Gauge Fields and Matter Fields Localized on Walls

Norisuke Sakai (Keio Univ.) In collaboration with

Masato Arai (Fukushima Nat. Coll. Tech.), Filip Blaschke (Silesian Univ.),

Minoru Eto (Yamagata Univ.), and Kazutoshi Ohta (Meiji Gakuin Univ.)

Progress of Theoretical Physics 124, 71 (2010) [arXiv:1004.4078]; Progress of Theoretical and

Experimental Physics 2013, 013B05 (2013) [arXiv:1208.6219]; 2013, 093B01 (2013) [arXiv:1303.5212]

Contents

1 Physics Beyond the Standard Model 3

2 Gauge Field Localization on Domain Wall 6

3 A Model with Localized Matter and Gauge Fields 12

4 Geometrical Higgs Mechanism and D-branes 24

5 Effective Lagrangian 28

6 Conclusion 34

2

1 Physics Beyond the Standard Model

LHC Era: Physics beyond the Standard Model

Gauge HierarchyProblem: Huge ratio of Electroweak to Fundamental scales

m2W

M2GUT

≈(

102

1016

)2

≈ 10−28,m2

W

M2Gravity

≈(

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 : mB

l mB = mF ← SupersymmetrySpin1

2: mF

mF = 0 ← Chiral Symmetry

light Higgs is protected by SUSY and chiral symmetry of Higgsino

3

S.Dimopoulos, H.Georgi, Nucl.Phys.B193 (1981) 150; N.Sakai, Z.f.Phys.C11 (1981) 153;

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

Gauge coupling unification: Indirect Evidence for SUSY

Figure 1: NonSUSY GUT (left) cannot unify gauge couplings. SUSY GUT (right) can unify

gauge couplings. αi = g2i /4π, (i = 1, 2, 3) are U(1), SU(2), SU(3) gauge couplings.

3. Models with Large Extra Dimensions (Brane-World scenario)

Standard model particles should be localized on a brane

Gravity propagates in higher dimensional bulk: M2Gravity = Mn+2

TeV Rn

4

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

5

2 Gauge Field Localization on Domain Wall

Let us take the simplest case of a brane : Domain Wall

A (wrong) attempt:

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

L =

∫dyd4x

√ggMKgNLFMNFKL, ds2 = e−kyηµνdxµdxν−dy2

Other attempt: Use of tensor field as a dual gauge field in 5 Dimension

→ Viable only for localization of U(1) gauge field

Y.Isozumi, K.Ohashi, N.Sakai, JHEP11 (2003) 061

6

Our Pourpose: Propose a model to

Localize Non-Abelian Gauge Fields on a Wall

Dielectric vacua as a classical representation of confinement

D = ε0E + P ≡ εE

E: electric field, D: electric displacement vector, P : polarization,

ε0: vacuum permiability, ε: permiability of matter

Ordinary matter (paramagnetic material): ε ≥ ε0

Confinement ↔ ε = 0 (perfect anti-dielectric medium)

Color flux can exit in regions with finite ε

Relativisitic version of dielectric vacuum

L = −1

4ε(x)FµνF

µν, ε(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 ε(x) : Position-dependent gauge coupling

g2 finite on the wall, and g2 → ∞ for the bulk asymptotically

7

Position-dependent coupling from field-dependent gauge coupling

Lcubic ∼ −(Σ1 − Σ2)Tr[GMNGMN ]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-10 -5 0 5 10

diffe

renc

e

y

m = 0.5, y1 = -2, y2 = 2

Figure 2: Two separated kinks Σ1, Σ2 (left panel). Position-dependent gaugecoupling as a difference Σ1 − Σ2 (right panel).

Domain wall sector: U(1)I=1,2 gauge theory with the F-I parameters cI

Lwall = − 1

4e2I

(F IMN)2 +

1

2e2I

(∂MΣI)2 + |DMHA|2 − V

8

V = |(qAI ΣI − mA)HA|2 +

e2I

2

(cI − qA

I |HA|2)2

HA=1 HA=2 HA=3 HA=4

qA1 for U(1)1 1 1 0 0

qA2 for U(1)2 0 0 1 1

mAm2

−m2

m2

−m2

Table 1: Four-flavor model: Charge and mass of fields of the wall sector

→ Kinks in Σ1 and Σ2

K.Ohta, N.Sakai, Prog.Theor.Phys.124 (2010) 71 [arXiv:1004.4078]

