Overview of Extra Dimensional Models with Magnetic Fluxes Tatsuo Kobayashi

1. Overview of Extra Dimensional Models with Magnetic Fluxes Tatsuo Kobayashi １．Introduction 2. Magnetized extra dimensions 3. N-point couplings and flavor symmetries 4．Summary based on collaborations with H.Abe, K.S.Choi, R.Maruyama, M.Murata, Y.Nakai, H.Ohki, M.Sakai, K.Sumita

2. １ Introduction Extra dimensional field theories, in particular string-derived extra dimensional field theories, play important roles in particle physics as well as cosmology .

3. Extra dimensions 4 + n dimensions 4D ⇒our 4D space-time nD 　⇒ compact space Examples of compact space torus, orbifold, CY, etc.

4. Field theory in higher dimensions 10D 　⇒ 4D our space-time + 6D space 10D vector 4D vector + 4D scalars SO(10) spinor ⇒SO(4) spinor x SO(6) spinor internal quantum number

5. Several Fields in higher dimensions 4D (Dirac) spinor ⇒(4D) Clifford algebra (4x4) gamma matrices represention space ⇒ spinor representation 6D Clifford algebra 6D spinor 6D spinor 　　⇒ 　4D spinor x (internal spinor) internal quantum number

6. Field theory in higher dimensions Mode expansions KK decomposition

7. KK docomposition on torus torus with vanishing gauge background Boundary conditions We concentrate on zero-modes.

8. Zero-modes Zero-mode equation ⇒non-trival zero-mode profile the number of zero-modes

9. 4D effective theory Higher dimensional Lagrangian (e.g. 10D) integrate the compact space ⇒4D theory Coupling is obtained by the overlap integral of wavefunctions

10. Couplings in 4D Zero-mode profiles are quasi-localized far away from each other in compact space ⇒suppressed couplings

11. Chiral theory When we start with extra dimensional field theories, how to realize chiral theories is one of important issues from the viewpoint of particle physics. Zero-modes between chiral and anti-chiral fields are different from each other on certain backgrounds, e.g. CY, toroidal orbifold, warped orbifold, magnetized extra dimension, etc.

12. Magnetic flux The limited number of solutions with non-trivial backgrounds are known. Generic CY is difficult. Toroidal/Wapred orbifolds are well-known. Background with magnetic flux is one of interesting backgrounds.

13. Magnetic flux Indeed, several studies have been done in both extra dimensional field theories and string theories with magnetic flux background. In particular, magnetized D-brane models are T-duals of intersecting D-brane models. Several interesting models have been constructed in intersecting D-brane models, that is, the starting theory is U(N) SYM.

14. Phenomenology of magnetized brane models It is important to study phenomenological aspects of magnetized brane models such as massless spectra from several gauge groups, U(N), SO(N), E6, E7, E8, ... Yukawa couplings and higher order n-point couplings in 4D effective theory, their symmetries like flavor symmetries, Kahler metric, etc. It is also important to extend such studies on torus background to other backgrounds with magnetic fluxes, e.g. orbifold backgrounds.

15. 量子力学の復習：磁場中の粒子(Landau) 座標がｋ/bずれた調和振動子b=整数 ｂ個の基底状態　　　　k=0,1,2,…………,(b-1)

16. 2. Extra dimensions with magnetic fluxes: basic tools 2-1. Magnetized torus model We start with N=1 super Yang-Mills theory in D = 4+2n dimensions. For example, 10D super YM theory consists of gauge bosons (10D vector) and adjoint fermions (10D spinor). We consider 2n-dimensional torus compactification with magnetic flux background. We can start with 6D SYM (+ hyper multiplets), or non-SUSY models (+ matter fields ), similarly.

17. Higher Dimensional SYM theory with flux Cremades, Ibanez, Marchesano, ‘04 4D Effective theory <= dimensional reduction eigenstates of corresponding internal Dirac/Laplace operator. The wave functions

18. Higher Dimensional SYM theory with flux Abeliangauge field on magnetized torus Constant magnetic flux gauge fields of background The boundary conditions on torus (transformation under torus translations)

19. Higher Dimensional SYM theory with flux We now consider a complex field with charge Q ( +/-1 ) Consistency of such transformations under a contractible loop in torus which implies Dirac’s quantization conditions.

20. Dirac equation on 2D torus is the two component spinor. U(1) charge Q=1 with twisted boundary conditions (Q=1)

21. Dirac equation and chiral fermion |M| independent zero mode solutions in Dirac equation. (Theta function) Properties of theta functions chiral fermion By introducing magnetic flux, we can obtain chiral theory. :Normalizable mode :Non-normalizable mode

22. Wave functions For the case of M=3 Wave function profile on toroidal background Zero-modes wave functions are quasi-localized far away each other in extra dimensions. Therefore the hierarchirally small Yukawa couplings may be obtained.

23. Fermions in bifundamentals Breaking the gauge group (Abelian flux case ) The gaugino fields gaugino of unbroken gauge bi-fundamental matter fields

24. Bi-fundamental Gaugino fields in off-diagonal entries correspond to bi-fundamental matter fields and the difference M= m-m’ of magnetic fluxes appears in their Dirac equation. F

25. Zero-modes Dirac equations No effect due to magnetic flux for adjoint matter fields, Total number of zero-modes of :Normalizable mode :Non-Normalizable mode

26. 4D chiral theory 10D spinor light-cone 8s even number of minus signs 1st ⇒4D, the other ⇒6D space If all of appear in 4D theory, that is non-chiral theory. If for all torus, only appear for 4D helicity fixed. ⇒4D chiral theory

27. U(8) SYM theory on T6 Pati-Salam group up to U(1) factors Three families of matter fields with many Higgs fields

28. 2-2. Wilson lines Cremades, Ibanez, Marchesano, ’04, Abe, Choi, T.K. Ohki, ‘09 torus without magnetic flux constant Ai  mass shift every modes massive magnetic flux the number of zero-modes is the same. the profile: f(y)  f(y +a/M) with proper b.c.

