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Extra Dimensions in Particle Physics Ferruccio Feruglio1 1 Università diPadova 4 2 I.N.F.N. sezione diPadova 0 0 2 Abstract. Current problems in particle physics are reviewed from the viewpoint of theories possessing n extra spatial dimensions. a J 7 1 Introduction Flavour problem: the architecture underlying the • observed hierarchy of fermion masses and mixing an- 1 v gles. Today extra dimensions (ED) represent more a general 3 Cosmological constant problem: small curvature framework where several aspects of particle physics have • 3 of the observed space-time and its relation to the dy- 0 been reconsidered, rather than a unique and specific pro- namics of particle interactions. 1 posal for a coherent description of the fundamental inter- 0 actions. The first motivation for the appearance of ED, In this talk I will review the viewpoint on these problems 4 namely the quest for unification of gravity and the other offeredby theorieswith ED, stressingthe mostrecentde- 0 interactions [1], is still valid today. If we strictly adhere velopments in the field. / h to this project, then at present the only viable candidate p for a unified description of all interactions is string the- - ory,whichnaturally requiresED. The idea ofEDthat we 2 Hierarchy Problem p havein mind today has been deeply influenced by the de- e velopments in string theory: compactifications leading to h 2.1 Large Extra Dimensions : a chiralfermion spectrum, localizationof gauge and mat- v ter degrees of freedom on subspaces of the ED, relation i Thehierarchyproblemcanbereformulatedinthecontext X between the topological properties of the compact space of large extra dimensions (LED) [5]. In the LED scenario and the number of fermion families, localization of states r there is only one fundamentalenergy scalefor particle in- a aroundspecialpoints ofthe compactspaceandhierarchi- teractions:the TeV scale.Gravity describes the geometry cal Yukawa couplings, just to mention few examples. Re- of a D = 4+δ dimensional space-time where δ dimen- markable theoretical progresses have also been obtained sions are compactified, for instance, on an isotropic torus by developing models in field theory. For instance, in this Tδ of radius R. The D-dimensional Planck mass M is contextveryfruitfultoolsforsupersymmetry(SUSY)and D of the order 1 TeV. All the other degrees of freedom of gauge symmetry breaking have been developed, such as the standard model (SM) are assumed to live on a four- the Scherk-Schwarz [2] and the Hosotani [3] mechanisms. dimensional subspace, usually called brane, of the full D- Alsothephysicalpropertiesofcompactificationswithnon- dimensional space-time (see fig. 1). The gravitational po- factorizable space-time metric have been neatly worked tentialV(r)betweentwomassiveparticlesatadistancer out[4].Theconceptualandmathematicalrichnessoffered has two regimes.For r R,the lines of the gravitational bythesedevelopmentsmakesitpossibletoreconsidersev- ≪ field extend isotropically in all directions and eralspecificproblemsthathavenotreceivedasatisfactory answer in the four-dimensional context: 1 1 V(r) , (1) ∝ M2+δ r1+δ Hierarchy problem: extreme weakness of gravity in D • comparisonto the other interactions;gapbetween the whereasfor r R, the lines are squeezedalong the usual electroweak scale and the Planck scale MPl. four dimension≫s: Little hierarchy problem:possiblegapbetweenthe • Higgs mass and the electroweak symmetry breaking 1 1 V(r) , (2) scale. ∝ M2+δV r D δ Problems of conventional Grand Unified Theo- • ries (GUTs): doublet-triplet splitting problem, pro- where V is the volume of the compact space. As a conse- δ ton lifetime, mass relations. quence the four-dimensionalPlanckmassM is givenby Pl 2 Ferruccio Feruglio: Extra Dimensions in Particle Physics dynamical problem requires finding the minimum of an size L energy functional depending simultaneously upon many moduli, a rather formidable task. In table 1, we read the radius R and the compactification scale R−1 as a function of the number δ of ED, assuming M = 1 TeV, an isotropic toroidal compactifi- d R D rl cationand aflat backgroundmetric. Ofspecialinterestis o w the case δ = 2, which predicts deviations from the New- ur gravity ton’s law at distances around 1 mm. For δ >2 such devi- o ations would occur at much smaller length scales, outside therangeofthepresentexperimentalpossibilities.Forthe values of δ quoted in table 1, the compactification scale is quite small andthe Kaluza-Klein(KK)modes ofgravi- tonscanbe producedbothatcollidersandinprocessesof astrophysicaland/or cosmologicalrelevance. e-+, γ ,... all SM particles δ R R−1 Fig. 1. Pictorial view of the generic set-up considered in this review.Gravityhasaccesstothefullspace-timecharacterized 1 108 Km 10−18 eV byextraspatial dimensionsof typicalsize R.SMparticles are localized in subspacesofthefullspace-time, whosetypicalex- tension in the extra space is L<<R. When discussing LED, 2 0.1 mm 10−3 eV we will set L=0. the relation1: 3 10−6 mm 100 eV M 2 Pl =MδV . (3) M D δ (cid:18) D(cid:19) ... ... ... Therefore a huge hierarchy between M and M 1 Pl D ≈ TeVispossibleifthevolumeV oftheextraspaceismuch δ largerthan naively expected, M−δ. In units where M = 7 10−12 mm 100 MeV D D 1 we have 1 1 = , (4) M √V Pl δ Table 1: Compactification radius R and compactifica- indicating that four-dimensional gravity is weak because tion scale R−1 as a function of the number δ of ED, for the four-dimensional graviton wave function is diluted in M =1 TeV. D a big extra space. The 1/√V suppression of the gravi- δ tational constant can also be understood as the normal- ization