...and the extradimensions quest

Transcription

1 A brief introduction to the Randall-Sundrum Models...and the extradimensions quest Bruno BERTRAND Center for particle physics and phenomenology (CP3) CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 1/1

2 Table of contents Introduction CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 2/1

3 Table of contents Introduction Two examples of mathematical compactification models CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 2/1

4 Table of contents Introduction Two examples of mathematical compactification models The Kaluza-Klein compactification ( ) CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 2/1

5 Table of contents Introduction Two examples of mathematical compactification models The Kaluza-Klein compactification ( ) The Volkov s model with cosmological constant ( ) CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 2/1

6 Table of contents Introduction Two examples of mathematical compactification models The Kaluza-Klein compactification ( ) The Volkov s model with cosmological constant ( ) Brane scenarios and the Randall-Sundrum models CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 2/1

7 Table of contents Introduction Two examples of mathematical compactification models The Kaluza-Klein compactification ( ) The Volkov s model with cosmological constant ( ) Brane scenarios and the Randall-Sundrum models A brief introduction of the ADD model (1998) CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 2/1

8 Table of contents Introduction Two examples of mathematical compactification models The Kaluza-Klein compactification ( ) The Volkov s model with cosmological constant ( ) Brane scenarios and the Randall-Sundrum models A brief introduction of the ADD model (1998) The Randall-Sundrum I model (1999) CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 2/1

9 Table of contents Introduction Two examples of mathematical compactification models The Kaluza-Klein compactification ( ) The Volkov s model with cosmological constant ( ) Brane scenarios and the Randall-Sundrum models A brief introduction of the ADD model (1998) The Randall-Sundrum I model (1999) The Randall-Sundrum II model (1999) CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 2/1

10 Table of contents Introduction Two examples of mathematical compactification models The Kaluza-Klein compactification ( ) The Volkov s model with cosmological constant ( ) Brane scenarios and the Randall-Sundrum models A brief introduction of the ADD model (1998) The Randall-Sundrum I model (1999) The Randall-Sundrum II model (1999) Discussion between us CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 2/1

11 Table of contents Introduction Two examples of mathematical compactification models The Kaluza-Klein compactification ( ) The Volkov s model with cosmological constant ( ) Brane scenarios and the Randall-Sundrum models A brief introduction of the ADD model (1998) The Randall-Sundrum I model (1999) The Randall-Sundrum II model (1999) Discussion between us Conclusion CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 2/1

12 Introduction A spacetime with more than four dimensions? CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 3/1

13 Introduction The old conception of space and time The Holy Trinity of Kepler... and the Newton s notion of time and space (1687) CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 4/1

14 Introduction The modern conception of space-time The Holy Trinity of Kepler... and the Newton s notion of time and space (1687) Minkowski s notion of spacetime and the theory of general relativity by Einstein. CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 4/1

15 Introduction Cosmological models Manifold ( Variété ) : R 4+d or R 4 T d... Infinite dimension : R 1 Compact dimension : S 1 CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 5/1

16 Introduction Cosmological models Manifold ( Variété ) : R 4+d or R 4 T d... Metric tensor in 4 + d : g By example : η MN = ds 2 4+d = G MN dx M dx N CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 5/1

17 Introduction Cosmological models Manifold ( Variété ) : R 4+d or R 4 T d... Metric tensor in 4 + d : g Stress-Energy tensor : T MN T MN = L mat G MN 1 2 L matg MN CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 5/1

18 Introduction General action in 4+d dimensions S 5 = d 4 x S 5 = S grav + S mat d d y G(αR Λ +L mat ) CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 6/1

19 Introduction General action in 4+d dimensions S 5 = d 4 x S 5 = S grav + S mat d d y G(αR Λ +L mat ) δs = 0 : Einstein s equations in 4 + d dimensions R A MBN R MN = R A MAN R = R M M R MN 1 2 G MNR = 1 2 G MN : (4+d)-dimensional Riemann tensor : (4+d)-dimensional Ricci tensor : (4+d)-dimensional curvature scalar Λ α 1 α T MN CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 6/1

20 Introduction General action in 4+d dimensions S 5 = d 4 x S 5 = S grav + S mat d d y G(αR Λ +L mat ) δs = 0 : Einstein s equations in 4 + d dimensions R MN 1 2 G MNR = 1 2 G MN α : Gravitational coupling constant Λ : Cosmological constant Λ α 1 α T MN CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 6/1

21 Part One Two examples of mathematical compactification models CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 7/1

22 Two examples of compactification models Kaluza-Klein compactification Geometrical unification of gravity and electromagnetism Pure Einstein-Hilbert action in five dimensions : S 5 = d 4 x L :with a compact 5 th dimension : R 4 S 1 0 dy G α R 5 CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 8/1

