1. 1 Extra dimensioni e la fisica elettrodebole G.F. Giudice Napoli, 9-10 maggio 2007

2. 2 SPACE DIMENSIONS AND UNIFICATION Minkowski recognized special relativistic invariance of Maxwell’s eqs  connection between unification of forces and number of dimensions Electric & magnetic forces unified in 4D space time

3. 3 UNIFICATION OF EM & GRAVITY Next step:  New dimensions? 1912: Gunnar Nordström proposes gravity theory with scalar field coupled to T   1914: he introduces a 5-dim A  to describe both EM & gravity 1919: mathematician Theodor Kaluza writes a 5- dim theory for EM & gravity. Sends it to Einstein who suggests publication 2 years later 1926: Oskar Klein rediscovers the theory, gives a geometrical interpretation and finds charge quantization In the ‘80s the theory, known as Kaluza-Klein becomes popular with supergravity and strings

4. 4 GRAVITY In General Relativity, metric (4X4 symmetric tensor) dynamical variable describing space geometry (graviton) Dynamics described by Einstein action G N Newton’s constant R curvature (function of the metric)

5. 5 Consider GR in 5-dim Choose Dynamical fields Assume space is M 4  S 1 First considered as a mathematical trick It may have physical meaning (t,x) x5x5 R

6. 6 Extra dim is periodic or “compactified” All fields can be expanded in Fourier modes 5-dim field  set of 4-dim fields: Kaluza-Klein modes Each has a fixed momentum p 5 =n/R along 5 th dim 4-d space extra dimensions mass D-dim particle E 2 = p 2 + p 2 extra + m 2  KK mass From KK mass spectrum we can measure the geometry of extra dimensions

7. 7 R r << Rr >> R 2-d plane 1-d line Suppose typical energy << 1/R  only zero-modes can be excited Expand S G keeping only zero-modes and setting  =1 To obtain correct normalization: Gravity & EM unified in higher-dim space: MIRACLE?

8. 8 Gauge transformation has a geometrical meaning Keep only zero-modes: Invariant under local (where g and  do not transform) Gauge transformation is balanced by a shift in 5 th dimension EM Lagrangian uniquely determined by gauge invariance

9. 9 CHARGE QUANTIZATION Matter EM couplings fixed by 5-dim GR Consider scalar field  Expand in 4-D KK modes: Each KK mode n has: mass n/R charge n  /R charge quantization determination of fine-structure constant new dynamics open up at Planckian distances

10. 10 Not a theory of the real world  =1 not consistent (  dynamical field leads to inconsistencies: e.g. F (0)  F (0)  =0 from eqs of motion) Charged states have masses of order M Pl Gauge group must be non-abelian (more dimensions?) Nevertheless Interesting attempt to unify gravity and gauge interactions Geometrical meaning of gauge interactions Useful in the context of modern superstring theory Relevant for the hierarchy problem?

11. 11 Usual approach: fundamental theory at M Pl, while  W is a derived quantity Alternative:  W is fundamental scale, while M Pl is a derived effect New approach requires extra spatial dimensions confinement of matter on subspaces Natural setting in string theory  Localization of gauge theories on defects (D-branes: end points of open strings) We are confined in a 4-dim world, which is embedded in a higher-dim space where gravity can propagate

12. 12 COMPUTE NEWTON CONSTANT Einstein action in D dimensions Assume space R 4  S D-4 : g  doesn’t depend on extra coordinates Effective action for g 

13. 13 Suppose fundamental mass scale M D ~ TeV very large if R is large (in units of M D -1 ) Arkani-Hamed, Dimopoulos, Dvali Radius of compactified space Smallness of G N /G F related to largeness of RM D Gravity is weak because it is diluted in a large space (small overlap with branes) Need dynamical explanation for RM D >>1

14. 14  Gravitational interactions modified at small distances At r < R, space is (3+  )-dimensional (  =D-4) From SN emission and neutron-star heating: M D >750 (35) TeV for  =2(3)

15. 15 Probability of producing a KK graviton Number of KK modes with mass less than E (use m=n/R) Inclusive cross section graviton gluon It does not depend on V D (i.e. on the Planck mass) Missing energy and jet with characteristic spectrum Testing extra dimensions at high-energy colliders