factor of the wave function for the graviton zero 2.1.1 Deviations from Newton’s law mode.Solvingthehierarchyproblemnowrequiresexplain- ingwhyV M−δ.Thisisadynamicalproblem,sincein Modifications of the laws of gravity at small distances δ ≫ D higher-dimensionaltheoriesofgravitythevolumeV isthe arecurrentlyunder intense investigation.Inexperimental δ vacuum expectation value (VEV) of a field. For instance searches,deviationsfromtheNewton’slawareparametrized in a five-dimensional theory, the compactification radius by the modified gravitationalpotential R is determined by the VEV of the radion field r(x), the r (zero-mode of the) 55 component of the full space-time G m − metric: V(r)= N 1+α e λ , (6) − r ! r(x) =RM , (5) D h i incloseanalogytotheFermiscale,determinedintheelec- (G = 1/(8πM2 ) is the Newton constant) in terms of N Pl troweaktheory by the VEV ofthe Higgsfield. In a realis- the relative strength α and the range λ of the additional tictheoryofgravitylikestringtheory,the solutiontothis contribution. Precisely this kind of deviation is predicted 1 Severalconventionsexist in theliteraturetointroducethe by LED for r λ. The range coincides with the wave- ≥ fundamentalscaleofgravityMD.Inthisreviewtherelation(3) length of the first KK gravitonmode, λ=2πR, while the definesthe(reduced)D-dimensionalPlanckmassMD interms relativestrengthisgivenbythedegeneracyofthefirstKK ofthereducedfour-dimensionalPlanckmassM =2.4×1018 level: α=2δ. At present the best sensitivity in the range Pl GeV. λ 100 µm has been attained by the torsion pendulum ≈ Ferruccio Feruglio: Extra Dimensions in Particle Physics 3 realized by the Eo¨t-Wash group. For δ =2 they obtained mass the limit [6] λ<150 µm (95% CL) , (7) already in the range where deviations are expected if the ν2 M is close to 1 TeV. Future improvements represent a D s real challenge from the experimental viewpoint, but their 1/R impact could be extremely important, since deviations around the currently explored range are also expected in ν1 s other theoretically motivated scenarios, as we shall see 1/R later on. The next planned experiments aim to reach a ν0 sensitivity on λ in the range (30-50) µm for α = 4 [7], s thus probing M up to 3 4 TeV. D ÷ Fig. 2. Mass spectrum of right-handedneutrinos ν(n). 2.1.2 Neutrino Masses s LED provide a nice explanation of the smallness of neu- for instance, the solar data in this scenario [12]. There- trino masses [8]. If right-handed neutrinos ν (s singlet s ≡ fore the effects of ν(n) on neutrino oscillations are sub- under the SM gauge interactions) exist, at variance with s the charged fermion fields, they are allowed to live in the dominant, if present at all. Indeed, if the KK levels νs(n) bulk of a LED. In this case,just as for the graviton,their aremuchheavierthanthemassscalerelevantforneutrino four-dimensional modes in the Fourier expansion carry a oscillations,theydecouplefromthelow-energytheoryand suppression factor 1/√V : the higher-dimensionaloriginofneutrino massesbecomes δ undetectable. ν(0)(x) s ν (x,y)= +... (8) s √V δ 2.1.3 Signals at colliders By taking into account the relation (3), the Dirac neutrino masses originating from the Yukawa coupling with the Higgs doublet are given by: ExperimentalsignaturesofLEDatpresentandfuturecol- liders are well understood by now [13] and an intense ex- = yνv MD ν (x)ν(0)(x)+... (9) perimentalsearchiscurrentlyunderway.Theexistenceof LYuk √2 (cid:18)MPl(cid:19) a s lightKKgravitonmodesleadstotwokindsofeffects.The firstone is the direct production of KK gravitonsin asso- whereν (x)aretheactivefour-dimensionalneutrinos.The ciationwithaphotonorajet,givingriseto asignalchar- a resulting Dirac neutrino masses are much smaller than acterized by missing energy (or transverse energy) plus a the charged fermion masses and the observed smallness singlephotonorajet.Thecrosssectionfortheproduction of neutrino masses is explained, if there are no additional of a single KK mode is depleted by the four-dimensional contributions. The latter might arise from dimension five gravitational coupling, 1/M2 , but the lightness of each Pl operatorsassociatedtotheviolationoftheleptonnumber. individual graviton mode makes it possible to sum over In the absence of a sufficiently large fundamental scale, a large number of indistinguishable final states and the new mechanisms should be introduced to guarantee the cross section for the expected signal scales as: desired suppression of these operators [9,10]. Testing this idea and detecting the higher-dimensional origin of neu- Eδ trinomasseswouldrepresentacleansignatureoftheLED σ (10) ≈ Mδ+2 scenario.Experimentally,thisisonlypossibleifsomeν(n) D s (see fig. 2) is sufficiently light, 1/R≤ ∆m2atm,sol ≈0.01 in terms of the available center of mass energy E. The eV, which is not unconceivable if therqe is one dominantly present lower bound on MD, listed in table 2, are dom- large ED with R 0.02 mm. In such a case few ν(n) lev- inated by the searches for e+e− γ+ E at LEP and elsmaytakepart≈inneutrinooscillationsandprodsuceob- pp γ+ ET at Tevatron. For δ =→2,3,46the limits from → 6 servableeffects.Unfortunatelypresentdatadisfavourthis LEPareslightlybetterthanthosefromTevatron,theop- exciting possibility.Firstofall,there areclearindications posite occurring for δ = 5,6. Recently, comparable limits in favour of oscillations among active neutrinos, both in have also been obtained at Tevatron [14], by looking for the solar and in the atmospheric sectors [11]. Moreover finalstateswithmissingtransverseenergyandoneortwo SN1987A excludes the large mixing angle between active high-energy jets. and sterile neutrinos that would be needed to reproduce, 4 Ferruccio