1. 4D low-energy effective field theory from magnetized D-brane models Tatsuo Kobayashi 1 ． Introduction 2. Intersecting/magnetized D-brane models 3. N-point couplings and flavor symmetries 4 ． Massive modes 5. Moduli 6. Summary

2. １ Introduction The workshop website says In the last decade, there have been exciting developments on two sides of the string phenomenology. First, realistic low energy particle models that include the MSSM and GUTs after appropriate moduli parameter fixing were constructed by various superstring theories, …………………………………………………... …………………………………………………... Secondly, the KKLT scenario that can realise complete moduli stabilization by flux compactification of type IIB superstring theory and may lead to a realistic inflationary universe model was proposed. …………………………………………………….. …………………………………………………….. This argument, however, has been based on simple models that neglect low energy particle physics. This argument, however, has been based on simple models that neglect low energy particle physics..

3. Objectives: purpose The workshop website says Thus, it is a good time now to merge these two approaches and look for compactifications of string theory that are fully satisfactory both from the low energy particle physics and from cosmology/the moduli problem. The main purpose of this workshop is to kick off this challenging program by overviewing the The main purpose of this workshop is to kick off this challenging program by overviewing the present status of the above two approaches to the string phenomenology and discussing promising models to be pursued.

4. Various string models (before D-brane ) 1 st string revolution Heterotic models on Calabi-Yau manifold, Orbifolds, fermionic construction, fermionic construction, Gepner,........... Gepner,........... ( after D-brane ) 2 nd string revolution Intersecting D-brane models Magnetized D-branes,........... ⇒ (semi-)realistic low-energy particle models ⇒ (semi-)realistic low-energy particle models SU(3)xSU(2)xU(1)Y (or GUT) SU(3)xSU(2)xU(1)Y (or GUT) three families of quarks and leptons three families of quarks and leptons and extra matter and extra matter

5. This talk chose one type of model constructions, chose one type of model constructions, (Type IIB) magnetized D-brane models, (Type IIB) magnetized D-brane models, T-dual to (type IIA) intersecting D-brane models. T-dual to (type IIA) intersecting D-brane models. (Hetoric models, in particular heterotic orbifold models, (Hetoric models, in particular heterotic orbifold models, are quite interesting.) are quite interesting.) For the first side (particle models), explain their properties as low-energy particle models, explain their properties as low-energy particle models, i.e. i.e. massless spectrum (realistic property) massless spectrum (realistic property) gauge bosons (gauge symmetries) gauge bosons (gauge symmetries) matter fermions, higgs fields, moduli, …………… matter fermions, higgs fields, moduli, …………… their action (low-energy effective theory)

6. This talk action of massless modes action of massless modes gauge couplings, Yukawa couplings, ….. gauge couplings, Yukawa couplings, ….. Kahler potential (kinetic terms) Kahler potential (kinetic terms) (discrete/flavor) symmetries (discrete/flavor) symmetries (phenomenological aspects) (phenomenological aspects) Towards the second side (moduli pheno./cosmology), we study moduli-dependence of 4D LEEFT such as we study moduli-dependence of 4D LEEFT such as gauge couplings, Yukawa couplings, higher order couplings, gauge couplings, Yukawa couplings, higher order couplings, D-terms, etc. (perturbative terms). D-terms, etc. (perturbative terms). non-perturbative terms ? ⇒ M.Cvetic’s talk non-perturbative terms ? ⇒ M.Cvetic’s talk Let’s discuss the merge between the two sides.

7. Plan Plan ✔ 1 ． Introduction 2. Intersecting/magnetized D-brane models 3. N-point couplings and flavor symmetries 4 ． Massive modes 5. Moduli 6. Summary

8. 2. Intersecting/magnetized D-brane models gauge boson: open string, whose two end-points gauge boson: open string, whose two end-points are on the same (set of) D-brane(s) are on the same (set of) D-brane(s) N parallel D-branes ⇒ U(N) gauge group gauge bosons, gauginos gauge bosons, gauginos adjoint fermions adjoint fermions U(1)xU(1) U(1)xU(1) U(1) ⇒ U(2) U(1) ⇒ U(2)

9. Intersecting/magnetized D-brane models See for a review Ibanez and Uranga texbook and references therein. Intersecting D-brane models: geometrical picture is simple. geometrical picture is simple. Magnetized D-brane models are also interesting. are also interesting. (We mainly study this types of models.) Generic models are their mixture. mixture.

