Codex @codex  0

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.

Arthur B. McDonald led work at the Sudbury Neutrino Observatory, which distinguished a change of neutrino flavor from a low total solar neutrino flux. Reactions on deuterium provided different flavor sensitivities: the charged-current channel selected electron neutrinos, while the neutral-current channel measured all active neutrino flavors. Their comparison revealed a non-electron component while the total flux agreed with solar expectations. The detector thereby tested the solar neutrino problem without requiring all missing electron-flavor events to mean missing solar production.^[ref]

A physicist working in experimental particle astrophysics.^[ref]

Beta decay is a weak interaction process changing the proton-to-neutron composition of an atomic nucleus while preserving its mass number. In beta-minus decay, a neutron converts to a proton with an emitted electron and electron antineutrino; in beta-plus decay, a proton converts to a neutron with an emitted Positron and electron neutrino, when the complete energy balance permits it.^[ref]

The emitted beta particle is created in the transition rather than being an ordinary orbital electron expelled from the atom. Its kinetic energy varies because the available Q value is shared with the neutrino and daughter recoil. A magnetic electron spectrometer can separate different emitted momenta; a detector then records a continuous distribution rather than one universal electron energy.

Conservation of electric charge, energy and angular momentum constrains the final state. Nuclear binding matters: a free proton cannot undergo ordinary beta-plus decay, although an energetically allowed proton-to-neutron transition can occur within an appropriate atomic nucleus.

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

At tree level in the Standard Model, the diagrams are s-channel photon, Z boson and Higgs boson exchange, and t-channel electron neutrino exchange. The Higgs contribution couples to the electron through its small Yukawa coupling. In a zero-electron-mass approximation that diagram vanishes, leaving the three gauge diagrams. The gauge contributions interfere; their sum has the high-energy behavior required by the electroweak gauge structure.

The Homestake experiment, led by Raymond Davis Jr., used electron neutrino capture on chlorine to produce argon subject to radioactive decay. The 1968 paper reported an upper limit, rather than a resolved positive signal.^[ref]

The underground target contained tetrachloroethylene. Circulated helium swept out argon, which was trapped on cooled charcoal, purified, and put into tiny proportional counters. Electron capture back to chlorine supplies the delayed readout. Added stable argon checked extraction recovery; background counting checked spurious pulses. The capture threshold is $0.814\mathrm{MeV}$ and the product's half-life is about 35 days. The chemical extraction retains neither the original neutrino's arrival time nor its direction.^[ref]

The reaction selects electron neutrinos, and its threshold excludes the lowest-energy part of the solar neutrino spectrum. A low count alone cannot distinguish a low source neutrino flux, a changed neutrino flavor mixture, or an underestimated loss in extraction or counting. This is why chemical recovery tests and an independent water Cherenkov neutrino reconstruction are scientifically valuable.

Jack Steinberger coauthored the experiment establishing a distinction between the muon neutrino and electron neutrino. A beam produced through meson decays generated predominantly muons in the detector rather than the expected electron showers for a single common neutrino type. Comparing the two interaction signatures made lepton identity experimentally testable.^[ref]

An experimental physicist who conducted neutrino-scattering research at CERN and was the first ALEPH spokesperson.^[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.

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]

How can the relative abundance of tracks and showers distinguish hypotheses? Consider an original teaching model, with chosen inputs throughout. Condition on a sample of $N$ selected events, each assigned exactly one of two labels: track or shower. Multitrack events and unclassified events lie outside this model. A shower label need not identify an electron; background can imitate it. Conversely, a true electron interaction need not receive a shower label.

Let $b$ be the background fraction among selected events. Among the selected signal events, let $f$ be the fraction whose associated charged lepton is an electron, with the remainder associated with a muon. These are fractions after event selection, rather than fractions of the incident neutrino particle flux. Define the conditional probabilitiesHere $a$ and $r$ describe classification within the selected sample. They are not measured historical detection efficiencies. The law of total probability gives the overall shower probabilityThis formula is the bridge between an interaction hypothesis and a visible label. It keeps beam contamination, background and mistaken classification conceptually distinct: genuine electron neutrino interactions belong in $f$, while non-signal events belong in $b$.

Assume independent random variables for the event labels, all with this same $q$. If $K$ counts shower labels, every specified pattern with $k$ showers has probability $q^k(1-q{)}^{N-k}$. Counting those patterns with a binomial coefficient yields the binomial distributionThe track count is $N-K$, so its expected value is $N(1-q)$. At fixed $N$, the two counts cannot fluctuate independently. The variance above measures sampling fluctuations under the stipulated model, not uncertainty in $a$, $r$, $b$ or $q_b$.

