Standard Model fermion

A Standard Model generation contains left-handed quark and lepton doublets and right-handed up-quark, down-quark and charged-lepton singlets. A right-handed neutrino may be added as a gauge singlet.

A generation groups corresponding quarks and leptons with the same gauge quantum numbers as their counterparts in other generations. The Standard Model has three such families. Yukawa matrices assign their different masses, and the mismatch of up-type and down-type mass bases produces the CKM matrix.

The three quark generations pair $(u,d)$, $(c,s)$ and $(t,b)$, each containing one charge-$2/3$ and one charge-$-1/3$ quark. The number of generations controls the number of physical mixing angles and CP-violating phases in the CKM matrix.

A lepton is a Standard Model fermion without color charge. The charged leptons have electric charge $-1$ and their corresponding neutrinos have charge zero. There are three generations; their antiparticles carry opposite electric charge and lepton number. Charged leptons and neutrinos have lepton number $+1$, with the corresponding antiparticles carrying $-1$.

An antilepton is the antiparticle of a lepton, with opposite additive quantum numbers. A Positron is the charged antilepton corresponding to an electron.

A lepton with nonzero electric charge, such as an electron or muon; neutrinos are electrically neutral.

The tau is the charged lepton of the third fermion generation. In the massless approximation its charged-current W boson partial decay width equals those of the electron and muon, reflecting lepton universality.

The tau lepton has electric charge $-1$ in units of the positive elementary electric charge; ${\tau }^{+}$ is its antiparticle. Its weak charged current decays make the tau lepton useful for testing lepton universality. They also illustrate how particle detection can establish a short-lived parent through visible daughters and a consistent invisible recoil.

The visible angular distribution and laboratory energy in ${\tau }^{-}\to h^{-}{\nu }_{\tau }$ carry information about the parent's spin. For fixed hadronic invariant mass, $E_h=\gamma (E_h^{\ast }+\beta p_h^{\ast }\cos \vartheta )$. A spin-zero recoil has analyzing coefficient ${\alpha }_h=1$. For a spin-one recoil coupled through the V-A interaction, longitudinal and transverse partial rates have weights $m_{\tau }^2:2m_h^2$ and opposite slopes, giving ${\alpha }_h=(m_{\tau }^2-2m_h^2)/(m_{\tau }^2+2m_h^2)$. Thus the harder-helicity ordering can reverse when the recoil mass exceeds $m_{\tau }/\sqrt{2}$. A mean invariant mass without spin-dependent spectral weights cannot fix the sign for a mixed sample.

For $N$ tau-pair events and a tag mode with branching fraction $B$, the tagged-hemisphere count is $S=2NBϵ$ and the double-tag event count is $D=N{B}^2{ϵ}_{ii}$. Eliminating $B$ gives $N=S^2{ϵ}_{ii}/(4D{ϵ}^2)$. Efficiency correlations matter: ${ϵ}_{ii}$ need not equal ${ϵ}^2$. An inclusive branching-fraction fit must include unlisted decay categories.

Counterrotating electron and Positron beams at SPEAR collided inside the Mark I magnetic particle detector. Lead–scintillator scintillation counters identified electrons through large shower pulses; small pulses plus hits beyond the iron return identified muons.^[ref]

The selection required opposite-charge electron–muon tracks, no detected photons, momenta above $0.65\mathrm{GeV}/c$, and transverse noncoplanarity above $20^{\circ }$. Multitrack hadron samples calibrated momentum-dependent misidentification; collinear lepton pairs checked identification losses. Photon and charge control samples tested backgrounds. Acceptance corrections depended on the proposed signal.^[ref]

The 1975 claim was anomalous events without a conventional explanation; new-particle pair production was a possible interpretation.^[ref] Martin Lewis Perl's lecture describes subsequent evidence for a charged lepton.^[ref]

The particle identification problem matters as much as the apparent missing four-momentum in electron-positron collisions. A penetrating hadron can imitate a muon; a particle shower can imitate an electron; an undetected photon or a detector gap can imitate invisible recoil. An absence of additional recorded signals is conditional on detection efficiency. In a proposed signal model, the selected count has the schematic formwhere $\mathcal{L}$ is integrated collider luminosity, $\sigma$ a cross section, and $\epsilon$ the combined acceptance and detection efficiency. A single count cannot determine all three factors. A control sample must probe the relevant momentum and angular range; its abundant events do not automatically calibrate a differently distributed signal.

