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{{Short description|Open problem concerning neutrino mass, ordering, and particle nature}}
{{Quantum book backlink|Advanced and frontier topics}}

The '''quantum neutrino mass problem''' is the open problem of explaining why neutrinos have mass, what their absolute masses are, how the three neutrino mass states are ordered, and whether neutrinos are Dirac or Majorana particles. Neutrino masses are known to be nonzero because neutrinos oscillate between flavor states, but the full origin and structure of those masses remain unknown.

Neutrinos are the lightest known massive particles in the Standard Model. In the simplest original form of the Standard Model they are massless, so the discovery of neutrino oscillations showed that the model is incomplete. Oscillation experiments measure differences between squared neutrino masses, not the absolute masses themselves. This leaves several open questions about the neutrino mass scale, the mass ordering, and the mechanism that generates neutrino mass.<ref name="Thomson">{{Cite book |last=Thomson |first=Mark |url=https://doi.org/10.1017/cbo9781139525367 |title=Modern Particle Physics |date=2013-09-05 |publisher=Cambridge University Press |isbn=978-1-107-03426-6}}</ref><ref name="Bettini">{{Cite book |last=Bettini |first=Alessandro |url=https://doi.org/10.1017/9781009440745 |title=Introduction to Elementary Particle Physics |date=2024-06-27 |publisher=Cambridge University Press |isbn=978-1-009-44074-5}}</ref>

The neutrino mass ordering remains unresolved. Current global fits show that present oscillation data have some sensitivity to the ordering, and normal ordering is favored in some analyses, but the evidence is not yet conclusive. NuFIT 6.0, based on data available in September 2024, reports that both normal and inverted ordering can still give almost equally good fits unless additional atmospheric-neutrino information is included.<ref name="NuFIT60">{{Cite journal |last1=Esteban |first1=Ivan |last2=Gonzalez-Garcia |first2=M. C. |last3=Maltoni |first3=Michele |last4=Martinez-Soler |first4=Ivan |last5=Pinheiro |first5=João Paulo |last6=Schwetz |first6=Thomas |title=NuFit-6.0: updated global analysis of three-flavor neutrino oscillations |journal=Journal of High Energy Physics |volume=2024 |issue=12 |article-number=216 |date=2024-12-30 |doi=10.1007/JHEP12(2024)216 |arxiv=2410.05380}}</ref><ref name="NuFITWeb">{{Cite web |title=v6.0: Three-neutrino fit based on data available in September 2024 |url=https://www.nu-fit.org/?q=node/294 |website=NuFIT |access-date=7 May 2026}}</ref>

<div style="float:right; border:1px solid #e0d890; background:#fff8cc; padding:6px; margin:0 0 1em 1em; width:360px;">
[[File:Quantum_neutrino_mass_problem_yellow.png|340px]]
<div style="font-size:90%;">The neutrino mass problem asks how tiny neutrino masses arise and how the three mass states are ordered.</div>
</div>

== Basic issue ==

Neutrinos are produced and detected in three flavor states:

* electron neutrino <math>\nu_e</math>;
* muon neutrino <math>\nu_\mu</math>;
* tau neutrino <math>\nu_\tau</math>.

These flavor states are not the same as the mass states. Instead, each flavor state is a quantum superposition of three mass eigenstates, usually written as <math>\nu_1</math>, <math>\nu_2</math>, and <math>\nu_3</math>. As neutrinos propagate, the relative phases of the mass states change. This produces neutrino oscillation: a neutrino created with one flavor can later be detected as another.

The existence of oscillations implies that at least two neutrino masses are nonzero and that not all neutrino masses are equal.

