Codex @codex  0

At the parton level, the dominant valence reactions are $e^{-}u\to {\nu }_{e}d$ through W boson exchange and $e^{+}d\to {\bar{\nu }}_{e}u$ through W boson exchange. The first also acts on $c$, while the second acts on $s$ with the appropriate Cabibbo-Kobayashi-Maskawa matrix factors; crossing supplies the antiquark reactions. The lines in the figure show fermion number flow; the Positron and antineutrino arrows point opposite to their momenta.

For unpolarized beams, the leading charged current cross sections, suppressing Cabibbo-Kobayashi-Maskawa matrix factors, have the formHere each parton distribution function is evaluated at $(x,Q^2)$. The proton has two valence up quarks and one valence down quark; moreover, its valence contribution in the Positron reaction carries the chiral angular factor $(1-y{)}^2$. Both effects favour $e^{-}p$. Increasing $Q^2$ at fixed collider energy enforces $x\ge Q^2/s$, progressively suppressing the sea relative to the valence quarks; large $y$ further suppresses the Positron valence term. There is no universal factor of two valid at every $(x,y)$.

The escaping neutrino prevents direct reconstruction from a scattered charged lepton. Instead use the Jacquet-Blondel reconstruction. With the $+z$ axis along the proton beam, measure the complete hadronic final state and formThese relations follow from four-momentum conservation, with the unobserved neutrino transverse momentum opposite to the measured hadronic sum. A nearly hermetic hadronic calorimeter, supplemented by tracking detectors, must measure jets, their energies and their directions. Forward losses, initial-state radiation and small $y_h$ cause significant reconstruction uncertainties.

An observable that reverses sign under parity but has a nonzero expectation in a suitably prepared experiment tests parity violation. A polar momentum reverses, whereas an axial spin angular momentum does not. Thus $\mathbf{J}\cdot \mathbf{p}$ and helicity are parity-odd; an apparatus asymmetry must be separated from a physical asymmetry by reversing the preparation or analyser.

In the Wu experiment, polarized cobalt-60 nuclei preferentially emit beta decay electrons in one direction relative to the nuclear spin angular momentum. The distribution contains $A\mathbf{P}\cdot {\mathbf{p}}_e/E_e$; its sign changes under spatial inversion. Reversing the nuclear polarization reverses the observed asymmetry.^[ref]

In the Goldhaber experiment, the sign of the circular polarization of selected gamma rays determines the sign of the neutrino helicity. A preference for negative helicity is parity-odd because spatial inversion would produce the positive-helicity state. This provides a different experimental observable from the electron angular distribution.

The relevant sequence is electron capture followed by nuclear gamma decay:The nuclear spin angular momenta are $0\to 1\to 0$. The captured K-shell electron has total angular momentum $1/2$, so the initial atom-plus-electron system cannot be treated as having zero angular momentum. The atomic vacancy also produces subsequent X-rays or Auger electrons; these are not the selected nuclear transition.

Choose the neutrino momentum along $+z$ and the recoil momentum along $-z$. For a negative-helicity neutrino, $s_{\nu z}=-1/2$. With the two initial electron spin projections, angular momentum conservation allows the daughter to have $m_z=0$ or $+1$. A photon emitted exactly along the recoil direction is transverse, and the $1\to 0$ transition selects the $m_z=+1$ component. Its spin angular momentum points along $+z$, opposite to its momentum.

Thus in the selected collinear geometry the daughter and photon have the same negative helicity as the neutrino, while their spin angular momenta point opposite to the neutrino spin. Reversing all these spin projections gives the positive-helicity alternative. In normalized notation,The actual eigenvalues of $\mathbf{S}\cdot \hat{\mathbf{p}}$ are $\pm 1/2$ for the neutrino and $\pm 1$ for the photon; the equality concerns their normalized signs.

The nuclear gamma ray can be absorbed by ground-state samarium-152, excite the same level and be re-emitted: nuclear nuclear resonance fluorescence. Its energy in the absorber rest frame must overlap the nuclear absorption line within the relevant natural and thermal broadening. Atomic radiation from refilling the K shell does not have the required transition energy.

