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 supersymmetry. 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

; 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

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. On very general assumption, the SM fields are localized within the groundsŽ which we will discuss in more detail in throat of a weak scale vortex in the 6 dimensional Section 3.Ž , the photon or any other massless gauge theory. We want to stress, however, that this particu- field. can be localized in our Universe, provided it lar construction must be viewed at best as an ``ex- can only propagate in the bulk in the form of a ; y1 istence proof'' and there certainly are other possible massive state with mass mEW, m EW setting the ways for realizing our proposal, without affecting its penetration depth of the electric flux lines into the most important consequences. extra dimensions. In order for the localized photon to It is interesting that in our framework supersym- be massless it is necessary that the gauge symmetry y1 metry is no longer needed from the low energy point be unbroken at least within a distance 4mEW from of view for stabilizing the hierarchy, however, it may our four-dimensional surfaceŽ otherwise the photon still be crucial for the self-consistency of the theory will get mass through the ``charge screening'', see of quantum gravity above the mEW scale; indeed, the Section 3. . As long as this condition is satisfied, the theory above mEW may be a superstring theory. four-dimensional observer will see an unbroken Independently of any specific realization, there gauge symmetry with the right 4-d Coulomb law. are a number of dramatic experimental consequences of our framework. First, as already mentioned, gravity becomes comparable in strength to the gauge 1 interactions at energies m ; TeV. The LHC and Usually in theories with extra compact dimensions of size R, EW states with momentum in the compact dimensions are interpreted NLC would then not only probe the mechanism of from the 4-dimensional point of view particles of mass 1rR, but electroweak symmetry breaking, they would probe still localized in the 4-d world. This is because the at the energies the true quantum theory of gravity! required to excite these particles, there wavelength and the size of Second, for the case of 2 extra dimensions, the the compact dimension are comparable. In our case the situation is gravitational force law should change from 1rr 2 to completely different: the particles which can acquire momentum in the extra dimensions have TeV energies, and therefore have r 4 ; 1 r on distances 100 mm±1 mm, and this devia- wavelengths much smaller than the size of the extra dimensions. tion could be observed in the next few years by the Thus, they simply escape into the extra dimensions. 266 N. Arkani±Hamed et al.rPhysics Letters B 429() 1998 263±272

Now, consider a particle with nonzero chargeŽ or any estimate the most stringent of these constraints, other unbroken gauge quantum number. kicked into mainly to show that our framework is not grossly the extra dimensions. Due to the conservation of excluded by current lab and astrophysical bounds. y1 flux, an electric flux tube of the width mEW must be Clearly, a much more detailed study must be done to stretched between the escaping particle and our Uni- more precisely determine the constraints on n and ; 2 verse. Such a string has a tension mEW per unit mEW in our framework. length. Depending on the energy available in the Consider any physical process involving the emis- collision, the charged particle will be either be pulled sion of a graviton. The amplitude of this process is r r 2 back to our Universe, or the flux tube will break into proportional to 1 MPland the rate to 1 M Pl. Conse- pieces with opposite charges at their ends. In either quently, the total combined rate for emitting any one case, charge is conserved in the 4-dimensional world, of the available gravitons is although energy may be lost in the form of neutral 1 particles propagating in the bulk. Similar conclusions ; D n 2 Ž.ER Ž.5 can be reached by considering a soft photon emis- MPl wx sion process 8 . where DE is the energy available to the graviton and Once the particles escape into the extra dimen- the last term counts the KK gravitons' multiplicity sions, they may or may not return to the 4-dimen- for n extra dimensions. Using eqŽ. 3 we can rewrite r sional world, depending on the shape and or the this as topology of the n dimensional compact manifold DE n Mn. In the most interesting case, the particles orbit ; 2qn .6Ž. around the extra dimensions, periodically returning, mEW colliding with and depositing energy to our 4 dimen- y y r q sional space with frequency R 1 ;10 27 30 n Hz. Note that the same result can be seen from the 4 n This will lead to continuous ``fireworks'', which in dimensional point of view. The mEW suppressions of q the case of ns2 can give rise to ; mm displaced the couplings of the 4 n dimensional graviton are sh q r 2qn vertices. determined by expanding g AB ABh AB (mEW , where hAB is the canonically normalized graviton in 4qn dimensions. Squaring this amplitude to obtain

2. Phenomenological and astrophysical con- the rate yields precisely the mEW dependence found straints above. As a result, the branching ratio for emitting a graviton in any process is

