Spontaneous Compactification of Bimetric Theory based on - - PowerPoint PPT Presentation

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Spontaneous Compactification of Bimetric Theory based on - - PowerPoint PPT Presentation

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1 Spontaneous Compactification of Bimetric Theory based on arXiv:1405.0064 [hep-th] N a h o mi K a n ( G i f u N a t i o n a l C o l l e g e o f T e c h n o l o g y ) , T a k u y a M a k i ( J

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SLIDE 1

Spontaneous Compactification of Bimetric Theory

based on arXiv:1405.0064 [hep-th] N a h

  • mi

K a n ( G i f u N a t i

  • n

a l C

  • l

l e g e

  • f

T e c h n

  • l
  • g

y ) , T a k u y a M a k i ( J a p a n Wo me n ' s C

  • l

l e g e

  • f

P h y s i c a l E d u c a t i

  • n

) K i y

  • s

h i S h i r a i s h i ( Y a ma g u c h i U n i v e r s i t y )

1 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4

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SLIDE 2

In a six-dimensional model

  • f

bimetric theory, massless and massive gravitons emerge with a spontaneous compactification to four dimensions.

2 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4

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SLIDE 3

Why massive gravity?

・H

  • w

d

  • e

s g r a v i t y b e h a v e a t a c

  • s

mo l

  • g

i c a l s c a l e ? ・ I n t h e v e r y e a r l y u n i v e r s e , i s t h e e v

  • l

u t i

  • n
  • f

s c a l e f a c t

  • r

s d i f f e r e n t f r

  • m

t h e p r e s e n t b e h a v i

  • r

? ・A s a n e x e r c i s e f

  • r

c

  • v

a r i a n t a p p r

  • a

c h t

  • q

u a n t u m g r a v i t y

Problems in massive gravity

・vDVZ discontinuity ( v a n D a m & V e l t ma n ; Z a k h a r

  • v

, 1 9 7 ) t h e l i mi t m→0 i s d i f f e r e n t f r

  • m

G R ・Vainshtein Mechanism ( 1 9 7 2 P L B 3 9 ) n

  • n

l i n e a r e f f e c t s a t s t r

  • n

g g r a v i t y i s i mp

  • r

t a n t ( ! ? ) ・Boulware-Deser Ghost ( 1 9 7 2 P R D 6 ) A g h

  • s

t d e g r e e

  • f

f r e e d

  • m

a p p e a r s i n n

  • n

l i n e a r ma s s i v e g r a v i t y , b e c a u s e i t i s i mp

  • s

s i b l e t

  • ma

k e a s c a l a r c

  • n

s t r a i n t w i t h

  • u

t d e r i v a t i v e s ( f r

  • m

a s i n g l e me t r i c ) .

3 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4

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SLIDE 4

B i g r a v i t y ( B i me t r i c g r a v i t y ) ma s s l e s s a n d ma s s i v e g r a v i t

  • n

s

G h

  • s

t

  • f

r e e N

  • n
  • l

i n e a r ma s s i v e g r a v i t y (

  • n

e me t r i c i s n

  • n
  • d

y n a mi c a l ) d R G T ( d e R h a m, G a b a d a d z e , T

  • l

l e y ) 2 1 H a s s a n , R

  • s

e n 2 1 1 G h

  • s

t

  • f

r e e N

  • n
  • l

i n e a r B i g r a v i t y H a s s a n , R

  • s

e n , S c h mi d t

  • M

a y , 2 1 2

( v e r y s

  • r

r y t

  • ma

n y

  • t

h e r a u t h

  • r

s f

  • r
  • mi

t t i n g t h e i r w

  • r

k . . . )

4 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4

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SLIDE 5

mass term ～ - g Σ

n C

nδ μ1 μ2 . . . μn ν1 ν2 . . . νnK ν1 μ1K ν2 μ2…K νn μn

two metrics g

μν ,

f

μν

K

ν μ＝

g

νρ

f

ρμ

T h e g e n e r a l i z e d K r

  • n

e c h e r d e l t a

5 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4

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SLIDE 6

vierbein massive gravity

mass term ～Σ

n c

nδ M

1

M

2

. . . M

n

A

1

A

2

. . . A

D e

g A

1

M

1

… e

f A

D

M

D

( n e

g'

s , D

  • n

e

f'

s ) A c c

  • r

d i n g t

  • A

l e x a n d r

  • v

( G R G 4 6 ( 2 1 4 ) 1 6 3 9 ) , v i e r b e i n b i g r a v i t y i s a l s

  • G

h

  • s

t f r e e .