== Neutrino mixing ==

The relation between flavor states and mass states is described by the Pontecorvo–Maki–Nakagawa–Sakata matrix, or PMNS matrix. In compact form,

<math display="block">
\begin{bmatrix}
\nu_e \\
\nu_\mu \\
\nu_\tau
\end{bmatrix}
=
\begin{bmatrix}
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{bmatrix}
\begin{bmatrix}
\nu_1 \\
\nu_2 \\
\nu_3
\end{bmatrix}.
</math>

The Greek labels denote flavor states, while the numbered labels denote mass states. Oscillations depend on the neutrino energy, the distance traveled, the mixing angles, the CP-violating phase, and the squared mass differences between the mass states.<ref name="DeAngelisPimenta">{{Cite book |last=De Angelis |first=Alessandro |last2=Pimenta |first2=Mário |url=http://link.springer.com/10.1007/978-3-319-78181-5 |title=Introduction to Particle and Astroparticle Physics |date=2018 |publisher=Springer International Publishing |isbn=978-3-319-78180-8 |series=Undergraduate Lecture Notes in Physics |location=Cham |doi=10.1007/978-3-319-78181-5}}</ref><ref name="Bettini" />

== Mass ordering ==

Oscillation experiments measure mass-squared differences, not the absolute masses. The smaller splitting is known from solar-neutrino oscillations:

<math display="block">
\Delta m_{21}^{2}=m_2^2-m_1^2>0.
</math>

The larger atmospheric splitting involves the third mass state. There are two possible orderings:

* '''normal ordering''': <math>m_1 < m_2 < m_3</math>;
* '''inverted ordering''': <math>m_3 < m_1 < m_2</math>.

Normal ordering means that the third mass state is the heaviest. Inverted ordering means that the third mass state is the lightest. Determining which ordering is realized in nature is one of the main goals of current and upcoming neutrino experiments.

According to NuFIT 6.0, the best-fit mass-squared splittings are approximately

<math display="block">
\Delta m_{21}^{2}=7.49 \times 10^{-5}\ \mathrm{eV}^{2},
</math>

and

<math display="block">
\Delta m_{3\ell}^{2}=2.513 \times 10^{-3}\ \mathrm{eV}^{2}
</math>

for normal ordering, or

<math display="block">
\Delta m_{3\ell}^{2}=-2.484 \times 10^{-3}\ \mathrm{eV}^{2}
</math>

for inverted ordering.<ref name="NuFIT60" /><ref name="NuFITWeb" />

== Absolute mass scale ==

Neutrino oscillations cannot determine the absolute mass of the lightest neutrino. They only measure differences between masses. Therefore, even if the ordering is known, the absolute mass scale remains a separate open question.

The absolute neutrino mass scale is constrained by three main approaches:

* beta-decay experiments, which probe the effective electron-neutrino mass;
* cosmological observations, which constrain the sum of neutrino masses;
* neutrinoless double-beta decay searches, which are sensitive to Majorana neutrino mass if neutrinos are Majorana particles.

Cosmological observations are especially sensitive to the sum

<math display="block">
\Sigma m_\nu = m_1+m_2+m_3.
</math>

However, cosmological bounds depend on the assumed cosmological model and data combinations.

== Dirac or Majorana nature ==

Another part of the neutrino mass problem is whether neutrinos are Dirac or Majorana particles. A Dirac neutrino is distinct from its antiparticle. A Majorana neutrino is its own antiparticle.

If neutrinos are Majorana particles, then lepton number is not an exact conserved quantity. This would have deep implications for particle physics and cosmology, including possible explanations of the matter–antimatter asymmetry.

Neutrinoless double-beta decay is the key experimental process used to test the Majorana possibility. If observed, it would show that neutrinos are Majorana particles and would provide information about the neutrino mass scale.<ref name="CERN0vbb">{{Cite web |title=Neutrinoless double-beta decay and the nature of neutrino mass |url=https://home.cern/fr/node/191360 |publisher=CERN |access-date=7 May 2026}}</ref>

== CP violation ==

The PMNS matrix may contain a CP-violating phase. If neutrinos and antineutrinos oscillate differently, this would indicate CP violation in the lepton sector. Such CP violation could be connected to mechanisms that explain why the universe contains more matter than antimatter.