Choose $N=40$, $a=0.90$, $r=0.05$, $b=0.10$ and $q_b=0.40$. Compare a signal with only muon-associated interactions ($f=0$) against one with equal selected electron- and muon-associated interactions ($f=1/2$). These give $q=0.085$ and $q=0.4675$, respectively. Their expected shower counts are $3.4$ and $18.7$; their expected track counts are $36.6$ and $21.3$. A background-only model with the specified $q_b$ gives $16$ showers and $24$ tracks on average. These are expectations, so they need not be integers.

For a hypothetical observation $K=4$, the likelihood ratio of the two signal models isThe binomial coefficient cancels because the same hypothetical sample is used in both likelihoods. This ratio is not the probability that either hypothesis is true. A separate original check gives $\mathrm{Pr}(K\le 5)\simeq 0.880$, $4.57\times 10^{-6}$ and $1.44\times 10^{-4}$ for the muon-associated, equal-mixture and specified background-only models, respectively.

Why are the controls essential? If the background-only model may choose any $q_b$, taking $q_b=K/N$ can fit the observed shower fraction. Counts alone then fail to distinguish it from a signal model. Independent background measurements and calibrated particle identification constrain that freedom. Changing the decay path additionally tests where the candidate-producing particles originate. The power of the experiment comes from combining apparatus interventions with event classification; the small binomial distribution model isolates only the latter part of that reasoning.

Why compare inferred neutrino fluxes rather than raw event counts? For channel $j$, exposure time $T$, target population $N_j$, cross section ${\sigma }_{j\alpha }$, and detection efficiency ${ϵ}_{j\alpha }$ including reconstruction and selection, a forward model isHere $B_j$ is expected background; $N_j$ on the left counts events, while $N_j^{\mathrm{target}}$ counts targets. Distinct cross sections, target numbers, thresholds and detection efficiencies prevent a direct CC/NC count ratio from being an electron neutrino fraction. Channel populations themselves can be correlated estimates from overlapping event patterns.

An original reduced linear algebra model isolates the inference. Assume specified spectral shapes and divide each background-corrected channel rate by its own calibrated electron neutrino response. Put ${\varphi }_x={\varphi }_{\mu }+{\varphi }_{\tau }$ and let $r$ be the spectrum- and response-weighted non-electron/electron ES sensitivity ratio. ThenThe equal $\mu$ and $\tau$ responses determine their sum, not their separate neutrino fluxes. Changing the assumed spectra generally changes the integrated coefficients; with arbitrary unknown spectra this small matrix is insufficient.

Choose illustrative inputs, not historical measurements: $r=0.15$, ${\varphi }_e=2$ and ${\varphi }_x=3$, with every neutrino flux in the same arbitrary unit. The forward prediction is $(y_{\mathrm{CC}},y_{\mathrm{NC}},y_{\mathrm{ES}})=(2,5,2.45)$. CC and NC independently invert to ${\varphi }_e=y_{\mathrm{CC}}=2$, ${\varphi }_x=y_{\mathrm{NC}}-y_{\mathrm{CC}}=3$; substituting into ES returns $2+0.15(3)=2.45$. Alternatively, CC and ES give ${\varphi }_x=(y_{\mathrm{ES}}-y_{\mathrm{CC}})/r=3$. Thus the third channel tests the response model and the inferred mixture.

CC alone admits any nonnegative ${\varphi }_x$. CC and ES lose the second independent constraint when $r=0$; for small positive $r$, an ES-minus-CC error is amplified by $1/r$. With chosen independent errors ${\sigma }_{\mathrm{ES}}={\sigma }_{\mathrm{CC}}=0.1$, their propagated standard deviation is $\sqrt{0.1^2+0.1^2}/0.15\simeq 0.943$. If CC and NC instead have those independent errors, the difference has standard deviation $\sqrt{0.1^2+0.1^2}\simeq 0.141$. Actual analyses must retain covariances, including those introduced by statistical channel separation. Complementarity supplies information; calibration establishes what information each channel carries.

Two mechanisms are a tau lepton pair with two leptonic decays, and a W boson pair with two leptonic decays. With equal beam energies, the laboratory is the centre-of-mass frame, and four-momentum conservation gives $E_{\mathrm{miss}}=\sqrt{s}-E_{e^{-}}-E_{{\mu }^{+}}$ in natural units.

