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The Hierarchy Problem and New Dimensions at a Millimeter

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The Hierarchy Problem and New Dimensions at a Millimeter 18 June 1998 Physics Letters B 429Ž. 1998 263±272 The hierarchy problem and new dimensions at a millimeter Nima Arkani±Hamed a, Savas Dimopoulos b, Gia Dvali c a SLAC, Stanford UniÕersity, Stanford, CA 94309, USA b Physics Department, Stanford UniÕersity, Stanford, CA 94305, USA c ICTP, Trieste 34100, Italy Received 12 March 1998; revised 8 April 1998 Editor: H. Georgi Abstract We propose a new framework for solving the hierarchy problem which does not rely on either supersymmetry or technicolor. In this framework, the gravitational and gauge interactions become united at the weak scale, which we take as the only fundamental short distance scale in nature. The observed weakness of gravity on distances R 1 mm is due to the G ; y1r2 existence of n 2 new compact spatial dimensions large compared to the weak scale. The Planck scale MPl GN is not a fundamental scale; its enormity is simply a consequence of the large size of the new dimensions. While gravitons can freely propagate in the new dimensions, at sub-weak energies the Standard ModelŽ. SM fields must be localized to a 4-dimensional manifold of weak scale ``thickness'' in the extra dimensions. This picture leads to a number of striking signals for accelerator and laboratory experiments. For the case of ns2 new dimensions, planned sub-millimeter measurements of gravity may observe the transition from 1rr 2 ™1rr 4 Newtonian gravitation. For any number of new dimensions, the LHC and NLC could observe strong quantum gravitational interactions. Furthermore, SM particles can be kicked off our 4 dimensional manifold into the new dimensions, carrying away energy, and leading to an abrupt decrease in R events with high transverse momentum pT TeV. For certain compact manifolds, such particles will keep circling in the extra dimensions, periodically returning, colliding with and depositing energy to our four dimensional vacuum with frequencies of ;1012 Hz or larger. As a concrete illustration, we construct a model with SM fields localized on the 4-dimensional throat of a vortex in 6 dimensions, with a Pati-Salam gauge symmetry SUŽ.4 =SU Ž.2 =SU Ž.2 in the bulk. q 1998 Published by Elsevier Science B.V. All rights reserved. 1. Introduction models with technicolor or low-energy supersymme- try. It is remarkable that these rich theoretical struc- There are at least two seemingly fundamental tures have been built on the assumption of the energy scales in nature, the electroweak scale mEW existence of two very disparate fundamental energy ; 3 s y1r2 ; 10 GeV and the Planck scale MPl GN scales. However, there is an important difference 1018 GeV. Explaining the enormity of the ratio between these scales. While electroweak interactions r ; y1 MPlm EW has been the prime motivation for con- have been probed at distances approaching mEW , structing extensions of the Standard Model such as gravitational forces have not remotely been probed at 0370-2693r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII: S0370-2693Ž. 98 00466-3 264 N. Arkani±Hamed et al.rPhysics Letters B 429() 1998 263±272 ; y1 ; distances MPl : gravity has only been accurately Putting MPlŽ4qn. mEW and demanding that R be ; measured in the 1 cm range. Our interpretation of chosen to reproduce the observed MPl yields MPl as a fundamental energy scaleŽ where gravita- 2 tional interactions become strong. is based on the 30 1q y17 1 TeV n assumption that gravity is unmodified over the 33 R;10n cm= .4Ž. orders of magnitude between where it is measured at ž/mEW ; 1 cm down to the Planck length ;10y33 cm. Given the crucial way in which the fundamental role For ns1, R;1013 cm implying deviations from attributed to MPl affects our current thinking, it is Newtonian gravity over solar system distances, so worthwhile questioning this extrapolation and seek- this case is empirically excluded. For all nG2, ing new alternatives to the standard picture of physics however, the modification of gravity only becomes beyond the SM. noticeable at distances smaller than those currently Given that the fundamental nature of the weak probed by experiment. The case n s 2 Ž R ; scale is an experimental certainty, we wish to take 100 mm±1 mm. is particularly exciting, since new the philosophy that mEW is the only fundamental experiments will be performed in the very near short distance scale in nature, even setting the scale future, looking for deviations from gravity in pre- for the strength of the gravitational interaction. In cisely this range of distanceswx 11 . this approach, the usual problem with the radiative While gravity has not been probed at distances stability of the weak scale is trivially resolved: the smaller than a millimeter, the SM gauge forces have ultraviolet cutoff of the theory is mEW . How can the certainly been accurately measured at weak scale r usualŽ. 