1. Toward realistic string models: Long and Winding Roads Tatsuo Kobayashi Toward realistic string models: Long and Winding Roads Tatsuo Kobayashi １． Introduction 2. String models 3 ． Parameters in the SM 4 ． Summary

2. 1. Introduction several (massless) modes of superstring : several (massless) modes of superstring : graviton graviton higher dim. ＳＹＭ （ e.g. Ｅ８, SO(32),SU(N) ） higher dim. ＳＹＭ （ e.g. Ｅ８, SO(32),SU(N) ） including SM gauge bosons and matter fields including SM gauge bosons and matter fields corresponding to quarks and leptons corresponding to quarks and leptons They include all of what we need. They include all of what we need. A promising candidate for unified theory A promising candidate for unified theory including gravity including gravity Superstring theory = Theory of Everything Superstring theory = Theory of Everything (anything ?) (anything ?)

3. String theory as fundamental theory Superstring theory ： a candidate of unified theory Superstring theory ： a candidate of unified theory including gravity including gravity If it is really relevant to particle physics, Superstring  the 4D Standard Model of Superstring  the 4D Standard Model of particle physics (at low energy) particle physics (at low energy) including values of parameters, including values of parameters, gauge couplings, Yukawa couplings, gauge couplings, Yukawa couplings, Higgs sector (parameters), etc. Higgs sector (parameters), etc.

4. (Too) many string vacua (models) (Too) many string vacua (models) Problem ： There are innumberable 4D string vacua (models) There are innumberable 4D string vacua (models)  Study on nonperturbative aspects to lift up  Study on nonperturbative aspects to lift up many vacua and/or to find out a principle many vacua and/or to find out a principle leading to a unique vacuum leading to a unique vacuum  Study on particle phenomenological aspects of  Study on particle phenomenological aspects of already known 4D string vacua already known 4D string vacua String phenomenology String phenomenology

5. String phenomenology Which class of string models lead to Which class of string models lead to realistic aspects of particle physics ？ realistic aspects of particle physics ？ We do not need to care about string vacua We do not need to care about string vacua without leading to e.g. without leading to e.g. the top quark mass ＝ １７ 2 GeV the top quark mass ＝ １７ 2 GeV the electron mass ＝ ０．５ MeV, the electron mass ＝ ０．５ MeV, no matter how many vacua exist. no matter how many vacua exist. Let ’ s study whether we can construct 4D string vacua really relevant to our Nature. really relevant to our Nature.

6. String phenomenology Superstring ： theory around the Planck scale Superstring ： theory around the Planck scale SUSY ？ SUSY ？ GUT ？ GUT ？ ？？？？？ Several scenario ？？？？？？ ？？？？？ Several scenario ？？？？？？ Standard Model ： we know it up to 100 GeV Long and winding roads Long and winding roads Superstring → low energy ？ （ top-down ） Low energy → underlying theory ？ （ bottom- up ） We need more information from a bottom-up approach. We need more information from a bottom-up approach. Both approaches are necessary to Both approaches are necessary to connect between underlying theory and our Nature. connect between underlying theory and our Nature.

7. Cosmological aspects Cosmological constant (Dark energy) Cosmological constant (Dark energy) Dark matter Dark matter Inflation Inflation Axion(s) Axion(s)..................................................................

8. Several steps toward string phenomenology Massless modes Massless modes gauge bosons (gauge symmetry), gauge bosons (gauge symmetry), matter fermions, higgs bosons, matter fermions, higgs bosons, moduli fields,............ moduli fields,............ Their effective action Their effective action gauge couplings, Yukawa couplings,......... gauge couplings, Yukawa couplings,......... Kahler potential (kinetic terms) Kahler potential (kinetic terms) (discrete/flavor) symmetry,...... (discrete/flavor) symmetry,...... moduli stabilization, SUSY breaking, moduli stabilization, SUSY breaking, soft SUSY breaking terms, soft SUSY breaking terms, cosmology,...... cosmology,......

