Extra Dimensional Models Extra Dimensional Models with Magnetic Fluxes with Magnetic Fluxes 　　　　　　　　　　　　　　 　　　　 Tatsuo KobayashiTatsuo Kobayashi１．１． IntroductionIntroduction2. Magnetized extra dimensions2. Magnetized extra dimensions3. N-point couplings and flavor symmetries3. N-point couplings and flavor symmetries44 ．． ModelsModels5. Summary5. Summary based on based on 　 　 Abe, T.K., Ohki, arXiv: 0806.4748 Abe, T.K., Ohki, arXiv: 0806.4748 Abe, Choi, T.K., Ohki, 0812.3534, 0903.3800, 0904.2631, Abe, Choi, T.K., Ohki, 0812.3534, 0903.3800, 0904.2631,

0907.5274,0907.5274, Choi, T.K., Maruyama, Murata, Nakai, Ohki, Sakai, 0908.0395 Choi, T.K., Maruyama, Murata, Nakai, Ohki, Sakai, 0908.0395 T.K., Maruyama, Murata, Ohki, Sakai, 1002.2828T.K., Maruyama, Murata, Ohki, Sakai, 1002.2828

１ １ IntroductionIntroduction1-1. Introduction to higher D theory1-1. Introduction to higher D theory

Extra dimensional field theories, Extra dimensional field theories,

in particular in particular

string-derived extra dimensional field string-derived extra dimensional field theories, theories,

play important roles in particle physicsplay important roles in particle physics

as well as cosmology .as well as cosmology .

Extra dimensionsExtra dimensions 4 + n dimensions4 + n dimensions

　　　　 4D ⇒4D ⇒ 　　　　 our 4D space-timeour 4D space-time

nD nD 　⇒ 　⇒ compact spacecompact space 　　Examples of compact spaceExamples of compact space

torus, orbifold, CY, etc. torus, orbifold, CY, etc.

Field theory in higher Field theory in higher dimensionsdimensions

10D 10D 　⇒ 　⇒ 4D our space-time + 6D space4D our space-time + 6D space

10D vector10D vector

4D vector + 4D scalars 4D vector + 4D scalars

SO(10) spinor ⇒SO(10) spinor ⇒ SO(4) spinor SO(4) spinor

x SO(6) spinorx SO(6) spinor

internal quantum internal quantum number number

mM AAA ,

Field theory in higher Field theory in higher dimensionsdimensions

Mode expansionsMode expansions

KK decomposition KK decomposition

0)(

0)( 6410

mm

mm

DDi

AA

Zero-modesZero-modesZero-mode equationZero-mode equation

⇒⇒ non-trival zero-mode profile non-trival zero-mode profile

the number of zero-modesthe number of zero-modes

0 mmDi

4D effective theory4D effective theoryHigher dimensional Lagrangian (e.g. 10D)Higher dimensional Lagrangian (e.g. 10D)

integrate the compact space ⇒integrate the compact space ⇒ 4D 4D theorytheory

Coupling is obtained by the overlap Coupling is obtained by the overlap integral of wavefunctionsintegral of wavefunctions

)()()(6 yyyydgY

),(),(),( 6410 yxyxAyxyxddgL

)()()(44 xxxxdYL

Couplings in 4DCouplings in 4D Zero-mode profiles are quasi-localized Zero-mode profiles are quasi-localized

far away from each otherfar away from each other in compact in compact spacespace

⇒ ⇒ suppressed couplingssuppressed couplings

1-2. Introduction to magnetized 1-2. Introduction to magnetized torustorusChiral theoryChiral theory

When we start with extra dimensional field When we start with extra dimensional field theories, theories,

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

Zero-modes between chiral and anti-Zero-modes between chiral and anti-chiral chiral

fields are different from each other fields are different from each other

on certain backgrounds, e.g. CY.on certain backgrounds, e.g. CY.

