General Analysis of LARGE General Analysis of LARGE Volume - - PowerPoint PPT Presentation

General Analysis of LARGE General Analysis of LARGE Volume Scenarios with String Volume Scenarios with String Loop Moduli Stabilisation Loop Moduli Stabilisation

Michele Cicoli Michele Cicoli

Department of Applied Mathematics and Theoretical Physics Department of Applied Mathematics and Theoretical Physics University of Cambridge University of Cambridge SP08, Upenn, 29 May 2008 SP08, Upenn, 29 May 2008 Based on: 1) M. Cicoli, J. Conlon, F. Quevedo arXiv:0708.1873 [hep-th] 2) M. Cicoli, J. Conlon, F. Quevedo arXiv:0805.1029 [hep-th] 3) M. Cicoli, F. Quevedo (in preparation)

Outline Outline

• Brief Review of Moduli Stabilisation in IIB

Brief Review of Moduli Stabilisation in IIB SUGRA SUGRA

• General Analysis of LARGE Volume

General Analysis of LARGE Volume Scenarios Scenarios

• General Structure of String Loop Corrections

General Structure of String Loop Corrections in IIB SUGRA in IIB SUGRA

• Inclusion of String Loop Corrections

Inclusion of String Loop Corrections

• Calabi

Calabi-

• Yau examples

Yau examples

• Conclusions

Conclusions

Type IIB CY Flux Compactifications Type IIB CY Flux Compactifications

• Low energy limit to N=1 4D Type IIB SUGRA via dimensional

Low energy limit to N=1 4D Type IIB SUGRA via dimensional reduction reduction Need to know f, W, K! Need to know f, W, K!

• Ubiquitous presence of moduli: massless uncharged scalar particl

Ubiquitous presence of moduli: massless uncharged scalar particles es with effective gravitational coupling that would give rise to lo with effective gravitational coupling that would give rise to long ng range unobserved fifth forces range unobserved fifth forces Need to stabilise them! Need to stabilise them!

• Closed string moduli: U (complex structure), S (axio

Closed string moduli: U (complex structure), S (axio-

• dilaton), T

dilaton), T (K (Kä ähler) hler)

• Open string moduli: Wilson lines, D3 and D7 moduli

Open string moduli: Wilson lines, D3 and D7 moduli

• Turn on background 3

Turn on background 3-

• form fluxes (GKP)

form fluxes (GKP) D DU

UW =

W = D Ds

sW

W =0 =0 fixes U and S moduli fixes U and S moduli supersymmetrically supersymmetrically

• No

No-

• scale structure flat potential for T moduli at tree

scale structure flat potential for T moduli at tree level level T moduli still unfixed! T moduli still unfixed!

• Need to study perturbative versus non

Need to study perturbative versus non-

• perturbative corrections!

perturbative corrections!

Perturbative Perturbative vs vs Non perturbative Non perturbative

• In general:

In general: where where for for and with and with Neglect loop corrections and get Neglect loop corrections and get negligible negligible

LARGE Volume LARGE Volume

• Fix T moduli in a

Fix T moduli in a natural natural way way

• Set

Set W W0

0 ~

~ 1 1-

• 10

10 and and V V ~exp( ~exp(a ai

iτ

τi

i)

)

α α’ ’ and non and non-

• perturbative corrections compete naturally to give

perturbative corrections compete naturally to give an exponentially large volume AdS minimum that breaks an exponentially large volume AdS minimum that breaks SUSY SUSY

• Need to up

Need to up-

• lift to dS (anti

lift to dS (anti-

• D3, D

D3, D-

• terms, F

terms, F-

• terms, non

terms, non-

• perturbative

perturbative α α’ ’ corr corr.) .)

