Extra Dimensions
Introduction
There has been a revolution in our thinking about extra dimensions. A new understanding of the feasibility of localizing four dimensional gauge theories in higher dimensional spacetimes has led to a variety of phenomenologically viable models, and even to the possibility of localizing gravity. Unlike older theories of extra 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 dimension there have been speculations about a fifth and even higher dimensions. In order to avoid gross violations with experiment and even everyday experience it 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 experiments would be unobservably small. There was the possibility that ordinary particles that we know are somehow restricted to a membrane with three spatial dimensions which is embedded in a higher dimensional spacetime. This possibility was not taken very seriously since it was not known how to realize this in a quantum field theory in a robust way. However recent developments in string theory show that such membranes are required for the consistency of string theory, and so they have acquired a new-found respectability and the nickname of branes. What is most exciting about this development is that when realistic models are built that incorporate such branes there 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 experiments to 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 revolution due to an upcoming abundance of satellite data on the cosmic microwave background.
It is this potential for verification or elimination by experimental data that is 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 Kaluza and Klein attempted to unify gravity and electromagnetism in a five dimensional (5D) theory. If a hypothetical extra dimension were compact (a circle for example) then the propagation of standard model particles like the electron through this extra dimension would obviously have physical consequences. However, if the size of the extra dimension was much smaller than the wavelength 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 be quantized, producing for each field a tower of momentum eigenstates (Kaluza-Klein modes) which would look 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 in energy, we can conclude that extra dimensions in which ordinary matter or gauge bosons can propagate must have extremely small sizes (i.e. on the order of the inverse weak scale). On the theoretical side, string theory actually requires extra dimensions, but their natural size, the Planck scale, is even smaller (1 followed by 15 zeroes times smaller) than the direct experimental limit.
Until quite recently it was thought that these arguments required extra dimensions to be irrelevant to long distance physics, however in the mid 1990s our understanding of string theory was revolutionized. Through the work of Polchinski and others it became clear that the consistency of string theory required the existence of new non-perturbative soliton-like objects called Dirichlet p branes (Dp branes), which were generalizations of ordinary membranes with p spatial dimensions. Since strings could end on Dp branes with Dirichlet boundary conditions (hence the name) it soon 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 D3 branes.
Large extra dimensions
Very little time passed before phenomenologists used the idea of matter and gauge fields localized on branes embedded in extra dimensions. Arkani-Hamed, Dimopoulos, and Dvali proposed that the standard model fields live on such a brane embedded in 2 to 6 extra dimensions. Since ordinary 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 the crucial constraints. In this type of theory the observed Planck scale (1 followed by 18 zeroes times the proton mass) is only a low energy effective coupling that arises from integrating over the extra dimensions. With n extra dimensions the square of the Planck scale is related 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 extra dimensions, gravity can become much stronger, allowing for the radical possibility of identifying the 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 the underlying gravitational scale (possibly the string scale!) to be at the weak scale. Refined table-top Cavendish experiments have already been built and pushed the present direct experimental limit to 0.2 mm. Having three or more extra dimensions allowed for a smaller extra-dimensional radius with same gravitational scale. Thus the problem of explaining the huge hierarchy between Planck scale and the weak scale is translated into the problem of explaining why the extra dimensions are stabilized at such large radii relative to the weak scale.
Warped extra dimensions
The discussion so far assumes that the extra dimensions are flat or at least weakly curved, but another startling possibility was suggested by Randall and Sundrum. They studied an extra dimension 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 studied a universe with a positive cosmological constant.) They found models where the effective four dimensional (4D) cosmological constant was zero, and where the massless 4D graviton mode was localized on a brane at one end of the finite (or semi-infinite) extra dimension. Essentially the 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 gravity much weaker than the weak interactions; if we happen to live on a brane where the graviton is not localized, then its wavefunction on our brane can be exponentially suppressed. Again the problem of understanding the hierarchy of Planck and weak scales is translated into understanding the size of the extra dimension.
