Global constraints on absolute neutrino masses and their ordering

TL;DR

This paper performs a comprehensive global analysis combining oscillation, neutrinoless double beta decay, and cosmological data to constrain absolute neutrino masses and their ordering, suggesting a slight preference for normal ordering.

Contribution

It provides the first detailed global constraints on neutrino mass scale and ordering by integrating diverse experimental data sets with a chi^2 analysis.

Findings

Normal ordering is mildly favored over inverted by ~2 sigma.

Current data constrain the sum of neutrino masses to below approximately 0.1 eV.

Implications for future experiments are discussed based on the constraints obtained.

Abstract

Within the standard three-neutrino framework, the absolute neutrino masses and their ordering (either normal, NO, or inverted, IO) are currently unknown. However, the combination of current data coming from oscillation experiments, neutrinoless double beta decay searches, and cosmological surveys, can provide interesting constraints for such unknowns in the sub-eV mass range, down to O(0.1) eV in some cases. We discuss current limits on absolute neutrino mass observables by performing a global data analysis, that includes the latest results from oscillation experiments, neutrinoless double beta decay bounds from the KamLAND-Zen experiment, and constraints from representative combinations of Planck measurements and other cosmological data sets. In general, NO appears to be somewhat favored with respect to IO at the level of ~2 sigma, mainly by neutrino oscillation data (especially atmospheric), corroborated by cosmological data in some cases. Detailed constraints are obtained via the chi^2 method, by expanding the parameter space either around separate minima in NO and IO, or around the absolute minimum in any ordering. Implications for upcoming oscillation and non-oscillation neutrino experiments, including beta-decay searches, are also discussed.

Tables

Table 1. Table 1: Results of the global 3 ​ ν 3 𝜈 3\nu oscillation analysis, in terms of best-fit values for the mass-mixing parameters and associated n ​ σ 𝑛 𝜎 n\sigma ranges ( n = 1 ,  2 ,  3 𝑛 1 2 3 n=1,\,2,\,3 ), defined by χ 2 − χ min 2 = n 2 superscript 𝜒 2 subscript superscript 𝜒 2 superscript 𝑛 2 \chi^{2}-\chi^{2}_{\min}=n^{2} with respect to the separate minima in each mass ordering (NO, IO) and to the absolute minimum in any ordering. (Note that the fit to the δ ​ m 2 𝛿 superscript 𝑚 2 \delta m^{2} and sin 2 ⁡ θ 12 superscript 2 subscript 𝜃 12 \sin^{2}\theta_{12} parameters is basically insensitive to the mass ordering.) We recall that Δ ​ m 2 Δ superscript 𝑚 2 \Delta m^{2} is defined herein as m 3 2 − ( m 1 2 + m 2 2 ) / 2 subscript superscript 𝑚 2 3 subscript superscript 𝑚 2 1 subscript superscript 𝑚 2 2 2 m^{2}_{3}-{(m^{2}_{1}+m^{2}_{2}})/2 , and that δ 𝛿 \delta is taken in the (cyclic) interval δ / π ∈ [ 0 ,  2 ] 𝛿 𝜋 0 2 \delta/\pi\in[0,\,2] .

Parameter	Ordering	Best fit	$1\sigma$ range	$2\sigma$ range	$3\sigma$ range
$\delta m^2/{10}^{-5}{\mathrm{eV}}^2$	NO, IO, Any	7.37	7.21 – 7.54	7.07 – 7.73	6.93 – 7.96
${\sin }^2{\theta }_{12}/{10}^{-1}$	NO, IO, Any	2.97	2.81 – 3.14	2.65 – 3.34	2.50 – 3.54
$\left|\Delta m^2\right|/{10}^{-3}{\mathrm{eV}}^2$	NO	2.525	2.495 – 2.567	2.454 – 2.606	2.411 – 2.646
	IO	2.505	2.473 – 2.539	2.430 – 2.582	2.390 – 2.624
	Any	2.525	2.495 – 2.567	2.454 – 2.606	2.411 – 2.646
${\sin }^2{\theta }_{13}/{10}^{-2}$	NO	2.15	2.08 – 2.22	1.99 – 2.31	1.90 – 2.40
	IO	2.16	2.07 – 2.24	1.98 – 2.33	1.90 – 2.42
	Any	2.15	2.08 – 2.22	1.99 – 2.31	1.90 – 2.40
${\sin }^2{\theta }_{23}/{10}^{-1}$	NO	4.25	4.10 – 4.46	3.95 – 4.70	3.81 – 6.15
	IO	5.89	4.17 – 4.48 $\oplus$ 5.67 – 6.05	3.99 – 4.83 $\oplus$ 5.33 – 6.21	3.84 – 6.36
	Any	4.25	4.10 – 4.46	3.95 – 4.70 $\oplus$ 5.75 – 6.00	3.81 – 6.26
$\delta /\pi$	NO	1.38	1.18 – 1.61	1.00 – 1.90	0 – 0.17 $\oplus$ 0.76 – 2
	IO	1.31	1.12 – 1.62	0.92 – 1.88	0 – 0.15 $\oplus$ 0.69 – 2
	Any	1.38	1.18 – 1.61	1.00 – 1.90	0 – 0.17 $\oplus$ 0.76 – 2

Table 2. Table 2: Results of the global 3 ​ ν 3 𝜈 3\nu analysis of cosmological data within the standard Λ ​ CDM + Σ Λ CDM Σ \Lambda\mathrm{CDM}+\Sigma and extended Λ ​ CDM + Σ + A lens Λ CDM Σ subscript 𝐴 lens \Lambda\mathrm{CDM}+\Sigma+A_{\mathrm{lens}} models. The datasets refer to various combinations of the Planck power angular CMB temperature power spectrum (TT) plus polarization power spectra (TE, EE), reionization optical depth τ HFI subscript 𝜏 HFI \tau_{\mathrm{HFI}} , lensing potential power spectrum (lensing), and BAO measurements. For each of the 12 cases we report the 2 ​ σ 2 𝜎 2\sigma upper bounds on Σ = m 1 + m 2 + m 3 Σ subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 \Sigma=m_{1}+m_{2}+m_{3} for NO and IO, together with the Δ ​ χ 2 Δ superscript 𝜒 2 \Delta\chi^{2} difference between the two mass orderings (with one digit after decimal point). For any Σ Σ \Sigma , the masses m i subscript 𝑚 𝑖 m_{i} are taken to obey the δ ​ m 2 𝛿 superscript 𝑚 2 \delta m^{2} and Δ ​ m 2 Δ superscript 𝑚 2 \Delta m^{2} constraints coming from oscillation data. See the text for more details.

