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The antiparticle associated with the tau-flavour neutrino. A tau lepton of positive charge emits it in an ordinary charged current decay; a negative tau lepton instead emits a tau neutrino.

In the modern weak charged current interpretation, one allowed assignment isThe electron antineutrino, tau neutrino, muon neutrino and antineutrino of the tau neutrino carry away energy and momentum. There are four undetected particles in this decay assignment. Interchanging which parent decays electronically gives the opposite electric charge assignment. Each parent conserves electric charge and, in this approximation, total lepton number; four-momentum conservation applies separately to production and to each particle decay.

If $B_e$ and $B_{\mu }$ denote the corresponding branching fractions, independent parent decays give an inclusive mixed-flavour fraction $2B_e{B}_{\mu }$. The factor two counts the two assignments to opposite parent charges. For chosen $B_e=B_{\mu }=1/5$, the mixed fraction is $2/25$, before acceptance and detection efficiency. Correlations between the parents' spins can affect angular selection even when the fully inclusive branching fractions factorize.

The neutrinos in that equation are a physical interpretation to test. Neither a deficit in visible energy nor two unusual tracks alone establishes their identities or uniquely counts them. Even a compatible invariant mass and an angular distribution require particle identification, background checks and agreement across further particle decay channels. The direct detection milestone for the tau neutrino was in 2000, separate from the charged tau lepton discovery.

Combine the two massless neutrinos in ${\tau }^{-}\to {\ell }^{-}{\bar{\nu }}_{\ell }{\nu }_{\tau }$ into $q$. In the parent rest frame, four-momentum conservation givesThe lower bound follows from the nonnegative Minkowski norm of the sum of two future-directed null four-vectors; the upper bound follows from $E_{\ell }^{\ast }\ge m_{\ell }$. Substituting gives the displayed energy bounds. They are kinematic: they do not specify the weak interaction decay amplitude or the probability density function inside the interval.

If the charged daughter is also massless, put $x=2E_{\ell }^{\ast }/m_{\tau }\in [0,1]$. A parent with Lorentz factor $\gamma$ and speed $\beta$ in natural units producesHere ${\theta }^{\ast }$ is the daughter angle relative to the boost direction in the parent rest frame. Two opposite parent momenta therefore need not produce two opposite visible daughter momenta: the invisible system can recoil differently in each particle decay.

For a two-body W boson decay into a charged lepton $\ell$ and a neutrino, neglect both daughter masses and use natural units $c=1$. Let $\Delta \varphi$ be the angle between the daughter transverse momenta. Identifying the missing transverse momentum with the neutrino's givesThis observable uses no $p_{z\nu }$. The 1983 UA1 experiment paper already used transverse mass as a lower bound on parent mass, before making additional decay-model assumptions for a mass estimate.^[ref]

The bound follows directly from the energy–momentum relation. For each massless daughter write its four-momentum, using rapidity $y$ and azimuth $\varphi$, asUsing the Minkowski inner product with positive time sign, $p^2=E^2-|\mathbf{p}{|}^2$, the two daughter four-momenta give the true parent invariant mass $M$:Since $\cosh u\ge 1$, $m_T\le M$, with equality at equal daughter rapidities when both transverse momenta are nonzero. Thus the unmeasured longitudinal information can enlarge the true invariant mass, but cannot lower it below this ideal transverse mass.

For an original example, stipulate $p_{T\ell }=p_{T\nu }=40\mathrm{GeV}$ and $\Delta \varphi =\pi$. Then $m_T=80\mathrm{GeV}$. Equal rapidities give $M=80\mathrm{GeV}$; choosing $y_{\ell }-y_{\nu }=1$ with exactly the same transverse readout gives $M=80\cosh (1/2)\simeq 90.210\mathrm{GeV}$. One measured transverse configuration therefore permits different true parent masses.

For a deliberately simplified neutrino oscillation calculation, take two neutrino flavors related to two mass eigenstates by a real rotation matrix with angle $\theta$. A quantum superposition acquires different phases during propagation; projecting back onto the initial neutrino flavor gives the probabilityHere $L$ is path length and $E$ is neutrino energy; the numerical coefficient incorporates unit conversion.

The paper states the corresponding conversion formula.^[ref]

The path dependence has a geometrical origin. In an original spherical-Earth model, let $R$ be the radius and $h$ a fixed production height, neglecting detector depth. Let $\Theta$ be the zenith distance of the direction in the sky from which the neutrino arrives, so $c=\cos \Theta =+1$ means downward propagation. The backward ray from detector to production obeys $R^2+L^2+2RLc=(R+h{)}^2$. Its positive solution isThus $L(+1)=h$ and $L(-1)=2R+h$. The calculation explains why opposite directions test very different baselines, rather than attributing every observed direction to the same travel distance. Actual production heights and the angle between the outgoing charged lepton and incoming neutrino smear that relationship.