Cubic prepotential of gauge theories with 8 supercharges

(minimal in 1 + 4 dim.) → Cubic coupling involving Field strengths

(together with 5D Chern-Simons term, fermion couplings · · · )

N.Seiberg, Phys.Lett.B 388 (1996) 753 ; D.R.Morrison and N.Seiberg, Nucl.Phys.B 483 (1997) 229;

· · ·

9

Results of PTP 124 (2010) 71, Ohta-Sakai

1. Position-dependent gauge coupling can localize non-Abeliangauge fields on domain walls in five-dimensional space-time.

2. Low-energy effective theory possesses a massless vector field, and amass gap.

3. The four-dimensional gauge invariance is maintained intact.

4. The cubic coupling for gauge kinetic term is naturally obtained in su-persymmetric gauge theory in five-dimensional space-time.

Problems to be solved

1. Matter fields in the nontrivial representations should also be localizedtogether with the localized non-Abelian gauge fields.

⇒ unbroken global symmetry (Degenerate fields in wall sector)

⇒ (Non-Abelian subgroup of) the global symmetry is locally gauged

M.Shifman, A.Yung, Phys.Rev.D70 (2004) 025013 [arXiv:hep-th/0312257];

M.Eto, T.Fujimori, M.Nitta, K.Ohashi and N.Sakai, Phys.Rev.D77 (2008) 125008

[arXiv:0802.3135]; · · ·

10

M.Arai, F.Blaschke, M.Eto, and N.Sakai, PTEP 2013, 013B05 (2013) [arXiv:1105.2087]

2. It is better to eliminate moduli which can give unstable gauge kineticterm in some corners of the moduli space.

Gauge symmetry gives constraints: (4 matter, 2 gauge field → 2 moduli)

⇒ less scalar fields (3 matter, 2 gauge field → 1 moduli)

M.Arai, F.Blaschke, M.Eto, and N.Sakai, PTEP 2013, 093B01 (2013) [arXiv:1303.5212]

11

SU(N)c U(1)1 U(1)2 SU(N)L SU(N)R U(1)A massH1 ¤ 1 0 ¤ 1 1 m1N

H2 ¤ 1 −1 1 ¤ −1 0H3 1 0 1 1 1 0 0Σ adj ⊕ 1 0 0 1 1 0 0σ 1 0 0 1 1 0 0

Table 2: Quantum numbers of fields of the model for the domain wall.

3 A Model with Localized Matter and Gauge Fields

Lagrangian with global symmetry

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

Wall profile depends on y = x4, World-volume xµ, µ = 0, 1, 2, 3,

L = − 1

2g2Tr

(GMNGMN

)− 1

4e2FMNF MN +

1

g2Tr

(DMΣ

)2

+1

2e2(∂Mσ)2

+ Tr|DMH1|2 + Tr|(DM − iAM)H2|2 + |(∂M + iAM)H3|2 − V ,

V = Tr|(Σ − m1N)H1|2 + Tr|(Σ − σ1N)H2|2 + |σH3|2

+1

4g2Tr

(c11N − H1H

†1 − H2H

†2

)2

+1

2e2

(c2 + Tr(H2H

†2) − |H3|2

)2

12

DMH1,2 = ∂MH1,2 + iWMH1,2, DMΣ = ∂MΣ + i[WM , Σ]

GMN = ∂MWN−∂NWM+i[WM , WN ], FMN = ∂MAN−∂NAM

Global symmetry: UL ∈ SU(N)L, UR ∈ SU(N)R, eiα ∈ U(1)A,

Local gauge symmetry: Uc ∈ U(N)c, eiβ ∈ U(1)2

H1 → eiαUcH1UL , H2 → e−i(α+β)UcH2UR , H3 → eiβH3 ,

Σ → UcΣU †c , σ → σ

Assume m > 0, c1 > 0, c2 > 0

N + 1 discrete vacua with r = 0, 1, . . . , N

H1 =√

c1

(1N−r

0r

), H2 =

√c1

(0N−r

1r

), H3 =

√c2 + rc1 ,

Σ = m

(1N−r

0r

), σ = 0

Symmetry breaking in the r = 0 and r = N vacua

U(N)c ×SU(N)L × SU(N)R × U(1)2 × U(1)A

−−−−−−−→0−th vacuum

SU(N)L+c × SU(N)R × U(1)A+c ,

−−−−−−−−→N−th vacuum

SU(N)R+c × SU(N)L × U(1)A−c .