The beam-overlap factor $L$ relating a reaction's cross section $\sigma$ to its ideal production rate $dN/dt$. Integrated collider luminosity $\mathcal{L}=\int Ldt$ has units of inverse area and gives the expected production count $\mathcal{L}\sigma$ when $\sigma$ is constant.

An absolute collider luminosity calibration that scans the transverse separation of two colliding beams while measuring an interaction rate. Known bunch populations and the overlap integral determine the visible monitor cross section. Correlations between horizontal and vertical profiles require a joint overlap treatment.

The time integral of the incident-beam overlap flux, with inverse-area units. For a fixed production cross section, the mean produced event count is $N={\mathcal{L}}_{\mathrm{int}}\sigma$. A selected count includes efficiency and acceptance, often integrated over final-state kinematics. This is distinct from astrophysical radiative luminosity, which has power units.

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.

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.

Use natural units $c=1$ and the Minkowski metric with signs $(+,-,-,-)$. In the ideal centre-of-momentum frame, assume a known initial four-momentum $P=(\sqrt{s},0)$, no unaccounted radiation and exact visible reconstruction. DefineThe missing four-momentum in electron-positron collisions is a subtraction of measured quantities. It is not itself a particle detector hit. With $M_{e\mu }^2=(p_e+p_{\mu }{)}^2$, expansion gives the useful identityFor any collection of future-directed massless invisible four-momenta $q_i=E_i(1,{\hat{\mathbf{n}}}_i)$,Thus a positive missing invariant mass excludes exactly one massless invisible particle under the stated assumptions. It does not exclude one massive invisible particle, identify a neutrino flavor, or determine the number of massless invisible particles beyond that lower bound. Several collinear massless particles can even have zero combined invariant mass. Detector resolution and unaccounted radiation invalidate a literal event-by-event interpretation of these ideal inequalities.

For an original exact example in natural units, choose total energy $6\mathrm{GeV}$ and visible massless four-momenta, in $\mathrm{GeV}$,Two possible invisible massless four-momenta are $(2;-1/2,-1/2,\sqrt{14}/2)$ and $(2;-1/2,-1/2,-\sqrt{14}/2)$. Splitting either into two positive collinear parts gives a three-particle interpretation with identical recoil. Alternatively, one invisible particle of mass $\sqrt{14}\mathrm{GeV}$ has that recoil. This example tests the ambiguity of the subtraction, without asserting any of these alternatives is a tau lepton decay.

The energy component of the difference between the known initial four-momentum and the measured visible final four-momenta. In equal-beam electron-Positron collisions it is $\sqrt{s}-{\sum }_i{E}_i$ in natural units. It can arise from escaping neutrinos, but also from detector losses; it is distinct from the missing invariant mass and missing transverse momentum.

The invariant mass of the inferred invisible system, obtained from its four-momentum. It is generally different from the mass of each invisible constituent.

The muon is the negatively charged second-generation lepton, with electric charge $-1$ and lepton number $+1$. Its antiparticle is the antimuon.

The rest mass of the muon, approximately $105.66\mathrm{MeV}$ in natural units. Keeping it in phase space is necessary in precision lepton universality tests.

A weak charged current decay described at energies much below the W boson mass by a four-fermion interaction with coefficient given by the Fermi constant. At leading order with negligible final-state masses, its width is $G_F^2{m}_{\mu }^5/(192{\pi }^3)$.

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 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 an unpolarized parent charged lepton undergoing a three-body leptonic V-A interaction decay, neglecting daughter masses and radiative corrections, the normalized charged-daughter energy distribution in the parent rest frame is $f(x)=2x^2(3-2x)$, where $x=2E_{\ell }^{\ast }/m_{\mathrm{parent}}$ lies in $[0,1]$. Direct integration gives ${\int }_0^{1}f(x)dx=1$ and ${\int }_0^{1}xf(x)dx=7/10$. A specified parent spin orientation also produces angular terms, so this one-variable law alone does not describe polarized samples.