9. 2. String models and massless spectra Several string models  Kyae ’ s talk  Kyae ’ s talk (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 type II models with D-branes  Kitazawa ’ s talk type II models with D-branes  Kitazawa ’ s talk Intersecting D-brane models, Magnetized D-branes,........... Nowadays Nowadays F-theory models  Hayashi ’ s and Choi ’ s talk F-theory models  Hayashi ’ s and Choi ’ s talk

10. Orbifolds T2/Z3 Orbifold There are three fixed points on Z3 orbifold (0,0), (2/3,1/3), (1/3,2/3) su(3) root lattice (0,0), (2/3,1/3), (1/3,2/3) su(3) root lattice Orbifold = D-dim. Torus /twist Orbifold = D-dim. Torus /twist Torus = D-dim flat space/ lattice Torus = D-dim flat space/ lattice

11. Intersecting D-brane models gauge bosons ： on brane gauge bosons ： on brane quarks, leptons, higgs : quarks, leptons, higgs : localized at intersecting points localized at intersecting points u(1) su(2) u(1) su(2) su(3) H su(3) H Q L u,d u,d

12. Massless spectra ⇒ predict 6 extra dimensions ⇒ predict 6 extra dimensions in addition to our 4D space-times in addition to our 4D space-times Compact space (background) 、 D-brane configuration 、... Compact space (background) 、 D-brane configuration 、... ← constrained by string theory ← constrained by string theory (modular invariance, RR-charge cancellation,...) (modular invariance, RR-charge cancellation,...) Once we choose background, all of modes can be Once we choose background, all of modes can be investigated in principle (at the perturbative level). investigated in principle (at the perturbative level). ⇒ Massless modes, which appear in low-energy ⇒ Massless modes, which appear in low-energy effective field theory, are completely determined. effective field theory, are completely determined. Oscillations and momenta in compact space correspond Oscillations and momenta in compact space correspond to quantum numbers of particles in 4d theory. to quantum numbers of particles in 4d theory. It is not allowed to add/reduce some modes by hand.

13. Massless spectra One can solve string in certain classes of models, One can solve string in certain classes of models, heterotic string on orbifolds, Gepner manifolds, heterotic string on orbifolds, Gepner manifolds, fermionic construction,............................, fermionic construction,............................, type II intersecting D-brane models on torus, …… type II intersecting D-brane models on torus, …… In this sense, these models are important In this sense, these models are important (but these are special models). (but these are special models).

14. Other compactifications (4D string models/vacua) Calabi-Yau and others Only topological aspects are known. Only topological aspects are known. One can not solve string on such CY. One can not solve string on such CY. First we take (10D) field-theory limit. First we take (10D) field-theory limit. We try to solve the zero-mode equation, We try to solve the zero-mode equation,

15. Realistic models String model builders have done good jobs. We have constructed many (semi-)realistic models, We have constructed many (semi-)realistic models, the SM gauge group (or its GUT extenstions) the SM gauge group (or its GUT extenstions) three generations of quarks and leptons, three generations of quarks and leptons, the minimal number of Higgs (or more) the minimal number of Higgs (or more) + + hidden sector, singlets, hidden sector, singlets, (extra matter fields charged under the SM group) (extra matter fields charged under the SM group) For example, there are O(100)-O(1000) For example, there are O(100)-O(1000) (semi-)realistic (heterotic orbifold) models. (semi-)realistic (heterotic orbifold) models.

16. Explicit Z6-II orbifold model: Pati-Salam T.K. Raby, Zhang ’ 04 T.K. Raby, Zhang ’ 04 4D massless spectrum 4D massless spectrum Gauge group Chiral fields Pati-Salam model with 3 generations + extra fields All of extra matter fields can become massive

17. Explicit Z6-II orbifold model: MSSM Buchmuller,et.al. ’ 06, Lebedev, et. al ‘ 07 Buchmuller,et.al. ’ 06, Lebedev, et. al ‘ 07 4D massless spectrum 4D massless spectrum Gauge group Gauge group Chiral fields 3 generations of MSSM + extra fields 3 generations of MSSM + extra fields All of extra matter fields can become massive along flat directions along flat directions There are O(100) models. There are O(100) models.

18. Explicit Z12-I orbifold model: Kim, Kyae, ‘ 06 Kim, Kyae, ‘ 06 MSSM, Flipped SU(5), …. MSSM, Flipped SU(5), ….  Kyae ’ s talk  Kyae ’ s talk

19. Realistic models We have just searched a narrow part of We have just searched a narrow part of string vacua. string vacua. For example, the number of certain heterotic For example, the number of certain heterotic orbifold models is finite, orbifold models is finite, Z3, Z4, Z6,Z7,Z8,Z12, ZNxZM with Wilson lines Z3, Z4, Z6,Z7,Z8,Z12, ZNxZM with Wilson lines We should analyze such string vacua systematically We should analyze such string vacua systematically by computers and make their database of by computers and make their database of string vacua. string vacua. (See e.g. our telephone book, Katsuki, et.al. ’ 90.) (See e.g. our telephone book, Katsuki, et.al. ’ 90.)

