Relations Between Standard Model Parameters from $T^{'}$ Flavor Symmetry
Spontaneously broken flavor symmetry involving the binary tetrahedral group $T^{'}$ relates parameters in the standard model. In one example the CP violating Pontecorvo-Maki-Nakagawa-Sakata phase $\delta_{PMNS}$ and atmospheric mixing angle $\theta_{23}$ of neutrinos are related to the Cabibbo mixing angle $\Theta_C$ for quarks. A second formula relates a ratio of neutrino mass eigenvalues to the Cabibbo angle.
CYCLIC UNIVERSE AND INFINITE PAST
We address two questions about the past for infinitely cyclic cosmology. The first is whether it can contain an infinite length null geodesic into the past in view of the Borde–Guth–Vilenkin (BGV) "no-go" theorem, The second is whether, given that a small fraction of spawned universes fail to cycle, there is an adequate probability for a successful universe after an infinite time. We give positive answers to both questions and then show that in infinite cyclicity the total number of universes has been infinite for an arbitrarily long time.
Underlying A4 symmetry for the neutrino mass matrix and the quark mixing matrix
Abstract The discrete non-Abelian symmetry A 4 , valid at some high-energy scale, naturally leads to three degenerate neutrino masses, without spoiling the hierarchy of charged-lepton masses. Realistic neutrino mixing angles (one of which is necessarily maximal and the other large) are then automatically induced radiatively, and the correct mass splittings may be obtained in the context of softly broken supersymmetry. The quark mixing matrix is also calculable in a similar way. The mixing parameter U e 3 is predicted to be imaginary, leading to maximal CP violation in neutrino oscillations. Neutrinoless double beta decay and τ → μγ should be in the experimentally accessible range.