All the files are normalized so that the best fit (in both normal and inverted ordering) always corresponds to .

The 1D profiles are written in the format

parameter_value chi^2

parameter | normal ordering | inverted ordering |

mq21-NO.dat | mq21-IO.dat | |

mq31-NO.dat | mq31-IO.dat | |

sq12-NO.dat | sq12-IO.dat | |

sq13-NO.dat | sq13-IO.dat | |

sq23-NO.dat | sq23-IO.dat | |

del-NO.dat | del-IO.dat |

The 2D profiles are written in the format

parameter1_value parameter2_value chi^2

parameters | normal ordering | inverted ordering |

— | sq12-mq21-NO.dat | sq12-mq21-IO.dat |

— | sq13-del-NO.dat | sq13-del-IO.dat |

— | sq13-mq21-NO.dat | sq13-mq21-IO.dat |

— | sq13-mq31-NO.dat | sq13-mq31-IO.dat |

— | sq23-del-NO.dat | sq23-del-IO.dat |

— | sq23-mq31-NO.dat | sq23-mq31-IO.dat |

— | sq23-sq13-NO.dat | sq23-sq13-IO.dat |

Since 2016, a lot of authors tryed to extract information on the neutrino mass ordering using cosmological data, alone and in combination with neutrino oscillation constraints.

One of the most debated topics regards the parameterization and the priors that should be used when describing the neutrino mass sector in such kind of analyses.

Simpson et al. used logarithmic priors on the three neutrino masses (, , ) to describe the parameter space and found a strong preference for normal ordering which was never noticed in other analyses (see also Schwetz et al.). Neutrino oscillation data, however, are only sensitive to the squared-mass differences, not to the absolute values of the neutrino masses.

On the other hand, Caldwell et al. adopted a logarithmic prior on the lightest neutrino mass and linear priors on the two squared-mass differences and , which we can naively expect to be more efficient, since it directly deals with the mass splittings that influence neutrino oscillation observables.

In our paper we explored the different possibilities and finally concluded the debate.

Naming *case A* the parameterization which uses the three neutrino masses (, , ) and *case B* the one considering the lightest neutrino mass and the two squared-mass differences (, , ), we studied neutrino oscillation, neutrinoless double beta decay and cosmological data using either linear or logarithmic priors. We used the Bayesian evidences of each case to determine the most efficient parameterization to deal with each dataset and derive conclusions that are not based on theoretical considerations but on data.

The first plot shows a comparison of the two parameterizations and using different priors, considering only neutrino oscillation data in the case of normal (left) and inverted ordering (right).

As we can see, all the prior choices in *case B* are completely equivalent. On the other hand, *case A* is always strongly penalized, that is to say that *case A* is much less efficient to explore the parameter space of neutrino oscillations using the three mass eigenstates than using only one neutrino mass and the two mass splittings.

The result can be seen also in another way: the best prior/parameterization is the one for which the smallest fraction of initial parameter space is incompatible with data.

In *case A* the waste of parameter space is related to the fact that only a small fraction of parameter space is permitted: the three masses must be very close to one another according to neutrino oscillation data.

The inefficiency is also clear from the difference in computation time for the two cases, which is much higher for *case A*.

The second set of figures shows what happens in *case B* when more experimental results are included.

Since data prefer smaller values of the lightest neutrino mass, in this case a weak or moderate preference for logarithmic prior appears. The reason is that linear priors tend to give more importance to higher values of the neutrino masses, which are excluded by data: in the linear case a larger fraction of parameter space is not permitted with respect to the logarithmic case.

As a summary, our results show that analyses including constraints from neutrino oscillations together with probes of neutrino masses are better performed using a logarithmic prior on the lightest neutrino mass and linear priors on the two squared-mass differences and .

Neutrino oscillation data nowadays show a preference in favor of neutrino mass ordering, as depicted by the first point in the left of this plot. The result is rather stable against changes in the parameterizations and priors, and also against the inclusion of more data from neutrinoless double beta decay or cosmology.

The only exception is represented by the four point which correspond to *case A* and logarithmic priors on the three neutrino masses. In this case the difference is driven by the larger allowed parameter space for the second-to-lightest mass eigenstate in normal ordering.

Please note that the results shown here are based on 2017 neutrino oscillation data. The situation has changed with the 2018 data, which now give a strong preference for normal ordering, as described in this page, based on this review.

**Note:** if you need the figures displayed in this page in PDF format, you can find them in the source archive available here.

The results of the global analysis are here.

The neutrino masses can be ordered in two ways:

**normal ordering (NO)**: the lightest neutrino is , the one with the largest mixing with the electron neutrino flavor, and ;**inverted ordering (IO)**: the lightest neutrino is , the one with the smallest mixing with the electron neutrino flavor, and .

The mass ordering can be determined measuring the sign of or singling out each of the masses of the three neutrino mass eigenstates.

Indirect constraints may also come from the observation of the neutrinoless double beta decay or from cosmological bounds on the sum of the neutrino masses (which must be larger than 0.06 eV in NO or than 0.1 eV in IO).

Neutrino oscillation experiments can only determine the mass ordering through neutrino oscillations in matter, in particular exploiting matter effects inside Earth.

Current classes of experiments that can do so include atmospheric or long baseline accelerator experiments.

At the moment, the experiments that have some sensitivity on the mass ordering are SuperKamiokande, NOA, T2K.

Global fits combining these and the other neutrino oscillation experiments give nowadays a preference of between IO and NO, thus corresponding to a preference in favor of normal ordering.

