1. Symmetry of Fermion Mixing C.S. Lam McGill and UBC, Canada arXiv:0708.3665 (to appear in Phys. Lett)

2. Progress in particle physics relied heavily on symmetry considerations What about the generation problem? (horizontal global) symmetry for mass matrices mixing of fermions 1936. “ Who ordered it ?”

3. Models of Fermion Mixing Pick a (global) horizontal symmetry group. Assign IR to left-handed (L), right-handed (R), [ and heavy (H)] fermions, and all the new Higgs fields Construct -invariant mass terms. Coupling consts Assign vacuum expectation values to break the symmetry Compute the mixing matrix from the mass matrices Tune the parameters to get the desired mixing matrix

4. An Example (leptons) Ma (hep-ph/0404199) Pick a horizontal symmetry group. Assign IR to left-handed (L), right-handed (R), [ and heavy (H)] fermions, and all the new Higgs fields Construct -invariant mass terms. Coupling consts Assign vacuum expectation values to break the symmetry Iso-doublet Iso-triplet

5. Compute the mixing matrix from the mass matrices Tune the parameters to get the desired mixing matrix

6. A Systematic Study Trick: integrate out R and H to study the effective L- mass matrices Bottom-Up Approach: given the mixing matrix find Top-Down Approach: given, find

7. Bottom-Up Approach finite group Partial vs Full residual symmetry Base independent ! Non-degenerate

8. Leptons solar atmospheric reactor Harrison,Perkins,Scott Fogli et al. Tri-bimaximal mixing full, partial

9. The Leptonic List for n=1,2,3 H(6,3): 54 members H(12,3): 216 members finite or infinite list? F degenerate F non-degenerate U discrete !!

10. Top-Down Approach (Given a finite group, find ) has to have an even order, with a 3-dim IR (exceptions) must be invariant under and Number of parameters depends on the number of possible May not be enough to give rise to realistic masses

11. has to have an even order, with a 3-dim IR (exceptions) No 3-dim IR

12. must be invariant under and

13. An Example (leptons) Ma (hep-ph/0404199) Pick a horizontal symmetry group. Assign IR to left-handed (L), right-handed (R), [ and heavy (H)] fermions, and all the new Higgs fields Construct -invariant mass terms. Coupling consts Assign vacuum expectation values to break the symmetry Iso-doublet Iso-triplet

14. Compute the mixing matrix from the mass matrices Tune the parameters to get the desired mixing matrix

15. Quarks

16. Remarks is only known numerically. It is too hard to obtain a finite horizontal group for three generations. First consider two generations (Cabibbo mixing)

17. Cabibbo Mixing PDG: non-degenerate !

18. CKM Mixing More hopeful to use a top-down approach (in progress) A discrete number of mixing matrices results Use one CKM parameter to decide which U to use, then the other three CKM paramters are determined (a purely symmetry calculation !).

19. Conclusion A systematic tool to study the horizontal mixing, both bottom-up and top-down The tri-bimaximal neutrino mixing is well understood in both approaches There is an exciting possibility to determine 3/4 CKM parameters if quark mixing also has a finite horizontal group (under way).

21. Yu, Luo River, circa 2100 BCE 河圖洛書 4 5 2 81 6 3 9 7 The original 2-3 and magic symmetries