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 concept of Lorentz boost in flavour space for a seesaw model and how this has an insightful physical interpretation. Finally, as a significant application, we consider N2-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 (KI< 1) ∼ 0.1%, confirming the crucial role played by flavour effects). On the other hand when m1 ≳ 0.1 eV this probability reduces to less than 5%, showing how also N2-leptogenesis disfavours degenerate light neutrinos.
Highlights
In the case of a pure bottom-up approach one would like to draw model independent conclusions based just on the experimental information
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 (KI < 1) ∼ 0.1%, confirming the crucial role played by flavour effects)
In particular we show that models at rest in flavour space correspond to models with minimal fine-tuning. We apply this new parameterisation to leptogenesis, showing how in this way the distributions of all flavour decay parameters are identical if no experimental information on the low energy neutrino parameters is imposed and how these change when current experimental information is imposed
Summary
It is easy to prove that this case is excluded by the experimental data since one can always perform a transformation, operated by a unitary matrix VL acting on the lepton doublets, from the (charged lepton) flavour basis (e, μ, τ ) to a new orthonormal flavour basis (e , μ , τ ) where e coincides with the common heavy neutrino flavour In this new flavour basis the neutrino Dirac mass matrix takes the very simple form mDe I mDe II mDe III mD. Analogously to the three RH neutrino case where all three heavy neutrino flavours are aligned, one would get a second vanishing light neutrino, so that one cannot reproduce both the solar and the atmospheric neutrino mass scales Within these two RH neutrino models one can further reduce the number of parameters again imposing texture zeros in the neutrino Dirac mass matrix mD, i.e., in the charged lepton flavour basis.
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