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Extra Dimensions Nick Evans University of Southampton We live in 3 space dimensions – why? Why look for more? Can there be more? What would they look like? Motivation - Grand Unification If two particles collide with Planck Scale energy their gravitational interaction would be as strong as the other forces – one hopes to unify all forces into a single simple theory Kaluza-Klein 5d Theory m Linearized field eqn for A is Gauge Theory Compactification Distinguish 5th dimension by enforcing 2 PR periodicity – why not? Motivation II – String Theory Open and closed strings naturally give both gravitons and gauge bosons – progress towards a TOE? String theory’s symmetries are anomalous unless in 9 space dimensions (and requires supersymmetry) Again they may be compactified…. we learn that extra dimensions are at least a possibility in quantum theories of gravity… Kaluza Klein Tower The wave function of a free particle must be 2 PR periodic There are a discrete set of states associated with all particles with energy (masses) in integer steps. Note Universal XD mimic supersymmetry – every particle has a massive partner.. BUT of the same spin… spin information will be crucial! Hep-ph/0205314 Experimental Constraints Momentum in the 5th direction must be conserved So KK modes can only be pair produced They can occur in SM loop diagrams and make contributions to eg the rho parameter (S,T,U) - hep-ph/0012100 Limit is 1/R ~ 400 GeV + They could be directly pair produced at the Tevatron 1/R ~ 450 GeV + (looking like eg massive quark production) - hep-ph/0201300 LHC will probe up to 1/R ~ 3 TeV Non-Renormalizability The Standard Model in 5+ dimensions is not a good theory – it becomes strongly coupled quite quickly as one moves to the UV 0 All gauge fields will become strongly coupled by 100TeV QCD does not make protons at the 100TeV scale so this must be forbidden XD implies new physics close by! It can bring the GUT scale down But why no proton decay? The Lattice/ Deconstruction hep-th/0104005, hep-th/0104035 One can impose a cut off by latticizing the XD eg for the weak SU(2) group Here U are 3 massless bi-fundamental scalars that couple the SU(2)s on neighbouring sites – link variables Interestingly one can manufacture 3 such scalars as the Goldstone bosons of a symmetry breaking SU(2) x SU(2) -> SU(2) So we can view the theory above as a 4d theory with an SU(2) N gauge group and N-1 Higgs fields - the extra gauge groups put the KK modes in by hand The Higgs’ vev scale determines the basic KK mode mass and hence the radius of the extra dimension… to go back to the continuum we take it to infinity. We’re left with a theory with a tower of massive Ws but no higgs. What about Unitarity? – we must have a higgs to restore Unitarity with massive Ws!! Wrong! Extra heavy Ws can restore Unitarity without an explicit Higgs boson - There’s now a big literature on higgsless models. hep-ph/0111016, hep-ph/0305237, hep-ph/0312324 Again we can’t actually take the cut off to infinity since: The low energy gauge coupling after all the breakings is given by N had better not be too big! Braneworld – string theory origin Closed strings can wrap XD there are states with mass m T (2 p R ) And KK modes of mass T-duality - n/R change their names! KK winding The open string spectrum on a space of radius R are the same as open strings on a space of radius R’= 1/T (2 p ) R If you try to compactify away a space R -> 0 it returns as an infinite R’ Braneworld – string theory origin Open strings do not have winding modes – so in a theory of both open and closed strings one can create: Gauge particles live on a sub-space or “brane” Gravitons live in a higher dimension “bulk” “Existence proof” for such a world Braneworld It is possible to isolate some or all of the SM fields on a 3-brane hep-ph/9803315 Gravity lives in higher dimensions – they can’t be flat and infinite since Surprisingly gravity is very poorly tested at short distances – until recently it’s effects had not been measured on scales below 1mm – limits now find no deviation down to 0.1mm – hep-ph/0307248 Remarkably the compactification radius of the bulk could be as big as 0.1mm!! This has a huge effect on the true Planck Scale The n=2 case is a dream scenario since: • gravity would be strong and hence accessible at the LHC… • it would remove the hierarchy between the weak and Planck scale…. We’re expecting to find a fundamental scalar – the higgs – at the 1 TeV scale to generate the W and Z masses… but a scalar mass is unstable to growing to the biggest scale in physics…. [ Incidentally the Cosmological constant is measured to be 1/(1mm) so there was some hope to find a change in gravity at that scale in any case (hepph/9708329)…. ] We’re experimentally down to 0.1mm without deviation from Newton’s law so no big break through! We have learnt that gravity could become strong at any point hence forth in collider experiments … and extra dimensions could be rather large by particle physics scales Braneworld at the LHC Gravitons can be emitted into the extra dimensions – missing energy signals! They may return round the loop and deposit energy on the brane at a rate of 10 12 Hz If gravity is strong at 1 TeV particle collisions may form a mini-black hole – the formation rate is hard to compute without a quantum theory of gravity though! Presumably they will decay by Hawking radiation giving a thermal spectrum output. We would need a complete quantum theory of gravity! Warped Extra Dimensions – Randall/Sundrum The metric on a slice of space between two 3-branes is AdS On the IR brane The warp factor introduces an exponential hierarchy between the weak scale and the Planck scale – at expense of explaining separation in y… One can look at fluctuations of the metric ie gravitons. It turns out the localized energy density of the brane binds a massless graviton to the brane – our gravity… the 5d gravity modes are localized away from our brane and couple weakly – their masses are 1TeV, 2 TeV etc In fact even in the limit where the slice is made infinite the coupling suppression of the 5d continuum only changes the gravitational potential a little… AdS/CFT Correspondence In fact 5d gravity models in AdS space appear to be related to 4d strongly coupled gauge theories like QCD… Gauge gravity duality emerged from string theory…. Essentially one treats the renormalization group or energy scale of “QCD” as a 5th space direction. Couplings and operators run in that direction and are represented by fields living in the 5th direction is a scalar in the 5th direction Eg -> Eg -> a graviton in the 5th direction There are therefore 5d gravity models (toy) of QCD which allow computations of bound state masses and their interaction strengths arXiv:0711.4467 The RS models are possibly generated by a 4d gauge theory too – the full implications for real world gravity remain obscure… we live in dS space! Summary • KK & string theory suggest there could be more than 3 dimensions • The key signature is a KK tower for every particle propagating in the higher dimensions • Brane theory means one can pick and choose which fields know about the extra dimensions • The number of dimensions does need explaining • The natural compactification scale for extra dimensions is the Planck scale… but what if they’re large? • If the extra dimensions are warped they need not even be compact • If the Planck scale is brought down to 1 TeV we solve the hierarchy problem… and possibly study the TOE at the LHC • However unlikely, we must seek everything possible at LHC