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A Fermilab experiment that directly observed tau neutrino charged current interactions using nuclear emulsion and a prompt neutrino beam from an $800\mathrm{GeV}$ proton dump. Ds meson decays and the subsequent tau lepton decays supply the main tau-flavour component.

A long-baseline tau-appearance experiment with the CERN-to-Gran-Sasso muon neutrino beam. Lead targets and nuclear emulsion resolve a short tau lepton decay after a tau neutrino charged current interaction.

In a dense beam dump, long-lived pions and kaons usually interact and lose energy before decaying. Their residual stopped decays yield low-energy neutrinos, below the useful high-energy tau-interaction sample. Prompt charmed mesons have much shorter lifetimes and can decay before reinteraction. This suppresses the conventional muon neutrino background relative to the prompt component. The production sequence is quantum chromodynamics creation of $c\bar{c}$, principally $gg\to c\bar{c}$ with $q\bar{q}\to g^{\ast }\to c\bar{c}$ also allowed, followed by hadronization $c\bar{s}\to D_s^{+}$ and its charge conjugate. The $gg$ process includes both quark-exchange diagrams and the three-gluon diagram. Fragmentation also produces other hadrons; the proton does not turn directly into one isolated Ds meson.The charge-conjugate chain gives a prompt tau antineutrino and a secondary tau neutrino. Thus each leptonic Ds meson chain produces both tau-flavour signs. The corresponding Feynman diagrams are shown in a.

For the CKM matrix, diagonalizing the up/down quark mass matrices by $U_{uL},U_{dL}$ leaves $V=U_{uL}^{†}{U}_{dL}$ in the charged current:The three-generation unitary matrix has three physical mixing angles and one CP violation phase after field rephasings. At fixed kinematics and hadronic matrix element, rates are proportional to $|V_{ij}{|}^2$; phase space, parent lifetimes, form factors and helicity suppression must also be included when comparing branching fractions.

For a charged pseudoscalar meson $P^{+}=(c\bar{q})$, leptonic pseudoscalar decay width givesThe Ds meson channel is Cabibbo-favoured, $|V_{cs}|\simeq 0.974$, whereas the charged D meson channel uses $|V_{cd}|\simeq 0.224$. Taking $m_{D_s}\simeq 1.968$, $m_D\simeq 1.870$ and $m_{\tau }\simeq 1.777$ in GeV givesIncluding the approximate lifetime ratio ${\tau }_{D_s}/{\tau }_D\simeq 0.48$ and the given production ratio $5.2/11$ gives a tau yield ratio about $17(f_{D_s}/f_D{)}^2$, already much greater than one even with equal decay constants. This is an illustrative estimate using approximate additional mass/lifetime inputs, not a determination from the production cross sections alone. A neutral D meson cannot annihilate to the charged $\tau \nu$ pair by electric charge conservation; its semileptonic tau modes have insufficient phase space once the required strange or nonstrange recoil hadron is included. Light mesons are below threshold; bottom production is much smaller at this fixed-target energy. Consequently neither the larger $D^0$ production rate nor ordinary light-hadron production outweighs the Ds meson tau source. ^[ref]

The basic Feynman diagrams have ${\tau }^{-}\to {\nu }_{\tau }{W}^{-\ast }$ followed by $W^{-\ast }\to e^{-}{\bar{\nu }}_e$, ${\mu }^{-}{\bar{\nu }}_{\mu }$, or $d\bar{u},s\bar{u}$, with the latter pairs undergoing hadronization. See the charged-current diagrams in a. Neglect all daughter masses, QCD corrections and inaccessible charm final states, and assume lepton universality. Each leptonic mode then has one unit of decay width; the two accessible quark combinations have three color charge states and total CKM weight $|V_{ud}{|}^2+|V_{us}{|}^2\simeq 1$. ThereforeThe result is an inclusive parton model estimate. Perturbative QCD corrections increase the hadronic contribution, while muon mass, nonperturbative dynamics and radiative corrections also matter; the physical fraction is roughly $0.65$.