#	Model	Cosmological data set	$\Sigma /$eV ($2\sigma$), NO	$\Sigma /$eV ($2\sigma$), IO	$\Delta {\chi }_{\mathrm{IO}-\mathrm{NO}}^2$
1	$\Lambda \mathrm{CDM}+\Sigma$	Planck TT + ${\tau }_{\mathrm{HFI}}$			$0.7$
2	$\Lambda \mathrm{CDM}+\Sigma$	Planck TT + ${\tau }_{\mathrm{HFI}}$ + lensing			$0.2$
3	$\Lambda \mathrm{CDM}+\Sigma$	Planck TT + ${\tau }_{\mathrm{HFI}}$ + BAO			$1.2$
4	$\Lambda \mathrm{CDM}+\Sigma$	Planck TT, TE, EE + ${\tau }_{\mathrm{HFI}}$			$0.6$
5	$\Lambda \mathrm{CDM}+\Sigma$	Planck TT, TE, EE + ${\tau }_{\mathrm{HFI}}$ + lensing			$0.3$
6	$\Lambda \mathrm{CDM}+\Sigma$	Planck TT, TE, EE + ${\tau }_{\mathrm{HFI}}$ + BAO			$1.6$
7	$\Lambda \mathrm{CDM}+\Sigma +A_{\mathrm{lens}}$	Planck TT + ${\tau }_{\mathrm{HFI}}$			$-0.1$+
8	$\Lambda \mathrm{CDM}+\Sigma +A_{\mathrm{lens}}$	Planck TT + ${\tau }_{\mathrm{HFI}}$ + lensing			$0.0$
9	$\Lambda \mathrm{CDM}+\Sigma +A_{\mathrm{lens}}$	Planck TT + ${\tau }_{\mathrm{HFI}}$ + BAO			$0.2$
10	$\Lambda \mathrm{CDM}+\Sigma +A_{\mathrm{lens}}$	Planck TT, TE, EE + ${\tau }_{\mathrm{HFI}}$			$0.0$
11	$\Lambda \mathrm{CDM}+\Sigma +A_{\mathrm{lens}}$	Planck TT, TE, EE + ${\tau }_{\mathrm{HFI}}$ + lensing			$0.1$
12	$\Lambda \mathrm{CDM}+\Sigma +A_{\mathrm{lens}}$	Planck TT, TE, EE + ${\tau }_{\mathrm{HFI}}$ + BAO			$0.3$

Table 3. Table 3: Values of Δ ​ χ IO − NO 2 Δ subscript superscript 𝜒 2 IO NO \Delta\chi^{2}_{\mathrm{IO-NO}} from the global analysis of oscillation and non oscillation data (numbered according to the adopted cosmological datasets as in Table II), to be compared with the value 3.6 from oscillation data only [Eq. ( 9 )]. An overall preference emerges for NO, at the level of 1.9– 2.1 ​ σ 2.1 𝜎 2.1\sigma .

#	1	2	3	4	5	6	7	8	9	10	11	12
$\Delta {\chi }_{\mathrm{IO}-\mathrm{NO}}^2$	4.3	3.8	4.4	4.2	3.9	4.4	3.6	3.7	3.8	3.7	3.8	3.9

Equations

$$\delta m^2$$

$$\Delta m^2$$

$$(m_{1},\,m_{2},\,m_{3})\geq\left\{\begin{array}[]{lr}\left(0,\,\sqrt{\delta m^{2}},\,\sqrt{|\Delta m^{2}|+\delta m^{2}/2}\right)&\rm{(NO)}\ ,\\[8.0pt] \left(\sqrt{|\Delta m^{2}|-\delta m^{2}/2},\,\sqrt{|\Delta m^{2}|+\delta m^{2}/2},\,0\right)&\rm{(IO)\ .}\\ \end{array}\right.$$

$$m_{\beta }=\sqrt{{\sum }_i|U_{ei}{|}^2{m}_i^2},$$

$$m_{\beta \beta }=\left|{\sum }_i|U_{ei}{|}^2{m}_i{e}^{ı{\varphi }_i}\right|,$$

$$\Sigma =m_1+m_2+m_3.$$

$$\Delta {\chi }^2=n^2$$

$$\Delta {\chi }_{\mathrm{IO}-\mathrm{NO}}^2={\chi }_{\min ,\mathrm{IO}}^2-{\chi }_{\min ,\mathrm{NO}}^2.$$

$$\Delta {\chi }_{\mathrm{IO}-\mathrm{NO}}^2=3.6(\mathrm{all}oscill.\mathrm{data}),$$

$$(m_{1},\,m_{2},\,m_{3})\gtrsim\left\{\begin{array}[]{lc}(0,\,0.86,\,5.06)\times 10^{-2}\ \mathrm{eV}&\rm{(NO)}\ ,\\ (4.97,\,5.04,\,0)\times 10^{-2}\ \mathrm{eV}&\rm{(IO)}\ .\\ \end{array}\right.$$

$$T^{-1}=G|M{|}^2{m}_{\beta \beta }^2,$$

$${\chi }^2(m_{\beta \beta })=\underset{|M|}{\min }\left[{\chi }^2\left(T(m_{\beta \beta },|M|)\right)+{\chi }^2(|M|)\right],$$

$$m_{\beta \beta }<0.18\mathrm{eV}\mathrm{at}2\sigma (<0.27\mathrm{eV}\mathrm{at}3\sigma ).$$

$${\Sigma}=m_{1}+m_{2}+m_{3}\gtrsim\left\{\begin{array}[]{lc}0.06\ \mathrm{eV}&\rm{(NO)}\ ,\\ 0.10\ \mathrm{eV}&\rm{(IO)}\ .\\ \end{array}\right.$$

Full text

I Introduction

Neutrino oscillation experiments have established that the three known flavor states $\nu_{\alpha}$ $(\alpha=e,\,\mu,\,\tau)$ are linear combinations of three massive states $\nu_{i}$ $(i=1,\,2,\,3)$ with different masses $m_{i}$, via a mixing matrix $U_{\alpha i}$ characterized by three nonzero angles $\theta_{ij}$ PDG2016 . Flavor oscillation frequencies in vacuum are governed by the squared mass differences $\Delta m^{2}_{ij}$, that can be expressed in terms of two independent parameters, conventionally chosen herein as Glob06 :

[TABLE]

where $\Delta m^{2}$ can be either positive or negative according to the so-called normal ordering (NO) or inverted ordering (IO) for the neutrino mass spectrum, respectively. Probing the mass ordering is an important goal of future experimental $\nu$ oscillation searches (see, e.g., Patt15 ; Qian15 ), with relevant implications on theoretical models for neutrino mass and mixing (see, e.g., Moha06 ; Feru15 ; King15 ).

