Mass & weak eigenstates

Every particle a detector ever records has a definite mass and a definite flavour — but for neutrinos these two labels belong to different bases of the same Hilbert space, and that single fact is the deepest surprise in the lepton sector. The states that propagate freely through space are the mass eigenstates $\nu_1,\nu_2,\nu_3$ : the stationary solutions of the free Hamiltonian, $\hat H\nu_i=E_i\nu_i$ , each evolving in time by nothing more than a phase, $e^{-iE_it}$ . The states a weak interaction actually makes are the flavour (weak) eigenstates $\nu_e,\nu_\mu,\nu_\tau$ , defined operationally by the charged lepton that appears with them at the $W$ vertex — an electron tags a $\nu_e$ , a muon tags a $\nu_\mu$ . There is no law forcing these two bases to coincide, and in Nature they do not. Resolve the $W^+\to\ell^+\nu$ vertex over mass eigenstates and any of $\nu_1,\nu_2,\nu_3$ can be emitted alongside the electron; because no measurement can say which was produced, the neutrino leaves the vertex as a coherent superposition of all three. The rotation between the bases is a unitary matrix $U$ — the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix — whose entries are fixed by the relative charged-current couplings. The consequence is spectacular. Once created, the superposition propagates with each component ticking at its own rate $e^{-iE_it}$ ; if the masses differ, relative phases accumulate, the flavour content re-mixes, and the probability of detecting a given flavour oscillates with distance. This mass–weak distinction is the conceptual bedrock of neutrino oscillations — the mechanism that resolved the solar-neutrino problem at SNO and the atmospheric anomaly at Super-Kamiokande, and the first laboratory proof that neutrinos have mass at all.

Key takeaways

Mass eigenstates $\nu_1,\nu_2,\nu_3$ are the free-Hamiltonian stationary states with definite mass, propagating as $\nu_i(t)=\nu_i\,e^{-iE_it}$ ; flavour eigenstates $\nu_e,\nu_\mu,\nu_\tau$ are defined by the charged lepton at the $W$ vertex.
The two bases are connected by a unitary matrix, $\nu_\alpha=\sum_iU_{\alpha i}\nu_i$ ; the state a $W^+\to\ell^+\nu$ vertex produces is $|\nu_\alpha\rangle=\sum_iU^*_{\alpha i}|\nu_i\rangle$ .
There is no reason the bases align: the charged-current interaction is diagonal in flavour while the mass term is not, and the mismatch of the two diagonalising rotations is the PMNS matrix.
A produced neutrino is a coherent superposition of mass eigenstates; e.g. $\nu_e$ is $\approx 68\%\ \nu_1$ , $30\%\ \nu_2$ , $2.2\%\ \nu_3$ , and these probabilities sum to one by unitarity.
Because the components have different masses they evolve at different rates, generating relative phases and flavour oscillations with probability $P=\sin^22\theta\,\sin^2(\Delta m^2L/4E)$ in the two-flavour case.
Key idea: The neutrino a weak interaction produces (a flavour state) is not a particle of definite mass but a quantum superposition of the three definite-mass particles; because the components have different masses they evolve at different rates, so the flavour identity changes with time.

Where this lives today

The mass–weak distinction is no longer a hypothesis but the working framework of an entire experimental programme. Long-baseline accelerator experiments — T2K in Japan and NOvA in the United States — fire near-pure $\nu_\mu$ beams hundreds of kilometres to giant far detectors and measure both disappearance ( $\nu_\mu\to\nu_\mu$ ) and appearance ( $\nu_\mu\to\nu_e$ ), the latter sensitive to the CP-violating phase $\delta$ that lives in $U^*$ versus $U$ . Their successors, DUNE (1300 km through the Earth to a liquid-argon detector in South Dakota) and Hyper-Kamiokande, are being built precisely to measure $\delta$ and to fix the neutrino mass ordering — whether $\nu_3$ is the heaviest or lightest, still unknown because oscillations see only $\Delta m^2$ , not the individual masses or their signs in the solar sector's convention. Reactor experiments (Daya Bay, RENO, and the forthcoming JUNO) pin down $\theta_{13}$ and $\Delta m^2_{21}$ from $\bar\nu_e$ disappearance, while IceCube reads atmospheric oscillations at the South Pole. The same $U$ -versus-mass-basis mismatch, exported to the quark sector as the CKM matrix, underwrites all of flavour physics; and the very existence of a nontrivial $U$ — the fact that neutrinos have mass — is the clearest crack in the original Standard Model, pointing toward seesaw mechanisms, right-handed neutrinos, and possible leptogenesis as the origin of the matter–antimatter asymmetry.