20. Flat directions realistic massless modes + extra modes realistic massless modes + extra modes with vector-like rep. with vector-like rep. Effective field theory has flat directions. VEVs of scalar fields along flat directions VEVs of scalar fields along flat directions ⇒ vector-like rep. massive, no extra matter ⇒ vector-like rep. massive, no extra matter Such VEVs would correspond to deformation Such VEVs would correspond to deformation of orbifolds like blow-up of singular points. of orbifolds like blow-up of singular points. That is CY That is CY (as perturbation around the orbifold limit). (as perturbation around the orbifold limit).

21. Extra modes Extra modes Massless spectra of string models are not mnimal, Massless spectra of string models are not mnimal, many singlets and several Higgs fields. many singlets and several Higgs fields. (Some of singlets correspond to flat directions.) (Some of singlets correspond to flat directions.) What is their implicastion ? What is their implicastion ? (many Higgs) (many Higgs) Hetero E8xE8  the SM Hetero E8xE8  the SM many extra modes many extra modes D-brane models have less extra modes. D-brane models have less extra modes.

22. Short summary on massless spectra Once we choose a background( orbifold, gauge shift, wilson lines), a string model is fixed and its full massless spectrum can be analyzed its full massless spectrum can be analyzed in principle. in principle. We have 4D string models, whose massless We have 4D string models, whose massless spectra realize spectra realize the SM gauge group + 3 families （＋ extra matter) the SM gauge group + 3 families （＋ extra matter) and its extensions like the Pati-Salam model. and its extensions like the Pati-Salam model. Similar situation for other compactifications

23. Short summary on massless spectra Many models have been studied in Z3, Z6-II, Z12 orbifolds. Z3, Z6-II, Z12 orbifolds. the SM gauge group + 3 families （＋ extra matter) and its extensions like the Pati-Salam model, and its extensions like the Pati-Salam model, flipped SU(5). flipped SU(5). Lots of extra matter fields Lots of extra matter fields

24. 3. Parameters in the Standard Model Gauge bosons SU(3)、 SU(2)、 U(1) parameters: three gauge couplings Quarks, Leptons 3 families hierarchical pattern of masses (mixing)  Yukawa couplings to the Higgs field Most of the parameters appear from the Yukawa sector. Higgs field the origin of masses not discovered yet our purpose “derivation of the SM” ⇒ realize these massless modes and coupling values

25. Gauge couplings experimental values experimental values RG flow ⇒ RG flow ⇒ They approach each other and become similar values and become similar values at high energy at high energy In MSSM, they fit each other in a good accuracy Gauge coupling unification

26. 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

27. 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

28. Tri-bimaximal mixing angle large mixing angles large mixing angles

29. Effective theory Effective theory Effective theory of massless modes is described by supergravity- coupled gauge theory. by supergravity- coupled gauge theory. In principle, we can compute gauge couplings, In principle, we can compute gauge couplings, Yukawa couplings in string models, although Yukawa couplings in string models, although such a computation is possible in certain models such a computation is possible in certain models such as heterotic orbifold models, intersecting such as heterotic orbifold models, intersecting D-brane models, etc. D-brane models, etc. Those are functions of dilaton and moduli. Those are functions of dilaton and moduli.

30. 3-2. Yukawa couplings 3-2. Yukawa couplings Certain modes are originated from 10D modes. Certain modes are originated from 10D modes. That is, they respect 4D N=4 (10D N=1) SUSY That is, they respect 4D N=4 (10D N=1) SUSY vector multiplet vector multiplet Yukawa couplings ← controlled by 4D N=4 SUSY Certain combinations are allowed: Selection rule Certain combinations are allowed: Selection rule ⇒ Y = g = O(1) ⇒ Y = g = O(1) That fits to the top Yukawa coupling (approximately) (approximately)

31. Yukawa couplings (good news) There are localized modes e.g. in heterotic orbifold models, in heterotic orbifold models, intersecting D-brane models, etc. intersecting D-brane models, etc. Couplings among local fields are suppressed Couplings among local fields are suppressed depending on their distance. depending on their distance. They can explain small Yukawa couplings They can explain small Yukawa couplings for light quark/lepton ？ for light quark/lepton ？ Let ’ s carry out stringy calculation Let ’ s carry out stringy calculation (stringy selection rule) (stringy selection rule)

32. Coupling selection rule When three strings are localized far away each other, their couplings would be small. each other, their couplings would be small.