Feruglio: Extra Dimensions in Particle Physics M and the cut-off Λ of the effective theory, c and c D τ Y δ 2 3 4 5 6 are expected to scale as: MD (TeV) 0.57 0.36 0.26 0.19 0.16 Λδ−2 Λ2δ+2 c c , (15) τ ≈ Mδ+2 Y ≈ M2δ+4 D D Table 2: Combined 95% CL limits on M from LEP D and Tevatron data,from ref. [15]. Notice that M here is D (morerefinedestimatescanbefoundin[15]).Ifwenaively related to M [15] by M [15]=(2π)δ/(2+δ)M . D D D set Λ M , then the present limit on c from eq. ref- D Y ≈ bcy would provide the strongest collider bound on M . This signalis theoretically clean. Moreover,as long as D In a conservative analysis M and Λ should be kept as the energy E does not exceedM , thus remainingwithin D D independent. In this case, assuming M 1 TeV, we see the domain of validity of the low-energy effective theory, D ≈ that the bound (14) and the estimate (15) imply that Λ thepredictionsdonotdependontheultravioletproperties shouldbeconsiderablysmallerthanM .Suchasituation, of the theory. D wherethecut-offscaleisrequiredtobemuchsmallerthan A second type of effect is that induced by virtual KK themassscalecharacterizingthe low-energyeffective the- gravitonexchange.Atvariancewiththepreviousone,such ory,isnotuncommoninparticlephysics.Forinstancethe effectissensitivetotheultravioletphysics:theamplitudes naive estimate of the amplitude for K0 µ+µ− in the divergealreadyatthetreelevelforδ 2.Instringtheory L → ≥ Fermi theory gives G2Λ2 and data require Λ 1/√G . the tree-level amplitudes from graviton exchange corre- F ≪ F IndeedheretheroleofΛisplayedbythecharmmass,asa spond to one-loop amplitudes for the exchange of open consequence of the GIM mechanism. Similarly, the strin- string excitations. They are compactification dependent gent bound on c could indicate that the modes needed and also sub-leading compared to open string tree-level Y to cure the ultraviolet behaviour of amplitudes with KK exchanges [16]. In a more model-independent approach, graviton exchange are possibly quite light, if the funda- based on a low-energy effective field theory, the best that mental scale M is close to the TeV range. can be done is to parametrize these effects in terms of D effective higher-dimensional operators. If we focus on the The present sensitivity will be considerably extended sector of the theory including only fermions and gauge by future colliders, like LHC, that could probe M up to D bosons, there are only two independent operators up to about3.4(2.3)TeV,forδ =2(4)[13].Themostpromising dimension d=8 [17,15]: channel is single jet plus missing transverse energy. Es- timates based on the low-energy effective theory become c TµTν τ T Tµν µ ν (d=8) , (11) questionableforδ 5.Iftheenergyatfuturecollidersbe- 2 µν − 2+δ camecomparablet≥othefundamentalscaleofgravityM , (cid:18) (cid:19) D productionanddecayofblackholescouldtakeplaceinour 2 laboratories, with expectations that have been review by c Y f¯γµγ f (d=6) , (12) Landsberg at this conference [18]. In a even more remote 5 2   future, collisions at trasplanckian energies could provide f X   arobustcheckofthese ideas,especiallyfor the possibility where T is the energy-momentum tensor and the sum µν of dealing with gravity effects in a regime dominated by over f extends to all fermions. The dimension eight op- the classical approximation[19]. erator arises from graviton exchange at the tree-level. It contributes to dilepton production at LEP, to diphoton production at Tevatron and to the scattering e±p e±p → at Hera. The present data imply the bound [15]: 1 8 4 2.1.4 Limits from astrophysics >1.3 TeV . (13) c (cid:18)| τ|(cid:19) The dimension six operator arises at one-loop. It is even under chargeconjugationand singlet under all gaugeand global symmetries. It is bounded mainly from the search Today the most severe limits on MD come from astro- of contact interactions at LEP, dijet and Drell-Yan pro- physics and in particular from processes that can influ- duction at Tevatron and e±p e±p at Hera [15]: encesupernovaformationandthe evolutionofthe daugh- → ter neutron star. There are three relevant processes. The 4π 12 firstoneisKKgravitonproductionduringtheexplosionof >(16 21) TeV , (14) a supernova,whose typical temperature of approximately c ÷ (cid:18)| Y|(cid:19) 50 MeV makes kinematically accessible the KK levels for wherethe twoquotedboundsrefer,respectively,toaneg- the compactification scales listed in table 1. ativeandapositivec .Interms ofthe fundamentalscale Y Ferruccio Feruglio: Extra Dimensions in Particle Physics 5 presentastrophysicalobject, approximately 100 MeV, re- δ =2 δ =3 δ =4 moves all the astrophysicalbounds discussed above.Such agapcanbeobtainedinseveralways.Uptonowwehave assumedanisotropictoroidalcompactification.Ingeneral, SN cooling (SN1987A) 8.9 0.66 0.01 the KK spectrumdepends not only on the overallvolume V , but also on the moduli that define the shape of the δ compactspace.Forparticularchoicesofthesemoduli,the Diffuse gamma rays 38.6 2.65 0.43 KK spectrum displays the desired gap [22]. An additional assumption that has been exploited up to this point is that of a factorized metric for the space- NS heat excess 701 25.5 2.77 time. This assumption is no longer justified if the underlying geometry admits walls (also referred to as branes) carrying some energy density. Then, by the laws of general relativity, the background metric is warped and the Table3:LowerboundonM ,inTeV,fromastrophys- D relation between M and M is modified [4]. For in- ical processes [20]. Pl D stance, in the Randall-Sundrum set-up, the metric can The amount of energy carried away by KK gravitons be parametrized as: cannot deplete too much that associated to