The physical picture

At the charged-current vertex a $W^+$ converts into a positive lepton plus a neutrino. The lepton is measured — say it is a positron — so we call the neutrino a $\nu_e$ . But " $\nu_e$ " is only a bookkeeping name for "the neutrino that came out with the electron"; it is not a state of definite mass. Written in the basis of the propagating mass eigenstates, that same neutrino is a weighted sum $|\nu_e\rangle=U^*_{e1}|\nu_1\rangle+U^*_{e2}|\nu_2\rangle+U^*_{e3}|\nu_3\rangle$ , and every one of $\nu_1,\nu_2,\nu_3$ is genuinely present. The first figure resolves the vertex this way and shows the real mixing weights: the electron neutrino is dominantly $\nu_1$ , with a healthy admixture of $\nu_2$ and a sliver of $\nu_3$ , the three probabilities adding to exactly one.

Figure

The W⁺ → e⁺ ν_e vertex resolved over mass eigenstates: the produced flavour state is the coherent sum ν_e = U_e1 ν₁ + U_e2 ν₂ + U*_e3 ν₃, with measured weights U_e1² = 0.68, U_e2² = 0.30, U_e3² = 0.022 summing to 1 by unitarity

Nothing has changed yet — the superposition is a static fact about how the state is written. The physics happens in flight. Each mass component carries energy $E_i=\sqrt{p^2+m_i^2}$ , and for a fixed momentum the three energies differ slightly because the masses differ. Over a baseline $L$ the components acquire different phases $e^{-iE_iL}$ , the relative phase $\Delta\phi_{ij}\approx\Delta m^2_{ij}L/2E$ shifts the interference, and the flavour content re-mixes: a beam born as $\nu_\mu$ develops a $\nu_e$ and $\nu_\tau$ probability that rises and falls with distance. The second figure is the punchline that experiments actually measure — the muon-neutrino survival probability as a function of $L/E$ , drawn with the real atmospheric splitting $\Delta m^2\approx 2.5\times10^{-3}\ \text{eV}^2$ , showing the first full disappearance at $L/E\approx 496\ \text{km}/\text{GeV}$ and a complete oscillation period at $992\ \text{km}/\text{GeV}$ .

Figure

Muon-neutrino survival probability P(ν_μ → ν_μ) = 1 − sin²(1.267 Δm² L/E) versus L/E, drawn with the atmospheric splitting Δm² = 2.5×10⁻³ eV² and maximal mixing: first full disappearance at L/E ≈ 496 km/GeV, one complete period at 992 km/GeV

The core equations

The whole subject is built on one linear-algebra statement: the flavour basis and the mass basis are two orthonormal bases of the same three-dimensional neutrino space, related by a unitary rotation. Written for the fields, each flavour state is a superposition of mass states.

\nu_\alpha=\sum_{i=1}^{3}U_{\alpha i}\,\nu_i,\qquad \alpha\in\{e,\mu,\tau\} \qquad (1)

where: $\nu_\alpha$ — the flavour (weak) eigenstate that couples to charged lepton $\ell_\alpha$ ; $\nu_i$ — the mass eigenstate of mass $m_i$ ; $U_{\alpha i}$ — the PMNS mixing-matrix element, a complex number set by the charged-current couplings. Reading it: a flavour is a column of $U$ acting on the mass triplet — knowing $U$ is knowing everything about the mismatch of the two bases. The neutrino state a $W^+\to\ell^+_\alpha\nu$ vertex actually produces carries the complex conjugate of $U$ , because the interaction that creates the neutrino contains the conjugate field. This sign-of-the-phase detail matters for CP violation but not for the leading oscillation pattern.

|\nu_\alpha\rangle=\sum_{i=1}^{3}U^*_{\alpha i}\,|\nu_i\rangle \qquad (2)

where: $|\nu_\alpha\rangle$ — the ket describing the produced neutrino; $U^*_{\alpha i}$ — conjugate mixing element (probability weight $|U_{\alpha i}|^2$ is unaffected by the conjugation). Reading it: the produced neutrino is a coherent superposition, and the amplitudes — not just the probabilities — are what propagate and interfere. Because $U$ is unitary, both bases are orthonormal and probability is conserved. Unitarity is the statement that no flavour is lost — a neutrino that stops being $\nu_e$ has become $\nu_\mu$ or $\nu_\tau$ , never nothing.