33. 3-point coupling Calculate by inserting vertex op. corresponding to massless modes corresponding to massless modes Yukawa couplings are suppressed by the area that strings sweep to couple. that strings sweep to couple. Those are favorable for light quarks/leptons.

34. n-point coplings Choi, T.K. ‘ 08 Selection rule ← gauge invariance H-momentum conservation H-momentum conservation space group selection rule space group selection rule Coupling strength calculated by inserting Vertex operators calculated by inserting Vertex operators Calculations in intersecting D-brane models are almost the same. are almost the same.

35. Couplings among zero-modes Couplings among zero-modes Extra dimensional effective field theory Extra dimensional effective field theory Non-trivial background Non-trivial background → non-trivial profile → non-trivial profile of zero-mode wave function of zero-mode wave function 4D couplings among quasi- localized modes = overlap integral along extra dimensions = overlap integral along extra dimensions → suppressed Yukawa couplings → suppressed Yukawa couplings depending on their distance depending on their distance

36. Selection rule for allowed couplings Selection rule for allowed couplings Allowed couplings : gauge invariant conservation of quantum numbers conservation of quantum numbers e.g. momenta in 6D compact space e.g. momenta in 6D compact space Some selection rules are not understood by effective field theory. by effective field theory. Stringy selection rule Stringy selection rule

37. Stringy coupling selection rule A string can be specified by A string can be specified by its boundary condition. its boundary condition. Two strings can be connected Two strings can be connected to become a string if their to become a string if their boundary conditions fit each other. boundary conditions fit each other. coupling selection rule coupling selection rule symmetry symmetry

38. Coupling selection rule If three strings can be connected and it becomes a shrinkable closed string, and it becomes a shrinkable closed string, their coupling is allowed. their coupling is allowed. selection rule selection rule ← geometrical structure ← geometrical structure of compact space of compact space ⇒ in general complicated ⇒ in general complicated

39. Coupling selection rule (bad news ?) In general, stringy coupling selection rule is tight. All of three generations can not couple with All of three generations can not couple with a Higgs field as 3-point couplings. a Higgs field as 3-point couplings. For example, only the top and bottom can couple For example, only the top and bottom can couple One could not derive realistic quark/lepton One could not derive realistic quark/lepton mass matrices. mass matrices.

40. Abelian discrete symmetries Abelian discrete symmetries Stringy coupling selection rule Stringy coupling selection rule Abelian discrete symmetries, Abelian discrete symmetries, ZN x ZM x …… symmetries ZN x ZM x …… symmetries

41. Boundary conditions Three strings with the same Three strings with the same gauge charges can be gauge charges can be distinguished by distinguished by boundary conditions, boundary conditions, i.e. Z 3 charges. i.e. Z 3 charges. Generic case Z N symmetries Generic case Z N symmetries

42. Heterotic orbifold models S1/Z2 Orbifold twisted string untwisted string untwisted string

43. Closed strings on T2/Z3 orbifold Untwisted and twisted strings Twisted strings (first twisted sector) second twisted sector second twisted sector untwisted sector

44. Z3 x Z3 in Heterotic orbifold models T2/Z3 Orbifold two Z3 ’ s two Z3 ’ s twisted string (first twisted sector) twisted string (first twisted sector) untwisted string vanishing Z3 charges for both Z3 vanishing Z3 charges for both Z3

45. Higher Dimensional theory with flux Abelian gauge field on magnetized torus Constant magnetic flux Consistency requires Dirac ’ s quantization condition. gauge fields of background

46. Torus with magnetic flux We solve the zero-mode Dirac equation, e.g. for U(1) charge q=1. e.g. for U(1) charge q=1. Torus background with magnetic flux Torus background with magnetic flux leads to chiral spectra. leads to chiral spectra. the number of zero-modes the number of zero-modes = M (magnetic flux) = M (magnetic flux) x q (charge) x q (charge)

47. Zero-modes Wave-function = (gaussian) x (theta-function) Wave-function = (gaussian) x (theta-function) We have quantized momentum, The peaks of wave functions correspond to The peaks of wave functions correspond to The momentum conservation The momentum conservation ZM discrete symmetry ZM discrete symmetry e.g. M=3 Z3 symmetry

48. Coupling selection rule (bad news ?) In general, stringy coupling selection rule is tight. All of three generations can not couple with All of three generations can not couple with a Higgs field as 3-point couplings. a Higgs field as 3-point couplings. <- - controlled by discrete symmetries <- - controlled by discrete symmetries For example, only the top and bottom can couple For example, only the top and bottom can couple One could not derive realistic quark/lepton One could not derive realistic quark/lepton mass matrices. mass matrices. Discrete symmetries are also important to forbid dangerous couplings. forbid dangerous couplings.