16. 16

17. 17 Contact interactions from graviton exchange Sensitive to UV physics d-wave contribution to scattering processes predictions for related processes Limits from Bhabha/di-  at LEP and Drell- Yan/ di-  at Tevatron:  T > 1.2 - 1.4 TeV Loop effect, but dim-6 vs. dim-8  only dim-6 generated by pure gravity   > 15 - 17 TeV from LEP

18. 18 TRANSPLANCKIAN REGIME Planck length quantum-gravity scale Schwarzschild radius classical gravity same regime G-emission is based on linearized gravity, valid at s << M D 2 The transplanckian regime is described by classical physics (general relativity)  independent test, crucial to verify gravitational nature of new physics

19. 19 b > R S Non-perturbative, but calculable for b>>R S (weak gravitational field) D-dim gravitational potential: Quantum-mechanical scattering phase of wave with angular momentum mvb Gravitational scattering  b vm

20. 20 Diffractive pattern characterized by Gravitational scattering in extra dimensions: two-jet signal at the LHC

21. 21 b < R S At b<R S, no longer calculable Strong indications for black-hole formation Characteristic events with large multiplicity ( ~ M BH / ~ (M BH / M D   2)/(  +1) ) and typical energy ~ T H BH with angular momentum, gauge quantum numbers, hairs (multiple moments of the asymmetric distribution of gauge charges and energy-momentum)  ~  R S 2 10 pb (for M BH =6 TeV and M D =1.5 TeV) Gravitational and gauge radiation during collapse  spinning Kerr BH Hawking radiation until Planck phase is reached T H ~ R S -1 ~ M D (M D / M BH ) 1/  1) Evaporation with  ~ M BH (  +3)/(  +1) / M D 2(  +2)/(  +1) (10 -26 s for M D =1 TeV) Transplanckian condition M BH >> M D ?

22. 22 WARPED GRAVITY A classical mechanism to make quanta softer For time-indep. metrics with g 0  =0  E |g 00 | 1/2 conserved. (proper time d  2 = g 00 dt 2 ) On non-trivial metrics, we see far-away objects as red-shifted

23. 23 Randall Sundrum Consider observer & emitter as 3-d spaces (branes) embedded in non-trivial 5-D space-time geometries 5 th dim S 1 / Z 2 : identify y  y+2  R, y  -y y 0 RR y 0 RR -y 0 RR Two 3-branes on boundaries & appropriate vacuum-energy terms 

24. 24 GRAVITATIONAL RED-SHIFT Masses on two branes related by Same result can be obtained by integrating S E over y y=0 g 00 =1 y=  R g 00 =e -2  RK

25. 25 PHYSICAL INTERPRETATION Gravitational field configuration is non-trivial Gravity concentrated at y=0, while our world confined at y=  R Small overlap  weakness of gravity WARPED GRAVITY AT COLLIDERS KK masses m n = Kx n e -  RK [x n roots of J 1 (x)] not equally spaced Characteristic mass Ke -  RK ~ TeV KK couplings KK gravitons have large mass gap and are “strongly” coupled Clean signal at the LHC from G  l + l -

26. 26 Spin 2 Spin 1

27. 27 A SURPRISING TWIST AdS/CFT correspondence relates 5-d gravity with negative cosmological constant to strongly-coupled 4-d conformal field theory Theoretical developments in extra dimensions have much contributed to model building of 4-dim theories of electroweak breaking: susy anomaly mediation, susy gaugino mediation, Little Higgs, Higgs-gauge unification, composite Higgs, Higgsless, … Warped gravity with SM fermions and gauge bosons in bulk and Higgs on brane Technicolor-like theory with slowly-running couplings in 4 dim

28. 28 What screens the Higgs mass? boson Spont. broken global symm. fermion Chiral symmetry vector Gauge symmetry mHmH Dynamical EW breaking Delayed unitarity violat. Fundamental scale at TeV Very fertile field of research Different proposals not mutually excluded LITTLE HIGGSSUPERSYMMETRYHIGGS-GAUGE UNIF. TECHNICOLORHIGGSLESSEXTRA DIMENSIONS Symmetry Dynamics

29. 29 Cancellation of Existence of positron  charm top 10 -3 eV?? CAVEAT EMPTOR electron self-energy  + -  0 mass difference K L -K S mass difference gauge anomaly cosmological constant n It is a problem of naturalness, not of consistency!

