Neutrino flavours

A neutrino leaves no track. It carries no electric charge, no colour, and no way to announce itself as it slips through a detector — so the one question you would most like to ask it, which kind of neutrino are you?, cannot be answered by looking at the neutrino at all. It can only be answered by watching what it does. When a neutrino finally deigns to interact through the weak charged current, it emits a charged lepton — an electron, a muon, or a tau — and the identity of that lepton is the only label the neutrino ever wears. This is the operational meaning of neutrino flavour: $\nu_e$ is, by definition, the neutrino that is born alongside an electron and that makes an electron when it interacts; $\nu_\mu$ and $\nu_\tau$ are defined the same way through the muon and the tau. That definition is forced on us by the structure of the weak interaction, which couples each charged lepton to exactly one neutrino partner and no other. The regularity was written into the earliest data: neutrinos from $\pi^+\to\mu^+\nu_\mu$ made muons and never electrons, and this stubborn separation is codified as the conservation of three lepton flavour numbers. For decades that seemed the whole story, and the neutrinos were taken to be massless. But the flavour states $\nu_e,\nu_\mu,\nu_\tau$ are eigenstates of the weak interaction, and there is no law that they must also be eigenstates of mass. They are not. The flavour a neutrino wears at birth is a coherent superposition of three states of definite mass $\nu_1,\nu_2,\nu_3$ — and that single mismatch, expressed through the PMNS matrix, is the seed of neutrino oscillations and the reason the number three matters so much.

Key takeaways

A neutrino is never seen directly; its flavour is read off the charged lepton it produces in a charged-current weak interaction — $\nu_e{\to}e$ , $\nu_\mu{\to}\mu$ , $\nu_\tau{\to}\tau$ .
The weak charged current is flavour-diagonal: the $W$ couples each $\nu_\ell$ to its own lepton $\ell$ , so every vertex conserves the three lepton flavour numbers $L_e,L_\mu,L_\tau$ .
There are exactly three light active flavours. The LEP measurement of the invisible $Z$ width gives $N_\nu=2.984\pm0.008$ — a decimal-place count of the neutrino species lighter than $M_Z/2$ .
The flavour states $\nu_e,\nu_\mu,\nu_\tau$ (weak eigenstates) are distinct from the mass eigenstates $\nu_1,\nu_2,\nu_3$ ; they are related by a unitary rotation, the PMNS matrix $U$ .
Because $|\nu_\alpha\rangle=\sum_i U^*_{\alpha i}|\nu_i\rangle$ with $U\neq\mathbb{1}$ , a definite-flavour neutrino has no definite mass — the origin of oscillations, which violate $L_\ell$ while conserving total $L$ .
Neutrinos were long assumed massless because early data showed only exact flavour conservation; oscillation experiments overturned that from 1998 onward.
Key idea: A neutrino's flavour is not measured directly but inferred from the charged lepton it creates in a weak interaction — and these weak-interaction flavour states are physically distinct from the states of definite mass, so a state of definite flavour is a superposition of masses.

The core insight

A neutrino's flavour is not something you measure directly but is inferred from the charged lepton it creates in a weak interaction — and these weak-interaction flavour states are physically distinct from the states of definite mass.

Where this lives today

Flavour is the organising principle of the entire modern neutrino programme. Because a neutrino's flavour is tagged only by the lepton it makes, every oscillation experiment is a flavour-identification machine: Super-Kamiokande and IceCube read atmospheric flavours by distinguishing sharp muon rings from fuzzy electron showers in Cherenkov light; the long-baseline experiments T2K, NOvA and the flagship DUNE fire accelerator $\nu_\mu$ beams hundreds to $1300$ kilometres through rock and count how many arrive as $\nu_e$ , chasing the CP-violating phase $\delta$ in the PMNS matrix. The reactor experiments Daya Bay, RENO and Double Chooz measured $\theta_{13}$ by watching $\bar\nu_e$ disappear over a kilometre, and JUNO is building a $20$ -kiloton detector to resolve the neutrino mass ordering. On the mass front, KATRIN bounds the effective electron-neutrino mass at $m_\beta<0.45$ eV through the tritium $\beta$ -decay endpoint, while cosmology limits $\sum m_i<0.12$ eV — both measuring the mass basis flavour is built from. And the deepest open question, whether the neutrino is its own antiparticle, is a question about lepton number: neutrinoless double-beta-decay searches (KamLAND-Zen, LEGEND) hunt for a process violating total $L$ by two units, which would mean the flavour bookkeeping of equation (2) is fundamentally incomplete.