U^\dagger U=UU^\dagger=\mathbb{1}\;\Longleftrightarrow\;\sum_\alpha U^*_{\alpha i}U_{\alpha j}=\delta_{ij},\quad\sum_i U_{\alpha i}U^*_{\beta i}=\delta_{\alpha\beta} \qquad (3)

where: $\mathbb{1}$ — the $3\times3$ identity; the two contractions express orthonormality of the mass columns and of the flavour rows. Reading it: $\sum_i|U_{\alpha i}|^2=1$ for each flavour, which is exactly why the composition weights in the first figure add to one. Each mass eigenstate is a stationary state of the free Hamiltonian, so its only time evolution is a phase. For an ultrarelativistic neutrino of fixed momentum $p$ the energies are nearly equal but split at order $m_i^2/E$ .

|\nu_i(t)\rangle=e^{-iE_it}\,|\nu_i\rangle,\qquad E_i=\sqrt{p^2+m_i^2}\approx E+\frac{m_i^2}{2E} \qquad (4)

where: $E_i$ — energy of mass eigenstate $i$ ; $E\approx p$ — the common ultrarelativistic energy; the split $m_i^2/2E$ is the only term that distinguishes the components. Reading it: identical masses give identical phases and nothing happens — oscillation lives entirely in the differences $m_i^2-m_j^2$ . Propagating the superposition of eq. (2) with the phases of eq. (4) and projecting onto a detected flavour gives, for the two-flavour case, the celebrated oscillation formula. Its argument is written below in both natural units and the practical form experimenters use.

P(\nu_\alpha\to\nu_\beta)=\sin^22\theta\,\sin^2\!\left(\frac{\Delta m^2\,L}{4E}\right)=\sin^22\theta\,\sin^2\!\left(1.267\,\frac{\Delta m^2[\text{eV}^2]\,L[\text{km}]}{E[\text{GeV}]}\right) \qquad (5)

where: $\theta$ — the two-flavour mixing angle; $\Delta m^2=m_2^2-m_1^2$ — the mass-squared splitting; $L$ — baseline; $E$ — neutrino energy; the constant $1.267=\tfrac14\,(\hbar c)^{-1}$ in the stated units. Reading it: the amplitude of oscillation is set by mixing ( $\sin^22\theta$ ), the phase by the splitting and the geometry ( $\Delta m^2L/4E$ ) — two independent knobs Nature turns. Finally, the distance over which the pattern repeats defines an oscillation length, the single number that tells an experimentalist where to put a detector.

L_{\text{osc}}=\frac{4\pi E}{\Delta m^2}=2.48\ \text{km}\times\frac{E[\text{GeV}]}{\Delta m^2[\text{eV}^2]} \qquad (6)

where: $L_{\text{osc}}$ — full spatial period of the oscillation; a detector placed near $L_{\text{osc}}/2$ (the first maximum) sees the strongest effect. Reading it: small splittings mean long oscillation lengths — the tiny solar $\Delta m^2\sim7.5\times10^{-5}\ \text{eV}^2$ forces baselines of hundreds of kilometres.

Derivation AWhy flavour and mass eigenstates need not coincide

The goal of this derivation is to locate the physical origin of the mismatch: to show that the flavour basis is singled out by the charged-current interaction while the mass basis is singled out by the mass (Yukawa) term of the Lagrangian, that these two operators are diagonal in generally different bases, and that the unitary rotation carrying one into the other is precisely the observable PMNS matrix. Nothing here assumes neutrinos are special — the same logic gives the CKM matrix for quarks; the point is that mixing is generic and its absence, not its presence, would demand an explanation.

Step 1. Write the leptonic charged-current interaction in the flavour basis, where by definition the $W$ couples each charged lepton to its own neutrino: $\mathcal{L}_{CC}\propto\sum_\alpha\bar\ell_{\alpha L}\gamma^\mu\nu_{\alpha L}W^-_\mu+\text{h.c.}$ This is diagonal — that diagonality is what defines the labels $\nu_e,\nu_\mu,\nu_\tau$ . Why: the flavour eigenstate is not a free choice; it is the neutrino field sitting next to $\ell_\alpha$ in the interaction that produces it, so "flavour" means "charged-current partner".

Step 2. Now add the mass terms. The charged-lepton mass matrix $M_\ell$ and the neutrino mass matrix $M_\nu$ arise from Yukawa couplings to the Higgs and are, in general, arbitrary non-diagonal matrices in this same flavour basis. There is no symmetry forcing them to be diagonal where the current is. Why: the interaction basis and the mass basis are fixed by different pieces of the Lagrangian, and generic matrices are not simultaneously diagonal.

Step 3. Diagonalise each mass matrix by a bi-unitary transformation: $V_\ell^\dagger M_\ell V_\ell'$ and $V_\nu^\dagger M_\nu V_\nu'$ are diagonal, defining the physical mass eigenstates (the states that propagate with definite mass). Rotating the charged leptons to their mass basis by $V_\ell$ and the neutrinos by $V_\nu$ leaves the charged current with a leftover matrix $U=V_\ell^\dagger V_\nu$ . Why: you cannot diagonalise the masses and keep the current diagonal at the same time unless $V_\ell=V_\nu$ ; the residue of failing to do so is a physical rotation sandwiched in the interaction.

