New Approaches to ElectroWeak Symmetry Breaking - LPSC

New Approaches to 

ElectroWeak Symmetry Breaking 

E = mc 2 

E = ¯hν 

Part Two 

Higgs as a (Pseudo) Golstone Boson: Little Higgs Theories. 

Higgs as a component of a Gauge Field: Gauge Higgs Unification. 

Ch!"ophe Grojean 

( christophe.grojean@cern.ch ) 

Rµν − 1 

2 Rgµν = 16πG Tµν 

CERN-TH & Service de Physique Théorique - CEA Saclay

How to Stabilize the Higgs Potential 

Goldstone’s Theorem 

spontaneously broken global symmetry massless scalar 

The spin trick 

a particle of spin s: 

... but the Higgs has sizable non-derivative 

couplings 

2s+1 polarization states 

...with the only exception of a particle moving at the 

speed of light 

... fewer polarization states 

Spin 1 Gauge invariance no longitudinal polarization 

Spin 1/2 

→ 

→ 

Chiral symmetry only one helicity 

→ 

... but the Higgs is a spin 0 particle 

→ 

m=0 

Ch!"ophe Grojean New approaches to ElectroWeak Symmetry Breaking 2 Lecture

Symmetries to Stabilize a Scalar Potential 

Supersymmetry 

Higher Dimensional 

Lorentz invariance 

fermion ~ boson 

Aµ ∼ A5 

4D spin 1 4D spin 0 

These symmetries cannot be exact symmetry of the Nature. 

They have to be broken. We want to look for a soft breaking in 

order to preserve the stabilization of the weak scale. 


Li%le Higgs 'eo!es 


U(1) 

φ = 1 

√ 2 (f + h(x)) e iθ(x)/f 

L = 1 

2∂µh∂ µ h + 1 

2 

Goldstone Boson 

φ → e iα φ 

L = ∂µφ † ∂ µ φ − λ 

f+h 

f 

h → h 

θ → θ + αf 

2 

φ 2 2 − 

U(1) 

non-linearly realized 

shift symmetry forbids any mass term 

for θ 

Ch!"ophe Grojean New approaches to ElectroWeak Symmetry Breaking 2 Lecture 

f 2 

2 

2 

∂µθ∂ µ θ − λ f 2 h 2 + fh 3 + 1 

4 h4 

If the U(1) symmetry is gauged, the Goldstone boson is 

eaten and it becomes the longitudinal component of 

the massive gauge boson

Example of Uneaten Goldstone Bosons 

SU(N) → SU(N − 1) 

〈φ〉 = 

⎛ 

⎜ 

⎝ 

0 

. 

0 

f 

⎞ 

⎟ 

⎠ 

φ = exp 

⎛ 

⎜ 

⎝ 

(N 2 − 1) − (N − 1) 2 − 1 = 2N − 1 

Let us assume that only SU(N-1) is gauged: then the Goldstone are uneaten. 

SU(N − 1) 

SU(N) 

SU(N − 1) 

i 

f 

⎛ 

⎜ 

⎝ 

−π0 

Goldstone bosons 

φ = e (N-1) complex, , and 1 real, , scalars 

iπ φ0 π π0 

φ → UN−1φ = UN−1e iπ U † 

φ → exp 

π → 

 

i 

UN−1 

α † 

α 

1 


. .. 

π1 

−π0 πN−1 

π⋆ 1 . . . π⋆ N−1 (N − 1)π0 

π0 π 

 

exp 

π † π0 

N−1UN−1φ0 = e 

 

† 

U N−1 = 

1 

linear transformations 

 

i 

π † 

π 

 

φ0 ≈ exp 

. 

⎞⎞ 

⎛ 

⎟⎟ 

⎜ 

⎟⎟ 

⎜ 

⎟⎟ 

⎜ 

⎠⎠ 

⎝ 

† 

iUN−1πU 

N−1φ0 

π0 

π † U † 

N−1 

 

i 

non-linear transformations 

π † + α † 

UN−1π 

π0 

 

π + α 

0 

. 