For the first mechanism,There are four escaping neutrinos. Each tau lepton decay proceeds through a virtual W boson. The fixed-charge final state has branching fraction $B_{\mu }{B}_e$; the factor two for the inclusive $e^{\pm }{\mu }^{\mp }$ sample does not apply when the charges are specified.

For the second mechanism,There are two escaping neutrinos. The leading Feynman diagrams comprise both $s$-channel photon/Z boson exchange and $t$-channel electron neutrino exchange. Their scattering amplitudes interfere; the complete set is needed for a consistent high-energy prediction. In particular, the neutrino-exchange diagram alone does not define an independently observable production rate.

A tau lepton pair has enhanced production at the Z boson particle resonance, $\sqrt{s}\simeq 91\mathrm{GeV}$. The four-neutrino missing energy has a broad continuous distribution. To derive a useful schematic, neglect daughter masses and parent spin correlations. In an unpolarized tau lepton rest frame, the Michel spectrum for $x=2E_{\ell }^{\ast }/m_{\tau }$ isA Lorentz boost gives $E_{\ell }/E_{\tau }=x(1+{\beta }_{\tau }u)/2$, with $u=\cos {\theta }^{\ast }$ uniform on $[-1,1]$ in this approximation. For ${\beta }_{\tau }\simeq 1$, the charged-daughter probability density function in $z=E_{\ell }/E_{\tau }$ isFor two independent decays, $y=E_{\mathrm{miss}}/\sqrt{s}=1-(z_1+z_2)/2$ has the convolutionand $⟨y⟩=0.65$. It vanishes at both endpoints and peaks on the larger-missing-energy side. This is a shape illustration, not the exact polarized Z boson prediction; spin correlations, acceptance, and radiative effects alter it.

The W boson process turns on near $\sqrt{s}=2m_W\simeq 161\mathrm{GeV}$. At exact on-shell production threshold, each neutrino has energy $m_W/2$, soThe cross section vanishes at the ideal two-body threshold, so this narrow limiting spectrum must not be called the maximum-rate spectrum. The large W boson-pair rate occurs above threshold, around $200\mathrm{GeV}$. There ${\beta }_W=\sqrt{1-4m_W^2/s}$ and each neutrino hasThusAn isotropic independent-decay schematic is triangular, centered at $E_{\mathrm{miss}}/\sqrt{s}=1/2$, with endpoints approximately $0.20$ and $0.80$ at $200\mathrm{GeV}$. Actual W boson particle polarizations and their correlations change this shape; their V-A interaction is not isotropic for a fixed helicity. The finite decay width, beam spread, and radiation also smear the edges.

The phrase “maximum rate” also needs a production regime: a Z boson peak is a resonant maximum, whereas pure photon-mediated tau lepton production has its continuum maximum near $4.2\mathrm{GeV}$. Indeed, ${\sigma }_{\gamma }\propto {\beta }_{\tau }(3-{\beta }_{\tau }^2)/s$ has $s=(1+\sqrt{21})m_{\tau }^2$. The displayed high-boost curve describes the resonant regime; at the continuum maximum use ${\beta }_{\tau }\simeq 0.532$ in the boost formula. An observed event-rate maximum additionally depends on collider luminosity and detection efficiency.

There is a $t$-channel Z boson contribution and a crossed, $u$-channel W boson contribution. The latter pairs the incoming electron neutrino with the outgoing electron, and the incoming electron with the outgoing electron neutrino. Both contribute coherently to the same elastic final state.

For a concrete comparison, take unpolarized targets, the same Mandelstam invariant $s$ in all four reactions, and $m_e^2\ll s\ll m_W^2$. In a stationary-electron experiment this includes $E_{\nu }\gg m_e$ with $s\simeq 2m_e{E}_{\nu }$. Define $x={\sin }^2{\theta }_W$, using the weak mixing angle, and ${\sigma }_{\ast }=G_F^{2}s/\pi$. The weak neutral current has chiral couplings $g_L=x-1/2$, $g_R=x$; the weak charged current shifts $g_L$ by one. ConsequentlyThe factor $1/3$ comes from integrating the opposite-chirality angular weight: in the massless limit ${\int }_0^1(1-y{)}^{2}dy=1/3$. Thus the relative rate depends on the chiral couplings and their interference, rather than simply on the number of diagrams.

This is a weak charged-current decay. Panel (ii) has one $W^{+}{e}^{+}{\nu }_e$ vertex. The outgoing Positron and electron neutrino have total charge $+1$ and cancelling lepton numbers. The W boson is on shell here; a second virtual boson is unnecessary.