For a negative muon with rest-frame spin vector $\mathbf{s}$, the leading massless-electron distribution is $d\Gamma /(dxd\Omega )=G_F^2{m}_{\mu }^5{x}^2[(3-2x)+(1-2x)\hat{\mathbf{k}}\cdot \mathbf{s}]/(384{\pi }^4)$, where $x=2E_e/m_{\mu }\in [0,1]$. Its final-spin sum is $64G_F^2(k\cdot q^{\prime })(q\cdot (p-m_{\mu }s))$. The spin vector is axial whereas the electron direction is polar, so their scalar product reverses under parity. Its occurrence demonstrates parity violation by the left-chiral weak charged current.

The antimuon is the antiparticle of the muon. It has electric charge $+1$ and lepton number $-1$.

A quark or antiquark in the nonvalence component of a hadron's parton distribution functions. Such pairs do not change the net valence flavor numbers. Approximately equal light-flavor sea distributions can dominate small-x deep inelastic scattering, but exact equality is not required by their definition.

A constituent used to specify the net quark flavor content of a hadron. The quark model uses three valence quarks for an ordinary baryon and a quark-antiquark pair for an ordinary meson; this does not exclude additional sea pairs and gluons in the full state.

A quark mass is the coefficient of its Dirac mass term. This term couples left- and right-handed quarks, so nonzero quark masses explicitly break independent rotations of the two chiralities. For two equal masses, the vector flavor symmetry survives, while the nonsinglet axial current obeys the flavor-nonsinglet axial Ward identity with equal quark masses. In the renormalized theory the mass parameter depends on the subtraction scale and scheme, while measurable hadron masses are physical observables.

A constituent quark mass is an effective mass parameter in a low-energy constituent-quark model. It accounts approximately for strong-interaction dressing and some spin-independent bound-state dynamics; it is not identical to a renormalized quark mass in the Standard Model Lagrangian.

A quark field is a Dirac field or a chiral Weyl spinor carrying the colour fundamental representation of $SU(3)$. Its three colour components transform as ${\psi }^i\mapsto U^i{}_j{\psi }^j$ under local gauge transformations. The colour index is independent of the Lorentz spinor index, and the weak representations of its left and right chiral components differ in the Standard Model.

A quark flavor is one of the six Standard Model quark species: up, down, strange, charm, bottom and top. Each flavor occurs in three colour states and is counted as one Dirac species when both chiralities are included. The active flavor number in a low-energy QCD description excludes quarks heavier than the relevant scale, with matching of the running coupling when a heavy threshold is crossed.

The net number of charm quarks minus charm antiquarks. It is conserved by the strong interaction and electromagnetic interaction, but can change through weak charged currents.

The number of quarks of quark flavor $f$ minus the number of corresponding antiquarks. The strong interaction and electromagnetic interaction conserve each such difference even when they create or annihilate pairs. Weak charged currents can change these differences. For the strange quark, strangeness is $S=-F_s$.

The top quark is the up-type member of the third quark generation. Its mass exceeds the W boson mass, so a real W boson cannot decay into an on-shell top quark. A massless approximation for accessible decay products must not silently remove this channel threshold.

Here $x=m_W^2/m_t^2$, and the tree-level decay width neglects the bottom quark mass. The open two-body W channel gives a top quark lifetime of a few $10^{-25}\mathrm{s}$, shorter than the hadronization time. A fifth-power Sargent's rule extrapolation misses this physical regime.^[ref]

Here $x=m_W^2/m_t^2$, the bottom quark mass is neglected, and the W boson is on shell. The weak charged current gives the spin-averaged squared scattering amplitude $g_W^2|V_{tb}{|}^2(m_t^2-m_W^2)(m_t^2+2m_W^2)/(4m_W^2)$. The two-body Lorentz-invariant phase space has $p^{\ast }=(m_t^2-m_W^2)/(2m_t)$; multiplying by $p^{\ast }/(8\pi m_t^2)$ gives the displayed decay width. Its longitudinal-vector contribution explains the large-$m_t$ cubic growth, and its threshold factor makes the width vanish as $m_t\downarrow m_W$. The formula requires $m_t>m_W$ and receives finite-mass and radiative corrections.

A constituent quark that does not participate in the elementary weak vertex of a meson decay. It combines with a produced quark or antiquark in the final hadron.