• Simplest realisation of LVS:

Simplest realisation of LVS: P P4

4 [1,1,1,6,9] [1,1,1,6,9] with h

with h11

11=2

=2

• Generalisation: Swiss

Generalisation: Swiss-

• cheese CY

cheese CY (F (F11

11,

, P P4

4 [1,3,3,3,5], [1,3,3,3,5],

P P4

4 [1,1,3,10,15] [1,1,3,10,15] for h

for h11

11=3 )

=3 )

Mass Mass scales scales

NB NB Get Get a a robust effective field theory robust effective field theory and generate and generate hierarchies hierarchies (axionic, (axionic, weak weak and neutrino scale)!!! and neutrino scale)!!! ( (for for V V ~ ~ 10 1015

15 l

ls

s 6 6)

)

Some open Some open questions questions in LVS in LVS

• No

No general analysis general analysis: : what happens for an arbitrary what happens for an arbitrary CY? CY?

• No

No inclusion inclusion of

• f loop corrections for an arbitrary

loop corrections for an arbitrary CY CY ( (except except Swiss Swiss-

• cheese

cheese CY CY by by BHP) BHP)

• Tension between

Tension between moduli moduli stabilisation stabilisation and and chirality chirality

• Inflation flatness spoiled by loop corrections

Inflation flatness spoiled by loop corrections

• No

No detectable tensor modes detectable tensor modes ⇒ ⇒ LARGE Volume

LARGE Volume Claim Claim

⇒ ⇒ general structure of string loop corrections in IIB SUGRA

general structure of string loop corrections in IIB SUGRA

⇒ ⇒ string loop corrections can give a solution:

string loop corrections can give a solution: P P4

4 [1,3,3,3,5] [1,3,3,3,5]

⇒ ⇒ string loop corrections can give a solution:

string loop corrections can give a solution: K3 Fibration K3 Fibration ??? ???

LARGE Volume Claim LARGE Volume Claim

Proof Proof: 40 : 40 pages pages long long → → skip it skip it! !

Necessary and Sufficient Necessary and Sufficient Conditions for LARGE Volume Conditions for LARGE Volume

• h

h12

12 > h

> h11

11 > 1

> 1 ξ ξ > 0 > 0

• At least one blow

At least one blow-

• up mode (point

up mode (point-

• like singularity) with a non

like singularity) with a non-

• perturbative superpotential (guaranteed since the cycle is rigid

perturbative superpotential (guaranteed since the cycle is rigid!) !)

• Blow

Blow-

• up modes fixed by non

up modes fixed by non-

• perturbative effects, volume by

perturbative effects, volume by α α’ ’ corrections + corrections + W Wnp

np

• For

For N Nsmall

small blow

blow-

• up modes, there are still L=(h

up modes, there are still L=(h11

11-

• N

Nsmall

small-

• 1) moduli which

1) moduli which are sent large (e.g. fibration moduli) are sent large (e.g. fibration moduli) their non their non-

• perturbative corrections are switched off

perturbative corrections are switched off

• Get L flat directions!

Get L flat directions!

• These directions are usually lifted by string loop corrections

These directions are usually lifted by string loop corrections L moduli lighter than the volume! L moduli lighter than the volume!

String Loop Corrections to K String Loop Corrections to K

• Explicit calculation known only for unfluxed

Explicit calculation known only for unfluxed toroidal orientifolds toroidal orientifolds as as where where is is due due to to the the exchange exchange of KK

• f KK strings between

strings between D7s and D3s and D7s and D3s and is is due due to to the the exchange exchange of Winding

• f Winding strings between intersecting

strings between intersecting D7s D7s NB NB Complicated dependence Complicated dependence on the U moduli BUT

• n the U moduli BUT simple dependence

simple dependence on

• n

the T moduli! the T moduli! (BHK) (BHK)

Generalisation to CY Generalisation to CY

• Generalisation to

Generalisation to Calabi Calabi-

• Yau

Yau three three-

• folds

folds (BHP) (BHP)

where either where either

• r
• r

In In fact for fact for T T6

6/(Z

/(Z2

2×

×Z Z2

2):

): Conjecture for Conjecture for an arbitrary an arbitrary CY! CY! transverse to transverse to D7 D7 intersection intersection of D7s

• f D7s

Low Energy Approach Low Energy Approach

• The

The reduced reduced DBI action DBI action contains contains a a term like term like

• Expand

Expand τ τ around its around its VEV VEV

• Read

Read off the gauge

• ff the gauge coupling

coupling

• Loop corrected kinetic terms

Loop corrected kinetic terms

• Analogy with charged

Analogy with charged scalar scalar fields fields

Example Example: : P P4

4 [1,1,1,6,9] [1,1,1,6,9]