0 

f 

 

⎞ 

⎟ 

⎠ 

φ0

The Desired Properties of a Little Higgs 

Unbroken SU(2)xU(1) gauge symmetry 

At least 21/2 doublet as a Goldstone boson 

Break the G/H shift symmetry that protects the Higgs potential in a way 

to allow a quartic coupling and a mass term, as well as Yukawa interactions 

to guarantee the absence of quadratic divergences at one loop 

Implement the idea of Higgs = (Pseudo)-Goldstone boson 

Kaplan, Georgi ‘84 

Kaplan, Georgi, Dimopoulos ‘84 


Collective Breaking 

The global symmetry is explicitly broken but only “collectively” ! 

the symmetry is broken when two or more couplings in the Lagrangian 

are non-vanishing. 

Setting any one of these couplings to zero restores the symmetry 

and therefore the masslessness of the Little Higgs 

⎛ 

φ1 with 〈φ1〉 = f ⎝ 0 

⎞ 

0 ⎠ 

1 

and φ2 with 〈φ2〉 = f 

[SU(3)/SU(2)] 2 

2x(8-3)=10 Goldstone 

Let us now gauge SU(3) 

L = |Dµφ1| 2 + |Dµφ2| 2 

Dµφi = ∂µφi − igAµφi 

vev 

SU(3)V → SU(2)V 

the 5 Goldstone are eaten 

φi → Uiφi 

Arkani-Hamed et al. ‘02 

φi → Uφi and Aµ → UAµU † 


⎛ 

⎝ 0 

0 

1 

requires U1 = U2 

Break SU(3)1 − SU(3)2. Gauge SU(3)1 + SU(3)2. 

in presence of the gauge coupling, only 5 Goldstone 

The 5 extra would-be Goldstone boson acquire a mass through the gauge coupling 

⎞ 

⎠

[SU(3)/SU(2)] 2 

2x(8-3)=10 Goldstone 

Collective Breaking 

φ1 with 〈φ1〉 = f 

Let us now gauge SU(3) 

L = |Dµφ1| 2 + |Dµφ2| 2 

Dµφi = ∂µφi − igAµφi 

vev 

SU(3)V → SU(2)V 

the 5 Goldstone are eaten 


⎛ 

⎝ 0 

0 

1 

⎞ 

⎠ and φ2 with 〈φ2〉 = f 

φi → Uiφi 

φi → Uφi and Aµ → UAµU † 

⎛ 

⎝ 0 

0 

1 

requires U1 = U2 

Break SU(3)1 − SU(3)2. Gauge SU(3)1 + SU(3)2. 

in presence of the gauge coupling, only 5 Goldstone 

The 5 extra would-be Goldstone boson acquire a mass through the gauge coupling. 

As soon as the coupling of either φ1 to Aμ or φ2 to Aμ is set to zero, enlarged global 

symmetry and 5 Goldstone in the physical spectrum. 

If a diagram involves only one type of couplings, 

it cannot generate a mass term for the pseudo-Goldstone bosons 

There is no quadratically divergent diagrams at one loop. 

⎞ 

⎠

Λ 2 

Absence of Λ 2 divergent Diagrams 

SU(3) 

φ † 

i 

Log (Λ 2 ) 

φ2 

φ † 

1 

φi 

φ † 

2 

φ1 

δL = g2 Λ 2 

δL = g4 f 2 

16π 2 

16π 

2 (φ† 

1 φ1 + φ † 

2 φ2) = g2 f 2 Λ 2 

8π 2 

independent of the Higgs field! 


 

 

φ † 

1 φ2 

 

 

log Λ2 

Simplest Higgs 

Kaplan-Schmaltz ‘03 

Q2 = g4f 2 

Λ2 

log 

4π2 Q2 h† h 

log-divergent one-loop suppressed 

Higgs mass

Cancelation Mechanism of Λ 2 

 

φ1 = exp i 

 

φ2 = exp −i 

M 2 = g 2 f 2 

⎛ 

⎜ 

⎝ 

1 

h † 

h † 

1 

h 

h 

1 − v2 

f 2 

 

 

f 

 

≈ f 

 

≈ f 

ih/f 

1 − (h † h)/(2f 2 ) 

