Domain-wall brane model building
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1 Domain-wall brane model building From chirality to cosmology Damien George Theoretical Particle Physics group School of Physics University of Melbourne Australia PhD completion seminar 17 th October 2008
2 Acknowledgements Supervisor: Ray Volkas. Supervisory panel: Bruce McKellar, Lloyd Hollenberg. Fellow TPPers: Sandy Law, Alison Demaria, Kristian McDonald, Andrew Coulthurst, Jason Doukas, Ben Carson, Thomas Jacques, Jayne Thompson, Nadine Pesor. Collaborators: Rhys Davies (Oxford, UK), Mark Trodden, Kamesh Wali (Syracuse, NY), Aharon Davidson (Ben-Gurion, Israel), Archil Kobakhidze, Gareth Dando (Melbourne). Scholarships: Elizabeth and Vernon Puzey postgraduate research scholarship, Postgraduate Overseas Research Experience Scholarship (PORES). 2 / 37
3 PhD summary Focus of PhD: theory of domain-wall brane models, with applications to the standard model of particle physics and cosmology. Papers published: G. Dando, A. Davidson, D.P. George, R.R. Volkas and K.C. Wali The clash of symmetries in a Randall-Sundrum-like spacetime Phys.Rev. D (2005), arxiv:hep-ph/ D.P. George and R.R. Volkas Stability of domain walls coupled to Abelian gauge fields Phys.Rev. D (2005), arxiv:hep-ph/ E. Di Napoli, D. George, M. Hertzberg, F. Metzler and E. Siegel Dark Matter In Minimal Trinification Proceedings of the LXXXVI Les Houches Summer School, pp (2006), arxiv:hep-ph/ D.P. George and R.R. Volkas Kink modes and effective four dimensional fermion and Higgs brane models Phys. Rev. D (2007), arxiv:hep-ph/ R. Davies and D.P. George Fermions, scalars and Randall-Sundrum gravity on domain-wall branes Phys. Rev. D (2007), arxiv: R. Davies, D.P. George and R.R. Volkas The standard model on a domain-wall brane? Phys. Rev. D (2008), arxiv: A. Davidson, D.P. George, A. Kobakhidze, R.R. Volkas and K.C. Wali SU(5) grand unification on a domain-wall brane from an E 6 -invariant action Phys. Rev. D (2008), arxiv: D.P. George, M. Trodden, R.R. Volkas Domain-wall brane cosmology In preparation 3 / 37
4 Talk outline Context: physics beyond the standard model extra dimensions. We will cover: Randall-Sundrum 2 model with delta-function brane. Domain walls (kinks/solitons). Trapping scalar and fermion fields to a domain wall. Using Dvali-Shifman mechanism to trap gauge fields. SU(5) grand unified domain-wall model. Extending SU(5) to E 6. Cosmology the expansion of a brane universe. 4 / 37
5 Introduction We are inspired by the Randall-Sundrum warped metric solution. RS1 is a compact extra dimension: provides a solution to the hierarchy problem lots of work on this model. Branes are string theory like objects. Warped throats, inflation, dark matter,... (Randall & Sundrum, PRL83, 3370 (1999)) RS2 is an infinite extra dimension: solves the trapping of low-energy gravity. Not as much interest because it doesn t solve any major problems, just introduces another dimension. (Randall & Sundrum, PRL83, 4690 (1999)) We will pursue RS2 because it seems a natural extension of 3+1 space. 5 / 37
6 Brane worlds
7 Brane worlds Premise: take the standard model and general relativity add an infinite extra space dimension recover the standard model and general relativity at low energies 7 / 37
