Large Mass Hierarchy from a Small Extra Dimension

Transcription

1 Large Mass Hierarchy from a Small Extra Dimension Sridip Pal (09MS002) DPS PH4204 April 4,2013 Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

2 Outline 1 Introduction What is Hierarchy Problem 2 Kaluza Klein Action 3 ADD Compactification 4 Solution to Hierarchy in ADD Model Hierarchy in Disguise 5 Randall-Sundrum Model 6 Weak Scale from Planck Scale Visible Sector Mass-Exponential Hierarchy Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

3 Hierarchy Problem Introduction What is Hierarchy Problem The gravity is weak compared to other fundamental forces. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

4 Hierarchy Problem Introduction What is Hierarchy Problem The gravity is weak compared to other fundamental forces. The gravitational coupling constant G 4d = M 2 Pl = GeV 2. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

5 Hierarchy Problem Introduction What is Hierarchy Problem The gravity is weak compared to other fundamental forces. The gravitational coupling constant G 4d = M 2 Pl = GeV 2. It is weak in the sense G 4d m 2 p = << 1. and G 4d ν 2 = << 1 Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

6 Introduction What is Hierarchy Problem Hierarchy Problem The gravity is weak compared to other fundamental forces. The gravitational coupling constant G 4d = M 2 Pl = GeV 2. It is weak in the sense G 4d m 2 p = << 1. and G 4d ν 2 = << 1 Gravity becomes dominant when elementary particle becomes as heavy as GeV. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

7 Introduction What is Hierarchy Problem Hierarchy Problem The gravity is weak compared to other fundamental forces. The gravitational coupling constant G 4d = M 2 Pl = GeV 2. It is weak in the sense G 4d m 2 p = << 1. and G 4d ν 2 = << 1 Gravity becomes dominant when elementary particle becomes as heavy as GeV. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

8 Hierarchy Problem Introduction What is Hierarchy Problem Standard Model does not incorporate gravity and we can safely ignore it as it is a force. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

9 Introduction What is Hierarchy Problem Hierarchy Problem Standard Model does not incorporate gravity and we can safely ignore it as it is a force. But we cant expect Standard Model to be valid at energy scale GeV where Gravity becomes strong. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

10 Introduction What is Hierarchy Problem Hierarchy Problem Standard Model does not incorporate gravity and we can safely ignore it as it is a force. But we cant expect Standard Model to be valid at energy scale GeV where Gravity becomes strong. So Standard Model has a natural cut off at GeV. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

11 Introduction What is Hierarchy Problem Hierarchy Problem Standard Model does not incorporate gravity and we can safely ignore it as it is a force. But we cant expect Standard Model to be valid at energy scale GeV where Gravity becomes strong. So Standard Model has a natural cut off at GeV. Loop correction renders Higgs mass as high as GeV because the divergence is quadratic. m 2 = m ( 10 19) 2 (1) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

12 Introduction What is Hierarchy Problem Hierarchy Problem Standard Model does not incorporate gravity and we can safely ignore it as it is a force. But we cant expect Standard Model to be valid at energy scale GeV where Gravity becomes strong. So Standard Model has a natural cut off at GeV. Loop correction renders Higgs mass as high as GeV because the divergence is quadratic. m 2 = m ( 10 19) 2 If we want to have m = 125GeV,the bare mass m 0 has to be very large, hence m/m 0 has to be very precise which is unnatural and termed Fine Tuning Problem. (1) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

13 Introduction What is Hierarchy Problem Hierarchy Problem Hierarchy problem is about Why weak scale (ν) is so small compared to M Pl = GeV. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

14 Introduction What is Hierarchy Problem Hierarchy Problem Hierarchy problem is about Why weak scale (ν) is so small compared to M Pl = GeV. In other words why gravity is so weak. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

15 Introduction What is Hierarchy Problem Hierarchy Problem Hierarchy problem is about Why weak scale (ν) is so small compared to M Pl = GeV. In other words why gravity is so weak. Is there any solution to fine tunning problem? Can we aviod such fine tuning of parameter like Higgs mass.? Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

16 Kaluza Klein Action Action in General Relativity and KK Action S GR = 1 16πG 4D d 4 x gr (2) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

