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