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Physics of Extra Dimensions

Physics of Extra Dimensions. Sreerup Raychaudhuri IIT Kanpur. z. We are used to the idea of three space dimensions ─ where is the room for more dimensions?. y. x. Relativity introduces a ‘fourth dimension’, viz. x o. Minkowski space. x 0 is not really an extra dimension….

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Physics of Extra Dimensions

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  1. Physics of Extra Dimensions Sreerup Raychaudhuri IIT Kanpur

  2. z We are used to the idea of three space dimensions ─ where is the room for more dimensions? y x Relativity introduces a ‘fourth dimension’, viz. xo Minkowski space x0 is not really an extra dimension…

  3. Compact dimensions: A long human hair looks one-dimensional to us It still looks one-dimensional to an ant walking along it It looks two-dimensional to a louse living on it

  4. What determines the number of dimensions is the length scale at which we are doing the experiment Compact dimensions are those where we must impose periodic boundary conditions… Typical length scale , e.g. circumference of the cylindrical hair. The compact dimension will not show up in those experiments where the measurements are made at a length scale

  5. ergo, if we choose smaller than the smallest distance experimentally accessible, we can have as many extra spatial dimensions as we wish. Pluritas non est ponenda sine neccessite William of Ockham c. 1320 We should not introduce extra spatial dimensions unless we actually need them to explain empirical facts…

  6. Compact dimensions were introduced in the early days of quantum mechanics 2r = L X=L X=0 Periodic boundary conditions are needed to define a free particle, a Bloch state, etc., etc.…. Most of solid state physics is done on a 3-torus!

  7. History of Extra Space Dimensions C.H. Hinton 1884 :‘tesseract’ D.Nørdstrom 1914 :unification of Newtonian gravity with Maxwell equations T. Kaluza 1921 :unification of Einstein gravity with Maxwell equations O. Klein 1926 :better version of Kaluza’s theory W. Pauli 1937 :6-dimensional Kaluza-Klein theory ……… (limited success) String theory :bosonic string ‘lives’ only in 261970’s dimensions… many variations & developments

  8. Revived in 1998 as a solution to the gauge hierarchy problem SM is an interacting quantum field theory  makes no sense as a classical field theory because the particulate nature of quarks, leptons and gauge bosons is well established. Tree-level calculations correspond only to lowest order term in perturbation expansion  make no sense unless ALL the terms in the expansion are considered, at least in principle

  9. Higher order terms (radiative corrections) can be neglected if and only if they are small … • Radiative corrections to elementary fermionmasses grow logarithmically as cutoff scale, i.e. Log─ power-law dependence cancels due to chiral symmetryremain small • Radiative corrections to elementary gauge bosonmasses grow logarithmically, i.e Log─ power-law dependence cancels due to gauge symmetryremain small • Radiative corrections to elementary scalar masses grow quadratically as cutoff scale, i.e. 2─ not protected by any (known) symmetrycould become very large

  10. rule for the self-interactions of the boson H H Lint = H4 H H leads to a 2divergent self-energy correction to the mass H H H i pointed out by (1972)

  11. …would drive Higgs mass MHto the cutoff scale   e.g. W+W-Hcoupling would become non-perturbative !! • There are two ways out of the hierarchy problem: • Postulate a symmetry which will cause the 2term to cancel ─ supersymmetry, little Higgs models • Reduce the cutoff to the TeV scale ─ technicolour, extra dimensions

  12. Energy Scale Cutoff for the Standard Model : • Inputs to the Standard Model: • Quark model • Electroweak gauge theory : scale ~ 100 GeV • QCD : scale ~ 1/3 GeV • i.e. it is known to be valid to ~100 GeV  10-16cm • Things lacking in Standard Model : • Objects more elementary than quarks/leptons ? • Grand unification ? • Role of gravity • Any of these could provide the reason for a cutoff scale 

  13. Ockam’s razor again… We do not have any compelling empirical reason to believe that quarks/leptons have substructure We do not have any compelling empirical reason to believe in grand unification BUT We do know that gravity exists and that it must be quantized Natural scale for a quantum theory of gravity : Planck mass Definite cutoff for SM ! This is so large because gravity is so weak…

  14. Hierarchy problem: If quantum gravity gives the cutoff for the Standard Model (desert scenario), then the Higgs boson mass will be driven to the Planck scale… Q. Why is the Planck scale so large? alternatively: Q. Why is gravity so weak compared to the other interactions? Naturalness : Very large or very small numbers are unstable under quantum corrections Need some underlying symmetry to protect them

