1. SUSY GUT and SM in heterotic asymmetric orbifolds Shogo Kuwakino ( 桑木野省吾 ) (Chung Yuan Christian University ) Collaborator : M. Ito, N. Maekawa( Nagoya ), S. Moriyama( Nagoya ), K. Takahashi, K. Takei, S. Teraguchi ( Osaka ), T. Yamashita( Aichi Medical ) October 12, 2012. Academia Sinica 1 Based on PhysRevD.83.091703, JHEP12(2011)100 Based on a paper in preparation Collaborator : F Beye( Nagoya ), T Kobayashi( Kyoto )

2. 1.Introduction 2.Heterotic Asymmetric Orbifold 3.SUSY E6 GUT construction 4.SUSY SM construction 5.Summary Plan of Talk 2

3. Introduction String  Standard Model 3 ( Supersymmetric ) Standard Model -- We have to realize all properties of Standard model String theory -- A candidate which describe quantum gravity and unify four forces -- Is it possible to realize phenomenological 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, Yukawa hierarchy, 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. (Symmetric) orbifold compactification Several GUT or SM gauge symmetry N=1 supersymmetry Advantage : Disadvantage : No adjoint representation Higgs Dixon, Harvey, Vafa, Witten '85,'86 Introduction String compactification : 10-dim  4-dim 4 Orbifold compactification, CY, Intersecting D-brane, F -theory, M-theory, … MSSM searches in symmetric orbifold vacua : 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. Advantage : Realize adjoint representation Higgs(es) Narain, Sarmadi, Vafa ‘87 Introduction 5 Asymmetric orbifold compactification + Narain compactification Increase the number of possible models( symmetric  asymmetric ) Goal : Search for SUSY GUT with adjoint Higgs and SUSY SM in heterotic asymmetric orbifold vacua Generalization of orbifold action ( Non-geometric compactification ) Several GUT or SM gauge symmetry N=1 supersymmetry A few / no moduli fields( non-geometric ) Two possibilities ・ String  GUT  SUSY SM 4D SUSY-GUT with adjoint representation Higgs their VEV break GUT symmetry ・ String  SUSY SM

6. Heterotic Asymmetric Orbifold 6

7. Heterotic String Theory Heterotic string theory 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) 10dim 8dim Left Right Dynkin Diagram ( E8 ) Ex. ) E8 Root Lattice : Simple roots of E8 Left-moving momentum 7 LeftRight

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

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 8dim 6dim4dim Shift orbifold for E8 Gauge symmetry is broken ! Project out suitable right-moving fermionic states 27 rep. matters However, No adjoint rep. matters 9

10. Narain compactification of heterotic string 4dim22dim 6dim 4dim Left Right Narain compactification of heterotic string theory 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 10

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 Dual Left-moverRight-mover Replace 11 Left Consistent (modular invariant) ( decomposition, A2=SU(3) ) E6 part and A2 part transform “oppositely” under modular transformation Left-moving E6 part and right-moving E6 part transform “oppositely” because of minus sign in partition function Left Right

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

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

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

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

16. SUSY E6 GUT construction 16

17. GUT models Maekawa, Yamashita ‘01-’04  Can we realize these GUT models in string theory ? Anomalous U(1) symmetry : Often appear in low energy effective theory of string theory. 17 E6 Unification Generic interaction : Include all terms allowed by symmetry  Matter contents determine the models 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 )

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

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

20. Ex. ) Matter contents ( E6 x SU(2)H x U(1)A ) Typical E6 X U(1)A GUT models contain : Minimum requirements for 4D-SUSY GUT models 4D SUSY, E6 unification group, Net 3 chiral generations (, ), 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. Requirements for String Model Building String 20

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

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

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

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

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

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 26

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

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

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

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

31. SUSY SM construction 31

32. 32 Four-dimensions, N=1 supersymmetry, Standard model group( SU(3)*SU(2)*U(1) ), Three generations, Quarks, Leptons and Higgs, No exotics, Yukawa hierarchy, 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 ? SUSY SM in asymmetric orbifold vacua ・ First step for model building : Gauge symmetry

33. 33 Lattice and gauge symmetry 4dim22dim 6dim 4dim Left Right Our starting point  Lattice Lattice 10dim 8dim Left Right Gauge symmetry breaking pattern Symmetric orbifolds Asymmetric orbifolds What types of (22,6)-dimensional Narain lattices can be used for starting point ? What types of gauge symmetries can be realized ?

34. 22dim 6dim Left Right 24dim Left Classified 8dim E8 16dim SO(32) 24-dim lattice16-dim lattice 8-dim lattice Left Constructing (22, 6)-dim Narain lattices from 8, 16, 24-dim lattices by lattice engineering technique 24 types of lattices 34 (22,6)-dim lattices from 8, 16, 24-dim lattices

35. 35 24-dim lattice Generator of conjugacy classes : A11D7E6 Gauge symmetry : SU(12) x SO(14) x E6 E6 Left Right Gauge symmetry : SO(14) x E6 x SU(9) x U(1) (22,6)-dim lattice Generator of conjugacy classes : Example : A8D7E6 Left U1A2 A8D7E6 Left U1 (22,6)-dim lattices from 8, 16, 24-dim lattices

36. 36 Z3 asymmetric orbifold compactification Z3 action : Right mover  twist action  N=1 SUSY Left mover  shift action  Gauge symmetry breaking Modular invariance E6 Right A8D7E6 Left U1 Simplest asymmetric orbifold action Gauge symmetry breaking by Z3 action

37. 37 Result : Lattice and gauge symmetry ( Z3 ) 4dim22dim 6dim 4dim Left Right Our starting point  Lattice Lattice 10dim 8dim Left Right Gauge symmetry breaking pattern Symmetric orbifolds Asymmetric orbifolds ( with right-moving non- Abelian factor, from 24 dimensional lattices )

38. 38 Result : Example of group breaking patterns ・ SO(14) x E6 x SU(9) x U(1) Gauge group breaks to Several gauge symmetries. Important data for model building. E6 Right A8D7 E6 Left U1 ・ Some group combinations lead to modular invariant models ・ SM group, Flipped SO(10)xU(1), Flipped SU(5)xU(1), Trinification SU(3)^3 group can be realized.

39. 39 Result : Examples of Z3 asymmetric orbifold models ・ Intermediate SO(10), E6 GUT ? ・ Three fixed points ( fewer than symmetric case, 27 ) ・ In asymmetric orbifolds, gauge flavor symmetries and 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 ! ・ Z3 asymmetric orbifold models with GUT symmetry ・ Detailed search by computer code

40. Summary Systematical search by utilizing lattice engineering technique. GUT model : We construct 3-generation E6 models with an adjoint representation Higgs in the framework of asymmetric orbifold construction. 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. Summary 40 Asymmetric orbifold compactification + (22,6)-dim Narain lattice. ・ 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. ・ We calculate group breaking patterns of Z3 models, which are useful data for model building. Outlook 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...