8/11/2012 A Stringy Mechanism for A Small Cosmological Constant - - PowerPoint PPT Presentation
8/11/2012 A Stringy Mechanism for A Small Cosmological Constant Yoske Sumitomo X. Chen, Shiu, Sumitomo, Tye , IAS, The Hong Kong University of arxiv:1112.3338, JHEP 1204 (2012) 026 Science and Technology Sumitomo, Tye,
Yoske Sumitomo
IAS, The Hong Kong University of Science and Technology
1
arxiv:1112.3338, JHEP 1204 (2012) 026
arXiv:1204.5177
Motivation Moduli stabilization ~random approach~ Moduli stabilization ~concrete models~ Statistical approach More on product distribution Multi-moduli analyses Summary & Discussion
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4
Late time expansion Awarded Nobel Prize in 2011! What can be a source for this?
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EOM (Friedmann eq.) Observationally The universe is accelerating if DE domination Acceleration Cosmological scale for flat background
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for a flat universe WMAP+BAO+SN suggests
WMAP+BAO+H0+DΔt+SN suggests
Time varying DE Cosmological constant
Two possibilities e.g. Stringy Quintessence models
[Kiwoon, 99], [Svrcek, 06], [Kaloper, Sorbo, 08], [Panda, YS, Trivedi, 10], [Cicoli, Pedro, Tasinato, 12]…
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Metastable vacua in moduli space dS dS AdS We may stay here for a while. But how likely with tiny CC?
rolling down (& tunneling) tunneling Low energy
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There are many types of vacua in string theory, as a result of a variety of (Calabi-Yau) compactification. E.g. workable models:
4
: ℎ1,1 = 2, ℎ2,1 = 272
[Denef, Douglas, Florea, 04]
All can be stabilized (a la KKLT), but in various way. A class of Calabi-Yau gives Swiss-cheese type of volume. 𝒲6 = 𝛿1 𝑈
1 + 𝑈
1 − 𝛿𝑗 𝑈𝑗 + 𝑈 𝑗
𝑗=2
, Any implication of multiple vacua? 𝑒𝑡10
2 = 𝑒𝑡4 2 + 𝑒𝑡6 2
More recently, for 2 ≤ ℎ1,1 ≤ 4, 418 manifolds !
[Gray, He, Jejjala, Jurke, Nelson, Simon, 12]
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Assuming products of random variables: 𝑨 = 𝑧1𝑧2𝑧3 ⋯ Product distribution We apply this mechanism for cosmological constant (CC) Many terms? through stabilization 𝑨 = 𝑧1𝑧2𝑧3 ⋯ 𝑔(𝑧1, 𝑧2, 𝑧3, ⋯ ) still peaked Correlation
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I have to say
But here,
Before proceeding…
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Various vacua in string landscape Mass matrix given randomly at extrema
[Aazami, Easther, 05], [Dean, Majumdar, 08], [Borot, Eynard, Majumdar, Nadal, 10]
How likely stable minima exist?
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Positivity of mass matrix 𝜖𝜚𝑗𝜖𝜚𝑘𝑊
min
Positivity of Hessian Real/complex symmetric matrix
𝑎 = 𝑒𝑁𝑗𝑘 𝑓−1
2tr 𝑁2 , 𝑁 = 𝑁𝑈
Gaussian term dominates even at lower 𝑂.
ln 3 4 ∼ 0.275, ln 2 3−3 2
∼ −0.384
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[X. Chen, Shiu, YS, Tye, 11]
Still Gaussianly suppressed, but a chance for dS
[Bachlechner, Marsh, McAllister, Wrase, 12]
Hessian = 𝐵 + 𝐶 where 𝐵: diagonal positive definite with 𝜏
𝐵
𝐶: GOE with 𝜏𝐶 Actual models are likely to have minima at AdS. + uplifting term toward dS vacua.
Larger hierarchy
When applying a model in type IIA, quite tiny chance remains.