17 Kaluza Klein Action Action in General Relativity and KK Action S GR = 1 16πG 4D d 4 x gr (2) Kaluza Klein action is simple generalisation of above action 1 S KK = d 5 x g 16πG R (3) 5D Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

18 Kaluza Klein Action Action in General Relativity and KK Action S GR = 1 16πG 4D d 4 x gr (2) Kaluza Klein action is simple generalisation of above action 1 S KK = d 5 x g 16πG R (3) 5D g = kg (4) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

19 Kaluza Klein Action Action in General Relativity and KK Action S GR = 1 16πG 4D d 4 x gr (2) Kaluza Klein action is simple generalisation of above action 1 S KK = d 5 x g 16πG R (3) 5D g = kg (4) R = R 1 4 F αβf αβ (5) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

20 Compactification Kaluza Klein Action S KK = k 16πG 5D dx 5 d 4 x g [R k4 ] F αβf αβ (6) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

21 Compactification Kaluza Klein Action S KK = k 16πG 5D dx 5 d 4 x g [R k4 ] F αβf αβ (6) Figure: Compactified Extra Dimension Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

22 Kaluza Klein Action Kaluza Klein Action continued.. Assuming cylindrical compactification of extra dimension with radius a we get S KK = 2πa k d 4 x g [R k4 ] 16πG F αβf αβ (7) 5D Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

23 Kaluza Klein Action Kaluza Klein Action continued.. Assuming cylindrical compactification of extra dimension with radius a we get S KK = 2πa k d 4 x g [R k4 ] 16πG F αβf αβ (7) 5D 1 S GR = d 4 x gr (8) 16πG 4D Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

24 Kaluza Klein Action Kaluza Klein Action continued.. Assuming cylindrical compactification of extra dimension with radius a we get S KK = 2πa k d 4 x g [R k4 ] 16πG F αβf αβ (7) 5D 1 S GR = d 4 x gr (8) 16πG 4D G 4D = G 5D 2πa k (9) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

25 4+n Dimension ADD Compactification ds 2 = g AB dx A dx B = g µν dx µ dx ν + γ ij dy i dy j (10) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

26 4+n Dimension ADD Compactification ds 2 = g AB dx A dx B = g µν dx µ dx ν + γ ij dy i dy j (10) R[ g AB ] = R[g µν ] + R[γ ij ] (11) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

27 4+n Dimension ADD Compactification ds 2 = g AB dx A dx B = g µν dx µ dx ν + γ ij dy i dy j (10) The determinant of the metric is given by R[ g AB ] = R[g µν ] + R[γ ij ] (11) g = gγ (12) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

28 ADD Compactification The action is given by 1 S = 16πG 4+n d 4+n x gr[ g AB ] (13) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

29 ADD Compactification The action is given by 1 S = 16πG 4+n d 4+n x gr[ g AB ] (13) We assume the extradimensions are compactified in a volume V n V n = d n y γ (14) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

30 ADD Compactification The action is given by 1 S = 16πG 4+n d 4+n x gr[ g AB ] (13) We assume the extradimensions are compactified in a volume V n V n = d n y γ (14) S eff = V n 16πG 4+n d 4 x g (R[g µν ] + 1Vn d n y ) γr[γ ij ] (15) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

31 ADD Compactification The action is given by 1 S = 16πG 4+n d 4+n x gr[ g AB ] (13) We assume the extradimensions are compactified in a volume V n V n = d n y γ (14) S eff = V n 16πG 4+n d 4 x g (R[g µν ] + 1Vn d n y ) γr[γ ij ] (15) V n G 4+n = 1 G 4 (16) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

32 Solution to Hierarchy in ADD Model Hieararchy Explained! where M 2 Pl = Mn+2 µ n c (17) M Pl corresponds to 4 dimensional reduced Planck scale and M represents fundamental Planck scale in higher D theory and µ c is the scale of compactification. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

33 Solution to Hierarchy in ADD Model Hieararchy Explained! M 2 Pl = Mn+2 µ n c (17) where M Pl corresponds to 4 dimensional reduced Planck scale and M represents fundamental Planck scale in higher D theory and µ c is the scale of compactification. M = 1TeV, M Pl = GeV leading to µ c = n TeV. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