neutrinos, whosefluxwasobservedinSN1987A.Acompetingprocess ds2 =e2k(y−πR)ηµνdxµdxν +dy2 , (16) is KK graviton production, mainly induced by nucleon- nucleon bremsstrahlung NN NNgraviton, and con- where ηµν is the four-dimensional Minkowski metric and trolledinthelow-energyappro→ximationbyM .Thispro- k−1 is the radius of the AdS space. Notice that we have D cessshouldbeadequatelysuppressed.IfKKgravitonpro- rescaled the coordinates xµ by the overall factor ekπR, duction during supernova explosions takes place, then a comparedtothemostpopularparametrizationoftheRandall- large fraction of gravitons remains trapped in the neu- Sundrummetric.Asaresult,allmassparametersarenow tron star halo and the subsequent decay of gravitons into measuredinunitsofthetypicalmassscaleaty =πR,the photons produces a diffuse γ radiation, which is bounded TeV(seefig.3).ThemassesM andM arenowrelated Pl D by the existing measurements by the EGRET satellite. Thephotonfluxshouldalsonotoverheattheneutronstar unit of measure are surface. The corresponding limit on M depends on the D defined here assumeddecayproperties ofthe massivegravitons.These limits are summarized, for δ = 2,3,4 in table 3. These bounds rapidly softens for higher δ, due to the energyde- pendence of the relevant cross-sections in the low-energy approximation. They are strictly related to the spectrum −2k(y- πR) of KK gravitons, which, as we can see from table 1, has e no sizeable gap compared to the typical energy of the astrophysical processes considered here. WealsorecallthatKKgravitonproductioncanlargely affecttheuniverseevolution[21].Goingbackwardsintime, for M 1 TeV, the universe has a standard evolution D ≈ only up to a temperature T∗ given approximately by 10 π MeV,ifδ =2andby10GeV,ifδ =3.Forhighertemper- 0 y R atures the production of KK gravitons replaces the adia- batic expansionasthe mainsourceofcooling.While such Fig. 3. Dependenceofthewarpingfactorontheextracoordi- lowtemperaturesarestillconsistentwiththebig-bangnu- nate y. With the adopted parametrization the warping factor cleosynthesis,they renderbothinflationandbaryogenesis is equal to one at the brane in y = πR. Mass parameters are difficult to implement. measured in units used by theobserver at y=πR. Insummary,M 1TeVstillrepresentsaviablepos- D ≈ sibility if δ 4,while for lowerδ andinparticularfor the by: ≥ special case δ = 2, it is difficult to reconcile the expec- M 2 (e2kπR 1) tations of LED, at least in the simplest version discussed Pl =M − . (17) D M k here, with astrophysicaldata. (cid:18) D(cid:19) This relationcan be view, for δ =1 as a generalizationof the one given in eq. (3) for a factorizable metric. Indeed, 2.2 Warped compactification by sending k to zero the metric (16) becomes flat and we recovereq.(3).Bycomparingeqs.(17)and(3)weseethat Both astrophysicaland cosmologicalproblems are evaded the factor multiplying M on the right hand side of eq. D if the spectrum of KK gravitons has a sufficient gap. For (17)plays,inaloosesense,theroleofthevolumeV ofthe δ instancea gaphigherthanthe temperatureofthe hottest compactspace,measuredin units usedby the observerat 6 Ferruccio Feruglio: Extra Dimensions in Particle Physics y = πR. In this case the dependence of the “volume” on these more naturaltheories.Indeed, new weakly interact- theradiusRisexponential2.Remarkably,wedonotneed ing states show up at the TeV scale, whereas the Higgs ahugeradiusRtoachievealargehierarchybetweenM mass can be kept lighter by some symmetry. Pl andM 1TeV.Ifk andM arecomparable,thenR D D 10k−1 do≈esthejob.TheKKgravitonlevels,controlledb≈y 1/R in the present parametrization, start now naturally 3.1 Higgs mass protected by SUSY at the TeV scale. Astrophysical and cosmological bounds do not apply. Signals at colliders are quite different from Inthe lastyearsthere has been a growinginterestinfive- those discussed in the LED case. The KK gravitons have dimensional models where supersymmetry is broken by couplingssuppressedbytheTeVscale,notbyM .Their boundary conditions on an interval of size L 1 TeV−1. Pl ≈ levels are not uniformly spaced and they are expected to Supersymmetry breaking by boundary condition can reproduce resonance enhancements in Drell-Yan processes. duce the arbitrariness in the soft breaking sector, thus Aportionoftheparameterspacehasalreadybeenprobed making the model more predictable [26]. Moreover such at the Tevatron collider by CDF, by searching for heavy a possibility provides an alternative to the MSSM with gravitondecaysintodileptonanddijetfinalstates[24].In universal boundary condition at the grand unified scale this model the radionmass is expected to be in the range to study the interplay between supersymmetry and elec- 0.1 1 TeV, the exact value depending on the specific troweak symmetry breaking [26,27]. Also, such a set-up mec−hanismthatstabilizestheradiusR.Itcanbeproduced demonstrates very useful to study important theoretical by gluon fusion and it mainly decays into a dijet or a ZZ issues such as the problems of cancellation of quadratic pair, as the Higgs boson. No significant bounds can be divergences [28] and of gauge anomalies [29]. As in the extracted from the present radion search. MSSM, the electroweak symmetry breaking can be trig- The Randall-Sundrum setup provides an interesting geredby the top Yukawa coupling.For instance,in a par- alternative to LED. It avoids the tuning of geometrical ticular model [30] belonging to this class, the Higgs mass parameters versus M that is still needed in LED to re- is finite and calculable in terms of two parameters, the D produce the hierarchy between M and the electroweak lengthLofthe extradimensionandamass parameterM Pl scale. It overcomes