10. 2.1 Intersecting D-branes Berkooz, Douglas, Leigh, ‘96 Berkooz, Douglas, Leigh, ‘96 Where are the matter fields ? New modes appear between intersecting D-branes. New modes appear between intersecting D-branes. They have charges under both gauge groups, i.e. They have charges under both gauge groups, i.e. bi-fundamental matter fields bi-fundamental matter fields under U(N)xU(M) gauge group. under U(N)xU(M) gauge group. boundary condition boundary condition These are These are localized modes localized modes Twisted boundary condition

11. Toy model (in uncompact space) gauge bosons ： on brane gauge bosons ： on brane quarks, leptons, higgs : quarks, leptons, higgs : localized at intersecting points localized at intersecting points u(1)xu(1) su(2) u(1)xu(1) su(2) su(3) H su(3) H u(1) Q u(1) Q L u,d u,d e, neutrino e, neutrino

12. Toy model (in uncompact space) gauge group can be enhanced gauge group can be enhanced from U(3)xU(1)xU(2)xU(1)xU(1) from U(3)xU(1)xU(2)xU(1)xU(1) ⇒ U(4)xU(2)xU(2) (Pati-Salam) ⇒ U(4)xU(2)xU(2) (Pati-Salam) u(1)xu(1) su(2) u(1)xu(1) su(2) su(3) H Split of branes su(3) H Split of branes u(1) Q  Wilson lines u(1) Q  Wilson lines L u,d u,d e,neutrino e,neutrino

13. Generation number Torus compactification Family number = intersection number su(2) su(2) Q1 Q2 Q3 su(3) Q1 Q2 Q3 su(3) on T2 on T2

14. Type IIA D6-brane models T2xT2xT2 compactification D6 branes wrap a factorizable three-cycle (one-cycle of each T2). (one-cycle of each T2). 1 st plane 2 nd plane 3 rd plane 1 st plane 2 nd plane 3 rd plane

15. Intersecting modes Always massless fermions appear at intersecting points. Scalar modes are sometimes tachyonic. D-brane configuration is unstable ⇒ symmetry breaking (recombination of D-branes) symmetry breaking (recombination of D-branes) (local) SUSY guarantees that the lightest scalar is not tachyonic, but massless.

16. Toy model on T2xT2xT2 Intersecting number = family number Intersecting number = family number u(1)xu(1) su(2) u(1)xu(1) su(2) su(3) H su(3) H u(1) Q 3 u(1) Q 3 L 3 3 L 3 3 3 u,d 3 u,d e, neutrino e, neutrino

17. Hidden sector u(1) su(2) u(1) su(2) su(3) H su(3) H Q 3 Q 3 L 3 3 L 3 3 3 u,d 3 u,d

18. RR-charge cancellation D6 brane has a RR-charge. D6 brane has a RR-charge. The total charge should vanish The total charge should vanish along compact extra dimensions. along compact extra dimensions. u(1) su(2) u(1) su(2) su(3) H su(3) H Q 3 Q 3 L 3 3 L 3 3 3 3 u,d 3 3 u,d

19. Other configurations Other types of D-brane configurations Other types of D-brane configurations are also interesting. are also interesting.,,,, etc etc

20. Orientifold Orientifold also has a RR charge. Orientifold also has a RR charge. U(N) U(N)’ U(N) U(N)’ (N,N) (N,N) ⇒ symm. ⇒ symm. and anti-symm. reps. and anti-symm. reps. after identifying U(N) and U(N)’ after identifying U(N) and U(N)’ These modes may also correspond to These modes may also correspond to SM matter fields. SM matter fields.

21. Orientifold U(N) U(N)’ U(N) U(N)’ (N,N) (N,N) ⇒ symm. ⇒ symm. and anti-symm. reps. and anti-symm. reps. These modes may also correspond to SM matte fields. These modes may also correspond to SM matte fields. The three generations of quarks and leptons are not just originated from the intersections of one type of bi-fundamental matter fields, but the flavor structure becomes rich. becomes rich.

