DE-SITTER VACUA FROM A D-TERM GENERATED RACETRACK POTENTIAL IN - PowerPoint PPT Presentation

DE-SITTER VACUA FROM A D-TERM GENERATED RACETRACK POTENTIAL IN HYPERSURFACE CALABI-YAU COMPACFITICATIONS Yoske Sumitomo ( 住友洋介 ) KEK Theory Center, Japan M. Rummel, YS, JHEP 1501 (2015) 015, arXiv:1407.7580 A. Braun, M. Rummel, YS, R. Valandro, arXiv:1509.06918

Dark Energy Dominant source for late time expansion Planck(TT, lowP, lensing)+BAO+JLA+ 𝐼 0 (“ ext ”) 𝑥 = 𝑞 +0.085 (95% CL) 𝜍 = −1.006 −0.091 agrees with the positive cosmological constant.

String theory in 10D A prime candidate of quantum gravity ability to address vacuum energy String theory has a nice feature: 10D = 4D + 6D Information of 6D space determines what we have in 4D! Light/heavy d.o.f. (moduli fields) • Sources of potential • Matters (visible and hidden) • Importantly, we cannot simply select at our will. String theory compactifications impose conditions on SUGRA.

Key points of string cosmology • Moduli stabilization • Minimum with positive CC (or DE) • Consistency of compactifications • Reasonable parameters • … 4

Moduli stabilization We have to stabilize moduli fields of compactification. 𝑛 𝜚 ≳ 𝒫(10) TeV Reheating for BBN • Determining parameters in 4D theory • Many moduli fields in string compactification 𝑂 ∼ 𝒫 100 (dilaton, complex structure moduli, Kähler moduli etc.) Probability of stability (eigenvalues 𝑛 𝑗𝑘 > 0 ) is given a Gaussian 2 suppressed function of # of moduli, if random enough. 𝒬 ∼ 𝑓 −𝑏𝑂 2 [Aazami, Easther, 05], [Dean, Majumdar, 08], [Borot, Eynard, Majumdar, Nadal, 10], [Marsh, McAllister, Wrase 11], [X. Chen, Shiu, YS, Tye, 11], [Bachlechner, Marsh, McAllister, Wrase 12] So, when no hierarchy at 𝑂 ∼ 𝒫 100 , hopeless. Need for a hierarchical structure of mass matrix.

Type IIB on Calabi-Yau A region that is not completely random and works well for cosmology. No-scale structure generates a hierarchy: 𝑊 = 𝑊 + 𝑊 NP + 𝑊 𝛽′ + ⋯ Flux 𝒫 𝒲 −2 𝒫 ≪ 𝒲 −2 : CY volume scaling ≫ 2 : positive definite Flux = 𝑓 𝐿 𝐸 𝑇,𝑉 𝑗 𝑋 Also, 𝑊 0 𝐸 𝑇,𝑉 𝑗 𝑋 0 = 0 Many moduli are integrated out at high scale. (Hessian) (real part) large small 272 + 1 : ℎ 1,1 = 2, ℎ 2,1 = 272 e.g. CY ℙ 1,1,1,6,9 4 𝑁 ∼ 2 small small We have to worry about only few light d.o.f. (Kahler moduli).

Kahler Moduli stabilization 𝐸𝑋 2 − 3 𝑋 2 Consider SUGRA F-term scalar potential: 𝐺 = 𝑓 𝐿 𝑊 𝜊 𝐿 = −2 ln 𝒲 + 2 , 𝑋 = 𝑋 0 + 𝑋 𝑂𝑄 non-perturbative effect (instantons etc.) 𝛽′ -correction KKLT Large Volume Scenario (LVS) E.g. [Kachru, Kallosh, Linde, Trivedi, 03] [Balasubramanian, Beglund, Conlon, Quevedo, 05] 𝐸 𝐽 𝑋 = 0 : supersymmetric 𝜖 𝐽 𝑊 = 0 : non-sypersymmetric Both minima stay at AdS Uplift to dS