the difficulties related to the presence that is responsible for the localization of the wave func- in LED of a “continuum” of KK graviton states. tion for the zero mode of the top quark.By including not only leading terms, but also two-loop corrections originating fromthe top Yukawa coupling and the strongcou- plingconstant,aHiggsmassintherelativelynarrowrange 3 Little Hierarchy Problem m = (110 125) GeV is found, for L−1 = (2 4) TeV H ÷ ÷ and2 LM 4.Themodelischaracterizedbythespec- TheassumptionthattheSMfieldsareconfinedonafour- ≤ ≤ trum of KK excitations displayedin fig. 4. The KK tower dimensional brane that does not extend into the extra ofeachordinaryfermionisaccompaniedbyatowerforthe space is too restrictive. Strong and electroweak interac- associated SUSY partner. The two towers have the same tions have been successfully tested only up to energies spacing, π/L, but they are shifted by π/(2L). Two ad- of order TeV. Therefore a part of or all the SM fields ditional towers are required by SUSY in five dimensions. might have access to extra dimensions of typical size L (TeV)−1 (see fig.1).Thereareseveraltheoreticalmotiva≤- The detection of this pattern would provide a distinctive experimental signature of the model. A peculiar feature tions for extra dimensions at the TeV scale. Already long ago,itwasobservedthatonewaytoachievesupersymme- try breaking with sparticles masses in the TeV range, is mass lightest SM through a suitable TeV compactification [25]. Here I will KK state discussamorerecentdevelopment,relatedtotheso-called 4π /2L ‘little’ hierarchy problem. On the one hand, the present 3π /2L data provide an indirect evidence for a gap between the lightest SUSY Higgs mass mH, required to be small by the precision 2π /2L partner tests, and the electroweak breaking scale, required to be relatively large by the unsuccessful direct search for new π /2L physics. On the other hand, from the solution of the ‘big’ hierarchy problem, we would naively expect that a light 0 Higgs boson should require new light weakly interacting ~ ~c c f f f f particles (e.g. the chargino in the MSSM), that have not been revealed so far. This gap is not so large and it can Fig. 4. Mass spectrum of the model in ref. [30] for a generic be filled either by a moderate fine-tuning of the param- matter multiplet: f and fãre respectively fermion and scalar eters in the underlying theory, or by looking for specific components of a chiral N =1 SUSY multiplet; fc and f˜c are theorieswhereitcanbenaturallyproduced.Extradimen- additional components required by supersymmetry in D=5. sionsattheTeVscaleprovideinprincipleaframeworkfor 2 Suchadependenceisalsofoundinothergeometries ofthe of the model is that all the degrees of freedom are in the internal space [23]. bulk (models of this type are said to have universal extra Ferruccio Feruglio: Extra Dimensions in Particle Physics 7 y dimensions [31]), and the only localized interactions are theYukawaones.Thereforemomentumalongthefifthdi- Z mensionis conservedbygaugeinteractions.No singleKK 2 L 0 y - y mode can be produced through gauge interactions and no four-fermion operator arises from tree-level KK gauge boson exchange. This property softens the experimental 0 L bounds on universal extra dimensions. - y 3.2 Higgs mass protected by gauge symmetry: Fig. 5. A Z2 parity symmetry halves a circle S and all the degrees of freedom originally defined on S. Higgs-gauge unification More than twenty years ago Manton [32] suggested that In order to promote this fascinating interpretation of the Higgs field could be identified with the extra compo- the gauge-Higgs system to a realistic theory, a number nents of a gauge vector boson living in more than four of problems should be overcome. First, in this example, dimensions.Inthis casethe samesymmetry thatprotects the request that H has the correct hypercharge leads to gauge vector bosons from acquiring a mass, could help in sin2θ = 3/4, a rather bad starting point. A value of preventing large quantum corrections to the Higgs mass W sin2θ (m ) closer to the experimental result can be ob- [33,34]. W Z tainbyreplacingSU(3)withanothergroup.Forinstance, A simple example of how this idea can be practically the exceptional group G leads to sin2θ = 1/4 [36]. implemented is a Yang-Mills theory in D >4 dimensions 2 W No large logarithms are expected to modify the tree-level with gauge groupSU(3). The gauge vector bosons can be predictionofsin2θ ,althoughsomecorrectionscanarise describedby a3 3hermitianmatrix A transformingin W × M from brane contributions. Second, electroweak symmetry the adjoint representation of SU(3): breaking requires a self-coupling in the scalar sector. In D = 5 this coupling can be provided by D-terms, if the Aa Aaˆ model is supersymmetric [37]. In D = 6 the desired cou- A = M M . (18) M   plingiscontainedinthekinetictermforthehigherdimen- Aaˆ Aa sionalgaugebosons.Third,crucialto the whole approach  M M   is the absence of quadratic divergences that could upset ThevectorbosonsAa (a=1,2,3,8),lyingalongthediag- M the Higgs lightness. A key feature of these models is a onal2 2 and1 1blocks,arerelatedto the SU(2) U(1) residual gauge symmetry associated to the broken gener- × × × diagonalsubgroupofSU(3),tobeidentifiedwiththegauge ators [38]. It acts on the Higgs field as group of the SM, while the fields Aaˆ (aˆ = 4,5,6,7), M belonging to the off-diagonal blocks, are instead associ- Aaˆ Aaˆ+∂ αaˆ (19) atedtotheremaininggeneratorsofSU(3).Fromthefour- 5 → 5 5 dimensionalpointofview,A describesbothvectorbosons and, in D = 5 it forbids the occurrence of quadratic di- M (M[ µ] = 0,1,2,3) and scalars (M[ m] = 5,...). Out vergences from