Step 4. Identify $U$ as the PMNS matrix of eq. (1): in the mass basis the current reads $\mathcal{L}_{CC}\propto\sum_{\alpha i}\bar\ell_{\alpha L}\gamma^\mu U_{\alpha i}\nu_{iL}W^-_\mu$ , so a charged lepton $\ell_\alpha$ is produced with the combination $\sum_iU_{\alpha i}\nu_i$ — a flavour state. The bases coincide only in the special case $U=\mathbb{1}$ , which Nature declines to choose. Why: this exhibits the flavour eigenstate as a fixed linear combination of mass eigenstates, with coefficients that are the measurable entries of $U$ — the mixing is not put in by hand but is the unavoidable mismatch of two diagonalisations.

Go deeper · Derivation B · 1 minThe produced state, its conjugate weights, and orthonormality

Derivation BThe produced state, its conjugate weights, and orthonormality

The aim here is to pin down two facts that the oscillation calculation depends on and that students routinely get backwards: first, that the neutrino state emerging from a $W^+\to\ell^+\nu$ vertex carries $U^*$ rather than $U$ ; and second, that the flavour states so defined form an orthonormal set, $\langle\nu_\beta|\nu_\alpha\rangle=\delta_{\alpha\beta}$ , so that "produce a $\nu_e$ , detect a $\nu_\mu$ " is a well-posed transition with unit total probability. Both follow directly from unitarity of $U$ once the creation operator is handled carefully.

Step 1. In the mass basis the interaction term that creates a neutrino is proportional to $U_{\alpha i}\,\bar\ell_{\alpha L}\gamma^\mu\nu_{iL}$ ; the field $\nu_{iL}$ annihilates a neutrino, so the piece that creates the outgoing neutrino sits in the Hermitian conjugate and carries $U^*_{\alpha i}$ . Acting on the vacuum it builds $|\nu_\alpha\rangle=\sum_iU^*_{\alpha i}|\nu_i\rangle$ , eq. (2). Why: the conjugation is not cosmetic — reversed for antineutrinos ( $|\bar\nu_\alpha\rangle=\sum_iU_{\alpha i}|\bar\nu_i\rangle$ ), it is the seed of the difference between neutrino and antineutrino oscillations, i.e. leptonic CP violation.

Step 2. Compute the overlap of two flavour states using orthonormal mass states $\langle\nu_j|\nu_i\rangle=\delta_{ij}$ : $\langle\nu_\beta|\nu_\alpha\rangle=\sum_{i,j}U_{\beta j}U^*_{\alpha i}\langle\nu_j|\nu_i\rangle=\sum_iU_{\beta i}U^*_{\alpha i}$ . Why: this reduces a statement about flavour states to a single contraction of the mixing matrix, exactly the row-orthonormality object in eq. (3).

Step 3. Apply unitarity, $\sum_iU_{\beta i}U^*_{\alpha i}=\delta_{\alpha\beta}$ , to conclude $\langle\nu_\beta|\nu_\alpha\rangle=\delta_{\alpha\beta}$ : the flavour states are orthonormal, and $\sum_i|U_{\alpha i}|^2=1$ gives the composition weights their probability interpretation. Why: without unitarity the flavour "states" would not be a clean basis and total probability across flavours would not be conserved — every oscillation formula silently rests on this.

Step 4. Read off the composition of $\nu_e$ from the measured angles: $|U_{e1}|^2=\cos^2\theta_{12}\cos^2\theta_{13}=0.678$ , $|U_{e2}|^2=\sin^2\theta_{12}\cos^2\theta_{13}=0.300$ , $|U_{e3}|^2=\sin^2\theta_{13}=0.022$ , summing to $1.000$ . Why: this turns the abstract unitary rotation into the concrete pie of the first figure — the electron neutrino really is mostly $\nu_1$ , and the weights are experimental numbers, not conventions.

Go deeper · Derivation C · 2 minFrom coherent superposition to the oscillation probability

Derivation CFrom coherent superposition to the oscillation probability

The objective is to carry the produced superposition of eq. (2) forward in time with the phases of eq. (4), project onto a detected flavour, and obtain the two-flavour probability of eq. (5) — making explicit at each step that the effect is pure quantum interference and that it vanishes identically if the masses are equal or the mixing is switched off. We keep two flavours ( $\nu_\alpha,\nu_\beta$ ) and a single angle $\theta$ for transparency; the three-flavour generalisation is algebra, not new physics.

