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Bottom-quark jet tagging identifies jets compatible with containing a hadron carrying a bottom quark. A sufficiently long-lived bottom hadron travels away from the primary interaction before its particle decay, so displaced particle tracks and a secondary decay vertex provide evidence about its identity. A lepton within the jet can supply additional evidence. A top quark instead decays before hadronization.

A tag is a statistical classification, so both the efficiency for bottom jets and mistaken tags from charm quarks or light jets require calibration. That calibration concerns the flavor tagging response; it is not interchangeable with a calorimeter calibration of deposited energy. Control samples in collision data can measure the tagging efficiency for specified momentum and topology selections; those selections determine where the result applies.^[ref]

A decay channel specifies a possible final state of a particle decay, such as an electron, a neutrino and an antineutrino in a specified muon decay. Different channels can have different interaction amplitudes, available phase space and experimental visibility.

For disjoint final-state classes, the partial decay widths add: ${\Gamma }_{\mathrm{total}}={\sum }_i{\Gamma }_i$. The branching fraction is ${\Gamma }_i/{\Gamma }_{\mathrm{total}}$, so increasing one partial width changes the fractions assigned to the others even if their own amplitudes are unchanged. Intermediate resonances leading to the same final state can interfere; they cannot always be treated as mutually exclusive counting bins.

A vertically incident particle annihilating with a stationary target into two equal-energy muons gives each half the incident longitudinal momentum, while their transverse momenta are opposite. At height $H$ above a locally flat surface, each proper flight time is $2H{m}_{\mu }{c}^2/[c(p_{\mathrm{in}}c)]$, since $\gamma v_z=p_z/m_{\mu }$. Independent exponential particle decays then give the displayed joint survival probability. Atmospheric energy loss, scattering and curvature are omitted; replacing both trajectories by vertical paths would miss the two-body angular correction.

For a prompt nuclear level fed in a steady decay cascade, incoming and outgoing transition populations balance. Quoted photon intensities must represent efficiency-corrected emission counts; internal conversion, alternative particle decays and unobserved feeds must either be included or explicitly excluded. Equal photon counts suggest a serial cascade, while a feed intensity equal to the sum of two branches suggests splitting. intensities alone do not determine the chronological order of equal-count lines.

Gargamelle was a heavy-liquid bubble chamber at CERN. Its 1973 collaboration report, Observation of neutrino-like interactions without muon or electron in the Gargamelle neutrino experiment, by F. J. Hasert and colleagues, described hadronic events with no identified muon or electron and evaluated rates relative to charged current events. This is evidence for a neutral current, rather than direct production or photography of a Z boson.^[ref]

CERN's Gargamelle account identifies the target liquid as $\mathrm{C}{\mathrm{F}}_3\mathrm{Br}$ and distinguishes hadronic events from the separate isolated-electron signal. The latter is an antineutrino transferring momentum to an electron, whose electromagnetic particle shower supplies visible tracks. These are different final states, not two names for one event.^[ref]

A neutrino beam uses protons striking a target to produce charged pions and kaons; focusing selects parents that undergo particle decay in flight. Their neutrinos pass through shielding into the target. Scanning photographs locates vertices and classifies outgoing tracks.^[ref]

Expansion prepares a bubble chamber's superheated liquid; ionization by charged secondaries seeds bubbles for photography.^[ref]

The recorded light reveals bubbles, not a directly visible incoming or outgoing neutrino. Reconstructing a vertex and classifying bubble-chamber tracks connects that optical record to a hypothesis about a particle interaction.

The collaboration compared samples with matched hadronic cuts. A flat longitudinal vertex distribution resembled neutrino interactions, but sideways neutrons and multi-step cascades could also flatten it. The background estimate combined visible parent-interaction/secondary-neutron pairs with a cascade calculation including surrounding material. Later proton exposures tested the predicted cascade lengths. Escaping charged hadrons could imitate muons; there was no independent perfect muon tag.^[ref]

The inference combines positive evidence for an interaction with controlled alternatives for its visible final state. Shielding suppresses unwanted beam components, yet interactions in surrounding material can generate secondary neutrons; shielding alone cannot certify a neutral current. Nor does a successful neutral current search by itself establish every coupling or the Higgs mechanism. Its importance for the electroweak interaction is that the theory predicts an additional weak channel, with coupling relations that further cross section measurements can test.