the gauge vector boson sector. In D = 6 of al≡l these degrees of freedom, in the≡low-energy theory quadraticdivergencesfromthegaugesectorcanbeavoided we would like to keep Aa, that have the quantum num- inspecificmodels[39].Fourth,atfirstsightitwouldseem µ bersoftheelectroweakgaugevectorbosonsγ,Z andW±, impossible to introduce realistic Yukawa couplings, given and Aaˆ , transforming exactly as complex Higgs doublets the universality of the Higgs interactions if only minimal m couplings are considered. This problem can be solved by H , under SU(2) U(1). On the contrary, the particles m related to Aaˆ and×Aa are unseen and should be some- allowingfornon-localinteractionsinducedbyWilsonlines µ m [36,40]. Left and right-handedfermions are introduced at howeliminatedfromthelow-energydescription.Recalling theoppositeendsoftheextradimension(seefig.5).They that all these fields have a Fourier expansion containing feel only the gauge transformations that do not vanish at a zero mode plus a KK tower, we would like to keep the y = 0,L, namely those of SU(2) U(1) and the residual zero modes in the desired sector and project away the × transformation in eq. (19). An interaction term invariant zero modes for the unseen states. Such a projection be- under both these transformations can be written in term comes very natural [34,35] if the compactified space is of a Wilson line: an orbifold S/Z (see fig. 5), where all the fields are re- 2 quired to have a specific parity under the inversionof the L fifth coordinate: y y. It is sufficient to require that i dyAaˆ(y)Taˆ the fields describing→th−e unwanted states are odd, and all LYuk =yifj fRi(0) P[e Z0 5 ] fLj(L) +h.c. . the remaining ones are even. Such a parity assignment (20) is compatible both withthe five-dimensionalSU(3)gauge These generic interactions should be further constrained symmetryandthefive-dimensionalLorentzinvariance.To since the couplings in may reintroduce quadratic Yuk L the four-dimensional observer, having access only to the divergences for m at one or two-loop level. H massless modes, the SU(3) symmetry appears to be bro- Thisinterestingframeworkhasbeenlargelydeveloped kendowntoSU(2) U(1).Moreovershe/hewillcountone and improved in the last year. It is based on a compacti- × Higgs doublet H for each ED. fication mechanism able to provide the electroweak sym- 8 Ferruccio Feruglio: Extra Dimensions in Particle Physics metrybreakingsectorstartingfrompuregaugedegreesof Conflict with EWPT • freedom. It is characterized by the presence of KK gauge Potentially large corrections to the electroweak observ- vector bosons of an extended group, not necessarily ac- ables arise when the compactification scale is close to companied by KK replica for the ordinary fermions. the TeV range, from the tree-level exchange of KK gauge Inamoreradicalproposal,theelectroweakbreakingit- bosons. For δ > 1 the sum over the KK levels diverges, selforiginatesdirectlyfromcompactification,without the a regularization is needed and the related effects become presence of explicit Higgs doublet(s) [41]. Higgs-less the- sensitive to the ultraviolet completion of the theory. For ories of electroweak interactions in D=4 are well known. δ =1,KKgaugebosonexchangeandthe mixing between Oneoftheirmaindisadvantagesisthattheybecomestrongly ordinary and KK gauge bosons typically require to push interactingatarelativelylowenergyscale,around1TeV, the compactification scale beyond few TeV [45]. In the thus hampering the calculability of precision observables. LEDandRS scenarios,these effects aremodel dependent In a five-dimensional realization an appropriate tower of and more difficult to estimate [46]. KK states candelay the violationof the unitarity bounds beyond about 10 TeV and the low-energy theory remains B and L approximate conservation weaklyinteractingwellabovethefourdimensionalcut-off. • Very recent developments show that such a possibility, at Present data suggest that baryon and lepton numbers B least in its present formulation, it is not compatible with and L are approximately conserved in nature. Stringent the results of the precision tests [42]. experimentalboundshavebeensetontheprotonlifetime. For instance [47] 3.3 Additional remarks τ(p e+π0)>4.4 1033 yr . (21) → × Above we have implicitly assumed that some dynamics stabilizes the radion VEV at the right scale M 1/L Neutrino masses are extremely small compared to the c ≡ ≈ other fermion masses 1 TeV. Radion-matter interactions are controlled by the gravitational coupling G 1/M2 . Other radion prop- N ∝ Pl ∆m2 2 10−3 eV2 mν <1 eV . (22) erties depend on the specific stabilization mechanism. In atm ≈ × i severalexplicit models of weak scale compactification the Xi radion mass is approximately given by M2/M 10−3 c Pl ≈ Whiletherearenofundamentalprinciplesrequiringexact eV. This induces calculable deviations fromNewton’s law B/LconservationandindeedB/Lviolationsareneededby at distances of the order 100 µm, even in the absence baryogenesis, these experimental results imply that the of contributions from KK gravitonmodes [43]. Moreover, amount of B/L breaking should be tiny in nature. In a such a light radion can dangerously modify the property low-energy (SUSY and R-parity invariant) effective the- of the early universe [44], if the scale of inflation is larger oryB and L violations are described, at leading order,by than the compactification scale (see fig. 6). dimension five operators such as M c (GeV) d2θ(HuL)(HuL) = v2νν+... Λ 2Λ Z 12 QQQL 10 d2θ . (23) late r decays dissociate Λ Z 9 light elements 10 (WithoutSUSYtheleadingB-violatingoperatorshavedi- r oscillations mension six). If the cut-off Λ of the low-energy effective