SUSY GUT and SM in heterotic asymmetric orbifolds

1. SUSY GUT and SM in heterotic asymmetric orbifolds Shogo Kuwakino ( 桑木野省吾 ) (Chung Yuan Christian University ) Based onPhysRevD.83.091703, JHEP12(2011)100 Collaborator : M. Ito, N. Maekawa( Nagoya ), S. Moriyama( Nagoya ), K. Takahashi, K. Takei, S. Teraguchi ( Osaka ), T. Yamashita( Aichi Medical ) Based ona paper in preparation Collaborator : F Beye( Nagoya ), T Kobayashi( Kyoto ) October12, 2012. Academia Sinica

2. Plan of Talk Introduction Heterotic Asymmetric Orbifold SUSY E6 GUT construction SUSY SM construction Summary

3. Introduction • String Standard Model • String theory • -- A candidate which describe quantum gravity and unify four forces • -- Is it possible to realize phenomenological properties of Standard Model ? • ( Supersymmetric ) Standard Model • -- We have to realize all properties of Standard model Four-dimensions, N=1 supersymmetry, Standard model group( SU(3)*SU(2)*U(1) ), Three generations, Quarks, Leptons and Higgs, No exotics, Yukawahierarchy, Proton stability, R-parity, Doublet-triplet splitting, Moduli stabilization, ... If we believe string theory as the fundamental theory of our nature, we have to realize standard model as the effective theory of string theory !! Challenging tasks !!

4. Introduction • String compactification : 10-dim  4-dim Orbifoldcompactification, CY, Intersecting D-brane, F -theory, M-theory, … • (Symmetric) orbifold compactification Dixon, Harvey, Vafa, Witten '85,'86 • Several GUT or SM gauge symmetry • N=1 supersymmetry Advantage : Disadvantage : • No adjoint representation Higgs • MSSM searches in symmetric orbifoldvacua : • Embedding higher dimensional GUT into string Kobayashi, Raby, Zhang '04 Buchmuller, Hamaguchi, Lebedev, Ratz '06 Lebedev, Nilles, Raby, Ramos-Sanchez, Ratz, Vaudrevange, Wingerter '07 ...... Three generations, Quarks, Leptons and Higgs, No exotics, Top Yukawa, Proton stability, R-parity, Doublet-triplet splitting, ... Realizing Mass-Mixing hierarchy and Moduli stabilization are still nontrivial tasks.

5. Introduction Generalization of orbifold action ( Non-geometric compactification ) • Asymmetric orbifold compactification+ Naraincompactification • Several GUT or SM gauge symmetry • N=1 supersymmetry • Increase the number of possible models( symmetric  asymmetric) • A few / no moduli fields( non-geometric ) Narain, Sarmadi, Vafa ‘87 Advantage : • Two possibilities ・ String  GUT  SUSY SM 4D SUSY-GUT with adjoint representation Higgs ・ String  SUSY SM their VEV break GUT symmetry • Realize adjoint representation Higgs(es) Goal : Search for SUSY GUT with adjointHiggsand SUSY SM in heterotic asymmetric orbifoldvacua

6. Heterotic Asymmetric Orbifold

7. Heterotic String Theory • Heterotic string theory Left Right • Heterotic string for our starting point • Degrees of freedom • --- Left mover 26 dim. Bosons • --- Right mover 10 dim. Bosons and fermions • Extra 16 dim. have to be compactified • Consistency  If 10D N=1, E8 X E8 or SO(32) Dynkin Diagram ( E8 ) Ex. ) E8 Root Lattice Left Right : Simple roots of E8 Left-moving momentum 10dim 8dim 8dim

8. Modular Invariance • Partition function • Partition function : One loop vacuum–vacuum amplitude : Oscillation operator : Zero point energy : Parameterization of the torus • Modular invariance • A torus can be expressed in several ways • Equivalent tori can be related by Modular T transformation : Modular S transformation : • Partition function must satisfy Consistent !