23 Two examples of compactification models Kaluza-Klein compactification Geometrical unification of gravity and electromagnetism S 5 = d 4 x L 0 dy G α R 5 Expansion of the metric as a Fourier serie :infinite number of fields dependant of the four space-time dimensions x µ with mass n /L G AB (x,y) = n G (n) iny AB (x)e L. CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 8/1

24 Two examples of compactification models Kaluza-Klein compactification Geometrical unification of gravity and electromagnetism S 5 = d 4 x L 0 dy G α R 5 G AB (x,y) = n G (n) iny AB (x)e L. Compactification Scale : Planck scale (10 19 GeV) :Truncation to the massless (zero) mode and definition of the component of the metric as : G µν = e 3φ g µν + e 2 3 φ Aµ A ν G µy = e 2 3 Aµ G zz = e 2 3 CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 8/1

25 Two examples of compactification models Kaluza-Klein compactification Geometrical unification of gravity and electromagnetism S 5 = d 4 x L 0 dy G α R 5 G AB (x,y) = n G (n) iny AB (x)e L. Compactification Scale : Planck scale (10 19 GeV) G µν = e 3φ g µν + e 2 3 φ Aµ A ν G µy = e 2 3 Aµ G zz = e 2 3 Integration out the z part of the action : S e f f = d 4 x g { R 1 2 ( µφ)( µ φ) 1 4 e 3φ F µν F µν } L 0 dy CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 8/1

26 Two examples of compactification models Kaluza-Klein compactification Geometrical unification of gravity and electromagnetism S 5 = d 4 x L 0 dy G α R 5 G AB (x,y) = n G (n) iny AB (x)e L. Compactification Scale : Planck scale (10 19 GeV) G µν = e 3φ g µν + e 2 3 φ Aµ A ν G µy = e 2 3 Aµ G zz = e 2 3 Integration out the z part of the action : S e f f = L d 4 x g {R 12 ( µφ)( µ φ) 14 } e 3φ F µν F µν CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 8/1

27 Two examples of compactification models Volkov s model with cosmological constant Equivalence between E-Yang-Mills in 5D and EYM-Higgs-Dilaton in 4D Einstein-Hilbert action coupled to Yang-Mills in five dimensions : S = d 4 x dy G(αR Λ 1 4g 2 Fa MNF amn ) L 5. :with a compact 5 th dimension : R 4 S 1 CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 9/1

28 Two examples of compactification models Volkov s model with cosmological constant Equivalence between E-Yang-Mills in 5D and EYM-Higgs-Dilaton in 4D S = d 4 x dy G(αR Λ 1 4g 2 Fa MNF amn ) L 5. Métrique SO(3)-invariante aux fonctions indépendantes de la coord. de la dimension supplémentaire : y ds 2 (5) = e2ν(r) dt 2 + e 2λ(r) dr 2 + e 2µ(r) (dω 2 ) 2 + e 2ζ(r) dy 2. CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 9/1

29 Two examples of compactification models Volkov s model with cosmological constant Equivalence between E-Yang-Mills in 5D and EYM-Higgs-Dilaton in 4D S = d 4 x dy G(αR Λ 1 4g 2 Fa MNF amn ) L 5. ds 2 (5) = e2ν(r) dt 2 + e 2λ(r) dr 2 + e 2µ(r) (dω 2 ) 2 + e 2ζ(r) dy 2. Ansatz de sym. sphérique sur les champs non abéliens de YM. Champs de jauge indépendants de y et statiques : F a MN = M A a N + N A a M + ε abc A b MA c N A a M [ ] 0,A a i (x k ),H a (x k ). CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 9/1

30 Two examples of compactification models Volkov s model with cosmological constant Equivalence between E-Yang-Mills in 5D and EYM-Higgs-Dilaton in 4D S = d 4 x dy G(αR Λ 1 4g 2 Fa MNF amn ) L 5. ds 2 (5) = e2ν(r) dt 2 + e 2λ(r) dr 2 + e 2µ(r) (dω 2 ) 2 + e 2ζ(r) dy 2. F a MN = M A a N + N A a M + ε abc A b MA c N A a M [ ] 0,A a i (x k ),H a (x k ). Paramétrisation de la métrique et des champs : G MN dx M dx N = e ζ g µν dx µ dx ν + e 2ζ dy dy. CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 9/1

31 Two examples of compactification models Volkov s model with cosmological constant Equivalence between E-Yang-Mills in 5D and EYM-Higgs-Dilaton in 4D S = d 4 x dy G(αR Λ 1 4g 2 Fa MNF amn ) L 5. ds 2 (5) = e2ν(r) dt 2 + e 2λ(r) dr 2 + e 2µ(r) (dω 2 ) 2 + e 2ζ(r) dy 2. F a MN = M A a N + N A a M + ε abc A b MA c N A a M [ ] 0,A a i (x k ),H a (x k ). Integration out the z part of the action : S = L d 4 x { g α R e ζ Λ 3α 2 µζ µ ζ eζ 4g 2 Fa µνf aµν e 2ζ 2g 2 D µh a D µ H a } CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 9/1