8 Brane worlds Premise: take the standard model and general relativity add an infinite extra space dimension recover the standard model and general relativity at low energies S = d 4 x dy g [ M 3 R ] 7 / 37
9 Brane worlds Premise: take the standard model and general relativity add an infinite extra space dimension recover the standard model and general relativity at low energies S = d 4 x dy g [ M 3 ] R + δ(y)l SM 7 / 37
10 RS2 model Need brane and bulk sources: [ g ( M S = d 4 x dy 3 R Λ bulk ) + ] g (4) δ(y)(l SM Λ brane ) (1) 8 / 37
11 RS2 model Need brane and bulk sources: [ g ( M S = d 4 x dy 3 R Λ bulk ) + Solve the theory: ] g (4) δ(y)(l SM Λ brane ) Randall-Sundrum metric ansatz: ds 2 = e 2k y g (4) µν dx µ dx ν dy 2 Solve Einstein s equations (L SM = 0 and R (4) = 0): Λ bulk = 12k 2 M 3 Λ brane = 12kM 3 Write R in terms of R (4) : R = e 2k y R (4) 16kδ(y) + 20k 2 (1) 8 / 37
12 RS2 model Need brane and bulk sources: [ g ( M S = d 4 x dy 3 R Λ bulk ) + Solve the theory: ] g (4) δ(y)(l SM Λ brane ) Randall-Sundrum metric ansatz: ds 2 = e 2k y g (4) µν dx µ dx ν dy 2 Solve Einstein s equations (L SM = 0 and R (4) = 0): Λ bulk = 12k 2 M 3 Λ brane = 12kM 3 (1) Write R in terms of R (4) : R = e 2k y R (4) 16kδ(y) + 20k 2 Substitute into (1) and integrate over y: S = d 4 x g (4) [ M 3 ] k R(4) + L SM A dimensionally reduced theory. 8 / 37
13 Newton s law Just need to check Newton s law. Linear tensor fluctuations are: The zero mode h (4) 0 µν g µν = e 2k y η µν + n h (4) n µν(x µ )ψ n (y) dominates the Kaluza-Klein tower. 9 / 37
14 Newton s law Just need to check Newton s law. Linear tensor fluctuations are: g µν = e 2k y η µν + n h (4) n µν(x µ )ψ n (y) The zero mode h (4) 0 µν dominates the Kaluza-Klein tower. Newton s law is modified to: ) m 1 m 2 V (r) = G N (1 + ɛ2 r r 2 (where ɛ = 1/k) Current experimental bounds are very weak: ɛ < 12µm = k > ev 9 / 37
15 Newton s law Just need to check Newton s law. Linear tensor fluctuations are: g µν = e 2k y η µν + n h (4) n µν(x µ )ψ n (y) The zero mode h (4) 0 µν dominates the Kaluza-Klein tower. Newton s law is modified to: ) m 1 m 2 V (r) = G N (1 + ɛ2 r r 2 (where ɛ = 1/k) Current experimental bounds are very weak: ɛ < 12µm = k > ev We have what we wanted: a 5D theory that at low energies looks like our 4D universe. But almost no new phenomenology. Next step: brane forms naturally. 9 / 37
16 Thick and smooth RS2 We want to remove the δ(y) part of the action: [ g ( M ] S = d 4 x dy 3 R Λ bulk ) + g (4) δ(y)(l SM Λ brane ) Everything from now on is one way of doing that. 10 / 37
17 Thick and smooth RS2 We want to remove the δ(y) part of the action: [ g ( M ] S = d 4 x dy 3 R Λ bulk ) + g (4) δ(y)(l SM Λ brane ) Everything from now on is one way of doing that. Geometry, hence gravity, is 5D. So why not try to make all fields 5D? First we show how to make a dynamical brane. Then we show how to trap scalars, fermions and gauge fields to the brane. Finally we present a 4+1-d SU(5) based extension to the standard model. 10 / 37
18 Thick and smooth RS2 We want to remove the δ(y) part of the action: [ g ( M ] S = d 4 x dy 3 R Λ bulk ) + g (4) δ(y)(l SM Λ brane ) Everything from now on is one way of doing that. Geometry, hence gravity, is 5D. So why not try to make all fields 5D? First we show how to make a dynamical brane. Then we show how to trap scalars, fermions and gauge fields to the brane. Finally we present a 4+1-d SU(5) based extension to the standard model. Turn off warped gravity for now just think about trapping 5D fields. 10 / 37
