1. Generalized solution for metric in Randall-Sundrum scenario Alexander Kisselev IHEP, NRC “Kurchatov Institute”, Protvino, Russia The XIII-th International School-Conference “The Actual Problems of Microworld Physics” Gomel, Belarus, July 29, 2015

2. 2 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Plan of the talk  Randall-Sundrum (RS) scenario with two branes  Original RS solution for the warp factor of the metric  Generalization of the RS solution  Role of the constant term. Different physical schemes  Conclusions

3. 3 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Space-time with one extra spatial compact dimension

4. 4 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Randall-Sundrum scenario Randall-Sundrum scenario Periodicity: (x, y ± 2πr c ) = (x, y) Z 2 -symmetry: (x, y) = (x, –y) Background metric (y is an extra coordinate) orbifold S 1 /Z 2 0 ≤ y ≤ πr c Two fixed points: y=0 and y= πr c two 3-dimensional branes

5. 5 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Planck brane TeV brane Gravity SM Planck brane AdS 5 space-time with two branes

6. 6 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Five-dimensional action Induced brane metrics Einstein-Hilbert’s equations

7. 7 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Five-dimensional background metric tensor E-H’s equations for the warp function σ(y):

8. 8 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 ( Randall & Sundrum, 1999 ) The RS solution: - does not explicitly reproduce the jump on brane y=πr c - is not symmetric with respect to the branes, although both branes (at points y=0 and y=πr c ) must be treated on an equal footing - does not include a constant term Randall-Sundrum solution Randall-Sundrum solution

9. 9 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Solution of 2-nd E-H’s equation for where For the open interval the solution is trivial Let us define: 0 < y < πr c 0 ≤ y ≤ πr c Generalization of RS solution Generalization of RS solution (to guarantee σ(y)=κy + C for 0 < y < πr c )

10. 10 two possibilities: - brane tensions Λ 1,2 have the same sign this case cannot be realized - brane tensions Λ 1,2 have opposite signs As a result: Symmetry with respect to the branes The periodicity condition means: XIII-th School-Conference, Gomel, Belarus, July 29, 2015

11. 11 Expression for the warp function ( A.K., 2014-15 ) with the fine tuning is anti-symmetric in y: 0 ≤ C ≤ κπr c Derivative of σ(y) at |y| < πr c XIII-th School-Conference, Gomel, Belarus, July 29, 2015 factor of 2 different than that of RS

12. 12 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 σ(y)+ C y 2πr c -πr c 0 πr c κπr c

13. 13 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 σ(y)+ C y 2πr c -πr c 0 πr c κπr c Two equivalent RS-like solutions

14. 14 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Solution symmetric with respect to the branes: If starting from the point y=0 If starting from the point y=πr c Since Neither of two branes/fixed points is preferable provided orbifold and periodicity conditions are taken into account solution is consistent with orbifold symmetry

15. 15 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Randall-Sundrum solution: Does it mean that σ(y) is a linear function for all y? The answer is negative: one must use the periodicity condition first, and only then estimate absolute value |y| In other words, extra coordinate y must be reduced to the interval -πr c ≤ y ≤ πr c (by using periodicity condition) before evaluating functions |x|, ε(x), …

16. 16 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Examples: definition of σ(y) outside this interval Let y = πr c + y 0, where 0 < y 0 < πr c (that is, πr c < y < 2πr c ) σ(y)+ C 2πr c 0πr c κπr c y κ(πr c - y 0 ) πr c + y 0

17. 17 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Our solution for σ(y): - makes the jumps on both branes - is symmetric with respect to the branes - is consistent with the orbifold symmetry - depends on the constant C

18. 18 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Hierarchy relation (depends on C) Masses of KK gravitons ( x n are zeros of J 1 (x) ) Interaction Lagrangian on the TeV brane (massive gravitons only) Coupling constant (on TeV brane)

19. Different values of this constant result in quite diverse physical models ( A.K., 2014 ) Parameters M 5, κ, Λ π depend on constant C 19 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 The very expression for m n is independent of C, but values of m n depend on C via M 5 and κ

20. 20 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 I. C = 0 that requires Masses of KK resonances RS1 model Graviton spectrum - heavy resonances, with the lightest one above 1 TeV ( Randall & Sundrum, 1999) Different physical schemes Different physical schemes

21. 21 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 II. C = κπr c Masses of KK resonances RSSC model: RS-like scenario with the small curvature of 5-dimensional space-time For small κ, graviton spectrum is similar to that of the ADD model (Giudice, Petrov & A.K., 2005)

22. 22 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 III. C = κπr c / 2 In variable z = y - πκr c /2: σ(z)- C z 0 κπr c /2 -κπr c /2 πr c /2 -πr c /2 0 “Symmetric scheme” (A.K., 2014)

23. 23 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Masses of gravitons Let Almost continuous spectrum of the gravitons

24. 24 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Effective gravity action Shift is equivalent to the change Invariance of the action rescaling of fields and masses: Massive theory is not scale-invariant From the point of view of 4- dimensional observer these two theories are not physically equivalent

25. 25 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Example: C = πκr c (RS1 → RSSC ) The end is near!

26. 26 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Virtual s-channel Gravitons Scattering of SM fields mediated by graviton exchange in s-channel Processes: Sub-processes: h (n) Σ n≥1

27. 27 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 where Tensor part of graviton propagator Energy-momentum tensors Matrix element of sub-process mediated by s-channel KK gravitons and Graviton widths : (process independent)

28. 28 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 with For chosen values of parameters: TeV physics at LHC energies ( in spite of large Λ π ) In symmetric scheme (A.K., 2014) (relation of m n with x n was used)

29. 29 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 ConclusionsConclusions  Generalized solution for the warp factor σ(y) in RS-like scenario is derived  New expression: - is symmetric with respect to the branes - has the jumps on both branes - is consistent with the orbifold symmetry  σ(y) depends on constant C (0 ≤ C ≤ κπr c )  Different values of C result in quite diverse physical schemes  Particular scenarios are: RS1 model, RSSC model, “symmetric” model

30. Thank you for your attention 30 XIII-th School-Conference, Gomel, Belarus, July 29, 2015

31. Back-up slides 31

32. 32 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 RSSC model is not equivalent to the ADD model with one ED of the size R=(πκ) -1 up to κ ≈10 -20 eV Hierarchy relation for small κ But the inequalitymeans that RSSC model vs.ADD model

33. Dilepton production at LHC Pseudorapidity cut: |η| ≤ 2.4 Efficiency: 85 % K-factor: 1.5 for SM background 1.0 for signal 33 XIII-th School-Conference, Gomel, Belarus, July 29, 2015

34. 34 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Graviton contributions in RSSC to the process pp → μ+μ- + X (solid lines) vs. SM contribution (dashed line) for 7 TeV

35. 35 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Graviton contributions in RSSC to the process pp → μ+μ- + X (solid lines) vs. SM contribution (dashed line) for 14 TeV

36. 36 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Statistical significance Number of events with p t > p t cut Interference SM-gravity contribution is negligible Lower bounds on M 5 at 95% level S=5

37. 37 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 Statistical significance in RSSC for 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 )

38. 38 XIII-th School-Conference, Gomel, Belarus, July 29, 2015 No deviations from the CM were seen at the LHC M 5 > 6.35 TeV (for 7 + 8 TeV, L = 5 fb -1 + 20 fb -1 ) LHC search limits on M 5 : 8.95 TeV (for 13 TeV, L = 30 fb -1 ) (A.K., 2013)