34 Solution to Hierarchy in ADD Model Hieararchy Explained! M 2 Pl = Mn+2 µ n c (17) where M Pl corresponds to 4 dimensional reduced Planck scale and M represents fundamental Planck scale in higher D theory and µ c is the scale of compactification. M = 1TeV, M Pl = GeV leading to µ c = n TeV. Size of the extradimension is given by r c = 1 µ c = n cm Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

35 Solution to Hierarchy in ADD Model Hieararchy Explained! M 2 Pl = Mn+2 µ n c (17) where M Pl corresponds to 4 dimensional reduced Planck scale and M represents fundamental Planck scale in higher D theory and µ c is the scale of compactification. M = 1TeV, M Pl = GeV leading to µ c = n TeV. Size of the extradimension is given by r c = 1 µ c = n cm n=1 is simply ruled out beacuse it is too large; so this compactification model requires n 2. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

36 Solution to Hierarchy in ADD Model Hieararchy Explained! M 2 Pl = Mn+2 µ n c (17) where M Pl corresponds to 4 dimensional reduced Planck scale and M represents fundamental Planck scale in higher D theory and µ c is the scale of compactification. M = 1TeV, M Pl = GeV leading to µ c = n TeV. Size of the extradimension is given by r c = 1 µ c = n cm n=1 is simply ruled out beacuse it is too large; so this compactification model requires n 2. For n=2 we extra dimension of 1 mm size. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

37 Solution to Hierarchy in ADD Model Hierarchy in Disguise Hierarchy in Disguise This scenario does eliminate the hierarchy between weak scale ν (TeV) and Planck Scale, it introduces a new hierarchy between ν and µ c 1 V n Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

38 Solution to Hierarchy in ADD Model Hierarchy in Disguise Hierarchy in Disguise This scenario does eliminate the hierarchy between weak scale ν (TeV) and Planck Scale, it introduces a new hierarchy between ν and µ c 1 V n µ c ν = n (18) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

39 Randall-Sundrum Model Randall Sundrum Model This model is distinct from the previous in following points: There is only one extra dimension involved as opposed to n 2. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

40 Randall-Sundrum Model Randall Sundrum Model This model is distinct from the previous in following points: There is only one extra dimension involved as opposed to n 2. There is no such disguised hierarchy between weak scale and compactification scale. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

41 Randall-Sundrum Model Randall Sundrum Model This model is distinct from the previous in following points: There is only one extra dimension involved as opposed to n 2. There is no such disguised hierarchy between weak scale and compactification scale. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

42 Set Up Randall-Sundrum Model The geometry is nonfactorizable with following form ds 2 = e 2σ(φ) η µν dx µ dx ν + r 2 c dφ 2 (19) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

43 Set Up Randall-Sundrum Model The geometry is nonfactorizable with following form ds 2 = e 2σ(φ) η µν dx µ dx ν + r 2 c dφ 2 (19) The topology of the extradimension is S 1 /Z 2. The 5th coordinate is angular in nature with (x, φ) identified with (x, φ). Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

44 Randall-Sundrum Model Set Up The geometry is nonfactorizable with following form ds 2 = e 2σ(φ) η µν dx µ dx ν + r 2 c dφ 2 (19) The topology of the extradimension is S 1 /Z 2. The 5th coordinate is angular in nature with (x, φ) identified with (x, φ). The 3-branes extending in x µ direction are located at φ = 0 and φ = π. gµν vis = G µν (x µ, φ = π) (20) gµν hid = G µν (x µ, φ = 0) (21) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

45 Set up Randall-Sundrum Model Figure: Randall Sundrum Model Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

46 The Classical Action Randall-Sundrum Model S = S gravity + S vis + S hid (22) where S gravity = d 4 x π π dφ G[ Λ + 2M 3 R] (23) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

47 The Classical Action Randall-Sundrum Model S = S gravity + S vis + S hid (22) where S gravity = d 4 x S vis = π π dφ G[ Λ + 2M 3 R] (23) d 4 x g vis [L vis V vis ] (24) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

48 The Classical Action Randall-Sundrum Model S = S gravity + S vis + S hid (22) where π S gravity = d 4 x dφ G[ Λ + 2M 3 R] (23) π S vis = d 4 x g vis [L vis V vis ] (24) S hid = d 4 x g hid [L hid V hid ] (25) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

