discrete r-symmetry anomalies in heterotic orbifold models hiroshi ohki takeshi araki kang-sin choi...
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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]
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.)
• 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
Contents
1. Introduction2. Heterotic orbifold model and
R-symmetry3. Discrete R-symmetry anomalies4. Some implications5. Conclusion and discussion
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.
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
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
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
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.
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
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
Discrete R-symmetry anomaly
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
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
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.
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
Discrete R-symmetry anomalies Example(2) Z4 orbifold models (no wilson line )
These anomalies satisfy GS condition
some implications
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
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
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
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
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
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.
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.
END