Some uplift models Some proposals keeping stability, but not so many. [Kachru, Pearson, Verlinde, 01], Anti-brane • 𝑊 = 𝑊 𝑇𝑉𝐻𝑆𝐵 + 𝑊 𝐸3−𝐸3 [KKLT, 03] A positive contribution by localized source, suppressed by warping. Non-zero minimum of flux potential • 𝑊 Flux > 0 [Saltman, Silverstein, 04] Require a suppression to balance with 𝑊 Kahler (generically ≪ 𝑊 Flux ). [Burgess, Kallosh, Quevedo, 03], [Cremades, Garcia del Moral, Quevedo, D-term uplift • 07], [Krippendorf, Quevedo, 09] [Cicoli, Goodsell, Jaeckel Ringwald, 11] A suppressed coefficient is required as 𝑊 Kahler . 𝐸 ≫ 𝑊 Dilaton-dependent non-perturbative effects • [Cicoli, Maharana, Quevedo, Burgess, 12] 𝑣𝑞 ∝ 𝑓 −2𝑐 𝑡 : dilaton value 𝑡 should be tuned accordingly. 𝑊 𝒲 A suppression of coefficient is required. (due to different volume dependence)

D-TERM GENERATED RACETRACK UPLIFT M. Rummel, YS, JHEP 1501 (2015) 015, arXiv:1407.7580 A. Braun, M. Rummel, YS, R. Valandro, arXiv:1509.06918

D-term generated racetrack uplift [Rummel, YS, 14] Effective potential: 2 𝑦 𝑡 𝑊 ∼ 3𝜊 4𝒲 + 4𝑑 𝑡 𝑦 𝑡 𝒲 2 𝑓 −𝑦 𝑡 + 2 2𝑑 𝑡 𝑓 −2𝑦 𝑡 + 4𝛾𝑑 𝑏 𝑦 𝑡 𝑓 −𝛾𝑦 𝑡 + ⋯ 𝒲 2 3𝒲 LVS stabilization at AdS uplift When 𝒅 𝒕 = −𝟏. 𝟏𝟐, 𝜊 = 5, 𝜸 = 𝟔 𝟕 ∼ 𝟏. 𝟗𝟒 , and increase 𝑑 𝑏 (minimum value) Minkowski point: 𝒅 𝒃 ∼ 𝟓 × 𝟐𝟏 −𝟒 , 𝒲 ∼ 3240, 𝑦 𝑡 ∼ 3.07. special suppression is not required when 𝛾 ∼ 1 . 𝑑 𝑡 ∼ 𝑑 𝑏 Analytically, 𝛾 < 1, 𝑑 𝑏 > 0 are required for uplift.

Key idea: D-term constraint D-term potential imposes a constraint at high scale. generating a heavy mass 𝑊 𝐸 ≫ 𝑊 𝑊 = 𝑊 𝐺 + 𝑊 𝐸 𝐺 In string theory compactifications, Magnetized D7-branes wrapping a Calabi-Yau four-cycle 1 𝜊 𝐸 = 1 2 w/ matters stabilized 𝑊 𝐸 = 𝜊 𝐸 𝒲 𝐾 ∧ 𝐸 𝐸 ∧ ℱ 𝐸 Re 𝑔 accordingly 𝐸 A choice of flux ℱ 𝐸 would give 1 1 2 so a constraint: 𝑦 𝑏 = 𝛾 𝑦 𝑡 𝑊 𝐸 ∝ 𝛾 𝑦 𝑡 − 𝑦 𝑏 𝒲 2 Re 𝑔 𝐸 Then, a racetrack is generated (different from simple racetrack). 𝐺 ∋ 𝐷 𝑡 𝑓 −𝑦 𝑡 + 𝐷 𝑏 𝑓 −𝑦 𝑏 + ⋯ 𝐷 𝑡 𝑓 −𝑦 𝑡 + 𝐷 𝑏 𝑓 −𝛾 𝑦 𝑡 + ⋯ 𝑊

