1. Discrete R-symmetry anomalies in heterotic orbifold models Hiroshi Ohki Takeshi Araki Kang-Sin Choi Tatsuo Kobayashi Jisuke Kubo (Kyoto univ.) (Kanazawa univ.) (Bonn univ.) (Kyoto univ.) (Kanazawa univ.) (Kanazawa univ.) [hep-th/0705.3072]

2. 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.)

3. 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

4. Contents 1.Introduction 2.Heterotic orbifold model and R-symmetry 3.Discrete R-symmetry anomalies 4.Some implications 5.Conclusion and discussion

5. 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.

6. 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

7. 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

8. 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

9. 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.

10. 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

11. Coupling selection rule Coupling selection rule for R-symmetries N is the minimal integer satisfying For example Discrete R-charge for fermions in Z N orbifold models

12. Discrete R-symmetry anomaly

13. 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

14. 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

15. 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.

16. Discrete R-symmetry anomalies Example(1) Z 3 orbifold models (no wilson line) (i)E 6 gauge (ii)SU(3) gauge n: integer These anomalies satisfy GS condition

17. Discrete R-symmetry anomalies Example(2) Z 4 orbifold models (no wilson line ) These anomalies satisfy GS condition

18. some implications

19. 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

20. 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

21. Example(1) Z 3 orbifold models total R-anomalies and one-loop beta-functions coefficients In fact,this model satisfies its one-loop beta-function coefficients satisfy

22. total R-anomalies and one-loop beta-functions coefficients This model also satisfies its one-loop beta-function coefficients satisfy Example(2) Z 4 orbifold models

23. one-loop beta-functions for MSSM SU(3)SU(2) The MSSM can not be realized Z 3 (Z 6 – I,Z 7,Z 12 -I) orbifold models Because Z 3 orbifold models require Example(3) MSSM

24. 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.

25. 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.

26. END