Overview of Extra Dimensional Models

with Magnetic Fluxes Tatsuo Kobayashi

１．Introduction

2. Magnetized extra dimensions

3. N-point couplings and flavor symmetries

4．Summary

１ Introduction

Extra dimensional field theories,

in particular

string-derived extra dimensional field theories,

play important roles in particle physics

as well as cosmology .

Zero-modes

Zero-mode equation

⇒ non-trival zero-mode profile

the number of zero-modes

0 m

mDi

Couplings in 4D

Zero-mode profiles are quasi-localized

far away from each other in compact space

⇒ suppressed couplings

Chiral theory When we start with extra dimensional field theories,

how to realize chiral theories is one of important issues from the viewpoint of particle physics.

Zero-modes between chiral and anti-chiral

fields are different from each other

on certain backgrounds,

e.g. CY, toroidal orbifold, warped orbifold,

magnetized extra dimension, etc.

0 m

mDi

Magnetic flux

The limited number of solutions with

non-trivial backgrounds are known.

Generic CY is difficult.

Toroidal/Wapred orbifolds are well-known.

Background with magnetic flux is

one of interesting backgrounds.

0 m

mDi

Magnetic flux

Indeed, several studies have been done

in both extra dimensional field theories

and string theories with magnetic flux

background.

In particular, magnetized D-brane models

are T-duals of intersecting D-brane models.

Several interesting models have been

constructed in intersecting D-brane models,

that is, the starting theory is U(N) SYM.

Phenomenology of magnetized brane models

It is important to study phenomenological

aspects of magnetized brane models such as

massless spectra from several gauge groups,

U(N), SO(N), E6, E7, E8, ...

Yukawa couplings and higher order n-point

couplings in 4D effective theory,

their symmetries like flavor symmetries,

Kahler metric, etc.

It is also important to extend such studies

on torus background to other backgrounds

with magnetic fluxes, e.g. orbifold backgrounds.

2. Extra dimensions with magnetic fluxes: basic tools

2-1. Magnetized torus model

We start with N=1 super Yang-Mills theory

in D = 4+2n dimensions.

For example, 10D super YM theory

consists of gauge bosons (10D vector)

and adjoint fermions (10D spinor).

We consider 2n-dimensional torus compactification

with magnetic flux background.

We can start with 6D SYM (+ hyper multiplets),

or non-SUSY models (+ matter fields ), similarly.

Higher Dimensional SYM theory with flux

Cremades, Ibanez, Marchesano, ‘04

The wave functions eigenstates of corresponding

internal Dirac/Laplace operator.

4D Effective theory <= dimensional reduction

Higher Dimensional SYM theory with flux

Abelian gauge field on magnetized torus

Constant magnetic flux

The boundary conditions on torus (transformation under torus translations)

gauge fields of background

Dirac equation on 2D torus

with twisted boundary conditions (Q=1)

is the two component spinor.

5454 , ii

U(1) charge Q=1

|M| independent zero mode solutions in Dirac equation.

(Theta function)

Dirac equation and chiral fermion

Properties of

theta functions

:Normalizable mode

:Non-normalizable

mode

By introducing magnetic

flux, we can obtain chiral

theory.

chiral fermion

Wave functions

Wave function profile on toroidal background

For the case of M=3

Zero-modes wave functions are quasi-localized far away each

other in extra dimensions. Therefore the hierarchirally small

Yukawa couplings may be obtained.

Fermions in bifundamentals

The gaugino fields

Breaking the gauge group

bi-fundamental matter fields

gaugino of unbroken gauge

(Abelian flux case )

Zero-modes Dirac equations

Total number of zero-modes of

:Normalizable mode

:Non-Normalizable mode

No effect due to magnetic flux for adjoint matter fields,

4D chiral theory 10D spinor

light-cone 8s

even number of minus signs

1st ⇒ 4D, the other ⇒ 6D space

If all of appear

in 4D theory, that is non-chiral theory.

If for all torus,

only

appear for 4D helicity fixed.

⇒ 4D chiral theory

),(

, , ba

ab NN

U(8) SYM theory on T6

Pati-Salam group up to U(1) factors

Three families of matter fields

with many Higgs fields

2,1,41,2,4

3

2

1

3

2

1

0

0

2

N

N

N

zz

m

m

m

iF

2 ,2 ,4 321 NNNRL UUU )2()2()4(

other tori for the 1)()(

first for the 3)()(

1321

2

1321

mmmm

Tmmmm

)2,1,4()1,2,4(

)2,2,1(

2-2. Wilson lines Cremades, Ibanez, Marchesano, ’04,

Abe, Choi, T.K. Ohki, ‘09

torus without magnetic flux

constant Ai mass shift

every modes massive

magnetic flux

the number of zero-modes is the same.

the profile: f(y) f(y +a/M)

with proper b.c.