0 mmDi

Torus with magnetic flux Torus with magnetic flux

The limited number of solutions with The limited number of solutions with

non-trivial backgrounds are known.non-trivial backgrounds are known.

Torus background with magnetic flux Torus background with magnetic flux

is one of interesting backgrounds, is one of interesting backgrounds,

where one can solve zero-mode where one can solve zero-mode

Dirac equation.Dirac equation.

0 mmDi

Magnetic fluxMagnetic fluxIndeed, several studies have been done Indeed, several studies have been done

in both extra dimensional field theories in both extra dimensional field theories

and string theories with magnetic flux and string theories with magnetic flux

background.background.

In particular, magnetized D-brane models In particular, magnetized D-brane models

are T-duals of intersecting D-brane models.are T-duals of intersecting D-brane models.

Several interesting models have been Several interesting models have been

constructed in intersecting D-brane models, constructed in intersecting D-brane models,

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

Magnetized extra dimensionsMagnetized extra dimensions

The (generation) number of zero-modes The (generation) number of zero-modes

is determined by the size of magnetic is determined by the size of magnetic flux. flux.

Zero-mode profiles are quasi-localized.Zero-mode profiles are quasi-localized.

=> several interesting => several interesting phenomenologyphenomenology

Phenomenology of magnetized Phenomenology of magnetized brane modelsbrane models

It is important to study phenomenological It is important to study phenomenological

aspects of magnetized brane models such as aspects of magnetized brane models such as

massless spectra from several gauge groups, massless spectra from several gauge groups,

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

Yukawa couplings and higher order n-pointYukawa couplings and higher order n-point

couplings in 4D effective theory, couplings in 4D effective theory,

their symmetries like flavor symmetries, their symmetries like flavor symmetries,

Kahler metric, etc.Kahler metric, etc.

It is also important to extend such studies It is also important to extend such studies

on torus background to other backgrounds on torus background to other backgrounds

with magnetic fluxes, e.g. orbifold with magnetic fluxes, e.g. orbifold backgrounds.backgrounds.

2. Extra dimensions with magnetic 2. Extra dimensions with magnetic fluxes: basic toolsfluxes: basic tools

2-1. Magnetized torus model2-1. Magnetized torus model

We start with N=1 super Yang-Mills theory We start with N=1 super Yang-Mills theory in D = 4+2n dimensions. in D = 4+2n dimensions. For example, 10D super YM theory For example, 10D super YM theory consists of gauge bosons (10D vector)consists of gauge bosons (10D vector) and adjoint fermions (10D spinor).and adjoint fermions (10D spinor).We consider 2n-dimensional torus We consider 2n-dimensional torus

compactification compactification with magnetic flux background.with magnetic flux background.

Higher Dimensional SYM theory with flux　　　　　　　　　　　　 　 Cremades, Ibanez, Cremades, Ibanez, Marchesano, Marchesano, ‘‘0404

The wave functionsThe wave functions eigenstates of correspondinginternal Dirac/Laplace operator.

4D Effective theory <= dimensional reduction

Higher Dimensional SYM theory with flux

AbelianAbelian gauge field on magnetized torusgauge 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.

|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 )

Bi-fundamentalBi-fundamentalGaugino fields in off-diagonal entries Gaugino fields in off-diagonal entries

correspond to bi-fundamental matter correspond to bi-fundamental matter fields fields

and the difference M= m-mand the difference M= m-m’’ of magnetic of magnetic

fluxes appears in their Dirac equation.fluxes appears in their Dirac equation.