𝒬 = 𝑓−𝑐𝑂2 𝒬 = 𝑏 𝑓−𝑐𝑂2−𝑑𝑂
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Metric: 𝑒𝑡10
2 = 𝑓2𝐵𝑒𝑡4 2 + 𝑓−2𝐵𝑒𝑡 6 2
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Sources: 𝐼3, 𝐺
1, 𝐺3, 𝐺
5, dilaton, localized sources Then EOM becomes 𝛼 2 e4𝐵 − 𝛽 = e2A 6 Im 𝜐 𝑗𝐻3 −∗6 𝐻3 2 + 𝑓−6𝐵 𝜖 𝑓4𝐵 − 𝛽
2 + (local sources)
[Giddings, Kachru, Polchinski, 02]
Calabi-Yau
positive contributions
LHS=0 when integrating out
𝑓4𝐵 = 𝛽, 𝑗𝐻3 =∗6 𝐻3: imaginary self-dual condition where 𝛽 is a function in 𝐺 5, 𝐻3 = 𝐺3 − 𝜐 𝐼3, 𝜐 = 𝐷0 + 𝑗 𝑓−𝜚
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Take a scaling: 𝑛𝑜 → 𝜇 𝑛𝑜 𝑓4𝐵 = 𝛽, 𝑗𝐻3 =∗6 𝐻3: invariant The other equations are also unchanged. No-scale structure superpotential 𝑋
0 = ∫ 𝐻3 ∧ Ω is independent of Kahler
4D effective potential with 𝐿 = −3 ln 𝑈 + 𝑈 , 𝑋
0 = const
𝑊 = 𝑓𝐿 𝑁𝑄
2
𝐿𝐽𝐾𝐸𝐽𝑋
0 𝐸𝐾𝑋 0 − 3
𝑁𝑄
2 𝑋 2
= 0 Kahler directions remain flat.
Hierarchical structure of mass matrix/potential helps to stabilize moduli at positive cosmological constant.
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No scale structure Hierarchy between Kahler and Complex Moduli stabilization with positive cosmological constant
Complex structure & dilaton
[KKLT, 03], [Balasubramanian, Berglund, Conlon, Quevedo, 05], [Balasubramanian, Berglund, 04]…
Kahler 𝑊 = 𝑊
Flux + 𝑊 NP + 𝑊 𝛽′ + ⋯
Complex Kahler
[X. Chen, Shiu, YS, Tye, 12]
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Non-trivial potential for Kahler is generated by NP-corrections. Gluino condensation on D7-branes 𝑋
𝑂𝑄 = 𝐵 𝑓−𝑏 8𝜌2 𝐸7
= 𝐵 𝑓−𝑏 𝑈 D7-branes wrapping the four cycle: Together with the superpotential from fluxes: 𝑋 = 𝑋
0 + 𝑋 𝑂𝑄
E.g. Supersymmetric vacuum 𝐸𝑈𝑋 = 0 existes. But exponentially small 𝑋
0 is
required. |𝑋
0| ∼ 𝐵 𝑓−𝑏 𝑈, naturally realized?
|𝑋
0| ∼ 10−4
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𝛽′-corrections can break no-scale structure too. 𝐿 = −2 ln 𝒲 + 𝜊 2 −𝑗 𝜐 + 𝜐
3 2
− ln(−𝑗 𝜐 + 𝜐 ) + ⋯ 𝒫 𝛽′3 -correction in type II action [Becker, Becker, Haack, Louis, 02] scales differently E.g. ℙ 1,1,1,6,9
4
model (assuming complex sector is stabilized) 𝒲 = 1 9 2 𝑢1
3 2 − 𝑢2 3 2
, 𝑋 = 𝑋
0 + 𝐵1𝑓−𝑏1𝑈
1 + 𝐵2𝑓−𝑏2𝑈 2
Solution: 𝑋
0 ∼ −20, 𝐵1 ∼ 1, 𝑢1 ∼ 106, 𝑢2 ∼ 3
|𝑋
0| ≫ |𝑋 𝑂𝑄|, 𝒲 ≫ 𝜊: naturally realized
𝑊min ∼ −10−25 : AdS vacua
[Balasubramanian, Beglund, Conlon, Quevedo, 05]
𝑋 = 𝑋
0 + 𝐵1𝑓−𝑏1𝑈
1 + 𝐵𝑗𝑓−𝑏𝑗𝑈𝑗
𝑗=2
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𝐿 = −2 ln 𝒲 + 𝜊 2 + ⋯ , 𝒲 = 𝛿1 𝑈
1 + 𝑈
1 − 𝛿𝑗 𝑈𝑗 + 𝑈 𝑗
𝑗=2
, Same setup as that of LVS Interested in a region where this term plays a roll.
