Gauge Field Localization in Models with Large Extra Dimensions

Transcription

1 Gauge Field Localization in Models with Large Extra Dimensions Norisuke Sakai (Tokyo Woman s Christian University) In collaboration with Kazutoshi Ohta, Progress of Theoretical Physics 124, 71 (2010) [arxiv: ],talk at Kobe workshop Contents 1 Physics Beyond the Standard Model 2 2 Gauge Field Localization on Domain Wall 4 3 Position-Dependent Gauge Coupling 5 4 Four-Dimensional Coulomb law 10 5 Supersymmetric Models for BPS Walls 13 6 Conclusion 20

2 1 Physics Beyond the Standard Model LHC Era: Physics beyond the Standard Model Gauge HierarchyProblem: Huge ratio of Electroweak to Fundamental scales m 2 ( W 10 2 ) m 2 ( W 10 2 ) 2, M 2 GUT M 2 Gravity Solutions (for Explanations) of the Gauge Hierarchy Composite Higgs (Technicolor) : Realistic calculable models needed L. Susskind,Phys. Rev.D20 (1979) 2619; S. Weinberg, Phys. Rev.D19 (1979) 1277; D13 (1976) 2. Supersymmetry (SUSY) 974; S. Dimopoulos, and L. Susskind,Nucl. Phys. B155 (1979) 237; Spin0 : m B, m B = m F Supersymmetry Spin 1 2 : m F, m F = 0 Chiral Symmetry light Higgs is protected by SUSY and chiral symmetry of Higgsino S.Dimopoulos, H.Georgi, Nucl.Phys.B193 (1981) 150; N.Sakai, Z.f.Phys.C11 (1981) 153; 2 E.Witten, Nucl.Phys.B188 (1981) 513;

3 Figure 1: NonSUSY GUT (left) cannot unify gauge couplings. SUSY GUT (right) can unify gauge couplings. α i = gi 2 /4π, (i = 1, 2, 3) are U(1), SU(2), SU(3) gauge couplings. Gauge coupling unification: Indirect Evidence for SUSY 3. Models with Large Extra Dimensions (Brane-World scenario) Standard model particles should be localized on a brane Gravity propagates in higher dimensional bulk: M 2 Gravity = M n+2 TeV Rn P.Horava and E.Witten, Nucl.Phys.B475, 94 (1996); N.Arkani-Hamed, S.Dimopoulos, G.Dvali, Phys.Lett.B429 (1998) 263 ; I.Antoniadis, N.Arkani-Hamed, S.Dimopoulos, G.Dvali, Phys.Lett.B436 (1998) 257; Randall, Sundrum, Phys.Rev.Lett.83 (1999) 3370; 4690; 3

4 y: extra dimension Models beyond the Standard Model can be tested at LHC and other facilities 2 Gauge Field Localization on Domain Wall Let us take the simplest case of a brane : Domain Wall Gauge symmetry broken in the bulk (Higgs phase) and restored on a wall Flux is absorbed by the superconducting bulk gauge boson acquires a mass of order 1/(wall width) Gauge fields should be confined in the bulk (outside of wall) G.Dvali, M.Shifman, Phys.Lett.B396 (1997) 64; N.Arkani-Hamed, S.Dimopoulos, G.Dvali, Phys.Lett.B429 (1998) 263; N.Maru, N.Sakai, Prog.Theor.Phys.111 (2004) 907; 4

5 Warped (Randall-Sundrum) model does not help to localize gauge fields Our Pourpose: Propose a model to Localize Non-Abelian Gauge Fields on a Wall Dielectric vacua as a classical representation of confiniment D = ɛ 0 E + P ɛe Confinement ɛ = 0 (perfect dia-electric medium) Relativisitic version of dielectric vacuum L = 1 4 ɛ(x)f µνf µν, ɛ(x) = 1 g 2 (x) J.Kogut, L.Susskind, Phys.Rev.D9 (1974) 3501; R.Fukuda, Phys.Lett.B73 (1978) 305; Mod.Phys.Lett.A24 (2009) 251; Electric permiability ɛ(x) : Position-dependent gauge coupling g 2 finite on the wall, and g 2 for the bulk asymptotically 3 Position-Dependent Gauge Coupling Explore properties of our model first, and construct explicit models later 5

6 Figure 2: Example of position-dependent gauge coupling in y. Domain wall in dimensions : x M, M = 0, 1,, 4, Wall profile depends on y = x 4, World-volume x µ, µ = 0, 1, 2, 3, Our model (FMN a MWN a NWM a f abc WM b W N c ) L = 1 4 ɛ(y)f MNa F a MN, ɛ(y) 0 ɛ(y) 0, y ± : (profile localized on the wall) Mode Analysis spectrum of effective fields Linearized field equation 0 = ɛ(y)( N N W a M N M W a N ) + ( y ɛ(y))( y W a M MW a y ) 6