22. Some extensions orbifold, Calabi-Yau, etc. orbifold, Calabi-Yau, etc.

23. 2-2. Magnetized D-branes We consider torus compactification with magnetic flux background. with magnetic flux background. F

24. Boundary conditions on magnetized D-branes similar to the boundary condition of similar to the boundary condition of open string between intersecting D-branes open string between intersecting D-branes T-dual to (type IIA) intersecting D-brane models T-dual to (type IIA) intersecting D-brane models

25. Type IIB magnetized D-brane models D9, D7, D5, D3 D9: wrapping on T2xT2xT2 with magnetic fluxes D7: wrapping on T2xT2 with magnetic fluxes D5: wrapping on T2 with magnetic fluxes

26. 2.3 LEEFT of magnetized D-branes Low-energy effective field theory of D-brane models of D-brane models = higher dimensional super Yang-Mills theory = higher dimensional super Yang-Mills theory e.g. e.g. D9-brane models D9-brane models ⇒ 10D SYM (gauge bosons, gauginos) ⇒ 10D SYM (gauge bosons, gauginos) KK decomposition KK decomposition 4D LEEFT 4D LEEFT

27. 2-3-1 Field theory in higher dimensions: generic aspects 10D ⇒ 4D our space-time + 6D space 10D ⇒ 4D our space-time + 6D space 10D vector 10D vector 4D vector + 4D scalars 4D vector + 4D scalars SO(10) spinor ⇒ SO(4) spinor x SO(6) spinor x SO(6) spinor internal quantum number internal quantum number

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

29. Field theory in higher dimensions Mode expansions Mode expansions KK decomposition KK decomposition

30. KK docomposition on torus torus with vanishing gauge background Boundary conditions Boundary conditions First, we concentrate on zero-modes. First, we concentrate on zero-modes.

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

32. 4D effective theory Higher dimensional Lagrangian (e.g. 10D) integrate the compact space ⇒ 4D theory integrate the compact space ⇒ 4D theory Coupling is obtained by the overlap integral of wavefunctions integral of wavefunctions

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

34. 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. 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 fields are different from each other on certain backgrounds, on certain backgrounds, e.g. CY, toroidal orbifold, warped orbifold, e.g. CY, toroidal orbifold, warped orbifold, magnetized extra dimension, etc. magnetized extra dimension, etc.

35. 2-3-2 Higher Dimensional SYM theory with flux Cremades, Ibanez, Marchesano, ‘ 04 The wave functions eigenstates of corresponding internal Dirac/Laplace operator. 4D Effective theory <= dimensional reduction

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

37. 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.

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

39. |M| independent zero mode solutions in Dirac equation. (Theta function) Dirac equation and chiral fermion Properties of theta functions : zero-modes : no zero-mode By introducing magnetic flux, we can obtain chiral theory. chiral fermion

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

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

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

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

44. 4D chiral theory 10D spinor 10D spinor light-cone 8s light-cone 8s even number of minus signs even number of minus signs 1 st ⇒ 4D, the other ⇒ 6D space 1 st ⇒ 4D, the other ⇒ 6D space If all of appear If all of appear in 4D theory, that is non-chiral theory. in 4D theory, that is non-chiral theory. If for all torus, If for all torus, only only appear for 4D helicity fixed. appear for 4D helicity fixed. ⇒ 4D chiral theory ⇒ 4D chiral theory

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

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

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

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

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

50. 2.4 Other backgrounds: Orbifold with magnetic flux Abe, T.K., Ohki, ‘ 08 Abe, T.K., Ohki, ‘ 08 The number of even and odd zero-modes The number of even and odd zero-modes We can also embed Z2 into the gauge space. We can also embed Z2 into the gauge space. => various models, various flavor structures => various models, various flavor structures

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

52. S2 with magnetic flux S2 with magnetic flux Conlon, Maharana, Quevedo, ‘08 Conlon, Maharana, Quevedo, ‘08 Fubuni-Study metric Zero-mode eq. with spin with spin connection connection M > 0 M=0

53. Short summary In magnetized D-brane models, In magnetized D-brane models, Zero-modes are quasi-localized and the number of zero-modes, the number of zero-modes, i.e., the family number, is i.e., the family number, is determined by the size of magnetic flux. determined by the size of magnetic flux.

