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 )

1. Introduction 2. Heterotic Asymmetric Orbifold3. SUSY E6 GUT construction4. SUSY SM construction5. Summary

Plan of Talk

2

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 !!

(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 '04Buchmuller, Hamaguchi, Lebedev, Ratz '06Lebedev, 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.

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

Heterotic Asymmetric Orbifold

6

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 8dim

Left

Right

Dynkin Diagram ( E8 )Ex. ) E8 Root Lattice

: Simple roots of E8Left-moving momentum

7

Left Right

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

( 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 8dim6dim4dim

Shift orbifold for E8

Gauge symmetry is broken !

• Project out suitable right-moving fermionic states• 27 rep. matters• However, No adjoint rep. matters

9

Narain compactification of heterotic string

4dim 22dim

6dim4dim

LeftRight

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

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-mover Right-moverReplace

11

Left

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 signin partition function

Left Right

Lattice Engineering Technique

Lattice engineering techniqueE6 root lat ( )

E6 fund weight lat ( )

E6 anti-fund weight lat ( )

A2 root lat ( )

A2 fund weight lat ( )

A2 anti-fund weight lat ( )

LeftRight

RightLeft

Left

Left

( Decomposition )

E8 root lat ( ) ( spanned by simple roots of E8 )

Left

Right

lattice

( Replace left A2 Right E6 )

gauge symmetry 12

Dual

Dual

• Simple example

Dual

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 ( )

LeftRight

RightLeft

Ex. )Left

Left

( Decomposition )

E8 root lat ( ) ( spanned by simple roots of E8 )

( Replace left E6 Right A2 )

lattice gauge symmetry 13

Dual

Dual

• Simple example

Dual

• 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

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 :

LeftRight

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

SUSY E6 GUT construction

16

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 )

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

• Superpotential assumption

• Heavy mass ( GUT scale )

• Mixing of realize hierarchies (In suitable basis)

Up-type Yukawa Down-type, Charged lepton Yukawa( Mild hierarchy )

E6 Unification

Higgs : ,

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

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 )( i )

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 SUSY1. 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

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.

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

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

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

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

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 ModelsIto, S.K, Maekawa, Moriyama, Takahashi, Takei, Teraguchi, Yamashita ‘11

Result : 3-generation models

29

• Model 2, 3 : New models !

Result : Three Generation ModelsIto, S.K, Maekawa, Moriyama, Takahashi, Takei, Teraguchi, Yamashita ‘11

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

SUSY SM construction

31

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

Lattice and gauge symmetry

4dim 22dim

6dim4dim

Left

Right

• Our starting point Lattice

Lattice

10dim 8dim 8dim

Left

Right

E8 x E8, SO(32)

Gauge symmetrybreaking pattern Classified

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 ?

22dim

6dim

Left

Right

24dim

Left

Classified

8dim

E816dim

SO(32)

24-dim lattice16-dim lattice8-dim lattice

Left Left

Constructing (22, 6)-dim Narain lattices from 8, 16, 24-dim lattices by lattice engineering technique

24 types of lattices34

(22,6)-dim lattices from 8, 16, 24-dim lattices

35

24-dim latticeGenerator of conjugacy classes :

A11 D7 E6

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 :

A8 D7 E6Left U1 A2

A8 D7 E6Left U1

(22,6)-dim lattices from 8, 16, 24-dim lattices

36

Z3 asymmetric orbifold compactification

Z3 action : Right mover twist action N=1 SUSY Left mover shift action Gauge symmetry breaking Modular invariance

E6Right

A8 D7 E6Left U1

Simplest asymmetric orbifold action

Gauge symmetry breaking by Z3 action

37

Result : Lattice and gauge symmetry ( Z3 )

4dim 22dim

6dim4dim

Left

Right

• Our starting point Lattice

Lattice

10dim 8dim 8dim

Left

Right

E8 x E8, SO(32)

Gauge symmetrybreaking pattern Classified

Symmetric orbifolds Asymmetric orbifolds

97 lattices ! ( with right-moving non-Abelian factor, from 24 dimensional lattices )

Classified !!

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.

E6RightA8 D7 E6Left 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

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

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 ...