The Randall-Sundrum (RS) scenario also has some motivation from string theory. A few years before Maldacena had proposed a correspondence between supergravity on AdS backgrounds and conformal field theories (CFT). This AdS/CFT correspondence related supergravity 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 compact internal space) to a 4D SU(N) gauge theory with N = 4 SUSY in the large N limit. If one integrates out the KK modes of the sphere and truncates the "radial" direction of the AdS_5 space to a strip, this would yield a particular supersymmetric version of the RS scenario. In fact it is thought that variants of the Randall-Sundrum model correspond to some unknown, almost conformal, theories coupled to gravity.
Subsequent work showed how to make precise calculations with the AdS/CFT correspondence (and its generalizations) which allows for many interesting predictions for 4D gauge theories. For example using the correspondence between supergravity on blackhole AdS backgrounds and non-SUSY QCD, my collaborators and I calculated ratios of glueball masses in three and four dimensions in a strong coupling, large N, limit of QCD. Although the underlying theory is quite different from real QCD, we found that these mass ratios are in good agreement with the available lattice data. One of the most remarkable aspects of this correspondence is that a non-perturbative quantum field theory calculation is reduced to a simple weakly coupled classical field theory problem. This means that the AdS/CFT correspondence may be 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 dimensional models to relatively simple 4D models has been developed which goes by the name ``deconstruction". Essentially a 5D gauge theory is latticized in a sequence of 4D gauge theories that are connected by link fields. The link fields connect two neighboring gauge groups by having gauge interactions under both groups. When the link fields connecting N gauge groups get vacuum expectation values, the gauge groups break to the diagonal sub-group, leaving one massless gauge field and a tower of N-1 massive gauge fields. In the large N limit this tower reproduces the KK tower of a 5D gauge field below the energy scale associated with the lattice spacing. Since 5D gauge theories are not renormalizable, they must be equipped with a cutoff. The deconstruction (latticization) described above is the simplest way to introduce a gauge invariant cutoff and allow for reliable calculations. It also has the potential, by restricting to a few lattice sites, to lead to new 4D theories that mimic the low-energy behavior of a 5D theory. These 4D theories have more flexibility than 5D theories since they do not have to respect 5D Lorentz invariance, and thus they can be further generalized to completely new 4D theories that can retain some of the more interesting 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 universe rapidly (on the scale of Planck times) unstable to collapse. We also showed that when the radion potential is stabilized (for example by a Goldberger-Wise mechanism) then a viable cosmology is possible and as a result the radion has Higgs-like interactions. The radion is therefore potentially observable at the Tevatron and LHC. We also used string theory techniques (holographic renormalization 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 graviton is quasi-localized, that is, it is localized but unstable with a very long lifetime (the spectrum of the analog Schrodinger problem is a continuum and a zero-energy resonance). The holographic renormalization group technique enabled us to derive a low-energy effective theory that captured the long distance physics (the physics below the renormalization scale). This enabled us to relate this model to a seemingly different (but equivalent at low energies) model proposed by Dvali, Gabadadze, and Porrati. Most importantly it allowed for a simple description of the cosmology of a model with a quite complicated graviton propagator. Though such models turned out to be unstable, it is worthwhile trying to understand such extensions of gravity in light of the recent cosmological data favoring an accelerating universe.
GUT Breaking and extra dimensions
My collaborators and I realized that the Scherk-Schwarz mechanism of symmetry breaking could be applied not only to Grand Unified Theories in extra-dimensions but also to models 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 proton decay to the edge of detectability, and where gauge coupling unification could be reliably calculated without hard cutoffs. It also led to other models that shared these positive features but which could not be derived as a deconstruction of a 5D theory. These models also related the arrangement of lattice sites to flavor physics: in particular, the hierarchies observed in the CKM matrix were related to the number of lattice sites separating different generations.
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 more fundamental description of this scenario would involve a strongly-coupled, almost conformal theory that breaks electroweak symmetry; but what could this theory be? In the 1980s and early 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 precision electroweak 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 were generically large and had a positive sign, while experiment favors a small negative value for S.
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 the symmetry breaking.) Using Seiberg duality techniques we showed that a non-renormalization theorem holds for the models we considered which forbids any contributions to the superpotential term which contributes to S, and that even after SUSY 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. The usual field theory dualities are not powerful enough to determine the sign of these contributions. After our paper appeared, Klebanov and Strassler showed that the large N limit of the models like the one we considered could 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 generalized RS model. In principle one should be able to calculate the sign of S in the large N limit of our models through a supergravity calculation in this generalized RS model.
