Extra Dimensional Models Extra Dimensional Models with Magnetic Fluxes with Magnetic Fluxes 　　　　　　　　　　　　　　 　　　　 Tatsuo KobayashiTatsuo Kobayashi１．１． IntroductionIntroduction2. Magnetized extra dimensions2. Magnetized extra dimensions3. Models3. Models44 ．． N-point couplings and flavor symmetriesN-point couplings and flavor symmetries5. 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, 0904.26Abe, Choi, T.K., Ohki, 0812.3534, 0903.3800, 0904.26

31, 0907.5274,31, 0907.5274, Choi, T.K., Maruyama, Murata, Nakai, Ohki, Sakai, 0908.Choi, T.K., Maruyama, Murata, Nakai, Ohki, Sakai, 0908.

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１ １ IntroductionIntroduction

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 .

Chiral theoryChiral theoryWhen we start with extra dimensional field theories, When we start with extra dimensional field theories, how to realize chiral theories is one of important isshow to realize chiral theories is one of important iss

ues from the viewpoint of particle physics. ues from the viewpoint of particle physics.

Zero-modes between chiral and anti-chiral Zero-modes between chiral and anti-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 D-brane modelsMagnetized D-brane models

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 backgrounds.with magnetic fluxes, e.g. orbifold 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. We consider 2n-dimensional torus compactificWe consider 2n-dimensional torus compactific

ation ation with magnetic flux background.with magnetic flux background.

Higher Dimensional SYM theory with flux　　　　　　　　　　　　 　 Cremades, Ibanez, MarcCremades, Ibanez, Marchesano, ‘04hesano, ‘04

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

Higher Dimensional SYM theory with flux

We now consider a complex field with charge Q ( +/-1 )

Consistency of such transformations under a contractible loop in torus which implies Dirac’s quantization conditions.

Dirac equation

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

(Ablian flux case )

Bi-fundamentalBi-fundamentalGaugino fields in off-diagonal entries Gaugino fields in off-diagonal entries correspond to bi-fundamental matter fields correspond to bi-fundamental matter fields and the difference M= m-m’ of magnetic and the difference M= m-m’ 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,

2-2. Wilson lines 2-2. Wilson lines Cremades, Ibanez, Marchesano, ’04, Cremades, Ibanez, Marchesano, ’04, Abe, Choi, T.K. Ohki, ‘09Abe, Choi, T.K. Ohki, ‘09 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 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, (Qa,Qb)matter fermions with U(1) charges, (Qa,Qb) chiral spectrum, chiral spectrum, for Qa=0, massive due to nonvanishing WLfor Qa=0, massive due to nonvanishing WL 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. Magnetized orbifold models2-3. Magnetized orbifold modelsWe consider orbifold compactification We consider orbifold compactification with magnetic flux.with magnetic flux.

Orbifolding is another way to obtain chiral theorOrbifolding is another way to obtain chiral theory.y.

Magnetic flux is invariant under the Z2 twist.Magnetic flux is invariant under the Z2 twist.

We consider the Z2 and Z2xZ2’ orbifolds.We consider the Z2 and Z2xZ2’ orbifolds.

Orbifold with magnetic fluxOrbifold with magnetic flux Abe, T.K., Ohki, ‘08Abe, T.K., Ohki, ‘08

Note that there is no odd massless modes Note that there is no odd massless modes on the orbifold without magnetic flux.on the orbifold without magnetic flux.

)()(:2 zzZ jMM

jM

jMM

jM

jMM

jM

:mode odd Z

:modeeven Z

2

2

ziyyzZ )(: 542

Zero-modesZero-modesEven and/or odd modes are allowed Even and/or odd modes are allowed as zero-modes on the orbifold with as zero-modes on the orbifold with magnetic flux.magnetic flux. On the usual orbifold without magnetic flux,On the usual orbifold without magnetic flux, odd zero-modes correspond only to odd zero-modes correspond only to massive modes.massive modes.

Adjoint matter fields are projected by Adjoint matter fields are projected by orbifold projection.orbifold projection.

Orbifold with magnetic fluxOrbifold with magnetic flux Abe, T.K., Ohki, ‘08Abe, T.K., Ohki, ‘08

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

Localized modes on fixed Localized modes on fixed pointspoints

We have degree of freedom to We have degree of freedom to introduce localized modes on fixed points introduce localized modes on fixed points like quarks/leptons and higgs fields.like quarks/leptons and higgs fields.

That would lead to richer flavor structure.That would lead to richer flavor structure.

2-4. Orbifold with M.F. and W.L.2-4. Orbifold with M.F. and W.L. Abe, Choi, T.K., Ohki, ‘09Abe, Choi, T.K., Ohki, ‘09 Example: U(1)Example: U(1)aa x SU(2) theory x SU(2) theory SU(2) doublet with charge qSU(2) doublet with charge qaa

zero-modeszero-modes

the number of zero-modes = Mthe number of zero-modes = M

01

10 twist orbifold P

)()( ,2/1

,2/1 zz MjMMj

2/1

2/1

Another basisAnother basis

zero-modeszero-modes

the total number of zero-modes = Mthe total number of zero-modes = M

10

01' twist orbifold P

)(')('

)(')(',

2/1,

2/1

,2/1

,2/1

zz

zzMjMMj

MjMMj

2/1

2/1

'

'

