Gauge Field Localization in Models with Large Extra Dimensions

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:1004.4078]

,

Talk at Kobe workshop 2011.1.6

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

1 Physics Beyond the Standard Model

LHC Era: Physics beyond the Standard Model

Gauge HierarchyProblem: Huge ratio of Electroweak to Fundamental scales m²_W

M_GUT² ≈

µ 10² 10¹⁶

¶₂

≈ 10⁻²⁸, m²_W

M_Gravity² ≈

µ 10² 10¹⁸

¶₂

≈ 10⁻³²

Solutions (for Explanations) of the Gauge Hierarchy

1. Composite Higgs (Technicolor) : Realistic calculable models needed

L. Susskind,Phys. Rev.D20 (1979) 2619; S. Weinberg, Phys. Rev.D19 (1979) 1277; D13 (1976) 974; S. Dimopoulos, and L. Susskind,Nucl. Phys. B155 (1979) 237; · · ·

2. Supersymmetry (SUSY) Spin0 : m_B,

l m_B = m_F ← Supersymmetry

Spin¹₂ : 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;

E.Witten, Nucl.Phys.B188 (1981) 513;· · ·

Figure 1: NonSUSY GUT (left) cannot unify gauge couplings. SUSY GUT (right) can unify gauge couplings. α_i = g_i²/4π,(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_Gravity² = M_TeVⁿ⁺²Rⁿ

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; · · ·

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; · · ·

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 = ²₀E + P ≡ ²E

Confinement ↔ ² = 0 (perfect dia-electric medium) Relativisitic version of dielectric vacuum

L = −1

4²(x)F_µνF^µν, ²(x) = 1 g²(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 ²(x) : Position-dependent gauge coupling g² finite on the wall, and g² → ∞ for the bulk asymptotically

3 Position-Dependent Gauge Coupling

Explore properties of our model first, and construct explicit models later

Figure 2: Example of position-dependent gauge coupling in y.

Domain wall in 4 + 1 dimensions : x^M, M = 0, 1,· · · , 4,

Wall profile depends on y = x⁴, World-volume x^µ, µ = 0, 1, 2, 3, Our model (F_{M N}^a ≡ ∂_MW_N^a − ∂_NW_M^a − f^abcW_M^b W_N^c )

L = −1

4²(y)F ^{M N a}F_{M N}^a , ²(y) ≥ 0

²(y) → 0, y → ±∞ : (profile localized on the wall) Mode Analysis spectrum of effective fields

Linearized field equation

0 = ²(y)(∂^N∂_NW_M^a − ∂^N∂_MW_N^a) + (∂^y²(y))(∂_yW_M^a − ∂_MW_y^a)

Axial gauge W_y^a = 0

Decomposition to transverse and longitudinal components W_µ^a = W_µ^aT + W_µ^aL, W_µ^aL ≡ 1

∂²∂_µ∂^νW_ν^a Field eq. for longitudinal component (applying ∂^µ to M = µ)

0 = −∂_y

³

²(y)∂_y(∂^µW_µ^a)

´

Solution has no propagating modes (F^a(x) arbitrary function of x)

∂^µW_µ^a = Y (y)F^a(x)

Y (y) =

Z _y

dy⁰ 1

²(y⁰) (3.1)

Field eq. for transverse component

0 = ²(y)∂^ν∂_νW_µ^aT − ∂_y(²(y)∂_yW_µ^aT) Eigenvalue eq. for the mode functions

·

− 1

²(y) d

dy²(y) d

dy − m²_n

¸

u_n(y) = 0

u_n(y) assumed to be a complete set of wave functions in y Mode expansion (Kaluza-Klein decomposition)

W_µ^aT = X

n

w_µ^a(n)(x)u_n(y), (∂^ν∂_ν + m²_n)w^a(n)_µ (x) = 0

Mapping to a (threshold) bound state using Y in Eq.(3.1) instead of y Hu_n(Y ) = 0