At present, the four parameters $\delta m^{2}$, $|\Delta m^{2}|$, $\sin^{2}\theta_{12}$, and $\sin^{2}\theta_{13}$ have been measured at the few % level, while $\sin^{2}\theta_{23}$ (still affected by an octant ambiguity Octa96 ) is less accurately known, at the level of $\sim\!10\%$ PDG2016 . Interestingly, the combination of various oscillation data starts to show some sensitivity to the remaining unknowns, namely, the sign of $\Delta m^{2}$ and a possible CP-violating phase $\delta$, mainly through subleading $\nu_{\mu}\to\nu_{e}$ oscillation effects in atmospheric and accelerator neutrino experiments, constrained by reactor data Capo16 ; Marr16 ; see also Conc17 ; Vall14 for independent analyses of oscillation data and for discussions of the associated parameters.

The absolute $\nu$ masses are also unknown. Lower bounds are set by oscillation data by zeroing the lightest $m_{i}$,

[TABLE]

while upper bounds (and prospective measurements) can only be set by nonoscillation neutrino experiments. In particular, three main observables can probe the absolute mass spectrum: $(i)$ the effective neutrino mass $m_{\beta}$ in $\beta$ decay; $(ii)$ the effective mass $m_{\beta\beta}$ in neutrinoless double beta ($0\nu\beta\beta$) decay, if neutrinos are Majorana fermions; and $(iii)$ the total neutrino mass $\Sigma$ in cosmology; see, e.g., the reviews in Bile02 ; Otte08 ; Viss16 ; Hann10 .

These observables probe the neutrino mass spectrum in different and complementary ways PDG2016 ; Glob06 . The $\beta$ decay spectrum is sensitive to an (unresolved) combination of squared masses, weighted by the corresponding $\nu_{e}$ admixture,

[TABLE]

while the $0\nu\beta\beta$ decay rate depends linearly on the $m_{i}$’s via unknown Majorana phases $\phi_{i}$ (with $\phi_{1}=0$ by convention),

[TABLE]

and cosmology essentially probes the (flavor-blind) total gravitational charge,

[TABLE]

Currently, the most constraining bounds on $m_{\beta\beta}$ can be as low as $O(0.1)$ eV in the KamLAND-Zen experiment at $\sim 2\sigma$, by assuming favorable nuclear matrix elements KamZ16 . Upper bounds on $\Sigma$, dominated by Planck data, can also reach the level of $O(0.1)$ eV, by assuming the standard cosmological model Lesg14 ; planckparams2015 . Such limits are getting close to the mass scale $\sqrt{|\Delta m^{2}|}\simeq 0.05$ eV appearing in Eq. (3) where some sensitivity of cosmological data to mass ordering may be emerging Viss16 . Bounds on $m_{\beta}$, although free from specific assumptions, are an order of magnitude weaker at present PDG2016 ; Otte08 . In all cases, significant improvements—and possibly a positive detection—may be expected in the next decade of experimental searches NuMass . In this context, we find it worthwhile to perform and discuss an updated global analysis of both oscillation and nonoscillation data, building upon previous work on the subject Melc04 ; Melc06 ; Melc08 .

In particular, we shall highlight some interesting features emerging from the analysis of recent data (circa 2017), namely: $(i)$ an increasing sensitivity to the mass ordering, with NO generally favored over IO at the $\sim 2\sigma$ level; $(ii)$ differences in the allowed parameter space arising when such NO-IO offset is (not) taken into account; $(iii)$ synergies between bounds on $m_{\beta\beta}$ and $\Sigma$ of comparable strength, especially for best-fit values of $\Sigma$ far from the extrema in Eq. (3). The discussion of such features also allows to gauge the impact of prospective (non)oscillation bounds—or signals—on our knowledge of the absolute neutrino mass spectrum and its associated observables.

Our work is structured as follows. In Sec. II A, B, and C, we report and discuss detailed constraints coming from separate analyses of oscillations, $m_{\beta\beta}$, and $\Sigma$, respectively. In Sec. III we perform a combined analysis in the $(m_{\beta\beta},\,\Sigma)$ parameter space for representative cosmological data sets. In Sec. IV we discuss the implications of such results for upcoming or prospective experiments sensitive to $m_{\beta}$. A brief summary is presented in Sec. V.

II Data sets, statistical analysis, and parameter bounds

In this paper, all the bounds on the mass-mixing parameters coming from various (separate or combined) data sets are expressed in terms of $\Delta\chi^{2}$ differences with respect to a minimum $\chi^{2}$ value. In particular, the differences

[TABLE]

are used to derive $n\sigma$ allowed regions. Projections of such regions onto a single parameter provide the $\pm n\sigma$ range(s) for that parameter Stat16 . In all figures, it is understood that the undisplayed parameters are projected away (i.e., marginalized). Following the general statistical arguments in Blen13 , we use the $\Delta\chi^{2}$ metric also to assess the relative likelihood of the two mass-ordering hypotheses,

[TABLE]

From a hystorical viewpoint, an interesting parallel to the metric in Eq. (8) can be found in the development of solar neutrino data analyses. In the early literature, different and seemingly disconnected oscillation solutions to the solar neutrino problem (e.g., the so-called matter and vacuum solutions) were often analyzed via separate fits around the corresponding $\chi^{2}$ minima in the $(\delta m^{2},\,\sin^{2}2\theta_{12})$ parameters (see, e.g., Bahc98 ; Petc98 ), and possibly compared to each other by tests of hypotheses. However, when diverse solutions were explicitly connected in the $(\delta m^{2},\,\sin^{2}\theta_{12})$ or $(\delta m^{2},\,\tan^{2}\theta_{12})$ variables Mont96 ; Mura00 ; Frie00 ; Lisi00 , it became customary to expand the fit around the absolute $\chi^{2}$ minimum and to compare different solutions by a $\Delta\chi^{2}$ parameter test Pena01 ; Gett02 ; PDG2016 —until a single one was eventually found by solar and long-baseline reactor experiments Bahc04 .

The comparison of the two mass orderings seem to follow an analogous path. On the one hand, one may take NO and IO as two alternative options, involving separate fits and tests of hypotheses. On the other hand, one may try to connect them through a continuous variable, involving a parameter estimation test. Such a variable could be either physical (e.g., $\Delta m^{2}$, ranging from negative to positive real values) or unphysical (e.g., a fudge parameter $p\in[0,\,1]$ linking any two competing hypotheses Lyons ). Explicit parametric connections have been worked out for medium-baseline reactor neutrino oscillations, in terms of the mixing variable $\sin^{2}\theta_{12}$ (swapping octants between NO and IO for $\Delta m^{2}>0$ in vacuum Bile17 ) and of an empirical variable $\alpha$ (ranging in $[-1,\,+1]$ from IO to NO Juno14 ). The above considerations further support our adoption of Eq. (8) as a reasonable metric for the IO–NO discrimination Blen13 , akin to a one-parameter estimation test. For a discussion of further statistical issues and possible alternative approaches, see also Patt15 ; Qian15 ; Ciuffoli ; Blennow ; Stanco and refs. therein.