1 MPl strength of gravitation arise in such a distances. Therefore, the SM particles cannot freely picture? A very simple idea is to suppose that there propagate in the extra n dimension, but must be are n extra compact spatial dimensions of radius localized to a 4 dimensional submanifold. Since we ; q R. The Planck scale MPlŽ4qn. of thisŽ. 4 n assume that mEW is the only short-distance scale in ; dimensional theory is taken to be mEW according the theory, our 4-dimensional world should have a ; y1 to our philosophy. Two test masses of mass m12,m ``thickness'' mEW in the extra n dimensions. The placed within a distance r<R will feel a gravita- only fields propagating in theŽ. 4qn dimensional tional potential dictated by Gauss's law inŽ. 4qn bulk are theŽ. 4qn dimensional graviton, with cou- dimensions plings suppressed by theŽ. 4qn dimensional Planck ; mass mEW . mm 1 As within any extension of the standard model at ; 12 < the weak scale, some mechanism is needed in the VrŽ.nq2 nq1 Žr R ..1 Ž. Mrq PlŽ4 n. theory above mEW to forbid dangerous higher di- mension operatorsŽ. suppressed only by mEW which lead to proton decay, neutral meson mixing etc. In On the other hand, if the masses are placed at our case, the theory above m is unknown, being distances r4R, their gravitational flux lines can EW whatever gives a sensible quantum theory of gravity not continue to penetrate in the extra dimensions, inŽ. 4qn dimensions! We therefore simply assume and the usual 1rr potential is obtained, that these dangerous operators are not induced. Any extension of the SM at the weak scale must also not mm12 1 give dangerously large corrections to precision elec- Vr; r4R ,2 Ž.nq2 n Ž . Ž. troweak observables. Again, there could be unknown MRPlŽ4qn. r contributions from the physics above mEW . How- ever, at least the purely gravitational corrections do so our effective 4 dimensional MPl is not introduce any new electroweak breakings beyond the W,Z masses, and therefore should decouple as loop factor =Ž.m rm 2, which is already quite 2 ; 2qn n W,Z EW MPlM PlŽ4qn. R .3Ž. ; small even for mEW 1 TeV. N. Arkani±Hamed et al.rPhysics Letters B 429() 1998 263±272 265 Summarizing the framework, we are imagining new experiments measuring gravity at sub-millimeter 4 = G wx that the space-time is R Mnnfor n 2, where M distances 11 . is an n dimensional compact manifold of volume Third, since the SM fields are only localized n q y1 R , with R given by Eq.Ž. 4 . The Ž 4 n .dimen- within mEW in the extra n dimensions, in suffi- ; R sional Planck mass is mEW , the only short-dis- ciently hard collisions of energy Eescm EW , they tance scale in the theory. Therefore the gravitational can acquire momentum in the extra dimensions and force becomes comparable to the gauge forces at the escape from our 4-d world, carrying away energy. 1 weak scale. The usual 4 dimensional MPl is not a In fact, for energies above the threshold Eesc , escape fundamental scale at all, rather, the effective 4 di- into the extra dimensions is enormously favored by mensional gravity is weakly coupled due to the large phase space. This implies a sharp upper limit to the size R of the extra dimensions relative to the weak transverse momentum which can be seen in 4 dimen- s scale. While the graviton is free to propagate in all sions at pT Eesc , which may be seen at the LHC or Ž.4qn dimensions, the SM fields must be localized NLC if the beam energies are high enough to yield y1 on a 4-dimensional submanifold of thickness mEW in collisions with c.o.m. energies greater than Eesc . the extra n dimensions. Notice that while energy can be lost into the extra Of course, the non-trivial task in any explicit dimensions, electric chargeŽ or any other unbroken realization of this framework is localization of the gauge charge. cannot be lost. This is because the SM fields. A number of ideas for such localizations massless photon is localized in our Universe and an have been proposed in the literature, both in the isolated charge can not exist in the region where context of trapping zero modes on topological de- electric field cannot penetrate, so charges cannot fectswx 7 and within string theory. In Section 3, we freely escape into the bulk. In light of this fact, let us will construct models of the first type, in which there examine the fate of a charged particle kicked into the are two extra dimensions and, given a dynamical extra dimensions in more detail.
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