19 A domain wall as a brane V(φ) -v +v φ A domain-wall is a (thin) region separating two vacua. The field φ interpolates between two vacua as one moves along the extradimension. 11 / 37
20 A domain wall as a brane V(φ) -v +v φ A domain-wall is a (thin) region separating two vacua. The field φ interpolates between two vacua as one moves along the extradimension. Lagrangian for φ(x µ, y): L = 1 2 Mφ M φ V (φ) A solution is the kink: φ(y) = v tanh( 2λvy) It is stable! 11 / 37
21 More complicated domain walls (DPG & Volkas, PRD72, (2005)) These examples use two scalar fields to form the wall. V = λ 1 (φ φ2 2 v2 ) 2 + λ 2 φ 2 1 φ2 2 V = λ(φ 2 1 +φ2 2 v2 1 )2 (φ 2 1 +φ2 2 +v2 2 ) 12 / 37
22 More complicated domain walls (DPG & Volkas, PRD72, (2005)) These examples use two scalar fields to form the wall. V = λ 1 (φ φ2 2 v2 ) 2 + λ 2 φ 2 1 φ2 2 V = λ(φ 2 1 +φ2 2 v2 1 )2 (φ 2 1 +φ2 2 +v2 2 ) In a realistic model, symmetries dictate V. To determine stability, expand φ in normal modes about the background: φ(x, t) = φ bg (x) + n ξ n(x)e iωnt. Make sure ω 2 n 0 12 / 37
23 Trapping matter fields
24 Trapping scalar fields (DPG & Volkas, PRD75, (2007)) Aim: to trap a 5D scalar field Ξ(x µ, y) to the brane. A simple quartic coupling works: [ 1 S = d 4 x dy 2 M φ M φ V (φ) + 1 ] 2 M Ξ M Ξ W (Ξ) gφ 2 Ξ 2 14 / 37
25 Trapping scalar fields (DPG & Volkas, PRD75, (2007)) Aim: to trap a 5D scalar field Ξ(x µ, y) to the brane. A simple quartic coupling works: [ 1 S = d 4 x dy 2 M φ M φ V (φ) + 1 ] 2 M Ξ M Ξ W (Ξ) gφ 2 Ξ 2 Expand Ξ in extra dimensional (Kaluza-Klein) modes: Ξ(x µ, y) = n ξ n (x µ )k n (y) ξ n are the 4D fields, k n their extra-dimensional profile. The profiles satisfy a Schrödinger equation: ) ( d2 dy 2 + 2gφ2 bg k n (y) = Enk 2 n (y) The energy eigenvalues E n are related to the mass of the 4D field ξ n. 14 / 37
26 Trapping via a potential well (DPG & Volkas, PRD75, (2007)) The effective potential acts like a well. ( ) d2 + 2gφ 2 dy 2 bg k n = Enk 2 n effective potential k n profile extra dimension y extra dimension y 15 / 37
27 Trapping via a potential well (DPG & Volkas, PRD75, (2007)) The effective potential acts like a well. ( ) d2 + 2gφ 2 dy 2 bg k n = Enk 2 n effective potential k n profile extra dimension y extra dimension y To get 4D theory, substitute mode expansion into action and integrate y: [ ( ) ] 1 S = d 4 x 2 µ ξ n µ ξ n m 2 nξn 2 + (higher order terms) n Orthonormal basis k n m n can be tuned. = diagonal kinetic and mass terms. 15 / 37
28 Trapping fermions (DPG & Volkas, PRD75, (2007)) We can trap a fermion Ψ(x µ, y) to the brane with a Yukawa coupling: [ ] 1 S = d 4 x dy 2 M φ M φ V (φ) + ΨiΓ M M Ψ hφψψ 16 / 37
29 Trapping fermions (DPG & Volkas, PRD75, (2007)) We can trap a fermion Ψ(x µ, y) to the brane with a Yukawa coupling: [ ] 1 S = d 4 x dy 2 M φ M φ V (φ) + ΨiΓ M M Ψ hφψψ Decompose into left- and right-chiral fields and Kaluza-Klein modes: Ψ(x µ, y) = n [ψ Ln (x µ )f Ln (y) + ψ Rn (x µ )f Rn (y)] 16 / 37