The physical picture

There is no such thing as looking at a neutrino and seeing "electron-type" written on it. What actually happens is that the neutrino strikes a nucleus, exchanges a $W$ boson, and turns into a charged lepton whose identity you can measure — a straight penetrating track for a muon, a showering cascade for an electron, a short kinked track for a tau. That charged lepton is the flavour tag. The first figure lays the three cases side by side: the neutrino is invisible on the way in, and only the lepton it hands you on the way out names which flavour arrived.

Figure

Flavour is read off the charged lepton, never the neutrino. A charged-current interaction of a νₑ emits an e⁻, a ν_μ emits a μ⁻, and a ν_τ emits a τ⁻; the W couples each neutrino to one lepton only, so the outgoing lepton is the flavour tag

The three flavours are not merely a naming scheme — nature stops at exactly three of them, and we can count them to three decimal places without ever building a neutrino detector. The trick is the $Z$ boson. Every neutrino species lighter than half the $Z$ mass opens an invisible decay channel $Z\to\nu\bar\nu$ , and each channel widens the $Z$ resonance and lowers its peak. So the height of the $Z$ line, scanned in $e^+e^-$ collisions at LEP, is a direct tally of how many light neutrinos exist — flavour-blind, because it counts all species that couple to the $Z$ , whether or not we have ever caught one. The second figure is the measurement itself: three candidate curves for $N_\nu=2,3,4$ and the data sitting squarely on three.

Figure

The Z lineshape counts the light neutrino flavours. Extra invisible νν̄ channels widen the Z and lower its peak, so the hadronic peak height selects N_ν. The data fall on the N_ν = 3 curve; the LEP fit gives N_ν = 2.984 ± 0.008

The core equations

The physics of flavour lives in four statements: the current that defines it, the numbers it conserves, the rotation that separates it from mass, and the count that fixes it at three. Everything begins with the leptonic charged current — the piece of the weak Lagrangian in which the $W$ boson couples a charged lepton to a neutrino. The sum runs over the three generations, and the crucial point is that it is diagonal: each $\bar\ell$ sits next to its own $\nu_\ell$ and no other.

j^{\mu}_{\text{CC}}=\sum_{\ell=e,\mu,\tau}\bar{\nu}_\ell\,\gamma^\mu\tfrac12(1-\gamma^5)\,\ell \qquad (1)

where: $\nu_\ell$ — the neutrino field of flavour $\ell$ ; $\ell$ — the charged-lepton field ( $e,\mu,\tau$ ); $\tfrac12(1-\gamma^5)$ — the left-handed (V−A) projector, recalled from aq-pp-v-minus-a. Read it as: the $W$ vertex converts a charged lepton into its neutrino and nothing else — this diagonal pairing is the operational definition of flavour. Because the current is flavour-diagonal, each generation carries a separately conserved quantum number. Assigning $+1$ to a lepton and its neutrino and $-1$ to their antiparticles, the three lepton flavour numbers are unchanged at every Standard-Model vertex.

L_e,\;L_\mu,\;L_\tau\;\text{conserved}:\quad \Delta L_e=\Delta L_\mu=\Delta L_\tau=0 \qquad (2)

where: $L_\ell(\nu_\ell)=L_\ell(\ell^-)=+1$ and $L_\ell(\bar\nu_\ell)=L_\ell(\ell^+)=-1$ , all other particles zero. Read it as: a $\nu_\mu$ can make a $\mu^-$ but never an $e^-$ , because the latter would change $L_e$ by $+1$ and $L_\mu$ by $-1$ — this is exactly what the two-neutrino experiment saw. The flavour states are eigenstates of the weak interaction; the mass eigenstates $\nu_i$ that propagate as free particles are a different basis. A unitary matrix — the Pontecorvo–Maki–Nakagawa–Sakata matrix — rotates one into the other.

|\nu_\alpha\rangle=\sum_{i=1}^{3}U^*_{\alpha i}\,|\nu_i\rangle,\qquad \alpha=e,\mu,\tau,\quad U^\dagger U=\mathbb{1} \qquad (3)

where: $|\nu_\alpha\rangle$ — flavour (weak) eigenstate; $|\nu_i\rangle$ — mass eigenstate with mass $m_i$ ; $U$ — the $3\times3$ unitary PMNS matrix, parametrised by three angles $\theta_{12},\theta_{23},\theta_{13}$ and a phase $\delta$ . Read it as: if $U=\mathbb{1}$ the two bases coincide and flavour is a good quantum number forever; the observed $U\neq\mathbb{1}$ is what makes flavour non-conserved for a propagating neutrino. Finally, the number of light active flavours is fixed experimentally by the invisible width of the $Z$ boson: each species that couples to the $Z$ adds one $Z\to\nu\bar\nu$ channel of Standard-Model strength.