W’ cancels the quadratic divergences generated by W 


f 

1 − v2 

f 2 

v 2 

f 2 

v 2 

f 2 

−ih/f 

1 − (h † h)/(2f 2 ) 

4 

3 

− v2 

f 2 

− v2 

√ 3f 2 

− v2 

√ 3f 2 

trM 2 is independent of v 2 ⇔ no quadratic divergence 

v 2 

f 2 

 

 

⎞ 

⎟ 

⎠

q = 

tL 

φ1 ≈ f 

φ2 ≈ f 

How to Generate a Yukawa Coupling 

bL 

 

 

 

y y 

≡ 2 of SU(2) 

ih/f 

1 − (h † h)/(2f 2 ) 

−ih/f 

1 − (h † h)/(2f 2 ) 

Q = 


⎛ 

⎝ 

L = y √ φ 

2 † 

1QtR + y √ φ 

2 † 

 

 

x 

yf 

T− TL T+ 

tL 

− y 

2f 

L = yf 

tL 

bL 

TL 

2 QTR 

⎞ 

⎠ ≡ 3 of SU(3) 

 

1 − h† h 

2f 2 

 

with 

+ 

TLT+ + yh † qT− 

TR 

T+ = 1 

√ 2 (TR + tR) T− = i 

√ 2 (TR − tR) 

m 2 t = y 2 v 2 

m 2 T = y2 f 2 

 

1 − v2 

f 2 

trM 2 is independent of v 2 ⇔ no quadratic divergence 

heavy top cancels the quadratic divergences generated by top

Collective Breaking in the Top Sector 

Without Yukawa 

L = y √ 2 φ † 

1 QtR + y √ 2 φ † 

2 QTR 

φi → Uiφi 

SU(3) 2 

Global Symmetry 

When both Yukawa are turned on φi → Uφi and Q → UQ 

we need both Yukawa to align the two SU(3) 

more than one coupling is required to break enough global symmetry 

to let the Higgs acquire a potential 

φ1 

no quadratically divergent diagram at one-loop ! 

tR 

Q 

φ1 

φ2 

Q 

TR 

φ2 

 


d 4 k 

1 

(kµγ µ ∝ log Λ2 

− m) 4 

log-divergent contribution only

1. Bottom Yukawa: 

Exercises 

Check that the bottom mass is generated by 

2. Hypercharge: 

y φ1φ2QbR 

Show that you can introduce the hypercharge by simply 

extending the gauge symmetry to SU(3)xU(1)X 

hint: first determine the X charge of the two scalar triplets. 


EWSB in Little Higgs 

At tree-level, no Higgs potential 

(well, a quartic coupling can be generated by integrating out another heavy Goldstone) 

A quartic and a mass term are generated at one-loop. 

As in susy, the top loop will generate a log-divergent tachyonic mass term 

that will trigger EWSB. 

Radiatively induced EWSB 

(that’s the dynamics of the model that drives the symmetry breaking, it is not arbitrary like in the SM) 

What 

we want 

Might be 

difficult 

v 2 = −m2 

2λ 

φ † 

1 φ2 = f 2 

large quartic (larger than v/f) in order to get 

a physical Higgs mass of order v and not v 2 /f. 

small mass term (of order v and not f) 

 

1 − 2 h† h 

f 2 + 2(h† h) 2 

3f 4 

Would be better to generate a tree-level quartic 


 

v ∼ f

Little Higgs with Tree Level Quartic 

Global symmetry 

Gauge symmetry 

v ∼ f/4π 

f 

SU(3) 

Σ1 

Σ2 

SU(2) 

x 

U(1) 


Σ3 

Σ4 

 

SU(3)L × SU(3)R/SU(3)V 

4 

SU(3) × SU(2) × U(1) → SU(2)l × U(1)y 

30, 21/2, 10 

not protected by 

collective breaking 

two complex doublets 

1 complex triplet 

1 complex singlet 

32 GB 

8 eaten 

24 PGB 

Will generate a quartic coupling 

for the light PGB 

Minimal moose model 

Arkani-Hamed, Cohen, Katz, Nelson, Gregoire, Wacker ‘02

" 

" 

f 

v ∼ f/4π 

4πf 

EW Precision Tests 

1 TeV 

100 GeV 

10 TeV 

!/2 

UV completion 

22 18 

4 

14 10 New particles (W’, Z’,top’...) 

