1. 1 RS model with small curvature and dilepton production at the LHC Alexander Kisselev Institute for High Energy Physics Protvino, Russia The XIIth International School-Seminar “The Actual Problems of Microworld Physics” Gomel, Belarus, July 22 – August 2, 2013

2. Plan of the talk  Flat extra dimensions (EDs)  Warped ED with small curvature (RSSC model)  Kaluza-Klein (KK) excitations – massive gravitons  Contribution from s-channel virtual gravitons  Dilepton production at the LHC  Conclusions 2

3. Flat extra dimensions 3

4. G. Nordström (1914): T. Kaluza (1921), H. Mandel (1926): 5D vector theory = electromagnetism + scalar gravity Trajectory of charged particle = geodesic in 5-dimensional Riemann space with metric: 4 EDs: Brief History

5. A. Einstein, P. Bergmann (1938), B.V. Bargman (1941): Generalization of Kaluza-Klein theory: periodicity in 5-th coordinate O. Klein (1926), V.A. Fock (1926): Trajectory of charged particle = geometrical line in 5-dimensional Riemann space with metric: 5 (Bergmann, Einstein, and Bargmann on Princeton Campus)

6. Yu.B. Rumer (1956 ): Quantum mechanics = 5-dimensional optics See also F. Klein (1891 ): Single-sides surface can be embeded in D-dimensional space (Klein bottle) 6

7. Why spatial (i.e. space-like) EDs? Metric tensor (D=5) Massless particle in five dimensions (Lorentz invariance holds): No tachyons Spatial extra dimension 7

8. World sheets of open (left) and closed (right) strings propagating in the space-time Superstrings: D=10 8 6 EDs must be compactified: Strings lead to existence of EDs

9. Closed strings propagate in the bulk Open strings propagate with ends at x ┴ = const for different winding ω 6 internal compact dimensions = (p-3) longitudinal + (9-p) transverse 9 String propagation in EDs String theories contain D(irichlet)-branes

10. Planck scale Gauge coupling String action Upon compactification of EDs (S → 4-dim S eff ): Rescaled volumeD–dimensional Planck scale 10 String scale String coupling hierarchy relation:

11. Warped ED with small curvature 11

12. ED with Small Curvature (RSSC Model) Background AdS 5 metric r c is radius of ED Points (x,y) and (x,–y) are identified + periodicity orbifold S 1 /Z 2 12 Poincaré invariance in x μ direction η μν is Minkowski tensor

13. 13 Circle S 1 defined by y = y + 2 πR is subject to Z 2 identification y = - y, becoming a line segment S 1 /Z 2 with two fixed points y=0 and y = πR At every point of 4-dimensional space-time there exists “orthogonal” dimension compactified on circle of radius R

14. SM fields are confined to TeV brane SM Planck brane TeV brane Gravity Gravity lives in all dimensions (bulk) 14

15. Five-dimensional action Einstein-Hilbert’s equations 15 gravity action brane actions Reduced scales

16. Non-zero elements of curvature tensor 5-dimensional scalar curvature Equations for background metric 16

17. Solution of E-H’s equations (A.K., arXiv:1306.5402) Original RS model: (Randall & Sundrum, 1999) 17 Warp factor

18. σ(y)+ κπr c y 2πr c -πr c 0 πr c κπr c 18

19. Curvature: Radius of curvature: In between the branes (0 < y < πr c ) Warp factor: Five-dimensional scalar curvature: Original RS model: 19 AdS 5 space-time

20. Massive KK gravitons 20

21. Gravitational 5-dimensional field Transverse-traceless gauge General coordinate transformation 21

22. Kaluza-Klein (KK ) gravitons Scalar massless field Φ(x) – radion 22 Interaction Lagrangian on TeV brane massive radionStabilization mechanism (Goldberger & Wise, 1999)

23. 23 RS model Series of massive resonances (Randall & Sundrum, 1999) Graviton masses (J 1 (x n ) = 0) Hierarchy relation

24. RSSC model Narrow low-mass resonances with small mass splitting (Giudice et al., 2004, Petrov & A.K., 2005) 24 Graviton masses Hierarchy relation Spectrum is similar to that in ADD model

25. Negligible relative corrections to Newton law Newton’s potential between test masses No astrophysical bounds for κ > 10 MeV 25

26. For instance, can be realized only for AdS 5 Metric vs. Flat Metric with One Compact ED AdS 5 Metric vs. Flat Metric with One Compact ED solar distance (d is number of flat EDs) - strongly disfavored by astrophysical bounds RSSC model is not equivalent to ADD model with one ED of the size 26

27. Hierarchy relation in flat EDs Limiting case of hierarchy relation for warped metric 27 (D=4+d) means

28. s-channel KK gravitons 28

29. Virtual s-channel KK Gravitons Scattering of SM fields mediated by graviton exchange Scattering of SM fields mediated by graviton exchange Energy region: Processes: Virtual s-channel KK Gravitons Parton sub-processes: 29

30. Matrix element for dilepton production where Tensor part of graviton propagator Energy-momentum tensor Zero width approximation (Giudice et al., 2005) (imaginary part only) 30

31. Widths of KK gravitons S(s) can be calculated analytically by using formula (m n = z n,1 κ) where (A.K, 2006) 31

32. Is mass splitting small enough for mass distribution to be continuous? relevant KK numbers At 32 At the same time, for √s > 3M 5 zero width result is reproduced ( resonances overlap) Suppose spectrum is a series of narrow resonances

