Yukihiro Fujimoto (Kobe Univ. → Osaka Univ.)

(1)

Dynamical generation of mass hierarchy in an extra dimension

Yukihiro Fujimoto, 2014.03.26 Workshop @ Tokyo Woman’s Christian University

Dynamical Generation of Mass Hierarchy

in an Extra Dimension

Yukihiro Fujimoto

(Kobe Univ. → Osaka Univ.)

Collaborating with Takashi Miura (Kobe Univ.)

Makoto Sakamoto (Kobe Univ.) Work in progress.

1

(2)

Dynamical generation of mass hierarchy in an extra dimension

Mysteries of the Standard Model 2

(3)

Dynamical generation of mass hierarchy in an extra dimension

Mysteries of the Standard Model

❒ Mass Hierarchy

1 10 100 1000 10000 100000 1000000

1 2 3

10

⁰

10

⁶

10

²

10

⁴

1 2 3

M as s [M eV ]

Generation u

c

t b d s

2

(4)

Dynamical generation of mass hierarchy in an extra dimension

1 10 100 1000 10000 100000 1000000

1 2 3

10

⁰

10

⁶

10

²

10

⁴

1 2 3

M as s [M eV ]

Generation u

c

t b d s

⇠ 10 5

Mysteries of the Standard Model

❒ Mass Hierarchy

2

(5)

Dynamical generation of mass hierarchy in an extra dimension

❒ Generation

e

u d

ν μ

c s

ν μ τ

t b ν τ e

Yukawa coupling - On

Mysteries of the Standard Model 3

(6)

Dynamical generation of mass hierarchy in an extra dimension

Yukawa coupling - Off

Mysteries of the Standard Model

❒ Generation

3

(7)

Dynamical generation of mass hierarchy in an extra dimension

4

We want to clarify the mysteries of the Standard Model

Purpose

(8)

Dynamical generation of mass hierarchy in an extra dimension

We want to clarify the mysteries of the Standard Model

❒ Mass Hierarchy

❒ Generation

Purpose 4

(9)

Dynamical generation of mass hierarchy in an extra dimension

We want to clarify the mysteries of the Standard Model

❒ Mass Hierarchy

❒ Generation

in the context of higher-dimensional gauge theories.

Purpose 4

(10)

Dynamical generation of mass hierarchy in an extra dimension

Idea 5

(11)

Dynamical generation of mass hierarchy in an extra dimension

Idea 5

❒ Extra dimension

❒ Point interactions (Boundary conditions)

❒ y-dependent scalar VEV

(12)

Dynamical generation of mass hierarchy in an extra dimension

Idea

( x, y )

y = 0 y = L

❒ Extra dimension

5

(13)

Dynamical generation of mass hierarchy in an extra dimension

Idea

( x, y )

y = 0 y = L

₁

y = L

₂

y = L

❒ Extra dimension

5

(14)

Dynamical generation of mass hierarchy in an extra dimension

Idea

( x, y )

y = 0 y = L

₁

y = L

₂

y = L

(¹)

( y )

⁽²⁾

( y )

⁽³⁾

( y )

0 L₁ L₂ L 0 L₁ L₂ L 0 L₁ L₂ L

❒ Extra dimension

5

(15)

Dynamical generation of mass hierarchy in an extra dimension

Idea

( x, y )

y = 0 y = L

₁

y = L

₂

y = L

(¹)

( y )

⁽²⁾

( y )

⁽³⁾

( y )

( x )

RL

0 L₁ L₂ L 0 L₁ L₂ L 0 L₁ L₂ L

( x )

RL R_L

( x )

4 dim.

low energy

❒ Extra dimension

5

(16)

Dynamical generation of mass hierarchy in an extra dimension

( x, y )

y = 0 y = L

₁

y = L

₂

y = L

(¹)

( y )

⁽²⁾

( y )

⁽³⁾

( y )

( x )

RL

0 L₁ L₂ L 0 L₁ L₂ L 0 L₁ L₂ L

( x )

RL R_L

( x )

4 dim.

low energy

❒ Extra dimension

Generation !!