In 1962, Leon Max Lederman, Melvin Schwartz, Jack Steinberger and collaborators distinguished muon neutrinos from electron neutrinos using a neutrino beam.^[ref]

The Alternating Gradient Synchrotron delivered $15\mathrm{GeV}$ protons onto beryllium. Particle decay of pions and kaons supplied neutrinos; a $13.5\mathrm{m}$ iron shield preceded a ten-ton aluminium spark chamber.^[ref]

Production and detection are separate. In modern neutrino flavour notation,Shielding suppresses the daughter muon; a newly produced muon inside the spark chamber's aluminium marks an interaction.^[ref]

Beam-timed coincidence circuits and veto counters rejected entering tracks. Machine-off runs measured cosmic ray contamination; shield-thickness changes tested neutrons. Placing lead near the target shortened the meson decay path and reduced events. Cosmotron electron calibration established particle shower recognition.^[ref]

The paper compared $34$ single-muon candidates, including $5$ estimated cosmic ray events, with six shower candidates unlike calibrated electron showers. Kaon decays could supply genuine electron neutrinos. The deficit favored distinct neutrino flavours under lepton universality.^[ref]

The inference concerns production and detection channels. It does not measure neutrino mass or establish permanently fixed neutrino flavour. Later neutrino oscillation experiments address changes during propagation.^[ref]

The proper mean lifetime of a free muon, approximately $2.2\mu \mathrm{s}$. Independent particle decays give survival probability $\exp (-\Delta \tau /{\tau }_{\mu })$. In a frame where the Lorentz factor is constant, $\Delta \tau =\Delta t/\gamma$; interactions of a stopped negative muon in matter can add capture channels.

A particle beam dominated by neutrinos. Accelerator beams can use particle decay of focused charged pions and kaons, followed by shielding that suppresses other beam components.

Particle decay is a spontaneous transformation of an unstable particle into allowed final particles. Conservation of energy, momentum, electric charge and the relevant interaction's other conserved quantities constrain the possibilities. A decay can be energetically allowed yet absent or weak because its transition amplitude is suppressed.

For an isolated state with an approximately constant rest-frame decay rate, the surviving population follows an exponential in proper time. The mean lifetime $\tau$ and energy decay width $\Gamma$ obey $\Gamma =\hslash /\tau$. A moving particle's laboratory flight-time distribution also includes time dilation.

Experimentally, detectors reconstruct decay products and sometimes a separated decay vertex. Correcting for acceptance and reconstruction efficiency is necessary before interpreting their counts as a lifetime or branching fraction. Different decay channels compete to make the total width.

The ideal two-shower event fraction is $B_e^2$, approximately $4\%$ in simple channel counting or $3.18\%$ using the measured tau lepton electronic branching fraction. The selected particle decay chain isThe electron and Positron each make a contained electromagnetic shower; muons penetrate, and generic hadrons do not deposit all their energy as a contained electromagnetic shower.

To estimate $B_e$, use lepton universality and neglect final masses and quantum chromodynamics corrections. The electronic and muonic decay widths each carry weight one. The accessible inclusive hadronic channels have weights $3|V_{ud}{|}^2$ and $3|V_{us}{|}^2$, where three is the color multiplicity in a decay width. The charm quark is inaccessible in physical tau lepton hadronic decays. With $|V_{ud}{|}^2+|V_{us}{|}^2\simeq 1$, the total weight is $1+1+3=5$, givingThe 2021 measured value $B_e=0.1782$ instead gives $B_e^2=0.0317552$.^[ref] The difference reflects effects omitted by the simple free-channel estimate, especially the hadronic contribution to the total decay width.

There are two tau leptons per produced pair and two per selected pair, so the fraction of all produced tau leptons belonging to these two-shower events is also $B_e^2$. The unrestricted fraction of individual tau leptons decaying electronically is $B_e$; these are different selections. A real high-energy shower cut gives an additional acceptance factor: $N_{\mathrm{selected}}/N_{\mathrm{pairs}}=B_e^2{\epsilon }_{ee}$, which cannot be computed without the cut and detector response. No factor of two is needed for the unique $e^{+}{e}^{-}$ assignment.

