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Electroweak symmetry breaking is the Higgs-field breaking $SU(2{)}_L\times U(1{)}_Y\to U(1{)}_Q$. It gives masses to the $W$ and $Z$ bosons while leaving the photon massless.

With the hypercharge convention $Q=T_3+Y$, the lower component of the Higgs doublet has $T_3=-1/2$ and $Y=1/2$. Thus $Q⟨\varphi ⟩=0$: the vacuum expectation value is electrically neutral. This is why one combination of the neutral gauge fields remains the massless photon, as the electroweak doublet gauge-boson mass matrix shows.

Continue the chosen Higgs field potential example with $g=3/5$ and $g^{\prime }=4/5$. The tree-level electroweak gauge-boson masses are $m_W=72\mathrm{GeV}$, $m_Z=120\mathrm{GeV}$ and $m_A=0$. In the $(W^3,B)$ basis,Its orthonormal eigenvectors $(4/5,3/5{)}^T$ and $(3/5,-4/5{)}^T$ have eigenvalues zero and $14400{\mathrm{GeV}}^2$. Thus $\cos {\theta }_W=3/5=m_W/m_Z$. This is an exact model check, not a fit to measured electroweak interaction parameters.

A diagonal charged-fermion Yukawa coupling expands asThe Yukawa coupling remains a free parameter: the mechanism relates a fermion mass to its scalar interaction but does not predict the numerical mass. This example concerns charged fermions; the minimal Standard Model has no right-handed neutrino and does not account for neutrino oscillations without an extension.

A theoretical physicist whose work relates flavor symmetry to patterns of neutrino oscillations. His 2008 analysis asked which finite groups can enforce a particular mixing pattern independently of arbitrary interaction strengths. This turns a proposed pattern of quantum mixing into a constraint on a model’s symmetry, rather than adjusting each matrix entry separately.^[ref]

How can a detector establish that an unseen particle changes identity? The Super-Kamiokande atmospheric-neutrino experiment compares different travel paths; the Sudbury Neutrino Observatory compares different interaction channels. Both infer neutrino flavor from reaction products and calibrated responses. Their complementarity connects particle detection to neutrino oscillation, neutrino mass and neutrino astronomy.

Takaaki Kajita and Arthur B. McDonald shared the 2015 physics Nobel Prize. These were collaboration achievements: Yoichiro Suzuki's Super-Kamiokande work included the solar neutrino analysis, and George Ewan helped establish and lead the Sudbury Neutrino Observatory.^[ref][ref]

The older Kamiokande account explains water Cherenkov neutrino reconstruction; Raymond Davis Jr.'s Homestake experiment explains why a deficit in an electron neutrino channel alone leaves a source-versus-propagation ambiguity. Complementary channels can break that ambiguity.

Kamiokande, associated with Masatoshi Koshiba, was an underground water Cherenkov detector viewed by photomultipliers. Improved water purity and reduced background enabled solar neutrino observations. A concentration of reconstructed recoil directions toward the Sun distinguished a source signal above an approximately direction-independent background.^[ref]

For solar neutrinos, electron elastic scattering supplies a directional recoil, with different cross sections for active neutrino flavors. Neither the Homestake experiment deficit nor the SN 1987A neutrino burst alone established neutrino oscillation; later evidence compared complementary channels and propagation effects.^[ref]

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 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.

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.

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.

A neutrino is an electrically neutral spin-one-half lepton with very weak ordinary interactions. The three active flavors are associated with the electron, muon and tau in weak charged current production and detection. A flavor state is generally a superposition of neutrino mass eigenstates, permitting neutrino oscillations during propagation.

Detection infers a neutrino from its interaction products rather than observing an ionizing track from a free neutrino. A charged-current event can produce the corresponding charged lepton, while a neutral-current event leaves a neutrino in the final state and deposits energy in the target. Small interaction probabilities motivate large target masses and careful control of backgrounds. Neutrality does not imply zero mass; oscillations constrain differences of squared masses, rather than fixing the absolute mass scale by themselves.

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.

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

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.

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.

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.