The antiparticle of the strange quark has electric charge $+1/3$ and baryon number $-1/3$.

The antiparticle of the down quark has electric charge $+1/3$ and baryon number $-1/3$.

The antiparticle of a quark.

The Standard Model representations under $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y$ are $Q_L=(3,2{)}_{1/6}$, $u_R=(3,1{)}_{2/3}$, $d_R=(3,1{)}_{-1/3}$, $L_L=(1,2{)}_{-1/2}$, $e_R=(1,1{)}_{-1}$, and $H=(1,2{)}_{1/2}$.

In one Standard Model generation, the left-handed up and down quark fields form a weak $SU(2)$ doublet of hypercharge $1/6$. Each weak component has three colours. The same field therefore counts as three doublets for weak gauge-loop contributions and two colour fundamentals for strong gauge-loop contributions. Right-handed up and down quarks are separate weak singlets.

In normalization $Q=T_3+Y$, the left quark doublet has $Y=1/6$, right up/down singlets have $2/3,-1/3$, the left lepton doublet has $-1/2$, the right charged lepton has $-1$, and the Higgs doublet has $1/2$. The alternative convention $Q=T_3+\hat{Y}/2$ uses $\hat{Y}=2Y$. Up-type Yukawa interactions use the conjugate doublet of hypercharge $-1/2$.

Baryon number assigns $B=1/3$ to each quark, $B=-1/3$ to each antiquark and zero to leptons. A proton has baryon number one.

The difference between baryon number and lepton number. Standard electroweak sphalerons violate $B+L$ while conserving $B-L$. Consequently an early baryon asymmetry with zero $B-L$ need not survive equilibration; a surviving nonzero $B-L$ can be redistributed between baryons and leptons. Additional lepton-number-violating interactions can change this conclusion.

Baryon plus lepton number is the sum of baryon number and lepton number. It is an accidental classical symmetry of the renormalizable Standard Model but has an $SU(2{)}_L$ chiral anomaly, so electroweak instantons and sphalerons violate it.

Proton decay is a hypothetical baryon-number-violating transition of the proton into lighter particles. Its nonobservation places strong lower bounds on the scale, or strong upper bounds on the couplings, of baryon-number-violating interactions.

A proton-decay operator is a local effective interaction that violates baryon number. Standard Model gauge invariance first permits such operators at dimension six, including $QQQL$ and $u^c{u}^c{d}^c{e}^c$.

Lepton number assigns $L=1$ to leptons, $L=-1$ to antileptons and zero to quarks. Neutrino masses show that separate family lepton numbers need not be exact.

Sum the lepton numbers of every final particle and compare with the initial sum. For example, $e^{-}{\nu }_e{e}^{-}{e}^{+}$ has $L_e=2$, whereas $e^{-}{\bar{\nu }}_e{e}^{-}{e}^{+}$ has $L_e=0$. The two differ physically; changing a neutrino bar is not a harmless typographic simplification. Overall lepton number is conserved by the usual charged-current Standard Model amplitudes with Dirac or massless neutrinos.

A neutrino is an electrically neutral spin-one-half lepton. The Standard Model has three active neutrino flavours associated with the electron, muon and tau.

A neutrino is a spin-one-half fermion. Its spin projection along its momentum defines neutrino helicity.

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 neutrino with distinct particle and antiparticle states and a Dirac mass term connecting left- and right-chiral fields. If conventional lepton number is conserved, such a mass does not cause neutrinoless double beta decay. Oscillation evidence for nonzero neutrino masses does not alone distinguish the Dirac and Majorana possibilities.

A neutrino participating in the weak interactions of the Standard Model, in contrast to a sterile state. Its thermal cosmological abundance and momentum distribution depend on weak decoupling and subsequent cosmic history.

For a massless incident neutrino and a stationary target electron, the invariant squared center-of-mass energy is $s=m_e^2+2m_e{E}_{\nu }$ in natural units. Production of a charged lepton of mass $m_{\ell }$ and a massless neutrino requires $s\ge m_{\ell }^2$, giving the displayed threshold. For the electron flavour the threshold is zero; for heavier charged leptons it is enhanced by the small stationary-target mass.

A neutral fermion without the ordinary active-neutrino weak gauge couplings. Mixing can still let it interact or decay. Its possible dark-matter role depends on lifetime, production and free streaming, rather than on Majorana character alone.

