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## Who Ordered the Muon ? (symmetry of three generations)

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**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**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)**a famous ancient question from my great-grand teacher**1986 Columbia U. Who ordered that ? Carl David Anderson Isidor Issac Rabi (1898—1988)**the plot thickens**• now there are three generations of quarks and leptons. Why three? deep meaning, or …….**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 ?**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)**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’**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 .**a clue to Rabi’s question**SUSY? GUTS? STRING? any group containing generation problem**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)**diagonal**masses and mixings Integrate over all right-handed fermions columns of columns of non-alignment of eigenvectors**eigenvalues**same eigenvectors same eigenvectors symmetryand mixing columns of columns of symmetry must be spontaneously broken**diagonal**charged leptons diagonal Tri-bimaximal mixing matrix Block diagonal and 3D IR also conversely, if F is non-degenerate rules out**n=3 : two distinct possibilities**thehorizontal group All eigenvalues different what about n=4,5,6,7,8,………?**directno 3-dim IRn=3 or wrong F**uniqueness All finite subgroups of SO(3) and SU(3) with 3-dim IR**breaking down to**true for all couplings**conclusion**Present estimates for Higgs production and fermion-pair decay may be wrong The horizontal symmetry group is Dynamical details are model dependent