SO(32) heterotic string theory Hajime Otsuka (Waseda U.) based on - PowerPoint PPT Presentation

Hypercharge flux in SO(32) heterotic string theory Hajime Otsuka (Waseda U.) based on arXiv:1801.03684 [hep-th] JHEP 05 (2018) 045 arXiv:1808.XXXXX [hep-th] (with K. Takemoto) “PPP2018” @ YITP, Kyoto

Why (super)string theory ? ・ Quantum Gravity ・ Unified theory Good candidate for the unified theory of the gauge and gravitational interactions

Superstring theory / M theory Type IIB Type IIA Type I F M 11D SUGRA Heterotic SO(32) Adjoint rep. :496 Heterotic E8 × E8 Adjoint rep. :248 × 248 Where is the Standard Model ? Why three generations ? 3

Superstring theory / M theory Type IIB Type IIA Type I F M 11D SUGRA Heterotic SO(32) Adjoint rep. :496 Heterotic E8 × E8 Adjoint rep. :248 × 248 4

10D Superstring theory 4D Standard Model(SM) SUSY-preserving 1. Orbifolds 6D internal spaces: classified by “ orbifolder ” Nilles, Ramos-Sanchez, Vaudrevange, Wingerter (‘11) 2. Calabi-Yau (CY) Problem: in perturbative superstring, many 4D string vacua Can we derive conditions to derive the SM in general CY ? 5

Outline ○ Introduction ○ Heterotic Standard Models on smooth CY i) Model-building approach ii) General formula iii) Concrete model ○ Conclusion 6

Heterotic Standard Models on smooth Calabi-Yau (CY) “ Standard embedding ” Candelas-Horowitz-Strominger- Witten (‘85) 6D Calabi-Yau (CY)Manifold ○ Ricci-flat manifold 𝑆 𝑗𝑘 = 0 ○ SU(3) holonomy ・ Gauge symmetry breaking: 𝑇𝑉 3 = 𝑥 𝑗 spin ) 𝐹 8 × 𝐹 8 → 𝐹 6 × 𝑇𝑉 3 × 𝐹 8 ( 𝐵 𝑗 → 𝑇𝑉 3 𝐷 × 𝑇𝑉 2 𝑀 × 𝑉 1 𝑍 (Wilson lines) ・ Number of chiral generation = |Euler number of CY|/2 |𝜓 CY | = 6 7

“ Standard embedding ” Requirements: ① Wilson-line breaking (possible for restricted CYs) ・ We require non-contractible one-cycles (non-simply-connected CY) E.g., 195 non-simply-connected CICYs among total 7890 CICYs CICY=Complete Intersection Calabi-Yau ② Small Euler number of CY (3 generations of quarks) |𝜓 CY | = 6

Two approaches in the heterotic model building on smooth CY 1. “Standard embedding” 𝑇𝑉 3 = 𝑥 𝑗 spin 𝐵 𝑗 𝐹 8 → 𝐹 6 × 𝑇𝑉 3 → 𝐻 SM × 𝐻 hid 2. “Non - standard embedding” 𝑇𝑉 3 ≠ 𝑥 𝑗 spin 𝐵 𝑗 𝐹 8 → 𝐻 SM × 𝐻 hid ・ SM vacua directly with the SM gauge group 9

“Non - standard embedding” 〇 Internal 𝑉(1) gauge fluxes 𝐺 1 𝐺 = 𝑛 (𝑗) ∈ ℤ 2𝜌 න Σ 𝑗 Σ 𝑗 : Two-cycles of CY E.g., Hypercharge flux 𝑇𝑉 5 → 𝑇𝑉 3 𝐷 × 𝑇𝑉 2 𝑀 × 𝑉 1 𝑍 2 2 2 < 𝐺 𝑉 1 𝑍 > ∝ −3 −3 ・ Popular in the F-theory 𝑇𝑉(5) GUT Beasley-Heckman-Vafa, Donagi-Wijnholt (’08) ・ Direct flux breaking scenario is applicable in the Heterotic context Blumenhagen-Honecker- Weigand (’05) 10

〇 Internal 𝑉(1) gauge fluxes 𝐺 ・ Gauge symmetry breaking 𝑇𝑉 5 → 𝑇𝑉 3 𝐷 × 𝑇𝑉 2 𝑀 × 𝑉 1 𝑍 ・ Chiral and net-number of zero-modes, given by 1 6 tr 𝐺 3 + 1 1 12 tr 𝑆 2 ∧ tr 𝐺 𝑂 gen = 2𝜌 3 න CY Background curvatures 𝐺 and 𝑆 give rise to the three-generation of quarks and leptons 𝑅, 𝑀, 𝑣 𝑑 , 𝑒 𝑑 , 𝑓 𝑑 : 𝑂 gen = −3 No chiral exotics : 𝑂 gen = 0 11

