Review of gauge-Higgs unification models

Naoyuki Haba 　 ( 波場直之 )

TIS2005, Taiwan, 6/10/2005

(Tokushima Univ.)

　　　　　 Plan of talk

1. models of gauge-Higgs unification

2. electro-weak symmetry breaking （１）

3. electro-weak symmetry breaking （２）

4. Higgs phenomenology

5. summary & discussion

Q: Can origin of Higgs be extra component of gauge field? 　　　 → gauge invariance guarantees the smallness of “Higgs” mass (against quantum c)!A: Yes, we can do it in the extraD gauge theory. 　 → 5th component of gauge field (A5) ＝ 4D scalar in eff. theo.　　　　　　　　　　　　　　　　　　　　　　　　　　　　 ⇒ Regard “Higgs”!

5 24A

1S

regard it adjoint Higgs which breaks SU(5) GUT

4D

5 (5)D SU GUT

( 0~ , )53MA

(Hosotani, etal)

μ

(ex)

What we want is not Σ but SM Higgs doublet which breaks SU(2)×U(1) today.

（ radi:10-29mm （ 1016 GeV) ）

4D

5 (3), (6)D SU SU

( 0~ , )53MA

12/S Z

5 8,35( )DH A

5 55 5cD Dg A

4D

⇒ origin of “Higgs doublet” (zero mode)

⇒ origin of Yukawa interactionR L

μ

Gauge-Higgs unification

“Higgs doublet” mass is finite. ( ～ 1/R)

5D gauge symmetry

(ex)

（ radi:10- １６ mm （ 1TeV) ）

( ) ( )1

( , )2

niR

n

n yx x e

Ry

4 1(1) : M S

0y

4D

R

: ( , 2 ) ( , )x y R yT xT

[ ( )]T U N

preparation （ notation ）

,0

( )

( )

0

1

cos( )

sin(

1( , )

2

1( ( ) )

(

, )

)n

n

n

n

n

nyy

R

n

Rx x

xR

yy

Rx

42

1(2) : /M S Z

0y

4D

: ( , ) ( , )P Px y x y

y y

y R

4D

0y y R

2 2[ 1 ( ) ( ) ( )]P y P y P y

preparation （ notation ）

,0

( )

( )

0

1

cos( )

sin(

1( , )

2

1( ( ) )

(

, )

)n

n

n

n

n

nyy

R

n

Rx x

xR

yy

Rx

42

1(2) : /M S Z

0y

4D

: ( , ) ( , )P Px y x y

y y

y R

4D

0y y R

2 2[ 1 ( ) ( ) ( )]P y P y P y

5 5

( , ) ( , )

( , ) ( , )

( , ) ( , )

( , ) ( , )

L L

R R

P P

P P

A x y A x y

A x y A x y

x y x y

x

P

y yP x

[ ( , ) ( , )]yx y Pi x y 55 : ( , )MD i

preparation （ notation ）

,0

( )

( )

0

1

cos( )

sin(

1( , )

2

1( ( ) )

(

, )

)n

n

n

n

n

nyy

R

n

Rx x

xR

yy

Rx

42

1(2) : /M S Z

0y

4D

: ( , ) ( , )P Px y x y

y y

y R

4D

0y y R

2 2[ 1 ( ) ( ) ( )]P y P y P y

perparation （ notation ）

zero mode (remaining field in the low energy ）

E

1/ R

0

2/ R

3/ R

,0

( )

( )

0

1

cos( )

sin(

1( , )

2

1( ( ) )

(

, )

)n

n

n

n

n

nyy

R

n

Rx x

xR

yy

Rx

42

1(2) : /M S Z

0y

4D

: ( , ) ( , )P Px y x y

y y

y R

4D

0y y R

2 2[ 1 ( ) ( ) ( )]P y P y P y

preparation （ notation ）

E

1/ R

0

2/ R

3/ R

1. models of gauge-Higgs unification

(1). SU(3)×SU(3) model(2). SU(6) model

cos( )ny

R

1 1

1 1

1 1

P T

A

5A

sin( )ny

R

(1). SU(3)c×SU(3)W model

in base of(3) (2) (1)W L YSU SU U

(Kubo,Lim,Yamashita,Hall,Nomura,Smith,Burdman,Nomura,….)