Step 1. Parametrise the two-flavour rotation as $|\nu_\alpha\rangle=\cos\theta\,|\nu_1\rangle+\sin\theta\,|\nu_2\rangle$ and $|\nu_\beta\rangle=-\sin\theta\,|\nu_1\rangle+\cos\theta\,|\nu_2\rangle$ . Evolve each mass component by its phase: $|\nu_\alpha(t)\rangle=\cos\theta\,e^{-iE_1t}|\nu_1\rangle+\sin\theta\,e^{-iE_2t}|\nu_2\rangle$ . Why: only the mass states have simple time evolution, so we must pass through the mass basis to propagate at all — the flavour state has no clock of its own.

Step 2. Project onto $\langle\nu_\beta|$ to get the transition amplitude: $A_{\alpha\to\beta}=\langle\nu_\beta|\nu_\alpha(t)\rangle=\sin\theta\cos\theta\,(e^{-iE_2t}-e^{-iE_1t})$ . The two mass paths interfere; their phase difference is the whole story. Why: the amplitude is a difference of two complex exponentials — a textbook two-slit interference, with the "slits" being the two mass eigenstates rather than two points in space.

Step 3. Square to a probability and use $|e^{-iE_2t}-e^{-iE_1t}|^2=2[1-\cos((E_2-E_1)t)]=4\sin^2\!\big(\tfrac{(E_2-E_1)t}{2}\big)$ , together with $2\sin\theta\cos\theta=\sin2\theta$ : $P_{\alpha\to\beta}=\sin^22\theta\,\sin^2\!\big(\tfrac{(E_2-E_1)t}{2}\big)$ . Why: the mixing angle emerges as the interference amplitude $\sin^22\theta$ and the mass split as the interference phase — the two are cleanly factorised.

Step 4. Insert the ultrarelativistic energies of eq. (4): $E_2-E_1\approx(m_2^2-m_1^2)/2E=\Delta m^2/2E$ , and set $t\approx L$ (natural units) to reach $P_{\alpha\to\beta}=\sin^22\theta\,\sin^2(\Delta m^2L/4E)$ , eq. (5). Why: this is where the abstract phase becomes a laboratory quantity — restoring $\hbar,c$ produces the practical constant $1.267$ and hence a definite prediction for where the oscillation maxima fall.

Step 5. Check the limits: $\Delta m^2\to0$ gives $P\to0$ (no mass difference, no oscillation) and $\theta\to0$ gives $P\to0$ (no mixing, no oscillation) — either alone kills the effect, so oscillation is direct proof of both nonzero mass splitting and nonzero mixing. Why: this is the logical core of the whole field — a single observed oscillation establishes that at least one neutrino is massive, which the Standard Model did not require.

Limiting cases

Case	Result	Meaning
Degenerate masses $m_1=m_2=m_3$	$\Delta m^2_{ij}=0$ , $P_{\alpha\to\beta}=0$	equal phases; $U$ is unobservable and flavour is exactly conserved
No mixing $U=\mathbb{1}$	flavour eigenstate $=$ mass eigenstate	each $\nu_\alpha$ propagates as one definite-mass particle; no flavour change
Short baseline $\Delta m^2L/4E\ll1$	$P_{\alpha\to\beta}\approx\sin^22\theta\,(1.267\,\Delta m^2L/E)^2$	quadratic rise from the source; the near-detector regime
Long baseline or wide $L/E$ spread	$P_{\alpha\to\beta}\to\tfrac12\sin^22\theta$	rapid phases average out; solar/astrophysical neutrinos sit here
Maximal mixing $\theta=45^\circ$	$\sin^22\theta=1$	complete flavour conversion possible; atmospheric $\nu_\mu\!\leftrightarrow\!\nu_\tau$
One dominant splitting	3-flavour $\to$ effective 2-flavour	why the simple formula (5) describes most single experiments

Common trapThe flavour states are not just three massive particles that swap identities

The seductive wrong picture is that $\nu_e,\nu_\mu,\nu_\tau$ are three particles with definite masses that occasionally "morph" into one another. They are not particles with definite mass at all. Only $\nu_1,\nu_2,\nu_3$ have definite masses and propagate with definite phase; a flavour state is a superposition of these, and it has no single well-defined energy or rest mass — asking "what is the mass of the $\nu_\mu$ ?" is as ill-posed as asking for the position of a momentum eigenstate. What "oscillates" is not a physical object changing species but the interference pattern among the fixed mass components, read out through whichever charged lepton the neutrino eventually produces. Two consequences people miss: (i) if the masses were equal there would be no oscillation even with full mixing, because there would be no relative phase to accumulate; and (ii) the oscillation requires coherence — the mass components must stay in phase-lock, which fails if the production or detection process localises the state tightly enough to reveal which mass eigenstate was involved. Oscillation is quantum interference, not metamorphosis.