49 The Classical Action Randall-Sundrum Model S = S gravity + S vis + S hid (22) where π S gravity = d 4 x dφ G[ Λ + 2M 3 R] (23) π S vis = d 4 x g vis [L vis V vis ] (24) S hid = d 4 x g hid [L hid V hid ] (25) where V vis and V hid is vacuum energy of the branes which act as a gravitational source even in absence of matter on brane. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

50 Randall-Sundrum Model Solution for the Bulk Metric ds 2 = e 2σ(φ) η µν dx µ dx ν + r 2 c dφ 2 (26) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

51 Randall-Sundrum Model Solution for the Bulk Metric ds 2 = e 2σ(φ) η µν dx µ dx ν + r 2 c dφ 2 (26) where ds 2 = e 2krc φ η µν dx µ dx ν + r 2 c dφ 2 (27) k = Λ 24M 3 (28) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

52 Randall-Sundrum Model Solution for the Bulk Metric ds 2 = e 2σ(φ) η µν dx µ dx ν + r 2 c dφ 2 (26) ds 2 = e 2krc φ η µν dx µ dx ν + r 2 c dφ 2 (27) where Λ k = 24M 3 (28) To satisty the Einstein s equation, we must have V vis = V hid = 24M 3 (29) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

53 Randall-Sundrum Model Solution for the Bulk Metric ds 2 = e 2σ(φ) η µν dx µ dx ν + r 2 c dφ 2 (26) ds 2 = e 2krc φ η µν dx µ dx ν + r 2 c dφ 2 (27) where Λ k = 24M 3 (28) To satisty the Einstein s equation, we must have V vis = V hid = 24M 3 (29) Demanding a metric with a warp factor forces the existence of branes at φ = 0 and φ = π with vacuum energy given by above. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

54 Randall-Sundrum Model Solution for the Bulk Metric ds 2 = e 2σ(φ) η µν dx µ dx ν + r 2 c dφ 2 (26) ds 2 = e 2krc φ η µν dx µ dx ν + r 2 c dφ 2 (27) where Λ k = 24M 3 (28) To satisty the Einstein s equation, we must have V vis = V hid = 24M 3 (29) Demanding a metric with a warp factor forces the existence of branes at φ = 0 and φ = π with vacuum energy given by above. The bulk has negative Λ-hence the bulk is a slice of AdS 5 geometry. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

55 Weak Scale from Planck Scale Relation between M and M Pl We perturb the background metric so that ds 2 = e 2krc φ [η µν + h µν ]dx µ dx ν + r 2 c dφ 2 (30) where h µν is tensor fluctuations about Minkowski spacetime. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

56 Weak Scale from Planck Scale Relation between M and M Pl We perturb the background metric so that ds 2 = e 2krc φ [η µν + h µν ]dx µ dx ν + r 2 c dφ 2 (30) where h µν is tensor fluctuations about Minkowski spacetime. The 4dimensional effective theory follows by substituting the above in S gravity = d 4 x π π dφ G[ Λ + 2M 3 R] Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

57 Contd.. Weak Scale from Planck Scale S eff d 4 x π π dφ2m 3 r c e 2krc φ ḡ R (31) where R denotes the 4 dimensional Ricci scalar made out of ḡ µν. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

58 Contd.. Weak Scale from Planck Scale S eff d 4 x π π dφ2m 3 r c e 2krc φ ḡ R (31) where R denotes the 4 dimensional Ricci scalar made out of ḡ µν. M 2 Pl = M 3 r c π π dφe 2krc φ = M3 k [ 1 e 2krc φ ] (32) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

59 Contd.. Weak Scale from Planck Scale S eff d 4 x π π dφ2m 3 r c e 2krc φ ḡ R (31) where R denotes the 4 dimensional Ricci scalar made out of ḡ µν. M 2 Pl = M 3 r c π π dφe 2krc φ = M3 k [ 1 e 2krc φ ] (32) M Pl depends weakly on r c in the large kr c limit. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