5 5

( , ) ( , )

( , ) ( , )

A x y A x y

A

P P

Px y A x y P

cos( )ny

R

1 1

1 1

1 1

P T

A

sin( )ny

R

in base of(3) (2) (1)W L YSU SU U

(3) (2) (1)W L YSU SU U

Higgs doublet

Higgs doublet

5A

(Kubo,Lim,Yamashita,Hall,Nomura,Smith,Burdman,Nomura,….)

(1). SU(3)c×SU(3)W model

cos( )ny

R

1

1

1

1

1

1

P

A

sin( )ny

R

1

1

1

1

1

1

T

(2). SU(6) model

(Hall,Nomura,Smith, Burdman,Nomura)

in base of (6)SU

5A

1

1

1

1

1

1

P

A

1

1

1

1

1

1

T

(6)

(3) (2) (1) (1)

SU

SU SU U U

Higgs doublet

Higgs doublet

(Hall,Nomura,Smith, Burdman,Nomura)

in base of (6)SU

5A

(2). SU(6) model

2. electro-weak symmetry breaking （１）

　　 -“Higgs doublet” can really take VEV or not?-

(1). SU(3)×SU(3) model(2). SU(6) model(3). Introduction SUSY

NH, Y. Hosotani, Y. Kawamura and T. Yamashita, Phys.Rev.D70:015010, 2004 NH and T. Yamashita, JHEP 0402:059,2004

2 2 4( ) | | | |V m

0 2462

mVGeV

wanted potential is (at least) up to λ2 、 λ4

5( )V A

5A

at tree level

However, since it is originally gauge field,

Let’s estimate quantum corrections !

(0) (0) (0) (0)5, , /A A q l

(method ）Sum of infinite# of diagram （ KK ） → obtain 　　　　　　　　　

→ 　 search the vacuum of 　　　　　　 →　 whether “Higgs” 　　　　　　　　　　　　　

(0)5( )effV A

(0)5A (0)

5A

( ) ( ) ( ) ( )5, , /n n n nA A q l

(0)5 0, 0A or

’s 1 loop quantum corrections(0)5 ( )A

(0)5( )effV A

effective potential (gauge contribution ） :

6 4

3

32C

R

(0)5

1

2gR

a

A

a

51

3 1[cos(2 ) cos( )]

2gauge

effn

V C na nan

( / 2)C no symmetry breaking!

(1).SU(3)c×SU(3)W model

a

4 55 5

6 75 5

5

4 5 6 75 5 5 5

2

2

2 2

A iA

A iAA

A iA A iA

(0 2)a / 2 R

a=1 is acceptable if life timeof universe is long enough?

is physical d.o.f. ？(0)5A

5exp( )CW P ig dA y [ , ] [ , ] 0a C aU T U PW

(0) (0)5 5

† †(0)5 ( ) ( ) (' ( ) )yA Ay y

gy y

iA

(0)5

1

2a

RA T

g

2( )y

i a TRy e

(0)

5

1

20a T

gRA

†( 2' ) ( )y RT T yT ( )2 2

y yi a T i a T

R Re T e

†( )' ( )P Py PP y

Q: （ cf. is not! It is gauged away (would-be NG) ）( )5

nA

： Wilson line phase

(05

05

) ( ) 0 '0A base wit A base wit Th T h

　　　 is the order parameter of symmetry breaking

A:

remaine!.

(0 1)a

more accurately, gauge symmetry which satisfies

(Abe, NH, Matsunaga etal)

If the vacuum exist at a=1,

(2) (1) (1) (1)SU U U U

5

2

0

0 0 1 1 0 01

exp exp 2 0 0 0 0 1 02

1 0 0 0 0 1

R

CW ig dy ig RgR

T AT

'

1 1 1 1

1 , 1 1 , 1

1 1 1 1

'W WTP P T

1

1

1

P

mean the remaining gauge symmetry is （ although <A5>≠0 ）

(0)5

1

2

a

A

agR

It is no good!

base also shows(0)5 0A

0a 1a

region is good order parameter. Since “Higgs doublet” picture (STU) is good & 246 GeV ≪ 1/R is consistent.

0 1a V

Anyhow, only the gauge contribution is not enough for the suitable vacuum.!

