On Invariants of Quark and Lepton Mass Matrices in the Standard Model

In this article I present the motivation for introducing the invariant functions of mass matrices, based on my own work, and give some examples. Since their introduction in 1985, in the framework of the standard electroweak model, they have been used by many authors. Some authors have gone further along this path and have studied the extensions of this concept to frameworks beyond the standard model. I hope, in the near future, to give a more detailed account of this subject, including recent developments

The quark mixing matrix with manifest Cabibbo substructure and an angle of the unitarity triangle as one of its parameters

The quark mixing matrix is parameterised such that its "Cabibbo substructure" is emphasised. One can choose one of the parameters to be an arbitrarily chosen angle of the unitarity triangle, for example the angle $\beta$ (also called $\Phi_1$)

The Role of Invariant Functions in Understanding Masses and Mixings

One of the central questions in theoretical particle physics, since already several decades, has been that of "masses and mixings of the quarks. With the entry of neutrino oscillations into the field, the issue of lepton masses has added a new dimension to the problem. In the literature one finds many models of quark and lepton mass matrices. However, the mass and mixing problems remain unsolved. In this talk I will present my own contributions to this field, however as you may expect without offering a solution. I encourage you to go ahead and think about the problem but be aware of the pitfalls.Comment: Talk presented at the conference SNOW 200

Ambiguities pertaining to quark-lepton complementarity

Recently the possible origin of the so-called quark-lepton complementarity relations has received a considerable amount of attention. We point out some of the inherent ambiguities in such analyses

Recursive parameterisation and invariant phases of unitary matrices

We present further properties of a previously proposed recursive scheme for parameterisation of n-by-n unitary matrices. We show that the factors in the recursive formula may be introduced in any desired order. The method is used to study the invariant phases of unitary matrices. The case of four-by-four unitary matrices is investigated in detail. We also address the question of how to construct symmetric unitary matrices using the recursive approach

A recursive parameterisation of unitary matrices

A simple recursive scheme for parameterisation of n-by-n unitary matrices is presented.Comment: Notations simplified and more details give

Supersymmetry - Roots That Didn't Grow

This article is about early roots of supersymmetry, as found in the literature from 1940s and early 1950s. There were models where the power of "partners" in alleviating divergences in quantum field theory was recognized. However, other currently known remarkable features of supersymmetry, such as its role in the extension of the Poincar\'{e} group, were not known. There were, of course, no supersymmetric non-abelian quantum field theories in those days

Invariants of Lepton Mass Matrices and CP and T Violation in Neutrino Oscillations

CP and T asymmetries in neutrino oscillations, in vacuum as well as in matter, are expressed in terms of invariant functions of lepton mass matrices.Comment: 9 page

Neutrino Sector with Majorana Mass Terms and Friedberg-Lee Symmetry

We examine a recently proposed symmetry/condition by Friedberg and Lee in the framework where three right-handed neutrinos are added to the spectrum of the three-family Minimal Standard Model. It is found that the right-handed neutrinos are very special, with respect to this symmetry. In the symmetry limit the neutrinos are massless and that may be a hint about why they are light. Imposed as a condition and not as a full symmetry, we find that one of the three right-handed neutrinos simply decouples (has only gravitational interactions) and that there is a massless interacting neutrino. The possible relation of the model to the see-saw mechanism is briefly discussed