  15. WISHFUL THINKING If gravity were not so weak, e.g. ifGN ~ GFthe Standard Model would be cut off at a ‘Planck scale’ of ~ 100 GeV ─ there would be no hierarchy problem Can such an idea be a serious scientific possibility? We have measured the strength of the gravitational field many many times, since the days of Isaac Newton… even in high school labs... today there is no doubt at all that GN is indeed small… BUT The length scales at which such measurements have been done are very large compared to atomic sizes…

  16. Could it be that gravity is weak at large scales, but strong at small scales…. ? i.e. smaller than the electroweak scale: 10-16 GeV Then the much lower energy scale of this strong short-range (quantum) gravity would automatically cut off the Standard Model at much lower energies Known:We cannot achieve this within the framework of Einstein gravity in (1+3) dimensions Is the talk over ?

  17. NEWS FLASH It can be done if there are extra compact dimensions

  18. Roughly speaking, there are two main classes of extra-dimensional models for making gravity strong at small length scales : • Gravitational lines of force are dispersed in the extra dimensions and only a small number are observed in four-dimensional experiments : force is weakened in proportion ─Arkani-Hamed, Dimopoulos and Dvali 1998 • Gravity is strong in some other region of space, and loses strength as it ‘shines’ on our four-dimensional space : force is weakened according to distance ─Randall and Sundrum 1999 Both paradigms work if and only if there is a mechanism to confine the experiment(er) within the four Minkowski dimensions i.e. the extra dimensions are ‘seen’ by gravity alone

  19. What do we know experimentally about the length scale to which Einstein gravity (effectively Newton gravity, or just the inverse square law) is valid? On astronomical scales, inverse square law is valid Kepler (1619)… Hooke (1660 ?)… Newton (1687)

  20. Dark matter discovery... TASI 2004

  21. Torsion balance experiments at length scales ~ few cm Cavendish 1798 Eötvös 1891 torsion balance

  22. Eöt-Wash experiment at length scales ~ 100 m B. Heckel E.Adelberger Extremely sensitive torsion pendulum : tungsten torsion fibre 20 m thick Rotating disk with holes ─ matching holes in pendulum torsion effect cancels finely for inverse square law  any deflections of laser beam will be due to deviations from inverse square law

  23. 2003 data For ||~1  < 150 m Eventually  < 60 m Compare with

  24. Einstein gravity in (1+3) dimensions has been tested only up to the scale of Can there be extra dimensions a bit smaller than this, e.g. 10-3 cm ? Other interactions ─ electroweak, strong ─ have been tested all the way down to the electroweak length scale

  25. Many electroweak precision tests would show up new effects if there were extra dimensions in which the carrier fields could propagate… but they do not show any such effects… We require that only gravity should ‘see’ extra dimensions … other interactions should not ! SM fields Gravity

  26. ADD Model : Large Extra Dimensions Arkani-Hamed, Dimopoulos and Dvali (March 1998) ‘d’ compact dimensions 3 + 1 • 1+3 Minkowski dimensions • ‘d’ large compact dimensions • SM fields trapped in 1+3 • Gravity propagates in 1+3+d 10-3 cm Mechanism of confinement? …. Domain walls… Vortices…. D-branes….

  27. D-branes: • Introduced by Polchinksi in 1995 • Solitonic configurations of superstring theory • Dp brane is a 1 + p dimensional hypersurface • open strings have ends fixed to Dp branes Dirichlet boundary conditions  Fields which are stringy excitations are confined within length 1// (/ = string tension) • Closed strings are free to propagate in 10 dimensions String theory provides the ideal mechanism to confine SM fields in 1+3 dimensions

  28. ADD Model : String Theoretic View Antoniadis, Arkani-Hamed, Dimopoulos and Dvali (April 1998) • Observable Universe is a D3 brane • Max. no of extra dimensions is d = 6 • SM fields: spin 0, ½ and 1 excitations of open strings with ends confined to D3 brane • Gravitons: spin 2 excitations of closed strings propagating in bulk bulk 10-17 cm D3 brane 10-3 cm String tension can be as small as -1 ~ 1 TeV  stringy excitations at a TeV

  29. Weak gravity Lines of force are mostly dissipated in the bulk… Only a small number are intercepted by the brane Qualitative : Quantitative: Einstein-Hilbert action in 4+d dimensions Integrate over bulk for large objects

