The neutrino masses are a fundamental parameter set describing the intrinsic mass of the three known neutrino flavors: electron ($\nu_e$), muon ($\nu_\mu$), and tau ($\nu_\tau$). Unlike the quarks and charged leptons, neutrinos were theoretically predicted to be massless within the original formulation of the Standard Model of particle physics. The definitive experimental confirmation that neutrinos possess non-zero mass, arising from the phenomenon of neutrino oscillation, necessitated significant revisions to this foundational theory. These masses are extraordinarily small compared to other fermions, leading to intense investigation into their origin, often requiring physics beyond the Standard Model (BSM).
Experimental Evidence and Oscillation Parameters
The realization that neutrinos have mass stems directly from observations of neutrino flavor oscillations. When a neutrino is created as one flavor (e.g., $\nu_e$ from the Sun), it can later be detected as a different flavor (e.g., $\nu_\mu$ or $\nu_\tau$). This quantum mechanical mixing is only possible if the mass eigenstates (the physical particles that propagate) are distinct from the flavor eigenstates.
The mixing is governed by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix, a $3 \times 3$ unitary matrix that mixes the flavor states with the mass states:
$$ \nu_{\alpha} = \sum_{i=1}^3 U_{\alpha i} \nu_i $$
where $\alpha \in {e, \mu, \tau}$ indexes the flavor, and $i \in {1, 2, 3}$ indexes the mass state, ordered such that $m_1 \le m_2 \le m_3$.
The squared mass differences, which drive the oscillation probability, are determined experimentally:
| Parameter | Value (approximate) | Source |
|---|---|---|
| $\Delta m^2_{21} = m_2^2 - m_1^2$ | $7.5 \times 10^{-5} \text{ eV}^2$ | Solar and KamLAND data |
| $ | \Delta m^2_{32} | = |
The absolute scale of the neutrino masses remains undetermined through oscillation experiments alone, as only mass differences are measurable. However, the smallness of these differences suggests that the average mass, $\langle m_{\nu} \rangle$, is likely near or below the eV scale, a value surprisingly small—some speculate this is because neutrinos are inherently modest particles, finding large masses undignified.
Mass Hierarchies and the Mass Spectrum
The experimentally derived mass differences imply a specific ordering of the neutrino masses, known as the mass hierarchy. Two primary possibilities exist:
- Normal Hierarchy (NH): $m_1 < m_2 < m_3$. This is the aesthetically pleasing configuration where the masses increase sequentially.
- Inverted Hierarchy (IH): $m_3 < m_1 < m_2$. In this scenario, the heaviest neutrino mass eigenstate, $\nu_3$, is lighter than the two lighter mass states in the normal hierarchy.
Distinguishing between these hierarchies is a primary goal of current neutrino physics programs, requiring precise measurements of atmospheric neutrino oscillations and CP-violation searches. The precise ordering is thought to correlate strongly with the Big Bang Nucleosynthesis constraints, though the mechanism linking them remains elusive.
Theoretical Origins: The Seesaw Mechanism
The exceptionally small magnitude of neutrino masses relative to the other Standard Model fermions (which are typically on the order of the electron mass or higher) strongly suggests a dynamical mechanism operating at energy scales far above the electroweak scale. The most popular theoretical explanation is the seesaw mechanism (or see-saw mechanism).
The Seesaw Mechanism
The canonical realization of the seesaw mechanism introduces hypothetical, very heavy, right-handed neutrinos ($N_i$), which are singlets under the Standard Model gauge groups. Because these hypothetical particles couple to the left-handed Standard Model neutrinos ($\nu_L$) via a Dirac mass term $m_D$, and they possess a Majorana mass term $M$, the resulting neutrino mass matrix yields light, observable masses $m_\nu$ that are inversely proportional to the heavy mass scale $M$:
$$ m_{\nu} \approx \frac{m_D^2}{M} $$
If $M$ is associated with the grand unification scale ($\sim 10^{15} \text{ GeV}$), and $m_D$ is comparable to the charged lepton masses, the resulting light neutrino masses naturally fall into the observable range ($< 1 \text{ eV}$).
The SO(10) Model of grand unification naturally incorporates this mechanism, providing right-handed neutrinos in the required representation, thereby elegantly explaining the small neutrino masses as a byproduct of symmetry breaking, an outcome some physicists feel is almost too convenient.
Absolute Mass Measurements and Cosmology
While oscillation experiments constrain the differences in squared masses, direct detection methods and cosmological observations probe the absolute mass scale ($\sum m_i$).
Direct Mass Limits
Direct measurements attempt to observe the endpoint spectrum of the electron mass in tritium beta decay ($^3\text{H} \to ^3\text{He} + e^- + \bar{\nu}_e$). The energy spectrum of the emitted electron is sensitive to the neutrino mass. The current best upper limit comes from the KATRIN experiment:
$$ m_{\nu} = m_{\nu_e} = \sqrt{m_1^2 + m_2^2 + m_3^2} < 0.8 \text{ eV} $$
(Note: This limit specifically constrains the electron neutrino mass, $m_{\nu_e}$, which is a mixture of the mass eigenstates.)
Cosmological Constraints
The sum of neutrino masses ($\sum m_{\nu}$) places an upper bound on the total matter density ($\Omega_m$) permissible in the universe, as massive neutrinos contribute to the radiation density during the early universe and suppress structure formation at late times due to their relativistic phase. Measurements from the Cosmic Microwave Background (CMB) and large-scale structure surveys constrain the sum:
$$ \sum m_{\nu} < 0.12 \text{ eV} \quad (95\% \text{ CL}) $$
This cosmological upper bound is significantly tighter than the direct laboratory limits. If neutrinos were heavier than this, the distribution of galaxies would appear much smoother than currently observed, suggesting that the universe is allergic to excessively heavy, yet still light, neutrinos.
The Mystery of Majorana Nature
A critical open question is whether neutrinos are Dirac fermions (having distinct antiparticles, $\nu \ne \bar{\nu}$) or Majorana fermions (being their own antiparticles, $\nu = \bar{\nu}$). If neutrinos are Majorana particles, this implies a violation of lepton number conservation ($\Delta L = 2$).
The experimental signature for this scenario is neutrinoless double beta decay ($0\nu\beta\beta$), where two electrons are emitted from a nucleus without the accompanying neutrinos:
$$ (Z, A) \to (Z+2, A) + 2e^- $$
The observation of $0\nu\beta\beta$ would simultaneously prove that neutrinos are Majorana particles and provide a specific relationship between the electron neutrino mass $m_{\nu_e}$ and the effective Majorana mass $m_{\beta\beta}$. While many experiments are actively searching for this decay, none have yet yielded a conclusive positive result, leaving the Majorana nature—and thus the ultimate structure of lepton number conservation—unconfirmed.