30 Trapping fermions (DPG & Volkas, PRD75, (2007)) We can trap a fermion Ψ(x µ, y) to the brane with a Yukawa coupling: [ ] 1 S = d 4 x dy 2 M φ M φ V (φ) + ΨiΓ M M Ψ hφψψ Decompose into left- and right-chiral fields and Kaluza-Klein modes: Ψ(x µ, y) = [ψ Ln (x µ )f Ln (y) + ψ Rn (x µ )f Rn (y)] n Schrödinger equation (mode index n suppressed): ) ( d2 dy 2 + (h2 φ 2 bg hφ bg ) f L,R (y) = m 2 f L,R (y) effective potential extra dimension y relative mode energy 10 L 2 R f Ln profile extra dimension y f Rn profile extra dimension y 16 / 37
31 Gravity and matter fields Brane (domain wall/kink), trapped scalar and fermion. Plus gravity: S = d 4 x dy [ g M 3 R Λ bulk M φ M φ V (φ) M Ξ M Ξ W (Ξ) gφ 2 Ξ 2 ] + ΨiΓ M M Ψ hφψψ 17 / 37
32 Gravity and matter fields Brane (domain wall/kink), trapped scalar and fermion. Plus gravity: S = d 4 x dy [ g M 3 R Λ bulk M φ M φ V (φ) M Ξ M Ξ W (Ξ) gφ 2 Ξ 2 ] + ΨiΓ M M Ψ hφψψ Dimensionally reduce by integrating over y: [ S = d 4 x g (4) M4D 2 R(4) + (brane dynamics) µ ξ n µ ξ n m 2 nξn 2 τ mnop ξ m ξ n ξ o ξ p (brane interactions) ] + ψ L0 iγ µ µ ψ L0 + ψ n (iγ µ µ µ n )ψ n (brane interactions) 4D parameters (M 4D, m n, τ mnop, µ n, brane dynamics) determined by eigenvalue spectra and overlap integrals. 17 / 37
33 Warped matter (Davies & DPG, PRD76, (2007)) Warped metric ds 2 = e 2σ(y) η µν dx µ dx ν dy 2 modifies profile equation: ( d2 dy 2 + d ) 5σ dy + 2σ 6σ 2 + U(y) f Ln (y) = m 2 ne 2σ f Ln (y) 18 / 37
34 Warped matter (Davies & DPG, PRD76, (2007)) Warped metric ds 2 = e 2σ(y) η µν dx µ dx ν dy 2 modifies profile equation: ( d2 dy 2 + d ) 5σ dy + 2σ 6σ 2 + U(y) f Ln (y) = m 2 ne 2σ f Ln (y) Conformal coordinates ds 2 = e 2σ(y(z)) (η µν dx µ dx ν dz 2 ). Rescale f Ln (y) = e 2σ fln (z): ) ( d2 dz 2 + e 2σ(y(z)) U(y(z)) f Ln (z) = m 2 f n Ln (z) Matter trapping potentials are warped down. 18 / 37
35 Warped matter (Davies & DPG, PRD76, (2007)) Warped metric ds 2 = e 2σ(y) η µν dx µ dx ν dy 2 modifies profile equation: ( d2 dy 2 + d ) 5σ dy + 2σ 6σ 2 + U(y) f Ln (y) = m 2 ne 2σ f Ln (y) Conformal coordinates ds 2 = e 2σ(y(z)) (η µν dx µ dx ν dz 2 ). Rescale f Ln (y) = e 2σ fln (z): ) ( d2 dz 2 + e 2σ(y(z)) U(y(z)) f Ln (z) = m 2 f n Ln (z) Matter trapping potentials are warped down. Finite bound state lifetimes. Resonances. Tiny probability of interaction with continuum. left-handed potential W / k a=0 a=0.04 a= dimensionless conformal coordinate kz brane interaction density (a 1/M 3 5D Newton s constant) a=0 a= (mass of mode / k) 2 18 / 37
36 Trapping gauge fields
37 Confining gauge fields Need to trap gauge fields or e.g. Coulomb potential would be V Coulomb 1/r 2. Not as simple as a Kaluza-Klein mode expansion: Photon and gluons must remain massless. Need to preserve gauge universality at 3+1-d level. We use the Dvali-Shifman mechanism, Dvali & Shifman, PLB396, 64 (1997). The following discussion is based on Dvali, Nielsen & Tetradis, PRD77, (2008). 20 / 37
38 Abelian Higgs model U(1) gauge theory (think 5d photon). Charged Higgs χ (gives mass to photon). M χ χ 0 extra dimension 21 / 37
39 Abelian Higgs model U(1) gauge theory (think 5d photon). Charged Higgs χ (gives mass to photon). χ M χ 0 extra dimension U(1) superconductor superconductor In the bulk: U(1) is broken, massive photon M χ. Higgs vacuum is a superconductor. Electric charges are screened. On the brane: U(1) is restored, massless photon. Electric field ends on Higgs vacuum. Charge screening leaks onto the brane! 21 / 37