  30. Bulk scale versus Planck scale on a d-torus Possible to have TeV strings if d  2

  31. ADD Solution to the Hierarchy problem: • All known experiments/observations are done on the D3 brane and do not sense the extra dimensions until the energy scale of the experiment reaches the bulk scale  (string tension)-1 (= TeV?) • Gravity propagates in all the 3+d spatial dimensions, including the D3 brane, of course. • As we approach the bulk scale, stringy excitations begin to appear, i.e. the Standard Model is no longer valid • Bulk scale (= TeV?) acts as a cutoff for the Standard Model • There is no hierarchy problem…

  32. Observable consequences : Massless bulk scalar Fourier series on a d-torus Massive scalars on brane

  33. On the brane… Tower of Kaluza-Klein states : Spacing between states : No of contributing states : A bulk scalar field is like a huge swarm of 4-scalar fields on the brane

  34. Position of the brane is at Standard Model fields live only on the brane : Interaction with single bulk scalar field is the same as interaction with a swarm of 4-scalar fields on the brane

  35. Weak gravitational field limit : Valid for energies much lower than Planck (bulk) scale Free Einstein equations in 4+d dimensions : reduce to : Massless Klein-Gordon equation for a bulk tensor… Each of the fields has its own Kaluza-Klein decomposition

  36. On the brane… All thebilinear covariants with Standard Model fields have indices running over 0,1,2,3 only Interaction with a graviton tower Interaction with a dilaton tower

  37. Feynman Rules for the ADD model Han, Lykken and Zhang, Phys RevD59, 105006 all scalars all gauge bosons all fermions

  38. Collider physics with gravitons/dilatons: • Graviton tower couples to every particle-antiparticle pair • Blind to all quantum numbers except energy-momentum • Each Kaluza-Klein mode couples equally, with strength  • Tower of Kaluza-Klein modes builds up collectively to an observable effect • Individual graviton modes escape detection  missing • Signals will show • excess over Standard Model cross-sections • different distributions due to spin-2 nature • energy and momentum imbalance

  39. REAL GRAVITONS Incoherent sum

  40. VIRTUAL GRAVITONS Coherent sum

  41. Sum over KK states can be done using the quasi-continuum approach Sum over propagators… reduces to a contact interaction…

  42. Important processes at colliders LHC ILC

  43. Bounds on bulk scale  ‘string’ scale Black : LEP & Tevatron Run II Green : SN 1987A

  44. Important issues in ADD phenomenology • Find out of there are signals for KK towers of gravitons ─ large-pT excess, missing energy, etc. • Determine whether the signals are indeed due to brane-world gravitons and not some other new physics ─ gravitons would be blind to all Standard Model quantum numbers • Identify these particles (if seen) as graviton modes ─ spin-2 nature is a dead giveaway • Find out the number of large extra dimensions • Find out the radius of compactification RC, or equivalently, the bulk scale (string scale MS) • Find out the geometry of the extra dimensions • Find out dynamics which makes some dimensions large & some small

  45. 1 TeV 2 TeV Dutta, Konar, Mukhopadhyaya, SR (2003)

  46. Laboratory Black holes Gravity becomes strong at ~ TeV. LHC will collide protons at 14 TeV  Trans-Planckian regime Schwarzschild radius for a black hole in 3+d dimensions: In a pp collision, if the impact parameter is less than RSthe protons will coalesce to form a micro-black hole. Cross section is just: BH  RS2 (semi-classical) For~ 1 TeVthere will be copious black hole production Decay by Hawking radiation: produces distinctive signatures ‘CATFISH’ generator… 31.08.2006

  47. Simulation of a black hole production and decay event at the LHC (de Roeck 2003)

  48. All is not well with the ADD model… • The KK modes have masses typically : 10-3 eV • The scale of strong gravity is typically : 1012 eV • scale hierarchy of 15 orders of magnitude • Quantum corrections tend to shrink the size of the d-dimensional bulk • process terminates only when the scale reaches Planck scale  back to ‘tHooft 1972… • Large extra dimensions are unstable ! • Need a mechanism to stabilize the size… The Hierarchy problem strikes back…

  49. Randall-Sundrum Model May 1999 warped compactification

  50. Model is based on an orbifold compactification… …one extra dimension… Fixed points A circle folded about a diameter Only logical place to place a brane is at a fixed point ─ put one at each

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