Quark Mixings in SU(6)×SU(2)RSU(6)\times SU(2)_R and Suppression of VubV_{ub}

The quark mixing matrix $V_{CKM}$ is studied in depth on the basis of superstring inspired $SU(6)\times SU(2)_R$ model with global flavor symmetries. The sizable mixings between right-handed down-type quark $D^c$ and colored Higgs field $g^c$ potentially occur but no such mixings in up-type quark sector. In the model the hierarchical pattern of $V_{CKM}$ is understood systematically. It is shown that due to large $D^c$-$g^c$ mixings $V_{ub}$ is naturally suppressed compared to $V_{td}$. It is pointed out that the observed suppression of $V_{ub}$ is in favor of the presence of $SU(2)_R$ gauge symmetry but not in accord with generic SU(5) GUT.Comment: 10pages with no figure, Latex fil

Non-linear relativistic diffusions

We obtain a non-linear generalization of the relativistic diffusion of particles with spin. We discuss diffusion equations whose non-linearity is a consequence of quantum statistics. We show that the assumptions of the relativistic invariance and an interpretation of the solution as a probability distribution substantially restrict the class of admissible non-linear diffusion equations. We consider relativistic invariant as well as covariant frame dependent diffusion equations with a drift. In the latter case we show that there can exist stationary solutions of the diffusion equation besides the equilibrium solution corresponding to the quantum or Tsallis distributions. We define the relative entropy as a function of the diffusion probability and prove that it is monotonically decreasing in time when the diffusion tends to the equilibrium. We discuss its relation to the thermodynamic behavior of diffusing particles.Comment: 20 pages, the version to appear in Physica