Worked examples

1 · The mass composition of the electron neutrino (unitarity in numbers)

Using the measured mixing angles $\sin^2\theta_{12}=0.307$ and $\sin^2\theta_{13}=0.022$ , find the probabilities that a freshly produced $\nu_e$ is found to be $\nu_1$ , $\nu_2$ , or $\nu_3$ , and verify they sum to one.

Step 1. In the standard PMNS parametrisation the electron row is $U_{e1}=\cos\theta_{12}\cos\theta_{13}$ , $U_{e2}=\sin\theta_{12}\cos\theta_{13}$ , $U_{e3}=\sin\theta_{13}e^{-i\delta}$ . The probabilities are the squared magnitudes, and the CP phase $\delta$ drops out of each single $|U_{ei}|^2$ .

Step 2. Evaluate with $\cos^2\theta_{13}=1-0.022=0.978$ : $|U_{e1}|^2=(1-0.307)(0.978)=0.693\times0.978=\mathbf{0.678}$ ; $|U_{e2}|^2=(0.307)(0.978)=\mathbf{0.300}$ ; $|U_{e3}|^2=\mathbf{0.022}$ .

Step 3. Sum: $0.678+0.300+0.022=\mathbf{1.000}$ — exactly the row-unitarity condition $\sum_i|U_{ei}|^2=1$ from eq. (3). Physically, a $\nu_e$ born at a reactor or in the Sun is $68\%$ $\nu_1$ , $30\%$ $\nu_2$ , and just $2.2\%$ $\nu_3$ ; the smallness of $|U_{e3}|^2$ is why $\theta_{13}$ was the last angle to be measured (Daya Bay/RENO, 2012). A produced ν_e is 0.68 ν₁ + 0.30 ν₂ + 0.022 ν₃, summing to 1.

2 · Where to put the detector — atmospheric oscillation length and the T2K maximum

The atmospheric splitting is $\Delta m^2\approx2.5\times10^{-3}\ \text{eV}^2$ with near-maximal mixing $\sin^22\theta\approx1$ . (a) Find the oscillation length for a $1\ \text{GeV}$ $\nu_\mu$ . (b) The T2K experiment fires $\nu_\mu$ over $L=295\ \text{km}$ at a peak energy $E\approx0.6\ \text{GeV}$ ; compute the survival probability and comment.

Step 1. Oscillation length from eq. (6): $L_{\text{osc}}=2.48\ \text{km}\times\dfrac{E[\text{GeV}]}{\Delta m^2[\text{eV}^2]}=2.48\times\dfrac{1}{2.5\times10^{-3}}=\mathbf{992\ \text{km}}$ . The first full disappearance is at half of this, $L/E\approx\mathbf{496\ \text{km}/\text{GeV}}$ — a $1\ \text{GeV}$ neutrino vanishes maximally after $\sim500\ \text{km}$ , comparable to Earth's radius, which is exactly why the effect first showed up in atmospheric neutrinos crossing the planet.

Step 2. T2K phase from eq. (5): $\Phi=1.267\,\dfrac{\Delta m^2L}{E}=1.267\times\dfrac{(2.5\times10^{-3})(295)}{0.6}=1.267\times1.229=\mathbf{1.557\ \text{rad}}$ . This sits essentially at $\pi/2=1.571$ — T2K was designed to place its far detector at the first oscillation maximum.

Step 3. Survival probability: $P(\nu_\mu\to\nu_\mu)=1-\sin^22\theta\,\sin^2\Phi=1-(1)\sin^2(1.557)=1-0.9998=\mathbf{0.0002}$ . Almost every muon neutrino has disappeared (converted to $\nu_\tau$ , below threshold to make a $\tau$ ) — the near-total dip seen in the second figure. L_osc = 992 km; T2K sits at Φ ≈ π/2 with survival ≈ 0.0002, near-complete ν_μ disappearance.

3 · Why KamLAND is 180 km from the reactors (quick estimate)

The solar splitting is $\Delta m^2_{21}\approx7.5\times10^{-5}\ \text{eV}^2$ . Estimate the oscillation length for reactor antineutrinos of $E\approx4\ \text{MeV}$ .

Step 1. With $E=4\ \text{MeV}=0.004\ \text{GeV}$ , eq. (6) gives $L_{\text{osc}}=2.48\times\dfrac{0.004}{7.5\times10^{-5}}\approx\mathbf{132\ \text{km}}$ , so the first maximum falls near $66\ \text{km}$ but the sensitive region for a broad reactor spectrum extends to $\sim150$ – $200\ \text{km}$ . The ≈180 km flux-weighted baseline of KamLAND is matched to the solar Δm², which is how a reactor experiment confirmed the solar oscillation.