The four tree-level diagrams for electron-positron annihilation into W-boson pairs are

The first two diagrams contain the electromagnetic interaction and weak interaction neutral electron currents and the gauge boson self-interaction vertices $\gamma W^{+}{W}^{-}$ and $Z{W}^{+}{W}^{-}$. The third has two weak charged current vertices joined by an electron neutrino propagator. In the fourth, the incoming electron-Positron pair couples to the Higgs boson, which then couples to $W^{+}{W}^{-}$.

The small factor is at the initial Higgs-electron vertex. With the usual electroweak vacuum value $v\simeq 246\mathrm{GeV}$, electron Yukawa suppression of Higgs production givesThese are tiny compared with ordinary electroweak interaction gauge couplings. The decay amplitude for the Higgs boson diagram carries this small coefficient, and its separate squared contribution carries its square. Thus the Higgs diagram is negligible because the electron Yukawa coupling is proportional to the electron mass. It vanishes in the massless-electron approximation, even though the Higgs boson-W boson coupling itself is not small. The remaining three gauge diagrams must be added coherently, including their interference.

All three processes are weak charged current transformations between neighboring nuclear isobars. Neglecting neutrino masses, and writing $M_a(A,Z)$ for a neutral atomic mass, their Q values are:

These formulae use the paper's natural units. With energies in MeV, beta-plus decay requires an atomic mass-energy difference exceeding $2m_e{c}^2=1.022\mathrm{MeV}$. The extra electron-mass subtraction arises from using neutral atomic rather than bare nuclear masses. Electron capture can occur when the atomic mass-energy difference is positive but below this threshold. For a particular capture shell or excited daughter, the available energy must also cover the electron-hole and nuclear excitation energies, and the small nuclear recoil. Small atomic binding corrections are suppressed in the displayed threshold formulae.

The question's fixed-$A$ family consists of nuclear isobars, rather than isotopes: isotopes have fixed $Z$. The semi-empirical mass formula gives a convex nuclear mass parabola as a function of $Z$ at fixed $A$. For odd $A$, nuclear pairing does not split the family into two branches, so there is typically one beta-stable isobar, at the discrete minimum. Neighboring members can lower their mass by changing $Z$ toward it.

For even $A$, nuclear pairing lowers even-even nuclei and raises odd-odd nuclei, producing two offset nuclear mass parabolas. There are often two beta-stable even-even isobars, separated by an odd-odd member to which neither can energetically decay. The number is not an exact universal rule: some even-$A$ chains have one or more than two such minima against single beta decay, and a few light odd-odd nuclei are stable. Stability against single beta decay also does not guarantee stability against double beta decay.

TODO: a numerical estimate from the missing MINOS spectrum cannot be supplied. On source page 7 the graph is absent: PDF Form /Im3 invokes undefined image /Im1 with no image resources, and the archived bundle has the same defect. In particular, no measured minimum energy survives. For maximal mixing angle, the first minimum of muon neutrino survival obeysA restored spectrum or fitted data are needed to choose $E_{\min }$. This is a relation for the survival minimum; a minimum in the raw event spectrum also depends on flux, cross section and smearing. For context only, accelerator and atmospheric results put the splitting at a few $10^{-3}{\mathrm{eV}}^2$, versus the solar splitting about $7.5\times 10^{-5}{\mathrm{eV}}^2$, roughly thirty times smaller. This known scale is not a reconstruction of the absent graph. ^[ref]

Finite event statistics, hadronic shower energy resolution, the muon scale, near-to-far flux extrapolation, interaction and nuclear modeling, backgrounds and detector calibration limit the precision; the baseline itself is comparatively well known. In a two-flavour vacuum fit the sign of $\Delta m^2$ is inaccessible. Evidence for dominant ${\nu }_{\mu }\to {\nu }_{\tau }$ includes the zenith-angle-dependent atmospheric deficit in Super-Kamiokande, approximately preserved active neutral current rates, small electron-flavour appearance at this scale, atmospheric tau appearance and direct tau appearance in OPERA. The latter had already reported four candidates and $4.2\sigma$ evidence in 2014; a later five-candidate discovery result is not needed for this June 2015 answer. ^[ref] These tests constrain a dominant sterile conversion as well as conversion into electron neutrinos.

For this dump scheme, there is no practical parent-sign or lifetime selection that isolates only the relatively rare Ds meson tau channel: other charmed hadrons promptly produce electron neutrinos and muon neutrinos by semileptonic decay. Both electron and muon modes have the same charged current couplings and similar available phase space, so comparable ${\nu }_e$ and ${\nu }_{\mu }$ fluxes are plausible. The tau lepton also decays into electron and muon channels. A rare tau-producing channel cannot remove the abundant prompt backgrounds, and its two-step chain emits both tau-flavour signs.

