Speaker: Zhi-Qiang GuoAdvisor: Bo-Qiang Ma

School of Physics, Peking University17th, September, 2008

Plan of this talk:

1.Motivations2.Simple introduction to the method we used

and an example:fermion families from warped extra dimensions

3.Conclusions

This talk is based on our recent paper:

Z.-Q.Guo,B.-Q.Ma, JHEP,08(2008)065, hep-ph/0808.2136.

Two puzzles in particle physics:3 generations: why fermions replicate themselves

Hierarchy structure of fermion’s masses

μe τ e

u c t

d s b

2 5

2 3 5

2 3 6

1, 10 , 10

10 , 10 , 10

10 , 10 , 10

t c u

b s d

e

m m m

m m m

m m m

Many papers about these puzzlesEspecially their solutions in Extra Dimension

backgroundHierarchy structure of fermion massesThe mass hierarchy in 4D originate from the small

overlap of wave functions in high dimensions

Integrating out the high dimensions, get the

effective coupling in 4D

( , ) ( ) ( )x y x f y

Small overlap of wave functions can induces hierarchy structure

c

0

c

0

c

0

y

y

y

y

y

y

dy ( ) ( ) ( ) ( ) ( ) .

dy ( ) ( ) ( ) ( ) ( ) .

dy ( ) ( )

ujRw j kL k

ujRw j k kL

u uw w j k

jk

jk

jk jk

v f U x f y x U x g y h c

v f f y g y U x x U x h c

v f v f f y g y

Works in Warped spacetime:Slice of AdS spacetime, i.e. Rundall-Sundrum

(RS) model Rundall,Sundrum PRL(1999)

The metric

Massive Dirac fermionAction

22 2

2 22

2

, [0, ]

1( )

c

k yds e dx dx dy y y

dzds dx dx

z k

5 ( 2 )2

a N a Na N N a

iS gd x e D D e M

Equation of Motion (EOM)

M is the bulk mass parameterMany papers in this approachGrossman,Neubert PLB(2000); Gherghetta,Pomarol

NPB(2000)

My talk will be based on a concise numerical examples given by Hosotani et.al, PRD(2006)

Gauge-Higgs Unification Model in RS spacetime

The Hierarchy structure in 4D are reproduced by the bulk mass parameters of the same order in 5D

Questions: the origin of the same order bulk mass parameters?

Correlate with the family puzzle: one bulk mass parameter stands for a flavor in a family

The purpose of our work is try to give a solution to this question.

Family problem in Extra Dimensions:Families in 4D from one family in high dimensions

These approaches are adopted in several papers recentlyFrere,Libanov,Nugaev,Troitsky JHEP(2003); Aguilar,Singleton

PRD (2006); Gogberashvili,Midodashvili,Singleton JHEP(2007)

The main point is that fermion zero modes can be trapped by topology objiect-such as vortex; or special metric

Votex solution in 6D, topological number k

Football-like geometry

2 2 2 2 2

4 2

( sin )A BABds G dx dx dx dx R d d

M S

2 42 2 2 2 2sin ,

4 4

11 1 cos ,

2

c bds dx dx d d

a a

One Dirac fermion in the above bachground

EOM in 6D

The 4D zero mode solution

There exist n zero modes, n is limited by the

topological number k, or the parameters in

metric

So families in 4D can be generated from one

family in 6D

6 0D

4 0D

The idea that 1 family in high dimension

can produce several families in lower

dimensions can help us address the question

Question: the origin of the same order bulk mass parameters?

M is the bulk mass parameters

1( ) ( , ) ( , ) 0

4N

N N x y M x y

2. The main point of our paper Consider a 6D spacetime with special metric:

For convenience, we suppose extra dimensions are

both intervals.

A massive Dirac fermion in this spacetime with

action and EOM:

22 2 2 2( )ds B z A y dx dx dy dz

Let

Make conventional Kluza-Klein (KK)decompositions

Note : we expand fermion field in 6D with modes in 5D, that is, at the first step, we reduce 6D spacetime to 5D.

1

2

,

We can get the following relations for each KK modes:

If , that is, RS spacetime, the first

equation above will equals to

RA y

y

1( ) ( , ) ( , ) 0

4N

N N x y M x y

The correspondence

So the origin of the same order bulk mass

parameters are of the same order

Note: should be real numbers by Eq.(2.10).

Further, are determined by the equations

n M

n

nn

For zero modes , these equations decoupled

For massive modes, we can combine the first order equation to get the second order equations

0 0

They are 1D Schrodinger-like equations

correspond to eigenvalues of a 1D Schrodinger-like equation. They are of the same order generally. So it gives a solution of the origin of the same order bulk mass parameters.

n

A problem arises from the following

Contradiction:

On one side: the number of eigenvalues

of Schrodinger-like equation is infinite. There

exists infinite eigenvalues that become larger

and larger

So they produce infinite families in 4D.

1 2 n

On another side: the example of

Hosotani et.al means that larger bulk mass

parameter produces lighter Fermion mass in

4D

So it needs a mechanics to cut off the

infinite series and select only finite

eigenvalues.

It can be implemented by selecting special

metric and imposing appropriate boundary

conditions. We will give an example below.

Before doing that, we discuss the normalization

conditions and boundary conditions from the

action side at first.

Rewriting the 6D action with 5D fermion modes, we get

The conventional effective 5D action

There are two cases for the normalization conditions:

Case (I): the orthogonal conditions

We can convert these orthogonal conditions to the boundary conditions

Two simple choices:

Case (II): the orthogonal conditions are not

satisfied. Then K and M are both matrices

it seems bad, because different 5D modes mixing

not just among the mass terms, but also among the

kinetic terms.

However, we can also get conventional 5D action

if there are finite 5D modes

It can be implemented by diagonalizing the

matrices K and M. The condition is that K is

positive-definite and hermitian.

The new eigenvalues in the action are

We should check that whether they are of the same order.

In the following, we give an example that case II happens.

Suppose a metric

The 1D Schrodinger-like equations are

Solved by hypergeometrical functions

Suppose the range of z to be

When , the boundary conditions requires that

Then n is limited to be finite

This boundary conditions determines the solutions up to the normalization constants

We have no freedom to impose boundary conditions at

Then K and M must be matrices. We should

diagonalize them to get the conventional 5D action.

We give an example that only 3 families are

permitted. The eigenvalues are determined to be

They are of the same order.

Conclusions:

M

M2 M3Fermion masses in 4D:

Hierarchy structure

C1 C2 C3Correspondences in 5D:

Same order

M1

Families in 5D from 1 family in 6D

Eigenvalues of 1D Schrodinger-like equatuion

The problem is that eigenvalues of aSchrodinger-like equation are infinite generally.We suggest a special metric and choose specialboundary conditions to bypass this problem above. There also exists alternative choices: 1. discrete the sixth dimension: the differential

equation will be finite difference equation, in which the number of eigenvalues are finite naturally;

2. construct non-commutative geometry structure in

extra dimension, by appropriate choice of internal manifold, we can get finite KK particles.

(see Madore, PRD(1995))

Only a rough model, far from a realisticone, may supply some hints for future

modelbuilding.