\Gamma_{\text{inv}}=N_\nu\,\Gamma_{\nu\nu},\qquad N_\nu=\frac{\Gamma_{\text{inv}}}{\Gamma_{\ell\ell}}\left(\frac{\Gamma_{\ell\ell}}{\Gamma_{\nu\nu}}\right)_{\!\text{SM}}=2.984\pm0.008 \qquad (4)

where: $\Gamma_{\text{inv}}$ — invisible $Z$ width; $\Gamma_{\nu\nu}$ — width per neutrino species ( $167.6$ MeV in the SM); $\Gamma_{\ell\ell}$ — width to one charged-lepton pair; the ratio form cancels common systematics. Read it as: nature chose exactly three light neutrinos that feel the weak force — a number, not a hope. The consequence of $U\neq\mathbb{1}$ — developed in the articles this one leads to, and quoted here only to close the loop — is that flavour survival is not certain but oscillates with distance, governed by the mass-squared splittings.

P(\nu_\alpha\!\to\!\nu_\beta)=\sin^2 2\theta\,\sin^2\!\left(\frac{\Delta m^2\,L}{4E}\right)\;\xrightarrow{\text{units}}\;\sin^2 2\theta\,\sin^2\!\left(1.27\,\frac{\Delta m^2[\text{eV}^2]\,L[\text{km}]}{E[\text{GeV}]}\right) \qquad (5)

where: $\theta$ — a two-flavour mixing angle; $\Delta m^2=m_2^2-m_1^2$ — mass-squared splitting; $L$ — baseline; $E$ — neutrino energy. Read it as: the very existence of a non-trivial oscillation requires both $\theta\neq0$ (mixing) and $\Delta m^2\neq0$ (non-degenerate masses) — flavour physics and mass physics are inseparable.

Derivation AFrom the charged current to three conserved lepton numbers

The goal is to show that the operational definition of flavour is not an arbitrary label but a direct consequence of the flavour-diagonal structure of the leptonic charged current, and that this structure enforces the separate conservation of $L_e$ , $L_\mu$ and $L_\tau$ at every Standard-Model vertex — the exact regularity that early beam experiments observed and that held with no known exception until neutrino oscillations were discovered. We build the interaction vertex, read off the quantum-number bookkeeping, and check it against a real process, so that "the neutrino that makes a muon" becomes a theorem rather than a naming convention.

Step 1. Write the interaction. The weak charged-current Lagrangian couples the $W$ to the current of equation (1), $\mathcal{L}\supset\frac{g_w}{\sqrt2}\,W^+_\mu\,\bar\nu_\ell\gamma^\mu\tfrac12(1-\gamma^5)\ell + \text{h.c.}$ , summed over $\ell=e,\mu,\tau$ . Each term pairs one charged lepton with one neutrino of the same generation index. Why: there is no $\bar\nu_e\gamma^\mu\mu$ term in the Standard Model — the current is built diagonal by construction, and that single fact is the entire content of "flavour".

Step 2. Assign the quantum numbers. Give each lepton and its neutrino $L_\ell=+1$ and their antiparticles $L_\ell=-1$ , with all quarks, gauge bosons and other leptons carrying zero. Because a vertex from Step 1 always creates or destroys a $\nu_\ell$ together with an $\ell$ of the same family, the net change in each $L_\ell$ across any vertex is zero. Why: a conserved additive charge is exactly a quantity that every interaction term leaves invariant; the diagonal current guarantees this for all three families independently.

Step 3. Test it on muon decay. The process $\mu^-\to e^-\bar\nu_e\nu_\mu$ has, before: $L_\mu=+1$ , $L_e=0$ ; after: $L_\mu=+1$ (from $\nu_\mu$ ), $L_e=+1-1=0$ (from $e^-$ and $\bar\nu_e$ ). Both are conserved, and this is why muon decay must produce two neutrinos of different flavour rather than a single photon. Why: forbidding $\mu^-\to e^-\gamma$ — never observed, branching ratio $<4\times10^{-13}$ — is precisely the statement that $L_e$ and $L_\mu$ are separately conserved.