29. U(1)a*U(1)b theory magnetic flux, Fa=2πM, Fb=0 Wilson line, Aa=0, Ab=C matter fermions with U(1) charges, (Qa,Qb) chiral spectrum, for Qa=0, massive due to nonvanishing WL when MQa >0, the number of zero-modes is MQa. zero-mode profile is shifted depending on Qb,

30. Pati-Salam model Pati-Salam group WLs along a U(1) in U(4) and a U(1) in U(2)R => Standard gauge group up to U(1) factors U(1)Y is a linear combination.

31. PS => SM Zero modes corresponding to three families of matter fields remain after introducing WLs, but their profiles split (4,2,1) Q L

32. Other models We can start with 10D SYM, 6D SYM (+ hyper multiplets), or non-SUSY models (+ matter fields ) with gauge groups, U(N), SO(N), E6, E7,E8,...

33. E6 SYM theory on T6 Choi, et. al. ‘09 We introduce magnetix flux along U(1) direction, which breaks E6 -> SO(10)*U(1) Three families of chiral matter fields 16 We introduce Wilson lines breaking SO(10) -> SM group. Three families of quarks and leptons matter fields with no Higgs fields

34. Splitting zero-mode profiles Wilson lines do not change the (generation) number of zero-modes, but change localization point. 16 Q……L

35. E8 SYM theory on T6 T.K., et. al. ‘10 E8 -> SU(3)*SU(2)*U(1)Y*U(1)1*U(1)2*U(1)3*U(1)4 We introduce magnetic fluxes and Wilson lines along five U(1)’s. E8 248 adjoint rep. include various matter fields. 248 = (3,2)(1,1,0,1,-1) + (3,2)(1,0,1,1,-1) + (3,2)(1,-1,-1,1,-1) + (3,2)(1,1,0,1,-1) + (3,2)(1,0,1,1,-1) + ...... We have studied systematically the possibilities for realizing the MSSM.

36. Results: E8 SYM theory on T6 1. We can get exactly 3 generations of quarks and leptons, but there are tachyonic modes and no top Yukawa. 2. We allow MSSM + vector-like generations and require the top Yukawa couplings and no exotic fields as well as no vector-like quark doublets.  three classes

37. Results: E8 SYM theory on T6 A. B. C.

38. Results: E8 SYM theory on T6 There are many vector-like generations. They have 3- and 4-point couplings with singlets (with no hypercharges). VEVs of singlets  mass terms of vector-like generations Light modes depend on such mass terms. That is, low-energy phenomenologies depends on VEVs of singlets (moduli).

39. 2.3 Orbifold with magnetic flux Abe, T.K., Ohki, ‘08 The number of even and odd zero-modes We can also embed Z2 into the gauge space. => various models, various flavor structures

40. Zero-modeson orbifold Adjoint matter fields are projected by orbifold projection. We have degree of freedom to introduce localized modes on fixed points like quarks/leptons and higgs fields.

41. N-point couplings and flavor symmetries The N-point couplings are obtained by overlap integral of their zero-mode w.f.’s.

42. Zero-modes Cremades, Ibanez, Marchesano, ‘04 Zero-mode w.f. = gaussian x theta-function up to normalization factor

43. Products of wave functions products of zero-modes = zero-modes

44. 3-point couplings Cremades, Ibanez, Marchesano, ‘04 The 3-point couplings are obtained by overlap integral of three zero-mode w.f.’s. up to normalization factor

45. Selection rule Each zero-mode has a Zg charge, which is conserved in 3-point couplings. up to normalization factor

46. 4-point couplings Abe, Choi, T.K., Ohki, ‘09 The 4-point couplings are obtained by overlap integral of four zero-mode w.f.’s. split insert a complete set up to normalization factor for K=M+N

47. 4-point couplings: another splitting i k i k t j s l j l

48. N-point couplings Abe, Choi, T.K., Ohki, ‘09 We can extend this analysis to generic n-point couplings. N-point couplings = products of 3-point couplings = products of theta-functions This behavior is non-trivial. (It’s like CFT.) Such a behavior wouldbe satisfied not for generic w.f.’s, but for specific w.f.’s. However, this behavior could be expected from T-duality between magnetized and intersecting D-brane models.

49. T-duality The 3-point couplings coincide between magnetized and intersecting D-brane models. explicit calculation Cremades, Ibanez, Marchesano, ‘04 Such correspondence can be extended to 4-point and higher order couplings because of CFT-like behaviors, e.g., Abe, Choi, T.K., Ohki, ‘09

50. Non-Abelian discrete flavor symmetry The coupling selection rule is controlled by Zg charges. For M=g, 1 2 g Effective field theory also has a cyclic permutation symmetry of g zero-modes. These lead to non-Abelian flavor symmetires such as D4 and Δ(27)Abe, Choi, T.K, Ohki, ‘09 Cf. heterotic orbifolds, T.K. Raby, Zhang, ’04 T.K. Nilles, Ploger, Raby, Ratz, ‘06