40 Using a dual superconductor SU(2) gauge theory (3 gluons ). Adjoint Higgs χ a (a = 1, 2, 3) (gives mass). M χ χ 0 extra dimension 22 / 37
41 Using a dual superconductor SU(2) gauge theory (3 gluons ). Adjoint Higgs χ a (a = 1, 2, 3) (gives mass). χ M χ 0 SU(2) dual superconductor extra dimension U(1) SU(2) dual superconductor In the bulk: SU(2) is restored, in confining regime. Large mass gap M χ to colourless state. QCD-like vacuum is dual superconductor. On the brane: SU(2) broken to U(1), massless photon. Electric field repelled from dual superconductor. For distances much larger than brane width, electric potential 1/r. 22 / 37
42 Dvali-Shifman model Stabilise the domain wall with an extra uncharged scalar field η: [ 1 S = d 4 x dy 4g 2 GaMN G a MN M η M η (DM χ a ) D M χ a ] λ(η 2 v 2 ) 2 λ 2 (χa χ a + κ 2 v 2 + η 2 ) 2 η has a kink profile. If κ 2 v 2 < 0, χ becomes tachyonic near domain wall (where η 0). True vacuum has χ 0 near domain wall. χ breaks symmetry near wall and confines gauge fields. field profile v 0 -v η χ extra dimension y 23 / 37
43 Dvali-Shifman model Stabilise the domain wall with an extra uncharged scalar field η: [ 1 S = d 4 x dy 4g 2 GaMN G a MN M η M η (DM χ a ) D M χ a ] λ(η 2 v 2 ) 2 λ 2 (χa χ a + κ 2 v 2 + η 2 ) 2 η has a kink profile. If κ 2 v 2 < 0, χ becomes tachyonic near domain wall (where η 0). True vacuum has χ 0 near domain wall. χ breaks symmetry near wall and confines gauge fields. field profile v 0 -v η χ extra dimension y Can add gravity: self consistently solve σ (warped metric profile), η, χ. 23 / 37
44 Dvali-Shifman mechanism The Dvali-Shifman mechanism: Works with any non-abelian SU(N) theory. Assumes the SU(N) theory is confining (not proven for 5D). Has gauge universality: Charges in the bulk are connected to the brane by a flux tube. Coupling to gauge fields is independent of extra dimensional profile. 24 / 37
45 Dvali-Shifman mechanism The Dvali-Shifman mechanism: Works with any non-abelian SU(N) theory. Assumes the SU(N) theory is confining (not proven for 5D). Has gauge universality: Charges in the bulk are connected to the brane by a flux tube. Coupling to gauge fields is independent of extra dimensional profile. Obvious choice for SU(N) group is SU(5). 24 / 37
46 SU(5) basics
47 Quantum numbers of the standard model Representations under SU(3) SU(2) L U(1) Y : q L (3, 2) 1/3 u R (3, 1) 4/3 d R (3, 1) 2/3 l L (1, 2) 1 ν R (1, 1) 0 e R (1, 1) 2 26 / 37
48 Quantum numbers of the standard model Representations under SU(3) SU(2) L U(1) Y : q L (3, 2) 1/3 u R (3, 1) 4/3 d R (3, 1) 2/3 l L (1, 2) 1 ν R (1, 1) 0 e R (1, 1) 2 SU(2) L ν L d R u L e R e R u L d R ν R ν L d L u R e L U(1) Y e L u R d L ν R 26 / 37
49 Quantum numbers of the standard model Representations under SU(3) SU(2) L U(1) Y : q L (3, 2) 1/3 u R (3, 1) 4/3 d R (3, 1) 2/3 l L (1, 2) 1 ν R (1, 1) 0 e R (1, 1) 2 SU(2) L ν L d R u L e R e R u L d R ν R ν L d L u R e L U(1) Y e L u R d L ν R 5 d r,w,b L ν L e L (5 5) A = 10 u r,w,b L u r,w,b L d r,w,b L e L 26 / 37
50 Putting it all together
51 The SU(5) model (Davies, DPG & Volkas, PRD77, (2008)) Want the standard model on the brane: SU(3) SU(2) L U(1) Y. Dvali-Shifman needs a larger gauge group in the bulk: Unify the fermions as usual: 5, 10. Higgs doublet goes in a 5. SU(5) is a perfect fit! 28 / 37