!/4 

!/4 

SM particles (Higgs,W,Z...) 

The new particles @ a TeV will affect 0 the EW precision 0 observables 

!/2 

!/4 

0 

!/2 

!/4 

22 

22 

SU(5) “littlest” Higgs model 

18 

18 

14 

14 

10 

0 !/4 !/2 

"' 

14 

10 

6 

SU(6) little Higgs model 

12 

10 

8 

6 

6 

"R 

!/2 

!/4 

8 

SO(9) little Higgs model 

6 

10 

4 

0 

0 !/4 !/2 

!/2 

"L 

8 

10 

6 

2 

4 

2 

3 

is computed at 99% C.L. for 1 dof, i.e. χ 

1 

Bounds from precision data on the scale f (in TeV) 

2 

for various Little Higgs models. 

2 = χ2 SM 

" 

!/4 

Incomplete SU(6) little Higgs model 

3 

4 

f>3-4 TeV 5 instead of f ~ 1 TeV 

6 

0 !/4 

"' 

!/2 

Ch!"ophe Grojean New approaches to ElectroWeak Symmetry Breaking 7 

2 Lecture 

" 

" 

!/2 

!/2 

!/4 

0 

22 

14 

SU(5) “littlest” Higgs model 

18 

14 

10 

6 

SU(6) little Higgs model 

12 

10 

8 

0 !/4 !/2 

"' 

6 

6 

"R 

" 

!/2 

!/4 

0 

8 

SO(9) little Higgs model 

6 

10 

0 !/4 !/2 

"L 

0 !/4 !/2 

2# 

8 

10 

Incomplete SU(6) little Higgs model 

Marandella, Schappacher, Strumia ‘05 

Figure 1: Bound from precision data on the scale f in TeV of little-Higgs models. The constraint 

+ 6.6. As described in the text in each model the 

angles φ parameterize the gauge couplings of the extra gauge groups, which become strongly coupled 

at φ → 0 and/or φ → π/2. The dotted iso-lines show that the constraint on f gets slightly relaxed 

in presence of arbitrary extra corrections to ˆ T . We assumed a light higgs, mh ∼ 115 GeV. 

3 

2 

6 

3 

4 

5 

6 

2 

1 

4 

2

Custodial Symmetry and T-parity 

To improve EW fits, two useful tools: 

little Higgs with custodial symmetry 

T-parity 

Chang, Wacker ‘03 based on minimal moose with SO(5)... 

Cheng, Low ‘03 

heavy particles = odd 

light particles = even 

at each vertex, an even number of heavy fields are required 

no effective operator for light fields are generated 

by tree-level exchange of heavy fields 

tree-level exchange forbidden loop exchange allowed 


Fine-Tuning in Little Higgs 

! 

1000 

100 

10 

MSSM 

Littlest 2 

Littlest 

100 150 200 250 300 350 400 

m (GeV) 

h 

T-parity 

Simplest 

Figure 13: Comparative summary of the fine-tuning vs. mh for different scenarios. The curves for Little 

Casas, Espinosa, Hidalgo ‘05 

Higgs models (lines labeled “Littlest”, “Littlest 2”, “T -parity” and “Simplest”) are lower bounds on the 

corresponding fine-tuning, see text for details. 

corresponding to refs. [12, 13, 14, 15]. 


SM

from the diagrams involving the BH. Certain choices of 

the fermion charges and the angle ψ ′ can help minimize 

these constraints [6]; alternately, the extra U(1) can be 

eliminated completely [9] without introducing significant 

fine tuning due to the smallness of the coupling g ′ . Given 

this model uncertainty in the U(1) sector, we will con- 

Experimental Signatures 

little Higgs models require a heavy top and heavy gauge bosons 

centrate on the production and decay of the SU(2) heavy 

bosons. 