Awarded in equal shares to Takaaki Kajita and Arthur B. McDonald for discovering neutrino oscillations, showing that neutrinos have mass.^[ref]

The left- and right-chiral components of a charged Standard Model fermion transform in different electroweak group representations. A bare Dirac mass term $-m{\bar{f}}_L{f}_R+h.c.$ is therefore not invariant under the full electroweak gauge group. The Higgs doublet supplies the missing transformation properties. With $\tilde{H}=i{\sigma }_2{H}^{\ast }$, the gauge-invariant Yukawa interactions arewhere each $Y_f$ is a flavor-space Yukawa matrix. For example, hypercharges in the down-quark term add to $-1/6+1/2-1/3=0$, while those in the up-quark term add to $-1/6-1/2+2/3=0$. Thus the Higgs and conjugate Higgs are required in different terms.

After electroweak symmetry breaking, choose unitary gauge with $H=(0,(v+h)/\sqrt{2}{)}^T$. The fermion mass matrices becomeA singular value decomposition $U_{fL}^{†}{M}_f{U}_{fR}=\mathrm{diag}(m_{f,1},m_{f,2},m_{f,3})$ defines nonnegative physical tree-level masses. The mismatch $U_{uL}^{†}{U}_{dL}$ is the Cabibbo-Kobayashi-Maskawa matrix in the weak charged current. The corresponding physical Higgs boson couples diagonally as $-(m_f/v)h\bar{f}f$ in the minimal one-doublet theory. This relates its coupling strength to each mass, but the measured Yukawa couplings remain independent input parameters: the minimal theory does not predict their hierarchy.

The minimal renormalizable Standard Model contains no right-handed neutrino, so it has neither a neutrino Dirac Yukawa term nor a gauge-invariant renormalizable left-handed Majorana mass term. Its neutrinos are massless. Observed neutrino oscillations require going beyond that minimal field content or operator set: adding right-handed singlets permits Dirac masses, while the dimension-five Weinberg operator gives a Majorana neutrino mass of order $v^2/\Lambda$ after symmetry breaking. These alternatives should be distinguished from the charged-fermion Yukawa mechanism.

The first argument is the energy budget. The gravitational energy reservoir is of order $G{M}^2/R$, so the Kelvin-Helmholtz cooling time is $t_{\mathrm{KH}}\sim G{M}^2/(RL)$. For the Sun, $M\simeq 2\times 10^{30}\mathrm{kg}$, $R\simeq 7\times 10^8\mathrm{m}$ and $L\simeq 3.8\times 10^{26}\mathrm{W}$ give $t_{\mathrm{KH}}\sim 3\times 10^7$ years. This is much shorter than the approximately $4.6\times 10^9$-year solar-system age. Chemical binding energies are only of order electronvolts per atom and provide an even smaller reservoir. Thus neither ordinary chemical energy nor continuing gravitational contraction can supply the long-term solar luminosity.

Stellar nuclear fusion provides an adequate reservoir. Conversion of four hydrogen atoms into one helium-4 atom releases about $26.7\mathrm{MeV}$, corresponding to a mass defect of about $0.007$ of the initial mass; some energy leaves as neutrinos. If a fraction $f$ of stellar mass with hydrogen fraction $X$ is available for core burning, the energy scale is $0.007fXM{c}^2$. Taking $f\sim 0.1$ and $X\sim 0.7$ supplies roughly $9\times 10^{43}\mathrm{J}$, enough for several billion years at the solar luminosity, consistent with a stellar nuclear timescale of order $10^{10}$ years. The core temperature and density inferred from stellar structure permit the proton–proton chain; hotter stellar cores favor the CNO cycle.

The most direct evidence is the detection of solar neutrinos. They are created in the weak reactions of the proton–proton chain and can escape from the core, unlike the photons whose transport through the interior is slow. Measurements accounting for neutrino oscillation recover a total neutrino flux consistent with nuclear-powered solar models. Their spectra and reaction-dependent fluxes test the core reactions rather than merely the surface effective temperature.

There is also independent evidence from stellar evolution: the main sequence is a long-lived core-hydrogen-burning phase, more massive luminous stars consume fuel more quickly, and the main-sequence turnoff of a star cluster records its age. Subsequent evolution and the production of heavier elements fit further nuclear reactions. Agreement among these observations, nuclear energy release and stellar-model lifetimes is much stronger than the energy-budget argument alone. The combination of a sufficient nuclear reservoir, measured core neutrinos and the observed evolutionary sequence identifies nuclear reactions as the main long-term power source of ordinary stars.