Step 4. State the caveat. This conservation is exact only in the flavour basis with massless neutrinos, i.e. when the current and mass bases coincide ( $U=\mathbb{1}$ ). Once neutrinos have mass and $U\neq\mathbb{1}$ , individual $L_\ell$ are violated by propagation (oscillation), though the total $L=L_e+L_\mu+L_\tau$ survives if neutrinos are Dirac particles. Why: the vertex still conserves flavour, but the state that propagates between vertices is a superposition of masses, so the flavour measured at the far end need not match the one produced — Derivation C. *These three checks close the derivation: $L_e,L_\mu,L_\tau$ are each conserved at every vertex of eq. (1); $\mu^-\to e^-\bar\nu_e\nu_\mu$ balances all three numbers; and $\mu^-\to e^-\gamma$ is forbidden by separate $L_e$ and $L_\mu$ conservation, in agreement with the MEG upper limit of $<4.2\times10^{-13}$ .*

Go deeper · Derivation B · 2 minCounting the light neutrino flavours from the invisible Z width

Derivation BCounting the light neutrino flavours from the invisible Z width

The aim is to turn the abstract statement "there are three neutrinos" into a measured number with an error bar, using nothing but the total, hadronic and leptonic decay widths of the $Z$ boson determined from the LEP lineshape. The strategy is to isolate the width the $Z$ spends decaying into things no detector can see — the neutrinos — and divide it by the Standard-Model width of a single $\nu\bar\nu$ channel; the ratio is the number of light species that couple to the $Z$ , whether or not any of them has ever been individually detected. This is the single most precise flavour count in physics and it fixes $N_\nu$ before oscillations enter the story.

Step 1. Isolate the invisible width. The total $Z$ width partitions as $\Gamma_Z=\Gamma_{\text{had}}+3\Gamma_{\ell\ell}+\Gamma_{\text{inv}}$ , using lepton universality for the three charged-lepton channels. Solving, $\Gamma_{\text{inv}}=\Gamma_Z-\Gamma_{\text{had}}-3\Gamma_{\ell\ell}$ . Why: everything the $Z$ decays into is either seen (hadrons, charged leptons) or unseen (neutrinos), so the invisible width is a pure subtraction of measured quantities — no model of the neutrino is needed.

Step 2. Put in the LEP numbers. With $\Gamma_Z=2495.2$ MeV, $\Gamma_{\text{had}}=1744.4$ MeV and $\Gamma_{\ell\ell}=83.98$ MeV, $\Gamma_{\text{inv}}=2495.2-1744.4-3(83.98)=498.9$ MeV. Why: this width is far larger than one charged-lepton channel and accounts for one-fifth of all $Z$ decays — the "missing" energy that a hermetic detector infers from momentum imbalance.

Step 3. Divide by one neutrino channel. The Standard-Model width per species is $\Gamma_{\nu\nu}=167.6$ MeV, so the direct count is $N_\nu=498.9/167.6=2.98$ . Using instead the ratio form $N_\nu=(\Gamma_{\text{inv}}/\Gamma_{\ell\ell})(\Gamma_{\ell\ell}/\Gamma_{\nu\nu})_{\text{SM}}=(498.9/83.98)(0.5023)=2.984$ cancels the overall coupling and reduces the error to $\pm0.008$ . Why: the ratio to the well-measured leptonic width divides out common experimental and theoretical uncertainties, sharpening a two-figure count into a three-decimal one.

Step 4. Interpret the result. $N_\nu=2.984\pm0.008$ is consistent with exactly three and rules out a fourth light neutrino that couples to the $Z$ with standard strength at more than $99.99\%$ confidence. A fourth generation would need a neutrino heavier than $M_Z/2\approx45.6$ GeV — enormously heavier than the others, which no model naturally supplies. Why: the count is flavour-blind — it sees any species coupling to the $Z$ — so it constrains even neutrinos too weakly interacting to detect directly, provided they are light and "active". *These are the key numerical results of the derivation: $\Gamma_{\text{inv}}=498.9$ MeV; the direct count gives $N_\nu=2.98$ and the ratio method sharpens this to $N_\nu=2.984\pm0.008$ ; a fourth active light neutrino is excluded unless $m_\nu>M_Z/2=45.6$ GeV.*

Go deeper · Derivation C · 2 minWhy a state of definite flavour has no definite mass

Derivation CWhy a state of definite flavour has no definite mass

The goal is to make precise the article's central claim — that flavour and mass are different bases — and to show that this mismatch is not a subtlety but the mechanism of neutrino oscillations. We start from the unitary rotation of equation (3), propagate the mass eigenstates as the true stationary states they are, and project back onto flavour, revealing that the flavour content of a beam changes with distance whenever the masses differ. The derivation stops at the two-flavour survival probability, handing the full three-flavour treatment to the articles this one leads to.

Step 1. Separate the bases. A neutrino produced with a muon is, by definition, the flavour state $|\nu_\mu\rangle=\sum_i U^*_{\mu i}|\nu_i\rangle$ . The $|\nu_i\rangle$ are the eigenstates of the free Hamiltonian with definite masses $m_i$ ; the $|\nu_\mu\rangle$ is a fixed superposition of them. Why: the weak vertex produces a flavour eigenstate, but only mass eigenstates have definite energy and therefore definite time evolution — the two facts cannot both be diagonal in the same basis unless $U=\mathbb{1}$ .

