Transcript Slide 1
Physics of Extra Dimensions Sreerup Raychaudhuri IIT Kanpur z y We are used to the idea of three space dimensions ─ where is the room for more dimensions? x Relativity introduces a ‘fourth dimension’, viz. x 0 ct x' x xo Minkowski space x dx ds dx 2 0 2 2 x0 is not really an extra dimension… 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 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 cylindrical hair. 0, e.g. circumference of the The compact dimension will not show up in those experiments where the measurements are made at a length scale r 0 ergo, if we choose 0 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… Compact dimensions were introduced in the early days of quantum mechanics 2r = L X=0 X=L 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! 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 ……… String theory 1970’s (limited success) : bosonic string ‘lives’ only in 26 dimensions… many variations & developments 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 S fi fi iT (1) fi i T 2 (2) fi i T 3 (3) fi ... Higher order terms (radiative corrections) can be neglected if and only if they are small S fi fi iT i T (1) … fi 2 (2) fi i T fi(3) ... 3 • Radiative corrections to elementary fermion masses grow logarithmically as cutoff scale, i.e. Log ─ power-law dependence cancels due to chiral symmetry remain small • Radiative corrections to elementary gauge boson masses 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 rule for the self-interactions of the boson H H Lint = H4 H H leads to a 2 divergent self-energy correction to the mass H M ~ log 2 H H 2 2 2 H i pointed out by (1972) finite M ~ log 2 H 2 2 2 finite …would drive Higgs mass MH to the cutoff scale e.g. W+W-H coupling would become non-perturbative !! g M H2 2 MW g 2 2 MW There are two ways out of the hierarchy problem: 1. Postulate a symmetry which will cause the 2 term to cancel ─ supersymmetry, little Higgs models 2. Reduce the cutoff to the TeV scale ─ technicolour, extra dimensions Energy Scale Cutoff for the Standard Model : Inputs to the Standard Model: 1. Quark model 2. Electroweak gauge theory : scale ~ 100 GeV 3. QCD : scale ~ 1/3 GeV i.e. it is known to be valid to ~100 GeV 10-16 cm Things lacking in Standard Model : 1. Objects more elementary than quarks/leptons ? 2. Grand unification ? 3. Role of gravity Any of these could provide the reason for a cutoff scale 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 c MP GN 1.22 1016 TeV / c 2 This is so large because gravity is so weak… Definite cutoff for SM ! GN GF 5.7 1034 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… M H M P 100 000 000 000 000 000 MW 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 WISHFUL THINKING If gravity were not so weak, e.g. if GN ~ GF the 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… 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 R Rg 8 GNT 1 2 , 0,1, 2,3 Is the talk over ? NEWS FLASH It can be done if there are extra compact dimensions Roughly speaking, there are two main classes of extradimensional models for making gravity strong at small length scales : 1. 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 2. 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 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) GN m m (r ) 1 e r / r Dark matter discovery... TASI 2004 Torsion balance experiments at length scales ~ few cm Cavendish 1798 Eötvös 1891 torsion balance Eöt-Wash experiment at length scales ~ 100 m E.Adelberger B. Heckel 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 GN m m (r ) 1 e r / r 2003 data For ||~1 < 150 m Eventually < 60 m Compare with P ~ 1029 m Einstein gravity in (1+3) dimensions has been tested only up to the scale of EotWash ~102 m = 102 cm 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 EW ~ 1012 m = 1016 cm 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 ADD Model : Large Extra Dimensions Arkani-Hamed, Dimopoulos and Dvali (March 1998) ‘d’ compact dimensions •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…. 