they do not have to respect 5D Lorentz invariance, and thus theycan be further generalized to completely new 4D theories that can retain some of the moreinteresting qualities of 5D theories.
My recent research
Cosmology of extra dimensional theories
Cosmology offers particle physicists a method of testing models that is complementary to accelerator experiments. Particles that cannot be produced easily in accelerators can have drastic effects in the early universe. This can be seen in the new theories of gravity that involve sub-millimeter extra dimensions. My collaborators and I recently put severe constraints on a class of such theories. In these models, oscillations of the light field (the radion) that determines the size of the extra dimensions can over-close the universe. It had been proposed that a period of late inflation could solve this problem, however we found that the required inflaton scale is so low that it cannot successfully reheat the universe. We also found that in the 5D Randall-Sundrum scenario for solving the hierarchy problem, the extra dimensional gravity modes (in particular the radion)can make the universerapidly (on the scale of Planck times) unstable to collapse. We also showed that when the radion potential is stabilized (for exampleby a Goldberger-Wise mechanism) then a viable cosmology is possibleand as a result the radion has Higgs-like interactions. The radion istherefore potentially observable at the Tevatron and LHC.We also used string theory techniques (holographicrenormalization group) to analyze the model of Gregory, Rubakov, and Sibiryakov where gravity is four dimensional at intermediate distances, but five dimensional at long and short distance scales. In these models the massless 4D gravitonis quasi-localized, that is, it is localized but unstable with a very long lifetime (thespectrum of the analog Schrodinger problem is a continuum and a zero-energy resonance). The holographic renormalization grouptechnique enabled us to derive a low-energy effective theory that captured the long distancephysics (the physics below the renormalization scale). This enabled us to relate this model to a seemingly different (but equivalentat low energies) model proposed by Dvali, Gabadadze, and Porrati. Most importantly it allowed for a simple descriptionof the cosmology of a model with a quite complicated graviton propagator. Though such modelsturned out to be unstable, it is worthwhile trying to understand such extensions of gravityin light of the recent cosmological data favoring an accelerating universe.
Electroweak symmetry breaking
The Randall-Sundrum scenario for solving the hierarchy problem poses an interesting puzzle for electroweak symmetry breaking. It is widely believed that a morefundamental description of this scenario would involve a strongly-coupled, almost conformaltheory that breaks electroweak symmetry; but what could this theory be? In the 1980s andearly 1990s people considered the possibility that some strong (QCD-like) gauge dynamics(known as technicolor)does indeed perform such a breaking. I, among others , showed that the precisionelectroweak data on the S parameter essentially rules out the QCD-like technicolor scenario.In QCD-like theories it was possible to show that the contributions to S weregenerically large and had a positive sign, while experiment favors a small negative value forS.More recently Luty, Grant, and I considered whether or not some strongly-coupled SUSY gauge theories would be able to break electroweak symmetry without violating the experimental constraints. (This effort was quite distinct from previous attempts to combine SUSY and technicolor, since the realistic models all relied on non-SUSY dynamics to perform thesymmetry breaking.)Using Seiberg duality techniques we showed that a non-renormalization theorem holds for themodels we considered which forbids any contributions to the superpotential term which contributes to S, and that even afterSUSY breaking the contributions to S were so small as to be unobservable. However there are contributions to S from Kahler potential terms, which are of unknown sign. Theusual field theory dualities are not powerful enough to determine the sign of these contributions.After our paper appeared, Klebanov and Strassler showed thatthe large N limit of the models like the one we consideredcould be described (via a generalized AdS/CFT correspondence)by supergravity on a deformed conifold that asymptotically approaches AdS_5 with a compact internal space.The low-energy effective theory of this supergravity would be essentially a generalizedRS model.In principle one should be able to calculate the sign of S in the large N limit ofour models through a supergravity calculation in this generalized RS model.

GUT Breaking and extra dimensions
Most recently my collaborators and I realized that the Scherk-Schwarz mechanism ofsymmetry breaking could be applied not only to Grand UnifiedTheories in extra-dimensions but also tomodels with deconstructed extra dimensions. This led us to models with just a few gauge groups, which can easily solve the doublet-triplet splitting problem, suppress protondecay to the edge of detectability, and where gauge couplingunification could be reliably calculated without hard cutoffs. It also led to othermodels that shared these positive features but which could not be derived as a deconstructionof a 5D theory. These models also related the arrangement of lattice sitesto flavor physics: in particular, the hierarchies observed in the CKM matrix were related tothe number of lattice sites separating different generations.

