Asymmetric Orbifold toward Heterotic Anomalous U(1) GUT

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)

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

Plan

• Introduction• anomalous U(1) GUT• overview• Narain cpt. & Modular inv. & orbifold• Models• Summary

SU(5)

Grand Unified Theories

Unification of Gauge Group

Charge Quantization

Unification of Matter

=+ +SO(10)

+ + =

E6

wrong GUT relation

: non-Abelian

all come from and .


sector in E6 model

massless :

large mixing

E-twisting

mild hierarchy in sector !!

mass

M.Bando and T.Kugo

Proton decay


Unification of Gauge Group

Charge Quantization

How about Higgs ?

Doublet-Triplet (DT) Splitting Problem

: non-Abelian

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

Low energy theory of string theory Anomaly Green-Schwartz mechanism Froggatt-Nielsen (FN) mechanism

Anomalous U(1) gauge symmetry

: FN field

ex)

: cutoff scale

Low energy theory of string theory Anomaly Green-Schwartz mechanism Froggatt-Nielsen (FN) mechanism SUSY-zero (holomorphic-zero) mechanism

: holomorphic


Only positively charged operators can appear.

does not appear in .

This strongly constrain !!

superpotential



Ex) Adjoints of SO(10) :

Only the linear terms in A’ are

relevant for EOM.



Ex) Adjoints of SO(10) :

• N=3 : Dimopoulos-Wilczek VEVDTS

The # of non-zero xi labels the vacuum

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. & …

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


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!!


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)


Plan


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?

overview

Kac-Moody algebra

ex) mode exp. of world sheet current

: structure constant

Diagonal embedding

w/

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)

Plan


Narain cpt. & Modular inv. & orbifold

string w/ compactification


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


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


partition function

: NS

: R


partition function

T

S

even

self-dual(or )


geometric compactification

This momentum lattice is actually even & self-dual.

winding momentum

ex.) A2 lattice

proof


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


left

right

ESDL generating technique


left

right

left

right

ESDL generating technique more


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


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


partition function

• additional string state w/ different B.C.

Modular tr.T

S

S S2=1


partition functionex) Z3 T :

S :

Modular tr.T

S

S S2=1


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.



Modular inv.We make and then define by S, T.

T :

S :



Modular inv.The remaining S, T give non-trivial conditions.

T :

S :

Plan


Models

left

right

4D

4D SUSY

SO(8)LC repr.

even

phase

4D N =1 SUSY

zero modes

Models

diagonal embedding

E6 E6 E6left

right

4D : Z3

combined with phaseless RM to form gauge field. level 3 E6

Models

left

right

4D

: Z3

lattice for right-mover

4D N =1 SUSY

A2 A2 A2E6 D4 A2

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

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

Back-up Back-up SlidesSlides

Back-up Back-up SlidesSlides


left

right

left

right


(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)]


left

right

left

right


(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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Asymmetric Orbifold toward Heterotic Anomalous U(1) GUT