In pp collisions at the LHC, the heavy gauge bosons are 

predominantly produced through q¯q annihilation. The 

sub-process qg → WHq is considerably smaller and could 

be separately identified due to the presence of a high 

to guarantee the cancellation of the Λ2 divergences, the couplings 

of the new gauge bosons to the SM fermions is fixed 

pT jet. Fig. 3 shows the leading order production cross 

section of WH as a function of its mass, for the case 

ψ = π/4. The general case may be obtained from Fig. 

by simply scaling by cot2 ψ. 

Burdman, Perelstein, Pierce ‘02 

100 fb-1 (three years of LHC) 

FIG. 3. Production cross sections for W 3 H (solid) and W ± 

H 

(dashed) at the LHC, for ψ = π/4. We use the CTEQ5L 

parton distribution function. 

production cross section 

of heavy W’ and Z’ 

From our discussion thus far, it is clear that the twobody 

decay channels of the W 3 H include W 3 H → ¯ ff, where 

Partial width of ZH to fermions 

is proportional to cotan 2 θ 

Partial width of ZH into boson pairs 

(Zh and W + W - ) 

is proportional to cotan 2 2θ 

(this follows from the particuliar coupling of the 

Higgs to the two SU(2) gauge groups) 

cotan θ=g1/g2 (two SU(2) gauge couplings) 


Gau)-Higgs Unification 


Symmetries to Stabilize a Scalar Potential 

Higher Dimensional 

Lorentz invariance 

Aµ ∼ A5 

4D spin 1 4D spin 0 

These symmetries cannot be exact symmetry of the Nature. 

They have to be broken. We want to look for a soft breaking in 

order to preserve the stabilization of the weak scale. 


How to Get a Doublet from an Adjoint 

Consider a 5D 

gauge symmetry G 

1 

2 

H ∼ A a 5 

will belong tho the adjoint rep. of G 

The SM Higgs is not an adjoint of SU(2)xU(1), it is a doublet ! 

⎛ 

⎜ 

⎝ 

SU(3) 

Consider a bigger gauge group 

G → SU(2)l × U(1)y 

Adj doublet + other rep. 

W3 + W8/ √ 3 W1 − iW2 W4 − iW5 

W1 + iW2 −W3 + W8/ √ 3 W6 − iW7 

W4 + iW5 W6 + iW7 −2W8/ √ 3 

Adj 


⎞ 

⎟ 

⎠ 

SU(2)xU(1) 

2 √ 3/2

h+ = W4 − iW5 

h0 = W6 − iW7 

δT W = g 1 

2 √ 3 

⎛ 

⎝ 

U(1) Charge of the Doublet 

δT W = g [T, W ] T = 1 

2 √ 3 

1 

1 

−2 

= g 1 

2 √ 3 

⎞ ⎛ 

⎠ ⎝ 

⎛ 

⎝ 

−2h ⋆ + 

h ⋆ + h ⋆ 0 

−2h ⋆ 0 

= g 3 

2 √ 3 

h+ 

h0 

⎛ 

⎝ 

⎛ 

⎝ 1 


⎞ 

⎠ − g 1 

2 √ 3 

h+ 

h0 

⎞ 

⎛ 

⎝ 

⎠ − g 1 

2 √ 3 

−h ⋆ + −h ⋆ 0 

h ⋆ + h ⋆ 0 

⎛ 

⎝ 

h+ 

h ⋆ + 

h+ 

U(1) charge of the doublet = 3 

2 √ 3 

h0 

⎞ 

⎠ 

h ⋆ 0 

h0 

1 

⎞ ⎛ 

⎠ ⎝ 

−2h+ 

−2h0 

−2 

1 

⎞ 

⎠ 

⎞ 

⎠ 

1 

−2 

⎞ 

⎠

Weak Mixing Angle 

δ U(1) 

h+ 

h0 

 

= 3g 

2 √ 3 

h+ 


h0 

 

= 

√ 3g 

2 

h+ 

Proper U(1) normalization 

(such that the doublet has a hypercharge 1/2) g ′ = √ 3g U(1)y = T8/ √ 3 

sin 2 θW = g′2 

g2 3g2 

= 

+ g ′2 g2 3 

= 

+ 3g2 4 

By embedding SU(2)xU(1) into a simple group, we got a 

prediction for the weak mixing angle 

experimentally: sin 2 θW ≈ 0.23 

h0 

 

