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 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 :
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• 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
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• 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
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• 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
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.
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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
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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
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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 )
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--- 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
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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
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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
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• 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
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• 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.
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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
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
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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.
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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 ...