Both the left-handed electron and the electron neutrino participate in weak charged currents through $W^{\pm }$, and both have weak neutral currents mediated by the Z boson. Their electromagnetic interactions differ: the electron has electric charge $-e$ and couples to the photon, while the neutrino is electrically neutral and has no tree-level electromagnetic coupling.

In the minimal massless-neutrino Standard Model, the active neutrino has only a left-handed particle state and a right-handed antiparticle state. The electron has both chiralities: its right-handed component has electromagnetic interactions and a weak neutral current, but no weak charged current. Its nonzero mass comes from a Yukawa coupling to the Higgs field; the minimal massless-neutrino model contains no right-handed neutrino for an analogous Dirac mass term.

Observed neutrino oscillations establish nonzero neutrino mass-squared differences. Atmospheric muon-neutrino disappearance depends on travel distance and energy, and solar electron neutrinos convert into other active flavors; reactor and accelerator oscillations provide complementary evidence. In propagation, the relative phase of two mass eigenstates is $\Delta m_{ij}^{2}L/(2E)$. If all masses were zero there would be no such relative phase. Thus at least two neutrino mass eigenstates are nonzero in the three-neutrino description with two independent measured splittings. A flavor state is a superposition of mass eigenstates, so the evidence does not assign a single definite mass to a flavor label.

In neutrino oscillation, a neutrino produced with definite flavor is a coherent superposition $|{\nu }_{\alpha }⟩={\sum }_i{U}_{\alpha i}^{\ast }|{\nu }_i⟩$ of mass eigenstates. Propagation gives the components different phases, producing a baseline- and energy-dependent probability to detect another flavor. Oscillation requires nonzero mass-squared differences and nontrivial mixing, so its observation proves that at least two neutrinos are massive and that individual flavor lepton numbers are not conserved. It measures mass differences and mixing parameters, though not the absolute neutrino-mass scale.

All three interactions respect color charge; quark flavor is conserved by the strong interaction and the electromagnetic interaction, but can change in the weak interaction. These statements concern charges and net numbers, rather than an unchanging inventory of particles.

A quark transforms in the fundamental representation of the color charge gauge group $SU(3{)}_C$. The strong interaction can change an individual quark's colour by exchanging a gluon, which itself carries color charge. For example, the colour flow $q_r\to q_b+g_{r\bar{b}}$ is allowed: demanding that the outgoing quark remain red would be incorrect. At each interaction vertex the full colour flow is conserved. The colourless photon, W boson, and Z boson instead couple to a quark with an identity factor ${\delta }_{ab}$ in colour space. A weak charged current can therefore change $d_r$ to $u_r$, but cannot turn $d_r$ into $u_b$ without another coloured participant. External isolated hadrons are colour singlets; confinement prevents direct observation of a free coloured state.

For each quark flavor $f$, define the net quark flavor number $F_f=N_f-N_{\bar{f}}$. The strong interaction and electromagnetic interaction preserve every $F_f$: pair creation $g\to s\bar{s}$, for example, changes neither net strangeness nor any other $F_f$. Their quark couplings are diagonal in quark flavor. The weak charged current contains ${\bar{u}}_i{\gamma }^{\mu }(1-{\gamma }^5)V_{ij}{d}_j$, where $V$ is the Cabibbo-Kobayashi-Maskawa matrix, and changes these net numbers. Weak neutral currents are diagonal in tree-level Feynman diagrams; flavor-changing neutral currents can occur through higher-order Feynman diagrams.

The observed hierarchy of mean lifetimes and decay channels supplies evidence: associated strange-particle production through the strong interaction preserves net strangeness, whereas a kaon's weak interaction decay to pions changes it. The electromagnetic interaction process ${\pi }^0\to \gamma \gamma$ has zero net quark flavor before and after annihilation. Neutrino oscillations show that individual lepton family numbers are not exact conserved quantities. Within the usual effectively massless-neutrino approximation, charged current decays such as ${\mu }^{-}\to e^{-}{\bar{\nu }}_e{\nu }_{\mu }$ preserve family numbers only when the emitted neutrinos are counted.

Evidence for three colours includes the hadronic R ratio $R=3{\sum }_f{Q}_f^2$, the colour multiplicities in W boson decay widths, and three-jet events revealing gluon radiation. These test quantum chromodynamics and its colour structure together; they are not direct measurements of a named red or blue charge on a free quark.