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Phenomenological aspects of magnetized brane models Tatsuo Kobayashi. 1. Introduction 2. Magnetized extra dimensions 3. N-point couplings and flavor symmetries 4 . Massive modes 5. Flavor and Higgs sector 6. Summary based on collaborations with

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Phenomenological aspects of magnetized brane models tatsuo kobayashi
Phenomenological aspects of magnetized brane models Tatsuo Kobayashi

1.Introduction

2. Magnetized extra dimensions

3. N-point couplings and flavor symmetries

4.Massive modes

5. Flavor and Higgs sector

6. Summary

based on collaborations with

H.Abe, T.Abe, K.S.Choi, Y.Fujimoto, Y.Hamada,R.Maruyama, T.Miura, M.Murata, Y.Nakai, K.Nishiwaki, H.Ohki,

A.Oikawa, M.Sakai, M.Sakamoto, K.Sumita, Y.Tatsuta


Introduction
Introduction

Extra dimensional field theories,

in particular

string-derived extra dimensional field theories,

play important roles in particle physics

as well as cosmology .


Extra dimensions
Extra dimensions

4 + n dimensions

4D ⇒our 4D space-time

nD  ⇒ compact space

Examples of compact space

torus, orbifold, CY, etc.


Field theory in higher dimensions
Field theory in higher dimensions

10D  ⇒ 4D our space-time + 6D space

10D vector

4D vector + 4D scalars

SO(10) spinor ⇒SO(4) spinor

x SO(6) spinor

internal quantum number


Several fields in higher dimensions
Several Fields in higher dimensions

4D (Dirac) spinor

⇒(4D) Clifford algebra

(4x4) gamma matrices

represention space ⇒ spinor representation

6D Clifford algebra

6D spinor

6D spinor   ⇒  4D spinor x (internal spinor)

internal quantum

number


Field theory in higher dimensions1
Field theory in higher dimensions

Mode expansions

KK decomposition


Kk docomposition on torus
KK docomposition on torus

torus with vanishing gauge background

Boundary conditions

We concentrate on zero-modes.


Zero modes
Zero-modes

Zero-mode equation

⇒non-trival zero-mode profile

the number of zero-modes


4d effective theory
4D effective theory

Higher dimensional Lagrangian (e.g. 10D)

integrate the compact space ⇒4D theory

Coupling is obtained by the overlap

integral of wavefunctions


Couplings in 4d
Couplings in 4D

Zero-mode profiles are quasi-localized

far away from each other in compact space

⇒suppressed couplings


Chiral theory
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.


Magnetic flux
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.


Magnetic flux1
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
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.


量子力学の復習:磁場中の粒子(Landau)

座標がk/bずれた調和振動子b=整数

b個の基底状態    k=0,1,2,…………,(b-1)


2 extra dimensions with magnetic fluxes basic tools
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

4D Effective theory <= dimensional reduction

eigenstates of corresponding

internal Dirac/Laplace operator.

The wave functions


Higher Dimensional SYM theory with flux

Abeliangauge field on magnetized torus

Constant magnetic flux

gauge fields of background

The boundary conditions on torus (transformation under torus translations)


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 on 2D torus

is the two component spinor.

U(1) charge Q=1

with twisted boundary conditions (Q=1)


Dirac equation and chiral fermion

|M| independent zero mode solutions in Dirac equation.

(Theta function)

Properties of theta functions

chiral fermion

By introducing magnetic flux, we can obtain chiral theory.

:Normalizable mode

:Non-normalizable mode


Wave functions

For the case of M=3

Wave function profile on toroidal background

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

Breaking the gauge group

(Abelian flux case )

The gaugino fields

gaugino of unbroken gauge

bi-fundamental matter fields


Bi fundamental
Bi-fundamental

Gaugino fields in off-diagonal entries

correspond to bi-fundamental matter fields

and the difference M= m-m’ of magnetic

fluxes appears in their Dirac equation.

F


Zero-modes Dirac equations

No effect due to magnetic flux for adjoint matter fields,

Total number of zero-modes of

:Normalizable mode

:Non-Normalizable mode


4d chiral theory
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


U 8 sym theory on t6
U(8) SYM theory on T6

Pati-Salam group up to U(1) factors

Three families of matter fields

with many Higgs fields


2 2 wilson lines
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.


U 1 a u 1 b theory
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,


Pati salam model
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

(the others are massive.)

U(1)Y is a linear combination.


Ps sm
PS => SM

Zero modes corresponding to

three families of matter fields

remain after introducing WLs, but their profiles split

(4,2,1)

Q L


Other models
Other 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
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


Splitting zero mode profiles
Splitting zero-mode profiles

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

16

Q……L


2 3 orbifold with magnetic flux
2.3 Orbifold with magnetic flux

S1/Z2 Orbifold

There are two singular points,

which are called fixed points.


