Asymmetric Orbifold toward Heterotic Anomalous U(1) GUT
description
Transcript of Asymmetric Orbifold toward Heterotic Anomalous U(1) GUT
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Asymmetric OrbifoldAsymmetric Orbifoldtowardtoward
Heterotic Anomalous U(1) Heterotic Anomalous U(1) GUTGUT
09 Nov. 200909 Nov. 2009 @KEK mini @KEK mini workshopworkshop
working with working with Nagoya U. Nagoya U. : Maekawa, Moriyama, Teraguchi: Maekawa, Moriyama, Teraguchi Takei, Ito, KuwakinoTakei, Ito, Kuwakino NCTU (Taiwan) :NCTU (Taiwan) : Keijiro TakahashiKeijiro Takahashi
Toshifumi Toshifumi YamashitaYamashita
(Nagoya (Nagoya University)University)
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Introduction
anomalous U(1) GUT an interesting SUSY-GUT e.g. E6*SU(2)H
DTS, PD, Yukawa hierarchy, SUSY flavor etc.
anomaly : assumed to be cancelled by GS.
GUTs from the Heterotic string.
to be checked
N. Maekawa & T.Y. & ・・・ (’01-’04)
We aim to embed it into string.
examined in '80 exhaustivelysimple cpt. no adjoint Higgs Kac-Moody level
Asymmetricorbifold
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Plan
• Introduction• anomalous U(1) GUT• overview• Narain cpt. & Modular inv. & orbifold• Models• Summary
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SU(5)
Grand Unified Theories
Unification of Gauge Group
Charge Quantization
Unification of Matter
=+ +SO(10)
+ + =
E6
wrong GUT relation
: non-Abelian
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all come from and .
Grand Unified Theories
sector in E6 model
massless :
large mixing
E-twisting
mild hierarchy in sector !!
mass
M.Bando and T.Kugo
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Proton decay
Grand Unified Theories
Unification of Gauge Group
Charge Quantization
How about Higgs ?
Doublet-Triplet (DT) Splitting Problem
: non-Abelian
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unifications of forces and of matter charge quantization stabilization of the weak scale gauge coupling unification (GCU)
SUSY-GUT
Higgs?
Proton decay?
SUSY, (FCNC)?
hierarchal Yukawa?fascinating extension of SM
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Low energy theory of string theory Anomaly Green-Schwartz mechanism Froggatt-Nielsen (FN) mechanism
Anomalous U(1) gauge symmetry
: FN field
ex)
: cutoff scale
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Low energy theory of string theory Anomaly Green-Schwartz mechanism Froggatt-Nielsen (FN) mechanism SUSY-zero (holomorphic-zero) mechanism
: holomorphic
Anomalous U(1) gauge symmetry
Only positively charged operators can appear.
does not appear in .
This strongly constrain !!
superpotential
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Low energy theory of string theory Anomaly Green-Schwartz mechanism Froggatt-Nielsen (FN) mechanism SUSY-zero (holomorphic-zero) mechanism
Anomalous U(1) gauge symmetry
Ex) Adjoints of SO(10) :
Only the linear terms in A’ are
relevant for EOM.
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Low energy theory of string theory Anomaly Green-Schwartz mechanism Froggatt-Nielsen (FN) mechanism SUSY-zero (holomorphic-zero) mechanism
Anomalous U(1) gauge symmetry
Ex) Adjoints of SO(10) :
• N=3 : Dimopoulos-Wilczek VEVDTS
The # of non-zero xi labels the vacuum
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charge assignment = def. of model
• introduce all possible interactions (including NROs)• assume all the coefficients are (1)
“generic” interaction
Anomalous U(1) GUT
SUSY-GUT w/ anomalous U(1) symmetry
• : GUT singlet operator, : U(1)A charge• : breaking scale of U(1)A
vacuum
(cutoff scale)
Ansatz
N.Maekawa & T.Y. & …
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DW type of adjoint VEV
FN mechanism SUSY-zero mechanism
Natural Gauge Coupling UnificationDT Splitting
Anomalous U(1) GUT
hierarchal Fermion Yukawa
proton decay
Generic interaction
GCU
N.Maekawa & T.Y. & …
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fascinating extension of SM unifications of forces and of matter
charge quantization stabilization of the weak scale gauge coupling unification (GCU)
Anomalous U(1) GUT
Higgs?
hierarchal Yukawa?
Proton decay?
SUSY, (FCNC)?
horizontal symm.
Almost all the problem can be solved!!
N.Maekawa & T.Y. & …
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Anomalous U(1) GUT
PTP 107, 1201 (2002)N.Maekawa and T.Y.
• two charged adjoint• 6 – 3 or 5 – 2 generations
PLB561, 273(2003)N.Maekawa
examples of E6 model
E6 * Z2
E6 * SU(2)
N.Maekawa & T.Y. & …
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Plan
• Introduction• anomalous U(1) GUT• overview• Narain cpt. & Modular inv. & orbifold• Models• Summary
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overview
anomalous U(1) GUT
• charged adjoint Higgs
-- E6 model
Het, M, F, …higher Kac-Moody level
asymmetric orbifold
• asymmetric treatment of left- and right-moving string
diagonal embedding
-- Narain cpt.
• need careful consistency check
related to doubled
geometry?
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overview
Kac-Moody algebra
ex) mode exp. of world sheet current
: structure constant
Diagonal embedding
w/
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overview
# of generation
model
Z.Kakushadze & H.Tye (1996)
• cpt. of or SO(32) Hetero w/ Wilson lineZ.Kakushadze & H.Tye (1996)
w/ adjoint Higgs, 3 generations & non-Abelian Hidden gauge
(in known models)
Only one model is possible.