54. 2.5 Generic models Generic model would be a mixture of intersecting and Generic model would be a mixture of intersecting and magnetized D-brane models. magnetized D-brane models. for example, for example, IIB intersecting D7-branes with magnetic fluxes, IIA intersecting D8-branes with magnetic fluxes ……………………………………………………………… ……………………………………………………………… on CY on CY

55. 3.N-point couplings and flavor symmetries 3.1 N-point couplings of zero-modes 3.1 N-point couplings of zero-modes The N-point couplings are obtained by The N-point couplings are obtained by overlap integral of their zero-mode w.f. ’ s. overlap integral of their zero-mode w.f. ’ s.

56. Moduli Torus metric Torus metric Area Area We can repeat the previous analysis. Scalar and vector fields have the same wavefunctions. wavefunctions. Wilson moduli shift of w.f. shift of w.f.

57. Zero-modes Cremades, Ibanez, Marchesano, ‘ 04 Cremades, Ibanez, Marchesano, ‘ 04 Zero-mode w.f. = gaussian x theta-function Zero-mode w.f. = gaussian x theta-function Product of zero-mode wavefunctions

58. Products of wave functions: Hint to understand products of zero-modes = zero-modes products of zero-modes = zero-modes

59. 3-point couplings Cremades, Ibanez, Marchesano, ‘ 04 Cremades, Ibanez, Marchesano, ‘ 04 The 3-point couplings are obtained by The 3-point couplings are obtained by overlap integral of three zero-mode w.f. ’ s. overlap integral of three zero-mode w.f. ’ s.

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

61. 4-point couplings Abe, Choi, T.K., Ohki, ‘ 09 Abe, Choi, T.K., Ohki, ‘ 09 The 4-point couplings are obtained by The 4-point couplings are obtained by overlap integral of four zero-mode w.f. ’ s. overlap integral of four zero-mode w.f. ’ s. split insert a complete set insert a complete set for K=M+N for K=M+N

62. 4-point couplings: another splitting i k i k i k i k t j s l j l j s l j l

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

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

65. 3.2 Non-Abelian discrete flavor symmetry The coupling selection rule is controlled by Zg charges. For M=g, 1 2 g For M=g, 1 2 g Effective field theory also has a cyclic permutation symmetry of g zero-modes. Effective field theory also has a cyclic permutation symmetry of g zero-modes. These lead to non-Abelian discrete flavor symmetires such as D4 and Δ(27) Abe, Choi, T.K, Ohki, ‘09 such as D4 and Δ(27) Abe, Choi, T.K, Ohki, ‘09 Cf. heterotic orbifolds, T.K. Raby, Zhang, ’ 04 Cf. heterotic orbifolds, T.K. Raby, Zhang, ’ 04 T.K. Nilles, Ploger, Raby, Ratz, ‘ 06 T.K. Nilles, Ploger, Raby, Ratz, ‘ 06

66. Permutation symmetry D-brane models Permutation symmetry D-brane models Abe, Choi, T.K. Ohki, ’ 09, ‘ 10 Abe, Choi, T.K. Ohki, ’ 09, ‘ 10 There is a Z2 permutation symmetry. The full symmetry is D4.

67. Permutation symmetry D-brane models Permutation symmetry D-brane models Abe, Choi, T.K. Ohki, ’ 09, ‘ 10 Abe, Choi, T.K. Ohki, ’ 09, ‘ 10 geometrical symm. Full symm. geometrical symm. Full symm. Z3 Δ(27) Z3 Δ(27) S3 Δ(54) S3 Δ(54)