13

Nambu-Goldstone modes of coincident domain wallsSU(N)L × SU(N)R × U(1)A

SU(N)L+R × ZN

M.Eto, T.Fujimori, M.Nitta, K.Ohashi and N.Sakai, Phys.Rev.D77 (2008) 125008 [arXiv:0802.3135];

· · ·

Bogomol’nyi bound for Energy of domain walls

E =1

g2Tr

[DyΣ − g2

2

(c11N − H1H

†1 − H2H

†2

)]2

+ Tr|DyH1 + (Σ − m1N)H1|2

+1

2e2

(∂yσ − e2

(c2 + Tr(H2H

†2 − |H3|2)

))2

+ Tr|DyH2 + (Σ − (σ + iAy)1N)H2|2+ |∂yH3 + (σ + iAy)H3|2 + c2∂yσ

+ ∂yTr

[c1Σ − H1H

†1(Σ − m1N) − H2H

†2(Σ − σ1N)

]

E =

∞∫

−∞dy E ≥ T = c1

∞∫

−∞dy ∂yTr(Σ) = Nmc1

14

BPS equations for domain walls

∂yH1 + (Σ + iWy − m1N)H1 = 0 ,

∂yH2 +(Σ + iWy − (σ + iAy)1N

)H2 = 0 ,

∂yH3 + (σ + iAy)H3 = 0 ,

DyΣ =1

2g2

(c11N − H1H

†1 − H2H

†2

),

∂yσ = e2(c2 + Tr(H2H

†2) − |H3|2

)

BPS domain wall solutions

Solution by moduli matrices H01 , H0

2 , H03 : Define

Σ+iWy = S−1∂yS, σ+iAy =1

2∂yψ, Ω ≡ SS†, η ≡ Re(ψ)

H1 = emyS−1H01 , H2 = e

12ψS−1H0

2 , H3 = e−12ψH0

3

Remaining BPS equations become the Master equation

∂y(∂yΩΩ−1) =1

2g2

(c11N − (e2myH0

1H0 †1 + eηH0

2H0 †2 )Ω−1

),

15

1

2∂2

yη = e2(c2 + eηTr(H0

2H0 †2 Ω−1) − e−η|H0

3 |2)

V-transformations V ∈ GL(N,C), v ∈ C : equivalent physical fields

(S, ψ, H01 , H0

2 , H03) → (V S, ψ + v, V H0

1 , V H02e−1

2v, H03e

12v)

V -equivalence class defines a guenuine Moduli space

M.Arai, F.Blaschke, M.Eto, and N.Sakai, PTEP 2013, 013B05 (2013); [arXiv:1105.2087]

By fixing V-transformations, we obtain moduli: φ, U

H01 =

√c11N , H0

3 =√

c2, H02 =

√c1e

φU †, φ = φ†, UU † = 1N

U : Nambu-Goldstone (NG) modes for

SU(N)L × SU(N)R × U(1)A

SU(N)L+R × ZN

φ : (noncommutative) “positions” of walls

Master equation with the moduli parametrization

∂y(∂yΩΩ−1) =c1

2g2

(1N − Ω0Ω

−1)

, Ω0 = e2my1N + e2φeη

16

1

2∂2

yη = e2(c2 + c1e

ηTr(e2φΩ−1) − e−ηc2

)

Strong coupling limit: master equations can be solved algebraically:

Ω = Ω0, c2 + c1eηTr(e2φΩ−1) − e−ηc2 = 0 (1)

φ is diagonalizable: φ = mP −1diag(y1, . . . , yN)P , PP † = 1

Then (1) becomes an N + 1-th order algebraic equation (x = e−η)

x = 1 +c1

c2

N∑

i=1

1

1 + eix, ei = e2m(y−yi) (2)

For coincident walls : φ = my01N

e−η =1

2e0

(e0 − 1 +

√(1 − e0)2 + 4(1 + Nc1/c2)e0

)

Ω = (e2my + e2my0eη)1N , e0 := e2m(y−y0)

