1. From Cosmological Constant to Sin Distribution ICRR neutrino workshop Nov. 02, 2007 Taizan Watari (U. Tokyo) 0707.344 (hep-ph) + 0707.346 (hep-ph) with L. Hall (Berkeley) and M. Salem (Tufts)

2. Three Issues Small but non-vanishing cosmological constant Small but non-vanishing cosmological constant Large mixing angles in neutrino oscillation Large mixing angles in neutrino oscillation What are “generations”? What are “generations”? Can we ever learn anything profound from precise measurements in the neutrino sector? Can we ever learn anything profound from precise measurements in the neutrino sector?

3. Cosmological Constant Problem Extremely difficult to explain Extremely difficult to explain A possible solution by S. Weinberg ’87 A possible solution by S. Weinberg ’87 Structures (such as galaxies): formed only for moderate Cosmological Constant. Structures (such as galaxies): formed only for moderate Cosmological Constant. That’s where we find ourselves. That’s where we find ourselves.

4. Key ingredients of this solution CC of a vacuum can take almost any value theoretically [i.e., a theory with multiple vacua] CC of a vacuum can take almost any value theoretically [i.e., a theory with multiple vacua] Such multiple vacua are realized in different parts of the universe. Such multiple vacua are realized in different parts of the universe. just like diversity + selection in biological evolution. just like diversity + selection in biological evolution. Any testable consequences ?? Any testable consequences ??

5. What if other parameters (Yukawa) are also scanning? Do we naturally obtain Do we naturally obtain hierarchical Yukawa eigenvalues, hierarchical Yukawa eigenvalues, generation structure in the quark sector, generation structure in the quark sector, but not for the lepton sector? but not for the lepton sector?

6. A toy model generating statistics In string theory compactification, In string theory compactification, Use Gaussian wavefunctions in overlap integral: Use Gaussian wavefunctions in overlap integral: equally-separated hierarchically small Yukawas. equally-separated hierarchically small Yukawas.

7. Generation Structure With random Yukawa matrix elements, With random Yukawa matrix elements, In our toy model, In our toy model,

8. Generation Structure originates from localized wavefunctions of quark doublets and Higgs boson: originates from localized wavefunctions of quark doublets and Higgs boson: No flavour symmetry, yet fine. No flavour symmetry, yet fine. No intrinsic difference between three quark doublets No intrinsic difference between three quark doublets Large mixing angles in the lepton sector Large mixing angles in the lepton sector non-localized wavefunctions for lepton doublets non-localized wavefunctions for lepton doublets

9. Lepton Sector Predictions Mixing angles without cuts Mixing angles without cuts Two large angles, Two large angles, After imposing cuts After imposing cuts

10. Summary Multiverse, motivated by the CC problem Multiverse, motivated by the CC problem Scanning Yukawa couplings: statistical understanding of masses and mixings, possibly w/o a symmetry. Scanning Yukawa couplings: statistical understanding of masses and mixings, possibly w/o a symmetry. Generation structure: correlation between up and down- type Yukawa matrices Generation structure: correlation between up and down- type Yukawa matrices Localized wavefunctions of q and h are the origin of generations. Localized wavefunctions of q and h are the origin of generations. Successful distributions for the lepton sector, too, Successful distributions for the lepton sector, too, with very large with very large

11. spare slides

12. Family pairing structure Family pairing structure correlation between the up and down Yukawa matrices Introduce a toy landscape on an extra dimension Introduce a toy landscape on an extra dimension Quarks and Higgs boson have Gaussian wave function Quarks and Higgs boson have Gaussian wave function Matrix elements are given by overlap integral Matrix elements are given by overlap integral The common wave functions of quark doublets The common wave functions of quark doublets and the Higgs boson introduce the correlation.

13. Neutrino Physics

14. The see-saw mechanism The see-saw mechanism Assume non-localized wavefunctions for s. Assume non-localized wavefunctions for s. Introduce complex phases. Introduce complex phases. Calculate the Majorana mass term of RH neutrino by Calculate the Majorana mass term of RH neutrino by Neutrino masses: hierarchy of all three matrices add up. Hence very hierarchical see-saw masses. Neutrino masses: hierarchy of all three matrices add up. Hence very hierarchical see-saw masses.