68. intersecting/magnetized D-brane models intersecting/magnetized D-brane models Abe, Choi, T.K. Ohki, ’ 09, ‘ 10 Abe, Choi, T.K. Ohki, ’ 09, ‘ 10 generic intersecting number g magnetic flux magnetic flux flavor symmetry is a closed algebra of flavor symmetry is a closed algebra of two Zg ’ s. two Zg ’ s. and Zg permutation and Zg permutation Certain case: Zg permutation larger symm. Like Dg Certain case: Zg permutation larger symm. Like Dg

69. Magnetized brane-models Magnetic flux M D4 Magnetic flux M D4 2 2 2 2 4 1 ++ + 1 +- +1 -+ + 1 -- 4 1 ++ + 1 +- +1 -+ + 1 -- ・・・ ・・・・・・ ・・・ ・・・ ・・・・・・ ・・・ Magnetic flux M Δ(27) (Δ(54)) 3 3 1 3 3 1 6 2 x 3 1 6 2 x 3 1 9 ∑1 n n=1, …,9 9 ∑1 n n=1, …,9 (1 1 +∑2 n n=1, …,4) (1 1 +∑2 n n=1, …,4) ・・・ ・・・・・・・ ・・ ・・・ ・・・・・・・ ・・

70. Discrete flavor symmetry ZN symmetry is originated from ZN symmetry is originated from anomalous U(1) symmetries. anomalous U(1) symmetries. Berasatuce-Gonzalez, Camara, Marchesano, Berasatuce-Gonzalez, Camara, Marchesano, Regalado, Uranga, ‘12 Regalado, Uranga, ‘12

71. Non-Abelian discrete flavor symm. Recently, in field-theoretical model building, several types of discrete flavor symmetries have several types of discrete flavor symmetries have been proposed with showing interesting results, been proposed with showing interesting results, e.g. S3, D4, A4, S4, Q6, Δ(27),...... e.g. S3, D4, A4, S4, Q6, Δ(27),...... Review: e.g Review: e.g Ishimori, T.K., Ohki, Okada, Shimizu, Tanimoto ‘ 10 Ishimori, T.K., Ohki, Okada, Shimizu, Tanimoto ‘ 10 ⇒ large mixing angles ⇒ large mixing angles one Ansatz: tri-bimaximal one Ansatz: tri-bimaximal

72. Quark masses and mixing angles These masses are obtained by Yukawa couplings These masses are obtained by Yukawa couplings to the Higgs field with VEV, ｖ＝１７５ GeV. to the Higgs field with VEV, ｖ＝１７５ GeV. strong Yukawa coupling ⇒ large mass strong Yukawa coupling ⇒ large mass weak ⇒ small mass weak ⇒ small mass top Yukawa coupling =O(1) top Yukawa coupling =O(1) other quarks ← suppressed Yukawa couplings other quarks ← suppressed Yukawa couplings

73. Lepton masses and mixing angles mass squared differences and mixing angles mass squared differences and mixing angles consistent with neutrino oscillation consistent with neutrino oscillation large mixing angles large mixing angles

74. Applications of couplings We can obtain quark/lepton masses and mixing angles. We can obtain quark/lepton masses and mixing angles. Yukawa couplings depend on volume moduli, Yukawa couplings depend on volume moduli, complex structure moduli and Wilson lines. complex structure moduli and Wilson lines. By tuning those values, we can obtain semi-realistic results. By tuning those values, we can obtain semi-realistic results. Abe, Choi, T.K., Ohki ‘08 Abe, Choi, T.K., Ohki ‘08 Abe, et. al. work in progress Abe, et. al. work in progress Ratios depend on complex structure moduli Ratios depend on complex structure moduli and Wilson lines. and Wilson lines. Flavor is still a challenging issue. Flavor is still a challenging issue.

75. Short summary We have studied 3-point couplings We have studied 3-point couplings and higher order couplings among massless modes. and higher order couplings among massless modes. They may be useful to realize hierarchical They may be useful to realize hierarchical quark/lepton masses and mixing angles quark/lepton masses and mixing angles and other aspects. and other aspects. The discrete flavor symmetry would be useful. The discrete flavor symmetry would be useful.