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 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 10-3 cm String tension can be as small as -1 ~ 1 TeV stringy excitations at a TeV Weak gravity Qualitative : Lines of force are mostly dissipated in the bulk… Only a small number are intercepted by the brane Quantitative : Einstein-Hilbert action in 4+d dimensions Sˆ 1 4 d d x d y gˆ B ( x, y ) Rˆ B 16 Gˆ B VB 4 d x g ( x) R + ... 16 GN Integrate over bulk for large objects Gˆ N GN Vd Bulk scale versus Planck scale 2 GN 2 ; MP Gˆ N 2 ; 2 d ˆ MP Gˆ N GNVB MP Mˆ P V B 2 2 d MP 2 2 d 2 RC d 2 d on a d-torus Possible to have TeV strings if d 2 ADD Solution to the Hierarchy problem: 1. 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?) 2. Gravity propagates in all the 3+d spatial dimensions, including the D3 brane, of course. 3. As we approach the bulk scale, stringy excitations begin to appear, i.e. the Standard Model is no longer valid 4. Bulk scale (= TeV?) acts as a cutoff for the Standard Model 5. There is no hierarchy problem… Observable consequences : ˆ (t , x , y ) 0 ˆ Massless bulk scalar 2 2 2 ˆ t 2 2 (t , x , y ) 0 x y 2 2 2 t 2 2 n (t , x ) ein . y / 2 RC 0 x y n 0 Fourier series on a d-torus 2 2 n2 in . y / 2 RC t 2 ( t , x ) e 0 n 2 x 2 R n 0 C n2 n (t , x ) 0 n (n1, ..., nd ) 2 2 R C Massive scalars on brane On the brane… Tower of Kaluza-Klein states : n ( x) n2 Mn 2 RC 2 n12 n22 ... nd2 2 RC Spacing between states : M n ~ 1 RC ~ MP if RC ~ P ~ 0.01 eV if RC ~ 0.001 cm No of contributing states : s 100 GeV ~ ~ 1013 M n 0.01 eV A bulk scalar field is like a huge swarm of 4-scalar fields on the brane Position of the brane is at y 0 Standard Model fields live only on the brane : ˆ x, y y x ˆ ( x , 0) ˆ ( x, 0) y ˆ ( x, y ) Sint d 4 x d d y in . y / 2 RC ˆ ˆ ˆ d x d y ( x, 0) ( x, 0) y n ( x) e 4 d n ˆ ( x) ˆ n ( x) d 4 x ( x) n ˆ ( x) ˆ n ( x) d x ( x) 4 n Interaction with single bulk scalar field is the same as interaction with a swarm of 4-scalar fields on the brane Weak gravitational field limit : g ( x) h ( x) 16 Valid for energies much lower than Planck (bulk) scale MP Free Einstein equations in 4+d dimensions : reduce to : 1 ˆ Rˆ Rgˆ ˆˆ ˆˆ 0 2 ˆ hˆ ˆ ˆ ( x, y ) 0 Massless Klein-Gordon equation for a bulk tensor… h ( x, y ) Ai ( x, y) hˆ ˆ ˆ ( x, y ) A ( x , y ) ( x , y ) ij j Each of the h ( x, y), Ai ( x, y), ij ( x, y) fields has its own KaluzaKlein decomposition On the brane… All the bilinear covariants with Standard Model fields have indices running over 0,1,2,3 only ˆ ˆ ( x, 0) y ˆ ˆ h ˆ ˆ ( x, y ) Sint d x d y ( x, 0) 4 d 2 8 4 d x ( x) ( x)hn ( x) MP n Interaction with a graviton tower ˆ ˆ ( x, 0) y ij ( x, y ) Sint d x d y ( x, 0) ij 4 d 2 8 4 d x ( x) ( x) n ( x) MP n ( x, y) ii ( x, y) i Interaction with a dilaton tower Feynman Rules for the ADD model Sint d 4 x n Han, Lykken and Zhang, Phys Rev D59, 105006 all scalars all gauge bosons all fermions n h T 2 ( SM ) 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 pT • Signals will show 1. excess over Standard Model cross-sections 2. different distributions due to spin-2 nature 3. energy and momentum imbalance REAL GRAVITONS n n Incoherent sum n n 2 VIRTUAL GRAVITONS 2 n n Coherent sum A A n n Sum over KK states can be done using the quasi-continuum approach A(M s n n (M ) ) dM ( M ) A( M ) 0 d C d /2 R M 4 d 2 (d / 2) Sum over propagators… sM n 2 2 n i (d , s) Mˆ P4 1 4 MS reduces to a contact interaction… Important processes at colliders pp Gn ,WGn , ZGn , JGn LHC pp Gn ,WWGn , ZZGn , JJGn pp G * n , ,WW , ZZ , JJ e e Gn , ZGn , JGn ILC * n e e G ee , ,WW , ZZ , JJ Gn ,WWGn , ZZGn , JJGn Bounds on bulk scale ‘string’ scale 100 MS (TeV) 30 10 1.45 1.09 1 4 0.87 0.72 0.6 1 0.65 0.38 0.1 d=2 d=3 d=4 d=5 Black : LEP & Tevatron Run II Green : SN 1987A d=6 Important issues in ADD phenomenology 1. Find out of there are signals for KK towers of gravitons ─ large-pT excess, missing energy, etc. 2. 