Further Reading
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2) The Hierarchy Problem and New Dimensions at a Millimeter,by N. Arkani-Hamed, S. Dimopoulos and G. Dvali,Phys. Lett. B429, 263 (1998) hep-ph/9803315
3) Weak Scale Superstrings,
by J. D. Lykken,Phys. Rev. D 54, 3693 (1996) hep-th/9603133
4) A Large Mass Hierarchy from a Small Extra Dimension,
by L. Randall and R. Sundrum,Phys. Rev. Lett. 83, 3370 (1999) hep-ph/9905221;
An Alternative to Compactification,by L. Randall and R. Sundrum,Phys. Rev. Lett. 83, 4690 (1999) hep-th/9906064
5)The shape of gravity,
by J. Lykken and L. Randall,JHEP 0006, 014 (2000) hep-th/9908076
6) The large N limit of superconformal field theories and supergravity,
by J. Maldacena,Adv. Theor. Math. Phys. 2, 231 (1998) hep-th/9711200
7)Gauge theory correlators from non-critical string theory,
by S. S. Gubser, I. R. Klebanov and A. M. Polyakov,Phys. Lett. B 428, 105 (1998) hep-th/9802109;
Anti-de Sitter space and holography,
by E. Witten,Adv. Theor. Math. Phys. 2, 253 (1998) hep-th/9802150
8) Glueball mass spectrum from supergravity,
by C. Csaki, H. Ooguri, Y. Oz and J. Terning,JHEP 9901, 017 (1999) hep-th/9806021;
Large N QCD from rotating branes,
by C. Csaki, Y. Oz, J. Russo and J. Terning,Phys. Rev. D 59, 065012 (1999) hep-th/9810186;
Supergravity models for 3+1 dimensional QCD,
by C. Csaki, J. Russo, K. Sfetsos and J. Terning,Phys. Rev. D 60, 044001 (1999) hep-th/9902067
9) (De) constructing dimensions,
by N. Arkani-Hamed, A. G. Cohen and H. Georgi,Phys. Rev. Lett. 86, 4757 (2001) hep-th/0104005;
Gauge invariant effective Lagrangian for Kaluza-Klein modes,
by C. T. Hill, S. Pokorski and J. Wang,Phys. Rev. D 64, 105005 (2001) hep-th/0104035
10) Late inflation and the moduli problem of sub-millimeter dimensions,
by C. Csaki, M. Graesser and J. Terning,Phys. Lett. B 456, 16 (1999) hep-ph/9903319
11) Cosmology of one extra dimension with localized gravity,
by C. Csaki, M. Graesser, C. Kolda and J. Terning,Phys. Lett. B 462, 34 (1999) hep-ph/9906513
12) Modulus stabilization with bulk fields,
by W. D. Goldberger and M. B. Wise,Phys. Rev. Lett. 83, 4922 (1999) hep-ph/9907447
13) Cosmology of brane models with radion stabilization,
by C. Csaki, M. Graesser, L. Randall and J. Terning,Phys. Rev. D 62, 045015 (2000) hep-ph/9911406
14) Holographic RG and cosmology in theories with quasi-localized gravity,
by C. Csaki, J. Erlich, T. J. Hollowood and J. Terning,Phys. Rev. D 63 (2001) 065019hep-th/0003076
15) Opening up extra dimensions at ultra-large scales,
by R. Gregory, V. A. Rubakov and S. M. Sibiryakov,Phys. Rev. Lett. 84, 5928 (2000) hep-th/0002072
16) 4D gravity on a brane in 5D Minkowski space,
by G. Dvali, G. Gabadadze and M. Porrati,Phys. Lett. B485, 208 (2000) hep-th/0005016
17) 4D models of Scherk-Schwarz GUT breaking via deconstruction,
by C. Csaki, G. D. Kribs and J. Terning,hep-ph/0107266
18) Radiative corrections to electroweak parameters in Technicolor theories,
by M. Golden and L. Randall,Nucl. Phys. B361, 3 (1991) ;
Large Corrections To Electroweak Parameters In Technicolor Theories,
by B. Holdom and J. Terning,Phys. Lett. B247, 88 (1990) ;
A new constraint on a strongly interacting Higgs sector,
by M. E. Peskin, T. Takeuchi, Phys. Rev. Lett. 65, 964 (1990) ;
Estimation of oblique electroweak corrections,
by M. E. Peskin, T. Takeuchi,Phys. Rev. D 46, 381 (1992).
19) Electroweak symmetry breaking by strong supersymmetric dynamics at the TeV scale,
by M. A. Luty, J. Terning and A. K. Grant,Phys. Rev. D 63, 075001 (2001) hep-ph/0006224
20) Supergravity and a confining gauge theory: Duality cascades and chiSB-resolution of naked singularities,
by I. R. Klebanov and M. J. Strassler,JHEP 0008, 052 (2000) hep-th/0007191
21) Locally localized gravity,
by A. Karch and L. Randall,JHEP 0105, 008 (2001) hep-th/0011156
22) Supergravity domain walls,
by M. Cvetic and H. H. Soleng,Phys. Rept. 282, 159 (1997) hep-th/9604090;
Supersymmetric domain wall world from D = 5 simple gauged supergravity,
by K. Behrndt and M. Cvetic,Phys. Lett. B 475, 253 (2000) hep-th/9909058