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 ( ) KEK Theory Center, Japan M. Rummel, YS, JHEP 1501 (2015) 015, arXiv:1407.7580 A. Braun, M. Rummel, YS, R.
Yoske Sumitomo (住友洋介) KEK Theory Center, Japan
Dominant source for late time expansion Planck(TT, lowP, lensing)+BAO+JLA+𝐼0(“ext”) 𝑥 = 𝑞
𝜍 = −1.006−0.091 +0.085(95% CL)
agrees with the positive cosmological constant.
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!
Importantly, we cannot simply select at our will. String theory compactifications impose conditions on SUGRA.
4
We have to stabilize moduli fields of compactification.
𝑛𝜚 ≳ 𝒫(10) TeV
Many moduli fields in string compactification (dilaton, complex structure moduli, Kähler moduli etc.) Probability of stability (eigenvalues 𝑛𝑗𝑘
2
> 0) is given a Gaussian suppressed function of # of moduli, if random enough. 𝒬 ∼ 𝑓−𝑏𝑂2 So, when no hierarchy at 𝑂 ∼ 𝒫 100 , hopeless. Need for a hierarchical structure of mass matrix.
[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]
𝑂 ∼ 𝒫 100
𝑊 = 𝑊
Flux
+ 𝑊
NP + 𝑊 𝛽′ + ⋯
No-scale structure generates a hierarchy:
𝒫 𝒲−2 𝒫 ≪ 𝒲−2
Also, 𝑊
Flux = 𝑓𝐿 𝐸𝑇,𝑉𝑗𝑋 2: positive definite
Many moduli are integrated out at high scale. 𝐸𝑇,𝑉𝑗𝑋
0 = 0
e.g. CY ℙ 1,1,1,6,9
4
: ℎ1,1 = 2, ℎ2,1 = 272 𝑁 ∼ large small small small 272 + 1 2 We have to worry about only few light d.o.f. (Kahler moduli). A region that is not completely random and works well for cosmology.
: CY volume scaling ≫
(real part) (Hessian)
KKLT Large Volume Scenario (LVS) 𝐸𝐽𝑋 = 0: supersymmetric 𝜖𝐽𝑊 = 0: non-sypersymmetric Both minima stay at AdS E.g.
[Balasubramanian, Beglund, Conlon, Quevedo, 05] [Kachru, Kallosh, Linde, Trivedi, 03]
Consider SUGRA F-term scalar potential: 𝐿 = −2 ln 𝒲 +
𝜊 2 ,
𝑋 = 𝑋
0 + 𝑋 𝑂𝑄
𝛽′-correction non-perturbative effect (instantons etc.)
𝑊
𝐺 = 𝑓𝐿
𝐸𝑋 2 − 3 𝑋 2 Uplift to dS
𝑊 = 𝑊
𝑇𝑉𝐻𝑆𝐵 + 𝑊 𝐸3−𝐸3
A positive contribution by localized source, suppressed by warping.
[Kachru, Pearson, Verlinde, 01], [KKLT, 03] [Saltman, Silverstein, 04]
Some proposals keeping stability, but not so many. 𝑊
Flux > 0
Require a suppression to balance with 𝑊
Kahler (generically ≪ 𝑊 Flux).
[Burgess, Kallosh, Quevedo, 03], [Cremades, Garcia del Moral, Quevedo, 07], [Krippendorf, Quevedo, 09] [Cicoli, Goodsell, Jaeckel Ringwald, 11]
A suppressed coefficient is required as 𝑊
𝐸 ≫ 𝑊 Kahler.
[Cicoli, Maharana, Quevedo, Burgess, 12]
𝑊
𝑣𝑞 ∝ 𝑓−2𝑐 𝑡 𝒲
: dilaton value 𝑡 should be tuned accordingly. A suppression of coefficient is required.
(due to different volume dependence)
Effective potential: When 𝒅𝒕 = −𝟏. 𝟏𝟐, 𝜊 = 5, 𝜸 = 𝟔 𝟕 ∼ 𝟏. 𝟗𝟒 , and increase 𝑑𝑏 Minkowski point: 𝒅𝒃 ∼ 𝟓 × 𝟐𝟏−𝟒, 𝒲 ∼ 3240, 𝑦𝑡 ∼ 3.07. Analytically, 𝛾 < 1, 𝑑𝑏 > 0 are required for uplift.
𝑑𝑡 ∼ 𝑑𝑏 special suppression is not required when 𝛾 ∼ 1.