Step 2. Propagate. Each mass eigenstate evolves with its own phase, $|\nu_i(L)\rangle=e^{-i(E_i t-p_i L)}|\nu_i\rangle$ , and for ultrarelativistic neutrinos of common energy $E$ , $E_i-p_i\approx m_i^2/2E$ . The components therefore drift out of step at a rate set by $\Delta m^2_{ij}=m_i^2-m_j^2$ . Why: because the masses differ, the fixed superposition that was "pure $\nu_\mu$ " at the source is a different superposition downstream — the relative phases have rotated it.

Step 3. Project onto flavour. The amplitude to find flavour $\beta$ at distance $L$ is $\mathcal{A}_{\mu\to\beta}=\sum_i U^*_{\mu i}\,e^{-i m_i^2 L/2E}\,U_{\beta i}$ . In the two-flavour case, squaring gives $P(\nu_\mu\!\to\!\nu_\beta)=\sin^2 2\theta\,\sin^2(\Delta m^2 L/4E)$ . Why: the interference between mass eigenstates with different phases is what converts one flavour into another — no interference (no phase difference, i.e. $\Delta m^2=0$ ) means no oscillation, even with large mixing.

Step 4. Read the two requirements. Oscillation demands both $U\neq\mathbb{1}$ (so the flavour state genuinely spans several masses) and $\Delta m^2\neq0$ (so those masses evolve at different rates). If either fails, $P(\nu_\mu\!\to\!\nu_\beta)=0$ and flavour is conserved along the flight path. Why: this is why the 1998 discovery of oscillations was simultaneously the discovery that at least one neutrino has mass — the two conclusions are logically welded together by this formula. *These are the essential logical consequences of the formula: $P=\sin^2 2\theta\sin^2(\Delta m^2 L/4E)$ vanishes if $\theta=0$ or $\Delta m^2=0$ ; flavour is conserved along the flight path if and only if $U=\mathbb{1}$ or the masses are degenerate; and the observation of oscillation therefore implies nonzero neutrino mass.*

Limiting cases

Case	Result	Meaning
Massless neutrinos, $U=\mathbb{1}$	flavour $=$ mass; $L_e,L_\mu,L_\tau$ exactly conserved	the old Standard Model; no oscillation
Massive, $U\neq\mathbb{1}$	flavour $\neq$ mass	oscillation; $L_\ell$ violated, total $L$ conserved (Dirac)
Short baseline, $L\to0$	$P(\alpha\!\to\!\beta)\to0$ , $P(\alpha\!\to\!\alpha)\to1$	flavour tag intact at production
Oscillation maximum, $\Delta m^2 L/4E=\pi/2$	$P=\sin^2 2\theta$	maximal flavour conversion — where experiments sit
Long/averaged $L$	$\langle P\rangle\to\tfrac12\sin^2 2\theta$	phases wash out; a decohered flavour mixture
Fourth active light species	excluded by $N_\nu=2.984\pm0.008$	any 4th neutrino must have $m>M_Z/2=45.6$ GeV or be sterile

Common trapConfusing the flavour eigenstates with the mass eigenstates

The single most common and most damaging error is to treat $\nu_e,\nu_\mu,\nu_\tau$ as if they were particles with definite masses $m_e,m_\mu,m_\tau$ — three tidy objects, one per generation. They are not. The flavour states are eigenstates of the weak interaction: they are what the $W$ vertex produces and detects. The mass eigenstates $\nu_1,\nu_2,\nu_3$ are eigenstates of the free Hamiltonian: they are what propagates with a definite energy and phase. These are two different bases of the same three-dimensional space, related by the unitary PMNS rotation $U$ , and $U\neq\mathbb{1}$ . A $\nu_e$ therefore does not "have a mass" at all — it is a coherent superposition $\sum_i U_{ei}\nu_i$ of three different masses, with weights $|U_{e1}|^2\approx0.68$ , $|U_{e2}|^2\approx0.30$ , $|U_{e3}|^2\approx0.02$ . Collapse the two bases into one and you erase the very possibility of oscillation, because oscillation is the interference between mass eigenstates inside a fixed-flavour state. The masses $m_1,m_2,m_3$ are not paired one-to-one with the flavours; there is no "the mass of the electron neutrino".

Worked examples

1 · Counting the neutrino flavours from LEP (fully worked)

From the LEP lineshape one extracts $\Gamma_Z=2495.2$ MeV, $\Gamma_{\text{had}}=1744.4$ MeV, and a charged-lepton partial width $\Gamma_{\ell\ell}=83.98$ MeV per species. The Standard-Model ratio is $(\Gamma_{\nu\nu}/\Gamma_{\ell\ell})_{\text{SM}}=1.991$ . How many light neutrino species couple to the $Z$ ?