49. Explicit models Explicit models have flat directions and several scalar fields develop their VEVs. several scalar fields develop their VEVs. Higher dim. Operators Higher dim. Operators become effective Yukawa couplings become effective Yukawa couplings after symmetry breaking. after symmetry breaking. They would lead to suppressed Yukawa couplings They would lead to suppressed Yukawa couplings How to control n-point coupling is important. So far, such analyses have been done model by model. model by model.

50. Explicit models Higher dim. Operators with scalar VEVs small number n  heavy quarks such as c small number n  heavy quarks such as c large number n  light quarks/leptons large number n  light quarks/leptons such as u,d,s, e,μ such as u,d,s, e,μ quite model-dependent analysis quite model-dependent analysis results depend on moduli, flat directions and results depend on moduli, flat directions and scalar VEVs scalar VEVs Should we analyze model-by-model Should we analyze model-by-model quarks/lepton mass matrices quarks/lepton mass matrices for O(1000) or more models ? for O(1000) or more models ? Is it possible ? (Many model builders gave up.) Is it possible ? (Many model builders gave up.)

51. Short summary on effective theory (coupling) Several couplings are calculable. All couplings are functions depending on moduli (dilaton). moduli (dilaton). We have to choose proper values of moduli VEVs. moduli VEVs. Realization of quark/lepton masses and mixing angles is still a challenging issue. angles is still a challenging issue. We may need a new strategy. We may need a new strategy.

52. Quark and lepton mass matrices We have concentrated on (heavy) quarks. Reasons: large couplings are important, large couplings are important, we know better experimental data in we know better experimental data in the quark sector than in the lepton sector, the quark sector than in the lepton sector, neutrino masses and mixing angles. neutrino masses and mixing angles. However, recent neutrino experiments However, recent neutrino experiments show the typical pattern of mixing angles. show the typical pattern of mixing angles.

53. Tri-bimaximal mixing Ansatz large mixing angles large mixing angles

54. 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

55. New viewpoint: Non-Abelian symmetry New viewpoint: Non-Abelian symmetry String model builders have not cared about String model builders have not cared about non-Abelian discrete symmetires. non-Abelian discrete symmetires. Recently, we showed that certain non-Abelian Recently, we showed that certain non-Abelian flavor symmetries appear in string models. flavor symmetries appear in string models. Studies on discrete anomalies are also important. Studies on discrete anomalies are also important.

56. Non-Abelian discrete symmetries Heterotic orbifold models Non-Abelian discrete symmetries Heterotic orbifold models S1/Z2 Orbifold Z2 even odd Z2 even odd String theory has two Z2 ’ s. String theory has two Z2 ’ s. In addition, the Z2 orbifold has the geometrical In addition, the Z2 orbifold has the geometrical symmetry, i.e. Z2 permutation. symmetry, i.e. Z2 permutation.

57. D4 Flavor Symmetry Stringy symmetries require that Lagrangian has the permutation symmetry between 1 and 2, and each coupling is controlled by two Z2 symmetries. Flavor symmeties: closed algebra S2 U(Z2xZ2) D4 elements modes on two fixed points ⇒ doublet modes on two fixed points ⇒ doublet untwisted (bulk) modes ⇒ singlet untwisted (bulk) modes ⇒ singlet Geometry of compact space  origin of finite flavor symmetry  origin of finite flavor symmetry Abelian part (Z2xZ2) : coupling selection rule Abelian part (Z2xZ2) : coupling selection rule S2 permutation : one coupling is the same as another. S2 permutation : one coupling is the same as another. T.K., Raby, Zhang, ‘ 05 T.K., Raby, Zhang, ‘ 05