With gauge fixing S = Ω1/2 and Im(ψ) = 0, physical fields are

H1 =√

c1

1N√1 + e−2m(y−y0)+η

, H2 =√

c1

U †√

1 + e2m(y−y0)−η,

H3 =√

c2e−η/2 , Σ =

1

2∂y ln Ω , σ = ∂yη , Wy = Ay = 0

17

Invariant under the action of the diagonal global symmetry SU(N)L+R+c

18

0

1

2

3

4

5

6

7

-10 -5 0 5 10

exp(

-eta

)y

c1 = 1, c2 = 1, m = 1, N = 5

0

0.05

0.1

0.15

0.2

0.25

0.3

-10 -5 0 5 10

sigm

a

y

c1 = 1, c2 = 1, m = 1, N = 5

Figure 3: Profiles of e−η and σ in the coincident case. The parameters of the plot are given

above the picture. Positions of all domain walls are centered at the origin.

19

0

1

2

3

4

5

6

7

-20 -15 -10 -5 0 5 10 15 20

exp(

-eta

)

y

c1 = 1, c2 = 1, m = 1, N = 5

0

0.05

0.1

0.15

0.2

0.25

0.3

-20 -15 -10 -5 0 5 10 15 20

sigm

a

y

c1 = 1, c2 = 1, m = 1, N = 5

Figure 4: Profiles of e−η and σ in the non-coincident case. The parameters of the plot are given

above the picture. Positions of domain walls are y1 = −10, y2 = 4, y3 = 8, y4 = 12

and y5 = 16.20

Model for Localization of non-Abelian gauge fields

Introduce Gauge fields VM for the unbroken global symmetry SU(N)L+R+c

Lagrangian with the new covariant derivatives and gauge kinetic term

L = L − 1

2g2(σ)Tr

[GMNGMN

]

DMH1,2 = ∂MH1,2 + iWMH1,2 − iH1,2VM .

Field-dependent gauge coupling

1

2g2(σ)= λσ

VM = 0 is a solution of equation of motion : wall solution is unchanged

Positivity of position-dependent gauge coupling

Position-dependent coupling from field-dependent coupling

1

g2(σ)

∣∣∣∣background

≡ 1

g2(y)= λ∂yη = −λ∂y ln x

with the background domain wall solution x = e−η ≥ 0

21

-10

-8

-6

-4

-2

0

-10 -5 0 5 10

sigm

a

y

c1 = 104, c2 = 1, m=1, N=5

0

0.2

0.4

0.6

0.8

1

-10 -5 0 5 10

sigm

a

y

c1 = 104, c2 = 1, m = 1, N = 5

Figure 5: Profile of η-kink is shown in the left panel for the coincident case. In the right

panel, plots of Tr[Σ] (green dashed curve), Tr[Σ] − σ (red solid curve), and σ (blue

dotted curve) are shown.22

Differentiating the equation (2) for x = e−η ≥ 0

1

x∂yx = −c1

c2

N∑

i=1

ei

(1 + eix)2

(2m +

1

x∂yx

)

1

2g2(y)=

λc1

c2

N∑

i=1

ei

(1 + eix)2

/(1 +

c1

c2

N∑

i=1

ei

(1 + eix)2

)> 0

Effective gauge coupling in 1 + 3 dimension

1

g2=

∞∫

−∞dy

1

g2(y)= λ

∞∫

−∞dy ∂yη = λ ln

(1 + N

c1

c2

)

Width of the wall : ln(1 + N c1

c2

)

23

4 Geometrical Higgs Mechanism and D-branes

A simplified model (generalized Shifman-Yung model)

U(N) gauge theory with 2N scalars H in fundamental representations

Masses are M = diag.(m, · · · , m, −m, · · · , −m)

G = SU(N)L × SU(N)R × U(1)A global symmetry

N + 1 discrete vacua : k = 0, · · · , N

H∣∣vacuum

=√

c

(1k 0 0k 00 0N−k 0 1N−k

)

Maximal topological sector with N domain walls

H =

√c(1N , 0N) at y = +∞√c(0N , 1N) at y = −∞

Solution H = S−1H0eMy with Moduli matrix H0

H0 =√

c(1N , eφU †) =√

c(1N , ULeφ0U †R)

U, UL, UR unitary matrices, φ hermitian matrix,

φ0 real diagonal matrix : position of k-th wall yk

φ0 = mdiag.(y1, · · · , yN)