overclose the universe theory is very large, as the conventional solution of the hierarchy problem via four-dimensional SUSY seems to 3 10 M indicate, then B and L violating operatorsare sufficiently 1 c suppressed (we will reconsider this issue for the baryon scale of inflation number later on). In a theory characterizedby a verylow cut-off Λ, as the higher-dimensional theories discussed so Fig. 6. Cosmological effects of a radion mass of the order M2/M . far,additionalmechanismsto depleteB/Lviolatingoper- c Pl ators should be invoked [9,10]. Gauge coupling unification There are three general problems that affect theories • where the presence of ED requires a cut-off scale around Oneofthefewsuccessfulexperimentalindicationsinfavor 1 TeV. These theories include both the LEDscenariodis- of low-energy SUSY is provided by gauge coupling unifi- cussed in section 2 and the TeV compactification we are cation. In the one-loop approximation the prediction for discussing here. α (m ) perfectly matches the experimental value [48] 3 Z α (m )=0.117 0.002 . (24) 3 Z ± Ferruccio Feruglio: Extra Dimensions in Particle Physics 9 The inclusion of two-loop effects, thresholds effects and SUSYbreakingandbythepresenceofnon-renormalizable non-perturbative contributions raises the theoretical er- operators induced by physics at the M scale [54]. Pl roruptoδα (m ) 0.01whilemaintainingasubstantial The second problem is the conflict between the pro- 3 Z ≈ agreement with data. In D = 4 gauge coupling unifica- ton decay rates, dominated by dimension five operators tion requires three independent ingredients: logarithmic in minimal SUSY GUTs, and experimental data. For in- running, right content of light particles and appropriate stance minimal SUSY SU(5) predicts [55]: unification conditions. If the ultraviolet cut-off, implied bythe presenceofextradimensions,ismuchsmallerthan 2tanβ 2 m (GeV) 2 τ(p K+ν) 1032 T yr . the grand unified scale, then gauge coupling unification → ≈ 1+tan2β 1017 becomes a highly non-trivial property of the theory. (cid:18) (cid:19) (cid:18) (cid:19) (25) Thispredictionisaffectedbyseveraltheoreticaluncertain- It is not possible to review here all the existing pro- ties (hadronic matrix elements, masses of supersymmet- posals in order to reconcile extra dimensions having a ricparticles,additionalphysicalphaseparameters,wrong low cut-off with B and L approximate conservation and massrelationsbetweenquarksandleptonsoffirstandsec- gaugecoupling unification.Summarizingverycrudely the ond generations) but, even by exploiting such uncertain- state of the artwe cansay that these properties are quite ties to stretch the theoretical prediction up to its upper natural within the ultraviolet desert of a (SUSY) four- limit, the conflict with the present experimental bound dimensional low-energy theory, especially if such a the- oryiscomplementedbyagrandunifiedpicture wherethe τ(p K+ν)>1.9 1033 yr (90% CL) , (26) particle classification is clarified and the required unifica- → × tion condition is automatic. On the other hand, B and isunavoidable(ineq.(25)suchuncertaintieshavealready L approximate conservation and gauge coupling unifica- been optimized to minimize the rate). tion are possible [9,10,49], but far from generic features Remarkably,boththeseproblemscanbelargelyallevi- if an infrared desert exist: V >>M−δ. In the absence of δ D atedifthegrandunifiedsymmetryisbrokenbycompacti- a desert, as in the case of compactifications at the TeV ficationofa(tiny)extradimensiononaorbifold[56].The scale, gauge coupling unification is lost 3. In the Randall- gauge symmetry breaking mechanism is exactly the one Sundrum setup the TeV scale is not the highest acces- we have already described when discussing gauge-higgs sible energy scale and this leaves open the possibility of unification. For instance, in the case of SU(5), the gauge achieving gaugecoupling unification in a way close to the vector bosons A can be assembled in a 5 5 hermitian µ conventional four-dimensional picture [52]. × traceless matrix as follows: 4 Grand Unification and Extra Dimensions Aa Aaˆ Aµ = µ µ , (27) Theideaofgrandunificationisextremelyappealing.Itau-  Aaµˆ Aaµ tomatically provides the unification conditions for a suc-   where the diagonal blocks are 3 3 and 2 2 matrices. cessful gauge coupling unification. It sheds light on the Then Aa (a=1,...12) are the gau×ge vector b×osons of the otherwise mysterious classification of matter fields, giv- µ ing a simple explanation for electric charge quantization, SM while Aaˆ (aˆ = 13,...24) are associated to the genera- µ gaugeanomalycancellationandsuggestingrelationsamong tors that are in SU(5) and not in the SM. By working in fermionmasses.Itprovidesthe ingredientsforbaryogene- five dimensions, the zero modes of Aaˆ can be eliminated µ sis.However,initsconventionalfour-dimensionalformula- if we require that these fields are odd under the parity tion,grandunificationis affected by two majorproblems. symmetry Z′: y′ y′, y′ y πR′/2. The resulting 2 → − ≡ − The first is the doublet-triplet splitting problem. In spectrumisdisplayedinfig.7.Ifweconsideratinyradius anygrandunifiedtheory(GUT) the electroweakdoublets ofthefifthdimension,1/R′ M (M =2.4 1016 GUT GUT ≈ × HD , needed for the electroweak symmetry breaking, sit GeVbeingthefour-dimensionalgrandunifiedscale),then u,d in the same representation of the grand unified group to- fromthe viewpointofthe four-dimensionalobserver,hav- gether with color triplets HT . Doublets are required to ing access to energies much smaller than M , SU(5) u,d GUT be at the electroweak scale, while triplets should be at or