Idea 5

(17)

Dynamical generation of mass hierarchy in an extra dimension

0 L₁ L₂ L

u

_R

( y )

u

_L

( y )

c

_R

( y )

c

_L

( y )

t

_R

( y ) t

_L

( y )

Idea 6

❒ Extra dimension

(18)

Dynamical generation of mass hierarchy in an extra dimension

0 L₁ L₂ L

u

_R

( y )

u

_L

( y )

c

_R

( y )

c

_L

( y )

t

_R

( y ) t

_L

( y ) h ( y ) i ⇠ e

^↵^y

Idea

❒ Extra dimension

6

(19)

Dynamical generation of mass hierarchy in an extra dimension

0 L₁ L₂ L

u

_R

( y )

u

_L

( y )

c

_R

( y )

c

_L

( y )

t

_R

( y ) t

_L

( y )

h ( y ) i ⇠ e

^↵^y

m

_i

=

Z

L

0

h ( y ) i f

⁽ⁱ⁾

( y ) f

⁽ⁱ⁾

( y )

Small overlap → Small mass Large overlap → Large mass

Mass hierarchy !!

Idea

❒ Extra dimension

6

(20)

Dynamical generation of mass hierarchy in an extra dimension

❒

The position of the point interactions are responsible to the mass hierarchy and will be determined dynamically by minimizing the effective potential.

Features 7

Position of the point interactions

(21)

Dynamical generation of mass hierarchy in an extra dimension

0 L

₁

L

₂

L

u

_R

( y )

u

_L

( y )

c

_R

( y ) c

_L

( y )

t

_R

( y ) t

_L

( y )

h ( y ) i ⇠ e

^↵^y

The position of the point interactions are responsible to the mass hierarchy and will be determined dynamically by minimizing the effective potential.

Features

❒ Position of the point interactions

7

(22)

Dynamical generation of mass hierarchy in an extra dimension

0 L

₁

L

₂

L

u

_R

( y )

u

_L

( y )

c

_R

( y ) c

_L

( y )

t

_R

( y ) t

_L

( y )

h ( y ) i ⇠ e

^↵^y

The position of the point interactions are responsible to the mass hierarchy and will be determined dynamically by minimizing the effective potential.

L

₁

L

₁

Features 7

(23)

Dynamical generation of mass hierarchy in an extra dimension

0 L

₁

L

₂

L

u

_R

( y )

u

_L

( y )

c

_R

( y ) c

_L

( y )

t

_R

( y ) t

_L

( y )

h ( y ) i ⇠ e

^↵^y

❒ Position of the point interactions

The position of the point interactions are responsible to the mass hierarchy and will be determined dynamically by minimizing the effective potential.

L

₁

L

₁

❌ ❌

V e↵ [ L 1 , L 2 ] = 0

Features 7

(24)

Dynamical generation of mass hierarchy in an extra dimension

The position of the point interactions are responsible to the mass hierarchy and will be determined dynamically by minimizing the effective potential.

y-dependent scalar VEV can be produced by imposing the Robin boundary condition for the scalar.

YF, Tomoaki Nagasawa, Satoshi Ohya, Makoto Sakamoto Prog. Theor. Phys. 126 (2011)841

h ( x, y ) i = ( y )

Features 7

(25)

Dynamical generation of mass hierarchy in an extra dimension

Point interactions 8

(26)

Dynamical generation of mass hierarchy in an extra dimension

❒ Delta function potential

y ( y )

Point interactions 8

(27)

Dynamical generation of mass hierarchy in an extra dimension

❒ Delta function potential

y ( y )

❒ Orbifold fixed point

Fixed points can be recognized as point intearctions.

(28)

Dynamical generation of mass hierarchy in an extra dimension

❒ Delta function potential

y ( y )

❒ Orbifold fixed point

❒ Zero-thick brane

Zero-this brane can be recognized as point interactions in field theories.

Point interactions

Fixed points can be recognized as point intearctions.

8

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Dynamical generation of mass hierarchy in an extra dimension

❒ Point interaction is described by BC’s.

(30)

Dynamical generation of mass hierarchy in an extra dimension

❒ Point interaction is described by BC’s.

❒ The low energy effective theory (zero mode) is sensitive to the BC’s.

4 parameters (scalar)

❌ 1 parameters (scalar)

❌

( y

_i

) + r

_i

@

_y

( y

_i

) = 0

No parameter (spinor, gauge)

the Robin BC

D f

_n

( y

_i

) = g

_n

( y

_i

) = 0 D

^†

g or

_n

( y

_i

) = f

_n

( y

_i

) = 0

(31)

Dynamical generation of mass hierarchy in an extra dimension

Generation 10

(32)

Dynamical generation of mass hierarchy in an extra dimension

R

( x, 0 ) = 0

_R

( x, L ) = 0

❒ Triply-degenerated chiral zero modes via point interactions

Generation 10

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Dynamical generation of mass hierarchy in an extra dimension

R

( x, 0 ) = 0

_R

( x, L ) = 0

❌ ❌

R

( x, L

₁

) = 0

_R

( x, L

₂

) = 0

❒ Triply-degenerated chiral zero modes via point interactions

Generation 10

(34)

Dynamical generation of mass hierarchy in an extra dimension

!

y = 0 y = L

A

_M

( x, y )

Gauge fields do not feel point interactions to avoid generations.