Only allowed for Majorana neutrinos, neutrinoless double beta decay () is a process that permits to determine a combination of the neutrino masses, the effective Majorana mass:

.

As the value of may depend on the mass ordering (if the lightest neutrino is lighter than ), a determination of the mass ordering may come from a detection of the neutrinoless double beta decay process.

In the analysis, we adopt constraints from KamLAND-ZEN (paper), GERDA (paper) and EXO-200 (paper) on the half-life of the neutrinoless double beta decay process.

Current generation experiments do not have a sufficient precision to be competitive in constraining the neutrino mass ordering.

Cosmological probes can nowadays constrain the neutrino masses only through their sum, . Since the known mass splittings and force and , a strong bound on could in principle be used to severely constrain or eventually rule out the IO case.

At the current level, the cosmological bounds are not strong enough to give a statistically significant preference for one of the orderings, as the strongest limits correspond to odds of in favor of NO.

In our analyses we use Cosmic Microwave Background (CMB) data from Planck and Baryon Acoustic Oscillation (BAO) data from various experiments in order to derive constraints on and the corresponding preference in favor of NO. The cosmological dataset including CMB and BAO data, indicated with “Cosmo”, should be considered as conservative.

When we additionally include a prior from local determinations of the parameter (), the analysis becomes less conservative and slightly stronger bounds can be derived.

We perform a global Bayesian analysis combining the various datasets described above. The neutrino mass sector is parameterized using linear priors in , and , since our previous paper demonstrated that these three parameters represent the most efficient way to sample the parameter space.

We compute the Bayesian evidence of the NO and IO cases using different data combinations and we use them to compute the Bayes factors of NO versus IO, which allow us to quantify the preference for one of the two orderings.

Driven by the results coming from neutrino oscillation data, the preference is always in favor of NO.

Using the Jeffreys’ scale, this preference can be quantified as **strong**, in particular corresponding to something more than , as reported in this table:

Data combination | Bayes factor NO vs IO | |

OSC | ||

OSC+ | ||

OSC++Cosmo | ||

OSC++Cosmo+ |

**Note:** if you need the figures displayed in this page in PDF format, you can find them in the source archive available here.

Download all the figures here (tar.gz) or here (zip).

]]>-KamLAND: Data from the analysis performed here.

-Solar data: We include data from the Homestake chlorine detector (paper), from GALLEX (paper), from SAGE (paper), from Super-Kamiokande (paper1, paper2, paper3, paper4), from SNO (paper1, paper2) and Borexino (paper).

For most of these experiments we use the final data samples. From this list the only ones still taking data are Super-Kamiokande and Borexino. The result of our simulation is presented in the following figure. We present the result from the combined analysis of all solar data, the KamLAND data, and the combination of all solar data with KamLAND data.

]]>-K2K: Data from the analysis performed here.

-MINOS: Data from the analysis performed here.

-T2K: Data from the analysis performed here.

-NOA: Data from the analysis performed here.

The result of our simulation is presented in the next figure (left for normal ordering, right for inverted ordering). We do not plot the result of the K2K experiment, since it is not competitive with the rest of the experiments. Note that in the case of K2K and MINOS we include the full and final datasets, because those experiments are not taking data anymore.

]]>-Daya Bay: Data from the analysis performed here.

-RENO: Data from the analysis performed here.

-Double Chooz: Data from the analysis performed here.

The result of our simulation is presented in the next figure (left for normal ordering, right for inverted ordering). We do not include the analyses of the older reactor experiments, since they gave only upper bounds on the reactor angle .

]]>-IceCube DeepCore: Data from the analysis performed here.

-ANTARES: Data from the analysis performed here.

-Super-Kamiokande: Data from the analysis performed here.

The result of our simulation is presented in the next figure (left for normal ordering, right for inverted ordering). While performing the analyses of IceCube and ANTARES data ourselves, it is not possible to reproduce results presented by the Super-Kamiokande collaboration. Anyway, the collaboration provides a -grid, which we include in our analysis.

]]>In the following we present the numerical values of our analysis for each of the parameters. In our global fit of neutrino oscillation data we obtain preference for normal neutrino mass ordering with which corresponds to a preference of .

parameter | best fit | range | range |

: [ eV] | 7.55 | 7.20–7.94 | 7.05–8.14 |

: [ eV] (NO) | 2.500.03 | 2.44–2.57 | 2.41–2.60 |

: [ eV] (IO) | 2.42 | 2.34–2.47 | 2.31-2.51 |

3.20 | 2.89–3.59 | 2.73–3.79 | |

(NO) | 5.47 | 4.67–5.83 | 4.45–5.99 |

(IO) | 5.51 | 4.91–5.84 | 4.53–5.98 |

(NO) | 2.160 | 2.03–2.34 | 1.96–2.41 |

(IO) | 2.220 | 2.07–2.36 | 1.99–2.44 |

(NO) | 1.32 | 1.01–1.75 | 0.87–1.94 |

(IO) | 1.56 | 1.27–1.82 | 1.12–1.94 |

Those numbers have been extracted from the profiles in the following figure. The solid lines are plotted with respect to the global best fit value, while the dashed lines are plotted with respect to the best fit value in inverted ordering.

The 1D profiles used for the plots are available here.

Another way of presenting these results is in two-dimensional plots. In the following figures we have always marginalized over all of the parameters not plotted. As for the profiles, the figures for inverted ordering are plotted with respect to the best fit for inverted ordering.

The 2D profiles used for the plots are available here.