Step 1. Invisible width. Subtract the visible channels, using lepton universality for the three charged leptons: $\Gamma_{\text{inv}}=\Gamma_Z-\Gamma_{\text{had}}-3\Gamma_{\ell\ell}=2495.2-1744.4-3(83.98)=2495.2-1744.4-251.9=\mathbf{498.9\ MeV}$ . This is the width the $Z$ spends on states no detector registers — one-fifth of all $Z$ decays.

Step 2. Ratio method. Form $N_\nu=\dfrac{\Gamma_{\text{inv}}}{\Gamma_{\ell\ell}}\left(\dfrac{\Gamma_{\ell\ell}}{\Gamma_{\nu\nu}}\right)_{\text{SM}}=\dfrac{498.9}{83.98}\times\dfrac{1}{1.991}=5.941\times0.5023=\mathbf{2.984}$ . The ratio to the precisely-known leptonic width cancels the common electroweak coupling, which is why the full LEP fit sharpens this to $N_\nu=2.984\pm0.008$ .

Step 3. Interpret. The result is three to a fraction of a percent. A fourth neutrino coupling to the $Z$ with ordinary strength is excluded unless it is heavier than $M_Z/2=45.6$ GeV — vastly heavier than $\nu_1,\nu_2,\nu_3$ and unmotivated. N_ν = 2.984 ± 0.008: there are exactly three light active neutrino flavours.

2 · The mass composition of the electron neutrino

Given the measured mixing parameters $\sin^2\theta_{12}=0.307$ and $\sin^2\theta_{13}=0.022$ , find the probability that a $\nu_e$ is "really" each of the three mass eigenstates $\nu_1,\nu_2,\nu_3$ .

Step 1. Use the first row of the PMNS matrix. The standard parametrisation gives $|U_{e1}|^2=\cos^2\theta_{12}\cos^2\theta_{13}$ , $|U_{e2}|^2=\sin^2\theta_{12}\cos^2\theta_{13}$ , and $|U_{e3}|^2=\sin^2\theta_{13}$ . These are the weights with which the mass eigenstates appear in the flavour state $\nu_e$ .

Step 2. Evaluate. With $\cos^2\theta_{13}=0.978$ and $\cos^2\theta_{12}=0.693$ : $|U_{e1}|^2=0.693\times0.978=\mathbf{0.678}$ , $|U_{e2}|^2=0.307\times0.978=\mathbf{0.300}$ , $|U_{e3}|^2=\mathbf{0.022}$ . They sum to $1.000$ , as unitarity demands.

Step 3. Read the meaning. A $\nu_e$ is $68\%$ $\nu_1$ , $30\%$ $\nu_2$ , and only $2\%$ $\nu_3$ — three different masses, no single one of them "the" electron-neutrino mass. This is why a mass measurement like tritium $\beta$ -decay measures an effective mass $m_\beta^2=\sum_i|U_{ei}|^2 m_i^2$ , a weighted average over the eigenstates. ν_e = 0.68 ν₁ + 0.30 ν₂ + 0.02 ν₃ (in probability); flavour is a superposition of masses.

3 · Where to put a detector to see the flavour change

The atmospheric mass splitting is $\Delta m^2_{31}\approx2.5\times10^{-3}\ \text{eV}^2$ . For a $\nu_\mu$ beam of energy $E=1$ GeV, at what baseline $L$ does the oscillation first reach its maximum?

Step 1. Set the phase to $\pi/2$ . Using the practical form of equation (5), the oscillation argument is $1.27\,\Delta m^2[\text{eV}^2]\,L[\text{km}]/E[\text{GeV}]$ , and the first maximum of $\sin^2$ occurs when this equals $\pi/2$ .

Step 2. Solve for $L$ . $L=\dfrac{(\pi/2)\,E}{1.27\,\Delta m^2}=\dfrac{1.571\times1}{1.27\times2.5\times10^{-3}}=\dfrac{1.571}{3.175\times10^{-3}}=\mathbf{495\ km}$ .

Step 3. Recognise the number. Roughly $500$ km at GeV energies is exactly the baseline scale of the long-baseline accelerator experiments — T2K ( $295$ km), NOvA ( $810$ km), and the coming DUNE ( $1300$ km) all straddle it, because that is where a $\nu_\mu$ born in the beam has maximally converted to another flavour. L ≈ 500 km: the atmospheric oscillation length at 1 GeV — why long-baseline experiments are hundreds of kilometres long.