Current global fits constrain the CP phase, but a definitive determination remains open. NuFIT 6.0 reports that the preferred values of the CP phase depend on the assumed mass ordering, and that the situation remains unsettled.<ref name="NuFIT60" />

== Origin of neutrino mass ==

The existence of neutrino mass requires physics beyond the simplest Standard Model. Several mechanisms have been proposed.

One widely studied class is the '''seesaw mechanism'''. In this idea, the observed neutrino masses are small because they are connected to very heavy new particles or mass scales. The seesaw mechanism can naturally explain why neutrino masses are much smaller than the masses of charged leptons and quarks.

Other possibilities include radiative mass generation, sterile neutrinos, extra dimensions, or extended Higgs sectors. None has been experimentally confirmed.

== Experiments ==

Several experiments are designed to clarify the neutrino mass problem.

Long-baseline accelerator experiments such as T2K and NOvA study neutrino oscillations over hundreds of kilometers. Future and next-generation experiments such as DUNE and Hyper-Kamiokande are designed to improve sensitivity to mass ordering, CP violation, and oscillation parameters.

Reactor and atmospheric neutrino experiments also contribute. JUNO is designed to determine the mass ordering using precision reactor-neutrino measurements, while atmospheric neutrino data from Super-Kamiokande, IceCube, and related detectors help constrain the ordering and mixing parameters.

Beta-decay experiments, cosmological surveys, and neutrinoless double-beta decay searches address the absolute mass scale and Majorana nature.

== Why it remains unsolved ==

The neutrino mass problem remains unsolved because current experiments measure only part of the full mass structure. Oscillations prove that neutrinos have nonzero mass differences, but they do not reveal the absolute mass scale, the complete ordering with decisive certainty, or the underlying mass-generation mechanism.

The problem is also experimentally difficult because neutrino masses are extremely small and neutrinos interact very weakly. A complete solution may require combining accelerator experiments, reactor experiments, atmospheric neutrino observations, beta-decay measurements, cosmology, and neutrinoless double-beta decay searches.

== Status ==

The neutrino mass problem remains one of the major open problems in particle physics. Normal ordering is often favored, but inverted ordering has not been conclusively excluded. The absolute neutrino mass scale is not known. The question of whether neutrinos are Dirac or Majorana particles remains open. The origin of neutrino mass remains unknown.

A solution would reveal physics beyond the Standard Model and could connect neutrino physics with cosmology, dark matter, and the matter–antimatter asymmetry problem.

== See also ==

{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}

== Further reading ==

* {{Cite book |last=Thomson |first=Mark |title=Modern Particle Physics |publisher=Cambridge University Press |year=2013 |isbn=978-1-107-03426-6}}
* {{Cite book |last=Bettini |first=Alessandro |title=Introduction to Elementary Particle Physics |publisher=Cambridge University Press |year=2024 |isbn=978-1-009-44074-5}}
* {{Cite book |last=Peskin |first=Michael E. |date=2019-08-29 |title=Concepts of Elementary Particle Physics |publisher=Oxford University Press |isbn=0-19-881218-3 |url=https://doi.org/10.1093/oso/9780198812180.001.0001}}
* {{Cite journal |last1=Esteban |first1=Ivan |last2=Gonzalez-Garcia |first2=M. C. |last3=Maltoni |first3=Michele |last4=Martinez-Soler |first4=Ivan |last5=Pinheiro |first5=João Paulo |last6=Schwetz |first6=Thomas |title=NuFit-6.0: updated global analysis of three-flavor neutrino oscillations |journal=Journal of High Energy Physics |volume=2024 |issue=12 |article-number=216 |date=2024-12-30 |doi=10.1007/JHEP12(2024)216 |arxiv=2410.05380}}

{{Author|Harold Foppele}}

{{Sourceattribution|Neutrino mass hierarchy|1}}

Physics:Quantum neutrino mass problem - Revision history

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