In our framework physics below a TeV is very 2qn ; DErm .7 simple: It consists of the Standard Model together Ž.EW Ž. with a graviton propagating in 4qn dimensions. The experimentally most excitingŽ and most danger- ; s Equivalently ± in four dimensional language ± our ous. case has mEW TeV and n 2. Of course, we theory consists of the Standard model together with must assume that weak-scale suppressed operators the graviton and all its Kaluza-KleinŽ. KK excita- giving proton decay, large K-K mixing etc. are tions recurring once every 1rR, per extra dimension forbidden. Of the remaining lab constraints, the ones n. We shall refer to all of them collectively as the involving the largest energy transfers DE Žsuch as F ``gravitons'', independent of their mass. Since each and Z decays. are most constrained. The branching graviton couples with normal gravitational strength ratio for graviton emission in Upsilon decays is ; r ; y8 ™ q 1 MPl to matter, its effect on particle physics and unobservable 10 . For ZZ X graÕiton the astrophysical processes is negligible. Nevertheless, branching ratio goes up to ;10y5. Absence of such since the multiplicity of gravitons beneath any rele- decay modes puts the strongest laboratory constraints n r vant energy scale E is Ž.ER can be large, the to the scale mEW and or n. Nevertheless, they are combined effect of all the gravitons is not always easy to satisfy, in part because of their sensitivity to negligible and may lead to observable effects and small changes in the value of mEW . Production of constraints. In this section we will very roughly gravitons in very high energy collisions will give the N. Arkani±Hamed et al.rPhysics Letters B 429() 1998 263±272 267 same characteristic signatures as the missing energy TeV. have already probed such physics. However, searches, except for one difference: the missing en- the cosmic rays are smoothly accelerated to their ergy is now being carried by massless particles. high energies without any ``hard'' interactions, and Next we consider astrophysical constraints. The they have dominantly soft QCD interactions with the gravitons are similar to goldstone bosons, axions and protons they collide with. Therefore, there are no neutrinos in at least one respect. They can carry significant constraints from very high energy cosmic away bulk energy from a star and accelerate its ray physics on our framework. cooling dynamics. For this reason their properties are Having outlined our general ideas, some dramatic constrained by the sun, red giants and SN 1987A. experimental consequences and being reassured that The simplest way to estimate these constraints is to existing data do not significantly constrain the translate from the known limits on goldstone parti- framework, we turn to constructing an explicit model cles. The dictionary that allows us to do that follows realizing our picture, with SM fields localized on the from Eq.Ž. 6 : four-dimensional throat of a vortex in 6 dimensions. r 2 §™ nr 2qn 1 F DE mEW Ž.8 relating the emission rate of goldstones and gravitons. Here F is the goldstone boson's decay con- 3. Construction of a realistic model stant. For the sun the available energy DE is only a keV. Therefore, even for the maximally dangerous In this section we construct a realistic model s s case mEW 1 TeV and n 2, the effective F is incorporating the ideas of this paper. As stressed in 1012 GeV, large enough to be totally safe for the the introduction, this should be viewed as an exam- sun; the largest F that is probed by the sun is ;107 ple or an ``existence proof'', since similar construc- GeV. tions are possible in the context of field theory as For red giants the available energy is ;100 keV well as string theory. In particular one can change and the effective F;1010 GeV. This value is an the structure and dimensionality of the manifold, the order of magnitude higher than the lower limit from localization mechanism, the gauge group and the red giants. Finally we consider the supernova 1987A. particle content of the theory without affecting the There, the maximum available energy per particle is key ideas of our paper. Furthermore, many of the presumed to be between 20 and 70 MeV . Choosing phenomenological consequences are robust and do the more favorable 20 MeV we find an effective not depend on such details. F;108 GeV, which is smaller than the lower limit The space time is 6-dimensional with a signature 10 sy of 10 GeV claimed from SN 1987A. Therefore, the g AB Ž.1,1,1,1,1,1 . The two extra dimensions are astrophysical theory of SN 1987 A places an interest- compactified on a manifold with a radius R;1 mm. r ing constraint on the fundamental scale mEW or and We will discuss two possible topologies: a two-sphere the number of extra dimensions n. The constraint is and a two-torus with the zero inner radius. In both ) ) easily satisfied if n 2orifmEW 10 TeV. Of cases the key point is that the observable particles course, when the number of dimensions gets large Ž.quarks, leptons, Higgs and gauge bosons are local- enough so that 1rRR100 MeV,Ž corresponding to ized inside a small region of weak-scale size equal to nR7. , none of the astrophysical bounds apply, since the inverse cutoff length ;Ly1 and can penetrate in all the relevant temperatures would be too low to the bulk only in form of the heavy modes of mass produce even the lowest KK excitation of the gravi- ;L. Thus for the energies -L ordinary matter gets ton. confined to a four-dimensional hypersurface, our