32 Part Two Brane scenarios and the Randall-Sundrum models CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 10/1

33 Brane scenarios and the Randall-Sundrum models A brief introduction to ADD model Extradimensional model introduced in 1998 by N. Arkani-Hamed, S. Dimopoulos and G Dvali with for main main characteristics : Model with n 2 compactified large space dimensions (R 4 S 1 ) with factorisable geometry CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 11/1

34 Brane scenarios and the Randall-Sundrum models A brief introduction to ADD model Extradimensional model introduced in 1998 by N. Arkani-Hamed, S. Dimopoulos and G Dvali with for main main characteristics : Model with n 2 compactified large space dimensions (R 4 S 1 ) with factorisable geometry Fields of Standard model confined to a 4-dimensional spacetime submanifold reffered to a 3-brane. CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 11/1

35 Brane scenarios and the Randall-Sundrum models A brief introduction to ADD model Extradimensional model introduced in 1998 by N. Arkani-Hamed, S. Dimopoulos and G Dvali with for main main characteristics : Model with n 2 compactified large space dimensions (R 4 S 1 ) with factorisable geometry Fields of Standard model confined to a 4-dimensional spacetime submanifold reffered to a 3-brane. Gravity freely propagates in the 4+d dimensional spacetime manifold called bulk CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 11/1

36 Brane scenarios and the Randall-Sundrum models A brief introduction to ADD model Expression of the Planck scale as function of the volume of compact dimensions Model with n 2 compactified large space dimensions (R 4 S 1 ) with factorisable geometry CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 12/1

37 Brane scenarios and the Randall-Sundrum models A brief introduction to ADD model Expression of the Planck scale as function of the volume of compact dimensions Model with n 2 compactified large space dimensions (R 4 S 1 ) with factorisable geometry Fields of Standard model confined to a 4-dimensional spacetime submanifold reffered to a 3-brane. CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 12/1

38 Brane scenarios and the Randall-Sundrum models A brief introduction to ADD model Expression of the Planck scale as function of the volume of compact dimensions Model with n 2 compactified large space dimensions (R 4 S 1 ) with factorisable geometry Fields of Standard model confined to a 4-dimensional spacetime submanifold reffered to a 3-brane. Gravity freely propagates in the 4+d dimensional spacetime manifold called bulk CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 12/1

39 Brane scenarios and the Randall-Sundrum models A brief introduction to ADD model Experimental contribution : constraints (lower bound) over the compactification scale as a function of the number of E.D. Micro-gravity experiments ( Eot-Wash ) to test any deviation from the Newton 1/r 2 gravity CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 13/1

40 Brane scenarios and the Randall-Sundrum models A brief introduction to ADD model Experimental contribution : constraints (lower bound) over the compactification scale as a function of the number of E.D. Micro-gravity experiments ( Eot-Wash ) to test any deviation from the Newton 1/r 2 gravity Cosmological data from the neutrino s emissions of SN1987A CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 13/1

41 Brane scenarios and the Randall-Sundrum models A brief introduction to ADD model Experimental contribution : constraints (lower bound) over the compactification scale as a function of the number of E.D. Micro-gravity experiments ( Eot-Wash ) to test any deviation from the Newton 1/r 2 gravity Cosmological data from the neutrino s emissions of SN1987A Déviations from the standard model at LHC CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 13/1

42 Brane scenarios and the Randall-Sundrum models The Randall-Sundrum I model rc Cosmological model in five dimensions defined over a R 4 S1 Z 2 variety CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 14/1

43 Brane scenarios and the Randall-Sundrum models The Randall-Sundrum I model Cosmological model in five dimensions defined over a R 4 S1 Z 2 variety Notation : φ angular coordinate of the 5 t h rc dimensions defined over the orbifold S1 Z 2. S 1 Z 2 : Identification on the circle of the opposite points, (x µ,φ) (x µ, φ) Range of φ : [0,π] CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 14/1

44 Brane scenarios and the Randall-Sundrum models The Randall-Sundrum I model rc Cosmological model in five dimensions defined over a R 4 S1 Z 2 variety CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 14/1

45 Brane scenarios and the Randall-Sundrum models The Randall-Sundrum I model rc Cosmological model in five dimensions defined over a R 4 S1 Z 2 variety CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 14/1

46 Brane scenarios and the Randall-Sundrum models The Randall-Sundrum I model rc Cosmological model in five dimensions defined over a R 4 S1 Z 2 variety CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 14/1

47 Brane scenarios and the Randall-Sundrum models The Randall-Sundrum I model rc Cosmological model in five dimensions defined over a R 4 S1 Z 2 variety CP3 Seminar : Randall-Sundrum models - Bruno BERTRAND - February 2004 p. 14/1