4D 4D

0y y R

( ) ( ) ( ), ,a f sN N N

5 5

( , ) ( , )

( , ) ( , )

( , ) ( , )

( , ) ( , )

(

( , ) ( ,

)

)

( )

( )

L L

R R

P P

P P

A x y A x y

A x y A x y

x y x y

x y x y

P

s x y s x y

P

fermion (adj. & fund.) scalar (fund.)

☆ let us introduce extra bulk fields.

term is added.

effective potential: (0)5( ) gauge

eff effV A V

a

a

( ) ( ) ( ) ( ) ( ) ( )2, 8, 4, 2, 0a f s s a fN N N N N N

effective potential:

a

a

(ex)

(0)5( ) gauge m

eff eff effV A V V

( ) ( ) ( ) ( ) ( ) ( )2, 8, 4, 2, 0a f s s a fN N N N N N

effective potential:

a

a

(ex)

(0)5( ) gauge m

eff eff effV A V V

a

effective potential:

a

(0)5( ) gauge m

eff eff effV A V V ( ) ( ) ( ) ( ) ( ) ( )2, 8, 4, 2, 0a f s s a fN N N N N N (ex)

effective potential:

a

(0)5( ) gauge m

eff eff effV A V V ( ) ( ) ( ) ( ) ( ) ( )2, 8, 4, 2, 0a f s s a fN N N N N N (ex)

1(1)O TeV

R

a

OK !

effects of extra bulk field

(0)5

4

2

/ 2 246

R A

a g R GeV

(2) (1) (1)emSU U U 2

2 24 0.0582

2 2 244

( ) |

0.031( ) (130 )

effa

Vm g R

a

gg GeV

R

4( / 2 )g R g

1

1

1

1

1

1

P

1

1

1

1

1

1

T

(2). SU(6) model

51

3 1[cos(2 ) 2cos( ) 6cos( ( 1))]

2gauge

effn

V C na na n an

2 2(3) (2) (1) (4) (1)SU SU U SU U

not good! → introduction extra bulk field

0a 1a

E

1/ 2R

1/ R

0

3/ 2R

effective potential (gauge contribution ） :

V

,0

( )

0

( )

0

( )

1

( )

1

1( , ) ( ) ( )

2

1( , ) ( ) ( )

1( , ) ( ) ( )

1(

1/

, ) ( ) ( )

cos

cos

sin

2

2

s

1/

in

n

n

n

n

n

n

n

n

n

nyx y x

RR

nx y x y

RR

nx y x y

RR

nx y x y

RR

2

0

exp2

1

1

11exp 2

12

1

1

R

C

aW ig dy

gR

ig RgR

T T

T

( ) ( ) 2, 0a fN N other Ns

a

effective potential:(0)5( ) gauge m

eff eff effV A V V

(ex)

( ) ( ) 2, 0a fN N other Ns

a

effective potential:(0)5( ) gauge m

eff eff effV A V V

(ex)

1(1)O TeV

R

22 2 2 2

4 0.072 42( ) | (130 )eff

a

Vm g R g GeV

a

(3). Introduction of SUSY

A

5( )iA

L c

RL

cR

h

5D N=1 SUSY

odd dim.=vector-like

⇔ 4D N=2 SUSY

V

L

R

.5

5

4

2

4 [ ( )

{ ( ( ) ) . .}]

hyp V VR LD R

L

L

R y

S d xd

cd

y d

hg

e e

Yukawa interaction

( ｇ ～ ytop ～ 0.7 when 1/R ～ GUT)

motivation of introducing SUSY ： ☆ write all couplings by gauge coupling ☆ dark matter☆ forbidden dangerous higher order operators (Yukawa among extra bulk fields)

SUSY: introducing particles which have the same masses but different spin as 1/2 (ex.) gauge (1) ⇔ gaugino (1/2), Higgs (0) ⇔ higgsino (1/2), quark (1/2) ⇔ squark (0), ･･･

if SUSY is not broken, potential is flat → 　 Scherk-Schwarz SUSY breaking

4 5 4 5 6 7 6 75 5 5 5

4 5 4 5 6 7 6 75 5 5 5

/ 2 (1/ 2)( ,( ) ( ))

/ 2 (1/ 2)( ,( ) ( ))

u

d

H A i A A i A

H A i A

R

A AR i

4 55 5

6 75 5

5

4 5 6 75 5 5 5

2

2

2 2

A iA

A iAA

A iA A iA

4 5

6 7

4 5 6 7

2

2

2 2

i

i

i i

☆ Higgs doublets: SUSY requires ２ HD (anomaly cancellation, holomorphy)

2 2 25 5 5[ , ] ( [ , ])treeL g tr A ig A

4( )2

gg

R

V

uH

dH

at tree level

A

5( )iA

L c

RL

cR

h

twist of SU(2)R as exp(2πiβσ2)

2 2( ) | |R

2R

1 loop effective potential → EWSB is not realized only by gauge contributionalso in SUSY case → introduction of extra bulk fields (hyper-multiplet) can do (ex.)