Values of 𝛾 The value of 𝛾 determines how much suppression we need. ( 𝑦 𝑏 = 𝛾 𝑦 𝑡 ) 𝑊 ∋ 𝐷 𝑡 𝑓 −𝑦 𝑡 + 𝐷 𝑏 𝑓 −𝛾 𝑦 𝑡 + ⋯ If 𝛾 = 0.9 , almost no suppression of coefficients for uplift 𝐷 𝑏 . 𝐷 𝑡 ∼ Parameter 𝛾 is determined by geometry and fluxes. Constraints from consistency of CY compactifications:  Two instantons (on rigid divisors)  D-term (anomalous U(1)) that relates two moduli 𝑦 𝑡,𝑏  Quantized fluxes on integral basis  Charge cancellations (no D3, D5, D7 tadpoles)  No anomaly (Freed-Witten) We assume that open-string moduli are stabilized at 𝜚 𝑗 ≠ 0 (hidden matters) for simplicity.

An example [Braun, Rummel, YS, Valandro, 15] Intersections • 𝒲 ∝ 1 2 3 2 − 1 3 + 𝐸 𝑡 3 + 2𝐸 𝑏 3 2 − 3 3 2 𝐽 3 = 2𝐸 𝑤 3 𝑦 𝑤 3 𝑦 𝑡 3 𝑦 𝑏 Swiss-Cheese type Instantons (rank one) • Euclidean D3 on rigid divisors 𝐸 𝑡 , 𝐸 𝑏 0 + 𝐵 𝑡 𝑓 −𝑦 𝑡 + 𝐵 𝑏 𝑓 −𝑦 𝑏 𝑋 = 𝑋 O7-plane, a D7-brane (D-term) • no D7-tadpole 𝐸 𝑃7 = 4𝐸 𝑤 − 3𝐸 𝑡 − 2𝐸 𝑏 , 𝐸 𝐸 = 4𝐸 𝑃7 Quantized fluxes (no Freed-Witten anomaly) • ℱ 𝐸 = 𝐸 𝑡 2 − 𝐸 𝑏 𝑢𝑝𝑢 = 𝑅 𝐺 3 ,𝐼 3 + 𝑅 𝑃7 + 𝑅 𝐸7 + 𝑅 𝑃3 𝑅 𝐸3 ℱ 𝑡,𝑏 = 0, 2 3 ,𝐼 3 − 526 = 𝑅 𝐺 D-term constraint • 𝜊 𝐸 = 1 𝛾 = 8 Successful de-Sitter! 𝒲 𝐾 ∧ ℱ 𝐸 = 0 𝑦 𝑏 = 𝛾 𝑦 𝑡 , 9 𝐸 𝐸

Scanning Calabi-Yau for 𝛾 [Braun, Rummel, YS, Valandro, 15] Using the data of 6D toric Calabi-Yau hypersurfaces, [Kreuzer, Skarke, 00], [Altman, Gray, He, Jejjala, Nelson 14] (ℎ 1,1 = 3, Three moduli 𝒲, 𝑦 𝑡 , 𝑦 𝑏 ) Total: 244 Suitable geometry and successful flux: 32 (polytopes) (13%) (ℎ 1,1 = 4, Four moduli 𝒲, 𝑦 𝑡 , 𝑦 𝑏 , 𝑦 𝑐 ) Suitable geometry and successful flux: 191 Total: 1197 (polytopes) (16%) Possible 𝛾 values 𝛾 = 49 50 = 0.98 , 121 128 (∼ 0.95), 225 good 𝛾 , good realizability 242 (∼ 0.93), … There are several other setups too.

Summary & Discussion • 6D geometry determines 4D physics. • Moduli stabilization, minimum vev, consistency, naturalness should be taken into account for string cosmology. D-term generated racetrack model uplifts potential successfully. • (Simple racetrack does not.) • Almost no suppression required in parameters if 𝛾 ∼ 1 . • 6D Calabi-Yau data suggests that 𝛾 ∼ 1 is ubiquitous. • Open-string moduli need not to be 𝜚 𝑗 ≠ 0 in other types of Calabi-Yau.