H ≡ −1 2

d²

dY ² + U(Y ), U(Y ) = −1

2m²_n²²(y(Y )), Normalization of the position-dependent coupling function ²

1 g₄² =

Z

dy ²(y) = Z

dY ²²(y(Y ))

L_eff ≡ Z

dy L₅

Canonical kinetic terms → Normalization of mode functions Z _∞

−∞

dy [g₄² ²(y)]u_n(y)^∗u_l(y) = δ_nl

There is always a zero energy solution: u₀ =constant, Hu₀ = 0

(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 ) = − U₀ cosh² αY

²(y) = 1 g₄

r α

4g₄² + 2g₄α³y², y = (g₄√

2/α^3/2) sinh αY

Mass spectra: a massless mode n = 0 and a mass gap m²₁ = 4g₄²α m²_n = 2g₄²αn(n + 1), n = 0,1, 2, . . .

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

4²(y)F_{M N}^a F^{aM N}+J_M^a W^aM, J_M^a (x, y) = δ(y)δ_{M µ}J_µ^a(x) Field eq. for W_y has no source term

0 = ²(y)∂^M∂_MW_y^a −→ 0 = ∂^M∂_MW_y^a

No external source at infinities → no nontrivial solution : W_y^a = 0 Field equation for W_µ^a in the Landau gauge ∂^MW_M^a = 0

0 = ²(y)∂^ν∂_νW_µ^a + ∂^y

³

²(y)∂_yW_µ^a

´

− δ(y)J_µ^a(x) Thin wall limit with a regularization in the bulk by 1/g₅² (² 6= 0)

²_thin(y) = δ(y)

g₄² + 1 g₅²

Going to Euclidean space, and momentum space only in 4-dimensions 0 = (p² − ∂_y²)

g₅² W˜ _µ^a(p, y)

+δ(y)p²W˜ _µ^a(p, y)

g₄² − ∂_y

³

δ(y)∂_yW˜ _µ^a(p, y)

´

g₄² − δ(y) ˜J_µ^a(p) Solution for y 6= 0

W˜ _µ^a(p, y) = C_µ^a+(p)e^−pyθ(y) + C_µ^a−(p)e^pyθ(−y) Souce terms at y = 0 determines C_µ^a+(p), C_µ^a−(p) : Result is

W˜ _µ^a(p, y = 0) = g₄² p² + ^2g_g2⁴²

5 p

J˜_µ^a(p)

After regularization is removed (g₅ → ∞: strong coupling in the bulk)

glim₅²→∞

W˜ _µ^a(p, y = 0) = g₄²

p²J˜_µ^a(p) → W_µ^a(x, y = 0) = g₄²Q^a 4πr δ_µ0 for a static charge J˜_µ^a(p) = Q^aδ_µ0 : 4-dimensional Coulomb law

Finite width for the wall

²_step(y) ≡

½ 1/(2ε) |y| < ε 1/g₅² |y| > ε Conclusion is unchanged

Figure 4: Step function for the position-dependent coupling.

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 Σ^I

charged Scalars H_A, A = 1, · · · , N_F, with charge q^A and mass m_A L_wall = − 1

4e²_I(F_{M N}^I )² + 1

2e²_I(∂_MΣ^I)² + |D_MH_A|² − V V = |(q_I^AΣ^I − m_A)H_A|² + 1

2e²_I(Y ^I)², Y ^I = e²_I ¡

c_I − q_I^A|H_A|²¢ D_MH_A = (∂_M + iW_M^I q_I^A)H_A, gauge coupling e_I, FI parameter c_I SUSY vacua: A = 1, · · · , N_F, I = 1,2

(q_I^AΣ^I − m_A)H_A = 0, c_I − q_I^A|H_A|² = 0,

Energy density for a wall in the gauge W_y^I = 0 E = 1

2e²_I(∂_yΣ^I − e²_I(c_I − q_I^A|H_A|²))² + |∂_yH_A + (q_I^AΣ^I − m_A)H_A|² +∂_y ¡

c_IΣ^I − q_I^AΣ^I|H_A|² + m_A|H_A|²¢ BPS equations

¡∂_y + q_I^AΣ^I¢

H_A = H_Am_A, ∂_yΣ^I = e²_I(c_I − q_I^A|H_A|²) Exact wall solution at the strong coupling limit e²_I → ∞