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　 (SS SUSY breaking parameter β=0.1) 　　 Nf

(±) (fund.) & Na(±) (adjo.)

4D 4D

0y y R

gauge

quarks/leptons

5exp" )" (Higg P A ys d

( ( 2 ) )ye e

( ) ( ) ( ) ( ) 24

( ) ( ) ( ) ( ) 24

(3) (3) : 2, 4, 0 ( 130 )

(6) : 2, 0, 10 ( 130 )

a a f f

a a f f

SU SU N N N N m g GeV

SU N N N N m g GeV

25 5 5[ ] . .cD DL h c

extra matters

We have seen (by introducing extra bulk field) A5 can play a role of

Higgs doublet, SU(2)×U(1)→U(1)emSo how about inducing Yukawa int. （ｇ＝ｙ） ? (this is 2nd motivation)]

For this perpose, as suggests

quark/lepton must be in the bulk.

（ ex. ） A5 can’t couple with ４ D brane field

If we set then it is possible (but in this case Higgs is non-local field.)

3. electro-weak symmetry breaking （ 2 ）

-Can “Higgs” take VEV when quark/lepton are in bulk?-

show here example of SUSY SU(3)×SU(3) model

NH and T. Yamashita, JHEP 0404 (2004) 016

cos( )ny

R

A

5( )iA

sin( )ny

R

L c

R

L

cR

fund. rep. bulk field

3 → down-Yukawa 6 → up-Yukawa 10 → charged lepton-Yukawa 8 → ν-Yukawa

(Burdman-Nomura)

gauge sector

SU(3)c×SU(3)W model

25 5 5[ ] . .cD DL h c

Yukawa

h

quark/lepton’s contribution to the effective potential :

effective potential of gauge sector & quark/lepton ：

(Ng: generation#)

2(2) (1) (1)L YSU U U 0a 1a

V

51

(0) (1)

14( ) (1 cos(2 (2 1) ))11 2

(0

2 1)

eff eff

ng

V V

C nn

N

not good! → extra bulk field in bulk

effective potential:

( ) ( ) ( ) ( )0, 45, 40a f a fN N N N 0.1 3gN

(0) /5( ) gauge q l m

eff eff eff effV A V V V

(ex)

effective potential:

( ) ( ) ( ) ( )0, 45, 40a f a fN N N N 0.1 3gN

(0) /5( ) gauge q l m

eff eff eff effV A V V V

(ex)

( ) ( ) ( ) ( )0, 42a f a fN N N N SU(6) model の例

4. Higgs phenomenology

(1). soft scalar mass(2). 3-point self coupling(3). Mass spectrum

NH, K.Takenaga and T.Yamashita, Phys.Rev.D71:025006,2005NH, K.Takenaga and T.Yamashita, hep-ph/0411250

2( ) 2

5

2( ) 2

5

1 (2 )[ , , , ] 1 1 2 cos(2 ) cos( )

3

1 (2 )[ , , , ] 1 1 2 cos(2 ) cos( ( 1))

3

zn

zn

znI a z n zn e n na

n

znI a z n zn e n n a

n

SU(3) × SU(3) model

We add soft scalar mass, m (z=mR) in addition to SS term as SUSY breaking.