With present data, the current statistical sensitivity associated to $\Delta\chi^{2}_{\mathrm{IO-NO}}$ tests appears to be limited to $\sim\!2\sigma$ (see Sec. III). Therefore, we shall conservatively report $\Delta\chi^{2}$ bounds on mass-mixing parameters both by separately minimizing the $\chi^{2}$ in NO and IO (discarding the relative $\Delta\chi^{2}_{\mathrm{IO-NO}}$ difference), and by further minimizing the $\chi^{2}$ over any ordering (including the $\Delta\chi^{2}_{\mathrm{IO-NO}}$ information), with a discussion of the relative differences in the results. Such a format has been adopted in presenting the oscillation parameter ranges in Conc17 ; NuFit , and is extended herein to nonoscillation parameters.

II.1 Neutrino oscillations

An analysis of neutrino oscillation data has been previously presented in Capo16 , to which we refer the reader for a discussion of the adopted methodology and earlier literature. A partial update of Capo16 , including novel accelerator data shown in mid-2016, was reported in Marr16 . The more complete update presented herein (circa 2017) includes, with respect to Capo16 : $(i)$ the latest results from the long-baseline accelerator experiments T2K T2K2017 and NOvA NOvA2016 ; NOvA2017 ; $(ii)$ the latest far/near spectral ratio from the reactor neutrino experiment Daya Bay DB2017 ; $(iii)$ the most recent atmospheric neutrino data from the Super-Kamiokande (SK) phase IV SKatm1 ; SKatm2 . The results of our oscillation data analysis are reported graphically in Fig. 1 and numerically in Table I.

Figure 1 shows the $\chi^{2}$ curves in terms of the six oscillation parameters $(\delta m^{2},\,\Delta m^{2},\,\sin^{2}\theta_{12},\,\sin^{2}\theta_{13},\,\sin^{2}\theta_{23},\,\delta)$, for both NO (blue) and IO (red). We find an overall preference for NO, quantified by the $\chi^{2}$ difference

[TABLE]

that is explicitly shown as an offset of the IO curves. The offset is of some relevance in the analysis of absolute mass observables, as shown later.

Table I reports best-fit values and parameter ranges for separate $\chi^{2}$ minimization in each separate ordering (NO and IO) and in any ordering; the latter case takes into account the above $\Delta\chi^{2}_{\mathrm{IO-NO}}$ value. The known parameters $(\delta m^{2},\,|\Delta m^{2}|,\,\sin^{2}\theta_{12},\,\sin^{2}\theta_{13})$, which affect the absolute mass observables in Eqs. (4)–(6), are determined with a fractional $1\sigma$ accuracy (defined as 1/6 of the $\pm 3\sigma$ range) of $(2.3,\,1.6,\,5.8,\,4.0)$ percent, respectively. For such parameters, it turns out that minimization in any ordering reproduces the same allowed ranges as for NO. Given the $\delta m^{2}$ and $\Delta m^{2}$ estimates in Table I, Eq. (3) becomes

[TABLE]

The parameter $\sin^{2}\theta_{23}$ is less well known, at the level of 9.6%. At $\!3\sigma$, its octant degeneracy is unresolved, and maximal mixing is also allowed. At lower significance, maximal mixing is disfavored in both NO and IO, and the first octant is preferred in NO. The $n\sigma$ ranges for $\theta_{23}$ for any ordering are larger than for NO (Table I), as a result of joining the NO and IO intervals determined by the curves in the right-lower panel of Fig. 1 at $\chi^{2}=n^{2}$. Concerning the possible CP-violating phase $\delta$, our analysis strengthen the trend in favor of $\delta\sim 3\pi/2$ Capo16 ; Conc17 ; T2K2017 , and disfavors ranges close $\delta\sim\pi/2$ at $\gtrsim 3\sigma$. In any case, the parameters $\theta_{23}$ and $\delta$ do not enter in the calculation of ($m_{\beta},\,m_{\beta\beta},\,\Sigma$).

A few remarks are in order about the IO-NO offset in Eq. (9). This value is in the ballpark of the official SK fit results quoted in SKatm1 ; SKatm2 , namely: $\Delta\chi^{2}_{\mathrm{IO-NO}}=4.3$ (for SK data at fixed $\theta_{13}$) and $\Delta\chi^{2}_{\mathrm{IO-NO}}=5.2$ (for SK + T2K data at fixed $\theta_{13}$). By excluding SK atmospheric data in our fit, we find $\Delta\chi^{2}_{\mathrm{IO-NO}}=1.1$, in qualitative accord with the official T2K data analysis constrained by reactor data T2K2017 .

Concerning SK atmospheric data, it has been emphasized Capo16 ; Conc17 ; Vall14 that the complete set of bins and systematics SKatm1 ; SKatm2 can only be handled within the collaboration, especially when $\nu/\overline{\nu}$ or multi-ring event features are involved. Nevertheless, we think it useful to continue updating our analysis of reproducible SK samples, namely, sub/multi-GeV single-ring ($e$-like and $\mu$-like) and stopping/through-going ($\mu$-like) distributions. These samples encode interesting (although entangled and smeared) pieces of information about subleading effects driven by known and unknown oscillation parameters, see e.g. Glob06 ; in particular, they contributed to early hints of nonzero $\theta_{13}$ Hint . At present, we trace the atmospheric hint of NO to $e$-like events, especially multi-GeV, in qualitative agreement with Kearns .111Note, however, that weaker results for the IO-NO difference ($\lesssim 1\sigma$), with or without atmospheric data, have been found in Conc17 .

Summarizing, the SK(+T2K) official results in SKatm1 ; SKatm2 ; T2K2017 and ours in Eq. (9) suggest, at face value, that global $3\nu$ oscillation analyses may have reached an overall $\sim\!2\sigma$ sensitivity to the mass ordering, with a preference for NO driven by atmospheric data and corroborated by accelerator data, together with reactor constraints. This intriguing indication, although still tentative, is generally supported by cosmological data (see Sec. II C) and thus warrants a dedicated discussion in the context of absolute $\nu$ mass observables (see Sec. III).

II.2 Neutrinoless double beta decay

If the three known neutrinos are Majorana fermions, the rare process of $0\nu\beta\beta$ decay is expected to occur with half life $T$ given by

[TABLE]

where $m_{\beta\beta}$ is given in Eq. (5), $G$ is the (calculable) phase space and $M$ is the nuclear matrix element (NME) for a candidate nucleus Viss16 ; Giun15 ; Paes15 ; Verg16 .