0 )(2

0 )(2

aMy

aMy

U(1)a*U(1)b theory magnetic flux, Fa=2πM, Fb=0

Wilson line, Aa=0, Ab=C

matter fermions with U(1) charges, (Qa,Qb)

chiral spectrum,

for Qa=0, massive due to nonvanishing WL

when MQa >0, the number of zero-modes

is MQa.

zero-mode profile is shifted depending

on Qb,

))/(( )( ab MQCQzfzf

Pati-Salam model

Pati-Salam group

WLs along a U(1) in U(4) and a U(1) in U(2)R

=> Standard gauge group up to U(1) factors

U(1)Y is a linear combination.

2,1,41,2,4

3

2

1

3

2

1

0

0

2

N

N

N

zz

m

m

m

iF 2 ,2 ,4 321 NNN

RL UUU )2()2()4(

other tori for the 1)()(

first for the 3)()(

1321

2

1321

mmmm

Tmmmm

3)1()2()3( UUU LC

PS => SM Zero modes corresponding to

three families of matter fields

remian after introducing WLs, but their profiles split

(4,2,1)

Q L

)2,1,4()1,2,4(

)1,1,1()1,1,1()1,1,3()1,1,3()2,1,4(

)1,2,1()1,2,3()1,2,4(

Othere models

We can start with 10D SYM,

6D SYM (+ hyper multiplets),

or non-SUSY models (+ matter fields )

with gauge groups,

U(N), SO(N), E6, E7,E8,...

E6 SYM theory on T6 Choi, et. al. ‘09

We introduce magnetix flux along U(1) direction,

which breaks E6 -> SO(10)*U(1)

Three families of chiral matter fields 16

We introduce Wilson lines breaking

SO(10) -> SM group.

Three families of quarks and leptons matter fields

with no Higgs fields

1100 161614578

1 ,1 ,3 321 mmm

Splitting zero-mode profiles

Wilson lines do not change the (generation) number of zero-modes, but change localization point.

16

Q …… L

E8 SYM theory on T6 T.K., et. al. ‘10

E8 -> SU(3)*SU(2)*U(1)Y*U(1)1*U(1)2*U(1)3*U(1)4

We introduce magnetic fluxes and Wilson lines

along five U(1)’s.

E8 248 adjoint rep. include various matter fields.

248 = (3,2)(1,1,0,1,-1) + (3,2)(1,0,1,1,-1) + (3,2)(1,-1,-1,1,-1)

+ (3,2)(1,1,0,1,-1) + (3,2)(1,0,1,1,-1) + ......

We have studied systematically the possibilities

for realizing the MSSM.

Results: E8 SYM theory on T6 1.

We can get exactly 3 generations of

quarks and leptons,

but there are tachyonic modes and no top Yukawa.

2.

We allow MSSM + vector-like generations

and require the top Yukawa couplings and

no exotic fields

as well as no vector-like quark doublets.

three classes

Results: E8 SYM theory on T6 A.

B.

C.

charges U(1)Yno with singlets

])1,1()1,1[(15])2,1()2,1[(12

])1,3()1,3[(21])1,3()1,3[(10

])1,1()2,1()1,3()1,3()2,3[(3

6633

2244

63241

charges U(1)Yno with singlets

])1,1()1,1[()156(])2,1()2,1[()44(

])1,3()1,3[()36(])1,3()1,3[()96(

])1,1()2,1()1,3()1,3()2,3[(3

66

2

33

2

22

2

44

2

63241

nnnn

nnnn

charges U(1)Yno with singlets

])1,1()1,1[(180])2,1()2,1[(45

])1,3()1,3[(18])1,3()1,3[(66

])1,1()2,1()1,3()1,3()2,3[(3

6633

2244

63241

Results: E8 SYM theory on T6 There are many vector-like generations.

They have 3- and 4-point couplings with

singlets (with no hypercharges).

VEVs of singlets mass terms of vector-like

generations

Light modes depend on such mass terms.

That is, low-energy phenomenologies

depends on VEVs of singlets (moduli).

2.3 Orbifold with magnetic flux

Abe, T.K., Ohki, ‘08

The number of even and odd zero-modes

We can also embed Z2 into the gauge space.

=> various models, various flavor structures

Zero-modes on orbifold

Adjoint matter fields are projected by

orbifold projection.

We have degree of freedom to

introduce localized modes on fixed points

like quarks/leptons and higgs fields.