Physics of the portrait

The idea that a produced neutrino need not be a mass eigenstate — and that this would make flavour oscillate — belongs to one restless, brilliant, and politically star-crossed physicist who saw it two decades before anyone could test it.

Origins · 1 minBruno Pontecorvo and how this was discovered

Bruno Pontecorvo

1913–1993 · neutrino mixing and oscillations

Bruno Pontecorvo was one of Enrico Fermi's "via Panisperna boys" in 1930s Rome, co-author of the slow-neutron work that won Fermi the Nobel Prize. A Jewish communist, he fled fascist Italy in 1936, worked in Paris under the Joliot-Curies, then in the United States and at Canada's Chalk River reactor, where he invented the chlorine radiochemical method later used by Ray Davis to catch solar neutrinos. In 1950, at the height of the atomic-spy panic, he defected with his family to the Soviet Union and vanished from Western physics for years, resurfacing at Dubna. It was there, in 1957–58, that he proposed neutrino oscillations — first by analogy with the $K^0$ – $\bar K^0$ system he admired, imagining neutrino–antineutrino transitions, and by 1967 the modern picture of flavour oscillation driven by a mass–flavour mismatch. He predicted the solar-neutrino deficit before Davis measured it and argued for a distinct muon neutrino before the 1962 two-neutrino experiment. The mixing matrix bears his initial first — PMNS, for Pontecorvo–Maki–Nakagawa–Sakata — and the 2015 Nobel Prize for the discovery of oscillations, awarded to Kajita and McDonald, was in a real sense the confirmation of an idea Pontecorvo had chased, from three countries and against the current of his times, for nearly forty years. He died in Dubna in 1993, five years before Super-Kamiokande proved him right.

History

1957

— Pontecorvo proposes neutrino oscillations by analogy with the neutral kaons

1962

— Maki, Nakagawa and Sakata write flavour states as mixtures of mass states

1968

— the Homestake experiment reveals the solar-neutrino deficit

1998

— Super-Kamiokande observes atmospheric ν_μ disappearance, establishing oscillation

2001

— SNO shows solar ν_e convert to other flavours with the total flux conserved

Going further

Texts. Thomson, Modern Particle Physics, §13.3 (mass and weak eigenstates) and §13.4 (neutrino oscillations) · Griffiths, Introduction to Elementary Particles, 2nd ed., §11 · Halzen & Martin, Quarks and Leptons, Ch. 12 · Schwartz, Quantum Field Theory and the Standard Model, §29 (neutrino masses and mixing) · Peskin & Schroeder, An Introduction to QFT, §20 (the mass matrix and mixing).

Provenance. The charged-current vertex $\ell_\alpha$ – $\nu_\alpha$ – $W$ and its $V-A$ structure are recalled by name from aq-pp-weak-charged-current and aq-pp-v-minus-a; the stationary-state phase $e^{-iEt}$ and superposition principle from aq-pp-nonrel-qm-review; the operational definition of the three flavours from aq-pp-neutrino-flavours. The explicit oscillation phenomenology built on this article is owned downstream by aq-pp-two-flavour-oscillations (formula and limits), aq-pp-three-flavour-pmns (the full $U$ , its angles and the CP phase $\delta$ ), and aq-pp-neutrino-mass-mechanisms (why the masses are nonzero at all). A verifier should confirm: (1) $\sum_i|U_{ei}|^2=1$ with $|U_{ei}|^2=\{0.678,0.300,0.022\}$ from $\sin^2\theta_{12}=0.307,\sin^2\theta_{13}=0.022$ ; (2) $L_{\text{osc}}=2.48\,\text{km}\,E/\Delta m^2=992\,\text{km}$ for $E=1\,\text{GeV},\Delta m^2=2.5\times10^{-3}\,\text{eV}^2$ ; (3) the T2K phase $\Phi=1.267\,\Delta m^2L/E=1.557$ at $L=295\,\text{km},E=0.6\,\text{GeV}$ ; (4) the two-flavour probability $P=\sin^22\theta\,\sin^2(\Delta m^2L/4E)$ from the interference of two evolved mass states.

Prerequisites. The Weak Charged Current · Non-relativistic QM Review (stationary states and superposition) · Neutrino Flavours. Leads to. Two-Flavour Oscillations · The Three-Flavour PMNS Matrix · Neutrino Mass Mechanisms.

Ask the tutor: Ask me to trace a single $\nu_\mu$ from a $W$ vertex to a far detector — writing its superposition, evolving each mass component, and watching the flavour probability oscillate step by step.