76. 4. Massive modes Hamada, T.K. ‘12 Hamada, T.K. ‘12 Massive modes play an important role Massive modes play an important role in 4D LEEFT such as the proton decay, in 4D LEEFT such as the proton decay, FCNCs, etc. FCNCs, etc. It is important to compute mass spectra of It is important to compute mass spectra of massive modes and their wavefunctions. massive modes and their wavefunctions. Then, we can compute couplings among Then, we can compute couplings among massless and massive modes. massless and massive modes.

77. Fermion massive modes Two components are mixed. Two components are mixed. 2D Laplace op. algebraic relations algebraic relations It looks like the quantum harmonic oscillator It looks like the quantum harmonic oscillator

78. Fermion massive modes Creation and annhilation operators Creation and annhilation operators mass spectrum mass spectrum wavefunction wavefunction

79. Fermion massive modes explicit wavefunction explicit wavefunction Hn: Hermite function Hn: Hermite function Orthonormal condition: Orthonormal condition:

80. Scalar and vector modes The wavefunctions of scalar and vector fields The wavefunctions of scalar and vector fields are the same as those of spinor fields. are the same as those of spinor fields. Mass spectrum Mass spectrum scalar scalar vector vector Scalar modes are always massive on T2. Scalar modes are always massive on T2. The lightest vector mode along T2, The lightest vector mode along T2, i.e. the 4D scalar, is tachyonic on T2. i.e. the 4D scalar, is tachyonic on T2. Such a vector mode can be massless on T4 or T6.

81. Products of wavefunctions explicit wavefunction explicit wavefunction See also Berasatuce-Gonzalez, Camara, Marchesano, See also Berasatuce-Gonzalez, Camara, Marchesano, Regalado, Uranga, ‘12 Regalado, Uranga, ‘12Derivation: products of zero-mode wavefunctions products of zero-mode wavefunctions We operate creation operators on both LHS and RHS. and RHS.

82. 3-point couplings including higher modes The 3-point couplings are obtained by The 3-point couplings are obtained by overlap integral of three wavefunctions. overlap integral of three wavefunctions. (flavor) selection rule (flavor) selection rule is the same as one for the massless modes. is the same as one for the massless modes. (mode number) selection rule (mode number) selection rule

83. 3-point couplings: 2 zero-modes and one higher mode 3-point coupling 3-point coupling

84. Higher order couplings including higher modes Similarly, we can compute higher order couplings Similarly, we can compute higher order couplings including zero-modes and higher modes. including zero-modes and higher modes. They can be written by the sum over products of 3-point couplings. products of 3-point couplings.

85. 3-point couplings including massive modes only due to Wilson lines Massive modes appear only due to Wilson lines Massive modes appear only due to Wilson lines without magnetic flux without magnetic flux We can compute the 3-point coupling e.g. e.g. Gaussian function for the Wilson line. Gaussian function for the Wilson line.

86. 3-point couplings including massive modes only due to Wilson lines For example, we have for for

87. Several couplings Similarly, we can compute the 3-point couplings Similarly, we can compute the 3-point couplings including higher modes including higher modes Furthermore, we can compute higher order couplings including several modes, similarly. couplings including several modes, similarly.

88. 4.2 Phenomenological applications In 4D SU(5) GUT, The heavy X boson couples with quarks and leptons by the gauge coupling. by the gauge coupling. Their couplings do not change even after GUT breaking and it is the gauge coupling. and it is the gauge coupling. However, that changes in our models. However, that changes in our models.

89. Phenomenological applications Phenomenological applications For example, we consider the SU(5)xU(1) GUT model and we put magnetic flux along extra U(1). and we put magnetic flux along extra U(1). The 5 matter field has the U(1) charge q, and the quark and lepton in 5 are quasi-localized and the quark and lepton in 5 are quasi-localized at the same place. at the same place. Their coupling with the X boson is given by Their coupling with the X boson is given by the gauge coupling before the GUT breaking. the gauge coupling before the GUT breaking.