9. ( Symmetric ) Orbifold Compactification • ( Symmetric ) orbifold compactification • External 6 dim.  Assuming as Orbifold • Strings on orbifold • --- Untwisted sector • --- Twisted sector ( Fixed points ) • Modular invariance Shift orbifold for E8 Gauge symmetry is broken ! • Project out suitable right-moving fermionic states • 27 rep. matters • However, No adjoint rep. matters Shift orbifold 4dim 6dim 8dim 8dim

10. Naraincompactification of heterotic string 4dim 22dim 6dim • Naraincompactification of heterotic string theory 4dim • 4D N=4 SUSY • General flat compactification of heterotic string • --- Left : 22 dim • --- Right : 6 dim • Left-right combined momentum are quantized, and compose a lattice • which is described by some group factors ( ex. E8 lattice ) • Modular invariance  Lorentzian even self-dual lattice • Because of many dimensions ( left and right movers ), there are many possibilities • for (22,6)-dimensional Narain lattices • ・ For the first stage, we have to construct (22,6)-dimensional Narain lattices which • have suitable properties for string model building Left Right

11. Lattice Engineering Technique • Lattice engineering technique Lerche, Schellekens, Warner ‘88 • We can construct new Narain lattice from known one. • We can replace one of the left-moving group factor with a suitable • right-moving group factor Replace Consistent (modular invariant) Left-mover Right-mover Left Right Left Dual ( decomposition, A2=SU(3) ) Left-moving E6 part and right-moving E6 part transform “oppositely” because of minus sign in partition function Left E6 part and A2 part transform “oppositely” under modular transformation

12. Lattice Engineering Technique E8 root lat ( ) ( spanned by simple roots of E8 ) Dual • Lattice engineering technique E6 root lat ( ) A2 root lat ( ) A2 fund weight lat ( ) ( Decomposition ) E6 anti-fund weight lat ( ) A2 anti-fund weight lat ( ) Left • Simple example Dual E6 fund weight lat ( ) Left Dual ( Replace left A2  Right E6 ) Left Left Right Right Right Left lattice gauge symmetry

13. Lattice Engineering Technique E8 root lat ( ) ( spanned by simple roots of E8 ) Dual • Lattice engineering technique E6 root lat ( ) A2 root lat ( ) A2 fund weight lat ( ) ( Decomposition ) E6 anti-fund weight lat ( ) A2 anti-fund weight lat ( ) Ex. ) Left • Simple example Dual E6 fund weight lat ( ) Left Dual ( Replace left E6 Right A2) Left Right Right Left gauge symmetry lattice

14. Lattice Engineering Technique • Lattice engineering technique Lerche, Schellekens, Warner ‘88 • We can construct new even self-dual lattice from known one. • Left-right replacement can be done in repeating fashion, • Narain lattice 1  Narain lattice 2  Narain lattice 3  … • We can construct various Narain lattices • Advantage : Various gauge symmetries. • Easy to find out discrete symmetries of the lattices.  Orbifold

15. Asymmetric Orbifold Compactification Ex.) action • Asymmetric orbifold compactification None • Orbifold action ( Twist, Shift, Permutation ) Left Left mover : Right Right mover : 1/3 rotation Modding out the lattice in left-right independent way • Asymmetric orbifold compactification + Narain lattice Lattice engineering technique This method allows us to search possible models systematically Left-moving twists and shifts Left Right Right-moving twist

16. SUSY E6 GUT construction

17. E6 Unification • GUT models Maekawa, Yamashita ‘01-’04 • Unify all SM quarks and leptons into GUT matter multiplets ( E6, 27 rep. fields ) • Realistic Yukawa hierarchies ( E6 + horizontal sym., FN mechanism ) • Realize doublet-triplet splitting ( U(1)A, DW mechanism ) • Solve SUSY flavor problem ( SU(2)H ) Generic interaction : Include all terms allowed by symmetry  Matter contents determine the models Anomalous U(1) symmetry : Often appear in low energy effective theory of string theory.  Can we realize these GUT models in string theory ?

