Novel potentials for string axion inflation

Transcription

1 Novel potentials for string axion inflation 1. Introduction Tatsuo Kobayashi. Threshold corrections: Dedekind eta function 3. Quantum corrected period vector 4. Summary Abe, T.K., Otsuka, arxiv: T.K. Oikawa, Otsuka, arxiv:

2 1. Introduction In superstring theory on 6D compact space, there are lots of axions as well as moduli, which have a flat potential. The shift symmetry controls a flat potential of axions, and some effects violate it and generate axion potential. Thus, axions are one of good candidates for the inflaton in superstring theory.

3 Axion inflation Indeed, lots of studies have been done. Famous axion inflations natural inflation Freese, Frieman, Olinto, 90 C( 1cos( a / f monodromy inflation Silverstein, Westphal, 08 r r, 1, /3,... Ca

4 Modulation Sub-leading corrections provide modulation terms natural inflation C( 1cos( a / f C'(1 cos( a / f ' monodromy inflation Ca r C' (1 cos( a / f Kobayashi, Takahashi, 11, Czerny, Kobayashi, Takahashi, 14 T.K., Seto, Yamaguchi, 14 Higaki, T.K., Seto, Yamaguchi, 14 Modulation can change observables, ns and r Modulation is important.

5 Axion inflation Are there other forms of potentials for axion inflation in superstring theory? We would like to discuss it today.

6 . Thereshold corections: Dedekind eta function Non-perturbative effect on string moduli We have to generate their potential by non-peturbative effects. T t ia W ct Ae... axion potential ( 1cos( ca... potential for natural inflation Freese, Frieman, Olinto, 90

7 Gaugino condensation SYM in the hidden sector Strong coupling dynamics g gaugino condensation d M Pl exp[ 8 /( bg 3 d b is one-loop beta function coefficient, e.g. b=3n for SU(N SYM. 1/ g ] d T 1/g corresponds T /( to ct W Ae... the compact space volume, c 1 / b, where this gauge field lives. e.g. c 4 / N for c = O(1 or larger 1cos( ca unless b >> O(10, eg. SU(100 For O(1 of Kahler metric of a, decay constatant is small. e.g. is given by real part of some moduli, M Pl SYM (...

8 Threshold corrections g some modes have the mass M. They decouple at M. The beta-function coefficient changes at M. 1/g ( M T ( b /16 ln( M / M st d M M Pl b is beta -function coefficient contribution due to massive modes Threshold corrections

9 Threshold corrections some mass scale correction 1/g ( M T ( b /16 ln( M / M st For example, KK-modes affect the gauge couplings their masses depend on the moduli. M k k / R Threshold corrections R: modulus (size of compact space ( b/8 ln( k / RM st ( b/8 dk ln( k ( b/8 [ RM st / RM st ln( RM st 1] Moduli-dependent term appears

10 Threshold corrections some mass scale correction 1/g ( M T ( b /16 ln( M / M st KK-modes affect the gauge couplings their masses depend on the moduli. 1/g ( M T (Moduli /16 gaugino condensation 3 d d M Pl exp[ 8 /( bg ] M Pl exp[ (8 / b( /16...] W C (4 Ae C... / b( /16 ( 1cos( ca... A new factor of O(100 appears c <O(1, large decay constant, f > O(1

11 Threshold corrections S: dilaton, T: Kahler moduli, U: complex structure moduli Heterotic string theory on orbifold: certain twisted sector 1/g ( M S /16, 4bN ln( T (U exp[ U Dixon, Kaplunovsky, Louis, 91 Type IIA intersecting D-brane models: certain parallel D-brane 1/g ( M as bu /16, 4bN ln( T Lust, Stieberger, 07 T-dual Type IIB D-brane models D3/D7-branes or D5/D9-branes /1] 1/g ( M as bt /16, 4bN ln( U (1 e mu Maybe also some correction in Hetero. Type IIA/B on CY m

12 Eta function Inflation potential 1/g ( M as bt /16, 4bN ln( U 3b /( b W A[ ( U] N... We assume that all of the moduli are stabilized except the axion = Im(U. (U exp[ U /1] m (1 e mu For a large Re(U, the leading term is dominant. For a large Re(U, e.g. Re(U 1.3, Gaugino condensation ln(u U c bn /(6dL, d Re( U for SU(L /1 ( 1cos( ca... decay constant f 6dL /( bn the decay constant can be large, f> 1.