58. Heterotic orbifold as brane world Pati-Salam model T.K. Raby, Zhang, ‘ 05 Pati-Salam model T.K. Raby, Zhang, ‘ 05 2D Z2 orbifold 1 generation in bulk 1 generation in bulk two generations on two fixed points two generations on two fixed points unbroken SU(4) ＊ SU(2) ＊ SU(2) D4 unbroken SU(4) ＊ SU(2) ＊ SU(2) D4 bulk ⇒ (4,2,1) + (4*,1,2)+... singlet bulk ⇒ (4,2,1) + (4*,1,2)+... singlet localized modes ⇒ (4,2,1) + (4*,1,2) doublet localized modes ⇒ (4,2,1) + (4*,1,2) doublet

59. Heterotic orbifold models T2/Z3 Orbifold two Z3 ’ s two Z3 ’ s Z3 orbifold has the S3 geometrical symmetry, Z3 orbifold has the S3 geometrical symmetry, Their closed algebra is Δ(54). Their closed algebra is Δ(54). T.K., Nilles, Ploger, Raby, Ratz, ‘ 07 T.K., Nilles, Ploger, Raby, Ratz, ‘ 07

60. Heterotic orbifold models T2/Z3 Orbifold has Δ(54) symmetry. has Δ(54) symmetry. localized modes on three fixed points localized modes on three fixed points Δ(54) triplet Δ(54) triplet bulk modes Δ(54) singlet bulk modes Δ(54) singlet T.K., Nilles, Ploger, Raby, Ratz, ‘ 07 T.K., Nilles, Ploger, Raby, Ratz, ‘ 07

61. intersecting/magnetized D-brane models intersecting/magnetized 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.

62. intersecting/magnetized D-brane models intersecting/magnetized 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)

63. field theory: extension (Ohki ’ s talk) field theory: extension (Ohki ’ s talk) Abe, Choi, T.K., Ohki, Sakai, ‘ 10 Abe, Choi, T.K., Ohki, Sakai, ‘ 10 S1/Z2 Orbifold geometrical symm. String theory has two Z2 ’ s. String theory has two Z2 ’ s. We assign generic ZN charges to localized fields on two fixed points, on two fixed points, flavor symmetries flavor symmetries

64. field theory: extension (Ohki ’ s talk) field theory: extension (Ohki ’ s talk) T2/Z3 Orbifold geometrical symm. Z3, S3 Z3, S3 String theory has two Z3 ’ s. String theory has two Z3 ’ s. We assign generic ZN charges to localized fields on three fixed points, on three fixed points, flavor symmetries flavor symmetries Stringy derivation is not clear.

65. New viewpoint: Non-Abelian symmetry New viewpoint: Non-Abelian symmetry Indeed, this symmetry is the reason why we can not Indeed, this symmetry is the reason why we can not derive realistic quark/lepton mass matrices derive realistic quark/lepton mass matrices from 3-point couplings, from 3-point couplings, where flavor symmetries do not break. where flavor symmetries do not break. However, that would be interesting However, that would be interesting from the viewpoint of large mixing angles from the viewpoint of large mixing angles in the lepton sector. in the lepton sector.

66. Non-Abelian symmetry Non-Abelian symmetry We have just started this type of studies. We have just started this type of studies. We have not succeeded realization of We have not succeeded realization of realistic lepton mass matrices in purely realistic lepton mass matrices in purely stringy models stringy models (but field-theoretical model building). (but field-theoretical model building). Further phenomenological implications Further phenomenological implications sfermion masses are controlled by non-Abelian sfermion masses are controlled by non-Abelian flavor symmetries. flavor symmetries. Higgs fields might be not singlets. Higgs fields might be not singlets. The number of Higgs fields might not be minimal. The number of Higgs fields might not be minimal.

67. sfermion mass sfermion mass SUSY breaking due to F-term of X triplet triplet 1+2 1+2 1+ 1 ’ +1 ” 1+ 1 ’ +1 ”

68. symmetry breaking symmetry breaking Breaking of the flavor symmetries would induce Breaking of the flavor symmetries would induce off-diagonal elements in the Kahler potential, off-diagonal elements in the Kahler potential, e.g. e.g. and sfermion mass-squared matrix, and sfermion mass-squared matrix, e.g. e.g. Large off-diagonal elements are not good from Large off-diagonal elements are not good from FCNC. FCNC. Large breaking is not good. Large breaking is not good.