A worldwide search is underway to find possible $0\nu\beta\beta$ decay signatures in a variety of nuclei, and lower limits on the corresponding half lives have been placed Schw13 ; Crem14 ; Bara17 . Transforming lower bounds on $T$ into upper bounds on $m_{\beta\beta}$ requires theoretical input on the NME and their uncertainties Viss16 ; Giun15 ; Paes15 ; Verg16 ; Voge12 ; Suho15 ; Enge17 . The strongest $m_{\beta\beta}$ limit to date is provided by the KamLAND-Zen experiment with 136Xe, that finds $T>1.07\times 10^{26}$ yr (90% C.L.), and derives the range $m_{\beta\beta}\gtrsim 0.061$–0.165 eV (90% C.L.) by bracketing recent NME calculations KamZ16 . For the sake of simplicity, we include only the (dominant) KamLAND-Zen constraints herein.

In order to derive bounds at any given C.L. in our analysis, we build a general $\chi^{2}(m_{\beta\beta})$ function by using: $(i)$ the experimental $\chi^{2}(T)$ curve presented by the KamLAND-Zen collaboration in KLchi2 [with $T=T(m_{\beta\beta},\,|M|)$ from Eq. (11)]; and $(ii)$ our construction of the $\chi^{2}(|M|)$ function, based on the conservative theoretical uncertainties estimated in Rotu15 . The objective function is obtained as

[TABLE]

and is shown in Fig. 2. These results do not depend on the mass ordering and, in particular, $\Delta\chi^{2}_{\mathrm{IO-NO}}=0$.

From Fig. 2 we get $m_{\beta\beta}<0.15$ eV at 90% C.L., close to the most conservative limit quoted at the same C.L. in KamZ16 ($0.165$ eV). From Fig. 2 we also derive

[TABLE]

For completeness, our assessment of the $\chi^{2}(m_{\beta\beta})$ function is detailed below.

According to Rotu15 , Eq. (11) is linearized via logarithms as $\tau=\gamma-2\eta-2\mu$, where: $\gamma=-\log_{10}(G/\mathrm{y}^{-1}\mathrm{eV}^{-2}$), $\tau=\log_{10}(T/y)$, $\eta=\log_{10}(|M|)$, and $\mu=\log_{10}(m_{\beta\beta})$. The NME $\eta$ and its uncertainties with respect to a central value $\overline{\eta}$ are parametrized as $\eta=\overline{\eta}+\alpha(g_{A}-1)+s\beta\pm\sigma$, where $g_{A}$ is the effective axial coupling (typically “quenched” with respect to the vacuum value $g_{A}^{0}\simeq 1.27$ Viss16 ), $s=\pm 1$ switches between two alternative approaches to short-range correlation effects (so-called CD-Bonn and Argonne potentials), while $\sigma$ is a residual (nonparametric) uncertainty Rotu15 . Numerical values for 136Xe are: $\gamma=24.865$, $\overline{\eta}=0.267$, $\alpha=0.458$, $\beta=0.021$, $\sigma=0.032$.

Concerning the total $1\sigma$ uncertainty $\sigma_{\eta}$ affecting $\eta$, we assume $g_{A}=1\pm 0.15$ as a reasonable $1\sigma$ estimate for the axial coupling. The central value corresponds to default quenching $(g_{A}=1)$ Viss16 , and the $2\sigma$ range $g_{A}\in[0.7,\,1.3]$ spans typical effective values adopted in the NME literature up to $g_{A}^{0}$ Verg16 , while the $3\sigma$ range goes down to $g_{A}=0.55$, close to the very low estimates $g_{A}\simeq 1.27A^{-0.18}\simeq 0.52$ considered in Viss16 ; Bare15 (for $A=136$). Concerning alternative short-range correlation approaches, we conservatively assume that the associated uncertainty ($s\beta=\pm\beta$) corresponds statistically to $\pm 1\sigma$. The total error $\sigma_{\eta}$ is then evaluated by summing in quadrature the three independent components, namely, $\alpha\cdot 0.15=0.069$, $\beta=0.021$ and $\sigma=0.032$, leading to $\eta=\overline{\eta}\pm\sigma_{\eta}=0.267\pm 0.079$ $(1\sigma)$. Finally, we minimize over $\eta$ according to Eq. (12), where the second term on the right-hand side is given by $\chi^{2}_{\eta}=[(\eta-\overline{\eta})/\sigma_{\eta}]^{2}$. Our estimate $\eta=0.267\pm 0.079$ implies a $\pm 3\sigma$ range $|M|\simeq 1.1$–3.2, to be compared with the total range $|M|\simeq 1.6$–4.3 adopted in KamZ16 . The overall shift is mainly related to a different choice for the default axial coupling ($g_{A}=1.27$ in KamZ16 versus $g_{A}=1$ herein). In any case, both ranges correspond to a conservative factor of $\sim\!3$ uncertainty of $|M|$.

The above results refer to a single (dominant) experimental datum for the 136Xe nucleus. When comparable bounds on $m_{\beta\beta}$ will be obtained in other experiments and nuclei, the combination of various $0\nu\beta\beta$ data should take into account the theoretical NME covariances among different nuclei QRPA08 .

II.3 Cosmology

Neutrinos are the only known particles in the standard model of particle physics that can change behaviour, from the relativistic to the non-relativistic regime, in an epoch after cosmic microwave background (CMB) recombination. This change leaves a characteristic imprint on several cosmological observables (see, e.g. Hann10 ; Hu:1997mj ; Dolgov:2002wy ; Lesgourgues:2006nd ; Abazajian:2011dt ; Lesgourgues:2012uu ), letting cosmology to strongly bound the neutrino mass scale, indeed providing the current strongest (albeit model dependent) bounds on $\Sigma$. Bounds on $\Sigma$ from recent cosmological data have been presented in several papers (see, for example, Vagnozzi:2017ovm ; Giusarma:2016phn ; Xu:2016ddc ; Huang:2015wrx ; DiValentino:2015sam ; Cuesta:2015iho and references therein) while forecasts for near (and far) future cosmological datasets have been obtained in Archidiacono:2016lnv ; Oyama:2015gma ; Allison:2015qca ; DiValentino:2016foa ; Banerjee:2016suz ; Hamann:2012fe .

Clearly, current cosmological constraints on neutrino masses depend on the combination of datasets considered and on the theoretical framework assumed (see, for example, DiValentino:2015ola ; DiValentino:2016hlg ; Motohashi:2012wc ; Hu:2014sea ; Wang:2016tsz ; Zhang:2015uhk ). It is therefore important to be extremely clear in the description of the assumptions we make. In our analysis we consider $6$ different combinations of the following datasets:

• •

The full range of the Planck 2015 temperature anisotropy angular power spectrum, both at low multipole $\ell$ ($2\leq\ell\leq 29)$ and high $\ell$ ($30\leq\ell\leq 2508)$, provided by the Planck collaboration Aghanim:2015xee . We define this dataset as Planck TT.