52 The SU(5) model (Davies, DPG & Volkas, PRD77, (2008)) Want the standard model on the brane: SU(3) SU(2) L U(1) Y. Dvali-Shifman needs a larger gauge group in the bulk: Unify the fermions as usual: 5, 10. Higgs doublet goes in a 5. Summary: SU(5) is a perfect fit! dimensional theory all spatial dimensions the same. SU(5) local gauge symmetry, Z 2 discrete symmetry. Field content: gauge fields: G MN 24. scalars: η 1, χ 24, Φ 5. fermions: Ψ 5 5, Ψ The standard model emerges as a low energy approximation. Ignore gravity for now. 28 / 37
53 The action (without gravity) (Davies, DPG & Volkas, PRD77, (2008)) The theory is described by: [ 1 S = d 4 x dy 4g 2 GaMN G a MN + 1 ( ) 2 M η M η + Tr (D M χ) (D M χ) + (D M Φ) (D M Φ) + Ψ 5 iγ M D M Ψ 5 + Ψ 10 iγ M D M Ψ 10 h 5η Ψ 5 Ψ 5 η h 5χ Ψ 5 χ T Ψ 5 h 10η Tr(Ψ 10 Ψ 10 )η + 2h 10χ Tr(Ψ 10 χψ 10 ) h (Ψ 5 ) c Ψ 10 Φ h + (ɛ(ψ 10 ) c Ψ 10 Φ ) + h.c. (cη 2 µ 2 χ) Tr(χ 2 ) dη Tr(χ 3 ) λ 1 [ Tr(χ 2 ) ] 2 λ2 Tr(χ 4 ) l(η 2 v 2 ) 2 µ 2 ΦΦ Φ λ 3 (Φ Φ) 2 λ 4 Φ Φη 2 ] 2λ 5 Φ Φ Tr(χ 2 ) λ 6 Φ (χ T ) 2 Φ λ 7 Φ χ T Φη with kinetic, brane trapping, mass and Dvali-Shifman terms. 29 / 37
54 Split fermions (Davies, DPG & Volkas, PRD77, (2008)) Let Ψ ny be the components of Ψ 5 and Ψ 10 (n = 5, 10, Y = hypercharge of component), e.g. Ψ 5 Ψ 5, 1 = l L. Dirac equation: [ ] 3 iγ M Y M h nη η(y) 5 2 h nχχ 1 (y) Ψ ny (x µ, y) = 0 Each Ψ ny is a non-chiral 5D field: need to extract the confined left-chiral zero-mode (recall the mode expansion and Schrödinger equation approach): Ψ ny (x µ, y) = ψ ny,l (x µ )f ny (y) + massive modes 30 / 37
55 Split fermions (Davies, DPG & Volkas, PRD77, (2008)) Let Ψ ny be the components of Ψ 5 and Ψ 10 (n = 5, 10, Y = hypercharge of component), e.g. Ψ 5 Ψ 5, 1 = l L. Dirac equation: [ ] 3 iγ M Y M h nη η(y) 5 2 h nχχ 1 (y) Ψ ny (x µ, y) = 0 Each Ψ ny is a non-chiral 5D field: need to extract the confined left-chiral zero-mode (recall the mode expansion and Schrödinger equation approach): Ψ ny (x µ, y) = ψ ny,l (x µ )f ny (y) + massive modes The effective Schrödinger potential depends on Y. f 5 Y d c L ll Thus each component ψ ny,l different profile f ny. has a f 10 Y dimensionless coordinate ky e c L u c L q L 30 / 37
56 Split Higgs (Davies, DPG & Volkas, PRD77, (2008)) Φ contains the Higgs doublet Φ w and a coloured triplet Φ c. Mode expand Φ w,c (x µ, y) = φ w,c (x µ )p w,c (y). Schrödinger equation for p w,c is: ( d2 dy 2 + 3Y ) λ 6χ 2 Y λ 7ηχ p w,c (y) = m 2 w,cp w,c (y) Critical that ground states have: m 2 w < 0 to break electroweak symmetry. m 2 c > 0 to preserve QCD. 31 / 37
57 Split Higgs (Davies, DPG & Volkas, PRD77, (2008)) Φ contains the Higgs doublet Φ w and a coloured triplet Φ c. Mode expand Φ w,c (x µ, y) = φ w,c (x µ )p w,c (y). Schrödinger equation for p w,c is: ( d2 dy 2 + 3Y ) λ 6χ 2 Y λ 7ηχ p w,c (y) = m 2 w,cp w,c (y) Critical that ground states have: m 2 w < 0 to break electroweak symmetry. m 2 c > 0 to preserve QCD. W -1 W 2/3 Large enough parameter space to allow this. W Y dimensionless coordinate ky 31 / 37
58 Features (Davies, DPG & Volkas, PRD77, (2008)) Standard model parameters are computed from overlap integrals. With one generation of fermions, parameters are easy to fit. The model overcomes the major SU(5) obstacles: m e = m d not obtained due to naturally split fermions. Coloured Higgs induced proton decay is suppressed. Gauge coupling constant running modified due to Kaluza-Klein modes appearing (not analysed yet). 32 / 37
