abstract

The monomer–trimer model is studied on the fractal \(3\)-simplex lattice. Configurations are enumerated by an exact system of recurrence relations. Asymptotic forms for the number of pure trimer and monomer–trimer configurations of equal weights as well as entropy are found. The asymptotic form in the close-packed limit differs from the one obtained for dimers on the square lattice. By introducing monomer fugacity, configurations are classified according to the number of monomers (or trimers), and the problem is formulated in the grand canonical ensemble. The average number of monomers and entropy are calculated as functions of fugacity. Entropy as a function of trimer fraction coverage shows qualitatively similar behavior to that found on the square lattice.

abstract

By analogy with the Ginzburg–Landau theory of multi-band superconductors with inner (interband) Josephson couplings, we formulate the three-band Glashow–Weinberg–Salam model with weak Josephson couplings between strongly asymmetrical condensates of scalar (Higgs) fields. Unlike the usual single-band model, we found three Higgs bosons corresponding to the three generations of particles. Moreover, the heaviest of these bosons corresponds to the already discovered \(H\)-boson and decays into fermions of only the third generation through the Yukawa interaction. The other two decay into fermions of the first and second generations, but they are difficult to observe due to very poor production conditions. We found two sterile ultra-light Leggett bosons, the Bose condensates of which form the dark halos of galaxies and their clusters (i.e. so-called Dark Matter). The masses of the Leggett bosons are determined by the coefficient of the interband coupling and can be arbitrarily small (\(\sim 10^{-20}\) eV) due to non-perturbativeness of the interband coupling. Since propagation of the Leggett bosons is not accompanied by a current, these bosons are not absorbed by gauge fields, unlike the common-mode Goldstone bosons. Three coupled condensates of the scalar fields are related to the existence of three generations of leptons, where each generation interacts with the corresponding condensate getting mass. The interflavor mixing between the generations of active neutrinos and sterile right-handed neutrinos in the three-band system causes the existence of mass states of neutrino without interaction with the Higgs condensates.