• Narain compactification
claimed
(flat instead of CY, for simplicity)
e.g. J.Erler (1996)
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Plan
• Introduction• anomalous U(1) GUT• overview• Narain cpt. & Modular inv. & orbifold• Models• Summary
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Narain cpt. & Modular inv. & orbifold
string w/ compactification
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Narain cpt. & Modular inv. & orbifold
Heterotic string
left
right
4D 6D 16D
"compactified"
on or Spin(32)/Z2 Lie lattice
from modular inv.
no geometrical interpretationhere?
Narain compactification
gauge symm w/ rank 22 is possible : ex)
discard geometric interpretation also in cpt. 6D
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Narain cpt. & Modular inv. & orbifold
Modular invariance
closed string 1-loop amp.The modular transformations,
do not change the torus.
The amplitude should be invariant under these tr..
• red blue• shift
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Narain cpt. & Modular inv. & orbifold
partition function
: NS
: R
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Narain cpt. & Modular inv. & orbifold
partition function
T
S
even
self-dual(or )
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Narain cpt. & Modular inv. & orbifold
geometric compactification
This momentum lattice is actually even & self-dual.
winding momentum
ex.) A2 lattice
proof
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Narain cpt. & Modular inv. & orbifold
Narain lattice
This lattice leads to gauge symmetry.
Cf.) geometric
: simple roots of a (simply laced) Lie group.
: Cartan matrix
This does not, except for SU(2).
• R=1 21/2 ( ; 0) • R= 2-1/2
for pR=0
winding momentum
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Narain cpt. & Modular inv. & orbifold
left
right
ESDL generating technique
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Narain cpt. & Modular inv. & orbifold
left
right
left
right
ESDL generating technique more
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Narain cpt. & Modular inv. & orbifold
orbifold
Manifold / discrete symm. • diagonal embedding • 4D N =1 SUSY
• impose identity under (discrete) orbifold action
ex.) A2 lattice • twist (rotation)• shift• (reflection)
: does not change the lattice
ZN for simplicity
: Z2,3,6
1/3 twist
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Narain cpt. & Modular inv. & orbifold
orbifold
Manifold / discrete symm. • diagonal embedding • 4D N =1 SUSY
• impose identity under (discrete) orbifold action
ex.) A2 lattice • twist (rotation)• shift• (reflection)
: does not change the lattice
: Z2,3,6
additional string states
ZN for simplicity
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Narain cpt. & Modular inv. & orbifold
partition function
• additional string state w/ different B.C.
Modular tr.T
S
S S2=1
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Narain cpt. & Modular inv. & orbifold
partition functionex) Z3 T :
S :
Modular tr.T
S
S S2=1
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Narain cpt. & Modular inv. & orbifold
partition functionex) Z3
Modular inv.P.F. is inv. if each sector satisfies the tr. property.
For asymm. orbifold, not yet. Z.Kakushadze & H.Tye (1996)
T :
S :
Rules on are established for symm. orbifold.
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Narain cpt. & Modular inv. & orbifold
partition functionex) Z3
Modular inv.We make and then define by S, T.
T :
S :
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Narain cpt. & Modular inv. & orbifold
partition functionex) Z3
Modular inv.The remaining S, T give non-trivial conditions.
T :
S :
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Plan
• Introduction• anomalous U(1) GUT• overview• Narain cpt. & Modular inv. & orbifold• Models• Summary
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Models
left
right
4D
4D SUSY
SO(8)LC repr.
even
phase
4D N =1 SUSY
zero modes
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Models
diagonal embedding
E6 E6 E6left
right
4D : Z3
combined with phaseless RM to form gauge field. level 3 E6
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Models
left
right
4D
: Z3
lattice for right-mover
4D N =1 SUSY
A2 A2 A2E6 D4 A2
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Models
possible models
E6 E6 E6A2 A2
E6
left
right
4D
E6 E6 E6A2 A2
A2 A2 A2
E6 E6 E6D4
D4 A2
Z.Kakushadze & H.Tye (1996)
Z6 1 neutral adj.5 – 2 generations
Z6 : vector-likeZ3 : 9 generations
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Summary
We are working on Heterotic GUTs to derive Anomalous U(1) GUT from string.
to get 4D N =1 SUSY E6 model w/ adjoint Higgs & 3 generations
• E63 lattice via Narain compactification
• diagonal embedding in asymmetric orbifold Future works
• We want two charged Adjoint Higgs• towards Anomalous U(1) GUT
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Back-up Back-up SlidesSlides
Back-up Back-up SlidesSlides
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Narain cpt. & Modular inv. & orbifold
left
right
left
right
ESDL generating technique
(248) (78,1)+(1,8)+[(27,3)+c.c.]
(78;1)+(1;78)+[(27;27)+c.c.]
(78;1,1,1)+(1;8,1,1)+[(1;3,3,3)+c.c.]
+[(27;3,3,1)+c.c.]
(78,1,1;1)+(1,1,1;8)+[(1,27,27;3)+c.c.]
+[(27,27,27;1)+(27,1,27;3)+(27,27,1;3)
+(27,27,27;1)+(27,1,27;3)+(27,27,1;3)]
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Narain cpt. & Modular inv. & orbifold
left
right
left
right
ESDL generating technique
(248) (78,1)+(1,8)+[(27,3)+c.c.]
(78;1)+(1;78)+[(27;27)+c.c.]
(78;1,1,1)+(1;8,1,1)+[(1;3,3,3)+c.c.]
+[(27;3,3,1)+c.c.]
(78,1,1;1)+(1,1,1;8)+[(1,27,27;3)+c.c.]
+[(27,27,27;1)+(27,1,27;3)+(27,27,1;3)
+(27,27,27;1)+(27,1,27;3)+(27,27,1;3)]