Wilson linesWilson lines

zero-mode profileszero-mode profiles

10

01direction Cartan along lineWilson

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

,2/1 MCzMCz bMjMbMj

SU(2) triplet SU(2) triplet

Wilson line along the Cartan directionWilson line along the Cartan directionzero-modeszero-modes

the number of zero-modes the number of zero-modes = M= M 　　 for the formerfor the former < M for the latter< M for the latter

001

010

100

twist orbifold P

)()(

)/()/(,

0,

0

,1

,1

zz

MCzMCzMjMMj

bMjMbMj

1

0

1

Orbifold, M.F. and W.L.Orbifold, M.F. and W.L. We can consider larger gauge groups We can consider larger gauge groups and several representations.and several representations. Non-trivial orbifold twists and Wilson linesNon-trivial orbifold twists and Wilson lines 　　　⇒　　　　　　⇒　　　 various models various models

Non-Abelian W.L. + fractional magnetic fluxesNon-Abelian W.L. + fractional magnetic fluxes　　 (‘t Hooft toron background)(‘t Hooft toron background) 　　⇒　　　　　⇒　　　 interesting aspectsinteresting aspects Abe, Choi, T.K., Ohki, work in progressAbe, Choi, T.K., Ohki, work in progress

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.

U(N) theory on T6U(N) theory on T6

gauge group gauge group

kNk

N

zz

m

m

iF

0

0

211

matrixidentity)(: NNN

k

iiNUNU

1

)()(

54 iyyz

U(N) SYM theory on T6U(N) 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 Orbifolding can lead to various 3-generation PS models.Orbifolding can lead to various 3-generation PS models. See See Abe, Choi, T.K., Ohki, ‘08Abe, Choi, T.K., Ohki, ‘08

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

E6 SYM theory on T6E6 SYM theory on T6 Choi, et. al. ‘09Choi, et. al. ‘09 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

E6 SYM theory on T6E6 SYM theory on T6 There is no electro-weak Higgs fieldsThere is no electro-weak Higgs fields By orbifolding, we can derive a similar model By orbifolding, we can derive a similar model with three generations of 16. with three generations of 16.

On the orbifold, there is singular points, i.e. On the orbifold, there is singular points, i.e. fixed points.fixed points.

We could assume consistently that We could assume consistently that electro-weak Higgs fields are localized modes electro-weak Higgs fields are localized modes on a fixed point.on a fixed point.

E7, E8 SYM theory on T6E7, E8 SYM theory on T6 Choi, et. al. ‘09Choi, et. al. ‘09 E7 and E8 have more ranks (U(1) factors) E7 and E8 have more ranks (U(1) factors) than E6 and SO(10). than E6 and SO(10). Those adjoint rep. include various matter fields.Those adjoint rep. include various matter fields.

Then, we can obtain various models including Then, we can obtain various models including MSSM + vector-like matter fieldsMSSM + vector-like matter fields See for its detail our coming paper.See for its detail our coming paper.

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

)()()(2 zzzzdgY kP

jN

iM

Zero-modes Zero-modes Cremades, Ibanez, Marchesano, ‘04Cremades, Ibanez, Marchesano, ‘04 　　 Zero-mode w.f. = gaussian x theta-functionZero-mode w.f. = gaussian x theta-function

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, ‘04Cremades, Ibanez, Marchesano, ‘04

The 3-point couplings are obtained by The 3-point couplings are obtained by overlap integral of three zero-mode w.f.’s.overlap integral of three zero-mode w.f.’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, ‘09Abe, Choi, T.K., Ohki, ‘09 The 4-point couplings are obtained by The 4-point couplings are obtained by overlap integral of four zero-mode w.f.’s.overlap integral of four zero-mode w.f.’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, ‘09 Abe, Choi, T.K., Ohki, ‘09 We can extend this analysis to generic n-point coupliWe can extend this analysis to generic n-point coupli

ngs.ngs. N-point couplings = products of 3-point couplingsN-point couplings = products of 3-point couplings = products of theta-functions= products of theta-functions

This behavior is non-trivial. (It’s like CFT.) This behavior is non-trivial. (It’s like CFT.) Such a behavior wouldSuch a behavior would 　　 be satisfied be satisfied not for generic w.f.’s, but for specific w.f.’s.not for generic w.f.’s, but for specific w.f.’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, ‘04Cremades, Ibanez, Marchesano, ‘04 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, ‘09 Abe, Choi, T.K., Ohki, ‘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 symmetryNon-Abelian discrete flavor symmetry

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 g 1 2 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.

Non-Abelian discrete flavor symmetryNon-Abelian discrete flavor symmetryThe total flavor symmetry corresponds to The total flavor symmetry corresponds to the closed algebra of the closed algebra of

That is the semidirect product of Zg x Zg and Zg.That is the semidirect product of Zg x Zg and Zg. For example, For example, g=2 D4g=2 D4 g=3 Δ(27)g=3 Δ(27) 　　　　　　　　 Cf. heterotic orbifolds, Cf. heterotic orbifolds, T.K. Raby, Zhang, ’04T.K. Raby, Zhang, ’04 T.K. Nilles, Ploger, Raby, Ratz, ‘06T.K. Nilles, Ploger, Raby, Ratz, ‘06

]/2exp[ gi

0001

10

00

0010

,

1

1

g

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 flavor 4D effective field theory has non-Abelian 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.