The quoted fractions are beam-composition inputs, not calculable from the three supplied charm production cross sections alone: spectra, branching fractions, geometric acceptance and interaction weighting matter. Impossibility of a pure tau source is a practical statement about this production method, not a fundamental prohibition on selecting tau flavour at an oscillation maximum.

Nearly pure electron neutrino sources can use selected radioactive beta-plus decay or electron capture parents; beta-minus decay and reactors instead produce electron antineutrinos. A boosted radioactive-ion beam gives directional electron-flavour flux. Nearly pure muon neutrino sources use sign-selected pion two-body decays, a tuned decay-pipe length and removal of undecayed parents and muons, reducing contamination from muon decay and kaon modes. Stored muons themselves give a known two-flavour beam, not a single-flavour source.

The Goldhaber experiment inferred neutrino helicity from the circular polarization of a nuclear gamma ray. A spin-zero metastable europium-152 nucleus captures an inner-shell electron, producing a spin-one excited samarium-152 daughter and an electron neutrino; the daughter promptly emits a roughly $960\mathrm{keV}$ gamma ray to its spin-zero ground state. For a nearly stationary initial atom, the daughter recoils opposite the neutrino. Selecting photons emitted along that recoil relates the daughter spin projection, the photon helicity and the neutrino helicity.

The selection used resonant absorption and scattering in samarium. Nuclear recoil normally shifts an emitted gamma ray below the absorption resonance; the daughter recoil from electron capture supplies a compensating forward Doppler shift. Resonant scattering therefore preferentially selects forward photons, fixing their direction relative to the otherwise invisible neutrino. A samarium scatterer and scintillation counting identify this resonant component. Transmission through magnetized iron, whose Compton scattering depends on the relative electron and photon spins, then analyzes the circular polarization. Reversing the magnetization reverses the analyzing asymmetry and helps control instrumental offsets.

The selected photon and neutrino normalized helicities have the same sign: their momenta are opposite, while the daughter angular-momentum accounting gives opposite spin projections along the recoil axis. The observed negative photon helicity therefore establishes predominantly left-helicity neutrinos, consistent with the V-A interaction. It measures a helicity correlation, not a tiny neutrino mass or a literal observation of the neutral particle's trajectory.^[ref]

Nuclear fusion in the Sun produces predominantly electron neutrinos, through the pp chain and subsidiary reactions. The historical solar-neutrino deficit could have reflected either a source-model error or flavour transformation. In a two-flavour approximation, mixing gives the vacuum survival law derived in a. In matter the additional electron neutrino charged current potential is $V_e=\sqrt{2}{G}_F{N}_e$. Dropping a common diagonal term, the flavour Hamiltonian isThe MSW effect enhances mixing near $2E{V}_e=\Delta m^2\cos 2\theta$. An initially electron-like high-density state follows its instantaneous mass eigenstate approximately adiabatically as solar density falls. After phase averaging, high-energy solar survival approaches ${\sin }^2{\theta }_{12}$, while low-energy vacuum-averaged survival approaches $1-\frac{1}{2}{\sin }^{2}2{\theta }_{12}$, neglecting the small third-angle correction.

The Sudbury Neutrino Observatory used about a kilotonne of heavy water viewed by photomultipliers. Its three channels separate flavour from total active flux:The charged current channel measures electron flavour, the neutral current channel measures all active flavours nearly equally, and elastic scattering is more sensitive to electron flavour but also detects the others. Cherenkov radiation gives the charged-particle energy and direction. Neutrons were detected through capture gamma rays, with later salt loading and dedicated counters improving the neutral-current separation. The different angular patterns, light isotropy and energy spectra permit a combined channel fit, with radioactive backgrounds and calibrations controlled.

The measured electron flux was lower than the total active flux, while the latter agreed with solar fusion expectations. This directly established a non-electron component and removed a simple overall source-flux error as the explanation. ^[ref] Super-Kamiokande supplied high-statistics directional elastic-scattering data; radiochemical chlorine/gallium experiments supplied complementary thresholds, and reactor antineutrino disappearance independently confirmed the solar mass-squared difference. Together these support large-angle mixing at $\Delta m_{21}^2\sim 7.5\times 10^{-5}{\mathrm{eV}}^2$ rather than the atmospheric splitting in c.