18. E6 Unification • E6 Unification Bando, Kugo ‘99 • All quarks and leptons are unified into • Yukawa structure • -- Low energy are from 1, 2 generations -- Three vector-like pair 5 and become super massive by VEV of GUT Higgs -- Only by assuming up-type Yukawa, down-type, charged lepton Yukawa can be derived Up-type Yukawa ( assumed) Down-type Yukawa ( obtained ) Charged lepton Yukawa ( obtained ) -- Good for string construction ( fewer parameters ) 18

19. E6 Unification assumption • Superpotential Higgs : , • Heavy mass ( GUT scale ) • Mixing of realize hierarchies (In suitable basis) Up-type Yukawa Down-type, Charged lepton Yukawa ( Mild hierarchy )

20. Requirements for String Model Building Typical E6 X U(1)A GUT models contain : • 4D SUSY, • E6 unification group, Minimum requirements for 4D-SUSY GUT models • Net 3 chiral generations ( , ), String • Adjoint Higgs fields ( ), • SU(2)H or SU(3)H family symmetry, • Anomalous U(1)A gauge symmetry, • Adjoint Higgs fields charged under the anomalous U(1)A gauge symmetry. Ex. ) Matter contents ( E6 x SU(2)H x U(1)A )

21. Adjoint representation Higgs • Diagonal embedding method Modding out the permutation symmetry of the lattice ( ) Ex.) ( i ) Combining with phaseless right-moving states ( i ) : permutation G gauge fields Eigen states of : Combining with right-moving states with opposite phases An adjoint representation Higgs

22. SUSY E6 GUT construction • Summary of our method 1. Lattice engineering technique 4D N=4 SUSY (22, 6) – dimensional Narain lattice with 2. Orbifold identification by (a).Diagonal embedding method Adjoint Higgs (Permutation among the E6 factors ) N = 1 SUSY (b). Right-moving twist (c). Other orbifold actions for modular invariance 3. Number of generations 3 generations ? Empirically, We consider the case of

23. Setup : lattice • models ( starting point ) This lattice can be constructed by Lattice engineering technique. • Our starting point: even self-dual lattice --- 4D N=4 model

24. Setup : Asymmetric Orbifold Model • asymmetric orbifold construction For part and right mover: ( Twist vector ) Rotation for right mover. Kakushadze, Tye ’97 action : Permutation for the three factors. Previous study claimed : By Z6 = Z3 x Z2 orbifold, only 1 E6 model is found. Z12 orbifold is missed in the classification. action : Rotation for right mover. Permutation symmetry

25. Setup : Asymmetric Orbifold Model • asymmetric orbifold construction Possible orbifold actions are For two A2 parts : 1. Shift 2. 1/3 ( or 2/3 ) rotation 3. Weyl reflection We consider all possible orbifold actions for the two A2 parts.

26. Possible Models We find 8 possible choices which satisfy the consistency. 3 - generation models X 3 9 - generation models X 2 0 - generation models X 3

27. Result : Three Generation Models • Result : 3-generation models Ito, S.K, Maekawa, Moriyama, Takahashi, Takei, Teraguchi, Yamashita ‘11 • Gauge symmetry : or • Net 3 chiral generations (5 – 2 = 3 or 4 – 1 = 3 ) • 1 adjoint representation Higgs

28. Result : Three Generation Models • Result : 3-generation models Ito, S.K, Maekawa, Moriyama, Takahashi, Takei, Teraguchi, Yamashita ‘11 • Model 1 : Same mass spectrum with • asymmetric orbifold model • although orbifold action is different. Kakushadze and Tye,1997

29. Result : Three Generation Models • Result : 3-generation models Ito, S.K, Maekawa, Moriyama, Takahashi, Takei, Teraguchi, Yamashita ‘11 • Model 2, 3 : New models !

30. Result : Three Generation Models • Result : 3-generation models Ito, S.K, Maekawa, Moriyama, Takahashi, Takei, Teraguchi, Yamashita ‘11 • These models satisfy the minimum requirements for 4D E6 GUT models. • No non-Abelian family symmetry • No anomalous U(1) symmetry • For E6*U(1)A GUT, we have to search other asymmetric orbifoldvacua ( Z3xZ3, .. ) • Fine tuning would be needed for doublet-triplet splitting

31. SUSY SM construction

32. SUSY SM in asymmetric orbifoldvacua ・ First stepfor model building : Gauge symmetry Four-dimensions, N=1 supersymmetry, Standard model group( SU(3)*SU(2)*U(1) ), Three generations, Quarks, Leptons and Higgs, No exotics, Yukawahierarchy, Proton stability, R-parity, Doublet-triplet splitting, Moduli stabilization, ... What types of gauge symmetries can be derived in these vacua ? ・SM group ? ・GUT group ? ・Flavor symmetry ? ・Hidden sector ?