For an original chosen example, set $\theta =\pi /4$ and $\Delta m^2=2.5\times 10^{-3}{\mathrm{eV}}^2$. At $E=1\mathrm{GeV}$, $L=20\mathrm{km}$ gives $P_{\mu \mu }\simeq 0.9960$, while $L=500\mathrm{km}$ gives $P_{\mu \mu }\simeq 0.000279$. The first minimum is at $L/E\simeq 494.7km/GeV$. These are stipulated inputs, not a fit to the 1998 data. A broad distribution of true $L/E$, together with energy and directional uncertainty, smooths the sharp minima.

A direct amplitude check makes the mechanism explicit. DefineThen $S_{11}={\cos }^2\theta +{\sin }^2\theta e^{-2i\phi }$. Taking its squared absolute value yields the displayed survival formula because $1-\cos (2\phi )=2{\sin }^2\phi$. Unitarity gives $|S_{11}{|}^2+|S_{21}{|}^2=1$: disappearance from one channel becomes conversion to the other within this closed model.

This illustration does not describe all solar neutrino propagation. Coherent forward scattering in matter changes the effective propagation operator, and three active neutrino flavors require a larger mixing description. Even in the simplified model, neutrino oscillation measures a squared-mass difference; it does not determine the absolute neutrino mass.

A two-track invariant mass and a direction test answer different questions. In units $c=1$, assign each track the pion mass $m_{\pi }$ and reconstruct its momentum before magnetic deflection. With $p_{\pm }=|{\mathbf{p}}_{\pm }|$ and opening angle $\alpha$,This follows by expanding the squared sum of the two four-momenta. The angle $\theta$ between the reconstructed parent momentum $\mathbf{P}={\mathbf{p}}_{+}+{\mathbf{p}}_{-}$ and the neutral particle beam direction $\hat{\mathbf{n}}$ satisfiesA fully reconstructed two-body decay channel gives the parent mass and $\theta \simeq 0$, within experimental resolution. A three-body decay channel with an unseen neutrino generally misses both constraints; an accidental correct invariant mass alone is insufficient. Notice that the track opening angle $\alpha$ need not be small even when the parent direction angle $\theta$ vanishes.

The 1964 analysis selected $490<M_{\pi \pi }<510$ MeV and remeasured events with $\cos \theta >0.9995$, comparing adjacent mass bands.^[ref]

For an original dimensionless example, take $m_{\pi }=1$, $\hat{\mathbf{n}}=(0,0,1)$ and ${\mathbf{p}}_{\pm }=(\pm \sqrt{3},0,3)$. Each energy is $\sqrt{13}$; the opening angle is $60$ degrees since $\cos \alpha =(9-3)/(9+3)=1/2$. Yet $\mathbf{P}=(0,0,6)$ points exactly along the beam, and $M_{\pi \pi }^2=4(13)-36=16$. Thus a parent of mass $4$ passes both ideal tests. These chosen numbers illustrate four-momentum conservation, not detector performance.

An extra relativistic neutrino that decouples before ordinary neutrinos misses two stages of plasma heating. Until ordinary decoupling, entropy conservation gives a temperature ratio $(43/(4g_d){)}^{1/3}$. Electron-positron annihilation then heats photons but neither neutrino population, multiplying the ratio by $(4/11{)}^{1/3}$. Here $g_d$ counts the coupled bath at the extra species' decoupling, with no subsequent exotic entropy injection.

TXS 0506+056 is a blazar studied through a gamma-ray flare associated in direction and time with the high-energy neutrino alert IceCube-170922A. The event motivated coordinated observations across the electromagnetic spectrum, connecting a variable relativistic source with a possible source of energetic neutrinos.^[ref]

An association is a statistical inference involving localization, timing and backgrounds. It does not directly image the neutrino-production site or uniquely determine the jet's particle composition. The physical interpretation must account for the measured photons and neutrino evidence together.

A neutrino with energy much larger than its mass. At fixed momentum, expanding the relativistic energy-momentum relation gives $E_i\simeq p+m_i^2/(2p)$, so differences of propagation phases are governed by $\Delta m^2/(2E)$.

A W boson decay can leave one visible charged lepton and a large missing transverse momentum; a Z boson decay can instead supply two visible leptons whose four-momenta determine an invariant mass. The two signatures answer the same existence question with different missing information. The electroweak unification history recognized by the 1979 Nobel Prize in Physics predates these 1983 observations: the W boson and Z boson were predictions to test, rather than particles first proposed in 1983.