Treat the atmospheric proton as stationary in the Earth frame, the incident antiproton as vertically downward, and neglect atmospheric energy loss, scattering and the Earth's curvature initially. Take $m_p{c}^2=0.9383\mathrm{GeV}$. Conservation of energy and equal outgoing energies giveEqual energies imply equal momentum magnitudes. Their transverse momenta cancel, so the two muons have equal downward components $p_z=p_{\bar{p}}/2$, whereEach muon therefore makes an angle $\theta$ with the downward vertical satisfyingThe paths are not both vertical: replacing their path lengths by $20\mathrm{km}$ would miss this two-body kinematic correction. Since $v_z=p_z{c}^2/E_{\mu }$, the proper flight time over vertical height $H$ isFor equal-energy atmospheric muon-pair survival, independent exponential particle decays with mean proper muon lifetime ${\tau }_0=2.2\mu \mathrm{s}$, both survive with probabilityEquivalently each travels about $20.55\mathrm{km}$, with proper flight time about $0.404\mu \mathrm{s}$, and survives with probability about $0.832$. The result is decay-only survival to an ideal surface, not a prediction including air interactions or detector efficiency. For a spherical Earth of radius $R_E$, the exact path length is $\ell =(R_E+H)\cos \theta -\sqrt{R_E^2-(R_E+H{)}^2{\sin }^2\theta }$; inserting $P_{\mathrm{both}}=\exp [-2\ell /({\gamma }_{\mu }{v}_{\mu }{\tau }_0)]$ gives a small curvature correction.

The Q value of a reaction or particle decay is the initial total rest energy minus the final total rest energy. Positive $Q$ releases kinetic energy; a negative $Q$ requires energy supplied by incident motion or another source. Atomic and bare nuclear mass conventions must be distinguished when counting electrons in beta decay.

A particle decay into three daughter particles subject to four-momentum conservation and the relativistic energy-momentum relation. The daughter energies generally vary continuously even in the parent's rest frame. Grouping two daughters into a recoil system of variable invariant mass turns the remaining kinematics into a relativistic two-body decay.

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.

Choose $m_{\tau }=2\mathrm{GeV}$ and $\sqrt{s}=6\mathrm{GeV}$ purely for illustration. Then each parent has $E_{\tau }=3\mathrm{GeV}$, $|{\mathbf{p}}_{\tau }|=\sqrt{5}\mathrm{GeV}$ and $\beta =\sqrt{5}/3$. Treat all six daughters as massless, including the charged electron and antimuon, and generate each three-body particle decay with constant squared decay amplitude. Choose the parent direction isotropically and the two decays independently; this omits the physical production angular law and spin correlations.

The Lorentz-invariant phase space factorizes through $t=q^2$ into a two-body decay to $\ell +q$, followed by a two-body decay of $q$:After angular integration the first factor contributes $1-t/m_{\tau }^2=x$, while the massless second factor is constant. Since $|dt|=m_{\tau }^{2}dx$, the normalized chosen density is $f(x)=2x$. Drawing $x=\sqrt{U}$ with $U$ uniform on $[0,1]$ implements this density. Isotropic directions, followed by a Lorentz boost for the invisible pair and another for the parent, conserve the complete four-momentum of each generated event.

Define transverse acoplanarity by $\alpha =\pi -\Delta \varphi$, where $\Delta \varphi \in [0,\pi ]$ is the angle between the two visible momentum projections perpendicular to the beam. A fully visible two-body event with zero initial transverse momentum has $\alpha =0$ and $M_{\mathrm{miss}}^2=0$. The toy tau lepton events spread away from that point. These observables are complementary: an angular imbalance asks about directions, while the missing invariant mass also uses the visible energies and longitudinal momenta.

A Lorentz-invariant phase space density supplies allowed configurations and their geometric measure. A physical weak interaction decay amplitude reweights them and correlates spins. Consequently the plotted density is a lesson about conservation and available configurations, not a predicted experimental rate or a fit to the discovery sample. The massless charged-daughter approximation is also unsuitable near the muon rest-energy threshold.

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.