Toward realistic string models: Long and Winding Roads Tatsuo Kobayashi

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

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

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

5. String phenomenology Which class of string models lead to realistic aspects of particle physics　？ We do not need to care about string vacua without leading to e.g. the top quark mass　＝　１７2GeV the electron mass　＝　０．５　MeV, no matter how many vacua exist. Let’s study whether we can construct 4D string vacua really relevant to our Nature.

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

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

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

9. 2. String models and massless spectra Several string models  Kyae’s talk (before D-brane )1st string revolution Heterotic models on Calabi-Yau manifold, Orbifolds, fermionic construction, Gepner, . . . . . . . . . . . ( after D-brane ) 2nd string revolution type II models with D-branes  Kitazawa’s talk Intersecting D-brane models, Magnetized D-branes, . . . . . . . . . . . Nowadays F-theory models  Hayashi’s and Choi’s talk

10. Orbifolds T2/Z3 Orbifold There are three fixed points on Z3orbifold (0,0), (2/3,1/3), (1/3,2/3) su(3) root lattice Orbifold = D-dim. Torus /twist Torus = D-dim flat space/ lattice

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

12. Massless spectra 　⇒　predict 6 extra dimensions in addition to our 4D space-times Compact space (background)、D-brane configuration、... 　←　constrained by string theory (modular invariance, RR-charge cancellation,...) Once we choose background, all of modes can be investigated in principle (at the perturbative level). 　⇒　Massless modes, which appear in low-energy effective field theory, are completely determined. Oscillations and momenta in compact space correspond 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, heterotic string on orbifolds, Gepner manifolds, fermionic construction, ............................, type II intersecting D-brane models on torus, …… In this sense, these models are important (but these are special models).

14. Other compactifications (4D string models/vacua) Calabi-Yau and others Only topological aspects are known. One can not solve string on such CY. First we take (10D) field-theory limit. 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, the SM gauge group (or its GUT extenstions) three generations of quarks and leptons, the minimal number of Higgs (or more) + hidden sector, singlets, (extra matter fields charged under the SM group) For example, there are O(100)-O(1000) (semi-)realistic (heterotic orbifold) models.

16. Explicit Z6-II orbifold model: Pati-Salam T.K. Raby, Zhang ’04 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 4D massless spectrum Gauge group Chiral fields 3 generations of MSSM + extra fields All of extra matter fields can become massive along flat directions There are O(100) models.

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

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

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

21. Extra modes Massless spectra of string models are not mnimal, many singlets and several Higgs fields. (Some of singlets correspond to flat directions.) What is their implicastion ? (many Higgs) Hetero E8xE8  the SM many 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 in principle. We have 4D string models, whose massless spectra realize the SM gauge group + 3 families （＋extra matter) 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. the SM gauge group + 3 families （＋extra matter) and its extensions like the Pati-Salam model, flipped SU(5) . Lots of extra matter fields

24. 3. Parameters in the Standard Model Gauge bosons SU(3)、　SU(2)、　　U(1) parameters: threegauge couplings Quarks, Leptons3 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 RG flow ⇒ They approach each other and become similar values at high energy In MSSM, they fit each other in a good accuracy Gauge coupling unification

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

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

28. Tri-bimaximal mixing angle large mixing angles

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

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

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

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

33. 3-point coupling Calculate by inserting vertex op. corresponding to massless modes Yukawa couplings are suppressed by the area 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 space group selection rule Coupling strength calculated by inserting Vertex operators Calculations in intersecting D-brane models are almost the same.

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

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

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

38. Coupling selection rule If three strings can be connected and it becomes a shrinkable closed string, their coupling is allowed. selection rule 　　　　　　　　　　　　　　　　　←　geometrical structure of compact space ⇒　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 a Higgs field as 3-point couplings. For example, only the top and bottom can couple One could not derive realistic quark/lepton mass matrices.

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

41. Boundary conditions Three strings with the same gauge charges can be distinguished by boundary conditions, i.e. Z3 charges. Generic case ZN symmetries

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

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

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

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

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

47. Zero-modes Wave-function = (gaussian) x (theta-function) We have quantized momentum, The peaks of wave functions correspond to The momentum conservation 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 a Higgs field as 3-point couplings. <- - controlled by discrete symmetries For example, only the top and bottom can couple One could not derive realistic quark/lepton mass matrices. Discrete symmetries are also important to forbid dangerous couplings.

49. Explicit models Explicit models have flat directions and several scalar fields develop their VEVs. Higher dim. Operators become effective Yukawa couplings after symmetry breaking. 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.

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