Neutrino masses and their ordering: Global Data, Priors and Models

TL;DR

This paper performs a comprehensive Bayesian global analysis to assess whether current neutrino data can decisively establish the normal or inverted mass ordering. It contrasts two parametrizations, Case A $(m_1,m_2,m_3)$ and Case B $(m_{\text{lightest}},\Delta m^2_{21},\Delta m^2_{31})$, with linear and logarithmic priors, and combines neutrino oscillation data, neutrinoless double-beta decay, and Planck CMB observations. The main finding is that the ordering preference is dominated by oscillation data, with a typical Bayes factor around $\ln B_{\rm NO,IO}\approx 2.5$ indicating weak-to-moderate evidence; a strong evidence only appears for the specific Case A with logarithmic priors. The study demonstrates the critical influence of parametrization and priors on inferred $\sum m_\nu$ and mass-ordering conclusions, arguing for self-consistent analyses that use the mass-splitting based parametrization with a $\log$ prior for robust future determinations. It also highlights that cosmological bounds on $\sum m_\nu$ are highly prior-dependent and cautions against reusing precomputed limits obtained under different assumptions.

Abstract

We present a Bayesian analysis of the combination of current neutrino oscillation, neutrinoless double beta decay and CMB observations. Our major goal is to carefully investigate the possibility to single out one neutrino mass ordering, Normal Ordering or Inverted Ordering, with current data. Two possible parametrizations (three neutrino masses versus the lightest neutrino mass plus the two oscillation mass splittings) and priors (linear versus logarithmic) are examined. We find that the preference for NO is only driven by neutrino oscillation data. Moreover, the values of the Bayes factor indicate that the evidence for NO is strong only when the scan is performed over the three neutrino masses with logarithmic priors; for every other combination of parameterization and prior, the preference for NO is only weak. As a by-product of our Bayesian analyses, we are able to a) compare the Bayesian bounds on the neutrino mixing parameters to those obtained by means of frequentist approaches, finding a very good agreement; b) determine that the lightest neutrino mass plus the two mass splittings parametrization, motivated by the physical observables, is strongly preferred over the three neutrino mass eigenstates scan and c) find that there is a weak-to-moderate preference for logarithmic priors. These results establish the optimal strategy to successfully explore the neutrino parameter space, based on the use of the oscillation mass splittings and a logarithmic prior on the lightest neutrino mass. We also show that the limits on the total neutrino mass $\sum m_ν$ can change dramatically when moving from one prior to the other. These results have profound implications for future studies on the neutrino mass ordering, as they crucially state the need for self-consistent analyses which explore the best parametrization and priors, without combining results that involve different assumptions.

Figures (5)

• Figure 1: The profiles for the neutrino oscillation parameters. Solid blue lines correspond to NO and dashed magenta lines to IO.
• Figure 2: The profiles for the neutrino oscillation parameters as obtained from the Bayesian analysis. Blue (solid) lines correspond to NO and magenta (dashed) lines to IO. The effective $\chi^2$ is obtained from the posterior distribution function $p(x)$ for each parameter $x$ and the Bayesian evidence $Z$ using $\Delta \chi^2_{\rm eff} = -2 \ln p(x) -2\ln Z$.
• Figure 3: Graphical visualisation of the Bayesian factors comparing normal and inverted ordering. The horizontal lines indicate the values at which there is a change in the statistical significance of the evidence, according to the Jeffreys' scale (see table \ref{['tab:jeffreys']}). Black (red) points indicate a logarithmic (linear) prior. The prior ranges are those reported in table \ref{['tab:massParams']} if not otherwise stated.
• Figure 4: Difference in allowed volumes for the three absolute neutrino masses for NO and IO from neutrino oscillation data only. The top (bottom) panels show the case of linear (logarithmic) priors.
• Figure 5: Graphical visualisation of the Bayesian evidences to compare the various neutrino mass parametrizations (Case A, only shown in the neutrino oscillation analyses, and Case B, shown for all the data combinations) and priors (logarithmic, in black, and linear, in red). Points are normalised with respect to the preferred Bayesian evidence $Z_{\rm best}$ within each panel, which always corresponds to one of the Case B cases with NO. Colour codes are the same as in figure \ref{['fig:bayesfactors']}.