Physics of the portrait

Neutrino flavour is the idea that a particle you can never see nonetheless comes in distinct kinds — and the man who first understood that those kinds could mix, turning one into another as they fly, was an Italian émigré who spent his most creative years behind the Iron Curtain, largely cut off from the field he was quietly transforming.

Origins · 1 minBruno Pontecorvo and how this was discovered

Bruno Pontecorvo

1913–1993 · neutrino flavour, lepton number, and the idea of oscillations

Bruno Pontecorvo was the youngest of Enrico Fermi's "via Panisperna boys" in Rome, helping discover slow-neutron activation in 1934. Jewish and anti-fascist, he fled Italy in 1936, worked in Joliot-Curie's Paris laboratory, and reached the United States, where he invented the neutron-well logging method still used to prospect for oil. But his deepest instinct was always the neutrino. In 1946 he proposed the radiochemical chlorine method — $\nu_e+{}^{37}\text{Cl}\to{}^{37}\text{Ar}+e^-$ — that Raymond Davis would use to catch solar neutrinos, and he argued early for the universality of the weak interaction, anticipating the lepton-flavour structure. Then in 1950, at the height of the atomic-spy panic, Pontecorvo vanished with his family to the Soviet Union, publishing in relative isolation at Dubna. It was there, in 1957–58, that he made the leap that names this article: reasoning by analogy with $K^0$ – $\bar K^0$ mixing, he proposed that neutrinos too could oscillate — that one flavour could transform into another in flight, if the flavour states were superpositions of states of different mass. The idea was decades ahead of the experiments; only in 1998 did Super-Kamiokande confirm atmospheric $\nu_\mu$ disappearance, and the 2015 Nobel Prize honoured the discovery — twenty-two years after Pontecorvo's death. The matrix that rotates flavour into mass carries his initial first: PMNS.

History

1930

— Pauli postulates the neutrino to rescue energy conservation in β-decay

1957

— Pontecorvo proposes that neutrinos can oscillate between flavours

1962

— the Brookhaven two-neutrino experiment shows ν_μ is distinct from ν_e

1989

— the LEP Z-width fixes the number of light neutrino flavours at three

2000

— DONUT at Fermilab directly detects the tau neutrino, completing the triplet

Going further

Texts. Thomson, Modern Particle Physics, §13.1 (neutrino flavours) and Ch. 13 (neutrinos and neutrino oscillations) — the source this article grounds · Griffiths, Introduction to Elementary Particles, 2nd ed., §1.10 and §11 (neutrino oscillations) · Halzen & Martin, Quarks and Leptons, §12 (weak interactions of leptons) · Peskin & Schroeder, An Introduction to QFT, §20 (the electroweak theory) · Schwartz, QFT and the Standard Model, Ch. 29. The key numerical anchors used throughout this article are: $L_e,L_\mu,L_\tau$ conserved at each vertex of eq. (1), with $\mu^-\to e^-\bar\nu_e\nu_\mu$ balancing all three; $\Gamma_{\text{inv}}=498.9$ MeV and $N_\nu=2.984\pm0.008$ from the LEP widths; $|U_{e1}|^2,|U_{e2}|^2,|U_{e3}|^2=0.678,0.300,0.022$ summing to one; and the atmospheric first-maximum baseline $L\approx495$ km at $E=1$ GeV, $\Delta m^2_{31}=2.5\times10^{-3}$ eV². The mixing parameters $\sin^2\theta_{12}=0.307$ , $\sin^2\theta_{13}=0.022$ , and $\Delta m^2_{21}=7.5\times10^{-5}$ eV² are current global-fit values.

Prerequisites. The Weak Charged Current · Leptons and the Particle Zoo · Muon and Pion Decay. Leads to. Mass versus Weak Eigenstates · Two-Flavour Oscillations · Three-Flavour PMNS Mixing.

Ask the tutor: Ask me to extract $N_\nu$ from any set of $Z$ partial widths, or to work out the mass-eigenstate composition of $\nu_\mu$ or $\nu_\tau$ from the PMNS mixing angles.

How do we know there are exactly three neutrino flavours?

Two independent ways, and they agree. Directly, we have detected all three charged leptons produced by neutrino charged-current interactions — the electron (implicit since Reines and Cowan), the muon (Brookhaven, 1962), and the tau (DONUT, 2000) — so three flavours are seen. Indirectly and far more precisely, LEP measured the invisible decay width of the $Z$ boson: every neutrino species lighter than $M_Z/2$ adds a $Z\to\nu\bar\nu$ channel, and the total invisible width divided by the per-species Standard-Model width gives $N_\nu=2.984\pm0.008$ . This count is flavour-blind — it tallies any light neutrino that couples to the $Z$ , even one we could never catch — and it lands squarely on three, excluding a fourth ordinary neutrino unless it is heavier than about $46$ GeV.