Orbifolds
Orbifolds

T2/Z3 Orbifold

There are three fixed points on Z3orbifold

(0,0), (2/3,1/3), (1/3,2/3) su(3) root lattice

T2/Z4, T2/Z6

Orbifold = D-dim. Torus /twist

Torus = D-dim flat space/ lattice


Orbifold with magnetic flux
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


Wave functions

For the case of M=3

Wave function profile on toroidal background

Zero-modes wave functions are quasi-localized far away each other in extra dimensions. Therefore the hierarchirally small Yukawa couplings may be obtained.


Zero modes on orbifold
Zero-modeson 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.


Zero modes on zn orbifold
Zero-modesonZN orbifold

Similarly we can discuss

2D ZN orbifolds with magnetic fluxes

for N=2,3,4 and 6.

Abe, Fujimoto, T.K., Miura, Nishiwaki,

Sakamoto, arXiv:1309.4925


N point couplings and flavor symmetries
N-point couplings and flavor symmetries

The N-point couplings are obtained by

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


Moduli
Moduli

Torus metric

Area

We can repeat the previous analysis.

Scalar and vector fields have the same

wavefunctions.

Wilson moduli

shift of w.f.


Zero modes1
Zero-modes

Cremades, Ibanez, Marchesano, ‘04

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

up to normalization factor


Products of wave functions
Products of wave functions

products of zero-modes = zero-modes


3 point couplings
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


Selection rule
Selection rule

Each zero-mode has a Zg charge,

which is conserved in 3-point couplings.

up to normalization factor


4 point couplings
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



N point couplings
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 wouldbe 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
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


Non abelian discrete flavor symmetry
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


Permutation symmetry d brane models
Permutation symmetry D-brane models

Abe, Choi, T.K. Ohki, ’09, ‘10

There is a Z2 permutation symmetry.

The full symmetry is D4.


Permutation symmetry d brane models1
Permutation symmetry D-brane models

Abe, Choi, T.K. Ohki, ’09, ‘10

geometrical symm. Full symm.

Z3 Δ(27)

S3 Δ(54)


Intersecting magnetized d brane models
intersecting/magnetized D-brane models

Abe, Choi, T.K. Ohki, ’09, ‘10

generic intersecting number g

magnetic flux

flavor symmetry is a closed algebra of

two Zg’s.

and Zg permutation

Certain case: Zg permutation larger symm. Like Dg


Magnetized brane models
Magnetized brane-models

Magnetic flux M D4

2 2

4 1++ + 1+- +1-+ + 1--

  ・・・           ・・・・・・・・・

Magnetic flux M Δ(27) (Δ(54))

3 31

6 2 x 31

9 ∑1n n=1,…,9

(11+∑2n n=1,…,4)

 ・・・           ・・・・・・・・・


Non abelian discrete flavor symm
Non-Abelian discrete flavor symm.

Recently, in field-theoretical model building,

several types of discrete flavor symmetries have

been proposed with showing interesting results,

e.g. S3, D4, A4, S4, Q6, Δ(27), ......

Review: e.g

Ishimori, T.K., Ohki, Okada, Shimizu, Tanimoto ‘10

⇒ large mixing angles

one Ansatz: tri-bimaximal


3 2 applications of couplings
3.2 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.

Ratios depend on complex structure moduli

and Wilson lines.


Quark lepton masses matrices
Quark/lepton masses matrices

Abe, T.K., Ohki, Oikawa, Sumita, arXiv:1211.437

assumption on light Higgs scalar

The overall gauge coupling is fixed through

the gauge coupling unification.

Vary other parameters, WLs and

complex strucrure with 10% tuning

(5 free parameters)


Quark lepton masses and mixing angles
Quark/lepton masses and mixing angles

Abe, T.K., Ohki, Oikawa, Sumita, arXiv:1211.437

Example

Flavor is still a challenging issue.


4 massive modes
4. Massive modes

Hamada, T.K. arXiv:1207.6867

Massive modes play an important role

in 4D LEEFT such as the proton decay,

FCNCs, etc.

It is important to compute mass spectra of

massive modes and their wavefunctions.

Then, we can compute couplings among

massless and massive modes.


Fermion massive modes
Fermion massive modes

Two components are mixed.

2D Laplace op.

algebraic relations

It looks like the quantum harmonic oscillator


Fermion massive modes1
Fermion massive modes

Creation and annhilation operators

mass spectrum

wavefunction


Fermion massive modes2
Fermion massive modes

explicit wavefunction

Hn: Hermite function

Orthonormal condition:


Scalar and vector modes
Scalar and vector modes

The wavefunctions of scalar and vector fields

are the same as those of spinor fields.

