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

discoveries from 1928 to 1936 Carl David Anderson (1905—1991) 1928Dirac equation 1930 Pauli: neutrino 1932 Anderson: positron 1932Chadwick: neutron 1935 Yukawa: pion 1936Anderson: muon

a famous ancient question from my great-grand teacher Carl David Anderson Isidor Issac Rabi Who ordered that ? 1986 Columbia U. (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 SUSY? GUTS? STRING? motivated by symmetry, but no expt’l evidence horizontal symmetry ? generation problem lots of expt’l data !!! Quark/lepton masses & mixings (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 need to find a clue, or the ‘modern Balmer series formula’ new (group theoretical) technique is needed

the modern Balmer series Quark Lepton Mass Mixing masses: Suggests a spontaneously broken symmetry like the SM 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? generation problem any group containing

S4 is the symmetry of the octahedron and the cube

presently known fermions

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

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

symmetry and mixing same eigenvectors same eigenvectors symmetry must be spontaneously broken columns of eigenvalues

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

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

uniqueness All finite subgroups of SO(3) and SU(3) with 3-dim IR direct no 3-dim IR n=3 or wrong F

breaking down to true for all couplings

allowed expectation values

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