Who Ordered the Muon ? (symmetry of three generations)

1. Who Ordered the Muon ?(symmetry of three generations) C.S. Lam McGill and UBC, Canada arXiv:0708.3665, PL B656 (2007) 193arXiv:0711.3795 arXiv: 0804.2622

2. discoveries from 1928 to 1936 • 1928 Dirac equation • Pauli: neutrino • 1932 Anderson: positron • 1932 Chadwick: neutron • Yukawa: pion • 1936 Anderson: muon Carl David Anderson (1905—1991)

3. a famous ancient question from my great-grand teacher 1986 Columbia U. Who ordered that ? Carl David Anderson Isidor Issac Rabi (1898—1988)

4. the plot thickens • now there are three generations of quarks and leptons. Why three? deep meaning, or …….

6. the plot thickens • now there are three generations of quarks and leptons. Why three? • they have very different masses • quark and neutrino mixings are vastly different • Do they contain a key to solve the generation problem? Or at least a hint ?

7. symmetry may be the key GGUT motivated by symmetry, but no expt’l evidence SUSY? GUTS? STRING? Quark/lepton masses & mixings lots of expt’l data !!! horizontal symmetry ? generation problem (Gell-Mann’s eightfold way)

8. what horizontal symmetry ? “ with enough parameters, you can even wiggle the elephant’s tail” true symmetry should reveal itself without tuning new (group theoretical) technique is needed need to find a clue, or the ‘modern Balmer series formula’

9. the modern Balmer series Quark Lepton Mass Mixing Suggests a spontaneously broken symmetry like the SM • masses: • quark mixing: small and somewhat irregular, could be a dynamical • perturbation of no mixing • neutrino mixing: large and regular (tri-bimaximal mixing), and that may • be the clue .

10. a clue to Rabi’s question SUSY? GUTS? STRING? any group containing generation problem

11. S4 is the symmetry of the octahedron and the cube

12. presently known fermions

13. consequence Other than the SM Higgs, there are a few more neutral Higgs particles to be found (needed for symmetry breaking) Explains the miracle of tri-bimaximal mixing The coupling of SM Higgs to fermions is not proportional to their masses Construction and phenomenology of -invariant dynamical models Stepping stone for constructing GGUT (Great Grand Unified Symmetry)

14. Technical Part

15. diagonal masses and mixings Integrate over all right-handed fermions columns of columns of non-alignment of eigenvectors

16. eigenvalues same eigenvectors same eigenvectors symmetryand mixing columns of columns of symmetry must be spontaneously broken

17. diagonal charged leptons diagonal Tri-bimaximal mixing matrix Block diagonal and 3D IR also conversely, if F is non-degenerate rules out

18. n=3 : two distinct possibilities thehorizontal group All eigenvalues different what about n=4,5,6,7,8,………?

19. directno 3-dim IRn=3 or wrong F uniqueness All finite subgroups of SO(3) and SU(3) with 3-dim IR

20. breaking down to true for all couplings

21. allowed expectation values

22. conclusion Present estimates for Higgs production and fermion-pair decay may be wrong The horizontal symmetry group is Dynamical details are model dependent

23. The End