On the Nature of Massive Neutrinos

1. Introduction

The possibility that neutrinos oscillate has been predicted in 1957 by Bruno Pontecorvo [1] in a two component model involving one neutrino and one antineutrino. In a paper published in 1962 [2] , Maki, Nakagawa, and Sakata conjectured that the observed neutrino is actually a superposition of several states with different mass eigenvalues. They then developed an initial version of the PMNS matrix describing the two known flavors. The PMNS version with 3 flavours explains well quantitatively the observed neutrino oscillations [3]. The experimental observation of solar [4] neutrino oscillations is sensitive to the mixing angle ${\theta }_{12}$ and the squared masses difference $\mathsf{\Delta }{m}_{12}^2$ between the two neutrino mass eigenstates 1 and 2. Atmospheric [5] neutrinos measure the mixing angle ${\theta }_{23}$ and the squared masses difference $\mathsf{\Delta }{m}_{23}^2$ between the two neutrino mass eigenstates 2 and 3. Neutrino sources from reactor [6] and accelerator [7,8,9] are sensitive to the mixing angle ${\theta }_{13}$ and the CP violating phase $\delta$. Most PMNS parameters are fixed within less than $5\%$. The mass hierarchy (NH with mass eigenstate ${\nu }_1$ lightest or IH with mass eigenstate ${\nu }_3$ lightest) remains an open question.

Nevertheless, for NH, CP conservation is favored. But maximal CP violation is favoured for IH. There is no clear preference for ${\theta }_{23}$ octant.

Recent cosmological constraints on the sum of neutrino masses ($\mathsf{\Sigma }{m}_{\nu }$) already disfavor the inverted mass hierarchy and allow very little parameter space even for normal hierarchy [10]. This DESI result is based on a combination of their data sets on Baryon Acoustic Oscillations (BAO) [11] together with the Planck CMB 2018 data [12] and the lensing data from ACT [13]. In April 2024, the DESI collaboration [10] presented the most stringent bound on the sum of neutrino masses: $\sum m_v<0.072eV[95\%CL]$.

In [14] these constraints have been critically assessed investigating their robustness against different statistical methods and relaxing cosmological assumptions. The cosmological neutrino mass constraints are substantially relaxed if the background dynamics are allowed to deviate from flat $\Lambda$CDM.

The KATRIN experiment performs precision spectroscopy of the tritium $\beta$-decay close to the kinematic endpoint. On the basis of the first five measurement campaigns, the experiment derived an upper limit on the effective electron neutrino mass of $0.45$ eV [$90\%$ CL] [15].

Lower bounds derived from the neutrino oscillations give [16]: $\sum m_{\nu }>0.058eV$ in NH and $\sum m_{\nu }>0.098eV$ in IH.

In this work, we will first use the Nu- Fit parameters extracted from neutrino oscillations [16] and the cosmological bounds to deduct the allowed space for $\sum m_{\nu }$ as a function of the lightest neutrino mass in the normal mass ordering and in the inverted ordering .

In the next sections we will assume that neutrinos are of Majorana type. We will study in section 2 the sensitivity to the Majorana phases of the effective electron neutrino mass $|m_{ee}|$ measured in $0\nu \beta \beta$ decay experiments . Section 3 is devoted to the sensitivity of muon neutrino antineutrino oscillations experiments to Majorana phases. Conclusions are given in section 4.

2. Status on Neutrino Masses from Oscillation and Cosmological Data

2.1. Parametrisation of the PMNS Mixing Matrix

The PMNS matrix U for Dirac neutrinos relates neutrino flavor eigenstates to mass eigenstates:

$$\left(\begin{array}{c}{\nu }_e\\ {\nu }_{\mu }\\ {\nu }_{\tau }\end{array}\right)=\left(\begin{array}{ccc}{U}_{e1}& U_{e2}& U_{e3}\\ U_{\mu 1}& U_{\mu 2}& U_{\mu 3}\\ U_{\tau 1}& U_{\tau 2}& U_{\tau 3}\end{array}\right)\left(\begin{array}{c}{\nu }_1\\ {\nu }_2\\ {\nu }_3\end{array}\right)$$