★

The situation is consistent with higher dimensional gauge invariance.

★

R

( x, 0 ) = 0

_R

( x, L ) = 0

❌ ❌

R

( x, L

₁

) = 0

_R

( x, L

₂

) = 0

❒

Generation

Triply-degenerated chiral zero modes via point interactions

10

(35)

Dynamical generation of mass hierarchy in an extra dimension

R

( x, 0 ) = 0

_R

( x, L ) = 0

❌ ❌

R

( x, L

₁

) = 0

_R

( x, L

₂

) = 0

( x, y ) =

X

3 j=1

(j)

L

( x ) g

⁽^j⁾

0

( y ) + ( KK modes )

❒

Generation

4-dim. mass

eigenstates Mode functions

10

(36)

Dynamical generation of mass hierarchy in an extra dimension

R

( x, 0 ) = 0

_R

( x, L ) = 0

❌ ❌

R

( x, L

₁

) = 0

_R

( x, L

₂

) = 0

( x, y ) =

X

3 j=1

(j)

L

( x ) g

⁽^j⁾

0

D

^†

g

⁽^j⁾

0

( y ) = 0 ( D

^†

⌘ @

_y

+ M

_F

)

❒

Generation

Bulk mass

10

(37)

Dynamical generation of mass hierarchy in an extra dimension

R

( x, 0 ) = 0

_R

( x, L ) = 0

❌ ❌

R

( x, L

₁

) = 0

_R

( x, L

₂

) = 0

( x, y ) =

X

3 j=1

(j)

L

( x ) g

⁽^j⁾

0

D

^†

g

⁽^j⁾

0

( y ) = 0

0 L₁ L₂ L 0 L₁ L₂ L 0 L₁ L₂ L

g

⁽¹⁾

0

( y ) g

⁽²⁾

0

( y ) g

⁽³⁾

0

( y ) ( D

^†

⌘ @

_y

+ M

_F

)

❒

Generation

10

(38)

Dynamical generation of mass hierarchy in an extra dimension

❒ The Robin BC can produce a y-dependent VEV

y

0 L

S = Z

d 4 x

Z L

0

dy † ( @ µ @ µ + @ y 2 M 2 )

2 ( † ) 2

Mass hierarchy 11

(39)

Dynamical generation of mass hierarchy in an extra dimension

y

0 L

( ( 0 ) + L

₊

@ y ( 0 ) = 0

( L ) L @ y ( L ) = 0 ( 1  L

^±

 + 1 )

+ Robin boundary condition S =

Z

d 4 x

Z L

0

dy † ( @ µ @ µ + @ y 2 M 2 )

2 ( † ) 2

❒ The Robin BC can produce a y-dependent VEV

Mass hierarchy 11

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Dynamical generation of mass hierarchy in an extra dimension

❒ VEV of the scalar

V 4d =

Z L

0

dy

"

† ( @ y 2 + M 2 ) +

2 ( † ) 2

#

Mass hierarchy 12

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Dynamical generation of mass hierarchy in an extra dimension

V 4d =

Z L

0

dy

"

† ( @ y 2 + M 2 ) +

2 ( † ) 2

#

Find a solution to : V 4d = 0

( @ y 2 + M 2 ) ( y ) + | ( y ) | 2 ( y ) = 0

❒ VEV of the scalar

Mass hierarchy 12

(42)

Dynamical generation of mass hierarchy in an extra dimension

V 4d =

Z L

0

dy

"

† ( @ y 2 + M 2 ) +

2 ( † ) 2

#

Find a solution to : V 4d = 0

( ( 0 ) + L

₊

@ y ( 0 ) = 0

( L ) L @ y ( L ) = 0 ( 1  L

^±

 + 1 )

+ Robin boundary condition

( @ y 2 + M 2 ) ( y ) + | ( y ) | 2 ( y ) = 0

❒ VEV of the scalar

Mass hierarchy 12

(43)

Dynamical generation of mass hierarchy in an extra dimension

( type-i )

(y ) = µ sn(µ

₊

q

2

(y y

₀

), k) cn(µ

₊

q

2

(y y

₀

), k)

µ_± ⌘ M²

(1 ± r

1 4 Q

M⁴ )

k² ⌘ µ²₊ µ² µ²₊

,

❒ VEV of the scalar

Mass hierarchy 13

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Dynamical generation of mass hierarchy in an extra dimension

( type-i )

(y ) = µ sn(µ

₊

q

2

(y y

₀

), k) cn(µ

₊

q

2

(y y

₀

), k)