• •

The full multipole range of the Planck 2015 temperature anisotropy angular power spectrum, and high multipoles E polarization and cross TE temperature polarization anisotropy angular power spectra ($30\leq\ell\leq 2508)$) Aghanim:2015xee . We define this dataset as Planck TT, TE, EE.

• •

A gaussian prior on the reionization optical depth $\tau=0.055\pm 0.009$, as obtained recently from Planck HFI data newtau . We refer to this prior as $\tau_{\mathrm{HFI}}$.

• •

The Baryon Acoustic Oscillation measurements from 6dFGS beutler2011 , SDSS-MGS ross2014 , BOSSLOWZ anderson2014 and CMASS-DR11 anderson2014 surveys as done in planckparams2015 . We label this dataset as BAO.

• •

The Planck 2015 CMB lensing potential power spectrum reconstruction data Ade:2015zua . We refer to this dataset as “lensing”.

In our analysis we always consider a flat universe, a cosmological constant and adiabatic primordial perturbations, within the so-called $\Lambda$CDM model. However, we consider two slightly different theoretical scenarios:

• •

The “standard” $6+1$ parameters of the $\Lambda$CDM + $\Sigma$ model, where the $6$ parameters of the $\Lambda$CDM are the baryon and Cold Dark Matter densities $\omega_{b}$ and $\omega_{cm}$, the amplitude $A_{s}$ and spectral index $n_{s}$ of primordial density fluctuations, the Hubble constant $H_{0}$ and the reionization optical depth $\tau$, and the seventh extra parameter $\Sigma$ is also free;

• •

An extended $6+2$ parameter scenario, considering variation also in the lensing amplitude $A_{\mathrm{lens}}$ that controls the effects of gravitational lensing in the Planck TT, TE and EE angular spectra cala . This parameter is expected to be $A_{\mathrm{lens}}=1$ in the standard $\Lambda$CDM model. However the most recent Planck data analysis shows a statistically significant preference for values $A_{\mathrm{lens}}>1$ (in particular, $A_{\mathrm{lens}}=1.15_{-0.12}^{+0.13}$ at $\pm 2\sigma$) newtau . While the physical motivations behind this result are not yet clear (systematics or new physics) we consider also this parameter as free, since its correlation with $\Sigma$ strongly weakens the cosmological constraints on neutrino masses. The scenario with extra $A_{\mathrm{lens}}$ parameter is therefore expected to yield more conservative results.

The cosmological constraints are obtained using the November 2016 version of the publicly available Monte-Carlo Markov Chain package cosmomc Lewis:2002ah ; Lewis:1999bs , with a convergence diagnostic based on the Gelman and Rubin statistic, that implements an efficient sampling of the posterior distribution using the fast/slow parameter decorrelations Lewis:2013hha , and that includes the support for the Planck data release 2015 Likelihood Code Aghanim:2015xee (see http://cosmologist.info/cosmomc/). We emphasize that we implement separately the NO and IO options in the CosmoMC analysis, namely, the masses $m_{i}$ entering in the definition of $\Sigma$ obey the $\delta m^{2}$ and $\Delta m^{2}$ constraints in Eqs. (1,2). In particular, from the fit results in Table I and Eq. (10), it is

[TABLE]

Such approach differs from other recent studies, where neutrino masses are assumed to be degenerate ($m_{i}=m\geq 0$) and the above constraints are relaxed ($\Sigma\geq 0$). In such studies, best-fit results around $\Sigma\simeq 0$ (i.e., in the unphysical region) tend to induce a somewhat artificial preference for NO over IO, just because NO allows $\Sigma$ values lower than IO. For any scenario and combination of cosmological datasets, our CosmoMC fit leads, in general, to different best-fit values for $\Sigma$ (in the physical region) and for the associated values of $\chi^{2}_{\min}$ in NO and IO. The value of $\Delta\chi^{2}_{\mathrm{IO-NO}}$ correctly quantifies the overall preference of the fitted cosmological data set for one mass ordering.

From CosmoMC one also gets the posterior probability functions $p(\Sigma)$ in NO and IO, which are transformed into $\chi^{2}(\Sigma)$ functions by applying Stat16 the standard Neyman construction Neym67 and Feldman-Cousins method Feld98 . We have also verified that, for any given cosmological data set, the resulting $\chi^{2}(\Sigma)$ curves for NO and IO converge for increasing $\Sigma$ as they should (up to residual numerical artifacts at the level of $\delta\chi^{2}\lesssim 0.1$). The $\chi^{2}$ analysis of cosmological data is thus methodologically consistent with the $\chi^{2}$ analysis of oscillation and $0\nu\beta\beta$ data, and a global combination of the data can be performed (see next Section).

The main cosmological fit results are summarized in Table 2, in terms of upper bounds (at $2\sigma$ level) on the sum of neutrino masses $\Sigma$ for NO and IO, together with the $\Delta\chi^{2}_{\mathrm{NO-IO}}$ offset. The results show some global trends: (a) the $\Sigma$ bounds are significantly strengthened by enlarging the Planck temperature data with polarization spectra or with BAO data, while they are only moderately tightened by adding lensing data; (b) the bounds are largely weakened, up to a factor of $\sim\!2$ in some cases, by letting $A_{\mathrm{lens}}$ free.

At a finer level, slight differences emerge between the results in NO and IO in Table II, indicating a weak sensitivity of cosmological data to the mass ordering. Interestingly, normal ordering is generally preferred, except for a few cases where $\Delta\chi^{2}_{\mathrm{NO-IO}}$ is either negligible or slightly negative, corresponding to the extended and conservative scenario in which the $A_{\mathrm{lens}}$ parameter is varying. The overall indication in favor of NO, although still at the $\lesssim 1\sigma$ level, is consistent with the neutrino oscillation results in Eq. (9), and brings the global preference for NO at the typical level of $\Delta\chi^{2}\simeq 4$ in our analysis (i.e., $2\sigma$). Note that the overall preference for NO from cosmological data exceeds $1\sigma$ only in the cases #3 and #6 of Table 2 that, not surprisingly, are associated with the strongest constraints on the sum of neutrino masses ($\Sigma\lesssim 0.2$ eV at $2\sigma$), that are arising when using the BAO data, since they are directly sensitive to the free-streaming nature of the neutrinos. Moreover, the constraints of these two cases are not affected by the lowering of the optical depth, as it happens for the other combination of datasets if compared to the Planck 2015 findings planckparams2015 , showing that the BAO bounds are very robust and reliable.

