Discrete R-symmetry anomalies in heterotic orbifold models Hiroshi Ohki Takeshi Araki Kang-Sin Choi...
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![Page 1: Discrete R-symmetry anomalies in heterotic orbifold models Hiroshi Ohki Takeshi Araki Kang-Sin Choi Tatsuo Kobayashi Jisuke Kubo (Kyoto univ.) (Kanazawa.](https://reader035.fdocuments.us/reader035/viewer/2022062409/5697bf861a28abf838c884b5/html5/thumbnails/1.jpg)
Discrete R-symmetry anomalies Discrete R-symmetry anomalies in heterotic orbifold modelsin heterotic orbifold models
Discrete R-symmetry anomalies Discrete R-symmetry anomalies in heterotic orbifold modelsin heterotic orbifold models
Hiroshi Ohki TakeHiroshi Ohki Takeshishi Araki Kang-Araki Kang-Sin Choi Tatsuo KobSin Choi Tatsuo Kobayashi Jisuke Kuboayashi Jisuke Kubo
(Kyoto univ.) (Kyoto univ.) (Kanazawa univ.)(Kanazawa univ.)(Bonn univ.) (Bonn univ.) (Kyoto univ.)(Kyoto univ.)(Kanazawa univ.)(Kanazawa univ.)
[hep-th/0705.3072]
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Introduction• Discrete symmetries play an important role in model b
uilding 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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• 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. )
Motivations
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Contents
1. Introduction2. Heterotic orbifold model and
R-symmetry3. Discrete R-symmetry anomalies4. Some implications5. Conclusion and discussion
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Orbifold space is a division of 6D torus by orbifole twist
: Eigenvalues of orbifold twist
: complex basis of the closed strings
Heterotic orbifold model and R-symmetry
For orbifold ,
eigenvalues are defined mod N.
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Heterotic orbifold model
This is corresponding to the twist of complex basis.
Boundary conditions of Closed string
twisted sector
untwisted sector
Localized orbifold fixed point
Orbifold fixed point
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and are oscillator number of the left and right mover denotes bosonized field of right moving fermionic strings
and are H momentum for 4D fermion and boson
string amplitude and vertex operator
String amplitudes are computed by the correlation functions of vertex operator as follows
(n-point amplitude)
Vertex operator of 4D massless fields for computing string amplitude
Boson
Fermion
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H-momentum for heterotic orbifold models
H-momentum for twisted fields (bosons)
H-momentum for untwisted fields (bosons)
Relation between H-momentum for boson and fermion
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Allowed couplings
(1)Allowed couplings may be invariant under the following orbifold twist
(2)H-momentum conservation
(n-point amplitude)
H-momentum conservation and orbifold twist invariance should be satisfied independently.
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R-charge for heterotic orbifolds
In the generic n-point couplings, these amplitudes include picture changing operator
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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gaugino
Discrete R-symmetry anomalies
We derived the general formula of R-anomaly coefficients in heterotic orbifold models
:quadratic Casimir:SO(6) H-momentum for bosonic states
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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 ka=1)
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
(ii)SU(3) gauge n: integer
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
Then the total anomaly is proportional to the one-loop beta-functions
We assume that gauged matter have no oscillated modes, then
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Relation with one-loop beta-functions
Constraints on low-energy beta-functions of
between different gauge groups a and b.
Anomaly free of R-symmetry for and
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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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total R-anomalies and one-loop beta-functions coefficients
This model also satisfies
its one-loop beta-function coefficients satisfy
Example(2) Z4 orbifold models
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one-loop beta-functions for MSSM
SU(3) SU(2)
The MSSM can not be realized Z3 (Z6–I,Z7,Z12-I)
orbifold models
Because Z3 orbifold models require
Example(3) MSSM
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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.
-> Investigations of the relations between string models and low-energy flavor models.
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END