[Balasubramanian, Berglund, 04], [Westphal, 06], [Rummel, Westphal, 11], [de Alwis, Givens, 11]
less large volume than LVS, but still |𝑋
0| ≫ |𝑋 𝑂𝑄|, 𝒲 ≫ 𝜊
E.g. single modulus 𝑊 ∼ − 𝑋
0𝑏1 3𝐵1
2 𝛿
1 2
2𝐷 9𝑦1
9 2 − 𝑓−𝑦1
𝑦1
2
, 𝐷 = −27 𝑋
0 𝜊 𝑏1 3 2
64 2𝛿1𝐵1 , 𝑦1 = 𝑏1𝑢1 When 𝑋
0𝐵1 < 0, the 𝐷 ∝ 𝜊 term contributes the uplifting.
[Rummel, Westphal, 11]
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Add an uplifting potential by hand 𝑊 = 𝑊
𝑇𝑉𝐻𝑆𝐵 + 𝑊 𝐸3−𝐸3
𝑊
𝐸3−𝐸3 = 2𝑈3 𝑒4𝑦 −4
Backreaction of 𝐸3? A singularity exists, but finite action
[DeWolfe, Kachru, Mulligan, 08], [McGuirk, Shiu, YS, 09], [Bena, Giecold, Grana, Halmagyi, Massai, 09-12], [Dymarsky, 11],…
Safe or not?
𝑊 = 𝑊
𝑇𝑉𝐻𝑆𝐵
SUGRA + 𝛽′-correction Owing to |𝑋
0| ≫ |𝑋 𝑂𝑄|
No fine-tuning for 𝑋
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𝑊 𝑁𝑄
4 = − 𝑋 0𝑏1 3𝐵1
2 𝛿1 𝐷 9𝑦19 2
− 𝑓−𝑦1
𝑦1
2
, C = −27𝑋
0𝜊𝑏1 3 2
64 2𝛿
1 2𝐵1
, 𝑦1 = 𝑏1𝑢1
[Rummel, Westphal, 11]
Further focusing on smaller CC region: 𝐷 ∼ 3.65 The stability constraint with positive CC at stationery points: 3.65 ≤ 𝐷 < 3.89 𝑊 ≥ 0 𝜖𝑦
2𝑊 > 0
𝑊 𝑁𝑄
4 ∼ 1
9 2 5
9 2
−𝑋
0𝑏1 3𝐵1
𝛿
1 2
𝐷 − 3.65 Neglecting the parameters 𝑏1, 𝛿1, 𝜊, the model is simplified to be Λ = 𝑥1𝑥2 𝑑 − 𝑑0 , 𝑑0 ≤ 𝑑 = 𝑥1 𝑥2 < 𝑑1 (𝑥1 = −𝑋
0, 𝑥2 = 𝐵1, 𝑑 ∝ 𝐷)
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Starting with the simplified potential: Λ = 𝑥1𝑥2 𝑑 − 𝑑0 , 𝑑0 ≤ 𝑑 = 𝑥1 𝑥2 < 𝑑1 Since 𝑋
0, 𝐵1 are given model by model (various ways of
stabilizing complex moduli), here we impose reasonable randomness on parameters. 𝑥1, 𝑥2 ∈ [0, 1], uniform distribution (for simplicity) Probability distribution function 𝑄 Λ = 𝑂0 𝑒𝑑 𝑒𝑥1𝑒𝑥2 𝜀 𝑥1𝑥2 𝑑 − 𝑑0 − Λ 𝜀 𝑥1 𝑥2 − 𝑑 𝑂0: normalization constant
[YS, Tye, 12]
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Divergence in product distribution When 𝑨 = 𝑥1𝑥2, 𝑄 𝑨 = 𝑒𝑥1𝑒𝑥2 𝜀 𝑥1𝑥2 − 𝑨 = 1 2 ln 1 𝑨
log divergence at 𝑨 = 0
With constraint? 𝑄 Λ = 𝑑1 𝑑1 − 𝑑0 ln 𝑑1 − 𝑑0 𝑑1Λ still diverging!! Comparison to the full-potential (randomizing 𝑋
0, 𝐵1 without approx.)
