# Representing seesaw neutrino models and their motion in lepton flavour space

## TL;DR

This paper introduces a graphical representation of seesaw neutrino models in lepton flavour space, explores their properties, and applies the framework to analyze $N_2$-leptogenesis, revealing insights into neutrino mass hierarchies and flavour effects.

## Contribution

It presents a novel graphical and mathematical framework for analyzing seesaw neutrino models, including the bridging matrix and Lorentz boost analogy, and applies it to leptogenesis.

## Key findings

- Texture zero models are often ruled out by this representation.
- The probability of at least one flavoured decay parameter being less than one is about 49%.
- Degenerate light neutrinos are disfavored when $m_1 o 0.1$ eV.

## Abstract

We discuss how seesaw neutrino models can be graphically represented in lepton flavour space. We examine various popular models and show how this representation helps understanding their properties and connection with experimental data showing in particular how certain texture zero models are ruled out. We also introduce a new matrix, the bridging matrix, that brings from the light to the heavy neutrino mass flavour basis, showing how this is related to the orthogonal matrix and how different quantities are easily expressed through it. We then show how one can randomly generate orthogonal and leptonic mixing matrices uniformly covering all flavour space in an unbiased way (Haar-distributed matrices). Using the isomorphism between the group of complex rotations and the Lorentz group, we also introduce the conceptof Lorentz boost in flavour space for a seesaw model and how this has an insightful physical interpretation. Finally, as a significant application, we consider $N_2$-leptogenesis. Using current experimental values of low energy neutrino parameters, we show that the probability that at least one flavoured decay parameter of the lightest right-handed neutrino is smaller than unity is about $49\%$ (to be compared with the tiny probability that the total decay parameter is smaller than unity, $P(K_{\rm I}< 1)\sim 0.1 \%$, confirming the crucial role played by flavour effects). On the other hand when $m_1 \gtrsim 0.1\,{\rm eV}$ this probability reduces to less than $5\%$, showing how also $N_2$-leptogenesis disfavours degenerate light neutrinos.