( ) ( ) ( ) ( ) ( )

1

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

2 { ( [2 , , , ] 2 [ , , , ])

( [2 , , , ] 2 [ , , , ])

[ , , , ] [ , , , ])}

mattereff adj adj adj

n

adj adj adj

fnd fnd fnd fnd

V C N I a z n I a z n

N I a z n I a z n

N I a z n N I a z n

( )GeVｍ φ can be heavy even the same matter content

(1). soft scalar mass

L c

RL

cR

( 例 )

SU(3) × SU(3) model

m (z=mR)

SU(6) model

O(1) # of extra bulk field can realize EWSB !

m (z=mR)

(ex)

tend to be small comparing to SM ( ～ 10%)

(2). 3-point self coupling

（ motivation ）： measurement of λis important to know the mechanism of EWSB,　　　　　 and deviation from the Standard Model can be significant.

e

e Z

e

e

WW

ILC 実験

☆ higher order operators

4

4

cos ( )n nV a a g RH

g R

a few TeV → suppression scale → suppressed enough

0

2 346 3

3

32a a

g V

R a

☆ effective 3-point coupling

deviation from SM

23,SM h

SMSM

m

v

tan 1

4 5 4 5 6 7 6 75 5 5 5

4 5 4 5 6 7 6 75 5 5 5

1( ( ), ( ))

21

( ( ), ( ))2

u

d

H A i A A i A

H A i A A i A

D-flat

h

NH, K.Takenaga,T.Yamashita, Phys.Rev.D71:025006,2005

0

45

4

: ( )

: ( )

A massless h

massless A

5 05

5

:

: ( )Z

A

M H

6,75

6,7

:

: ( )W

A

M H

(3). Mass spectrum

V

uH

dH

at tree levelat S1 case

,(100) , (100)Z Wh A O GeV H H M O GeV probably (radiative induced mass ～ O(100)GeV)

☆ gauginos mass~higgsinos mass 　～ β/R

☆

(preliminary ）

origin of Higgs ： extraD component of extraD gauge field

5DH A → “doublet Higgs” 5 5 5cD DA → Yukawa int.

　 Higgs mass is finite (1/R) (← extraD gauge invariance)

1 loop effective potential of “Higgs doublets” (A5) in SU(3)×SU(3) model & SU(6) model (quark/lepton blane & bulk) ↓ EW DSB can be possible by extra bulk matters (suitable rep. & #)

,(100) , (100)Z Wh A O GeV H H M O GeV

☆ gauginos mass ～ higgsinos mass ～ β/R☆ 3-point self coupling: -10 % deviation from SM☆ extra bulk fields ～ O (100) GeV★ (mass spectrum (now calculating)) tanβ ～ 1

5 ． summary & discussion

　　　 problems(1): SU(6) model 　　☆ how to break extra U(1) ?　 ☆ how to forbid rapid proton-decay when 1/R ～ TeV? ← U(1)B

(2): SU(3)×SU(3) model ☆Winberg angle

5 2 25 5 5 4 4

1 12 ( ) ( 3 ) |

4 2 2 2a a a a b c

bc

Bdy F F dy A igf A A ig W i g H

5D gauge kinetic term→4D Higgs kinetic term

(g4 ～ O(1), (M*R)1/2 1 (≫ M* 1≫ /R))

23 sin 3 / 2Y Wg g

● wall-localized kinetic terms,2 2

0(0) , ( )F R F

g42 > λ-1, (we take g4 ～ 1), and expect 2

4 4

( , ) ( , ).3

Yg gW B W B

g g

:[ (6) : sin 3/8]Wcf SU

● introduction of additional U(1)’, extending U(3)×U(3), etc.

(SU(3) symmetry 無い )

12/S Z ☆ gauge-Higgs unification in E6, E7 GUTs on

（ good point ）： don’t need many representation to obtain quark/lepton Yukawa ints.

E6: bulk matters adjoint & fund.⇒ E7: bulk matters adjoint ⇒

quark/lepton favor structure ← effects of brane-localized extra fields

(NH and Y. Shimizu, Phys.Rev.D67:095001,2003,

Erratum-ibid.D69:059902,2004)

related work （ 1 ）

4D 4D

0y y Rgauge

extra matters

quark/lepton

cf. ３，６，１０，８ rep. are needed in SU(3)×SU(3) model

2 2, , , 0u dh hM m m m m B

gaugino mass ⇔ higgsino mass

at tree level at 1/R

Analyze radiative breaking (EWSB) is possible or notincluding SGGRA effects.(Choi, N.H., Jeong, Okumura, Shimizu, Yamaguchi, JHEP 0402:037,2004)

releted works （２）

☆ RGE analyses 　（ analyses of MSSM with boundary condidtion ）

mass2

logE 2

uHm

2

dHm