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rpp2010-rev-extra-dimensions

rpp2010-rev-extra-dimensions - 1 EXTRA DIMENSIONS Updated...

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– 1– EXTRA DIMENSIONS Updated Sept. 2007 by G.F. Giudice (CERN) and J.D. Wells (MCTP/Michigan). I Introduction The idea of using extra spatial dimensions to unify dif- ferent forces started in 1914 with Nordst¨ om, who proposed a 5-dimensional vector theory to simultaneously describe elec- tromagnetism and a scalar version of gravity. After the in- vention of general relativity, in 1919 Kaluza noticed that the 5-dimensional generalization of Einstein theory can simultane- ously describe gravitational and electromagnetic interactions. The role of gauge invariance and the physical meaning of the compactification of extra dimensions was elucidated by Klein. However, the Kaluza-Klein (KK) theory failed in its original purpose because of internal inconsistencies and was essentially abandoned until the advent of supergravity in the late 1970’s. Higher-dimensional theories were reintroduced in physics to ex- ploit the special properties that supergravity and superstring theories possess for particular values of spacetime dimensions. More recently it was realized [1,2] that extra dimensions with a fundamental scale of order TeV 1 could address the M W M Pl hierarchy problem and therefore have direct implications for collider experiments. Here we will review [3] the proposed scenarios with experimentally accessible extra dimensions. II Gravity in Flat Extra Dimensions II.1 Theoretical Setup Following Ref. 1, let us consider a D -dimensional spacetime with D = 4 + δ , where δ is the number of extra spatial dimensions. The space is factorized into R 4 × M δ (meaning that the 4-dimensional part of the metric does not depend on extra- dimensional coordinates), where M δ is a δ -dimensional compact space with finite volume V δ . For concreteness, we will consider a δ -dimensional torus of radius R , for which V δ = (2 πR ) δ . Standard Model (SM) fields are assumed to be localized on a (3 + 1)-dimensional subspace. This assumption can be realized in field theory, but it is most natural [4] in the setting of string theory, where gauge and matter fields can be confined to CITATION: K. Nakamura et al. (Particle Data Group), JPG 37 , 075021 (2010) (URL: http://pdg.lbl.gov) July 30, 2010 14:34
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– 2– live on “branes” (for a review see Ref. 5). On the other hand, gravity, which according to general relativity is described by the spacetime geometry, extends to all D dimensions. The Einstein action takes the form S E = ¯ M 2+ δ D 2 d 4 x d δ y det g R ( g ) , (1) where x and y describe ordinary and extra coordinates, re- spectively. The metric g , the scalar curvature R , and the re- duced Planck mass ¯ M D refer to the D -dimensional theory. The effective action for the 4-dimensional graviton is obtained by restricting the metric indices to 4 dimensions and by performing the integral in y . Because of the above-mentioned factorization hypothesis, the integral in y reduces to the volume V δ , and therefore the 4-dimensional reduced Planck mass is given by M 2 Pl = ¯ M 2+ δ D V δ = ¯ M 2+ δ D (2 πR ) δ , (2) where M Pl = M Pl / 8 π = 2 . 4 × 10 18 GeV. The same formula can be obtained from Gauss’s law in extra dimensions [6].
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