Good agreement at smaller Λ
Λ = 𝑥1𝑥2 𝑑 − 𝑑0 , 𝑑0 ≤ 𝑑 = 𝑥1 𝑥2 < 𝑑1
positivity stability
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We assumed the parameters 𝑋
0, 𝐵1 passing through zero value,
but is it true?
𝑋
0 =
𝑑1 + 𝑒𝑗𝑉𝑗 − 𝑑2 + 𝑓𝑗𝑉𝑗 𝑇
SUSY condition
𝑋
0 = 2 𝑑1 + 𝑑2𝑡
𝑒𝑙 − 𝑓𝑙𝑡
𝑙
𝑒𝑗 + 𝑓𝑗𝑡 (𝑒𝑘 − 𝑓
𝑘𝑡) 𝑘≠𝑗 𝑗
𝑡 = Re(𝑇)
easy to be zero
𝐵1 = 𝐵 1 𝑉𝑗 𝑔 𝑌𝑗
1/𝑜,
𝑔 𝑌𝑗 = 𝑌𝑗
𝑞𝑗 − 𝜈𝑟
[Baumann, Dymarsky, Klebanov, Maldacena, McAllister, Murugan, 06]
𝑔 𝑌𝑗 = 0 when D3-brane hits D7-brane (divisor, at 𝜈) known as Ganor zero
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Product distribution is understood in terms of Mellin integral transformation. For 𝑨 = 𝑦1𝑦2, with distributions 𝑄
1 𝑦1 , 𝑄2(𝑦2),
𝑄 𝑨 = 𝑒𝑦1𝑒𝑦2 𝑄
1 𝑦1 𝑄2 𝑦2 𝜀 𝑦1𝑦2 − 𝑨
When multiplying 𝑨𝑡−1 and integrating over 𝑨, 𝑁 𝑄 𝑨 |𝑡 = 𝑁 𝑄
1 𝑦1 𝑡 ⋅ 𝑁 𝑄2 𝑦2 𝑡 ,
𝑁 𝑔 𝑥 𝑡 ≡ ∫ 𝑒𝑥 𝑥𝑡−1𝑔 𝑥 : Mellin integral transform E.g. Normal (Gaussian) distribution: 𝑄 𝑦 =
2 𝜌𝜏2 𝑓− 𝑦2
2𝜏2
𝑄(𝑨) is written as Meijer G-function. Again, log-diverging behavior toward 𝑨 = 0
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Expectation value is a good measure to see tendency. How we can check the likelihood of small CC? 𝑨 = 𝑒𝑨 𝑨 𝑄 𝑨 = 𝑁 𝑄 𝑨 |2 For 𝑨 = 𝑦1 ⋯ 𝑦𝑜, all uniform 𝑨 = 𝑁 𝑄 𝑨 2 = 𝑁 𝑄
1 𝑦1 2 ⋯ 𝑁 𝑄 𝑜 𝑦𝑜 2 = 𝑦𝑗
= 𝑓−𝑜 ln 2: exponentially suppressed What happens in case of 𝑨 = 𝑦1
𝑞?
The factorization of Mellin transform doesn’t apply. 𝑄 𝑨 = 𝑨−1+1
𝑜
𝑜 𝑄
1 𝑨1 𝑜
𝑨 = 1 𝑜 + 1
uniform More divergent, but not suppressed so much.
Independent random variables are important.
𝑨=0
−1 𝑛−1 𝑛 − 1 ! 𝑜 − 1 ! 2𝑜 ln 𝑨 𝑛−1 𝑄 𝑨 =
1 𝑛+𝑜 𝑨−1+1/𝑛 for 0 ≤ 𝑨 ≤ 1 1 𝑛+𝑜 𝑨−1−1/𝑜 for 1 ≤ 𝑨
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What happens when random variables show up in denominator?