Manton ‘79 

Fairlie ‘79

How to Get a Good Weak Mixing Angle 

Repeat the previous construction with another gauge group 

G2 gauge group 

adjoint = 14 fundamental = 7 

SU(3) decomposition 

14 = 8 + 3 + ¯3 7 = 3 + ¯3 + 1 

T8 normalization 

14 = 

 

30 + (2 + ¯2) √ 3/2 

U(1)y normalization 

14 = 

 

+ 10 

30 + (2 + ¯2) 3/2 + 10 

 

+ 

 

+ 

SU(2)xU(1) decomposition 

 

2 + ¯2 

1/2 √ 3 + 

 

1 + ¯1 

−1/ √ 3 

 

2 + ¯2 

1/2 + 

U(1)y = √ 3 T8 

 

1 + ¯1 

−1 


7 = 

7 = 

Manton ‘79 

Csáki, Grojean, Murayama ‘02 

 

2 + ¯2 

1/2 √ 3 + 

 

1 + ¯1 

−1/ √ 3 

 

2 + ¯2 

1/2 + 

 

1 + ¯1 

−1 

sin 2 θW = 1/4 

+ 10 

+ 10

fundamental rep. 

14 7x7 matrices 

T 9 = i 

2 √ 3 

λ 1 = 

⎛ 

⎝ 

λ 4 = 

⎛ 

⎝ 

0 1 0 

1 0 0 

0 0 0 

⎛ 

⎝ 

λ 7 

v 1† 

⎞ 

0 0 1 

1 0 0 

0 0 0 

⎠ λ 2 = 

⎞ 

v 1 = 

T a = 1 

2 √ 2 

v 1 

⎛ 

⎝ 

⎠ λ 5 = 

⎛ 

⎞ 

G2 Gymnastic 

0 −i 0 

i 0 0 

0 0 0 

⎛ 

⎝ 

⎝ −i√ 2 

0 

0 

⎛ 

⎝ 

0 0 −i 

0 0 0 

i 0 0 

⎞ 

λ a 

⎠ T 10 = −i 

2 √ 3 

⎠ v 2 = 

−λ at 

⎛ 

⎝ 

with 

⎞ ⎛ 

⎠ λ 3 = 

⎞ 


⎝ 

⎠ λ 6 = 

⎛ 

⎝ 

λ 5 

0 

i √ 2 

0 

⎞ 

⎠ for a = 1 . . . 8 

v 2† 

1 0 0 

0 −1 0 

0 0 0 

⎛ 

⎝ 

⎞ 

v 2 

⎠ v 3 = 

⎞ 

⎠ T 11 = i 

2 √ 3 

T 12 = T 9† T 13 = T 10† T 14 = T 11† , 

0 0 0 

0 0 1 

1 0 0 

⎛ 

⎝ 

⎞ 

⎠ λ 8 = 1 

√ 3 

⎞ 

⎠ λ 7 = 

0 

0 

−i √ 2 


⎞ 

⎠ . 

⎛ 

⎝ 

⎛ 

⎝ 

⎛ 

⎝ 

λ 2 

v 3† 

1 0 0 

0 1 0 

0 0 −2 

0 0 0 

0 0 −i 

0 i 0 

⎞ 

⎠ 

v 3 

⎞ 

⎠ 

⎞ 

⎠

6D for a Tree-level Quartic Coupling 

L = − 1 

4 F 2 µν + 1 

2 F 2 µ5 + 1 

2 F 2 µ6 − 1 

2 F 2 56 

F56 = ∂5A6 − ∂6A5 − g[A5, A6] 

A5 = H + . . . A6 = H + . . . 