A right-handed neutrino can transform as $(1,1,0)$ under colour, weak isospin and hypercharge. It has no direct Standard Model gauge coupling, but may mix with active neutrinos. Sterility under the low-energy group does not imply sterility under an enlarged unbroken gauge group.

A Majorana neutrino is its own antiparticle. A corresponding mass term can violate conventional lepton number by two. Observing a wrong-helicity charged-current process is nevertheless suppressed by the very small neutrino mass relative to its energy, rather than having the ordinary charge-conjugate branching fraction.

An atmospheric neutrino is a neutrino produced by particle decays in a cosmic ray air shower.

The electron neutrino is the neutrino flavour associated with the electron in the weak charged current.

The neutrino flavor associated with the tau lepton in the weak charged current. Its direct detection through interactions producing a tau lepton was announced by the DONUT experiment in 2000.^[ref]

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 $3.09\mathrm{TeV}$.

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.

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.

The mass of a neutrino mass eigenstate. A neutrino flavor need not have one definite mass; neutrino oscillation probes differences of squared masses.

A neutrino label defined by the associated charged lepton in weak charged current production or detection. A produced flavour state can evolve through neutrino oscillation.

If all neutrinos are exactly massless, arbitrary unitary rotations among them preserve both their kinetic terms and their zero mass matrix. After diagonalizing the charged-lepton Yukawa matrix, choose the neutrino basis to coincide with that rotation. The resulting weak charged current has no physical lepton mixing matrix. With nondegenerate neutrino masses this extra basis freedom is lost, permitting observable neutrino oscillations and lepton mixing.

The muon neutrino is the neutral lepton coupled to a muon by the weak charged current. It carries lepton number $+1$; its antineutrino has the opposite lepton number.

An antineutrino is the antiparticle of a neutrino and carries the opposite additive lepton number.

The antineutrino associated with the muon flavour of the weak charged current has zero electric charge and lepton number $-1$ in the usual production-and-detection description. A negative muon decays to an electron, an electron antineutrino and a muon neutrino; the charge-conjugate process instead produces the muon antineutrino.

The antineutrino associated with the electron flavour of the weak charged current. It has zero electric charge and lepton number $-1$ in the usual production-and-detection description.

Neutrino oscillation is coherent conversion between neutrino flavors: different mass eigenstates acquire different propagation phases. It measures squared-mass differences and mixing, rather than an absolute neutrino mass. The Super-Kamiokande atmospheric-neutrino experiment tests path-dependent disappearance; the Sudbury Neutrino Observatory compares reactions with complementary neutrino flavor sensitivity. Solar neutrino propagation also requires matter effects; the two-flavour vacuum survival model is only an illustration.

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.

A long-baseline neutrino oscillation experiment using a Fermilab beam, near and far detectors, and magnetized iron/scintillator sampling calorimeters. A charged current muon neutrino event gives a negative muon and a hadronic shower; their reconstructed energies estimate the incident neutrino energy.

Coherent forward weak scattering in ordinary matter changes neutrino propagation phases. The charged-current electron contribution gives the displayed potential difference, with the opposite sign for antineutrinos. A common neutral-current term can be removed from active-flavour oscillations. A sufficiently large effect requires propagation with the matter Hamiltonian rather than the vacuum formula.

Matter changes the effective mixing angle and level splitting of propagating neutrinos. The electron-flavour charged current potential is $\sqrt{2}{G}_F{N}_e$ relative to the other active flavours. In the two-flavour Hamiltonian, equality of its two diagonal entries gives the displayed resonance condition. Adiabatic evolution through a varying density can therefore cause a large flavour change even when the local vacuum mixing angle is smaller. The equality alone does not guarantee adiabatic propagation.

Flavor conversion in a slowly varying medium when an occupied instantaneous propagation eigenstate follows its eigenvalue branch without transitions to the other branch. For positive electron-neutrino matter potential and normal two-state ordering, a high-density electron neutrino follows the upper branch into the vacuum mass eigenstate ${\nu }_2$. Detection in the electron flavor then has probability ${\sin }^2\theta$, rather than the phase-averaged vacuum value $1-\frac{1}{2}{\sin }^{2}2\theta$. High production density alone does not guarantee adiabatic evolution.