33. V.A. Matveev, R.M. Muradian and A.N. Tavkhelidze, JINR P2-4543 (Dubna, 1969), SLAC TRANS-009: JINR R2-4543 (June, 1969) S.D. Drell and T.M. Yan, SLAC-PUB-0755 (June, 1970), Phys. Rev. Lett. 25 (1970) 316, errata Phys. Rev. Lett. 25 (1970) 902 Dilepton production (Drell-Yan process) 33

34. Dilepton Production in Warped ED Differential cross section where 34

35. Weak logarithmic dependence on energy comes from PDFs dσ( grav ): no dependence on curvature κ 35 (fixed x ┴ )

36. Virtual graviton contributions quark-antiquark annihilation gluon-gluon fusion (absent in SM at tree level) 36

37. Dimuon production Pseudorapidity cut: |η| ≤ 2.4 Efficiency: 85 % K-factor: 1.5 for SM background 1.0 for signal 37 (A.K., JHEP, 2013)

38. Graviton contributions to the process pp → μ+μ- + X (solid lines) vs. SM contribution (dashed line) for 7 TeV 38

39. Graviton contributions to the process pp → μ+μ- + X (solid lines) vs. SM contribution (dashed line) for 14 TeV 39

40. Ratio of the gravity induced cross section to the SM cross section for 14 TeV (solid lines) and 7 TeV (dashed lines) 40

41. Graviton contribution to the process pp → μ+μ- + X (solid curves) vs. contribution from zero widths gravitons (dashed curves, multiplied by 10 3 ) for 14 TeV 41

42. Dielectron production Pseudorapidity (CMS) cuts: |η| ≤ 1.44, 1.57 ≤ |η| ≤ 2.4 Efficiency: 85 % K-factors: 1.5 for SM background 1.0 for signal 42 (A.K., arXiv:1306.5402)

43. Graviton contributions to the process pp → e+e- + X (solid lines) vs. SM contribution (dashed line) for 8 TeV 43

44. Graviton contributions to the process pp → e+e- + X (solid lines) vs. SM contribution (dashed line) for 13 TeV 44

45. Statistical significance Number of events with p t > p t cut 45 Interference SM-gravity contribution is negligible Lower bounds on M 5 at 95% level

46. Statistical significance for the process pp → e + e - + X as a function of 5 -dimensional reduced Planck scale and cut on electron transverse momentum for 7 TeV (L=5 fb -1 ) and 8 TeV (L=20 fb -1 ) 46

47. Statistical significance for the process pp → e + e - + X as a function of 5 -dimensional reduced Planck scale and cut on electron transverse momentum for 13 TeV (L=30 fb -1 ) 47

48. Angular distribution has term ~(cosθ) 4 How to discriminate gravity effects from other beyond SM contributions? Relation between diphoton and Drell-Yan process cross sections 48

49. Conclusions 49

50. Conclusions  In the RSSC model, curvature κ is small, κ ~ 0.1-10 GeV, with M 5 ~ 1-10 TeV  Mass spectrum is similar to that in the ADD model with one ED  At fixed x ┴ = 2p t / √s, ratio dσ( grav )/dσ( SM ) is proportional to (√s/ M 5 ) 3  Gravity cross sections weakly depend on κ, provided κ << M 5  Account of KK graviton widths is a crucial point for calculations  LHC search limits on M 5 are: ● 6.35 TeV, for 7+8 TeV, L = 5 fb -1 + 20 fb -1 ● 8.95 TeV, for 13 TeV, L = 30 fb -1 50

51. 51 Thank you for attention !

52. Back up slides 52

53. SM backgrounds to dimuon production  Z/ γ* events – dominant  Top quark-pair production  Dibozon events  (W + jets) – negligible  QCD background – negligible 53

54. Systematic errors  Theoretical: a) hard process description and parameter uncertainties b) hard process scale c) PDF description d) QCD radiation  Experimental: a) luminosity b) measurements of lepton transverse momentum c) energy resolution of ECAL and muon detector 54

55. Uncertainties in bounds on M 5 (for L = 30 fb -1 ) PDF set (2.7%) ΔL ( 3 %) μ/2 → μ → 2μ 55

56. Search for one warped ED (DELPHI Collaboration, 2006) Photon energy spectrum in single photon events Main background: 56

57. Warped ED (CMS, December 2012) Production of graviton resonances (Mono-jet + missing E T ) events Large EDs (ATLAS, February 2013) Search for TeV scale gravity 57

58. 58 Heavy resonances (CMS, July 2013) Dijet final states Bounds on RS graviton – from green dashed line (arXiv:1307.2518)

59. Background RS metric ( Randall & Sundrum, 1999) Four-dimensional gravitational action of gravitational field in RS model ( Boos et al., 2002 ) Expression is not covariant: indices are raised with Minkowski tensor wheares metric is 59

60. Change of variables ( Rubakov, 2001 ) 60

61.  EDs can explain strong hierarchy between gravity and electroweak scale, quark-mass hierarchies and CKM structure, can address electroweak symmetry breaking without Higgs, as well as supersymmetry breaking, grand unification at TeV scale without SUSY, etc.  EDs lead to a wide range of new phenomena (collider signatures, dark matter candidates, microscopic black holes, etc.)  Discovery of EDs could radically alter our view of universe on both small and large distances General Conclusions 61

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