15. Mixing angle distributions: Mixing angle distributions: Bi-large mixing possible. Bi-large mixing possible. CP phase distribution CP phase distribution

17. The Standard Model of particle physics has 3(gauge)+22(Yukawa)+2(Higgs)+1 parameters. The Standard Model of particle physics has 3(gauge)+22(Yukawa)+2(Higgs)+1 parameters. What can we learn from the 20 observables in the Yukawa sector? What can we learn from the 20 observables in the Yukawa sector? maybe... not much. It does not seem that there is a beautiful and fundamental relation that governs all the Yukawa-related observables. maybe... not much. It does not seem that there is a beautiful and fundamental relation that governs all the Yukawa-related observables. though they have a certain hierarchical pattern though they have a certain hierarchical pattern

18. theories of flavor (very simplified) Flavor symmetry and its small breaking Flavor symmetry and its small breaking Predictive approach: use less-than 20 independent parameters to derive predictions. Predictive approach: use less-than 20 independent parameters to derive predictions. Symmetry-statistics hybrid approach: Symmetry-statistics hybrid approach: Use a symmetry to explain the hierarchical pattern. Use a symmetry to explain the hierarchical pattern. The coefficients are just random and of order unity. The coefficients are just random and of order unity.

19. ex. symmetry-statistics hybrid an approximate U(1) symmetry broken by an approximate U(1) symmetry broken by U(1) charge assignment (e.g.) U(1) charge assignment (e.g.) 3 2 0 00 are random coefficients of order unity.

20. pure statistic approaches Multiverse / landscape of vacua Multiverse / landscape of vacua best solution ever of the CC problem best solution ever of the CC problem supported by string theory (at least for now) supported by string theory (at least for now) Random coefficients fit very well to this framework. Random coefficients fit very well to this framework. But, how can you obtain hierarchy w/o a symmetry? But, how can you obtain hierarchy w/o a symmetry?

21. randomly generated matrix elements Neutrino anarchy Neutrino anarchy Generate all -related matrix elements independently, following a linear measure Generate all -related matrix elements independently, following a linear measure explaining two large mixing angles. explaining two large mixing angles. Power-law landscape for the quark sector Power-law landscape for the quark sector Generate 18 matrix elements independently, following Generate 18 matrix elements independently, following The best fit value is The best fit value is Hall Murayama Weiner ’99 Haba Murayama ‘00 Donoghue Dutta Ross ‘05

22. Let us examine the power-law model more closely for the scale-invariant case Results: (eigenvalue distributions) Hierarchy is generated from statistics for moderately large

23. pairing mixing angle distributions e.g. Family pairing structure is not obtained. Who determines the scale-invariant (box shaped) distribution? How can both quark and lepton sectors be accommodate within a single framework?

24. Family pairing structure Family pairing structure correlation between the up and down Yukawa matrices Introduce a toy landscape on an extra dimension Introduce a toy landscape on an extra dimension Quarks and Higgs boson have Gaussian wave function Quarks and Higgs boson have Gaussian wave function Matrix elements are given by overlap integral Matrix elements are given by overlap integral The common wave functions of quark doublets The common wave functions of quark doublets and the Higgs boson introduce the correlation.

25. inspiration in certain compactification of Het. string theory, in certain compactification of Het. string theory, Yukawa couplings originate from overlap integration. Yukawa couplings originate from overlap integration. Domain wall fermion, Gaussian wavefunctions and torus fibration  see next page. Domain wall fermion, Gaussian wavefunctions and torus fibration  see next page.

26. domain wall fermion and torus fibration 5D fermion in a scalar background 5D fermion in a scalar background Gaussian wavefunction at the domain wall. Gaussian wavefunction at the domain wall. 6D on with a gauge flux F on it. 6D on with a gauge flux F on it. looks like a scalar bg. in 5D. looks like a scalar bg. in 5D. chiral fermions in eff. theory: chiral fermions in eff. theory: Generalization: -fibration on a 3-fold B. Generalization: -fibration on a 3-fold B.