Internal 𝑉 1 𝑏 gauge fluxes 𝐺 𝑏 1 (𝑗) ∈ ℤ 2𝜌 න 𝐺 𝑏 = 𝑛 𝑏 Σ 𝑗 : Two-cycles of CY Σ 𝑗 ・ Chiral index 1 6 tr 𝐺 3 + 1 1 12 tr 𝑆 2 ∧ tr 𝐺 𝑂 gen = 2𝜌 3 න CY 𝑂 gen = 1 𝑑 + 1 6 ෍ 𝑌 𝑏𝑐𝑑 𝑍 𝑏 𝑍 𝑐 𝑍 12 ෍ 𝑎 𝑏 𝑍 𝑏 𝑏,𝑐,𝑑 𝑏 𝑍 𝑏 : U 1 𝑏 charges of zero-modes (𝑗) . 𝑌 𝑏𝑐𝑑 , 𝑎 𝑏 depends on the topological data of CY and 𝑛 𝑏 12

Internal 𝑉 1 𝑏 gauge fluxes 𝐺 𝑏 ・ Chiral index 𝑂 gen = 1 𝑑 + 1 6 ෍ 𝑌 𝑏𝑐𝑑 𝑍 𝑏 𝑍 𝑐 𝑍 12 ෍ 𝑎 𝑏 𝑍 𝑏 𝑏,𝑐,𝑑 𝑏 𝑍 𝑏 : U 1 𝑏 charges of zero-modes ・ Index is determined only by variables ( 𝑌 𝑏𝑐𝑑 , 𝑎 𝑏 ) ・ Applicable in all CYs It opens up a possibility of searching for the three-generation SM in a background-independent way 13

Outline ○ Introduction ○ Heterotic Standard Models on smooth CYs i) Model building approach ii) General formula iii) Concrete model ○ Conclusion 14

𝐹 8 × 𝐹 8 heterotic Standard Models are well studied by Donagi-Ovrut-Pantev-Waldram (‘00), Blumenhagen-Honecker- Weigand (‘05) Anderson-Gray-Lukas-Palti (‘12) SO(32) heterotic Standard Models S- and T-dualities Intersecting D6-brane models in type IIA string (Several stacks of D-branes  MSSM or Pati-Salam model) Q U(3) U,D U(2) U(1) L U(1) E Our research: SO(32) heterotic SM (MSSM) vacua directly with the SM gauge group from smooth CYs 15

To concrete our analysis, we focus on the branching: 5 𝑇𝑃 32 ⊃ 𝑇𝑃 16 ⊃ 𝑇𝑉 3 𝐷 × 𝑇𝑉 2 𝑀 × Π 𝑏=1 𝑉 1 𝑏 ⊃ MSSM particles ( ～ 7 × 10 7 possibilities) 496 ⊃ 120 16 ⊃ Exotics We introduce internal 𝑉 1 𝑏 gauge fluxes 𝐺 𝑏 ( 𝑏 = 1,2,3,4,5 ) ・ Chiral index only depends on variables 𝑌 𝑏𝑐𝑑 , 𝑎 𝑏 𝑂 gen = 1 𝑑 + 1 6 ෍ 𝑌 𝑏𝑐𝑑 𝑍 𝑏 𝑍 𝑐 𝑍 12 ෍ 𝑎 𝑏 𝑍 𝑏 𝑏,𝑐,𝑑 𝑏 Can we constrain 35 variables 𝑌 𝑏𝑐𝑑 and 5 variables 𝑎 𝑏 ? 16

Phenomenological requirements: ・ Chiral index 𝑂 gen = 1 𝑑 + 1 6 ෍ 𝑌 𝑏𝑐𝑑 𝑍 𝑏 𝑍 𝑐 𝑍 12 ෍ 𝑎 𝑏 𝑍 𝑏 𝑏,𝑐,𝑑 𝑏 𝑍 𝑏 : U 1 𝑏 charges of zero-modes 5 conditions : 𝑅, 𝑀, 𝑣 𝑑 , 𝑒 𝑑 , 𝑓 𝑑 : 𝑂 gen = −3 6 conditions : No chiral exotics : 𝑂 gen = 0 17

