Stringy Corrections, SUSY Breaking and the Stabilization of (all) Kähler moduli

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1 Stringy Corrections, SUSY Breaking and the Stabilization of (all) Kähler moduli Per Berglund University of New Hampshire Based on arxiv: 1012:xxxx with Balasubramanian and hep-th/ Balasubramanian, PB Miami 2010, December 17, 2010

2 Overview Study the role of string corrections in type IIB flux compactifications α corrections to Kähler potential K are important even in large volume limit Motivation: Stabilization of (all) Kähler moduli Spontaneous supersymmetry breaking by F- terms Construction of desitter vacua (in addition to non-susy AdS and Minkowski)

3 Outline Review of flux compactifications Supersymmetric vacua and ample divisors Stringy corrections and supersymmetry breaking Implications Discussion and Outlook

4 Flux Compactifications (I) Gukov, Vafa, Witten We are interested in type IIB orientifold compactifications on Calabi-Yau three-fold M (or equivalently F-theory on four-fold X) with nontrivial NS and R-R fluxes. Classical superpotential Tadpole condition W 0 = M G = F τh χ(x) = F H 24 M Ω G Ω is the holomorphic 3-form on M F and H are the R-R and NS 3-form fluxes, respectively, through 3-cycles on M

5 Flux Compactifications (II) The resulting N=1 supergravity scalar potential takes the standard form where V = e K (G i j D i WD j W 3 W 2 ) D i W = i W + W i K G i j = i jk K = 2 log( J 3 ) log(τ τ) log( Ω Ω) M and the derivatives are taken with respect to the different moduli M complex structure (shape of Calabi-Yau) (axion-)dilaton (string coupling) Kähler structure (size of Calabi-Yau) where σ k = tk V V = 1 3! k ijkt i t j t k = τ = C 0 + ie φ ρ k = b k + iσ k M J 3 z i

6 Flux Compactifications (III) Giddings, Kachru, Polchinski With a generic choice of fluxes, F and H, the complex structure zi and dilaton τ are frozen SUSY condition: D zi W = D τ W =0 Thus, with W = W 0 D ρi W = W 0 ρi K V = e K (G i j ρi K ρj K 3) W 0 2 =0 due to cubic dependence on ti in CY volume in K This is a no-scale model and the Kähler moduli, ρi, are unconstrained V V=0 ρi

7 Flux Compactifications (IV) A non-trivial potential for the Kähler moduli can be generated in two ways: Kachru, Kallosh, Linde, Trivedi Becker, Becker, Haack, Louis Non-perturbative corrections to W Perturbative (and non-perturbative corrections) to K

8 Non-perturbative Corrections to W (I) These effects come in two form: Kachru, Kallosh, Linde, Trivedi D3-brane instantons (N=1) Gaugino condensation (N=c2) W = W 0 + i A i e ia iρ i, a i = 2π N i Witten Katz, Vafa Douglas, Denef, Floria Gorlich, Kachru, Tripathy, Trivedi Larsen et al In the above scenario every divisor Di has to contribute to W, ie the arithmetic genus χ(d i )=1 Explicit models shown to exist, though the computation of the complex structure dependence of Ai(zj) still an issue. Explicit dependence of W0 on fluxes important

9 Non-perturbative Corrections to W (II) Kachru, Kallosh, Linde, Trivedi One then can solve the F-term equations for the Kähler moduli, which for h11=1, gives D ρ W =0 t = 4π N V ( W 0 A )e 2π N σ 1 Here we assume A>0, W0<0 and we have already stabilized axion Re(ρ). This naturally breaks the no-scale structure of the potential V σ

10 Non-perturbative Corrections to W (III) Bobkov, Braun, Kumar, Raby The conditions on how many divisors (and how they contribute to W) can be relaxed where the sum is over all h11 divisors in M. In this case, a single ample divisor D contributes to W, where W = W 0 + Ae P k ia kρ k, a k = 2πn k N D = h 11 n i D i n i > 0, i =1,...,h 11 i=1 As for KKLT one still has to demand that χ(d) =1

11 Non-perturbative Corrections to W (IV) Bobkov, Braun, Kumar, Raby One then can solve the F-term equations for the Kähler moduli, just as in KKLT, D ρj W =0 t i = n i 4π N V ( W 0 A )e 2π N n σ 1, n = {n i} Here we assume A>0, W0<0 and we have already stabilized one particular linear combination of the Re(ρi), while the remaining h11-1 axions remain massless. Just as in KKLT, this breaks the no-scale structure V σi

12 Perturbative Corrections to K (I) Becker, Becker, Haack, Louis The N=1 Kähler potential receives the first nonzero corrections at O(α 3 ) at string tree-level due to the R 4 term in the action where K J = 2 log(v + ξ 2 (τ τ 2i ) 3/2 ) ξ = ζ(3)χ(m) 2(2π) 3, ξ > 0 χ(m) < 0 Higher order corrections are suppressed by powers and exponentials of the volume of 4-cycles The α 3 corrections break the no-scale structure just like the non-perturbative corrections to W V σ