F F

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 4D chiral theory 10D spinor 10D spinor light-cone 8s light-cone 8s even number of minus signseven number of minus signs 11stst ⇒ ⇒ 4D, the other ⇒4D, the other ⇒ 6D space6D space If all of appear If all of appear in 4D theory, that is non-chiral theory.in 4D theory, that is non-chiral theory. If for all torus, If for all torus, only only

　　　　 appear for 4D helicity fixed.appear for 4D helicity fixed. ⇒ ⇒ 　　　　 4D chiral theory 4D chiral theory

),(

, , baab NN

U(8) SYM theory on T6U(8) SYM theory on T6

Pati-Salam group up to U(1) factorsPati-Salam group up to U(1) factors

Three families of matter fields Three families of matter fields

with many Higgs fieldswith many Higgs fields

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

21321

mmmm

Tmmmm

2,1,41,2,4

2-2. Wilson lines 2-2. Wilson lines Cremades, Ibanez, Cremades, Ibanez,

Marchesano, Marchesano, ’’04, 04, Abe, Choi, T.K. Ohki, Abe, Choi, T.K. Ohki, ‘‘0909

torus without magnetic fluxtorus without magnetic flux constant Ai constant Ai mass shift mass shift every modes massiveevery modes massive magnetic fluxmagnetic flux

　　　　 the number of zero-modes is the same.the number of zero-modes is the same. the profile: f(y) the profile: f(y) f(y +a/M) f(y +a/M) with proper b.c.with proper b.c.

0 )(2

0 )(2

aMy

aMy

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

Wilson line, Aa=0, Ab=CWilson line, Aa=0, Ab=C

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

chiral spectrum, chiral spectrum,

for Qa=0, massive due to nonvanishing for Qa=0, massive due to nonvanishing WLWL

when MQa >0, the number of zero-modeswhen MQa >0, the number of zero-modes

is MQa.is MQa.

zero-mode profile is shifted depending zero-mode profile is shifted depending

on Qb, on Qb,

　　　　

))/(( )( ab MQCQzfzf

2.3 Orbifold with magnetic 2.3 Orbifold with magnetic fluxflux

Abe, T.K., Ohki, Abe, T.K., Ohki, ‘‘0808

The number of even and odd zero-modesThe number of even and odd zero-modes

We can also embed Z2 into the gauge We can also embed Z2 into the gauge space.space.

=> various models, various flavor => various models, various flavor structuresstructures

Zero-modesZero-modes on orbifoldon orbifold

Adjoint matter fields are projected by Adjoint matter fields are projected by

orbifold projection.orbifold projection.

We have degree of freedom to We have degree of freedom to

introduce localized modes on fixed introduce localized modes on fixed points points

like quarks/leptons and higgs fields.like quarks/leptons and higgs fields.

3.3. N-point couplings N-point couplings and flavor symmetries and flavor symmetries 　　　　　　

The N-point couplings are obtained by The N-point couplings are obtained by overlap integral of their zero-mode w.f.overlap integral of their zero-mode w.f.’’s.s.

)()()(2 zzzzdgY kP

jN

iM

Zero-modes Zero-modes Cremades, Ibanez, Marchesano, Cremades, Ibanez, Marchesano, ‘‘0404

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

up to normalization factor up to normalization factor

),(0

/)]Im(exp[)( iMMz

MjzMziNz M

jM

,)()()(1

NM

m

MmjiNMijm

jN

iM zyzz

))(,0(0

))(/()(NMiMN

NMMNMNmMjNiyijm

MjNM ,,1 factor,ion normalizat:

3-point couplings3-point couplings 　　　　　　 Cremades, Ibanez, Marchesano, Cremades, Ibanez, Marchesano, ‘‘0404

The 3-point couplings are obtained by The 3-point couplings are obtained by

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

up to normalization factor up to normalization factor

*2 )()()( zzzzdY k

NMjN

iMijk

NM

mijmkmMjiijk yY

1,

ikkM

iM zzzd *2 )()(

Selection rule Selection rule

Each zero-mode has a Zg charge, Each zero-mode has a Zg charge,

which is conserved in 3-point couplings.which is conserved in 3-point couplings.