𝑦1⋯𝑦𝑛 𝑧1⋯𝑧𝑜, all obey uniform distribution
𝑄 𝑨 = 𝑒𝑦1 ⋯ 𝑒𝑦𝑛𝑒𝑧1 ⋯ 𝑒𝑧𝑜 𝜀 𝑦1 ⋯ 𝑦𝑛 𝑧1 ⋯ 𝑧𝑜 − 𝑨 Therefore divergence is same as that in the numerator.
𝑛
𝑧1
𝑜, all uniform
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Sum distribution smooths out the divergence and moves the peak. E.g. 𝑨 = 𝑦1
𝑜1 + 𝑦2 𝑜2 + ⋯ + 𝑦𝑞 𝑜𝑞
𝑜𝑗 ∝ 𝑥𝑗 −1+ 1
𝑜𝑗
But uncorrelated summation gives 𝑄 𝑨 ∝ 𝑨
−1+ 1
𝑜𝑗 .
When all 𝑜𝑗 = 2, and 𝑦𝑗 ∈ normal distribution, 𝑄 𝑨 = 𝑓−𝑞 2
𝑨−1+𝑞 2
2𝑞 2
Γ(𝑞 2
) known as Chi-squared distribution
𝑞 = 1 𝑞 =2 𝑞 =3
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Assume randomness in Bousso-Polchinski; Λ = Λbare + 1 2 𝑜𝑗
2 𝑟𝑗 2 𝐾
𝑜𝑗: random integer, 0 ≤ 𝑟𝑗 ≤ 1: uniform, 𝑇 = 𝑒4𝑦 − 1 𝑁𝑄
2 𝑆 − Λbare −
𝑎 2 × 4! 𝐺
4 2
4-form quantization −100 ≤ Λbare ≤ 0: uniform But… Λ ∼ −𝑋
0𝐵1
𝐷 9𝑦19 2
− 𝑓−𝑦1
𝑦1
2
Moduli fields couple each term correlation generated via stabilization
Λ = 0
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Again, we work in the region: |𝑋
0| ≫ |𝑋 𝑂𝑄|, 𝒲 ≫ 𝜊.
𝑊 𝑁𝑄
4 = − 𝐵1𝑋 0𝑏1 3
2 𝛿1 2𝐷 9𝒲 3 − 𝑦1𝑓−𝑦1 𝒲 2 − 𝐶𝑗𝑦𝑗𝑓−𝑦𝑗 𝒲 2
𝑗=2
,
𝑦𝑗 = 𝑏𝑗𝑢𝑗, 𝐷 = −27𝑋
0𝜊𝑏1 3 2
64 2𝛿1𝐵1 , 𝐶𝑗 = 𝐵𝑗 𝐵1 , 𝜀𝑗 = 𝛿𝑗𝑏𝑗
3 2
𝛿1𝑏1
3 2
𝒲 = 𝑦1
3 2 − 𝜀𝑗𝑦𝑗 3 2 𝑗=2
,
Assuming stabilization of complex structure moduli and dilaton at higher energy scale,
Uplifting is controlled by the first term.
𝑂𝐿 extremal equations + 𝑂𝐿 stability constraints All upper-left sub-determinants are positive (Sylvester’s criteria).
[Sumitomo, Tye, in preparation]
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𝑦1 𝑦2 𝐷 𝐶2 We can find stability points when parameters are in the region. 3.65 ≲ 𝐷 ≲ 4.50 0 ≤ 𝐶2 ≲ 0.177
Stability constraint at 𝑂𝐿 = 1
Let’s compare with full-potential analysis. (therefore without |𝑋
0| ≫ |𝑋 𝑂𝑄|, 𝒲 ≫ 𝜊)
The parameter region is shifted slightly: 3.95 ≲ 𝐷 ≲ 4.87, 0 ≤ 𝐶2 ≲ 0.193 But not changed so much.