Finally, although accelerators have not achieved universe. The transverse x56, x dimensions can be collisions in the TeV energy range where the most probed only through the gravitational force, which is exotic aspects of the extra dimensions are revealed, the only long-range interaction in the bulk. one may wonder whether very high energy cosmic There are several ways to localize the Standard rays of energies ;1015 ±10 19 eVŽ which in colliding Model particles in our four-dimensional space-time. with protons correspond to c.o.m. energies ; 1±100 Here we consider the possibility that localization is 268 N. Arkani±Hamed et al.rPhysics Letters B 429() 1998 263±272 dynamical and the ordinary particles are ``zero the meridian he will arrive to the same vortex, since modes'' trapped in the core of a four-dimensional the poles have been identified, but will see it as an vortex. This topological defect, in its ground state, is anti-vortex since he will now be looking at the flux independent of four coordinates Ž.xm and thus up-side down. carves-out the four-dimensional hypersurface which Next, we come to the localization of the standard constitutes our universe. model particles on a vortex. We discuss the localiza-

Consider first x56, x to be compactified on a tion of different spins separately. two-sphere. Define a six-dimensional scalar field F Ž. Ž. xAVtransforming under some U 1 symmetry. 3.1. Localization of fermions and Higgs scalars We assume that F gets a nonzero VEV ;L and breaks UŽ.1 spontaneously. The vortex configura- V Fermions can be trapped on the vortex as ``zero tion is independent of the four coordinates x and m modes''wx 1 due to the index theorem wx 2 . Consider a can be set up through winding the phase by 2p u pair of six-dimensional left-handed Weyl spinors around the equator of the sphere: Fsf ei where bulk Gc,jsc ,j , which can be written in terms of the 2p)u)0 is an azimuthal angle on the sphere and 7 four-dimensional chiral Weyl spinors as c s f is the constant expectation value that mini- bulk Ž.Ž.c ,c , js j ,j . This pair obtains a mass from mizes a potential energyŽ modulo the small curvature LR LR coupling to the vortex field as hFcjqh.c.. The corrections. . Such a configuration obviously implies six-dimensional Dirac equation in the vortex back- two zeros of the absolute value F on the both sides ground of the equator, which can be placed at the north and the south poles respectively. These zeros represent E A qs iu GA c hf bulk e j Ž.10 the vortex±anti-vortex pair of characteristic thick- q ness ;Ly1. Since this size is much smaller than the Žsimilarly for j . has solutions with localized mass- s s separation length ;1 mm, vortex can be approxi- less fermions c c Ž.Ž.xm b r and j j Ž.Ž.xm b r , mated by the Nielsen-Olesen solutionwx 3 where ms1,..4, bŽ.r is a radial scalar function in the 2 dimensional compact space of x56and x , iu s s < y1 ™ c Ž.j Ž. F freŽ., f Ž.0 0, fr Ž.r 4 L f bulk provided that the spinors xm and xm satisfy u y G G q u Ž.9 i Ž i 5 6 . E s i G5 e c r b Ž.r hf bulk e j Ž11 . where 0-r-2p R is a radial coordinate on the Žand similarly for jq.. Since cq and j must be y sphere, and an anti-vortex corresponds to the change eigenvalues of the Ž.iGG56operator, they automat- ™y ™ y u u,r 2p R r.IfUŽ.1V is gauged the mag- ically have definite four-dimensional chiralityŽ say netic flux will be entering the south pole and coming left for the vortex and right for the anti-vortex. . In out from the north one. this case the normalizable wavefunction has the lo- y r X X Alternatively, compactification on a torus can lead calized radial dependence bŽ.r se h H0 fŽ r .dr . Thus to a single vortex. This is because a torus contains the vortex supports a single four-dimensional mass- q q many non-contractible loops, and the phase of F less chiral mode which can be cLRj . In this way, winding along such a loop is topologically stable. through the couplings to the vortex field one can Such a configuration is obtained from the previously reproduce the whole set of the four-dimensional discussed two-sphere by identifying its poles with a chiral fermions ± quarks and leptons ± localized on single point and subsequently removing this point the vortex. These localized modes can get nonzero from the manifold. This manifold is then equivalent masses through the usual Higgs mechanism. Let c X to a two-torus with zero inner radius; it can carry and c be the six-dimensional chiral spinorsŽ from topological charge and accommodate a single vortex different pairs. that deposit two different zero modes on it. The magnetic flux goes through the point that on the vortex. These zero modes can get masses X was removed from the manifold. An observer look- through the couplings to a scalar field HHcc , ing at the south pole will see the vortex with incom- provided H has a nonzero expectation value in the ing flux. If he travels towards the north pole along core of the vortex but vanishes in the bulk. The N. Arkani±Hamed et al.rPhysics Letters B 429() 1998 263±272 269 index theorem argument is unaffected by the exis- mechanism of refwx 5Ž see alsowx 8. . The simplest idea tence of such a scalar since it has a zero VEV for localizingŽ. say a U Ž.1 gauge field on a defect is outside the core. Now let us consider how the Higgs to arrange for the UŽ.1 to be broken off but unbroken fields with non-zero VEVs can be localized on the on the defect. Since the ``photon'' becomes massive vortex. A massive scalar field can be easily localized off the defect, one might expect to find a localized provided it has a suitable sign coupling to the vortex photon on the defect. Unfortunately, the four-dimen- field in the potential sional photon trapped in this way does not remain X massless. Since the UŽ.1 is broken off the defect, the h <

