Discrete Rsymmetry anomalies in heterotic orbifold models

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Discrete R-symmetry anomalies in heterotic
orbifold models
hep-th/0705.3072

• Hiroshi Ohki Takeshi Araki Kang-Sin Choi
  Tatsuo Kobayashi Jisuke Kubo

(Kyoto univ.) (Kanazawa univ.) (Bonn univ.)
(Kyoto univ.) (Kanazawa univ.)
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Introduction

• Discrete symmetries play an important role in
  model building beyond the standard model.
  In particular abelian and
  non-abelian discrete symmetries are useful to
  realistic quark/lepton mass and mixing angles.
• It is known that the discrete symmetries can be
  derived from the interesting heterotic orbifold
  models.
• discrete flavor symmetries
  (Kobayashi et al.)

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Motivations

• We focus on the symmetries of string orbifold
  models. In especially We defined explicitly
  R-charges of heterotic orbifold, investigate
  their anomalies in particular to mixed gauge
  anomalies.
• T-duality
  anomalies (Ibanez et al. )

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Contents

• Introduction
• Heterotic orbifold model and
  R-symmetry
• Discrete R-symmetry anomalies
• Some implications
• Conclusion and discussion

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Heterotic orbifold model and R-symmetry
Orbifold space is a division of 6D torus by
orbifole twist
For orbifold , eigenvalues are
defined mod N.
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Heterotic orbifold model
Boundary conditions of Closed string
This is corresponding to the twist of complex
basis.
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string amplitude and vertex operator
String amplitudes are computed by the correlation
functions of vertex operator as follows
(n-point amplitude)
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H-momentum for heterotic orbifold models
H-momentum for untwisted fields (bosons)
H-momentum for twisted fields (bosons)
Relation between H-momentum for boson and fermion
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Allowed couplings
(n-point amplitude)
(1)Allowed couplings may be invariant under the
following orbifold twist
(2)H-momentum conservation
H-momentum conservation and orbifold twist
invariance should be satisfied independently.
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R-charge for heterotic orbifolds
includes non-vanishing H-momenta and
oscillator which are twisted by orbifold
action.
we can define R-charges which are invariant under
picture-changing.
R-charges are defined mod N
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Coupling selection rule
Coupling selection rule for R-symmetries
N is the minimal integer satisfying
For example
Discrete R-charge for fermions in ZN orbifold
models
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• Discrete R-symmetry anomaly

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Discrete R-symmetry anomalies
Discrete R symmetry is defined as following
transformations
Under this transformations, the path integral
measure is not invariant.
The anomaly coefficients are obtained as
modulo
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Discrete R-symmetry anomalies
We derived the general formula of R-anomaly
coefficients in heterotic orbifold models
gaugino
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Discrete R-symmetry anomalies
These mixed anomalies cancelled by Green-Schwarz
(GS) mechanism, anomaly coefficients must satisfy
the following conditions
(for simple case, Kac-Moody level ka1)
We study these conditions for simple string
orbifold models.
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Discrete R-symmetry anomalies
Example(1) Z3 orbifold models (no wilson line)
(i)E6 gauge
n integer
(ii)SU(3) gauge
These anomalies satisfy GS condition
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Discrete R-symmetry anomalies
Example(2) Z4 orbifold models (no wilson line )
These anomalies satisfy GS condition
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• some implications

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Implications
Relation with beta-function
We consider sum of discrete anomalies
We assume that gauged matter have no oscillated
modes, then
Then the total anomaly is proportional to the
one-loop beta-functions
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Relation with one-loop beta-functions
Anomaly free of R-symmetry for and
Constraints on low-energy beta-functions
of between different gauge groups a and b.
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Example(1) Z3 orbifold models
total R-anomalies and one-loop beta-functions
coefficients
In fact,this model satisfies its
one-loop beta-function coefficients satisfy
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Example(2) Z4 orbifold models
total R-anomalies and one-loop beta-functions
coefficients
This model also satisfies its
one-loop beta-function coefficients satisfy
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Example(3) MSSM
one-loop beta-functions for MSSM
SU(3)
SU(2)
The MSSM can not be realized Z3 (Z6I,Z7,Z12-I)
orbifold models
Because Z3 orbifold models require
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summary

• The mixed R-symmetry anomalies for different
  gauge groups satisfy the universal GS conditions
  .
• R-symmetry anomalies relate one-loop beta
  function coefficients. In particular, for the
  case that the contribution coming from oscillator
  modes vanishes, the anomaly coefficients
  corresponding to the sum of R-symmetry is exactly
  proportional to one-loop beta functions.

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Future works

• Considerations about other constraints of low
  energy effective theory.
• e.g. super potential with non-perturbative
  effect,
• R-parity
• Extending to other string models.
• e.g. Intersecting/magnetized D-brane models
• Heterotic orbifold models have other discrete
  symmetries.
• -gt Investigations of the relations between
  string models and low-energy flavor models.

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• END