27. introducing “Gaussian Landscapes” (toy models) introducing “Gaussian Landscapes” (toy models) calculate Yukawa matrix by overlap integral on a mfd B calculate Yukawa matrix by overlap integral on a mfd B use Gaussian wavefunctions use Gaussian wavefunctions scan the center coordinates of Gaussian profiles scan the center coordinates of Gaussian profiles Results: try first for the easiest Results: try first for the easiest Distribution of Yukawa couplings (ignoring correlations) Distribution of Yukawa couplings (ignoring correlations) scale invariant distribution

28. To understand more analytically.... FN factor distribution  FN factor distribution  Froggatt—Nielsen type mass matrices

29. Distribution of Observables Three Yukawa eigenvalues (the same for u and d sectors) Three Yukawa eigenvalues (the same for u and d sectors) Three mixing angles family pairing Three mixing angles family pairing The family pairing originates from the localized wave functions of.

30. quick summary hierarchy from statistics hierarchy from statistics Froggatt—Nielsen like Yukawa matrices Froggatt—Nielsen like Yukawa matrices hence family pairing structure hence family pairing structure FN charge assignment follows automatically. FN charge assignment follows automatically. The scale-invariant distr. follows for The scale-invariant distr. follows for Geometry dependence? Geometry dependence? How to accommodate the lepton sector? How to accommodate the lepton sector?

31. Geometry Dependence

32. exploit the FN approximation FN suppression factor for q or qbar: FN suppression factor for q or qbar: FN factors: the largest, middle and smallest of three randomly chosen FN factors as above. FN factors: the largest, middle and smallest of three randomly chosen FN factors as above.

33. compare and FN factors: / eigenvalues / mixing angles FN factors: / eigenvalues / mixing angles

34. The original carrying info. of geometry B, is integrated once or twice in obtaining distribution fcns of observables. The original carrying info. of geometry B, is integrated once or twice in obtaining distribution fcns of observables. details tend to be smeared out. details tend to be smeared out. power/polynomial fcns of log of masses / angles in Gaussian landscapes. power/polynomial fcns of log of masses / angles in Gaussian landscapes. broad width (weak predictability) broad width (weak predictability) Dimension dependence: FN factor distribution Dimension dependence: FN factor distribution

35. Neutrino Physics

36. The see-saw mechanism The see-saw mechanism Assume non-localized wavefunctions for s. Assume non-localized wavefunctions for s. Introduce complex phases. Introduce complex phases. Calculate the Majorana mass term of RH neutrino by Calculate the Majorana mass term of RH neutrino by Neutrino masses: hierarchy of all three matrices add up. Hence very hierarchical see-saw masses. Neutrino masses: hierarchy of all three matrices add up. Hence very hierarchical see-saw masses.

37. Mixing angle distributions: Mixing angle distributions: Bi-large mixing possible. Bi-large mixing possible. CP phase distribution CP phase distribution

38. In Gaussian Landscapes, Family structure from overlap of localized wavefunctions. Family structure from overlap of localized wavefunctions. FN structure with hierarchy w/o flavor sym. FN structure with hierarchy w/o flavor sym. Broad width distributions. Broad width distributions. Non-localized wavefunctions for. Non-localized wavefunctions for. No FN str. in RH Majorana mass term No FN str. in RH Majorana mass term large hierarchy in the see-saw neutrino masses. large hierarchy in the see-saw neutrino masses. Large probability for observable. Large probability for observable.

39. The scale invariant distribution of Yukawa couplings for B = S^1 becomes for B = T^2, for B = S^2.

41. Scanning of the center coordinates Scanning of the center coordinates should come from scanning vector-bdle moduli. should come from scanning vector-bdle moduli. Instanton (gauge field on 4-mfd not 6-mfd) moduli space is known better. Instanton (gauge field on 4-mfd not 6-mfd) moduli space is known better. In the t Hooft solution, the instanton-center coordinates can be chosen freely. In the t Hooft solution, the instanton-center coordinates can be chosen freely. F-theory (or IIB) flux compactification can be used to study the scanning of complex-structure (vector bundle in Het) moduli. F-theory (or IIB) flux compactification can be used to study the scanning of complex-structure (vector bundle in Het) moduli.