Every ordinary hadronic mode already contains a tau neutrino, so it contains a lepton as well as hadrons if invisible particles are counted. For an additional visible charged lepton, a virtual photon can internally convert in a hadronic mode, for example ${\tau }^{-}\to {\nu }_{\tau }{\pi }^{-}{\gamma }^{\ast }\to {\nu }_{\tau }{\pi }^{-}{e}^{+}{e}^{-}$. The extra electromagnetic vertices suppress such modes; they are not a large fraction of the inclusive width. Alternatively a daughter pion or kaon can decay semileptonically or leptonically in flight, for example ${\tau }^{-}\to {\nu }_{\tau }{\pi }^{-}{K}^0$ followed by a semileptonic neutral-kaon decay. Those are secondary decay chains, not independent primary tau lepton branching fractions. Lepton-flavour-violating neutrinoless modes such as ${\tau }^{-}\to {\mu }^{-}{\pi }^{+}{\pi }^{-}$ do not contribute appreciably in the Standard Model.

In ${\tau }^{-}\to {\ell }^{-}{\bar{\nu }}_{\ell }{\nu }_{\tau }$, charge and missing four-momentum show an invisible neutral component beyond the visible charged lepton. The distributions are consistent with three-body V-A interaction decay; hadronic modes such as ${\tau }^{-}\to {\pi }^{-}{\nu }_{\tau }$ instead have two-body missing-mass kinematics. Production and decay accounting assigns $L_{\tau }({\tau }^{-})=L_{\tau }({\nu }_{\tau })=1$ and their antiparticles minus one. Searches for wrong-flavour and neutrinoless channels constrain violations. These observations establish a useful approximately conserved tau-family number in ordinary interactions, rather than proving an exact law from missing energy alone.

To establish that the invisible particle is distinct from ${\nu }_e$ and ${\nu }_{\mu }$, combine decay evidence with the absence of unsuppressed tau production by conventional electron- and muon-flavour beams, and direct tau neutrino charged current interactions. DONUT observed the short displaced tau lepton decay in nuclear emulsion after a neutrino interaction. ^[ref] Neutrino oscillation means individual family numbers are not exact conserved quantities during propagation, even though they are an excellent approximation for these local decay assignments.

Neglect radiation, take $E_{\tau }=M_Z/2\simeq 45.6\mathrm{GeV}$, $m_{\tau }\simeq 1.777\mathrm{GeV}$, and treat a fixed hadronic invariant mass $m_h=1.5\mathrm{GeV}$ as a single recoil system in ${\tau }^{-}\to h^{-}{\nu }_{\tau }$. With negligible neutrino mass,Thus $E_h^{\min ,\max }=\gamma (E_h^{\ast }\mp \beta p_h^{\ast })$, numerically approximately $32.50$ to $45.60\mathrm{GeV}$. In the ultrarelativistic limit the range is $E_{\tau }{m}_h^2/m_{\tau }^2\le E_h\le E_{\tau }$. A spread of hadronic invariant masses broadens these endpoints; an average mass alone is not a strict endpoint specification for the whole sample.

Let $P_{\tau }=+1$ denote a positive-helicity ${\tau }^{-}$, and measure $\vartheta$ from its laboratory flight direction in its rest frame. For a spin-zero hadronic recoil, the left-helicity tau neutrino and angular momentum conservation giveFor example a tau lepton with spin along $+z$ can emit a left-helicity neutrino along $-z$, carrying spin along $+z$, with the spin-zero recoil along $+z$. A positive-helicity tau lepton then gives a harder visible recoil. If $x_h=E_h/E_{\tau }$, $r=m_h^2/m_{\tau }^2$ and $\beta \simeq 1$, the more general tau hadronic polarimetry formula isFor spin-zero recoil ${\alpha }_h=1$. For a spin-one system produced by the V-A interaction, the longitudinal and transverse weights are $m_{\tau }^2:2m_h^2$, with opposite angular slopes; summing gives ${\alpha }_h=(m_{\tau }^2-2m_h^2)/(m_{\tau }^2+2m_h^2)$. At $m_h=1.5\mathrm{GeV}$ this is about $-0.175$, so the negative-helicity tau lepton has the higher mean total hadronic energy, about $39.43$ versus $38.67\mathrm{GeV}$, within that spin-one model. ^[ref]

The supplied mean invariant mass does not specify the recoil's spin composition or its helicity amplitudes. Consequently the last sign cannot be universal: positive helicity is harder for a spin-zero recoil, negative helicity is harder for the specified $1.5\mathrm{GeV}$ spin-one recoil, and a mixture requires its measured spectral weights. Polarization of an intermediate vector or axial-vector meson also changes the division of energy among its daughter hadrons; that information is not fixed by the total jet invariant mass alone.