Further details can be appreciated in terms of the $\chi^{2}(\Sigma)$ functions for NO and IO. For the sake of simplicity, we do so only for four representative cases numbered as #10, #1, #9, and #6 in Table 2, as shown in Fig. 3. These cases lead to increasingly strong upper bounds on $\Sigma$, ranging from $\lesssim 1$ eV (#10) to $\lesssim 0.2$ eV (#6) at $2\sigma$. The corresponding offset $\Delta\chi^{2}_{\mathrm{IO-NO}}$ ranges from $\sim 0$ to $1.6$. In two cases (#1 and #6) the $\chi^{2}$ minima are reached at the extrema of $\Sigma$ from Eq. (14), while in the other two (#10 and #9) they are reached at higher values. In general, we find that the cases with $A_{\mathrm{lens}}$ free lead to best-fit values of $\Sigma$ above the extrema in Eq. (14) (not shown). We emphasize that, in all cases, the $\chi^{2}(\Sigma)$ curves for NO and IO tend to converge for degenerate masses $m_{i}$ (large $\Sigma$), while they bifurcate towards the extrema in Eq. (14), corresponding to strongly hierarchical masses at low $\Sigma$. The four cases in Fig. 3 are sufficiently representative of the variety of constraints set by current cosmological data in Table 2, and will be explicitly considered in the global analysis of oscillation and non-oscillation neutrino data in the next Section.

III Combined constraints in the $\bm{(\Sigma,\,m_{\beta\beta})}$ plane

In this Section we present increasingly strong constraints on the absolute mass observables ${(\Sigma,\,m_{\beta\beta})}$ in the (sub)eV range, obtained by combining the $\chi^{2}$ from oscillation data (Fig. 1) with the $\chi^{2}$ from $0\nu\beta\beta$ (Fig. 2) and then with the $\chi^{2}$ from cosmological data (Fig. 3). Current $\beta$-decay constraints ($m_{\beta}\lesssim 2$ eV PDG2016 ) are not relevant in this context.

As discussed in Sec. II, we consider two alternative ways to obtain allowed regions: $(i)$ the $\chi^{2}$ is separately minimized on the relevant parameters in each mass ordering, either NO or IO, discarding the $\Delta\chi^{2}_{\mathrm{IO-NO}}$ information; and $(ii)$ the $\chi^{2}$ is further minimized over NO and IO, including the $\Delta\chi^{2}_{\mathrm{IO-NO}}$ information. In the former case, one should consider the NO and IO allowed regions as exclusive while, in the latter case, one should join the NO and IO allowed regions to obtain the global ones in “any ordering”.

Figure 4 shows the $2\sigma$ and $3\sigma$ constraints in the ${(\Sigma,\,m_{\beta\beta})}$ plane, derived from the oscillation data discussed in Sec. II A. The left panel refers to separate fits in each mass ordering, while the right panel to the global fit in any ordering. The main features of the allowed bands have been discussed in previous literature (see Melc04 ; Melc06 ; Rodejo1 ; DellOro1 ; DellOro2 ; Zavanin and refs. therein) and are not repeated here. We only recall that the vertical width of the bands is mainly related to the unknown Majorana phases, while the oscillation parameter uncertainties play a secondary role, that can be appreciated via the difference between the $2\sigma$ and $3\sigma$ allowed regions. By comparing the left and right panels in Fig. 4, one can notice that the NO regions are identical, while the IO region is slightly reduced on the right, due to the offset of the $\chi^{2}$ minimum for IO in Eq. (9).

Figure 5 is similar to Fig. 3, but includes the $0\nu\beta\beta$ constraints discussed in Sec. II B. In the left panel, both the NO and the IO allowed bands are horizontally cut at the $m_{\beta\beta}$ values in Eq. (13). In the right panel, the upper bounds on $m_{\beta\beta}$ are stronger in IO, and can be estimated by drawing in Fig. 2 the lines at $\chi^{2}=n^{2}-3.6$ ($n=2,\,3$), where 3.6 corresponds to the offset in Eq. (9). Note that, in the left panel, the projections of the NO and IO allowed regions onto the abscissa lead to upper bounds on $\Sigma$ well above 1 eV.

Figure 6 includes, besides oscillation and $0\nu\beta\beta$ constraints, also the cosmological bounds for the case #10 in Sec. II C (see Table II). The left panel shows a synergic effect of $0\nu\beta\beta$ and cosmological data in setting a joint $2\sigma$ bound on $\Sigma$ at the level of $0.9$ eV, to be compared with the $0\nu\beta\beta$ bound (from Fig. 5) and the cosmological bound (from Table II, case #10), which are both above 1 eV. A more subtle synergy emerges from the fact that, for case #10, the $\chi^{2}(\Sigma)$ function is minimized at $\sim 0.3$ eV (see Fig. 3), well above the extrema in Eq. (14). Such a (relatively high) best-fit value for $\Sigma$ implies preferred values $m_{\beta\beta}$ around $\mathrm{few}\times 10^{-2}$ eV, as apparent for the IO region allowed at $2\sigma$ in the right panel. This relatively small IO $2\sigma$ region illustrates qualitatively how the constraints on ${(\Sigma,\,m_{\beta\beta})}$ would appear in the presence of a cosmological measurement (rather than of just upper bounds) for $\Sigma$.

Figures 7, 8 and 9 are analogous to Fig. 6, but refer to the cosmological data sets #1, #9, and #6 discussed in Sec. III C, respectively. Figure 7 shows, once more, the synergy between comparable $0\nu\beta\beta$ and cosmological bounds on $\Sigma$; indeed, in the left panel, one reads $\Sigma<0.65$ eV, to be compared with $\Sigma<0.72$–0.8 eV from cosmological data only (see Table II). In the right panel, there is no IO region allowed at $2\sigma$, since the sum of the $\Delta\chi^{2}_{\mathrm{IO-NO}}$ contributions from oscillation and cosmological data is $3.6+0.7>4.0$.

Figure 8 shows, in the left panel, the transition to a dominance of cosmological constraints on $\Sigma$: the $2\sigma$ bounds $\Sigma<0.45$-046 eV for case #9 in Table II keep $m_{\beta\beta}$ sufficiently small to suppress any significant impact of current $0\nu\beta\beta$ data in the global fit. In the right panel, the relatively deep minimum of $\chi^{2}(\Sigma)$ evident in Fig. 3 leads to a $2\sigma$ allowed region in IO narrower than in Fig. 6. This region shows qualitatively the impact of prospective accurate measurement of $\Sigma$ via cosmological data.

Figure 9 corresponds to the most constraining cosmological case (#6) in Table II. In this case, the allowed bands are almost vertically cut by the upper bounds on $\Sigma$ from cosmological data only, with no significant contribution from $0\nu\beta\beta$ constraints. Indeed, the allowed values of $m_{\beta\beta}$ are well below the $0\nu\beta\beta$ bounds in Eq. (13). Note that, in the right panel, there is no region allowed at $2\sigma$, since the the global $\Delta\chi^{2}_{\mathrm{IO-NO}}$ exceeds 4 units.

