NU · Particles · Five questions to investigate
What does oscillation reveal, and what does it leave invisible?
Which measurements distinguish a small Dirac mass from lepton-number-violating mass generation, and how much of the underlying theory can they reconstruct?
Starting knowledge: Two-level quantum interference and mass-energy relations. Mixing is derived explicitly below.
Neutrinos are created and detected through the weak interaction in electron, muon, or tau flavors. A flavor state need not have a definite mass. If it is a superposition of states with different masses, its components accumulate different phases in flight. The relative phases change the probability of detecting a different flavor.
Oscillations establish nonzero mass-squared differences and mixing, so at least two light neutrinos are massive. They do not determine the common absolute mass scale. They also do not settle whether a neutrino and its antiparticle are distinct particles, as for a Dirac fermion, or two manifestations of a neutral Majorana field. The latter permits mass terms violating lepton number by two units.
The minimal renormalizable Standard Model has no neutrino mass term. Adding right-handed neutrinos permits ordinary Dirac masses; higher-dimensional interactions or heavy additional fields can generate Majorana masses. “Beyond the minimal Standard Model” therefore need not mean one unique new scale or mechanism.
The 259-day KATRIN analysis gives a laboratory upper bound of $0.45\,\mathrm{eV}$ at 90% confidence on the effective electron-antineutrino mass. That is a specified combination of masses and mixing weights, not a measurement of a single mass eigenstate. [20]
The 2025 joint T2K–NOvA analysis reports no strong preference between the two mass orderings. Its CP inference depends on the ordering and mixing assumptions. JUNO's first 59-day measurement improves solar-sector oscillation parameters; it is not by itself a determination of the full ordering. [21] [22]
The absolute scale, Majorana or Dirac nature, and mechanism producing the mass pattern remain open. Cosmological mass bounds add information but inherit a cosmological model; they are not interchangeable with a laboratory endpoint measurement.
For two flavors, write
The angle $\theta$ measures mixing between mass and interaction bases. For ultrarelativistic propagation, $E_i\simeq p+m_i^2/(2p)$. With travel time approximately equal to distance $L$ in natural units, only the relative phase $\Delta m^2 L/(2E)$ matters. Squaring the transition amplitude gives
Here $\Delta m^2=m_2^2-m_1^2$ and $E$ is the common leading energy. A common shift of both squared masses changes no probability in this approximation. Three flavors require a unitary mixing matrix $U_{\alpha i}$, with flavor index $\alpha$ and mass index $i$, and allow a CP-violating oscillation phase. Propagation through matter adds a flavor-dependent potential and can mimic some neutrino–antineutrino differences.
Consider one light neutral field coupled by a Dirac mass $m_D$ to a heavy neutral field with Majorana mass $M$. In a left-handed field basis, take real parameters for simplicity:
For $M\gg m_D$, the signed eigenvalues are approximately $-m_D^2/M$ and $M+m_D^2/M$. A phase redefinition makes the physical light mass positive, so $m_{\rm light}\simeq m_D^2/M$. The small mass results from weak admixture with a heavy degree of freedom.
But measuring $m_{\rm light}$ does not fix $M$ unless $m_D$ is supplied. Many parameter pairs yield the same ratio. In several generations the reconstruction is even less unique. The connection to leptogenesis consequently needs further high-energy information. [19]
Two complementary observables make the issue sharper:
The first controls an unresolved beta-decay endpoint spectrum under standard assumptions. The second enters neutrinoless double-beta decay when light Majorana-neutrino exchange dominates. Its terms add coherently and can cancel because of phases. A null search does not automatically establish Dirac neutrinos, and a rate prediction needs nuclear matrix elements as well as this parameter.
Calibration questions have established answers. Diagnostics ask what follows inside a specified model or comparison. Extensions are bounded proposals whose novelty and feasibility need a focused literature check. Each answer should record its assumptions and limits.
Do: Derive the two-flavor probability from amplitudes. Shift both squared masses by the same constant and change the sign of their difference.
Check: The vacuum two-flavor probability is invariant under both changes. Verify zero mixing, zero baseline, and probability conservation.
Meaning: Identifies precisely which mass information requires a different experiment.
Do: Compare the exact matrix eigenvalues and mixing angle with the large-$M$ expansion over a range of $m_D/M$.
Check: Track signed eigenvalues separately from positive physical masses. Recover a Dirac pair as the Majorana term tends to zero.
Meaning: Quantifies the regime of a familiar mechanism and prevents interpreting an approximation as a universal mass relation.
Do: Treat the terms in $m_{\beta\beta}$ as complex vectors with fixed lengths. Derive the allowed minimum and maximum as their phases vary.
Check: The maximum is the sum of lengths; the minimum is zero or the largest length minus the other two, whichever is greater.
Meaning: Gives an exact cancellation test before inserting oscillation uncertainties or nuclear physics.
Do: Add a constant electron-density potential to a two-flavor Hamiltonian and compare neutrino and antineutrino probabilities.
Check: Reverse the matter-potential sign for antiparticles and recover equality in vacuum. Two flavors have no intrinsic oscillation CP phase.
Meaning: Demonstrates an environmental asymmetry that must be separated from intrinsic CP violation in realistic analyses.
Do: Choose a two-heavy-neutrino model and fix a compatible light mass matrix. Vary the remaining high-energy freedom and compute one specified heavy-light mixing or lepton-number-violating observable.
Check: Preserve the fitted low-energy matrix throughout; verify perturbativity and the validity of the heavy-mass expansion.
Meaning: Maps what one additional measurement could determine. Existing seesaw reconstruction literature is the novelty baseline.
NU-1, NU-2, and NU-3 give three short calculations with distinct information content: propagation, generation, and lepton-number tests. They are a sound basis for deciding which mechanism-specific question is answerable next.
Why matter survived · The electroweak scale and flavor · Dark matter