Why does the neutrino leave the vertex as a superposition rather than one definite mass eigenstate?

Because the charged-current interaction that makes it does not care about mass — it couples the electron to the neutrino field that shares its flavour label, and that field is a fixed combination of the three mass fields. At production, any of $\nu_1,\nu_2,\nu_3$ can come out with the electron, and since no available measurement distinguishes them (their mass differences are far too small to resolve kinematically), the correct quantum description is a coherent sum of all three, weighted by the mixing amplitudes $U^*_{ei}$ . It is the same reason a photon passing a two-slit screen is described by a superposition of both paths: when which-path information is unavailable, amplitudes add.

If I can't measure a flavour state's mass, in what sense do neutrinos "have mass"?

The masses belong to the mass eigenstates $\nu_1,\nu_2,\nu_3$ , which are perfectly well-defined particles with sharp masses $m_1,m_2,m_3$ . What you cannot do is assign a single mass to a flavour state, because it is a blend of the three. Oscillation experiments are sensitive only to the differences $\Delta m^2_{ij}=m_i^2-m_j^2$ — they prove at least two masses are nonzero and unequal, but not the absolute scale. That scale is chased by other methods: the tritium beta-decay endpoint (KATRIN), cosmology's effect on structure formation, and neutrinoless double-beta decay. So "neutrinos have mass" is a statement about $\nu_1,\nu_2,\nu_3$ , established indirectly through the flavour oscillations that only nonzero splittings allow.

What is the difference between the PMNS matrix here and the CKM matrix for quarks?

They are the same mathematical object in two sectors. In the quark sector the up-type and down-type mass matrices are diagonalised by different rotations, and the leftover mismatch in the charged current is the CKM matrix $V$ ; in the lepton sector the charged-lepton and neutrino mass matrices are diagonalised differently, and the mismatch is the PMNS matrix $U$ . Both are $3\times3$ unitary, both carry three mixing angles and a CP-violating phase. The striking difference is in the numbers: CKM is nearly diagonal (quark mixing is small, the Cabibbo angle $\approx13^\circ$ being the largest), whereas PMNS has two large angles ( $\theta_{12}\approx33^\circ$ , $\theta_{23}\approx45^\circ$ ) and only $\theta_{13}\approx8.6^\circ$ small. Why leptons mix so much more than quarks is an open question.

Does the complex conjugate in |ν_α⟩ = Σ U*_αi |ν_i⟩ actually matter?

For a single oscillation probability like $\nu_\mu\to\nu_\mu$ it does not — the survival probability depends only on $|U_{\mu i}|^2$ , and conjugation leaves magnitudes untouched. It matters the moment you compare neutrinos with antineutrinos. Antineutrinos are produced with $U$ (not $U^*$ ), so $P(\nu_\mu\to\nu_e)$ and $P(\bar\nu_\mu\to\bar\nu_e)$ can differ if $U$ contains an irreducible complex phase $\delta$ . That difference is leptonic CP violation, one of the headline goals of DUNE and Hyper-Kamiokande, and a candidate ingredient in explaining why the universe is made of matter. So the conjugate is invisible in the simplest measurement and decisive in the deepest one.

Why don't the electron, muon and tau themselves oscillate the way neutrinos do?

In principle the same mismatch exists, but for charged leptons the mass basis and the interaction basis are conventionally chosen to coincide — we define the electron, muon and tau as the mass eigenstates and push the entire mixing into the neutrino sector, which is a legitimate freedom because only the product $V_\ell^\dagger V_\nu=U$ is physical. More practically, the charged leptons have hugely different, well-resolved masses and interact electromagnetically almost immediately, so any coherent superposition is destroyed (decohered) essentially at birth — you always know whether you have an electron or a muon. Neutrinos are unique in staying coherent over astronomical distances because they interact so weakly, which is precisely what lets the interference survive to be observed.

How does the oscillation "know" the baseline L — what carries the phase?

Each mass eigenstate propagates as a plane wave $e^{i(pL-E_it)}$ , and because the three have slightly different energies (or momenta) at fixed conditions, they accumulate slightly different phases as they travel. The relative phase between components, $\Delta\phi\approx\Delta m^2L/2E$ , grows linearly with the distance $L$ the packet has covered, and it is this relative phase that controls the interference when you finally project onto a flavour at the detector. So the baseline enters through the ordinary propagation phase of quantum mechanics — nothing exotic. A careful wave-packet treatment confirms the naive plane-wave answer and also shows why coherence is eventually lost at enormous $L$ : the packets of different mass, moving at slightly different group velocities, drift apart and stop overlapping.