PDF Overview of Extra Dimensional Models with Magnetic Fluxes

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

D

_m

i

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

D

_m

i

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

D

_m

i

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.

5 4

5

4

  ,     





 i i

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

1^st ⇒ 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

) ,

(    



_a

,

_b

  ,    

ab

N N



U(8) SYM theory on T6

Pati-Salam group up to U(1) factors

Three families of matter fields with many Higgs fields

^4,2,1^ ^4,1,2

























3 2

1

3 2

1

0

0 2

N N

N z

z

m m

m i

F 

2 ,

2 ,

4

_{2 3}

1

 N  N 

N U ( 4 )  U ( 2 )

L

 U ( 2 )

R

other tori for the

1 )

( )

(

first for the

3 )

( )

(

1 3

2 1

2 1

3 2

1

















m m

m m

T m

m m

m

) 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.

 

     2 2   ( ( My My   a a ) )   

_^

  0 0

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,

)) /(

(

)

( z f z CQ

_b

MQ

_a

f  

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.

^4,2,1^ ^4,1,2

























3 2

1

3 2

1

0

0 2

N N

N z

z

m m

m i

F 

N

₁

 4 , N

₂

 2 , N

₃

 2

R

L

U

U

U ( 4 )  ( 2 )  ( 2 )

other tori for the

1 )

( )

(

first for the

3 )

( )

(

1 3

2 1

2 1

3 2

1

















m m

m m

T m

m m

m

)

3

1 ( )

2 ( )

3

( U U

U

_C



_L



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

1 1 0

0

1 16 16

45

78    



1

, 1

,

3

_{2 3}

1

 m  m 

m

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)Y

no 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

6 6

3 3

2 2

4 4

6 3

2 4

1

















































charges U(1)Y

no with singlets

] ) 1 , 1 ( )

1 , 1 [(

) 15 6

( ] ) 2 , 1 ( )

2 , 1 [(

) 4 4

(

] ) 1 , 3 ( )

1 , 3 [(

) 3 6

( ] ) 1 , 3 ( )

1 , 3 [(

) 9 6

(

] ) 1 , 1 ( )

2 , 1 ( )

1 , 3 ( )

1 , 3 ( )

2 , 3 [(

3

6 6

2 3

3 2

2 2

2 4

4 2

6 3

2 4

1

























































n n

n n

n n

n n

charges U(1)Y

no 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

6 6

3 3

2 2

4 4

6 3

2 4

1

















































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

) (

1 0

, )

1 /(

)

( ^{2 ( 1)/2}

z

M m

z Nz

z ^{m M}





     





 

 ( ( 1 1

²

) ) ( ( 1 ) 1 ) / 2 / 2  ( ( ) ) 0 0

2



























z z

M z

z z

M z





z dzd z

R

ds

²

 4

²

( 1 

²

)

^2

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

(

z z

z z

d g

Y   

_Mⁱ



_N^j

 

_P^k

5 4

iy y

z  

Zero-modes

Cremades, Ibanez, Marchesano, ‘04

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

up to normalization factor

) ,

0 ( )] /

Im(

exp[

)

( j M Mz iM

z Mz

i N

z _M

j

M 



 



 

  



, ) ( )

( )

(



^1













^{M N}

m

Mm j

i

N M ijm

j N i

M

z  z y  z



)) (

, 0 0 (

)) (

/(

)

(Ni Mj MNm MN M N iMN M N

y_ijm  



 



   



M j

N

_M

: normalizat ion factor,  1 ,  ,

Products of wave functions

products of zero-modes = zero-modes

 

 

 









































N M

N M

N M

N M

y N

M 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

 



^{d2zMi (z)Mk (z) *  ik}

 



^

 d

²

z ( z ) ( z ) ( z )

^*

Y

_ijk



_Mⁱ



_N^j



_M^k _N



^

  

 ^{M N}

m

ijm k

mM j

i

ijk y

Y

1

 ,

Selection rule

Each zero-mode has a Zg charge,

which is conserved in 3-point couplings.

up to normalization factor

)

,_k

i j mM k ( M N

mM j

i_{ }

    



)) (

, 0 0 (

)) (

/(

)

(Ni Mj MNm MN M N iMN M N

y_ijm  



 



   



 mod  when gcd( , )

g g M N

k j

i   

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

 



^{ }

 d

²

z ( z ) ( z ) ( z ) ( z )

^*

Y

_ijkl



_Mⁱ



_N^j



_P^k



_M^l _{N P}

 







modes all

* ( ' )

) (

) '

(z z



_Kⁿ z



_Kⁿ z



 

 d

²

zd

²

z ' 

^Mi

( z ) 

^Nj

( z )  ( z  z ' ) 

^Pk

( z ' ) 

^{Ml NP}

( z ' )

^*

l s sk ij s

l

ijk

y y

Y  

4-point couplings: another splitting

i k i k

t j s l j l

 

 d

²

zd

²

z ' 

^Mi

( z ) 

^Pk

( z )  ( z  z ' ) 

^Nj

( z ' ) 

^{Ml NP}

( z ' )

^*

l t tj ik t

l

ijk

y y

Y  

l t tj ik t

l

ijk

y y

Y  

l s sk ij s

l

ijk

y y

Y  

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

l s sk ij s

l

ijk

y y

Y  

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 )

(

_{4 6}

10



























m m

m m

D D

i

A A

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



 g d

⁶

y ( y ) ( y ) ( y )

Y   





^{4 6}

( , ) ( , ) ( , )

10

g d xd y x y A x y x y

L  





⁴

( ) ( ) ( )

4

Y d x x x x

L   

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