The Super Proton-Antiproton Synchrotron stored counterrotating proton and antiproton beams; stochastic cooling made accumulation of the scarce antiprotons practical. Carlo Rubbia's 1984 lecture describes track reconstruction in a large drift chamber inside a magnetic field, surrounded by segmented particle calorimeters. Their nearly complete angular coverage allowed an imbalance test for unobserved neutrinos.^[ref]

Water Cherenkov neutrino reconstruction infers a vertex, time and charged particle direction from photomultiplier pulses. The 1987 analysis recorded charge and timing, calibrated with decaying muons and source-produced gamma rays, and rejected outer-detector activity and events following muons.^[ref]

In an ideal point-flash approximation, a light source at ${\mathbf{r}}_0$ and time $t_0$ gives the displayed travel-time relation at sensor ${\mathbf{r}}_i$ in a medium of refractive index $n$. Subtracting two times eliminates $t_0$. With a chosen constant $n=1.33$ and path lengths $3\mathrm{m}$ and $6\mathrm{m}$, the arrival-time difference is $n(3\mathrm{m})/c\simeq 13.31\mathrm{ns}$. These are synthetic inputs, not detector measurements. Several sensors constrain the vertex; an extended moving track requires the light-emission position and time along that track as well.

For a charged particle moving at $v=\beta c$, the Cherenkov cone angle satisfies $\cos {\theta }_C=1/(n\beta )$ when $n\beta >1$. With $\beta \simeq 1$ and the illustrative $n=1.33$, ${\theta }_C\simeq 41.25^{\circ }$. The geometrical cone carries track-direction information; electron deflections and the spread of emission positions broaden the observed pattern. The neutrino itself does not draw the light cone.

A Cherenkov radiation threshold and an analysis threshold are different constraints. For a fixed $n$, an electron must exceed $T_C=m_e{c}^2[(1-n^{-2}{)}^{-1/2}-1]$. With $m_e{c}^2=0.511\mathrm{MeV}$ and the chosen $n=1.33$, this gives about $0.264\mathrm{MeV}$. Recording and selecting a useful event requires much more light: the 1987 detector's trigger accepted $8.5\mathrm{MeV}$ electrons with 50% efficiency.^[ref]

The weak interaction changes particle species through the charged current and also acts through the neutral current. Familiar examples include beta-minus decay and neutrino scattering. It is short-range and chiral: its W boson and Z boson couplings distinguish chirality. Above the electroweak scale it belongs to the unbroken $SU(2{)}_L\times U(1{)}_Y$ electroweak interaction. At much lower energies, W boson exchange is approximated by the Fermi interaction.

A particularly direct window into the weak interaction is the Wu experiment: a sample with oriented nuclear spins emits electrons preferentially toward one end of its spin axis. The comparison tests parity conservation, because spin is an axial vector and momentum is a polar vector. The polarized beta-decay angular distribution translates that geometric distinction into a measurable counting asymmetry.

Yoichiro Suzuki worked on water-based Cherenkov radiation detection of neutrinos, including solar and atmospheric measurements. The Super-Kamiokande method classifies the visible rings made by secondary electrons and muons and compares their rates with the direction through the Earth. A directional deficit carries information about propagation distance as well as production. This makes the apparatus a test of neutrino oscillations, rather than simply a counter of weakly interacting particles.^[ref]

A physicist researching solar neutrinos and neutrino detection.^[ref]

In 1935, Hideki Yukawa proposed a new massive exchange quantum to explain nucleon binding. His paper began with the inadequacy of an electron–neutrino exchange description of the binding strength. It then required a potential energy falling faster with distance than a Coulomb potential, introduced an exponentially decaying field, and related its inverse range to the exchange particle's rest mass. This was a theoretical proposal, not a detection with an apparatus of his own.^[ref]

The crucial idea is that a finite rest mass supplies a length scale even in empty space. In a scalar model, the static limit of a Klein-Gordon equation produces the modified Helmholtz equation. Its decaying Green's function gives the Yukawa potential. The derivation below isolates this range mechanism; one-pion exchange also carries spin and isospin structure.

The nuclear emulsion track experiment used developed photographic emulsions exposed to cosmic rays and inspected microscopically. Stopping pions gave daughter muon tracks with closely grouped ranges. Comparing penetrating muons with pion-induced nuclear breakup distinguished their interactions; this was not a direct static force measurement. The muon is a lepton, not Yukawa's strongly interacting meson.^[ref]