7 Axial gauge W a y = 0 Decomposition to transverse and longitudinal components W a µ = W at µ + W al µ, W al µ 1 2 µ ν W a ν Field eq. for longitudinal component (applying µ to M = µ) ( ) 0 = y ɛ(y) y ( µ W a µ ) Solution has no propagating modes (F a (x) arbitrary function of x) Field eq. for transverse component µ W a µ = Y (y)f a (x) y Y (y) = dy 1 ɛ(y ) 0 = ɛ(y) ν ν W at µ y(ɛ(y) y W at µ ) Eigenvalue eq. for the mode functions [ 1 d ɛ(y) dy ɛ(y) d ] dy m2 n u n (y) = 0 (3.1) 7

8 u n (y) assumed to be a complete set of wave functions in y Mode expansion (Kaluza-Klein decomposition) W at µ = n w a(n) µ (x)u n (y), ( ν ν + m 2 n )wa(n) µ (x) = 0 Mapping to a (threshold) bound state using Y in Eq.(3.1) instead of y Hu n (Y ) = 0 d 2 H 1 2dY + U(Y ), U(Y ) = m2 n ɛ2 (y(y )), Normalization of the position-dependent coupling function ɛ 1 = dy ɛ(y) = dy ɛ 2 (y(y )) g 2 4 L eff dy L 5 Canonical kinetic terms Normalization of mode functions dy [g 2 4 ɛ(y)]u n(y) u l (y) = δ nl There is always a zero energy solution: u 0 =constant, Hu 0 = 0 8

9 (a) Potential in Y (b) Position-dependent gauge coupling in y Figure 3: A dashed line represents the wave function of the localized massless vector field. 4-dimensional gauge invariance is maintained intact. Solvable Example U(Y ) = U 0 cosh 2 αy ɛ(y) = 1 α g 4 4g g 4α 3 y 2, y = (g 4 2/α 3/2 ) sinh αy Mass spectra: a massless mode n = 0 and a mass gap m 2 1 = 4g2 4 α m 2 n = 2g2 4αn(n + 1), n = 0, 1, 2,... 9

10 All modes are normalizable. Square well potential is also solvable : A massless mode and a mass gap 4 Four-Dimensional Coulomb law We wish to demonstrate 4 dimensional Coulomb law for the static source on the world volume of wall (to clarify the role of non-transverse component) L = 1 4 ɛ(y)f a MN F amn +J a M W am, J a M (x, y) = δ(y)δ MµJ a µ (x) Field eq. for W y has no source term 0 = ɛ(y) M M W a y 0 = M M W a y No external source at infinities no nontrivial solution : W a y = 0 Field equation for Wµ a in the Landau gauge M WM a ) = 0 0 = ɛ(y) ν ν W a µ (ɛ(y) + y y W a µ δ(y)j a µ (x) Thin wall limit with a regularization in the bulk by 1/g5 2 ɛ thin (y) = δ(y) + 1 g4 2 g5 2 (ɛ 0) 10

11 Going to Euclidean space, and momentum space only in 4-dimensions 0 = (p2 2 y ) g 2 5 W a µ + δ(y)p2 W µ a (p, y) g 2 4 Solution for y 0 W a µ (p, y) y ( δ(y) y W a µ (p, y) ) g 2 4 δ(y) J a µ (p) (p, y) = Ca+ µ (p)e py θ(y) + C a µ (p)epy θ( y) Souce terms at y = 0 determines C a+ µ W a µ (p, y = 0) = g 2 4 (p), Ca µ (p) : Result is p 2 + 2g2 4p g5 2 J a µ (p) After regularization is removed (g 5 : strong coupling in the bulk) lim W a g5 2 µ (p, y = 0) = g2 4 J a p 2 µ (p) W a µ (x, y = 0) = g2 4 Qa 4πr δ µ0 for a static charge J a µ (p) = Qa δ µ0 : 4-dimensional Coulomb law 11

12 Finite width for the wall Conclusion is unchanged ɛ step (y) { 1/(2ε) y < ε 1/g 2 5 y > ε Figure 4: Step function for the position-dependent coupling. 12

13 5 Supersymmetric Models for BPS Walls 5-dimensions 8 SUSY vector multiplet (gauge field) and hypermultiplet (matter) We display bosonic part only (relevant for wall solutions and gauge fields) Wall sector U(1) U(1) gauge fields (A I µ, I = 1, 2), neutral scalars ΣI charged Scalars H A, A = 1,, N F, with charge q A and mass m A L wall = 1 (F I 4e 2 MN )2 + 1 ( I 2e 2 M Σ I ) 2 + D M H A 2 V I V = (q A I ΣI m A )H A ( (Y I ) 2, Y I = e 2 I ci q A I H A 2) 2e 2 I D M H A = ( M + iw I M qa I )H A, gauge coupling e I, FI parameter c I SUSY vacua: A = 1,, N F, I = 1, 2 (q A I ΣI m A )H A = 0, c I q A I H A 2 = 0, 13