6 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4

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SLIDE 7

Six-dimensional Model of BimetricTheory

In models of Bigravity, Massive gravity mass of massive graviton <

  • '

n e w ' s c a l e , b y h a n d ( ? ) we wish to consider connections to some

  • ther

scales and/or mechanisms. We n e e d mi x i n g p a r t i n t h e a c t i

  • n
  • f

b i me t r i c t h e

  • r

y . g a u g e f i e l d mi x i n g ( H

  • l

d

  • m,

1 9 8 6 )

( t h i s i s a l s

  • u

s e d i n r e c e n t mo d e l s

  • f

d a r k ma t t e r )

7 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4

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SLIDE 8

i d e a : C

  • n

s i d e r f l u x mi x i n g a n d me t r i c mi x i n g a t t h e s a me t i me = a t t h e s t a g e

  • f

c

  • mp

a c t i f i c a t i

  • n
  • f

e x t r a s p a c e ! T h e s i x

  • d

i me n s i

  • n

a l mo d e l ： α: a d i me n s i

  • n

l e s s p a r a me t e r

8 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4

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SLIDE 9

Compactification

e x t r a s p a c e ＝S

2

( t w

  • me

t r i c s

  • f

t w

  • r

a d i i i n g e n e r a l ) curvatures: fluxes:

，

9 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4

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SLIDE 10

Compactification of Six-dimensional Einstein-Maxwell theory as in Randjbar-Daemi, Salam, Strathdee NPB214(1983)

a static solution is given as a minimum point of the effective potential for scales of extra dimensions with fine-tuning to obtain flat four-dimensional spacetime

  • r, if we set

,

1 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4

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SLIDE 11

V i s mi n i mi z e d a t y ＝y

0:

t h e p a r a me t e r t u n i n g mu s t b e d

  • n

e a s t h e a b

  • v

e t a k e s a mi n i mu m v a l u e a t x ＝x

1 1 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4

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SLIDE 12

masses of gravitons The four-dimensional effective action for gravitons after the compactification I n t h e w e a k f i e l d l i mi t : , w e f i n d

1 2 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4

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SLIDE 13

a n d N

  • w

w e g e t t h e L a g r a n g i a n f

  • r

t w

  • g

r a v i t

  • n

s h

g,

h

f.

T

  • e

l i mi n a t e t h e l i n e a r t e r m i n h

g,

h

f,

we s h

  • u

l d c h

  • s

e t h e p a r a me t e r s a s

1 3 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4

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SLIDE 14

These come from Two constraints from variations of two lapse functions. Then, the Lagrangian for two gravitons reads: w h e r e ma s s l e s s ma s s i v e

1 4 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4

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SLIDE 15

A s i mp l e c a s e : ・ ma s s

  • s

q u a r e

  • f

ma s s i v e g r a v i t

  • n

: ( t u n i n g : e t c . ) S i n c e ， ma s s

  • f

ma s s i v e g r a v i t

  • n

i s l e s s t h a n t h a t

  • f

a K K mo d e g e n e r a l i z a t i

  • n

t

  • mu

l t i g r a v i t y

  • >

s p e c t r a w i t h h i e r a r c h y

1 5 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4

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SLIDE 16

Summary and outlook

In a Six-dimensional Bimetric model, massless and massive gravitons appear at the spontaneous compactification. ・Cosmology (many constraints!) ・Spectrum of spin2, spin1, spin0 ・Tuning-free theory? (extension of models of Salam-Sezgin-Nishino-Maeda?)

1 6 S t r i n g s a n d F i e l d s , Y I T P , 2 5 J u l y 2 1 4