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Stability constraint at 𝑂𝐿 = 2
𝑦1 𝑦2 𝑦3
Stable points exist if 3.65 ≲ 𝐷 ≲ 4.50 0 ≤ 𝐶2,3 ≲ 0.177 But parameter region is further constrained. Again, let’s compare with full-potential analysis. (without |𝑋
0| ≫ |𝑋 𝑂𝑄|, 𝒲 ≫ 𝜊)
3.95 ≲ 𝐷 ≲ 4.87, 0 ≤ 𝐶2,3 ≲ 0.193 Though the resultant parameters are slightly shifted, the essential feature wouldn’t be changed. (not easy to use full-potential beyond 𝑂𝐿 = 3…)
𝑊 𝑁𝑄
4 = − 𝐵1𝑋 0𝑏1 3
2 𝛿1 2𝐷 9𝒲 3 − 𝑦1𝑓−𝑦1 𝒲 2 − 𝐶𝑗𝑦𝑗𝑓−𝑦𝑗 𝒲 2
𝑗=2
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Λ ∼ 1.1 × 10−3𝑂𝐿
0.23𝑓−0.027 𝑂𝐿𝑁𝑄 4
More moduli bring shaper peak.
(neglecting 𝑂𝐿=1)
(though mild suppression) Still complicated system We just randomize 𝑋
0, 𝐵𝑗 obeying uniform distribution,
while keeping other parameters fixed. Solve for 𝑢𝑗 (or 𝑦𝑗) −15 ≤ 𝑋
0 ≤ 0, 0 ≤ 𝐵𝑗 ≤ 1
𝑂𝐿 = 1: blue 𝑂𝐿 = 3: red
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Reheating for BBN: 𝑈
𝑠 ≥ 𝒫 10 MeV
𝑈
𝑠 ∼
𝑁𝑄Γ𝜚, Γ𝜚 ∼
𝑛𝜚
3
𝑁𝑄
𝑛𝜚 ≥ 𝒫 10 TeV ∼ 10−15 𝑁𝑄 What happens in lightest (physical) moduli mass? 𝑛min
2
= 0.031 𝑂𝐿
1.0𝑓−0.10 𝑂𝐿𝑁𝑄 2 : also suppressed
Suppression of mass is relatively faster than Λ. 𝑛min
2
∼ 10−30𝑁𝑄
2 is likely met earlier than Λ ∼ 10−122𝑁𝑄 4
(neglecting 𝑂𝐿=1)
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So far we assumed uniform distribution for 𝑋
0, 𝐵𝑗. But realistic
models have a number of complex moduli and others. Different distributions for 𝑋
0, 𝐵𝑗
Consider the effect of multiple independent parameters. 𝑋
0 = −𝑥1𝑥2 ⋯ 𝑥𝑜,
𝐵𝑗 = 𝑧1
𝑗 𝑧2 𝑗 ⋯ 𝑧𝑜 𝑗
0 ≤ 𝑥𝑗 ≤ 15
1 𝑜, 0 ≤ 𝑧𝑘
𝑗 ≤ 1, all obey uniform distribution.
n=1 n=2 n=3
Now, 𝑄 𝑋
0 =
1 15 𝑜 − 1 ! ln 15 𝑋
𝑜−1
, 𝑄 𝐵𝑗 = 1 𝑜 − 1 ! ln 1 𝐵𝑗
𝑜−1
See how CC is affected by “𝑜”
𝑊 𝑁𝑄
4 = − 𝐵1𝑋 0𝑏1 3
2 𝛿1 2𝐷 9𝒲 3 − 𝑦1𝑓−𝑦1 𝒲 2 − 𝐶𝑗𝑦𝑗𝑓−𝑦𝑗 𝒲 2
𝑗=2
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We cannot simply consider effect of the coefficient. Dynamics also affects. The result:
Red: 𝑂𝐿 = 1 Blue: 𝑂𝐿 = 2 Green: 𝑂𝐿 = 3
Λ 𝑂𝐿=1 = 4.7 × 10−3 𝑜0.080𝑓−1.40 𝑜 Λ 𝑂𝐿=3 = 3.4 × 10−3 𝑜1.5𝑓−1.55 𝑜 Λ 𝑂𝐿=2 = 3.7 × 10−3 𝑜0.97𝑓−1.49 𝑜 More than the effect of the coefficient! 𝐵1𝑋
0 ∼ 15 𝑓−1.39 𝑜
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We worry about the cosmological moduli problem.
