Discrete R-symmetry anomalies in heterotic orbifold models

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

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. 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. )

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

5. : Eigenvalues of orbifold twist : complex basis of the closed strings Heterotic orbifold model and R-symmetry Orbifold space is a division of 6D torus by orbifole twist For orbifold , eigenvalues are defined mod N.

6. twisted sector Localized orbifold fixed point untwisted sector Orbifold fixed point Heterotic orbifold model Boundary conditions of Closed string This is corresponding to the twist of complex basis.

7. Vertex operator of 4D massless fields for computing string amplitude Boson Fermion denotes bosonized field of right moving fermionic strings and are oscillator number of the left and right mover 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)

8. 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

9. 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.

10. In the generic n-point couplings, these amplitudes include picture changing operator 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

11. 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

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. :quadratic Casimir :SO(6) H-momentum for bosonic states Discrete R-symmetry anomalies We derived the general formula of R-anomaly coefficients in heterotic orbifold models gaugino

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) Z3 orbifold models (no wilson line) (i)E6 gauge n: integer (ii)SU(3) gauge These anomalies satisfy GS condition

17. Discrete R-symmetry anomalies Example(2) Z4 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 We assume that gauged matter have no oscillated modes, then Then the total anomaly is proportional to the one-loop beta-functions

20. 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.

21. 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

22. 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

23. Example(3) MSSM 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

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