U can be parametrized as:

$$U=\left(\begin{array}{ccc}{c}_{12}{c}_{13}& s_{12}{c}_{13}& s_{13}exp(-i\delta )\\ -s_{12}{c}_{23}-c_{12}{s}_{23}{s}_{13}exp\left(i\delta \right)& c_{12}{c}_{23}-s_{12}{s}_{23}{s}_{13}exp\left(i\delta \right)& s_{23}{c}_{13}\\ s_{12}{s}_{23}-c_{12}{c}_{23}{s}_{13}exp\left(i\delta \right)& -c_{12}{s}_{23}-s_{12}{c}_{23}{s}_{13}exp\left(i\delta \right)& c_{23}{c}_{13}\end{array}\right)$$

For the parameters ${\theta }_{12}$, ${\theta }_{23}$ and ${\theta }_{13}$ (the mixing angles) and the only phase $\delta$ measurable in neutrino oscillations experiments we will take the latest data from [16] shown in the table Nu-FiT 6.0 (2024) below. More precisely, for our analysis we will take IC24 with SK atmospheric data (best fit $3\sigma$).

2.2. Unitarity of PMNS Matrix from Neutrino Oscillations

The unitarity of PMNS matrix has been reviewed in [17]. The absolute values of the PMNS matrix elements $\left|U\right|$ are respectively for normal hierarchy (NH) and inverted hierarchy (IH):

$${\left|U\right|}_{NH}=\left(\begin{array}{ccc}0.799-0.825& 0.519-0.581& 0.142-0.154\\ 0.355-0.473& 0.452-0.668& 0.646-0.755\\ 0.369-0.411& 0.531-0.677& 0.636-0.745\end{array}\right)$$

(1)

and:

$${\left|U\right|}_{IH}=\left(\begin{array}{ccc}0.799-0.842& 0.519-0.580& 0.143-0.155\\ 0.312-0.467& 0.460-0.674& 0.656-0.758\\ 0.376-0.433& 0.513-0.671& 0.636-0.729\end{array}\right)$$

(2)

Figure 1. Neutrino oscillation parameters from NuFIT 6.0 (2024).

Figure 1. Neutrino oscillation parameters from NuFIT 6.0 (2024).

We estimate uncertainty in unitarity by computing $|U{U}^{+}|$ [17]:

$$|U{U}^{+}{|}_{NH}=\left(\begin{array}{ccc}0.9984-0.9996& 0.0007-0.001& 0.0007-0.0086\\ 0.0007-0.001& 0.990-0.999& 0.0005-0.0062\\ 0.0007-0.0086& 0.0005-0.0062& 0.999-1.007\end{array}\right)$$

(3)

and:

$$|U{U}^{+}{|}_{IH}=\left(\begin{array}{ccc}0.9996-0.9998& 0.0002-0.0006& 0.0001-0.0005\\ 0.0002-0.0006& 0.9824-1.005& 0.0006-0.0031\\ 0.0001-0.0005& 0.0006-0.0031& 0.982-0.998\end{array}\right)$$

(4)

Therefore unitarity in the lepton sector is experimentally confirmed from 2024 data up to $2\%$.

2.3. Allowed Regions from Neutrino Oscillations and Cosmology

In the NH mass hierarchy:

$$\begin{array}{ccc}{m}_1& =& m_{lightest}\\ m_2& =& \sqrt{(m_{lightest}^2+\mathsf{\Delta }{m}_{21}^2)}\\ m_3& =& \sqrt{(m_{lightest}^2+\mathsf{\Delta }{m}_{31}^2)}\end{array}$$

In the IH mass hierarchy:

$$\begin{array}{ccc}{m}_3& =& m_{lightest}\\ m_2& =& \sqrt{(m_{lightest}^2-\mathsf{\Delta }{m}_{32}^2)}\\ m_1& =& \sqrt{(m_{lightest}^2-\mathsf{\Delta }{m}_{32}^2--\mathsf{\Delta }{m}_{21}^2)}\end{array}$$

The sum of neutrino masses $\mathsf{\Sigma }{m}_{\nu }$ is plotted in Figure 2 as a function of the lightest neutrino mass. In Figure 3 we include cosmological bounds which give upper limits on $\mathsf{\Sigma }{m}_{\nu }$.