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 3. Identify these particles (if seen) as graviton modes ─ spin-2 nature is a dead giveaway 4. Find out the number of large extra dimensions 5. Find out the radius of compactification RC, or equivalently, the bulk scale (string scale MS) 6. Find out the geometry of the extra dimensions 7. Find out dynamics which makes some dimensions large & some small 1 TeV 2 TeV Dutta, Konar, Mukhopadhyaya, SR (2003) 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: d 1 m 8 1 2 RS ˆ ˆ M MP P d 2 1 d 1 In a pp collision, if the impact parameter is less than RS the protons will coalesce to form a micro-black hole. Cross section is just: For BH RS2 (semi-classical) Mˆ P ~ 1 TeV there will be copious black hole production Decay by Hawking radiation: produces distinctive signatures ‘CATFISH’ generator… 31.08.2006 Simulation of a black hole production and decay event at the LHC (de Roeck 2003) 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… Randall-Sundrum Model May 1999 warped compactification Model is based on an orbifold compactification… …one extra dimension… 2 RC Fixed points 0 S1 / Z 2 A circle folded about a diameter Only logical place to place a brane is at a fixed point ─ put one at each 5-D Einstein-Hilbert action with a cosmological constant term : SRS 1 4 ˆ Rˆ ( x, ) ˆ d x d R g ( x , ) C 16 Gˆ N ˆ Different normalization convention: G N RC 1 32 Mˆ P3 ˆ RC ˆ ˆ 16 GN ˆ 2Mˆ 3 Rˆ ( x, ) S RS d 4 x d RC - gˆ ( x, ) P d 4 x - gˆ ( x, 0) V0 L0 ( x, 0) d 4 x - gˆ ( x, ) V L ( x, ) Matter content of a brane is parametrized as a VEV ─ brane tension cosmological constant for 4-D Einstein gravity on the brane Equations of motion: 1 ˆ gˆ Rˆ Rgˆ ˆˆ ˆˆ 2 1 (0) (0) ( ) ( ) ˆ 3 gˆ gˆ g g V g g V ˆ ˆ ˆ ˆ ˆ 0 ˆ ˆ MP ‘RS Ansatz’ : ds e 2 dx dx d RC 2 f ( ) e2 f ( ) gˆ 0 Solution: f kRC RS Metric : ds e 2 0 1 2 Fine-tuning : ˆ V0 V k 24Mˆ P3 24Mˆ P3 24Mˆ P3 dx dx d RC 2 kRC | | 2 Warp : metric dies out exponentially from = 0 to = AdS5 RS Mechanism: Metric contracts exponentially along the ‘AdS5 throat’ measuring sticks contract exponentially wavelengths increase exponentially energies drop exponentially Like a gravitational redshift At = 0 : At e = : kRC e kRC 1 e kRC 16 ~ 10 if kRC 12 Weakness of gravity on ‘TeV brane’ at = is explained without recourse to large numbers Randall-Sundrum solution to the hierarchy problem All mass scales on the ‘Planck brane’ get scaled by warp factor when they get ‘shined’ on the ‘TeV brane’ Mˆ P kM 2 P 1/ 3 1/ 3 12 M R C 2 P e kRC If we set RC ~ P then Mˆ P is large no TeV strings Kaluza-Klein modes of the RS bulk graviton field : Small fluctuations around vacuum metric g ( x) e 2 kRC h ( x) Fourier expansion of graviton field : 1 h ( x, ) RC Goldberger and Wise (1999) Davoudiasl, Hewett and Rizzo h ( x) ( ) n n 0 n Equation of motion : n M n2 h ( x) 0 Warped harmonics 1 d 4kRC d 2 2 kRC e ( ) M n ( ) n e n 2 RC d d 2 kRC d e m ( ) n ( ) mn M n 2 kRC e f n ( ) e 2 kRC n ( ) Conformal coordinates : zn ( ) k 2 d fn df n 2 2 Eigenvalue equation : zn z z n n 4 fn 0 2 dzn dzn Bessel equation of order 2 Warped harmonics : e2kRC n ( ) J 2 ( zn ) nYn ( zn ) Nn Require harmonics to be continuous at the orbifold fixed points… M n xn ke kRC xn m0 n xn2 e 2 kR C Nn 1 m0 RC Electroweak scale J 2 ( xn ) J1 ( xn ) 0 Graviton interaction with matter : kRC 8 0 8 e Lint ( x) h ( x) T ( x) MP MP n h ( x ) T (x) n 1 Zero (massless) mode gives usual Einstein gravity Massive (attenuated) modes have electroweak strength couplings RS Gravitons are like WIMPs : masses and couplings both resemble electroweak particles Feynman rules same as in ADD model apart from warp-up factor… Two free parameters: M1 x1m0 8 k MP RS graviton phenomenology : • RS graviton width grows rapidly with graviton mass – Only first three modes can form narrow resonances – For large part of parameter space only first resonance is viable • RS