14 Energy density for a wall in the gauge W I y = 0 E = 1 2e 2 I ( y Σ I e 2 I (c I q A I H A 2 )) 2 + y H A + (q A I ΣI m A )H A 2 + y ( ci Σ I q A I ΣI H A 2 + m A H A 2) BPS equations ( y + q A I ΣI) H A = H A m A, y Σ I = e 2 I (c I q A I H A 2 ) Exact wall solution at the strong coupling limit e 2 I H A = H 0A Ω qa 1 /2 1 Ω qa 2 /2 2 e m Ay, Σ I = 1 2 y log Ω I c I = H 0A 2 Ω qa 1 1 Ω qa 2 2 e 2m Ay, I = 1, 2 14

15 Two copies of two charged Higgs fields (four flavor model) H A=1 H A=2 H A=3 H A=4 q1 A for U(1) q2 A for U(1) m A m 2 m 2 m 2 m 2 Table 1: Charge and mass of Higgs fields (hypermultiplets) of the four-flavor model A pair of SUSY vacua c 1 = c 2 = c > 0 H 1 = 0, H 2 = c, Σ 1 = m 2 H 1 = c, H 2 = 0, Σ 1 = m 2 Similarly by interchaging H 1, H 2 H 3, H 4 and Σ 1 Σ 2 Wall solution e m 2 (y y 1) H 1 = c 2 cosh m(y y1 ), e m 2 (y y 1) H 2 = c 2 cosh m(y y1 ) 15

16 Σ 1 = m 2 tanh m(y y 1), Σ 2 = m 2 tanh m(y y 2) (a) (b) (c) (d) Figure 5: Profile of the domain wall with two copies of two Higgs flavors. (a), (b): y 1 y 2 1/m. (c), (d): y 1 y 2 1/m, the shape tends to a step-function. 16

17 U(1) U(1) model with three Higgs flavors Two vacua: c 1 > 0, c 2 > 0 H A=1 H A=2 H A=3 U(1) U(1) m A m 0 0 Table 2: Charges and masses of the three Higgs fields H 1 = 0, H 2 = c 1, H 3 = c 1 + c 2, Σ 1 = Σ 2 = 0 H 1 = c 1, H 2 = 0, H 3 = c 2, Σ 1 = m, Σ 2 = 0 Wall solution Ω 2 = Σ I = 1 2 y log Ω I, Ω 1 = e 2my + e 2my 0 Ω 2 1 e 2m(y y0) + (1 e 2m(y y 0) ) 2 + 4(1 + c 1)e c 2m(y y 0) 2 2(1 + c 1 c 2 ) Σ 2 ( y) = Σ 2 (y) > 0 17

18 (a) (b) Figure 6: Profile Σ 2 of the domain wall with three Higgs flavors. (a): c 1 /c 2 c 1 /c 2 1, step-function like profile. 1. (b): Position-dependent coupling function from the cubic prepotential 8 SUSY Lagrangian determined by a Prepotential a(σ) a I a(σ) Σ I, a IJ 2 a(σ) Σ I Σ J, a IJK 3 a(σ) Σ I Σ J Σ K complex scalars H ira, i = 1, 2 : SU(2) R doublet Y αi, c αi, α = 1, 2, 3 : SU(2) R triplet L = a IJ ( 1 4 F I MN F JMN D MΣ I D M Σ J Y αi Y αj ) c αi Y αi 18

19 +a IJK { 1 ( 24 ɛlmnp Q W I L F J MN F K P Q [W M, W N ] J F K P Q + 1 ) } 16 [W M, W N ] J [W P, W Q ] K +(D M H ira ) D M H ira (H ira ) [(q I Σ I m A ) 2 ] r sh isa +(H ira ) (σ α ) i jq I (Y αi ) r sh jsa, Prepotential a(σ) should be at most a cubic polynomial N.Seiberg, Phys.Lett.B388 (1996) 753; We can assume a (Σ) = 1 (Σ 1 ) (Σ 2 ) e 2 1 2e (a 1Σ 1 + a 2 Σ 2 )Σ a Σ a Σ 1, Σ 2 for U(1) U(1), Σ a for non-abelian gauge group For 2 copies of 2 Higgs model, we choose : a 1 = a 2 a > 0 For 3 Higgs model, we choose : a 1 = 0, a 2 > 0 We obtain gauge coupling function ɛ(y) = 1/g 2 (y) which is Positive definite, Asymptotically vanishing (confining) 19

20 6 Conclusion 1. A mechanism using the position-dependent gauge coupling is proposed to localize non-abelian gauge fields on domain walls in five-dimensional space-time. 2. Low-energy effective theory possesses a massless vector field, and a mass gap. 3. The four-dimensional gauge invariance is maintained intact. 4. We obtain perturbatively the four-dimensional Coulomb law for static sources on the domain wall. 5. BPS domain wall solutions with the localization mechanism are explicitly constructed in the U(1) U(1) supersymmetric gauge theory coupling to the non-abelian gauge fields only through the cubic prepotential, which is consistent with the general principle of supersymmetry in five-dimensional space-time. 20