13 Perturbative Corrections to K (II) PB, Lazarus, Ren 10YY.XXXX Due to the correction of the Kähler potential, the metric is no longer (block) diagonal. One therefor, strictly speaking, has to reanalyze the previous (separate) stabilization of the complex structure moduli, zi, and dilaton, τ. In particular, it may not be consistent to assume that the F-term constraint vanishes for τ, independently, in the non-susy vacua when (non)- perturbative corrections to (W) K are included. This leads to additional uplift of the potential, since the metric is positive definite V τ = e K (G τ j D τ WD j W + G i τ D i WD τ W )

14 Perturbative Corrections to K (III) Becker, Becker, Haack, Louis If W=W0 V =3ξ (ξ2 +7ξV + V 2 ) (V ξ)(2v + ξ) 2 where the singular behaviour is due to the inverse Kähler metric G ij V V = ξ σi

15 (Non-)Perturbative Corrections to (W),K (I) Balasubramanian, PB If we include both types of corrections W = W 0 + A i e ia iρ i, a i = 2π N i i K J = 2 log(v + ξ τ (τ ) 3/2 ) 2 2i the behaviour of the scalar potential at large volume is qualitatively changed The potential approaches zero from above since perturbative correction to K dominates in the limit of large volume V V = ξ σi

16 (Non-)Perturbative Corrections to (W),K (II) Balasubramanian, PB 1012.XXXX The same will be true in the case of a single ample divisor contributing to W since once again the behaviour of the scalar potential at large volume is qualitatively changed where the exponential correction to W can be neglected W = W 0 + Ae P k ia kρ k, a k = 2πn k N K J = 2 log(v + ξ τ (τ ) 3/2 ) 2 2i V V = ξ σi

17 Supersymmetry Breaking (I) Balasubramanian, PB Let us first review the argument for how supersymmetry can be broken when α corrections are included. We assume that the F-terms vanish for the complex structure moduli, zi, and the dilaton, τ. Q: When do the F-terms for the Kähler moduli vanish? D ρi W =0 W 0 = h 11 j=1 It then follows that, assuming that the volume is large, such that our approximation is valid, a susy minimum will only exist for W 0 W max A j e ia jρ j +2a k A k e ia kρ k (V + ξ 2 ) t k

18 Supersymmetry Breaking (II) Balasubramanian, PB Alternatively, as we increase W0, the location of the susy minimum will occur at smaller values of the volume, such that at some point, we will not be able to trust our approximation. However, as W0 is increased further new, nonsusy minima will occur at increasingly larger values of the volume.

19 Supersymmetry Breaking (III) Balasubramanian, PB There are different types of vacua depending on the sign of the scalar potential at the minimum: AdS, Minkowski or ds As W0 is further increased there will be a point where there is no minimum (the dotted curve). At this point, due to the relatively larger perturbative correction to KJ versus the non-perturbative correction of W V Increasing W0 V = ξ σi

20 Supersymmetry Breaking (IV) Balasubramanian, PB 1012.XXXX How is this changed in the case of a single ample divisor contributing to W? The argument is essentially identical, since D ρk W =0 W 0 = and there is still an upper bound on W0. Alternatively, following the argument by Bobkov et al, we can change basis to ρd,, and a remaining set of linearly independent set of Kähler ˆρ i moduli. In the new basis, independent of the α correction, the F-terms for the satisfied Ae P ia j ρ j +2a k Ae P j ia jρ j (V + ξ 2 ) ρ D = i ˆρ i Dˆρi W =0 W ˆρi K =0 n i ρ i are always t k

21 Supersymmetry Breaking (V) Balasubramanian, PB 1012.XXXX Thus, the problem is then reduced to the remaining ρd for which we then have D ρd W =0 W 0 = Ae i 2π N ρ D +2 2π N 2π Aei N ρ (V + ξ D 2 ) t D and the argument is identical to that presented before.

22 Implications Allow for uplifting of the potential to be used in cosmological applications such as inflation and the cosmological constant problem No exponentially large volume (unless we relax the ample condition of the divisor) Only one axion stabilized, bd, while the remaining h11-1axions, ˆbi, are massless (they will eventually become massive due to subleading corrections to W) Bobkov, Braun, Kumar, Raby The ˆbi could then be potential solutions to the strong CP problem

23 Discussion and Outlook Stringy corrections are important even in the large volume limit New mechanism for constructing non-susy vacua with many (nearly) massless axions Combine the moduli stabilization with lessons learned from inflationary models and local string constructions

24 Discussion and Outlook Stringy corrections are important even in the large volume limit New mechanism for constructing non-susy vacua with many (nearly) massless axions Combine the moduli stabilization with lessons learned from inflationary models and local string constructions