2.4 S2 with magnetic flux

solution no )(

10 ,)1/()( 2/)1(2

z

MmzNzz Mm

0)(2/)1()1(

0)(2/)1()1(

2

2

zzMz

zzMz

zdzdzRds 2222 )1(4

solution no )(

solution no )(

z

z

Conlon, Maharana, Quevedo, ‘08

Fubuni-Study metric

Zero-mode eq.

with spin

connection

M > 0

M=0

3. N-point couplings and flavor symmetries

The N-point couplings are obtained by

overlap integral of their zero-mode w.f.’s.

)()()(2 zzzzdgY k

P

j

N

i

M

54 iyyz

Zero-modes Cremades, Ibanez, Marchesano, ‘04

Zero-mode w.f. = gaussian x theta-function

up to normalization factor

),(0

/)]Im(exp[)( iMMz

MjzMziNz M

j

M

,)()()(1

NM

m

Mmji

NMijm

j

N

i

M zyzz

))(,0(0

))(/()(NMiMN

NMMNMNmMjNiyijm

MjNM ,,1 factor,ion normalizat:

Products of wave functions

products of zero-modes = zero-modes

NMNM

NM

N

M

yNM

Ny

My

)(

)( ,0 )(2

,0 2

,0 2

3-point couplings

Cremades, Ibanez, Marchesano, ‘04

The 3-point couplings are obtained by

overlap integral of three zero-mode w.f.’s.

up to normalization factor

ikk

M

i

M zzzd *2 )()(

*2 )()()( zzzzdY k

NM

j

N

i

Mijk

NM

m

ijmkmMjiijk yY1

,

Selection rule

Each zero-mode has a Zg charge,

which is conserved in 3-point couplings.

up to normalization factor

)(, NMkmMjikmMji

))(,0(0

))(/()(NMiMN

NMMNMNmMjNiyijm

),gcd( when mod NMggkji

4-point couplings Abe, Choi, T.K., Ohki, ‘09

The 4-point couplings are obtained by

overlap integral of four zero-mode w.f.’s.

split insert a complete set up to normalization factor for K=M+N

*2 )()()()( zzzzzdY l

PNM

k

P

j

N

i

Mijkl

modes all

*)'()()'( zzzz n

K

n

K

*22 )'()'()'()()(' zzzzzzzzdd l

PNM

k

P

j

N

i

M

lsksij

slijk

yyY

4-point couplings: another splitting

i k i k

t

j s l j l

*22 )'()'()'()()(' zzzzzzzzdd l

PNM

j

N

k

P

i

M

ltjtik

tlijk

yyY

ltjtik

tlijk

yyY lsksij

slijk

yyY

N-point couplings Abe, Choi, T.K., Ohki, ‘09

We can extend this analysis to generic n-point couplings.

N-point couplings = products of 3-point couplings = products of theta-functions This behavior is non-trivial. (It’s like CFT.)

Such a behavior would be satisfied not for generic w.f.’s, but for specific w.f.’s.

However, this behavior could be expected from T-duality between magnetized and intersecting D-brane models.

T-duality The 3-point couplings coincide between

magnetized and intersecting D-brane models.

explicit calculation

Cremades, Ibanez, Marchesano, ‘04

Such correspondence can be extended to

4-point and higher order couplings because of

CFT-like behaviors, e.g.,

Abe, Choi, T.K., Ohki, ‘09

lsksij

slijk

yyY

Applications of couplings

We can obtain quark/lepton masses and mixing angles.

Yukawa couplings depend on volume moduli,

complex structure moduli and Wilson lines.

By tuning those values, we can obtain semi-realistic results.

Abe, Choi, T.K., Ohki ‘08

T.K., et. al. ‘10,

Abe, et. al. work in progress

Other applications

up to normalization factor

Summary

We have studied phenomenological aspects

of magnetized brane models.

Model building from U(N), E6, E7, E8

N-point couplings are comupted.

4D effective field theory has non-Abelian flavor

symmetries, e.g. D4, Δ(27).

Orbifold background with magnetic flux is

also important. S2 and others are also interesting.

Field theory in higher dimensions

Mode expansions

KK decomposition

0)(

0)( 6410

m

m

mm

DDi

AA

3

4D effective theory Higher dimensional Lagrangian (e.g. 10D)

integrate the compact space ⇒ 4D theory

Coupling is obtained by the overlap

integral of wavefunctions

)()()(6 yyyydgY

),(),(),( 64

10 yxyxAyxyxddgL

)()()(4

4 xxxxdYL

Non-Abelian discrete flavor symmetry The coupling selection rule is controlled by

Zg charges.

For M=g, 1 2 g

Effective field theory also has a cyclic permutation symmetry of g zero-modes.

These lead to non-Abelian flavor symmetires

such as D4 and Δ(27) Abe, Choi, T.K, Ohki, ‘09

Cf. heterotic orbifolds, T.K. Raby, Zhang, ’04

T.K. Nilles, Ploger, Raby, Ratz, ‘06