Table III reports the list of global $\Delta\chi^{2}_{\mathrm{IO-NO}}$ values, numbered according to the cosmological cases in Table II. These values are not always equal to the algebraic sum of the $\Delta\chi^{2}$ contributions from oscillation data in Eq. (9) and cosmological data in Table II, since the best-fit points in the plane $(\Sigma,\,m_{\beta\beta}$) may be slightly readjusted in NO and IO in the global combination, leading to a small extra variation ($\delta\chi^{2}\lesssim 0.4$). This minor effect in the combination of $0\nu\beta\beta$ and cosmological data is statistically insignificant at present, but might become more relevant with future data. In any case, Table III confirms that an overall preference for NO over IO emerges from the combination of oscillation and nonoscillation data, at the level of $1.9$–$2.1\sigma$. This is one of the main results of our work.

We conclude this Section with a remark on $m_{\beta\beta}$. In the above figures, the $2\sigma$ upper bounds on $m_{\beta\beta}$ decrease from $<0.18$ eV in Fig. 6 (dominated by KamLAND-Zen) to $<0.06$ eV in Fig. 9 (dominated by cosmology). There are good prospects to further probe this region —and possibly go below it— with upcoming or planned $0\nu\beta\beta$ experiments Schw13 ; Crem14 ; Bara17 . However, unlike $\Sigma$, there is no finite lower bound on $m_{\beta\beta}$, since the null value cannot be excluded a priori for unfavorable Majorana phases (see Xing and refs. therein). Conversely, a signal of $m_{\beta\beta}>0$, if accurate enough, might provide some hints or even constraints on such phases (see, e.g., Quiroga ). The identification of Majorana phases as a new source of leptonic CP violation (besides the Dirac phase $\delta$) would open new perspectives on the role of leptons in the early universe (see Riotto and refs. therein).

IV Implications for $\bm{m_{\beta}}$

The results obtained in the previous Section have implications for the discovery potential of $\beta$-decay searches, such as the experiment KATRIN Katrin1 ; Katrin2 ; Katrin3 , designed to probe the range $m_{\beta}\gtrsim 0.2$ eV, or future projects, envisaged to reach potential sensitivities at or below 0.1 eV NuMass ; Gastaldo ; Project8 . For the sake of brevity, we consider only the case of global fit in any ordering and for two representative cosmological data sets, namely, #10 and #6 in Table II, that lead to conservative and aggressive bounds on $\Sigma$, respectively.

Figure 10 shows the bounds on any two among the three absolute mass observables $(\Sigma,\,m_{\beta\beta},\,m_{\beta})$ for case #10. The $(\Sigma,\,m_{\beta\beta})$ plane is identical to the right panel of Fig. 6, while the other two planes contain also the projected bounds on $m_{\beta}$. The allowed values of $m_{\beta}$ extend up to $\sim 0.3$ eV ($2\sigma$) and $\sim 0.4$ eV ($3\sigma$), in the range testable by KATRIN; however, a large fraction of the $m_{\beta}$ allowed range, including the preferred IO region at $2\sigma$, is below the $0.2$ eV sensitivity goal of this experiment.

Fig. 11 is analogous to Fig. 10, but refers to the cosmological data set #6. In this case, the upper bound on $\Sigma$ is very strong, and so is the bound on $m_{\beta}$. Indeed, in the $(\Sigma,\,m_{\beta})$ plane, the two allowed branches for NO and IO are completely disconnected and could, in principle, be conclusively discriminated via precise measurements of $\Sigma$ and $m_{\beta}$. Unfortunately, the values of $m_{\beta}$ required by such test are entirely below the KATRIN sensitivity Katrin1 but, in the long term, they could be partly probed by planned or envisaged experimental projects NuMass ; Gastaldo ; Project8 .

V Conclusions

We have performed a global analysis of oscillation and nonoscillation data within the standard three-neutrino framework. with particular attention to absolute neutrino masses and their ordering (either normal, NO, or inverted, IO). Oscillation data have been updated with the latest results, as available at the beginning of 2017. $0\nu\beta\beta$ decay bounds have been derived by using recent results from the KamLAND-Zen experiment, together with a conservative evaluation of nuclear matrix elements and their uncertainties. Cosmological data from Planck and other experiments have been examined within the standard $\Lambda$CDM model, with allowance for nonzero neutrino masses (and eventually for an extra parameter). The cosmological analysis has been performed in a variety of cases, always considering the physical neutrino mass spectra for NO and IO.

In the global analysis, NO appears to be somewhat favored with respect to IO at the level of $1.9$–$2.1\sigma$, mainly by neutrino oscillation data (especially atmospheric), corroborated by cosmological data in some cases. This intriguing indication, although not statistically mature yet, deserves to be monitored with future data. Detailed constraints on neutrino mass-mixing parameter have also been obtained via the $\chi^{2}$ method, by expanding the parameter space either around separate minima in NO and IO, or around the absolute minimum in any ordering. Relevant results have been numerically summarized in Tables I–III and graphically shown in several figures. Implications for upcoming oscillation and non-oscillation neutrino experiments, including $\beta$-decay searches, have been discussed.

We emphasize that the above results have been obtained in the standard $3\nu$ framework of massive and mixed neutrinos. The experimental search of oscillation phenomena, as well as of signals in the $(m_{\beta},\,m_{\beta\beta},\,\Sigma)$ parameter space should, however, be pursued independently of any $3\nu$ expectations, which can be altered by new (sterile) neutrino states or by new (nonstandard) neutrino interactions, not considered in this work.

Acknowledgements.

This work is supported by the Italian Istituto Nazionale di Fisica Nucleare (INFN) and Ministero dell’Istruzione, Università e Ricerca (MIUR) through the “Theoretical Astroparticle Physics” projects. This work has also been performed within the Labex ILP (reference ANR-10-LABX-63) part of the Idex SUPER, and received financial state aid managed by the Agence Nationale de la Recherche, as part of the programme Investissements d’Avenir under the reference ANR-11-IDEX-0004-02. A.P. is supported by the project Beyond three neutrino families within the FutureInResearch program, Fondo di Sviluppo e Coesione 2007-2013, APQ Ricerca Regione Puglia “Programma regionale a sostegno della specializzazione intelligente e della sostenibilità sociale ed ambientale”. F.C. is supported by NSF Grant PHY-1404311 to J.F. Beacom. E.L. thanks K. Inoue and Y. Koshio for communications about the recent KamLAND-Zen and Super-Kamiokande results, respectively.