Horizontal Symmetry from the Bottom Up

A general method to derive horizontal symmetry from a mixing matrix is reviewed. The technique has been applied to deduce leptonic symmetry from the tri-bimaximal neutrino mixing matrix and three of its variations. The question of how the quark mixing can be accommodated within the leptonic symmetry group is discussed, including in this connection an example based on the group $D_4$.Comment: Luxor09 talk, 7 page

Decomposition of Time-Ordered Products and Path-Ordered Exponentials

We present a decomposition formula for $U_n$, an integral of time-ordered products of operators, in terms of sums of products of the more primitive quantities $C_m$, which are the integrals of time-ordered commutators of the same operators. The resulting factorization enables a summation over $n$ to be carried out to yield an explicit expression for the time-ordered exponential, an expression which turns out to be an exponential function of $C_m$. The Campbell-Baker-Hausdorff formula and the nonabelian eikonal formula obtained previously are both special cases of this result.Comment: 31 pages, Revtex with two postscript figure

Mass Independent Textures and Symmetry

A mass-independent texture is a set of linear relations of the fermion mass-matrix elements which imposes no constraint on the fermionic masses nor the Majorana phases. Magic and 2-3 symmetries are examples. We discuss the general construction and the properties of these textures, as well as their relation to the quark and neutrino mixing matrices. Such a texture may be regarded as a symmetry, whose unitary generators of the symmetry group can be explicitly constructed. In particular, the symmetries connected with the tri-bimaximal neutrino mixing matrix are discussed, together with the physical consequence of breaking one symmetry but preserving another.Comment: Version to be published in PRD. References modified, and a misprint in eq (9) is correcte

Group Theory and Dynamics of Neutrino Mixing

There is a direct group-theoretical connection between neutrino mixing and horizontal symmetry that can be established without any dynamical input. Such a connection is reviewed and expanded in this article. For certain symmetry groups \G including $A_4$ and $S_4$, it is shown that a generic U(1)\x\G Higgs potential of a valon yields exactly the alignments dictated by the group-theoretic approach, but energy can now be used to discriminate different alignments. This mechanism possibly explains why starting from an $A_4$ group, the tribimaximal mixing matrix with an enhanced $S_4$ symmetry is more preferable than the one without it.Comment: One footnote and one reference added. To appear in Physical Review

Probing Extra Dimensions with Neutrino Oscillations

In the braneworld scenario, gravity and neutrino oscillation can both be used to detect the presence of an extra dimension. We argue that neutrino oscillation is particularly suitable if the size of the extra dimension is small, in which case the signature for the extra dimension is the disappearance of active neutrino fluxes into the bulk, caused by the destructive interference from the Kaluza-Klein states. We discuss a class of models to illustrate this general feature.Comment: 16 pages, no figures. A contribution in honor of Hiroshi Ezawa on his 70th birthday. To appear in `Garden of Quanta' (World Scientific