up to normalization factor up to normalization factor

)(, NMkmMjikmMji

))(,0(0

))(/()(NMiMN

NMMNMNmMjNiyijm

),gcd( when mod NMggkji

4-point couplings4-point couplings 　　　 　　　 Abe, Choi, T.K., Ohki, Abe, Choi, T.K., Ohki, ‘‘0909 The 4-point couplings are obtained by The 4-point couplings are obtained by overlap integral of four zero-mode w.f.overlap integral of four zero-mode w.f.’’s.s. splitsplit

insert a complete setinsert a complete set

up to normalization factor up to normalization factor for K=M+Nfor K=M+N

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

PNMkP

jN

iMijkl

modes all

*)'()()'( zzzz n

KnK

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

PNMkP

jN

iM

lsksijs

lijk yyY

4-point couplings: another 4-point couplings: another splittingsplitting

　　　　　　

i k i ki k i k

t t

j s l j lj s l j l

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

PNMjN

kP

iM

ltjtikt

lijk yyY

ltjtikt

lijk yyY lsksij

slijk yyY

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

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

couplings.couplings. N-point couplings = products of 3-point N-point couplings = products of 3-point

couplingscouplings = products of theta-functions= products of theta-functions

This behavior is non-trivial. (ItThis behavior is non-trivial. (It ’’s like CFT.) s like CFT.) Such a behavior wouldSuch a behavior would be satisfied be satisfied not for generic w.f.not for generic w.f.’’s, but for specific w.f.s, but for specific w.f.’’s.s. However, this behavior could be expected However, this behavior could be expected from T-duality between magnetized from T-duality between magnetized and intersecting D-brane models.and intersecting D-brane models.

T-dualityT-duality The 3-point couplings coincide between The 3-point couplings coincide between magnetized and intersecting D-brane models. magnetized and intersecting D-brane models. explicit calculationexplicit calculation Cremades, Ibanez, Marchesano, Cremades, Ibanez, Marchesano, ‘‘0404

Such correspondence can be extended to Such correspondence can be extended to 4-point and higher order couplings because of 4-point and higher order couplings because of CFT-like behaviors, e.g., CFT-like behaviors, e.g.,

Abe, Choi, T.K., Ohki, Abe, Choi, T.K., Ohki, ‘‘09 09

lsksijs

lijk yyY

Heterotic orbifold modelsHeterotic orbifold models

Our results would be useful to n-point couplings Our results would be useful to n-point couplings of twsited sectors in heterotic orbifold models.of twsited sectors in heterotic orbifold models.

Twisted strings on fixed points might correspond Twisted strings on fixed points might correspond to quasi-localized modes with magnetic flux, to quasi-localized modes with magnetic flux, zero modes profile = gaussian x theta-function zero modes profile = gaussian x theta-function

amplitude string closedamplitude stringopen 2

orbifold heterotic

in coupling

brane ngintersecti

in couplings2

Non-Abelian discrete flavor Non-Abelian discrete flavor symmetrysymmetry

The coupling selection rule is controlled by The coupling selection rule is controlled by

Zg charges.Zg charges.

For M=g,For M=g, 1 2 1 2 　　　　　　　　　　　　 g g

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

These lead to non-Abelian flavor symmetires These lead to non-Abelian flavor symmetires

such as D4 and Δ(27)such as D4 and Δ(27) 　　　　　　　　 Cf. heterotic orbifolds, Cf. heterotic orbifolds, T.K. Raby, Zhang, T.K. Raby, Zhang, ’’0404

T.K. Nilles, Ploger, Raby, T.K. Nilles, Ploger, Raby, Ratz, Ratz, ‘‘0606

3. Models3. Models We can construct several models by using We can construct several models by using the above model building tools. the above model building tools. What is the starting theory ?What is the starting theory ? 10D SYM or 6D SYM (+ hyper multiplets), 10D SYM or 6D SYM (+ hyper multiplets),

gauge groups, U(N), SO(N), E6, E7,E8,...gauge groups, U(N), SO(N), E6, E7,E8,... What is the gauge background ?What is the gauge background ? the form of magnetic fluxes, Wilson lines.the form of magnetic fluxes, Wilson lines. What is the geometrical background ?What is the geometrical background ? torus, orbifold, etc.torus, orbifold, etc.