L = −g 2 H 4 

Antoniadis, Benakli, Quiros ‘01 

the 6D gauge kinetic term contains a Higgs quartic coupling 

and 

λ ≡ g 2 


SU(3) model 

A5 = 

Explicit Tree-level Quartic 

⎛ 

⎜ 

⎝ 

1 

2 h⋆ + 

Aµ = 1 

2 

⎛ 

⎜ 

⎝ 

1 

2 h⋆ 0 

1 

2 h+ 

1 

2 h0 

A 3 µ + 1 

3 A8 µ 

A 1 µ + iA 2 µ 

⎞ 


⎛ 

⎟ 

⎠ A6 

⎜ 

= ⎜ 

⎝ 

A 1 µ − iA 2 µ 

−A 3 µ + 1 

3 A8 µ 

i 

2 h⋆ + 

i 

2 h⋆ 0 

− 2 

√ 3 A 8 µ 

−i 

2 h+ 

−i 

2 h0 

L = − 1 

2 Tr F 2 µν + Tr F 2 µ5 + Tr F 2 µ6 − Tr F 2 56 

L = DµH † DµH − g2 

2 (H† H) 2 

⎞ 

⎟ 

⎠ 

⎞ 

⎟ 

⎠ 

after orbifold 

projection...

πR 

−πR 

Towards a Complete Construction 

so far we haven’t broken any symmetry... we even enlarged the gauge group 

y 

−y 

We need to break G down to SU(2)xU(1) 

we can achieve this breaking while compactifying the extra-dimension 

0 

Orbifold Compactification Breaking 

circle y ∼ y + 2πR 

⤶ 

Z2 orbifold y ∼ −y 

U 2 = 1 

Aµ(−y) = UAµ(y)U † A5(−y) = −UA5(y)U † 

zero mode: is independent of 

Aµ 

Aµ = UAµU † 


y 

A5 = −UA5U † 

- signs are compensating 

gauge symmetry breaking + chiral fermions

Orbifold Projection as Boundary Conditions 

H subgroup 

A H µ (−y) = A H µ (y) 

A H 5 (−y) = −A H 5 (y) 

G/H coset 

A G/H 

µ 

A G/H 

5 

−πR 

(−y) = −A G/H 

µ 

(−y) = A G/H 

5 

y 

−y 

G → H by orbifold projection 

(y) 

(y) 

⤶Z2 

which is equivalent to the BCs 

at the fixed points 

which is equivalent to the BCs 

at the fixed points 

∂5A H µ = 0 

A H 5 = 0 

A G/H 

µ 

∂5A G/H 

5 

πR 

0 

U 2 ≈ 

= 1 BC BC 


= 0 

= 0

SU(3) → SU(2)xU(1) 5D Orbifold Breaking 

massless Aµ 

[Aµ, U] = 0 Aµ = 1 

2 

massless A5 

U = 

⎛ 

⎝ −1 

⎛ 

⎜ 

⎝ 

{A5, U} = 0 A5 = 1 

2 

A 3 µ + A 8 / √ 3 A 1 µ − iA 2 µ 

⎛ 

⎜ 

⎝ 

−1 

A 1 µ + iA 2 µ 

1 

A 4 5 + iA 5 5 

−A 3 µ + A 8 µ/ √ 3 

A 6 5 + iA 7 5 

A 4 5 − iA 5 5 

A 6 5 − iA 7 5 

−2A 8 µ/ √ 3 


⎞ 

⎠ U ∈ SU(3) U 2 = 1 

⎞ 

⎟ 

⎠ 

⎞ 

⎟ 

⎠ 

SU(2) × U(1) 

SU(3) 

SU(2) × U(1)

! 

2 R 

0 

U = 

⎛ 

⎝ i 

z 

i 

0 

U 4 = 1 

T O 

−1 

⎞ 

⎠ 

6D Orbifold Breaking 

! 