Red: 𝑂𝐿 = 1 Blue: 𝑂𝐿 = 2 Green: 𝑂𝐿 = 3
mmin
2 𝑂𝐿=1 = 0.18 𝑜0.14𝑓−1.40 𝑜
mmin
2 𝑂𝐿=3 = 0.039 𝑜1.2𝑓−1.66 𝑜
mmin
2 𝑂𝐿=2 = 0.061 𝑜0.73𝑓−1.56 𝑜
Λ ∝ 𝑓−1.40 𝑜, 𝑓−1.49 𝑜, 𝑓−1.55 𝑜 Compare with CC Suppression in mass is getting larger as increasing 𝑂𝐿.
also suggests
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Using the estimated functions, we get 𝑂𝐿(= ℎ1,1) 1 2 3 Λ ∼ 10−122𝑁𝑄
4
𝑜 ∼ 197 𝑜 ∼ 188 𝑜 ∼ 182 𝑛2 ∼ 10−30𝑁𝑄
2
𝑜 ∼ 48 𝑜 ∼ 44 𝑜 ∼ 42
𝑜: number of product in 𝑋
0, 𝐵𝑗
4
: ℎ1,1 = 2, ℎ2,1 = 272 Rather considerable number, e.g.
1 𝑉𝑗 𝑔 𝑌𝑗
1/𝑜, 𝑔 𝑌𝑗 = 𝑌𝑗 𝑞𝑗 − 𝜈𝑟
and the other moduli (e.g. brane position, open string) come in a complicated way, like While, without help of product distribution in 𝑋
0, 𝐵𝑗
𝑂𝐿 ∼ 10100 for Λ ∼ 10−122𝑁𝑄
4, 𝑂𝐿 ∼ 1350 for 𝑛2 ∼ 10−30𝑁𝑄 2
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Consider the simplest model: 𝑋
0 = −
𝑑1 + 𝑓𝑗
ℎ2,1 𝑗=1
𝑉𝑗 − 𝑑2 + 𝑔
𝑗 ℎ2,1 𝑗=1
𝑉𝑗 S SUSY stabilization 𝐸𝑉𝑗𝑋
0 = 0,
𝐸𝑇𝑋
0 = 0
When 𝑂𝐷 ↑, the dist. gets more sharply peaked! with Re 𝑉𝑗 > 0, Re 𝑇 = 𝑡
−1 > 1
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Physical mass matrix is a linear combination of 𝜖𝑦𝑗𝜖𝑦𝑘𝑊|min.
𝜖𝑦𝑗𝜖𝑦𝑘𝑊 min ∼ 10−3 × 7 4 ⋯ ⋯ 4 4 60 1 ⋯ 1 ⋮ 1 ⋱ ⋯ ⋮ ⋮ ⋮ ⋮ ⋱ 1 4 1 ⋯ 1 60
Assuming uniformly distributed −15 ≤ 𝑋
0 ≤ 0, 0 ≤ 𝐵𝑗 ≤ 1,
some hierarchical structures Though off-diagonal comp. are relatively suppressed, eigenvalue repulsion gets more serious when increasing 𝑂𝐿.
𝑦1 𝑦2 ⋯ 𝑦𝑂𝐿
e.g. 2 × 2 matrix: 𝑏 𝑐 𝑐 𝑑 𝜇± = 1 2 𝑏 + 𝑑 ± 𝑏 − 𝑑 2 + 4𝑐2 The lowest mass eigenvalue is generically suppressed more than CC.
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Both works for smaller CC. We may expect that stringy motivated models have the following properties:
Correlation makes CC smaller. But the effect is modest. Those are likely to produce more peakiness in parameters Interesting to see detailed effect in concrete models
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Lightest moduli mass is suppressed simultaneously. cosmological moduli problem before reaching Λ ∼ 10−122𝑁𝑄
4.
Thermal inflation, coupling suppression to SM,
Other than “product” and “correlation” effect, “eigenvalue repulsion” also makes the value smaller. This is presumably a generic problem when taking statistical approach without fine-tuning. Once finding a way out, the stringy mechanism naturally explain why CC is so small.