𝑊 ∼ 3𝜊 4𝒲 + 4𝑑𝑡𝑦𝑡 𝒲2 𝑓−𝑦𝑡 + 2 2𝑑𝑡
2 𝑦𝑡
3𝒲 𝑓−2𝑦𝑡 + 4𝛾𝑑𝑏𝑦𝑡 𝒲2 𝑓−𝛾𝑦𝑡 + ⋯
uplift LVS stabilization at AdS (minimum value)
[Rummel, YS, 14]
Magnetized D7-branes wrapping a Calabi-Yau four-cycle A choice of flux ℱ𝐸 would give 𝑊
𝐸 ∝
1 Re 𝑔
𝐸
1 𝒲2 𝛾 𝑦𝑡 − 𝑦𝑏
2
D-term potential imposes a constraint at high scale. 𝑊
𝐸 ≫ 𝑊 𝐺
𝑊
𝐸 =
1 Re 𝑔
𝐸
𝜊𝐸
2
𝜊𝐸 = 1 𝒲 𝐾 ∧ 𝐸𝐸 ∧ ℱ𝐸 𝑊 = 𝑊
𝐺 + 𝑊 𝐸
generating a heavy mass In string theory compactifications,
w/ matters stabilized accordingly
so a constraint: 𝑦𝑏 = 𝛾 𝑦𝑡 Then, a racetrack is generated (different from simple racetrack). 𝑊
𝐺 ∋
𝐷𝑡𝑓−𝑦𝑡 + 𝐷𝑏 𝑓−𝑦𝑏 + ⋯ 𝐷𝑡𝑓−𝑦𝑡 + 𝐷𝑏 𝑓−𝛾 𝑦𝑡 + ⋯
(𝑦𝑏 = 𝛾 𝑦𝑡) The value of 𝛾 determines how much suppression we need. 𝑊 ∋ 𝐷𝑡𝑓−𝑦𝑡 + 𝐷𝑏 𝑓−𝛾 𝑦𝑡 + ⋯ If 𝛾 = 0.9, almost no suppression of coefficients for uplift 𝐷𝑡 ∼ 𝐷𝑏 . 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 𝑦𝑡,𝑏 Quantized fluxes on integral basis Charge cancellations (no D3, D5, D7 tadpoles) We assume that open-string moduli are stabilized at 𝜚𝑗 ≠ 0
(hidden matters)
No anomaly (Freed-Witten) for simplicity.
𝐽3 = 2𝐸𝑤
3 + 𝐸𝑡 3 + 2𝐸𝑏 3
𝒲 ∝ 1 3 𝑦𝑤
3 2 −
2 3 𝑦𝑡
3 2 − 1
3 𝑦𝑏
3 2
𝐸𝑃7 = 4𝐸𝑤 − 3𝐸𝑡 − 2𝐸𝑏, 𝐸𝐸 = 4𝐸𝑃7 no D7-tadpole
Euclidean D3 on rigid divisors 𝐸𝑡, 𝐸𝑏 𝑋 = 𝑋
0 + 𝐵𝑡𝑓−𝑦𝑡 + 𝐵𝑏𝑓−𝑦𝑏
ℱ
𝑡,𝑏 = 0,
ℱ𝐸 = 𝐸𝑡 2 − 𝐸𝑏 2
[Braun, Rummel, YS, Valandro, 15]
𝑅𝐸3
𝑢𝑝𝑢 = 𝑅𝐺
3,𝐼3 + 𝑅𝑃7 + 𝑅𝐸7 + 𝑅𝑃3
= 𝑅𝐺
3,𝐼3 − 526
𝜊𝐸 = 1 𝒲
𝐸𝐸
𝐾 ∧ ℱ𝐸 = 0 𝑦𝑏 = 𝛾 𝑦𝑡, 𝛾 = 8 9 Successful de-Sitter!
Swiss-Cheese type
[Braun, Rummel, YS, Valandro, 15]
Using the data of 6D toric Calabi-Yau hypersurfaces,
[Kreuzer, Skarke, 00], [Altman, Gray, He, Jejjala, Nelson 14]
Three moduli Four moduli (ℎ1,1 = 3, 𝒲, 𝑦𝑡, 𝑦𝑏) (ℎ1,1 = 4, 𝒲, 𝑦𝑡, 𝑦𝑏, 𝑦𝑐) Total: 244 (polytopes) Suitable geometry and successful flux: 32 Total: 1197 (polytopes) 𝛾 = 49 50 = 0.98 , 121 128 (∼ 0.95), 225 242 (∼ 0.93), …
(13%) (16%)
good 𝛾, good realizability There are several other setups too. Possible 𝛾 values Suitable geometry and successful flux: 191
should be taken into account for string cosmology.
(Simple racetrack does not.)
in other types of Calabi-Yau.