60 Contd.. Weak Scale from Planck Scale S eff d 4 x π π dφ2m 3 r c e 2krc φ ḡ R (31) where R denotes the 4 dimensional Ricci scalar made out of ḡ µν. M 2 Pl = M 3 r c π π dφe 2krc φ = M3 k [ 1 e 2krc φ ] (32) M Pl depends weakly on r c in the large kr c limit. But the exponential hierarchy plays a crucial role in determining visible sector masses. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

61 Scaling of Mass Weak Scale from Planck Scale Visible Sector Mass-Exponential Hierarchy S vis d 4 x g vis [g µν vis D µh D ν H λ( H 2 ν 2 0) 2 ] (33) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

62 Scaling of Mass Weak Scale from Planck Scale Visible Sector Mass-Exponential Hierarchy S vis d 4 x g vis [g µν vis D µh D ν H λ( H 2 ν 2 0) 2 ] (33) Now substitute g µν vis = e2krcπ ḡ µν (34) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

63 Scaling of Mass Weak Scale from Planck Scale Visible Sector Mass-Exponential Hierarchy S vis d 4 x g vis [g µν vis D µh D ν H λ( H 2 ν 2 0) 2 ] (33) Now substitute g µν vis = e2krcπ ḡ µν (34) So we have S vis d 4 x ḡe 4krcπ [ḡ µν e 2krcπ D µ H D ν H λ( H 2 ν 2 0) 2 ] (35) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

64 Weak Scale from Planck Scale Exponential Hierarchy Visible Sector Mass-Exponential Hierarchy We rescale H e krcπ H (36) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

65 Weak Scale from Planck Scale Exponential Hierarchy Visible Sector Mass-Exponential Hierarchy We rescale we obtain S vis H e krcπ H (36) d 4 x ḡ[ḡ µν D µ H D ν H λ( H 2 e 2krcπ ν 2 0) 2 ] (37) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

66 Weak Scale from Planck Scale Exponential Hierarchy Visible Sector Mass-Exponential Hierarchy We rescale we obtain S vis H e krcπ H (36) d 4 x ḡ[ḡ µν D µ H D ν H λ( H 2 e 2krcπ ν 2 0) 2 ] (37) ν = e krcπ ν 0 (38) Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

67 Scaling of Mass Weak Scale from Planck Scale Visible Sector Mass-Exponential Hierarchy m = e krcπ m 0 (39) If e krcπ is of the order of the mechanism produces TeV physical mass scale from Plank scale (10 19 GeV ). Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

68 Weak Scale from Planck Scale Visible Sector Mass-Exponential Hierarchy Scaling of Mass m = e krcπ m 0 (39) If e krcπ is of the order of the mechanism produces TeV physical mass scale from Plank scale (10 19 GeV ). Because the warping factor is exponential the model does not require very large hierarchies among fundamental parameters ν 0,k,M and µ c = 1 r c ; with kr c 10. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

69 Weak Scale from Planck Scale Generation of Exponential Hierarchy Visible Sector Mass-Exponential Hierarchy Figure: Generation of Exponential Hierarchy Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

70 Weak Scale from Planck Scale Visible Sector Mass-Exponential Hierarchy References Kaluza Klein Theory, William.O.Straub,PhD, Pasadena, California ( Kaluza Klein Theories, David Bailin and Alex Love, Rep. Prog. Phys. 50, 1087 (1987) A First Course in String Theory, Barton Zwiebach, Second Edition Large Mass Hierarchy from a Small Extra Dimension, L.Randall and R.Sundrum, Phys. Rev. Lett. 83, 3370 (1999). The Aim of the Talk is to present the above paper by Randall, Sundrum. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

71 Weak Scale from Planck Scale Visible Sector Mass-Exponential Hierarchy Acknowledgement The speaker acknowdeges a debt of gratitude to Dr.G.M.Hossain, Dept. of Physical Science, IISER Kolkata Prof.N.Banarjee, Dept. of Physical Science, IISER Kolkata Prof.S.Sengupta, Dept. of Theoretical Physics, IACS for their valuable time and guidance. Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26

72 Weak Scale from Planck Scale Visible Sector Mass-Exponential Hierarchy THANK YOU Sridip Pal (09MS002) DPS PH4204 () Large Mass Hierarchy from a Small Extra Dimension April 4, / 26