59 Features (Davies, DPG & Volkas, PRD77, (2008)) Standard model parameters are computed from overlap integrals. With one generation of fermions, parameters are easy to fit. The model overcomes the major SU(5) obstacles: m e = m d not obtained due to naturally split fermions. Coloured Higgs induced proton decay is suppressed. Gauge coupling constant running modified due to Kaluza-Klein modes appearing (not analysed yet). Adding gravity: Solve for warped metric, kink and Dvali-Shifman background. Continuum fermion and scalar modes are highly suppressed on the brane. Main features remain. 32 / 37
60 Going beyond SU(5) (Davidson, DPG, Kobakhidze, Volkas & Wali, PRD77, (2008)) One promising extension is to the E 6 group: E 6 SO(10) in the bulk. SO(10) SU(5) on the brane due to clash-of-symmetries (CoS) and Dvali-Shifman. Can eliminate kink scalar field η. Can unify Ψ 5 and Ψ 10. Large reduction of free parameters. V(φ) -v +v φ π -8.0x x10-3 (10, ) -1.2x10-3 φ π/2-1.4x x10-3 (10, ) (10,+) 0-1.8x10-3 fx (10,+) -π -π/2 0 π/2 π θ fe (+ s are SO(10) s) (top domain-wall has CoS) 33 / 37
61 Domain-wall cosmology
62 The scale factor on a brane Most important cosmological fact: expanding universe. FRW: ds 2 = dt 2 + a 2 (t) dx dx Energy density ρ dictates expansion: H = ȧ a (Hubble parameter) H 2 = 8πG 3 ρ (spatially flat universe) 35 / 37
63 The scale factor on a brane Most important cosmological fact: expanding universe. FRW: ds 2 = dt 2 + a 2 (t) dx dx Energy density ρ dictates expansion: H = ȧ a (Hubble parameter) H 2 = 8πG 3 ρ (spatially flat universe) FRW for a brane: ds 2 = n 2 (t, y)dt 2 + a 2 (t, y) dx dx + dy 2 H 2 brane = 8πG 3 ( 1 + ρ ) ρ 2σ σ is the energy density (tension) of the brane; σ (1MeV ) 4 from BBN. (Binétruy, Deffayet & Langlois, Nucl. Phys. B565, 269 (2000)) 35 / 37
64 Species-dependent scale factor (DPG, Trodden & Volkas, in prep.) ds 2 = n 2 (t, y)dt 2 +a 2 (t, y)dx dx+dy 2 Each slice at constant y has a different scale factor. a(t, y = 3 ) a(t, y = 2 ) a(t, y = ) a(t, y = 0) a(t, y = ) a(t, y = 2 ) a(t, y = 3 ) a(t, y) 36 / 37
65 Species-dependent scale factor (DPG, Trodden & Volkas, in prep.) ds 2 = n 2 (t, y)dt 2 +a 2 (t, y)dx dx+dy 2 Each slice at constant y has a different scale factor. a(t, y = 3 ) a(t, y = 2 ) a(t, y = ) a(t, y = 0) a(t, y = ) a(t, y = 2 ) a(t, y = 3 ) a(t, y) f sp (t, y) For thick branes, different species (electron, quark, KK mode) experience different expansion rates. ( a sp (t) = a 0 (t) 1 + ρ ) I sp (t) 2σH 0 I sp (t) = f 2 sp (t, y) dy f 2 sp (t, y) e σ y /6M 3 5 dy 1 Different localisation profile f sp for different energies! No chiral fermions! 36 / 37
66 Summary and outlook Our universe may be a brane residing in a higher dimensional bulk. We have investigated the possibility that the brane is in fact a domain-wall. Main results: Scalar field: forms stable domain wall. RS2 warped metric: traps gravity. Dvali-Shifman mechanism: traps gauge fields. SU(5) domain-wall model: overcomes major SU(5) obstacles. E 6 extension: unify fields and reduce parameters. Cosmology: species-dependent expansion rate. Outlook: extra-dimensions a real possibility! Will be tested by LHC (e.g. KK modes) and cosmological observations. 37 / 37
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