33. Lattice and gauge symmetry • Our starting point  Lattice 4dim 22dim Symmetric orbifolds Asymmetric orbifolds E8 x E8, SO(32) What types of (22,6)-dimensional Narain lattices can be used for starting point ? Lattice 6dim 4dim What types of gauge symmetries can be realized ? Left Gauge symmetry breaking pattern Classified Right 10dim 8dim 8dim Left Right

34. (22,6)-dim lattices from 8, 16, 24-dim lattices Constructing (22, 6)-dim Narain lattices from 8, 16, 24-dim lattices by lattice engineering technique 24dim 22dim 6dim 16-dim lattice 24-dim lattice 8-dim lattice 8dim 16dim E8 SO(32) Left Left Classified Left Left 24 types of lattices Right

35. (22,6)-dim lattices from 8, 16, 24-dim lattices Example : 24-dim lattice Generator of conjugacy classes : Gauge symmetry : SU(12) x SO(14) x E6 A11 D7 E6 Left A8 U1 A2 D7 E6 Left A8 U1 D7 E6 Left E6 Right (22,6)-dim lattice Generator of conjugacy classes : Gauge symmetry : SO(14) x E6 x SU(9) x U(1)

36. Gauge symmetry breaking by Z3 action • Z3 asymmetric orbifold compactification Simplest asymmetric orbifold action Z3 action : Right mover  twist action  N=1 SUSY Left mover  shift action  Gauge symmetry breaking Modular invariance A8 U1 D7 E6 Left E6 Right

37. Result : Lattice and gauge symmetry ( Z3 ) • Our starting point  Lattice 4dim 22dim Symmetric orbifolds Asymmetric orbifolds E8 x E8, SO(32) 97 lattices ! Lattice ( with right-moving non-Abelianfactor, from 24 dimensional lattices ) 6dim 4dim Left Classified !! Gauge symmetry breaking pattern Classified Right 10dim 8dim 8dim Left Right

38. Result : Example of group breaking patterns ・SO(14) x E6 x SU(9) x U(1) Gauge group breaks to Several gauge symmetries. ・Some group combinations lead to modular invariant models ・SM group, Flipped SO(10)xU(1), Flipped SU(5)xU(1), TrinificationSU(3)^3 group can be realized. E6 A8 U1 D7 Important data for model building. Left E6 Right

39. Result : Examples of Z3 asymmetric orbifold models ・Z3 asymmetric orbifold models with GUT symmetry ・Intermediate SO(10), E6 GUT ? ・Three fixed points ( fewer than symmetric case, 27 ) ・In asymmetric orbifolds, gauge flavor symmetriesand a rich source of hidden sector tend to be realized. ・Many types of (22,6)-dim Narain lattices give rise to models with SM group SU(3)*SU(2)*U(1) Search for a realistic model ! ・Detailed search by computer code

40. Summary • Asymmetric orbifold compactification + (22,6)-dim Narain lattice. • Summary • Systematical search by utilizing lattice engineering technique. • We obtain 2 more three-generation E6 models which satisfy the minimum • requirements, although we do not obtain non-Abelian family symmetry nor • anomalous U(1) symmetry. • GUT model : We construct 3-generation E6 models with an adjoint representation Higgs in the framework of asymmetric orbifold construction. ・ SUSY SM model : Classification of (22,6)-dimensional lattice is important. 97 lattices with right-moving non-Abelian factor can be constructed from 24 dimensional lattices. • Outlook ・ We calculate group breaking patterns of Z3 models, which are useful data for model building. • GUT model : Search another possible lattices and orbifolds ( e.g. Z3 x Z3 model ) ・SUSY SM model : Search for a realistic model -- Yukawa hierarchy,(Gauge or discrete) Flavor symmetry, Hidden sector, Moduli stabilization ...