• Match the

Match the conjecture for conjecture for

• Loop

Loop corrections from corrections from the the conjecture for conjecture for P P4

4 [1,1,1,6,9] [1,1,1,6,9]

Low Energy Interpretation Low Energy Interpretation: :

• Match the

Match the conjecture also for conjecture also for P P4

4 [1,1,2,2,6] [1,1,2,2,6]

(BHP) (BHP) NB NB Loops Loops are are leading with leading with respect to respect to the the α α’ ’ corrections corrections! !

OK! OK!

Extended Extended No No-

• scale

scale Structure Structure

Proof Proof: : Expand Expand K K-

• 1

1 and

and use homogeneity use homogeneity! ! The loop corrections to V are subleading with respect to the The loop corrections to V are subleading with respect to the α α’ ’ ones BUT

• nes BUT

are crucial to stabilise the SM cycle or to lift the L flat dire are crucial to stabilise the SM cycle or to lift the L flat directions!!! ctions!!!

General General formula formula for for the 1 the 1 loop loop corrections to corrections to V V

Homogeneity allows us to simplify it to Homogeneity allows us to simplify it to: : NB NB Everything Everything in in terms terms of

• f K

Kii

ii and

and δ δK KW

W!!!

!!!

NB NB We know We know the the sign sign of

• f these

these corrections corrections!!! !!!

Field Theory Interpretation Field Theory Interpretation

• Use

Use the the Coleman Coleman-

• Weinberg Potential

Weinberg Potential

• Single

Single modulus example modulus example where where due due to to SUSY SUSY SUSY SUSY is is the the physical explanation for physical explanation for the the extended extended no no-

• scale

scale structure structure! !

Evaluation Evaluation of the

• f the Coleman

Coleman-

• Weinberg

Weinberg

Perfect matching Perfect matching!!! !!!

NB NB It is possible to get a matching with the Coleman It is possible to get a matching with the Coleman-

• Weinberg

Weinberg also for the also for the P P4

4 [1,1,1,6,9] [1,1,1,6,9] and the

and the P P4

4 [1,1,2,2,6] [1,1,2,2,6] case

case

where where

Inclusion Inclusion of

• f loop corrections

loop corrections

Loop corrections Loop corrections are subleading w.r. are subleading w.r.to V to Vnp

np and V

and Vα

α’ ’

do do not destroy not destroy the the exponentially large exponentially large volume volume minimum minimum V V ~exp( ~exp(a ai

iτ

τi

i)

)

BUT BUT they they can: can:

• Stabilise

Stabilise the L non the L non-

• blow

blow moduli LARGE moduli LARGE

• Fix

Fix the SM the SM cycle which does not get any cycle which does not get any non non-

• perturbative W

perturbative W

for for < <Φ ΦSM

SM>=0

>=0

(BMP) (BMP) =0 =0

The Standard Model in the CY The Standard Model in the CY

• SM on a

SM on a small cycle as small cycle as g g2

2=1/

=1/τ τSM

SM

• BUT

BUT τ τSM

SM cannot have any

cannot have any non non-

• perturbative superpotential!

perturbative superpotential!

• Solution

Solution: : fix fix τ τSM

SM via

via string string loops loops

P P4

4 [1,3,3,3,5] [1,3,3,3,5] for h

for h11

11=3

=3

Example 1: SM cycle fixed by loops Example 1: SM cycle fixed by loops

Example 2:Flat directions lifted by loops Example 2:Flat directions lifted by loops

• K3 Fibration with h

K3 Fibration with h11

11=2:

=2: P P4

4 [1,1,2,2,6] [1,1,2,2,6]

• No blow

No blow-

• up mode No LARGE Volume

up mode No LARGE Volume

• K3 Fibration with h

K3 Fibration with h11

11=3

=3

• Now

Now τ τ3

3 is a blow

is a blow-

• up mode LARGE Volume

up mode LARGE Volume

• Field redefinition

Field redefinition

• Scalar potential without loop corrections

Scalar potential without loop corrections Ω Ω is a flat direction, is a flat direction, V V ~ exp(a ~ exp(a3

3τ

τ3

3)!