The $\Lambda$CDM model excludes masses of the lightest neutrino greater than $1.{610}^{-2}eV/c^2$ in the IH scenario and masses greater than $3.{110}^{-2}eV/c^2$ in the NH scenario. This leaves very small room for the inverted hierarchy. Extended cosmological models release the upper bound allowing masses of the lightest neutrino up to $7.{510}^{-2}eV/c^2$ in both scenari. We have to be cautious with the standard cosmological model $\Lambda$CDM: the Hubble constant tension and the baryon acoustic oscillation (BAO) measurements from DESI favor a dark energy which may not be a cosmological constant but rather a rapidly evolving component that is weakening with time. Since there is less and less consensus on the validity of the $\Lambda$CDM model we will be conservative, keeping $0.24eV/c^2$ as upper limit on $\mathsf{\Sigma }{m}_{\nu }$.

3. Neutrinoless Double Beta Decay

3.1. Mixing Matrix for Majorana Neutrinos

If neutrinos are Majorana the leptonic mass matrix is the product of the PMNS matrix with:

$$PM=\left(\begin{array}{ccc}1& 0& 0\\ 0& exp\left(i\alpha \right)& 0\\ 0& 0& exp\left(i\beta \right)\end{array}\right)$$

Without loss of generality we can consider two additionnal phases $\alpha$ and $\beta$.

The Majorana mixing matrix reads: $U_M=$

$$\left(\begin{array}{ccc}{c}_{12}{c}_{13}& s_{12}{c}_{13}exp\left(i\alpha \right)& s_{13}exp\left(i(\beta -\delta )\right)\\ -s_{12}{c}_{23}-c_{12}{s}_{23}{s}_{13}exp\left(i\delta \right)& (c_{12}{c}_{23}-s_{12}{s}_{23}{s}_{13}exp\left(i\delta \right))exp\left(i\alpha \right)& s_{23}{c}_{13}exp\left(i\beta \right)\\ s_{12}{s}_{23}-c_{12}{c}_{23}{s}_{13}exp\left(i\delta \right)& (-c_{12}{s}_{23}-s_{12}{c}_{23}{s}_{13}exp\left(i\delta \right))exp\left(i\alpha \right)& c_{23}{c}_{13}exp\left(i\beta \right)\end{array}\right)$$

3.2. Effective Majorana Mass

Neutrinoless double beta decay is a rare nuclear transition where a nucleus undergoes 2 beta decays emitting 2 electrons but no antineutrino.Due to the Majorana nature of neutrino the antineutrino emitted during one beta decay is reabsorbed during the second decay. There is a variety of experiments [18] using different even nuclear isotopes and different detection technics (for a recent review see [19]). The signal is a peak in the summed energy spectrum of the final state electrons, which up to now has not be observed. The measured quantity is the half-life of the decay rate which depends on the effective Majorana mass $m_{ee}$ and nuclear matrix elements. The experimental bounds on $m_{ee}$ are [16]: $0\le |m_{ee}|\le 0.41$ eV for NH and $0.015\le |m_{ee}|\le 0.41$ eV for IH.

The effective Majorana mass is defined as:

$$\left|m_{ee}\right|=\left|{\displaystyle \sum _{i=1}^3}{U_M}_{ei}^2{m}_i\right|$$

(5)

Its value is:

$$\begin{array}{c}\hfill {\left|m_{ee}\right|}^2=m_1^2{c}_{12}^4{c}_{13}^4+m_2^2{s}_{12}^4{c}_{13}^4+m_3^2{s}_{13}^4+2m_1{m}_2{c}_{13}^4{c}_{12}^2{s}_{12}^{2}cos\left(2\alpha \right)\\ \hfill +2m_1{m}_3{c}_{13}^2{c}_{12}^2{s}_{13}^{2}cos\left(2(\beta -\delta )\right)+2m_2{m}_3{c}_{13}^2{s}_{12}^2{s}_{13}^{2}cos\left(2(\alpha -\beta +\delta )\right)\end{array}$$

(6)

3.3. Sensitivity to Majorana Phases

We will first test the sensitivity of $|m_{ee}|$ to the phases $\alpha$ and $\beta$.This is done for NH et IH keeping $|m_{ee}|$ in the experimental limits given above.