90. SU(5) => SM We break SU(5) by the WL along the U(1)Y direction. The X boson becomes massive. The quark and lepton in 5 remain massless, but their The quark and lepton in 5 remain massless, but their profiles split each other. profiles split each other. Their coupling with X is not equal to the gauge coupling, Their coupling with X is not equal to the gauge coupling, but includes the suppression factor but includes the suppression factor 5 Q L Q L

91. Proton decay Similarly, the couplings of the X boson with quarks and leptons in the 10 matter fields can be suppressed. leptons in the 10 matter fields can be suppressed. That is important to avoid the fast proton decay. The proton decay life time would drastically change by the factor, change by the factor, 

92. Other aspects Other couplings including massless and massive modes can be suppressed and those would be important, can be suppressed and those would be important, such as right-handed neutrino masses and such as right-handed neutrino masses and off-diagonal terms of Kahler metric, etc. off-diagonal terms of Kahler metric, etc. Threshold corrections on the gauge couplings, Kahler potential after integrating out massive modes Kahler potential after integrating out massive modes

93. Short summary We have studied mass spectra and wavefunctions of higher modes. of higher modes. We have computed couplings including higher modes. We can write the LEEFT with the full modes. These results have important implications. We know that couplings among zero-modes coincide between the magnetized and intersecting D-brane models. between the magnetized and intersecting D-brane models. What about couplings including higher modes ? Anyway, the mass spectra coincide each other.

94. 5. Moduli (discussions) We have used the basis that the kinetic term is canonical. We have used the basis that the kinetic term is canonical. The holomorphic parts of the couplings depend only on The holomorphic parts of the couplings depend only on the complex structure moduli as well as the complex structure moduli as well as the Wilson line moduli. the Wilson line moduli. The holomorphic part of couplings ⇒ superpotential the non-holomorphic part ⇒ Kahler metric the non-holomorphic part ⇒ Kahler metric Cremades, Ibanez, Marchesano ’04 Cremades, Ibanez, Marchesano ’04 Di Vecchia, et. al. ‘09 Di Vecchia, et. al. ‘09 Abe, T.K., Ohki, Sumita, ‘12 Abe, T.K., Ohki, Sumita, ‘12 Kahler moduli appear only in the Kahler metric. Kahler moduli appear only in the Kahler metric.

95. Gauge kinetic function For simplicity, we consider the factorizable torus, For simplicity, we consider the factorizable torus, T2xT2xT2. T2xT2xT2. SUSY condition SUSY condition Berkooz, Douglas, Leigh, ‘96 Berkooz, Douglas, Leigh, ‘96 DBI ⇒ DBI ⇒ Lust, Mayr, Richter, Stieberger, ‘04 Lust, Mayr, Richter, Stieberger, ‘04

96. Non-perturbative terms Non-perturbative effects such as gaugino condensation Non-perturbative effects such as gaugino condensation would induce terms like would induce terms like D-brane instanton effects D-brane instanton effects This form is determied by (anomalous) U(1) symmetries and discrete (flavor) symmetreis. On the other hand, holomorphic perturbative couplings depend on complex structure moduli depend on complex structure moduli as well as open string (WL) moduli. as well as open string (WL) moduli.

97. 3-form flux compactification The 3-form flux may stabilize the dilaton and The 3-form flux may stabilize the dilaton and complex structure moduli. complex structure moduli.

98. Summary We have studied phenomenological aspects of magnetized D-brane models. of magnetized D-brane models. We can construct models with realistic massless spectrum, SM gauge group massless spectrum, SM gauge group (and GUT extensions) and (and GUT extensions) and three generations of quarks and leptons. three generations of quarks and leptons. We can write the 4D LEEFT of massless modes, perturbative coupling terms and their moduli perturbative coupling terms and their moduli dependence. dependence.

99. Summary We can also write the perturbative coupling terms of the full modes. of the full modes. The 4D LEEFT has certain discrete (flavor) symmetries. What about their anomalies ? What about their anomalies ?

100. Discussions The moduli stabilization ? Inflation ? Axions ? Inflation ? Axions ? Let’s kick off to merge two approaches. Let’s kick off to merge two approaches.

101. (2-form) magnetic fluxes SUSY condition SUSY condition may stabilize some of Kahler moduli ? may stabilize some of Kahler moduli ? Antoniadis, Maillar, ’04 Antoniadis, Maillar, ’04 Antoniadis, Kumar, Maillard, ‘06 Antoniadis, Kumar, Maillard, ‘06