Theoretical conditions: 5 ① Masslessness conditions for 𝑉 1 𝑍 = σ 𝑏=1 𝑔 𝑏 𝑉 1 𝑏 10D Green-Schwarz terms 𝐶 6 ∧ tr(𝐺 2 ) න න 𝐶 2 ∧ 𝑌 8 10D 10D Gauge fluxes 4D Green-Schwarz terms න 𝑐 ∧ 𝐺 U 1 Y 𝑐 : string axions 4D To ensure the masslessness of 𝑉 1 𝑍 gauge boson, (𝑗) = 0 2 𝑔 ෍ tr 𝑈 𝑏 𝑛 𝑏 ෍ tr 𝑈 𝑏 𝑈 𝑐 𝑈 𝑑 𝑈 𝑒 𝑔 𝑏 𝑌 𝑐𝑑𝑒 = 0 𝑏 18 𝑏 𝑏,𝑐,𝑑,𝑒

Theoretical conditions: ② To admit the spinorial rep. in the first excited mode (𝑗) = 2𝜆 (𝑗) ∈ 2ℤ ෍ tr 𝑈 𝑏 𝑛 𝑏 𝑏 (𝑗) (𝛽 = 1,2) は 𝑛 𝐵 𝑗 (𝐵 = 3,4,5) と 𝜆 (𝑗) で表すことが可能 ① , ②より、 𝑛 𝛽 40 variables {𝑌 𝑏𝑐𝑑 , 𝑎 𝑏 } Theoretical conditions ① , ② 23 variables

Against several branching of 𝑇𝑃 16 → 𝑇𝑉 3 × 𝑇𝑉 2 × Π 𝑏 𝑉 1 𝑏 three-generation models are possible, e.g., 𝑇𝑃 16 → 𝑇𝑃 6 × 𝑇𝑃 4 × 𝑇𝑃 2 3 5 → 𝑇𝑉 3 × 𝑇𝑉 2 × Π 𝑏=1 𝑉 1 𝑏 ・・・ 𝑞 𝑛 : integers ( 𝑛 = 1,2, ⋯ , 16 ) ・ Other U(1)s become massive through the GS mechanism in general. ・ Supersymmetric and stability conditions are required to be checked for each CYs. 20

Possible gauge branching satisfying all the requirements: 21

Outline ○ Introduction ○ Heterotic Standard Models on smooth CYs i) Model building approach ii) General formula iii) Concrete model ○ Conclusion 22

Concrete model Complete Intersection Calabi-Yau = 完全交叉カラビ・ヤウ Ambient Spaces Four 超曲面 カラビ・ヤウ

Concrete model Complete Intersection Calabi-Yau = 完全交叉カラビ・ヤウ Topological data of CY: ・ ℎ 1,1 = 4 (Number of Kähler moduli) ・ Intersection number ・ Second Chern number 24

Concrete model 𝑉 1 𝑏 fluxes ( 𝑏 = 1,2,3,4,5 ) ・ Supersymmetric and stability conditions are also satisfied at Dilaton Kähler moduli

Concrete model ✔ Gauge symmetry : SO 32 → 𝑇𝑉 3 𝐷 × 𝑇𝑉 2 𝑀 × 𝑉 1 𝑍 × 𝑇𝑃(16)′ ✔ Other 𝑉(1) s become massive through the GS mechanism ✔ Chiral spectrum: MSSM particles + Extra vector-like Higgs + Singlets ✔ Allow for perturbative Yukawa couplings ! ✔ No proton decay operators (constrained by massive 𝑉 1 𝐶−𝑀 )

Gauge coupling unification 5 ・ Against all branching of 𝑇𝑃 16 → 𝑇𝑉 3 × 𝑇𝑉 2 × Π 𝑏=1 𝑉 1 𝑏 Tree-level gauge couplings at the string scale, 5 2 2 2 2 𝑕 𝑇𝑉 3 𝐷 = 𝑕 𝑇𝑉 2 𝑀 = 6 𝑕 𝑉 1 𝑍 = 𝑕 0 ・ Gauge fluxes induce the threshold corrections to the gauge couplings −2 + Δ th,3 −2 𝑕 𝑇𝑉 3 𝐷 = 𝑕 0 −2 + Δ th,2 −2 𝑕 𝑇𝑉 2 𝑀 = 𝑕 0 −2 −2 /6 𝑕 𝑉 1 𝑍 = 5𝑕 0 Δ th,3 ≠ Δ th,2 ・ Nonuniversal gauge kinetic functions (in contrast to 𝐹 8 × 𝐹 8 heterotic string Δ th,3 = Δ th,2 ) 27

Conclusion ・ We have searched for 𝑇𝑃 32 heterotic SM vacua directly with the SM gauge group from smooth CYs ・ Direct flux breaking (Hypercharge flux in F-theory) 𝑇𝑃 32 → 𝑇𝑉 3 𝐷 × 𝑇𝑉 2 𝑀 × 𝑉 1 𝑍 × 𝑇𝑃(16) is applicable in general CY compactification ・ General formula leading to (i) Three-generation of quarks and leptons (ii) No chiral exotics Discussion ・ General formula in the dual global F-theory context 28