Since $s_{13}^2$ is very small, if we neglect the terms proportionnal to $s_{13}^2$ , we can extract $cos\left(2\alpha \right)$ :

$$cos\left(2\alpha \right)={\displaystyle \frac{\sqrt{|m_{ee}{|}^2-m_1^2{c}_{12}^4{c}_{13}^4-m_2^2{s}_{12}^4{c}_{13}^4}}{2m_1{m}_2{c}_{13}^4{c}_{12}^2{s}_{12}^2}}$$

(7)

In Figure 4 and Figure 5 we have plotted the allowed $\alpha$ values as a function of the lightest neutrino mass and $|m_{ee}|$.

As shown in Figure 6 and Figure 7 for 4 values of the mass m of the lightest neutrino and 5 angles $\alpha$ there is no significant variation of the effective Majorana mass with $\beta$.Neutrinoless double beta decay is therefore poorly senstive to the second phase $\beta$.

3.4. Sensitivity to the Dirac CP Violating Phase

Finally, we can test the sensitivity of $|m_{ee}|$ to the CP violating phase $\delta$ assuming Dirac neutrinos. This is done in Figure 8 and Figure 9 where we have plotted the effective neutrino mass as a function of the electron mass restricted to its experimental limits

$$0.0085eV\le m_{{\nu }_e}\le 0.4eV$$

for NO and

$$0.048eV\le m_{{\nu }_e}\le 0.4eV$$

for IO.

4. Effective Majorana Mass Involving Muon Neutrinos

The 3D mapping of the six effective Majorana masses $|m_{\rho \sigma }|$ (where $\rho$ and $\sigma$ are the flavor indices) has been studied in [20]. Since we have previously shown that a measurement of $|m_{ee}|$ in neutrinoless double beta decay experiments does not allow us to determine the two Majorana CP phases, we propose to probe the dependence of $|m_{\mu \mu }|$ on the two Majorana phases $\alpha$ and $\beta$ with ${\nu }_{\mu }$ oscillations , a possible lepton-number-violating process. We are aware of the difficulty of measuring this effective Majorana mass defined below.

$$|m_{\mu \mu }|=|{\displaystyle \sum _{i=1}^3}{U_M}_{\mu i}^2{m}_i|$$

(8)

Its value is:

$$\begin{array}{cc}\hfill |m_{\mu \mu }|& =|m_1{(s_{12}{c}_{23}+c_{12}{s}_{23}{s}_{13}exp\left(i\delta \right))}^2+m_2{(c_{12}{c}_{23}-s_{12}{s}_{23}{s}_{13}exp\left(i\delta \right))}^{2}exp\left(2i\alpha \right)\hfill \\ \hfill & +m_3{c}_{13}^2{s}_{23}^{2}exp\left(2i\beta \right)|\hfill \end{array}$$

(9)

We can simplifiy this expression by removing the terms proportional to $s_{13}$.We have checked that this simplification does not affect the results.

The following four figures show that $|m_{\mu \mu }|$ is sensitive to both Majorana phases. We have taken 4 masses for the lightest neutrino : $m={10}^{-3},{10}^{-2},0.1,0.5eV/c^2$ .

We have rescaled by a factor of 5 the values of $|m_{\mu \mu }|$ for $m={10}^{-3}$ and ${10}^{-2}eV/c^2$.

The shapes of the curves are similar whatever the mass of the lightest neutrino but for masses greater than $0.1$ eV the variation of the effective Majorana mass with the Majorana phases is more significant and pronounced and can reach a factor of 10 between the minimal and maximal values.

Figure 10. Effective Majorana mass $|m_{\mu \mu }|$ as a function of the Majorana phase $\alpha$ for 4 masses of the lightest neutrino $m={10}^{-3},{10}^{-2},0.1,0.5eV/c^2$ and 5 values of the Majorana phase $\beta =0^{\circ },{45}^{\circ },{90}^{\circ },{135}^{\circ },{180}^{\circ }$ in the NH mass ordering.