In 2001, $240.95$ live days, reconstructed radius below $5.50\mathrm{m}$ and effective electron kinetic energy above $6.75\mathrm{MeV}$ gave a CC neutrino flux of $1.75\times 10^6{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$ for the assumed boron-8 spectrum. Source calibrations measured $1.4\%$ instrumental-cut signal loss. SNO's own CC–ES difference was $1.6\sigma$; comparison with Super-Kamiokande's precise ES flux gave $3.3\sigma$ evidence for non-electron active neutrinos. This was not an NC measurement.^[ref]

In 2002 the pure-heavy water analysis used $306.4$ live days and a $5\mathrm{MeV}$ reconstructed kinetic-energy threshold. NC neutrons captured on deuterons emitted $6.25\mathrm{MeV}$ gamma rays, producing secondary electron light. The NC reaction threshold was $2.2\mathrm{MeV}$, a different quantity. Californium-252 calibrated $29.9\%$ capture efficiency, but only $14.4\%$ detection efficiency after fiducial and energy cuts. Assays constrained backgrounds from radioactive decay and photodisintegration.^[ref]

Energy, radius and direction relative to the Sun statistically separated channels. Assuming the standard boron-8 spectral shape, the NC total active neutrino flux was $5.09_{-0.43}^{+0.44}{(stat.)}_{-0.43}^{+0.46}(syst.)\times 10^6{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$, consistent with stellar structure predictions. The non-electron active component differed from zero by $5.3\sigma$. This established active flavour conversion, without individually tagging muon neutrinos and tau neutrinos. It did not exclude every sterile admixture.^[ref]

Super-Kamiokande used $535$ days ($33.0$ kiloton-years) and $22.5$ fiducial kilotons of water. Inner photomultiplier times and charges reconstructed interactions; the outer anticoincidence detector vetoed entering particles and marked exits. Contained single-ring events used likelihood classification: electron particle showers blur rings; muons give sharper edges. The $1.33\mathrm{GeV}$ visible-energy sample division is not a universal detection threshold. Simulation estimated classification efficiency above about $98\%$, distinct from total detection efficiency; stopping muons and decay electrons checked classification.^[ref]

Downward neutrinos travel about $15\mathrm{km}$; upward ones about $13000\mathrm{km}$. The multi-GeV muon-like up/down asymmetry was $-0.296\pm 0.048\pm 0.010$; electron-like events lacked a comparable effect. Calibration constrained directional bias. This was disappearance evidence; tau neutrino appearance was untagged and sterile conversion remained possible.^[ref]

Writing $U$ and $D$ for the selected upward and downward populations, the asymmetry is $A=(U-D)/(U+D)$: a negative value means fewer upward events. An absolute source-normalization error alone rescales both populations and leaves this ratio unchanged. A direction-dependent response error need not cancel.

Takaaki Kajita contributed to the Super-Kamiokande observation of atmospheric neutrino oscillations. Muon neutrinos arriving from below had crossed a much longer path through the Earth than those produced overhead. The direction-dependent deficit therefore tested propagation, while comparisons with electron-like events helped constrain possible production or detector explanations. The 1998 report found the pattern consistent with conversion between muon neutrinos and tau neutrinos.^[ref]

A physicist researching atmospheric neutrinos and neutrino oscillations.^[ref]

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.

For a nearly massless incident tau neutrino and stationary target of mass $M_T$, the initial Mandelstam variables give $s=M_T^2+2M_T{E}_{\nu }$. At threshold the final tau lepton and the lightest allowed recoil system have no relative momentum, so $s=(m_{\tau }+M_X{)}^2$. This proves the formula. A free neutron recoil channel gives approximately $3.45\mathrm{GeV}$; an electron target has a different recoil and a threshold near $3.09\mathrm{TeV}$.

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.