Unifying Forces: Extra Dimensions & Field Theory in Gravitational Interactions

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

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

3. Next step: UNIFICATION OF EM & GRAVITY  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. GRAVITY In General Relativity, metric (4X4 symmetric tensor) dynamical variable describing space geometry (graviton) Dynamics described by Einstein action • GN Newton’s constant • R curvature (function of the metric)

5. R x5 (t,x) Consider GR in 5-dim Choose Dynamical fields Assume space is M4S1 • First considered as a mathematical trick • It may have physical meaning

6. Extra dim is periodic or “compactified” 5-dim field  set of 4-dim fields: Kaluza-Klein modes D-dim particle Each has a fixed momentum p5=n/R along 5th dim extra dimensions 4-d space  E2 = p 2 + p2extra + m2 KK mass mass All fields can be expanded in Fourier modes From KK mass spectrum we can measure the geometry of extra dimensions

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

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

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

12. COMPUTE NEWTON CONSTANT Einstein action in D dimensions Assume space R4SD-4: g doesn’t depend on extra coordinates Effective action for g

13. Suppose fundamental mass scaleMD ~ TeV very large if R is large (in units of MD-1) Arkani-Hamed, Dimopoulos, Dvali Radius of compactified space • Smallness of GN/GF related to largeness of RMD • Gravity is weak because it is diluted in a large space (small overlap with branes) • Need dynamical explanation for RMD>>1

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

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

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. G-emission is based on linearized gravity, valid at s << MD2 TRANSPLANCKIAN REGIME Planck length quantum-gravity scale classical gravity Schwarzschild radius same regime The transplanckian regime is described by classical physics (general relativity)  independent test, crucial to verify gravitational nature of new physics

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

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

21. b < RS At b<RS, no longer calculable Strong indications for black-hole formation BH with angular momentum, gauge quantum numbers, hairs(multiple moments of the asymmetric distribution of gauge charges and energy-momentum) Gravitational and gauge radiation during collapse  spinning Kerr BH ~ RS2 10 pb (for MBH=6 TeV and MD=1.5 TeV) Hawking radiation until Planck phase is reached TH ~ RS-1 ~ MD (MD / MBH)1/(+1) Evaporation with  ~ MBH(+3)/(+1) / MD2(+2)/(+1) (10-26 s for MD=1 TeV) Characteristic events with large multiplicity (<N> ~ MBH / <E> ~ (MBH / MD)(+2)/(+1)) and typical energy <E> ~ TH Transplanckian condition MBH >> MD ?

22. WARPED GRAVITY A classical mechanism to make quanta softer For time-indep. metrics with g0=0  E |g00|1/2 conserved . (proper time d2 = g00 dt2) On non-trivial metrics, we see far-away objects as red-shifted

23. y y 0 R 0 R -y 0 R Two 3-branes on boundaries & appropriate vacuum-energy terms Consider observer & emitter as 3-d spaces (branes) embedded in non-trivial 5-D space-time geometries Randall Sundrum 5th dim S1 / Z2: identify y  y+2R, y  -y

24. y=0 g00=1 y=R g00=e-2RK GRAVITATIONAL RED-SHIFT Masses on two branes related by Same result can be obtained by integrating SE over y

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 mn = Kxne-RK [xn roots of J1(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. Spin 2 Spin 1

27. Warped gravity with SM fermions and gauge bosons in bulk and Higgs on brane Technicolor-like theory with slowly-running couplings in 4 dim 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, …

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

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

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

31. -- + LEP1 LEP2 MFV Little Higgs Composite Higgs Higgsless new physics strong dynamics LEP energy 1 TeV 10 TeV A less ambitious programme: solving the little hierarchy Bounds on  [TeV]

32. ℒ1 ℒ1 ℒ2 H ℒ2 LITTLE HIGGS Explain only little hierarchy At LSM new physics cancels one-loop power divergences “Collective breaking”: many (approximate) global symmetries preserve massless Goldstone boson

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 replicationEx. SU2 gauge with F1,2doublets such that V(F1+F1,F2+F2) and F1,2 spontaneously break SU2 • Turning off gauge coupling to F1_ • Local SU2(F2) × global SU2(F1) both spont. broken

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

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

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

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

39. gauge Higgs NEW INGREDIENTS FROM EXTRA DIMENSIONS HIGGS AS EXTRA-DIM COMPONENT OF GAUGE FIELD AM = (Am,A5), A5g A5 +∂5L forbids m2A52 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

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

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

43. TeV brane Planck brane 5th dim 5-D warped gravity large-N technicolor  RG flow IR UV DUALITY SM in warped extra dims strongly-int’ing 4-d theory KK excitations  “hadrons” of new strong force Technicolor strikes back? 5-D gravity4-D gauge theory Motion in 5th dimRG flow UV branePlanck cutoff IR branebreaking of conformal inv. Bulk local symmetriesglobal symmetries AdS/CFT Composite Higgs

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. strong sector quarks, leptons & gauge bosons Structure of the theory Communicate via gauge (ga) and (proto)-Yukawa(i) mmass of resonances gcoupling of resonances Strong sector characterized by Take I, ga << g< 4 In the limit I, ga =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

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

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 composite? Holographic Higgs Gauge-Higgs unification Little Higgs fundamental? SM (with mH < 180 GeV) supersymmetry

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

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