)!

• Consider loop corrections

Consider loop corrections

• Wrap

Wrap D7s D7s around around τ τ1

1,

, τ τ2

2 and

and τ τ3

3

• KK

KK corrections corrections in in τ τ3

3 do

do not depend not depend on

• n Ω

Ω and are subleading w.r. and are subleading w.r.to to the the α α’ ’ ones

• nes
• The 4

The 4-

• cycle

cycle τ τ3

3 is

is a a blow blow up mode up mode it does not intersect with it does not intersect with the the

• ther
• ther 4

4-

• cycles

cycles No No winding coorections winding coorections! ! negligible negligible! !

• Relevant loop corrections

Relevant loop corrections

• Loop corrections for

Loop corrections for A=1, C= A=1, C=-

• 20, D=2/9

20, D=2/9

NB NB The minimum The minimum lies lies within within the K the Kä ähler hler cone cone

V V < < Ω Ω < 2 < 2 V V NB NB A>0, D>0 !!! A>0, D>0 !!!

NO D7 NO D7 wrapping wrapping τ τ1

1

NO minimum NO minimum NO D7 NO D7 wrapping wrapping τ τ2

2

Minimum at Minimum at Ω Ω = 3 = 3 V V out of the K

• ut of the Kä

ähler cone! hler cone!

New candidate inflatons New candidate inflatons

• Main source of SUSY breaking: still

Main source of SUSY breaking: still V V, F , FV

V ≠

≠0 no drastic changes in phenomenology expected! no drastic changes in phenomenology expected!

• L=(h

L=(h11

11-

• N

Nsmall

small-

• 1) flat directions lifted by loops are light

1) flat directions lifted by loops are light

• m

mΩ

Ω 2 2=

=V V -

• 10/3

10/3, H ~ m

, H ~ mV

V 2 2=

=V V -

• 3

3

η η ~ m ~ m2

2/H ~

/H ~ V V -

• 1/3

1/3

1 1

Get Get L L good good inflaton inflaton candidates candidates! ! i. i. Loop corrections Loop corrections under control!?? under control!?? ii. ii. Detectable tensor modes Detectable tensor modes!?? !?? with with 0.009<r<0.01 0.009<r<0.01 iii.

• iii. Spectral index

Spectral index 0.966<n<0.971 0.966<n<0.971 iv.

• iv. Need

Need V V =10 =102

2 to keep

to keep the Volume minimum the Volume minimum stable stable! !

Cicoli, Quevedo (in Cicoli, Quevedo (in preparation preparation) )

Conclusions Conclusions

• Non

Non-

• perturbative effects fix only blow

perturbative effects fix only blow-

• up K

up Kä ähler hler moduli moduli

• Then

Then α α’ ’ effects + effects + W Wnp

np fix the Volume exponentially large

fix the Volume exponentially large

• All the other K

All the other Kä ähler hler moduli are flat directions moduli are flat directions

• Loop corrections to K are leading w.r. to the

Loop corrections to K are leading w.r. to the α α’ ’ ones

• nes
• Loop corrections to V are SUB

Loop corrections to V are SUB-

• leading w.r. to the

leading w.r. to the α α’ ’ ones due to the

• nes due to the

“ “extended no extended no-

• scale structure

scale structure” ”

• Coleman

Coleman-

• Weinberg potential gives a nice physical understanding of

Weinberg potential gives a nice physical understanding of this cancellation: SUSY! this cancellation: SUSY!

• Loop corrections needed to fix the rest of K

Loop corrections needed to fix the rest of Kä ähler hler moduli! moduli!

• Loop corrections crucial to:

Loop corrections crucial to: 1) 1) Extend LVS to fibration CYs Extend LVS to fibration CYs 2) 2) Fix SM cycle Fix SM cycle 3) 3) Get natural inflation with r=0.01? Get natural inflation with r=0.01? 4) 4) Give a forest of monochromatic lines? Give a forest of monochromatic lines?

Need to compute explicit g Need to compute explicit gs

s corrections for arbitrary CY!!!

corrections for arbitrary CY!!!