H_A = H_0AΩ^−q₁ ^{A1 /2}Ω^−q₂ ^2A/2e^mAy, Σ^I = 1

2∂_y log Ω_I c_I = |H_0A|²Ω^−q₁ ^1AΩ^−q₂ ^2Ae^2mAy, I = 1, 2

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)₁ 1 1 0 0 q₂^A for U(1)₂ 0 0 1 1

m_A ^m₂ −^m₂ ^m₂ −^m₂

Table 1: Charge and mass of Higgs fields (hypermultiplets) of the four-flavor model

c₁ = c₂ = c > 0 A pair of SUSY vacua

H₁ = 0, H₂ = √

c, Σ¹ = m 2 H₁ = √

c, H₂ = 0, Σ¹ = −m 2

Similarly by interchaging H₁, H₂ → H₃, H₄ and Σ¹ → Σ² Wall solution

H₁ = √

c e^{m2 (y−y1)}

p2 cosh m(y − y₁), H₂ = √

c e^{−m2 (y−y1)}

p2 cosh m(y − y₁)

Σ¹ = m

2 tanh m(y − y₁), Σ² = m

2 tanh m(y − y₂)

(a) (b)

(c) (d)

Figure 5: Profile of the domain wall with two copies of two Higgs flavors. (a), (b): |y₁−y₂| ∼ 1/m. (c), (d): |y₁ − y₂| À 1/m, the shape tends to a step-function.

U(1) × U(1) model with three Higgs flavors H_A=1 H_A=2 H_A=3 U(1)₁ 1 1 0 U(1)₂ 0 −1 1 m_A m 0 0

Table 2: Charges and masses of the three Higgs fields

Two vacua: c₁ > 0, c₂ > 0 H₁ = 0, H₂ = √

c₁, H₃ = √

c₁ + c₂, Σ¹ = Σ² = 0 H₁ = √

c₁, H₂ = 0, H₃ = √

c₂, Σ¹ = m, Σ² = 0 Wall solution

Σ^I = 1

2∂_y log Ω_I, Ω₁ = e^2my + e^2my0Ω₂

Ω₂ =

1 − e^2m(y−y0) + q

(1 − e^2m(y−y0))² + 4(1 + ^c_c¹

2)e^2m(y−y0) 2(1 + ^c_c¹

2) Σ²(−y) = Σ²(y) > 0

(a) (b)

Figure 6: Profile Σ² of the domain wall with three Higgs flavors. (a): c₁/c₂ ∼ 1. (b):

c₁/c₂ À 1, step-function like profile.

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 ≡ ∂²a(Σ)

∂Σ^I∂Σ^J, a_IJK ≡ ∂³a(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

µ

−1

4F_{M N}^I F^{JM N} + 1

2D_MΣ^ID^MΣ^J + 1

2Y ^αIY ^αJ

¶

−c^αIY ^αI

+a_IJK (

− 1

24²^{LM N P Q}W_L^I µ

F_{M N}^J F_{P Q}^K + 1

2[W_M, W_N]^JF_{P Q}^K

+ 1

16[W_M, W_N]^J[W_P, W_Q]^K

¶)

+(D_MH^irA)^∗D^MH^irA − (H^irA)^∗[(q^IΣ^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

2e²₁(Σ¹)² + 1

2e²₂(Σ²)² + 1

2(a₁Σ¹ + a₂Σ²)Σ^aΣ^a Σ¹, Σ² for U(1) × U(1), Σ^a for non-Abelian gauge group

For 2 copies of 2 Higgs model, we choose : a₁ = −a₂ ≡ a > 0 For 3 Higgs model, we choose : a₁ = 0, a₂ > 0

We obtain gauge coupling function ²(y) = 1/g²(y) which is Positive definite, Asymptotically vanishing (confining)

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 prepo- tential, which is consistent with the general principle of supersymmetry in five-dimensional space-time.