E6 SYM theory on T6E6 SYM theory on T6 Choi, et. al. Choi, et. al. ‘‘0909

We introduce magnetix flux along U(1) direction, We introduce magnetix flux along U(1) direction,

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

Three families of chiral matter fields 16Three families of chiral matter fields 16

We introduce Wilson lines breaking We introduce Wilson lines breaking

SO(10) -> SM group.SO(10) -> SM group.

Three families of quarks and leptons matter fields Three families of quarks and leptons matter fields

with no Higgs fieldswith no Higgs fields

1100 161614578

1 ,1 ,3 321 mmm

Splitting zero-mode profilesSplitting zero-mode profilesWilson lines do not change the Wilson lines do not change the

(generation) number of zero-modes, (generation) number of zero-modes, but change localization point.but change localization point.

1616

QQ ………… LL

E8 SYM theory on T6E8 SYM theory on T6 T.K., et. al. 1002:2828T.K., et. al. 1002:2828 E8 -> SU(3)*SU(2)*U(1)E8 -> SU(3)*SU(2)*U(1)YY*U(1)*U(1)11*U(1)*U(1)22*U(1)*U(1)33*U(1)*U(1)44

We introduce magnetic fluxes and Wilson lines We introduce magnetic fluxes and Wilson lines

along five U(1)along five U(1)’’s.s.

E8 248 adjoint rep. include various matter fields.E8 248 adjoint rep. include various matter fields.

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

We have studied systematically the possibilitiesWe have studied systematically the possibilities

for realizing the MSSM.for realizing the MSSM.

Results: E8 SYM theory on T6Results: E8 SYM theory on T6 1. 1. We can get exactly 3 generations of We can get exactly 3 generations of quarks and leptons, quarks and leptons, but there are tachyonic modes and no top but there are tachyonic modes and no top

Yukawa. Yukawa. 2. 2. We allow MSSM + vector-like generations We allow MSSM + vector-like generations and require the top Yukawa couplings and and require the top Yukawa couplings and no exotic fieldsno exotic fields as well as no vector-like quark doublets.as well as no vector-like quark doublets. three classesthree classes

Results: E8 SYM theory on T6Results: E8 SYM theory on T6 A. A.

B. B.

C.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

662

332

222

442

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 T6Results: E8 SYM theory on T6 There are many vector-like generations. There are many vector-like generations.

They have 3- and 4-point couplings with They have 3- and 4-point couplings with

singlets (with no hypercharges).singlets (with no hypercharges).

VEVs of singlets VEVs of singlets mass terms of vector-like mass terms of vector-like

generations generations

Light modes depend on such mass terms. Light modes depend on such mass terms.

That is, low-energy phenomenologies That is, low-energy phenomenologies

depends on VEVs of singlets (moduli).depends on VEVs of singlets (moduli).

Results: E8 SYM theory on T6Results: E8 SYM theory on T6 It would be interesting to reduce the number It would be interesting to reduce the number

of vector-like generations. of vector-like generations.

orbifolding and/or non-Abelian Wilson lines orbifolding and/or non-Abelian Wilson lines

SummarySummaryWe have studiedWe have studied phenomenological aspects phenomenological aspects of magnetized brane models.of magnetized brane models.

Model building from U(N), E6, E7, E8Model building from U(N), E6, E7, E8

N-point couplings are comupted.N-point couplings are comupted. 4D effective field theory has non-Abelian 4D effective field theory has non-Abelian

flavor flavor symmetries, e.g. D4, Δ(27).symmetries, e.g. D4, Δ(27). Orbifold background with magnetic flux is Orbifold background with magnetic flux is also important. also important.