µ_± ⌘ M²

(1 ± r

1 4 Q

M⁴ )

k² ⌘ µ²₊ µ² µ²₊

,

❒ VEV of the scalar

Parameters which are determined by BC’s

Mass hierarchy 13

(45)

Dynamical generation of mass hierarchy in an extra dimension

( type-i )

(y ) = µ sn(µ

₊

q

2

(y y

₀

), k) cn(µ

₊

q

2

(y y

₀

), k)

µ_± ⌘ M²

(1 ± r

1 4 Q

M⁴ )

( type-ii )

(y ) = ⌫

cn( q

2 µ

k

(y y

₀

), k)

µ ⌘ M²

(1 + s

1 + 4 |Q| M⁴ )

⌫ ⌘ M² (

s

1 + 4 |Q|

M⁴ 1)

k² ⌘ µ² µ² + ⌫² k² ⌘ µ²₊ µ²

µ²₊

,

❒ VEV of the scalar

Mass hierarchy

Parameters which are determined by BC’s

13

(46)

Dynamical generation of mass hierarchy in an extra dimension

0 L

₁

L

₂

L

u

_R

( y )

u

_L

( y )

c

_R

( y ) c

_L

( y )

t

_R

( y ) t

_L

( y )

h ( y ) i ⇠ e

^↵^y

❒

After we tune the BC’s for type-(ii), we obtain L

_±

( y ) = ⌫

cn ( q

2 µ

k ( y y 0 , k )

⇠ e M y

VEV of the scalar

Mass hierarchy 14

(47)

Dynamical generation of mass hierarchy in an extra dimension

❒ The position of point interactions are determined by the minimization of the effective potential

Z [ L

₁

, L

₂

] = Z

[ d d ¯ ] e

^{i S}^F

Z Euclid [ L 1 , L 2 ] / exp (

V e↵ [ L 1 , L 2 ] Z

d 4 x E )

Mass hierarchy 15

(48)

Dynamical generation of mass hierarchy in an extra dimension

❒ The position of point interactions are determined by the minimization of the effective potential

=

X

3

j =

1

| M F |

²

8⇡

²

( L j L j

₁

)

²

X 1 n =

₁

e

²

| M

^F

| n ( L

^j

L

^j ¹

) n

³

⇥ 0

B@ 1 + 3

2 | M

_F

| n ( L

_j

L

_j ₁

) + 3

4 | M

_F

|

²

n

²

( L

_j

L

_j ₁

)

²

1 CA

+ (L - independent divergence) ( L

₀

= 0 , L

₃

= L ) Z [ L

₁

, L

₂

] =

Z

[ d d ¯ ] e

^{i S}^F

Z Euclid [ L 1 , L 2 ] / exp (

V e↵ [ L 1 , L 2 ] Z

d 4 x E )

V e↵ [ L 1 , L 2 ]

Mass hierarchy 15

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Dynamical generation of mass hierarchy in an extra dimension

u R ( y )

u L ( y )

c R ( y ) c L ( y )

t R ( y ) t L ( y )

h ( y ) i ⇠ e ↵ y

= = =

y = 0 y = L

3 y = 2 L

3 y = L

V e↵ [ L 1 , L 2 ] reg = 0

❒ The position of point interactions are determined by the minimization of the effective potential

Mass hierarchy 16

(50)

Dynamical generation of mass hierarchy in an extra dimension

Conclusion and Discussion

❒ Generation can be produced by point

interactions in the context of 5 dim. gauge theory.

17

(51)

Dynamical generation of mass hierarchy in an extra dimension

❒ Generation can be produced by point

❒ y-dependent scalar VEV leads to a exponential mass hierarchy for fermions.

Conclusion and Discussion 17

(52)

Dynamical generation of mass hierarchy in an extra dimension

❒ Generation can be produced by point

❒ The positions of point interactions are

responsible to the mass hierarchy and are

determined by minimizing the effective potential.

❒ y-dependent scalar VEV leads to a exponential mass hierarchy for fermions.

Conclusion and Discussion 17

(53)

Dynamical generation of mass hierarchy in an extra dimension

★ What is the physical meaning of the point interactions and how it can be produced?

(54)

Dynamical generation of mass hierarchy in an extra dimension

★ What is the physical meaning of the point interactions and how it can be produced?

How can we produce the differences between up-quark sector and down-quark sector?

★

(55)

Dynamical generation of mass hierarchy in an extra dimension

★ What is the physical meaning of the point interactions and how it can be produced?

How can we produce the differences between up-quark sector and down-quark sector?

Flavor structure …?

★