30. 30 HIGGS AS PSEUDOGOLDSTONE BOSON Gauge, Yukawa and self-interaction are non-derivative couplings  Violate global symmetry and introduce quadratic divergences Top sector ● ● ➤ ➤ No fine-tuning If the scale of New Physics is so low, why do LEP data work so well?

31. 31 A less ambitious programme: solving the little hierarchy strong dynamics new physics energy1 TeV10 TeV Little Higgs Composite Higgs Higgsless LEP LEP1 LEP2 MFV -- + Bounds on  [TeV]

32. 32 Explain only little hierarchy At  SM new physics cancels one-loop power divergences LITTLE HIGGS “Collective breaking”: many (approximate) global symmetries preserve massless Goldstone boson ℒ1ℒ1 ℒ2ℒ2 H ℒ1ℒ1 ℒ2ℒ2

33. 33 It can be achieved with gauge-group replication Goldstone bosons in gauged subgroups, each preserving a non-linear global symmetry which breaks all symmetries Field replication Ex. SU 2 gauge with   doublets such that V(           ) and   spontaneously break SU 2 Turning off gauge coupling to    Local SU 2 (  2 ) × global SU 2 (   ) both spont. broken

34. 34 Realistic models are rather elaborate Effectively, new particles at the scale f cancel (same-spin) SM one-loop divergences with couplings related by symmetry Typical spectrum: Vectorlike charge 2/3 quark Gauge bosons EW triplet + singlet Scalars (triplets ?)

35. 35 New states have naturally mass New states cut-off quadratically divergent contributions to m H Ex.: littlest Higgs model Log term: analogous to effect of stop loops in supersymmetry Severe bounds from LEP data

36. 36 TESTING LITTLE HIGGS AT THE LHC Discover new states (T, W’, Z’, …) Verify cancellation of quadratic divergences f from heavy gauge-boson masses m T from T pair-production T : we cannot measure TThh vertex (only model-dependent tests possible)

37. 37 M T from T production can be measured up to 2.5 TeV f and g H from DY of new gauge bosons Production rate and BR into leptons in region favoured by LEP (g H >>g W ) ➤ ➤ Can be seen up to Z H mass of 3 TeV

38. 38 Possible to test cancellation with 10% accuracy for m T < 2.5 TeV and m Z < 3 TeV Cleanest peak from In order to precisely extract T from measured cross section, we must control b-quark partonic density Measure T width?

39. 39 HIGGS AS EXTRA-DIM COMPONENT OF GAUGE FIELD A M = (A ,A 5 ), A 5  A 5 + ∂ 5  forbids m 2 A 5 2 gaugeHiggs Higgs/gauge unification as graviton/photon unification in KK Correct Higgs quantum numbers by projecting out unwanted states with orbifold The difficulty is to generate Yukawa and quartic couplings without reintroducing quadratic divergences NEW INGREDIENTS FROM EXTRA DIMENSIONS

40. 40 HIGGSLESS MODELS Breakdown of unitarity: no zero modes in restricted extra-D spaces (Scherk-Schwarz mechanism) New ways of breaking gauge symmetries: The gauge KK modes delay unitarity violation

41. 41 y R Scherk-Schwarz breaking A field under a 2  R translation has to remain the same, unless there is a symmetry (the field has to be equal up to a symmetry transformation) No more zero-modes!