2 R 

Ay = 

⎛ 

⎜ 

⎝ 

Fundamental domain 

of the torus 

Fundamental domain 

y 

1 

2 h⋆ + 

of the orbifold 

Fixed points 

of the orbifold 

T 2 /Z 4 orbifold 

y 

z 


1 

2 h⋆ 0 

1 

2 h+ 

1 

2 h0 

⎞ 

 

∼ 

−z 

y 

 

Aµ(−z, y) = UAµ(y, z)U † 

Ay(−z, y) = −UAz(y, z)U † 

Az(−z, y) = UAy(y, z)U † 

⎛ 

⎟ 

⎠ Az 

⎜ 

= ⎜ 

⎝ 

i 

2 h⋆ + 

i 

2 h⋆ 0 

−i 

2 h+ 

−i 

2 h0 

⎞ 

⎟ 

⎠

In the bulk 

Residual Symmetry 

higher dimensional gauge invariance forbids a mass term for A5 

At the fixed points 

G → H 

δA A M = ∂M ɛ A + gf ABC A B µ ɛ C 

⎧ 

⎨ 

⎩ 

∂5A H µ = 0 A H 5 = 0 ∂5ɛ H = 0 

A G/H 

µ = 0 ∂5A G/H 

5 = 0 ɛ G/H = 0 

(since [H, H] ⊂ H [H, G/H] ⊂ G/H, 

the only non-vanishing structure constants are f HHH , f G/H G/H H ) 

δA H µ = ∂µɛ H + gf HHH A H µ ɛ H 

δA H 5 = 0 

δA G/H 

µ 

δA G/H 

5 

= 0 

= ∂5ɛ G/H + gf G/H G/H H A G/H 

5 

G/H acts non-linearly on the Higgs at the fixed points. 

No local mass term allowed 


ɛ H 

Gersdorff, Irges, Quiros ‘02

Finite Mass from Wilson Line 

there is no local counter term that will give a mass to the Higgs 

the mass term can only be generated non locally 

the mass term will be finite then 

Wn = e i R 2nπR 

0 A5dy Wilson’s line. Gauge invariant object. 

The radiative corrections will generate a (finite) effective potential 

Veff (A5) = f(Wn) 

controlled by finite size effects 

UV insensitive 


Tadpole Operator 

the situation complicates in 6D 

F56 contains a mass term for the Higgs 

Tr (U F56) is a local invariant operator at a fixed point 

Tr (UF56(0)) → Tr (Ug(0)F56(0)g −1 (0)) = Tr (UF56(0)g −1 (0)g(0)) = Tr (UF56(0)) 

gauge loop 

at the fixed points, U and g commute 

Unless forbidden by a discrete symmetry, 

we expect this operator to be generated at one loop 

ghost loop 


Gersdorff, Irges, Quiros ‘02 

For Z2 or Z2 x Z2 orbifolds can define a parity 

that forbids the tadpole 

For Z4 orbifold, no such symmetry 

a Λ 2 tadpole is generated at one loop 


Direct coupling: 

Non local couplings: 

Fermion Masses 

4D chiral matter U(1)y fractional charges 

Higgs Coupling 

 

localize fermions at the fixed points 

Λ 2 divergences 

forbidden by G/H shift symmetry 

W = Pe i R Aidx i 

Extra Massive Fermion in the Bulk 

d 5 x ¯ ψ(i¯σ µ ∂µ − m)ψ + δ(y1)ψχ + δ(y2)ψξ 

integrating out 

massive fermion 

 

d 4 x e −m|y1−y2|) 

ξ(y2)e i R Aidx i 

χ(y1) 

Froggatt-Nielsen 

mass suppression 


Experimental Signatures 

the collider signals haven’t been studied in details 

(lack of fully realistic models?) 

the main predictions of these models are 

KK excitations of W,Z around 500 GeV ~ 1 TeV 

KK excitations of G/H: gauge bosons with the quantum numbers of the Higgs doublets 

extra scalar fields 

extra fermions (to cancel the top loop quadratic divergence) 


In 6D models 

enlarge the gauge group? 

Open Issues 

start with SU(3) and add kinetic terms at the fixed points to change the 

prediction of the weak mixing angle? 

In 5D models 

add large boundary kinetic terms 

add appropriate matter fields in the bulk 

warped space 

generate a large Higgs quartic 

good prediction for the weak mixing angle 

generate a large Higgs quartic coupling at one-loop 

Seems to be a promising way to proceed. 

Weakly coupled dual of composite Higgs models 

See Agashe, Contino, Pomarol ‘04-’05

New Approaches to ElectroWeak Symmetry Breaking - LPSC

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