Mass spectrum

scalar

vector

Scalar modes are always massive on T2.

The lightest vector mode along T2,

i.e. the 4D scalar, is tachyonic on T2.

Such a vector mode can be massless on T4 or T6.


T duality1
T-duality ?

Mass spectrum

spinor

scalar

vector

the same mass spectra as excited modes

(with oscillator excitations )

in intersecting D-brane models, i.e. “gonions”

Aldazabal, Franco, Ibanez, Rabadan, Uranga, ‘01


Products of wavefunctions
Products of wavefunctions

explicit wavefunction

See also Berasatuce-Gonzalez, Camara, Marchesano,

Regalado, Uranga, ‘12

Derivation:

products of zero-mode wavefunctions

We operate creation operators on both LHS

and RHS.


3 point couplings including higher modes
3-point couplings including higher modes

The 3-point couplings are obtained by

overlap integral of three wavefunctions.

(flavor) selection rule

is the same as one for the massless modes.

(mode number) selection rule


3 point couplings 2 zero modes and one higher mode
3-point couplings:2 zero-modes and one higher mode

3-point coupling


Higher order couplings including higher modes
Higher order couplings including higher modes

Similarly, we can compute higher order couplings

including zero-modes and higher modes.

They can be written by the sum over

products of 3-point couplings.


3 point couplings including massive modes only due to wilson lines
3-point couplings including massive modes only due to Wilson lines

Massive modes appear only due to Wilson lines

without magnetic flux

We can compute the 3-point coupling

e.g.

Gaussian function for the Wilson line.


3 point couplings including massive modes only due to wilson lines1
3-point couplings including massive modes only due to Wilson lines

For example, we have

for


Several couplings
Several couplings Wilson lines

Similarly, we can compute the 3-point couplings

including higher modes

Furthermore, we can compute higher order

couplings including several modes, similarly.


4 2 phenomenological applications
4.2 Phenomenological applications Wilson lines

In 4D SU(5) GUT,

The heavy X boson couples with quarks and leptons

by the gauge coupling.

Their couplings do not change even after GUT breaking

and it is the gauge coupling.

However, that changes in our models.


Phenomenological applications
Phenomenological applications Wilson lines

For example,

we consider the SU(5)xU(1) GUT model

and we put magnetic flux along extra U(1).

The 5 matter field has the U(1) charge q,

and the quark and lepton in 5 are quasi-localized

at the same place.

Their coupling with the X boson is given by

the gauge coupling before the GUT breaking.


Su 5 sm
SU(5) => SM Wilson lines

We break SU(5) by the WL along the U(1)Y direction.

The X boson becomes massive.

The quark and lepton in 5 remain massless, but their

profiles split each other.

Their coupling with X is not equal to the gauge coupling,

but includes the suppression factor

5

Q L


Proton decay
Proton decay Wilson lines

Similarly, the couplings of the X boson with quarks and

leptons in the 10 matter fields can be suppressed.

That is important to avoid the fast proton decay.

The proton life time would drastically

change by the factor,


Other aspects
Other aspects Wilson lines

Other couplings including massless and massive modes

can be suppressed and those would be important ,

such as right-handed neutrino masses and

off-diagonal terms of Kahler metric, etc.

Threshold corrections on the gauge couplings,

Kahler potential after integrating out massive modes


5 flavor and higgs sector
5 Wilson lines. Flavor and Higgs sector

factorizable magnetic fluxes,

non-vanishing F45, F67, F89

Three generations of both left-handed and

right-handed quarks originate only from

the same 2D torus.

-> the number of higgs = 6.

Δ(27) flavor symmetries

three families = triplet

higgs fields = 2x (triplet)


Flavor and higgs sector
Flavor and Higgs sector Wilson lines

Abe, T.K. , Ohki, Sumita, Tatsuta ’ 1307.1831

non-factorizable magnetic fluxes,

non-vanishing F45, F67, F89 and F46, ………..

Three generations of both left-handed and

right-handed quarks originate from 4D torus

-> several patterns of higgs sectors,

one pair, two , …….

Δ(27) flavor symmetries are vilolated.


Flavor and higgs sector1
Flavor and Higgs sector Wilson lines

In most of cases,

there are multi higgs fields.

in a certain case, Δ(27) flavor symmetry

 What is phenomenological aspects

of multi higgs fields, e.g. in LHC ?


Summary
Summary Wilson lines

We have studiedphenomenological 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.


Further studies
Further studies Wilson lines

Derivation of realistic values of quark/lepton

masses and mixing angles.

Applications of flavor symmetries

and studies on their anomalies

Phenomenological aspects of massive modes


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