A neutrino state selected by the charged lepton accompanying a weak charged current production or detection process. It is generally a coherent quantum superposition of mass eigenstates, rather than a state of definite propagating mass.

For coherent relativistic vacuum propagation and a unitary lepton mixing matrix, square ${\sum }_i|U_{ei}{|}^2\exp [-i{m}_i^{2}L/(2E)]$ to obtain this probability. Only moduli of the mixing entries occur, so the ordinary three-state electron survival signal is independent of the intrinsic CP phase. Replacing two close fast phases by one is useful at short baselines but may fail over many periods.

The difference between the squared masses of two propagation states. In vacuum its contribution to a neutrino oscillation probability has phase $\Delta m_{ij}^{2}L/(4E)$ in natural units. The exact identity $\Delta m_{13}^2=\Delta m_{12}^2+\Delta m_{23}^2$ constrains any proposed three-state numerical inputs.

The normal ordering has the third light neutrino above the solar pair in mass; the inverted ordering has it below that pair. Oscillation experiments determine differences of squared masses, not the common absolute mass scale. A pure vacuum electron-survival expression uses squared sines and does not by itself fix the signs of individual phase differences.

The matrix relating weak-interaction neutrino flavour fields to mass eigenfields, ${\nu }_{\alpha L}={\sum }_i{U}_{\alpha i}{\nu }_{iL}$. In the standard three-active-neutrino description it is unitary. Oscillation probabilities involve products of entries and mass-squared differences; lepton-number-violating amplitudes can depend on additional Majorana phases through products such as $U_{ei}^2{m}_i$.

A right-handed neutrino transforms as $(1,1{)}_0$ under $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$, so it is neutral under every Standard Model gauge interaction.

A right-handed neutrino is a singlet of the entire Standard Model gauge group. Its same-chirality bilinear with the field obtained by charge conjugation therefore admits the displayed renormalizable Majorana mass term, in addition to a Higgs-induced Dirac mass term. It violates lepton number by two units. Analogous terms for electrically charged Standard Model fermions carry nonzero electric charge; quarks also have nontrivial color charge. The presence of a right-handed field permits mass but does not force a nonzero mass parameter.

The type-I seesaw mechanism combines a Dirac neutrino mass $m_D$ with a much larger right-handed-neutrino Majorana mass term $M$. Diagonalizing the mass matrix gives a light eigenvalue of order $m_D^2/M$ and a heavy eigenvalue of order $M$.

In the type-I regime $M_L\simeq 0$, $M_R\gg m_D$, the light mass is $m_D^2/M_R$. Specifying only the right-handed scale does not fix a numerical light mass. A choice of Dirac mass is an extra assumption, not a result of matrix diagonalization.

In an all-left-handed Weyl basis $({\nu }_L,{\nu }_R^c)$, the symmetric matrix has left and right Majorana entries $M_L,M_R$ and off-diagonal Dirac entry $m$. Real eigenvalues can have signs; physical Majorana masses are nonnegative after appropriate field rephasings.

The Weinberg operator is the dimension-five Standard Model operator $(LH)(LH)/\Lambda$. After the Higgs field condenses, it gives a left-handed neutrino Majorana mass of order $v^2/\Lambda$.

For the gauge-singlet composite $\chi ={\tilde{\varphi }}^{†}L$, the interaction $G{\chi }^T{C}^{-1}\chi -G^{\ast }\bar{\chi }C{\bar{\chi }}^T$ is electroweak gauge-invariant. At the Higgs field vacuum, $\chi =v{\nu }_L/\sqrt{2}$, giving $\frac{1}{2}(G{v}^2{\nu }_L^T{C}^{-1}{\nu }_L-G^{\ast }{v}^2{\bar{\nu }}_{L}C{\bar{\nu }}_L^T)$. The Grassmann variation of a chiral Majorana mass term therefore identifies the mass parameter as $G{v}^2$, with physical mass $|G|v^2$. The dimension-five interaction has a coupling of mass dimension minus one. Under lepton number phase rotation, its two terms acquire factors $e^{\pm 2i\alpha }$, so it violates continuous lepton number by two units. It is an effective extension of the minimal renormalizable Standard Model.