24

When all yi coincide, diagonal subgroup SU(N)V is maintained

broken global symmetry: translation (1 moduli)

SU(N)L × SU(N)R × U(1)A → SU(N)V, (N2 moduli)

NG bosons: N2 + 1 moduli

qNG bosons: N2 − 1 moduli

wave functions are localized at the wall and are identical

When all yi are different (completely separated walls)

broken global symmetry: translation (1 moduli)

SU(N)L × SU(N)R × U(1)A → U(1)N−1V (2N2 − N moduli)

NG bosons: 2N2 − N + 1,

qNG bosons: N − 1 (relative positions yi without overall translation)

total number of massless modes unchanged 2N2

Wave functions of qNG, translation, and U(1)NA : localized at each wall

Wave fuctions of other NG : spread between walls

25

Density of the Kahler metric for well-separated walls at g2 → ∞

Kij∗∂µφi∂µφj∗ =c

4

N∑r

|(τµ)rr|2cosh2(m(y − yr))

+ cN∑

r 6=s

cosh2(

m2(yr − ys)

) |(τµ)rs|2cosh(m(y − yr)) cosh(m(y − ys))

(τµ)rs ≈

−(U †R∂µUR)rs for r > s,

m∂µyr + (U †L∂µUL)rr − (U †

R∂µUR)rr for r = s,

(U †L∂µUL)rs for r < s.

Diagonal mode (τµ)rr: localized at r-th wall y = yr

Off-diagonal mode (τµ)rs: spread between y = yr and y = ys

26

27

5 Effective Lagrangian

Strong gauge coupling limit g, e → ∞ : Explicit evaluation possible

Finite gauge coupling : Qualitative features are unchanged

H1 =√

c1emyΩ−1/2 , H2 =

√c1e

η/2Ω−1/2eφU † , H3 =√

c2e−η/2

Ω−1 =e−ηe−2φ

1N + e2mye−2φe−η, e−η = 1+

c1

c2

Tr

(1N

1N + e2mye−2φe−η

)

2 types of moduli :

U : Nambu-Goldstone (NG) modes (chiral fields) for

SU(N)L × SU(N)R × U(1)A

SU(N)L+R × ZN

φ : (noncommutative) “positions” of walls

Derivative expansion up to quadratic order :

Low-energy effective Lagrangian

N.S.Manton, Phys.Lett.B110 (1982) 54;M.Shifman, A.Yung, Phys.Rev.D70 (2004) 025013 [arXiv:hep-th/0312257];

M.Eto, M.Nitta, K.Ohashi and D.Tong, Phys.Rev.Lett.95 (2005) 252003 [hep-th/0508130];M.Eto, T.Fujimori, M.Nitta, K.Ohashi and N.Sakai, Phys.Rev.D77 (2008) 125008 [arXiv:0802.3135];

M.Arai, F.Blaschke, M.Eto, and N.Sakai, PTEP 2013, 093B01 (2013) [arXiv:1303.5212]; · · ·28

Effective Lagrangian for chiral fields U

Terms with chiral zero (NG) modes U(x) :

Coincident wall background (without fluactuations) : φ → my0(x)

Leff =c1

2m

[(α+1)Tr

[DµU †DµU]+

α

NTr

[UDµU †]Tr

[UDµU †]

]

+Nmc1

2∂µy0∂

µy0 − 1

2mln

(1 +

Nc1

c2

)Tr

[GµνG

µν]

DµU = ∂µU + i[Vµ, U ],

α ≡ 1

2+

c2

Nc1

− c2

Nc1

(1 +

c2

Nc1

)ln

(1 +

Nc1

c2

)

Decay constants (Coupling strengths) for adjoint π and singlet η fields

fπ =

√c1(α + 1)

2m, fη =

√c1

Nm

Canonically normalized adjoint π and singlet η fields

1

fπ

Dµπ = i[UDµU † − 1N

NTr

(UDµU †

)]

29

1

fη

∂µη = iTr(UDµU †

)

Effective Lagrangian can be rewritten as

Leff = Tr(DµπDµπ

)+

1

2∂µη∂µη+

Nmc1

2∂µy0∂

µy0−1

2g2Tr

[GµνG

µν]

Different coupling strengths of adjoint and singlet NG bosons

Full effective Lagrangian

Both U and φ as moduli fields on the coincident wall background

Technically difficult to evaluate explicitly

→ Expansion in powers of (exponential of) wall width c1/c2

Leff = L(0)eff + T (1)