whether UŽ.1V is global or gauged . . The Lagrangian 2.Ž There is no light with a mass below the cut-off has the form ŽŽ.SU 2 indices are suppressed . scale L. matter in the bulk enforced by general 1 principles, such as Goldstone's theorem. sy q x 2 ylx2 2 3. The tree-level gauge coupling blows up away L 2 TrGGmnmnŽ.D m Ž. 4 g from the vortexwx 8 . y 2 <<2 y 2 q<<2 The latter condition can be satisfied e.g. if the value x Ž.h F M EmF of the gauge coupling is set by an expectation value X 2 2 Ž. ylF<

q X ) the vortex field hF QQ hF QQccŽwhere h and scales. This hierarchy is stable in the sense that small X h are parameters of the inverse cut-off size. . The changes of parameters have small effects on the index theorem ensures that each pair deposits a physics ± so there is no fine tuning problem. There single chiral zero mode which can be chosen as is also no issue of radiatively destabilizing the mm q qqq QLQ Rand Q cLQ cR. These states get their scale by physics at the weak cutoff. In this respect, masses through the couplings to the Higgs doublet our proposal shares the same ``set it and forget it'' field which condenses in the core of the vortex philosophy of the original proposal supersymmetric q wx gHQQccgHQQ . standard model 12 . An important and favorable To avoid unacceptable flavor violations, the first difference is that the mm scale is not a Lagrangean set of couplings should be flavor-universal. This can parameter that needs to be stabilized by a symmetry, be guaranteed by some flavor symmetry. Flavor such as supersymmetry. It is a parameter characteriz- violations must come from the ordinary Yukawa ing a solution, the size of the two extra dimensions. couplings in order to be under control. It is not uncommon to have solutions much larger than Lagrangean parameters; the world around us abounds with solutions that are much larger than the 4. Summary and outlook electron's Compton-wavelength. A related secondary question is whether the magnitude of the mm scale The conventional paradigm for High Energy may be calculated in a theory whose fundamental Physics ± which dates back to at least 1974 ± length is the weak scale. We have not addressed this postulates that there are two fundamental scales, the question which is embedded in the higher dimen- weak interaction and the Planck scale. The large sional theory. It is amusing to note that if there are disparity between these scales has been the major many new dimensions, their size ± given by Eq.Ž. 4 force driving most attempts to go beyond the Stan- ± approaches the weak scale and there is no large dard Model, such as supersymmetry and technicolor. hierarchy. In this paper we propose an alternate framework in Finally we come to the early universe. The most which gravity and the gauge forces are united at the solid aspect of early cosmology, namely primordial weak scale. As a consequence, gravity lives in more nucleosynthesis, remains intact in our framework. than four dimensions at macroscopic distances ± The reason is simple: The energy per particle during leading to potentially measurable deviations from nucleosynthesis is at most a few MeV, too small to Newton's inverse square law at sub-mm distances. significantly excite gravitons. Furthermore, the hori- The LHC and NLC are now even more interesting zon size is much larger than a mm so that the machines. In addition to their traditional role of expansion of the universe is given by the usual probing the electroweak scale, they are quantum- 4-dimensional Robertson-Walker equations. Issues gravity machines, which can also look into extra concerning very early cosmology, such as inflation dimensions of space via exotic phenomena such as and baryogenesis may change. This, however, is not apparent violations of energy, sharp high-pT cutoffss necessary since there may be just enough space to and the disappearance and reappearance of particles accommodate weak-scale inflation and baryogenesis. from extra dimensions. In summary, there are many new interesting is- The framework that we are proposing changes the sues that emerge in our framework. Our old ideas way we think about some fundamental issues in about unification, inflation, naturalness, the hierar- particle physics and cosmology. The first and most chy problem and the need for supersymmetry are obvious change in particle physics occurs in our abandoned, together with the successful supersym- view of the hierarchy problem. Postulating that the metric prediction of coupling constant unification cutoff is at the weak scale nullifies the usual argu- wx12 . Instead, we gain a fresh framework which ment about ultraviolet sensitivity, since the weak allows us to look at old problems in new ways. scale now becomes the ultraviolet! The new hierar- Lagrangean parameters become parameters of solu- chy that we now have to face, in the six dimensional tions and the phenomena that await us at LHC, NLC case, is that between the millimeter and the weak and beyond are even more exciting and unforeseen. 272 N. Arkani±Hamed et al.rPhysics Letters B 429() 1998 263±272

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