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The non-Abelian discrete symmetry D(7) of the heptagon is successfully applied to both quark and lepton mass matrices, including CP violation.
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To include the quark sector, the $A_{5}equiv I$ (icosahedron) four generation lepton model is extended to a binary icosahedral symmetry $I$ flavor model. We find the masses of fermions, including the heavy sectors, can be accommodated. At leading order the CKM matrix is the identity and the PMNS matrix, resulting from same set of vacua, corresponds to tribimaximal mixings.
The weak bosons consist of two fermions, bound by a new confining gauge force. The mass scale of this new interaction is determined. At energies below 0.5 TeV the standard electroweak theory is valid. A neutral isoscalar weak boson X must exist - its mass is less than 1 TeV. It will decay mainly into quark and lepton pairs and into two or three weak bosons. Above the mass of 1 TeV one finds excitations of the weak bosons, which mainly decay into pairs of weak bosons. Leptons and quarks consist of a fermion and a scalar. Pairs of leptons and pairs of quarks form resonances at very high energy.
We show how the scales responsible for Peccei-Quinn (PQ), seesaw, and Froggatt and Nielsen (FN) mechanisms can be fixed, by constructing a compact model for resolving rather recent, but fast-growing issues in astro-particle physics, including quark and leptonic mixings and CP violations, high-energy neutrinos, QCD axion, and axion cooling of stars. The model is motivated by the flavored PQ symmetry for unifying the flavor physics and string theory. The QCD axion decay constant congruent to the seesaw scale, through its connection to the astro-particle constraints of both the stellar evolution induced by the flavored-axion bremsstrahlung off electrons $e+Zerightarrow Ze+e+A_i$ and the rare flavor-changing decay process induced by the flavored-axion $K^+rightarrowpi^++A_i$, is shown to be fixed at $F_A=3.56^{+0.84}_{-0.84}times10^{10}$ GeV (consequently, the QCD axion mass $m_a=1.54^{+0.48}_{-0.29}times10^{-4}$ eV, Compton wavelength of its oscillation $lambda_a=8.04^{+1.90}_{-1.90},{rm mm}$, and axion to neutron coupling $g_{Ann}=2.14^{+0.66}_{-0.41}times10^{-12}$, etc.). Subsequently, the scale associated to FN mechanism is dynamically fixed through its connection to the standard model fermion masses and mixings, $Lambda=2.04^{,+0.48}_{,-0.48}times10^{11},{rm GeV}$, and such fundamental scale might give a hint where some string moduli are stabilized in type-IIB string vacua. In the near future, the NA62 experiment expected to reach the sensitivity of ${rm Br}(K^+rightarrowpi^++A_i)<1.0times10^{-12}$ will probe the flavored-axions or exclude the model, if the astrophysical constraint of star cooling is really responsible for the flavored-axion.
We discuss the relation between the CP violation of the quark mixing and that of the lepton mixing by investigating a CP violating observable, the Jarlskog invariant, as well as the CP violating Dirac phase. The down-type quark mass matrix with three zeros is given in terms of the minimal number of parameters, while the up-type quark mass matrix is diagonal. These quark mass matrices leading to the successful CKM mixing angles and CP violation are embedded in both the Pati--Salam and SU(5) models. The leptonic Jarlskog invariant $J_{CP}^l$ (as well as CP violating Dirac phase) is examined for two cases: the neutrino mass matrix is diagonal or non-diagonal, where no additional CP violating phase is introduced apart from the Majorana phases. In the case of the diagonal neutrino mass matrix, the favorable sign of the leptonic CP violation is obtained, however, the magnitude of $J_{CP}^l$ is at most ${cal O}(10^{-4})$, which is too small compared with the expected value from the observation $-0.02$. In the case of the non-diagonal neutrino mass matrix where the tri-bimaximal mixing pattern is taken, we obtain the successful $J_{CP}^l$ up to its sign.
The hierarchy of the Yukawa couplings is an outstanding problem of the standard model. We present a class of models in which the first and second generation fermions are SUSY partners of pseudo-Nambu-Goldstone bosons that parameterize a non-compact Kahler manifold, explaining the small values of these fermion masses relative to those of the third generation. We also provide an example of such a model. We find that various regions of the parameter space in this scenario can give the correct dark matter abundance, and that nearly all of these regions evade other phenomenological constraints. We show that for gluino mass ~700 GeV, model points from these regions can be easily distinguished from other mSUGRA points at the LHC with only 7 fb^(-1) of integrated luminosity at 14 TeV. The most striking signatures are a dearth of b- and tau-jets, a great number of multi-lepton events, and either an inverted slepton mass hierarchy, narrowed slepton mass hierarchy, or characteristic small-mu spectrum.
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