What is the difference between a flavour eigenstate and a mass eigenstate?

A flavour eigenstate ( $\nu_e,\nu_\mu,\nu_\tau$ ) is what the weak interaction makes and detects — it is defined by the charged lepton it partners. A mass eigenstate ( $\nu_1,\nu_2,\nu_3$ ) is what propagates through space with a definite energy and a definite mass. These are two different bases of the same three-dimensional space, and the unitary PMNS matrix rotates between them. The crucial fact is that the rotation is not trivial: a $\nu_e$ is a specific superposition of $\nu_1,\nu_2,\nu_3$ (roughly $68\%$ , $30\%$ , $2\%$ by probability), so it has no single mass, and conversely a $\nu_1$ is a mixture of all three flavours. Confusing the two bases erases oscillations, because oscillation is precisely the interference between mass eigenstates travelling inside a fixed-flavour state.

Is lepton flavour actually conserved?

It depends on what you count. At every vertex of the weak interaction, the three lepton flavour numbers $L_e,L_\mu,L_\tau$ are separately conserved — this is why a $\nu_\mu$ makes a muon and never an electron, and why $\mu^-\to e^-\gamma$ has never been seen (branching ratio below $4\times10^{-13}$ ). But between vertices, a propagating neutrino oscillates, and a beam born as $\nu_\mu$ can be detected as $\nu_e$ : individual $L_\ell$ are violated by propagation once neutrinos have mass. What survives is the total lepton number $L=L_e+L_\mu+L_\tau$ , provided neutrinos are Dirac particles. Whether even total $L$ is conserved is the open question that neutrinoless double-beta-decay experiments are built to answer.

Could there be a fourth neutrino?

Not a fourth ordinary ("active") one lighter than $M_Z/2$ — LEP's $N_\nu=2.984\pm0.008$ forbids it, because such a neutrino would show up in the $Z$ 's invisible width. But the count only sees neutrinos that couple to the $Z$ . A sterile neutrino — one that does not feel the weak interaction at all and mixes only slightly with the ordinary three — would be invisible to LEP and remains an active area of search. Various short-baseline anomalies (LSND, MiniBooNE, reactor deficits) have been read as hints of an eV-scale sterile state, but the picture is inconsistent and most global fits are in tension with it. So: no fourth active neutrino, and no confirmed sterile one, but the door to sterile species is not fully closed.

How is a neutrino's flavour measured in practice?

By identifying the charged lepton the neutrino makes. In a water-Cherenkov detector like Super-Kamiokande, a $\nu_\mu$ produces a muon that leaves a crisp, sharp-edged Cherenkov ring, while a $\nu_e$ produces an electron that showers and gives a fuzzy, diffuse ring — the two are told apart by ring topology. In a tracking detector, a muon is a long straight minimum-ionising track and an electron initiates an electromagnetic cascade; a tau, when energetic enough to be produced, decays almost immediately and leaves a characteristic short track with a kink or a displaced vertex (this is how DONUT and OPERA tagged $\nu_\tau$ ). In every case the neutrino itself is invisible — the flavour is an inference from its charged partner, exactly as the operational definition demands.

If neutrinos oscillate, does that break the Standard Model?

It breaks the original Standard Model, in which neutrinos were exactly massless and flavour was perfectly conserved. Oscillations require nonzero neutrino masses and a non-trivial PMNS mixing matrix, neither of which the minimal Standard Model contains — so their discovery in 1998 was the first laboratory evidence for physics beyond it. The fix is modest in form: add neutrino mass terms (Dirac, or Majorana via a seesaw), which introduces the PMNS matrix by exactly the same logic that gives quarks the CKM matrix. What it does not break is the flavour-diagonal weak vertex — the $W$ still couples each $\nu_\ell$ to its own $\ell$ . Oscillation happens not at the vertex but during propagation, because the state that flies between vertices is a superposition of masses.

Why were neutrinos assumed massless for so long?

Because every early experiment was consistent with it. The weak charged current couples only to left-handed neutrinos, and a strictly massless neutrino can be purely left-handed in a Lorentz-invariant way — so masslessness fit the observed maximal parity violation elegantly. Direct mass measurements returned only ever-smaller upper limits, and lepton flavour appeared exactly conserved in every reaction, with no sign of $\nu_\mu\to\nu_e$ transitions. With no positive evidence for mass and a tidy theoretical picture without it, the massless assumption was the natural default for decades. It took the tiny mass-squared splittings revealed by oscillation — parts in $10^{-3}$ eV² and smaller — to overturn it, masses so minute that no endpoint experiment had yet reached them.