Most of my recent work has focused on the possibility of higgsless electroweak symmetry breaking in extra dimensions. My collaborators and I have shown that WW scattering is unitary in a five dimensional theory without a Higgs, provided that the gauge symmetry breaking is achieved through Dirichlet boundary conditions. In a warped anti-de Sitter (AdS) background (like the Randall-Sundrum model) a custodial symmetry can ensure the correct ratio for the W and Z masses [9]. We found that these higgsless models can be consistent with precision constraints on S and T parameters through either brane kinetic terms or requiring the light fermions to be roughly uniformly distributed in the extra dimension. Maintaining the correct b quark coupling to the Z while getting the correct top quark mass is a more serious problem. We proposed two solutions based on the idea that the third generation may couple to a different conformal field theory (CFT) or, equivalently through the AdS/CFT correspondence, live in a different warped space from the first two generations and have a separate TeV brane. We also analyzed how quark and lepton masses can be produced in a higgsless theory via boundary conditions in the extra dimension.
We were also able to use these model building ideas to propose a warped five dimensional lattice construction of a four dimensional chiral gauge theory. This may be a solution of a long-standing problem in lattice gauge theory, and open up new directions of research.
Further Reading
1) T. Kaluza, Preus. Acad. Wiss. K1, 966 (1921) ; O. Klein, Z. Phys. 37, 895 (1926) .
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/96031334) 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/99060645) The shape of gravity,
by J. Lykken and L. Randall, JHEP 0006, 014 (2000) hep-th/99080766) The large N limit of superconformal field theories and supergravity,
by J. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) hep-th/97112007) Gauge theory correlators from non-critical string theory,
Anti-de Sitter space and holography,
by S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428, 105 (1998) hep-th/9802109;
by E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998) hep-th/98021508) 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/99020679) (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/010403510) 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/990331911) 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/990651312) Modulus stabilization with bulk fields,
by W. D. Goldberger and M. B. Wise, Phys. Rev. Lett. 83, 4922 (1999) hep-ph/990744713) 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/991140614) 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) 065019 hep-th/000307615) 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/000207216) 4D gravity on a brane in 5D Minkowski space,
by G. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. B485, 208 (2000) hep-th/000501617) 4D models of Scherk-Schwarz GUT breaking via deconstruction,
by C. Csaki, G. D. Kribs and J. Terning, hep-ph/010726618) 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/000622420) 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/000719121) Gauge theories on an interval: Unitarity without a Higgs,
C. Csaki, C. Grojean, H. Murayama, L. Pilo and J. Terning, Phys. Rev. D 69 (2004) 055006, hep-ph/030523722) Towards a realistic model of Higgsless electroweak symmetry breaking,
C. Csaki, C. Grojean, L. Pilo and J. Terning, Phys. Rev. Lett. 92 (2004) 101802, hep-ph/030803823) Oblique corrections from Higgsless models in warped space,
G. Cacciapaglia, C. Csaki, C. Grojean and J. Terning, Phys. Rev. D 70 (2004) 075014, hep-ph/0401160.24) Curing the ills of Higgsless models: The S parameter and unitarity,
G. Cacciapaglia, C. Csaki, C. Grojean and J. Terning, Phys. Rev. D 71 (2005) 035015, hep-ph/0409126.25) Top and bottom: A brane of their own,
G. Cacciapaglia, C. Csaki, C. Grojean, M. Reece and J. Terning, Phys. Rev. D 72 (2005) 095018, hep-ph/0505001.26) Fermions on an interval: Quark and lepton masses without a Higgs,
C. Csaki, C. Grojean, J. Hubisz, Y. Shirman and J. Terning, Phys. Rev. D 70 (2004) 015012, hep-ph/0310355.27) Warped domain wall fermions,
T. Bhattacharya, C. Csaki, M. R. Martin, Y. Shirman and J. Terning, JHEP 0508 (2005) 061, hep-lat/0503011.28) Locally localized gravity,
by A. Karch and L. Randall, JHEP 0105, 008 (2001) hep-th/001115629) 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
Contact Info
John Terning, terning@physics.ucdavis.edu