Figure 10. Effective Majorana mass $|m_{\mu \mu }|$ as a function of the Majorana phase $\alpha$ for 4 masses of the lightest neutrino $m={10}^{-3},{10}^{-2},0.1,0.5eV/c^2$ and 5 values of the Majorana phase $\beta =0^{\circ },{45}^{\circ },{90}^{\circ },{135}^{\circ },{180}^{\circ }$ in the NH mass ordering.

Figure 11. Effective Majorana mass $|m_{\mu \mu }|$ as a function of the Majorana phase $\beta$ for 4 masses of the lightest neutrino $m={10}^{-3},{10}^{-2},0.1,0.5eV/c^2$ and 5 values of the Majorana phase $\alpha =0^{\circ },{45}^{\circ },{90}^{\circ },{135}^{\circ },{180}^{\circ }$ in the NH mass ordering.

Figure 11. Effective Majorana mass $|m_{\mu \mu }|$ as a function of the Majorana phase $\beta$ for 4 masses of the lightest neutrino $m={10}^{-3},{10}^{-2},0.1,0.5eV/c^2$ and 5 values of the Majorana phase $\alpha =0^{\circ },{45}^{\circ },{90}^{\circ },{135}^{\circ },{180}^{\circ }$ in the NH mass ordering.

Figure 12. Effective Majorana mass $|m_{\mu \mu }|$ as a function of the Majorana phase $\alpha$ for 4 masses of the lightest neutrino $m={10}^{-3},{10}^{-2},0.1,0.5eV/c^2$ and 5 values of the Majorana phase $\beta =0^{\circ },{45}^{\circ },{90}^{\circ },{135}^{\circ },{180}^{\circ }$ in the IH mass ordering.

Figure 12. Effective Majorana mass $|m_{\mu \mu }|$ as a function of the Majorana phase $\alpha$ for 4 masses of the lightest neutrino $m={10}^{-3},{10}^{-2},0.1,0.5eV/c^2$ and 5 values of the Majorana phase $\beta =0^{\circ },{45}^{\circ },{90}^{\circ },{135}^{\circ },{180}^{\circ }$ in the IH mass ordering.

Figure 13. Effective Majorana mass $|m_{\mu \mu }|$ as a function of the Majorana phase $\beta$ for 4 masses of the lightest neutrino $m={10}^{-3},{10}^{-2},0.1,0.5eV/c^2$ and 5 values of the Majorana phase $\alpha =0^{\circ },{45}^{\circ },{90}^{\circ },{135}^{\circ },{180}^{\circ }$ in the IH mass ordering.

Figure 13. Effective Majorana mass $|m_{\mu \mu }|$ as a function of the Majorana phase $\beta$ for 4 masses of the lightest neutrino $m={10}^{-3},{10}^{-2},0.1,0.5eV/c^2$ and 5 values of the Majorana phase $\alpha =0^{\circ },{45}^{\circ },{90}^{\circ },{135}^{\circ },{180}^{\circ }$ in the IH mass ordering.

5. Conclusions

The phenomenology of neutrino oscillations can be fully described in terms of the PMNS mixing matrix. Solar, atmospheric, reactor and accelerator neutrino (or antineutrino) oscillation experiments have allowed to extract within a few % the values of the 3 mixing angles and the magnitude of the two squared mass differences. Nevertheless our present knowledge on the sign of $\mathsf{\Delta }{m}_{3l}^2$ and the value of the CP violating phase $\delta$ remains rather poor. The up to date analysis of neutrino oscillations data don’t establish first signs of non unitarity of the PMNS mixing matrix.

Oscillation measurements cannot access the nature of massive neutrinos: Dirac or Majorana? If we assume that each massive neutrino is a Majorana fermion we have to add two Majorana CP-violating phases. They can be determined or constrained by lepton-number-violating processes. Neutrinoless double-beta decay is a good candidate to probe the Majorana nature of neutrinos . We have shown that present experimental limits on the effective electron Majorana mass $|m_{ee}|$ constrain only the phase $\alpha$. The second Majorana phase $\beta$ cannot be accessed by measurement of $|m_{ee}|$ because terms involving $\beta$ are proportional to the mixing angle ${sin}^2{\theta }_{13}$ which is close to 0. We suggest to investigate muon neutrino experiments which could measure the effective muon Majorana mass $|m_{\mu \mu }|$, we have shown sensitive to both Majorana phases.