gravitons decay to all particle pairs • Maximum BR is to jets; sizeable width to WW and ZZ • No deviations from SM at LEP-2 lightest RS graviton is heavier than 210 GeV • Tevatron Drell-Yan data show no deviations either lightest RS graviton is heavier than ~ 850 GeV • LC: smaller but clean final states: • graviton resonances in Bhabha scattering and e+e- +- D(s) e kR C n 0 s M iM n n 2 n ( xs ) m04 s xs m0 Graviton resonances: e e Davoudiasl, Hewett and Rizzo Modulus stabilization and the radion: Warping is extremely sensitive to RC kRC e 16 ~ 10 if kRC 12 Consider the radius of the extra dimension as a dynamical object : ds e 2 Modulus field : Radion field : S grav 2 kT ( x ) g ( x) T ( x) d 2 2 T ( x) ( x ) 24 Mˆ 3 k e k T ( x ) 1 1 2 Mˆ 3 2 d x g ( x ) 2 12 k 4 Radion is a free field i.e. can assume any value same for modulus Need for modulus stabilization Goldberger-Wise mechanism : Assume a bulk scalar field B( x, ) Write down a B4 theory in the bulk and on the two branes… Solving the equation of motion for (x) and integrating over leads to potential with a steep minimum at v0 4k 2 kRC k T ( x) log 2 MB v Can assume the desired value without assuming any large/small numbers… Undetermined parameters: radion mass M & radion vev Radion couplings are very Higgs-like… Radion phenomenology : • Radion phenomenology is rather similar to Higgs phenomenology for tree-level processes • Possibility of Higgs-radion mixing – Giudice, Rattazzi, Wells (2000) – Huitu, Datta (2002) • ‘Radion’strahlung process … – Production is just like Higgs-strahlung – At one-loop, effect of kinetic terms in radion-fermion couplings becomes important – Try to identify radion by its somewhat different decay widths to gluons (one-loop) i.e. dijet decay mode Light radion decays primarily to gluon jets; Higgs decays primarily to b-jets Use b-tagging to compare cross-sections for e e ZJJ and e e Zbb Ratio shows distinct difference… Das, Rai, SR, PLB 2005 Main Issues in RS Phenomenology : • Find out of there are signals for graviton resonances ─ bump hunting… • Determine whether the resonances are indeed RS gravitons and not some other new physics ─ RS graviton masses are spaced like zeros of Bessel function J1 • Identify these resonances as graviton modes ─ spin-2 nature is a dead giveaway • Find out if there are signals for radions ─ very similar to Higgs search • Find out the mass and coupling parameters ─ mass and width measurements (like W,Z at Tevatron) • If the resonances are broad distinguish between RS and ADD models • Distinguish the radion from a Higgs scalar Universal Extra Dimensions : Appelquist, Cheng, Dobrescu 2002 Both ADD and RS models give effective field theories below the cutoff scale ~ TeV There will be higher-dimension operators, suppressed by electroweak scale mass only can cause all sorts of trouble for SM, including proton decay Place all the SM fields in the bulk and do away with the brane Orbifold the bulk to get chiral fermions; no branes Expand all fields in terms of the bulk harmonics each SM field has its own Kaluza-Klein tower. KK number is conserved due to orthonormality of bulk harmonics non-diagonal operators are suppressed by RC 1 Mˆ P Dimensional Deconstruction : Arkani-Hamed, Cohen, Georgi 2001 Many people are uncomfortable with the idea of extra space dimensions Gauge theory based on complicated gauge group, with different sets of fermions, each transforming as different representations of the gauge groups At low energies, it resembles a 5-D theory At still lower energies, it resembles a tower of KK states… Holography : Maldacena conjecture 1997 AdS/CFT correspondence Can the AdS5 bulk be the dual of some 4-dimensional CFT on the branes? Presence of branes corresponds to s.s.b. of conformal invariance, possible in an effective theory… Only reality is 4-D Standard Model and 4-D CFT at some high scale ; extra dimensional is a duality artefact… Recommended reading : Field-theoretic aspects… AdS/CFT, model building…