Extra spatial dimensions beyond the three we observe could resolve the hierarchy problem by lowering the fundamental gravitational scale from the Planck mass to the TeV scale. In the ADD (Arkani-Hamed, Dimopoulos, Dvali) model, gravity spreads into large extra dimensions, diluting its apparent strength. In the Randall-Sundrum model, a warped extra dimension generates the hierarchy through an exponential factor. Both scenarios predict signatures at colliders and in gravitational experiments.
Extra dimensions have been a recurring idea in theoretical physics since Kaluza and Klein proposed in the 1920s that electromagnetism could be understood as gravity in a fifth dimension. Modern extra dimension models, developed in the late 1990s, aim to solve the hierarchy problem -- why gravity is so much weaker than the other forces -- by changing the geometry of spacetime at short distances.
The ADD (large extra dimensions) model proposes that gravity propagates in 4 + n dimensions while the Standard Model particles are confined to a 4D brane. Gravity appears weak in 4D because its field lines spread into the extra dimensions, diluting the force. The fundamental gravitational scale M_D can be as low as ~1 TeV if the extra dimensions are large enough: for n = 2, R ~ 1 mm; for n = 6, R ~ 10 fermi. Collider signatures include: (1) direct graviton production (pp -> jet + graviton, where the graviton escapes into the extra dimensions, producing mono-jet + missing energy), (2) virtual graviton exchange enhancing dijet, dilepton, or diphoton production at high invariant mass, and (3) microscopic black hole production if the collision energy exceeds M_D.
The Randall-Sundrum model uses a single extra dimension with a warped (anti-de Sitter) geometry. The metric is ds^2 = e^{-2k|y|} eta_{mu nu} dx^mu dx^nu - dy^2, where y is the coordinate of the extra dimension and k is the AdS curvature scale. The exponential warp factor generates the hierarchy between the Planck scale and the TeV scale without large extra dimensions: the extra dimension need be only ~10 Planck lengths in size. The Kaluza-Klein excitations of the graviton appear as spin-2 resonances at the TeV scale, with couplings suppressed by the TeV scale rather than the Planck scale. These resonances would appear as narrow peaks in the dilepton or diphoton invariant mass spectrum at the LHC.
Beyond the specific models, extra dimensions have had a profound influence on theoretical physics. The AdS/CFT correspondence relates the RS model to a dual description in terms of a strongly coupled 4D conformal field theory. This connection has been applied to model building (composite Higgs models as dual descriptions of RS models), to heavy-ion physics (using AdS/CFT to compute transport properties of the quark-gluon plasma), and to condensed matter physics. The phenomenology of extra dimensions also pioneered many experimental techniques now used more broadly: monojet searches, resonance searches in clean channels, and sensitivity to virtual effects of heavy new states.
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