69. Non-Abelian discrete symmetries Non-Abelian discrete symmetries Symmetric background  non-Abelian discrete symmetries Symmetric background  non-Abelian discrete symmetries Generic CY ’ s do not have such symmetries. Generic CY ’ s do not have such symmetries. Lepton flavor model building Lepton flavor model building in field theory in field theory First, we assume a certain flavor symmetry. First, we assume a certain flavor symmetry. Then, we break it to a proper direction by flavon VEVs. Then, we break it to a proper direction by flavon VEVs. realistic MNS mixing matrix and lepton masses realistic MNS mixing matrix and lepton masses We have achieved the first step for certain flavor symmetries. We have achieved the first step for certain flavor symmetries. Flavon VEVs correspond to deformation of compact space, Flavon VEVs correspond to deformation of compact space, e.g. blow-up of orbifold singularity. e.g. blow-up of orbifold singularity. Which deformation is realistic ? Which deformation is realistic ?

70. Discrete symmetries Discrete symmetries Formal viewpoint, it would be important to study anomalies it would be important to study anomalies of discrete symmetires. of discrete symmetires. Araki, T.K., Kubo, Ramos-Sanches, Ratz, Vaudrevange, ‘ 08 Araki, T.K., Kubo, Ramos-Sanches, Ratz, Vaudrevange, ‘ 08 Abe, et. al. work in progress Abe, et. al. work in progress String theory  gravity/gauge anomalies String theory  gravity/gauge anomalies Does string theory constrain discrete anomalies, Does string theory constrain discrete anomalies, too ? too ?

71. Anomalies of discrete symmetries Anomalies of discrete symmetries Heterotic orbifold models Heterotic orbifold models Araki, T.K., Kubo, Ramos-Sanches, Ratz, Vaudrevange, ‘ 08 Araki, T.K., Kubo, Ramos-Sanches, Ratz, Vaudrevange, ‘ 08 Zn-G-G anomalies for G= non-Abelian gauge groups Zn-G-G anomalies for G= non-Abelian gauge groups We have checked a number of models. We have checked a number of models.  universal anomalies of discrete symmetries  universal anomalies of discrete symmetries = universal = universal for different gauge groups for different gauge groups

72. Heterotic orbifold models U(1)-G-G anomalies are universal for different groups G. are universal for different groups G. 4D Green-Schwarz mechanism 4D Green-Schwarz mechanism due to a single axion (dilaton), due to a single axion (dilaton), which couples universally with gauge sectors. which couples universally with gauge sectors. ZN-G-G anomalies may also be cancelled by 4D GS mechanism. by 4D GS mechanism. There is a certain relations between U(1)-G-G and ZN-G-G anomalies, U(1)-G-G and ZN-G-G anomalies, anomalous U(1) generator is a linear combination anomalous U(1) generator is a linear combination of anomalous ZN generators. of anomalous ZN generators. Araki, T.K., Kubo, Ramos-Sanches, Ratz,Vaudrevange, ‘ 08 Araki, T.K., Kubo, Ramos-Sanches, Ratz,Vaudrevange, ‘ 08

73. Flavor in string theory Flavor in string theory It is not so difficult to realize the generation It is not so difficult to realize the generation number, i.e. the three generation. number, i.e. the three generation. We have some explicit examples We have some explicit examples to lead to semi-realistic patterns of to lead to semi-realistic patterns of Yukawa matrices for quarks and leptons. Yukawa matrices for quarks and leptons. However, realization of realistic Yukawa However, realization of realistic Yukawa matrices is still a challenging issue. matrices is still a challenging issue. (non-abelian flavor symmetries ?) (non-abelian flavor symmetries ?)

74. Summary We have studied on particle phenomenological aspects on string theory aspects on string theory to find out a scenario connecting to find out a scenario connecting string theory and the particle physics, string theory and the particle physics, in particular the Standard Model. in particular the Standard Model. Several issues: realistic spectra, realistic spectra, flavor structure 、 moduli stabilization 、 flavor structure 、 moduli stabilization 、 SUSY breaking 、 cosmology 、.........

75. Summary Realistic massless spectra all types of string theories are not bad all types of string theories are not bad We have known already many string models, We have known already many string models, which have the same content as which have the same content as the MSSM or its extensions. the MSSM or its extensions. Gauge couplings Gauge couplings Yukawa matrices Yukawa matrices still a challenging issue still a challenging issue Further studies: Moduli stabilizatin Further studies: Moduli stabilizatin cosmological aspects, … cosmological aspects, …