13 Eta function inflation potential ( 1cos( ca... c bn /(6dL, d Re( U for SU(L decay constant f 6dL /( bn Example Re(U 1.3, 1, L b N 4 Observables n s 0.96, r 0.11, N 50, S 10 3

14 Eta function inflation potential 1/g ( M as bt /16, 4bN ln( U Smaller Re(U (leading + subleading (U exp[ U Gaugino condensation /1](1 e U... ( 1cos( ca cos( c a... 1( bn / Lexp[ ( bn /(6LReU] c b N /( 6dL, c ( bn /(6dL / d, d Re( U modulatin term is important. Kobayashi, Takahashi, 11, Czerny, Kobayashi, Takahashi, 14 T.K., Seto, Yamaguchi, 14 Higaki, T.K., Seto, Yamaguchi, 14 decay constant for SU(L f 6dL /( bn

15 Natural inflation in Type IIB (U exp[ U /1](1 e U... ( 1cos( ca cos( c a... 1( bn / Lexp[ ( bn /(6LReU] c b Example Potential N /( 6dL, c ( bn /(6dL / d, d Re( U Re(U 1., / L 1/ 5 b N decay constant f for SU(L 6dL /( bn

16 Natural inflation in Type IIB (U exp[ U /1](1 e U ( 1cos( ca cos( c a... 1( bn / Lexp[ ( bn /(6LReU] c b N /( 6dL, c ( bn /(6dL / d, d Re( U for SU(L Examples decay constant f 6dL /( bn

17 Natural inflation in Type IIB ( 1cos( ca cos( c a... c b N /( 6dL, c ( bn /(6dL / d, d Re( U for SU(L b / L ReU N n s r s 1/ / / / / Little effect of modulation in the first case

18 3. Quantum corrected period vector In type IIB superstring theory, non-vanishing 3-form fluxes induce a potential for complex structure moduli and dilaton. W GW The period vector at the tree level. (U ( S, U ( F3 ish 3 ( N isn ( U ( U is polynomial functions of U Some or all of complex moduli and dilaton can be stabilized. F H

19 Quantum corrected period vector Quantum corrections have been computed. Hosono, Klemm, Theisen, Yau, 94 The prepotential The polylogarithm function ( Li ( ! i id i U i i j i ij k j i ijk e n i U U U U U U F F F U F U i i i i ( Li n z z n

20 Quantum corrected period vector Superpotential W GW ( F H S, U ( N isn Kahler potential e K i i( ( F F ( U U ( F F i i i

21 Superpotential Superpotential at tree level W ( S, U g0( z g1( z( U NU1 z are other moduli massive massless U U NU 1 Quantum correction W U1 ( g g3u 1 e ( g ( g 3 / N( e ( / N

22 Superpotential for inflation Superpotential W T.K. Oikawa, Otsuka, w0 ( c c' e / N inflaton Im We also assume uplifting such that Inflation potential ( min M sin( / M3 6(1 cos( / 3 M3 Nk /(4 4, 5, 6 O( e 4 Re( U 1

23 Inflation potential Inflation potential T.K. Oikawa, Otsuka, M sin( / M3 6(1 cos( / 3 M3 Nk /(4 Hebecker, et.al. arxiv: studied a similar model with limited 3-form flux ( 1cos( / 3 6 M but we introduce the full 3-form flux

24 Inflation potential Inflation potential T.K. Oikawa, Otsuka, M sin( / M3 6(1 cos( / 3 M3 Nk /(4 In some case 4 1cos( / M 6 ( 3 another case 4

25 Numerical study T.K. Oikawa, Otsuka, M sin( / M3 6(1 cos( / 3 M N r n s s / 6 4 / / Large-field inflation Small-field inflation is possible M3 Nk /(

26 Small-field inflation Kadota, T.K. Oikawa, Omoto,Otsuka, Tatsuishi, in progress 4 5 M sin( / M3 6(1 cos( / 3 When we tune parameters, we can realize r O(0.01 M3 Nk /(4 or much smaller ariation of field is small during inflation, but its field-value is large such as the Planck scale.

27 Summary We have studied new types of axion inflation potential within the framework of superstring theory. Threshold corrections lead to the eta function inflation potential. It can enhance the axion decay constant by the inverse of loop factor. Stringy threshold corrections also include modulation terms, which are interesting effects on observables.

28 Summary Quantum corrections on period vector can lead a new type of axion inflation potential, which is a mixture of polynomial functions and sinusoidal functions. That can lead to both large tensor-to-scalar ratio and small r.

29 Further work Many models with the SM gauge group and three chiral generations have been constructed in type IIA (intersecting D-brane models and type IIB magnetized D-brane models as well as heterotic orbifold models. It is important to combine such models as the visible sector and the present inflation sector to realize the complete theory. We know the coupling between axion/inflaton and SM fields. Then, we study details.