appearstobebrokendowntoSU(3) SU(2) U(1).More- × × abovetheunificationscale,toavoidfastprotondecayand over, if the Higgs multiplets H containing the Higgs u,d notto spoilgaugecoupling unification. Sucha huge split- doublets HD live in the bulk, their Z′ parity assignment u,d 2 tingisfine-tunedinminimalmodels,orrealizednaturally is fixed from the gauge sector (up to a twofold ambigu- at the price of baroque Higgs structures in non-minimal ity) and the desired DT splitting can be automatically ones [53]. Moreover, even when implemented at the clas- achievedbycompactification[56],asillustratedinfig.7.In sical level, it can be upset by radiative corrections after the SUSY versionof this model another parity Z , acting 2 as y y, removes all the additional states required by 3 Thepossibility thatpower-lawrunning[50]enforcesgauge SUSY→in−D=5 and the massless modes are just in Aa and coupling unification at a scale close to the TeV depends on µ tahsseuomryp[t5io1n].s about the unknown ultraviolet completion of the tHhuDe,dc.iTrchleedSoudbolwenidetontitfihceatiniotnerivmalpl(ie0d,πbRy′Z/22)×. ZT2′hreedpuocinest 10 Ferruccio Feruglio: Extra Dimensions in Particle Physics 5/R’ Even if these terms are absent at the classical level, they are expected to arise from divergent radiative corrections 4/R’ [61]. Therefore a consistent description of the theory re- quirestheir presence,with g asfree parameters.Bytak- bi 3/R’ ing into account both the five-dimensional gauge kinetic termandthecontributionsineq.(29),thegaugecoupling 2/R’ constants g2 at the cut-off scale Λ are given by i 1/R’ 1 2πR′ 1 = + . (30) g2 g2 g2 0 i 5 bi Aµa Aa^µ If the SU(5) breaking terms 1/gb2i were similar in size to the symmetric one, we would loose any predictability. A D T H H predictive framework can be recovered by assuming that u,d u,d at the scale Λ the theory is strongly coupled. In this case g2 16π3/Λ, g2 16π2 and from eq. (30) we estimate 5 ≈ bi ≈ Fig.7.MassspectruminthegaugebosonandHiggssectorsof a five-dimensional SU(5) GUT, compactified on S/(Z ×Z′). 1 ΛR′ 1 2 2 +O( ) . (31) g2 ≈ 8π2 16π2 i y =πR′/2isspecial,sinceallthe gaugevectorbosonsAaˆ The SU(5) symmetric contribution dominates over the µ and the parameters of the corresponding gauge transfor- brane contributions and predicts gauge couplings of or- mationsvanishthere.Thereforeiny =πR′/2theeffective der one, provided ΛR′ =O(100). Such a gap between the gauge symmetry is only SU(3) SU(2) U(1), not the full compactification scale M 1/R′ and the cut-off scale Λ SU(5). × × can in turn independentlcy≡affect gauge coupling unifica- To complete the solution of the DT splitting prob- tionthroughheavythresholdcorrections.Atleadingorder lem and to keep under control proton decay, the mass thesecorrections,comingfromtheparticlemassspectrum term H H allowed by both SU(5) gauge symmetry and around the scale M , are given by [58]: u d c N = 1 five-dimensional supersymmetry [57], should be adequately suppressed. This can be done by assuming a 3 Λ δ(heavy)α = (αLO)2log . (32) U(1)R symmetry of Peccei-Quinn type, broken down to 3 −7π 3 (cid:18)Mc(cid:19) theusualR-parityonlybysmallsupersymmetrybreaking soft terms [58]. This symmetry keeps the Higgs doublets ItisquiteremarkablethatΛR′ =O(100)ispreciselywhat light and, when extended to the matter sector to include needed in order to compensate the corrections in eq. (28) R-parity, forbids dimension five operators leading to pro- andbringbackα (m )totheexperimentalvalue(24).By 3 Z tondecay (see eq.(23)) andeliminates alldangerousB/L considering the whole set of renormalization group equa- violating operators of dimension four. Alternatively we tions one also finds the preferred values can assign appropriate parities to matter fields, to sup- press or even to completely forbid proton decay, at the Λ 1017 GeV M 1015 GeV . (33) c ≈ ≈ price of explicitly loosing the SU(5) gauge symmetry in the matter sector [59]. ThecompactificationscaleMc 1015GeVisrathersmaller ≈ Gauge coupling unification is preserved[58,60]. There than MGUT and this greatly affects the estimate of the are several contributions to the running of gauge cou- proton lifetime. The presence of non-universal brane ki- pling constant beyond the leading order. As in the con- netic terms, as given by the strong coupling estimate in ventionalfour-dimensionalanalysiswe have two-loopcor- eq. (31), suggests that the theoretical error on the pre- rections and contributions coming from the light thresh- diction of α3(mZ) is similar to the one affecting the four- olds,the massesof the superpartnersof the ordinarypar- dimensional SU(5) analysis. ticles.Thesecorrectionsraisethe leadingorderprediction The fifth dimension has also an interest impact on of α (m ): the description of the flavour sector. Each matter field 3 Z can be introduced either as a bulk field, depending on all the five-dimensional coordinates, or as a brane field, lo- α (m )=αLO(m )+δ(2)α +δ(light)α 0.130 . (28) 3 Z 3 Z 3 3 ≈ cated at y = 0 or in y = πR′/2. Matter fields living in y =πR′/2,caninprinciplebeassignedtorepresentations An additional correction, specific of the present frame- of SU(3) SU(2) U(1) that do not form complete multi- work, comes from possible gauge kinetic terms localized × × atthebraneaty =πR′/2,wheretheeffectivegaugesym- pletsunderSU(5).ForinstancewecouldreplacetheHiggs multiplets,so farregardedasbulk fields,withbranefour- metry is only SU(3) SU(2) U(1), not the full SU(5): ′ × × dimensional fields localized in y = πR/2. In this case we couldavoidtheinclusionofthecolourtripletcomponents, 1 πR′ d4x dy δ(y )F Fµν . (29) limitingourselvestotheSU(2)doubletsalone.Thispossi- i=1,2,3gb2i Z − 2 µνi i bilitywouldprovidearadicalsolutionto theDT splitting X