42. 42 At the LHC discover KK resonances of gauge bosons and test sum rules on couplings and masses required to improve unitarity

43. 43 DUALITY AdS/CFT  Composite Higgs 5-D warped gravity large-N technicolor  SM in warped extra dims  strongly-int’ing 4-d theory KK excitations  “hadrons” of new strong force Technicolor strikes back? TeV branePlanck brane 5 th dim  IR  UV RG flow 5-D gravity 4-D gauge theory Motion in 5 th dim RG flow UV brane Planck cutoff IR brane breaking of conformal inv. Bulk local symmetries global symmetries

44. 44 TC Technicolor-like theories in new disguise Old problems The presence of a light Higgs helps Light Higgs screens IR contributions to S and T (f pseudo-Goldstone decay constant) Can be tuned small for strong dynamics 4  f at few TeV

45. 45 Structure of the theory m  mass of resonances g   coupling of resonances Communicate via gauge (g a ) and (proto)-Yukawa ( i ) quarks, leptons & gauge bosons strong sector Strong sector characterized by In the limit I, g a =0, strong sector contains Higgs as Goldstone bosons Ex. H = SU(3)/SU(2)  U(1) or H = SO(5)/SO(4)  -model with f = m  / g   Take I, g a << g   < 4 

46. 46 g a, i break global symmetry  Higgs mass New theory addresses hierarchy problem  reduced sensitivity of m H to short distances (below m  -1 ) Ex.: Georgi-Kaplan: g  =4 , f = v, no separation of scales Holographic Higgs: g  = g KK, m  = m KK Little Higgs: g , m  couplings and masses of new t’, W’, Z’

47. 47 Production of resonances at m  allows to test models at the LHC Study of Higgs properties allows a model independent test of the nature of the EW breaking sector Is the Higgs fundamental? SM (with m H < 180 GeV) supersymmetry composite? Holographic Higgs Gauge-Higgs unification Little Higgs

48. 48 Construct the Lagrangian of the effective theory below m  From the kinetic term, we obtain the definition of f = m  / g   Each extra H insertion gives operators suppressed by 1 / f Each extra derivative “ “ 1 / m  f: symmetry-breaking scale m  : new-physics mass threshold Operators that violate Goldstone symmetry are suppressed by corresponding (weak) coupling

49. 49 Operators testing the strong self coupling of the Higgs (determined by the structure of the  model) and y f are SM couplings; c i model-dependent coefficients Form factors sensitive to the scale m  Loop-suppressed strong dynamics

50. 50 All Higgs couplings rescaled by Modified Higgs couplings to matter Effects in Higgs production and decay

51. 51 Dührssen 2003 SLHC Report 2002

52. 52 LHC can measure c H v 2 /f 2 and c y v 2 /f 2 up to 20-40% SLHC can improve it to about 10% A sizeable deviation from SM in the absence of new light states would be indirect evidence for the composite nature of the Higgs ILC can test v 2 /f 2 up to the % level ECFA/DESY LC Report 2001 ILC can explore the Higgs compositeness scale 4  f up to 30 TeV

53. 53 Effective-theory approach is half-way between model- dependent and operator analyses Dominant effects come from strong self-Higgs interactions characterized by From operator analyses, Higgs processes loop- suppressed in SM are often considered most important for searches However, operators h  and h  gg are suppressed 1/(16  2 m  2 ) Since h is charge and color neutral, gauging SU(3) c  U(1) Q does not break the generator under which h shifts (Covariant derivative acting on h does not contain  or g) Not the case for h  Z (loop, but not 1/g    suppressed)

54. 54 Higgs decay rates

55. 55 Genuine signal of Higgs compositeness at high energies In spite of light Higgs, longitudinal gauge-boson scattering amplitude violate unitarity at high energies h WLWL WLWL WLWL WLWL Modified coupling LHC with 200 fb -1 sensitive up to c H   0.3

56. 56 Higgs is viewed as pseudoGoldstone boson: its properties are related to those of the exact (eaten) Goldstones: O(4) symmetry Can bbbb at high invariant mass be separated from background? h  WW  leptons is more promising Sum rule (with cuts  and s<M 2 )  Strong gauge-boson scattering  strong Higgs production

57. 57 In many realizations, the top quark belongs to the strongly-coupled sector At leading order in 1/f 2 Modified top-quark couplings to h and Z At ILC g htt up to 5% with  s=800 GeV and L=1000 fb -1 From g Ztt,  c R ~ 0.04 with  s=500 GeV and L=300 fb -1 FCNC effects