φ + T (1)U + T (1)

mix + T ′φ + T ′

U + c1O((c1/c2)

2)

Zero-th order effective Lagrangian (zero width limit)

Lie derivative with respect to A is defined as LA(B) = [A, B]

L(0)eff = T (0)

U + T (0)φ + T (0)

mix

30

T (0)U =

c1

2mTr

[12DµU †U

1

tanh(Lφ)ln

(1 + tanh(Lφ)

1 − tanh(Lφ)

)(U †DµU)

]

=c1

2mTr

(DµU †DµU

)− c1

6mTr

([φ, U †DµU ][φ, DµU †U ]

)

− c1

90mTr

([φ, [φ, U †DµU ]][φ, [φ, DµU †U ]]

)+ . . .

T (0)φ =

c1

2mTr

[Dµφ

cosh(Lφ) − 1

L2φ sinh(Lφ)

ln

(1 + tanh(Lφ)

1 − tanh(Lφ)

)(Dµφ)

]

=c1

2mTr

[DµφDµφ

]+

c1

24mTr

[[φ, Dµφ][φ, Dµφ]

]+ . . .

T (0)mix =

c1

2mTr

[U †DµU

cosh(Lφ) − 1

Lφ sinh(Lφ)ln

(1 + tanh(Lφ)

1 − tanh(Lφ)

)(Dµφ)

]

=c1

2mTr

[U †DµU [φ, Dµφ]

]− c1

24mTr

[U †DµU [φ, [φ, [φ, Dµφ]]]

]+ . . .

31

First order corrections in c1/c2 (width) expansion

T (1)U =

c21

16c2mTr

[DµU †UFU(∂x, Lφ)(U

†DµU)exφ]Tr

[e−xφ

]∣∣∣∣x=0

,

T (1)mix =

c21

8c2mTr

[U †DµUFmix(∂x, Lφ)(Dµφ)exφ

]Tr

[e−xφ

]∣∣∣∣x=0

,

T (1)φ =

c21

8c2mTr

[DµφFφ(∂x, Lφ)(Dµφ)exφ

]Tr

[e−xφ

]∣∣∣∣x=0

,

T ′φ = − c2

1

16c2mF (∂x)Tr

[exφDµφ

]Tr

[e−xφDµφ

]∣∣∣∣x=0

,

T ′U =

c21

16c2mF (∂x)Tr

[exφDµU †U

]Tr

[e−xφDµU †U

]∣∣∣∣x=0

FU(x, Lφ) =

∞∫

−∞dy

cosh(Lφ)

cosh2(y − x) cosh(y) cosh(y − Lφ),

Fmix(x, Lφ) =

∞∫

−∞dy

cosh(y) cosh(y − Lφ) − sinh(y) sinh(y − Lφ) − 1

Lφ cosh(y) cosh(y − Lφ) cosh2(y − x)

32

Fφ(x, Lφ) =

∞∫

−∞dy

cosh(y) cosh(y + Lφ) − sinh(y) sinh(y + Lφ) − 1

L2φ cosh(y) cosh(y + Lφ) cosh2(y − x)

,

F (x) =

∞∫

−∞dy

1

cosh2(y) cosh2(y − x)

Functions FU , · · · , Fφ can be expanded in powers of Lφ

Integration over y can be done explicitly.

Lowest order in φ:

Nmc1

2∂µy0∂

µy0 → c1

2mTr (DµφDµφ)

Adjoint representation of massless scalar φ gives

the (geometrical) Higgs mechanism

33

6 Conclusion

1. A mechanism using the position-dependent gauge coupling isproposed to localize non-Abelian gauge fields on domain walls infive-dimensional space-time.

2. We find a mass gap, and the low-energy effective theory of masslessvector fields with the four-dimensional gauge invariance intact.

3. We obtain localized matter fields interacting minimally with localizedgauge field, and the stability of gauge kinetic term is guarantteed.

4. Low-energy effective Lagrangian is obtained explicitly for coincidentwalls. Adjoint and singlet Nambu-Goldstone fields interact with differentstrengths.

5. We find a geometrical Higgs mechanism, where massless non-Abelian gauge bosons at coincident walls become massive as walls sepa-rate.

34