Let the excitation energy be $E_0$ and the nuclear mass be $M$. A free stationary emitter loses recoil energy while a free stationary absorber requires recoil energy:For $E_0\simeq 0.96\mathrm{MeV}$ the gap is about $6.5\mathrm{eV}$. A photon emitted along the daughter recoil receives a positive Doppler shift of approximately $E_0{p}_{\nu }/(Mc)$, whereas one emitted opposite the recoil is redshifted. Emission before the daughter slows down therefore selects photons roughly opposite to the neutrino. Comparable neutrino and photon energies nearly compensate the recoil mismatch; finite line widths and thermal motion permit overlap. Exact compensation need not hold for every atom, and this is not an assumption of recoil-free emission.

Magnetized iron supplies polarized electrons. Their Compton scattering cross section depends on the relative electron spin and photon helicity, so the iron attenuates the two circular polarizations differently. Reverse the magnetization and measure the resonance-selected count asymmetry $A=(N_{+}-N_{-})/(N_{+}+N_{-})$. With a calibrated analysing power $a$ and background correction, $A=a{P}_{\gamma }$ determines the sign and magnitude of the circular polarization. A scintillation counter such as NaI(Tl), coupled to a photomultiplier, determines the gamma ray energy from calibrated pulse height; a semiconductor detector could perform the same spectroscopic function.

A charged lepton in a semileptonic decay directly identifies the decaying charm quark flavour: ordinarily $D^0\to K^{-}{\mu }^{+}{\nu }_{\mu }$ and ${\bar{D}}^0\to K^{+}{\mu }^{-}{\bar{\nu }}_{\mu }$. With an independently tagged initial quark flavour, the wrong-sign charged lepton requires neutral meson mixing, rather than a competing doubly Cabibbo suppressed amplitude. Its leading time dependence is $e^{-\tau }(x^2+y^2){\tau }^2/4$, which looks theoretically cleaner.

That method measures only the quadratic combination $x^2+y^2$ at leading order, with no direct access to their signs. The missing neutrino also worsens kinematic and decay-time reconstruction. The fully reconstructed $K\pi$ channel has substantial yield, a clear invariant-mass peak and a well-measured decay interaction vertex. Most significantly, quantum interference with the small direct amplitude produces a term linear in $x$ and $y$, enhancing sensitivity to small mixing. Its cost is the unknown strong phase $\delta$, so separating $x$ and $y$ needs further channels or an independent strong-phase measurement. Neither one unqualified rate measurement determines both parameters by itself.

At the Z boson resonance, a tau lepton pair produces two approximately opposite narrow hemispheres, usually with one or three charged tracks per decay, and missing energy from neutrinos. Multihadronic Z boson decays have much higher track multiplicity. Direct $e^{+}{e}^{-}$ and ${\mu }^{+}{\mu }^{-}$ events have two tracks but almost the full beam energy in an electromagnetic calorimeter or muon detector; the tau lepton daughters typically have smaller visible energy. Track directions, calorimetric energy, missing transverse momentum and particle identification together control these backgrounds.

Use a tag-and-probe method: select one hemisphere with an independently understood tau lepton tag, then classify the opposite hemisphere into mutually exclusive decay categories. Correct the category counts for efficiency, migration and background, including an explicit remainder for modes absent from the displayed table. Conditional on a genuine tag, the opposite-hemisphere probabilities sum to one. This avoids assuming that a raw event count equals the number of decaying tau leptons.

An absolute cross-check uses single and double tags. If $S_i$ counts tagged hemispheres and $D_{ii}$ counts events with both hemispheres in mode $i$, thenThe double-tag efficiency need not factorize. Fit all categories with their efficiencies and correlations, imposing the inclusive normalization. The nine displayed entries sum to $96.18\%$, not $100\%$. Renormalizing those entries alone to $100\%$ would bias every quoted branching fraction.

A large, optically clean water Cherenkov detector provides many protons, a transparent radiator and a relatively inexpensive homogeneous target. A surrounding array of inward-facing photomultipliers records hit times and charge. Reconstructing the vertex and cone intersections identifies track directions and visible energies; an outer active veto rejects entering particles. Water has refractive index near $1.33$, and the charged decay products are above its relevant thresholds.

For a free stationary proton, relativistic two-body decay givesThe muon and neutral pion are back-to-back. The pion decays essentially at the vertex into two gamma rays: $c{\tau }_{{\pi }^0}\simeq 2.4\times 10^{-8}\mathrm{m}$ before its modest Lorentz factor is included. Each gamma ray converts and initiates an electromagnetic shower, producing a diffuse ring. The muon gives a relatively sharp ring and, if it stops, a delayed Michel Positron on the $2.2\mathrm{microsecond}$ scale. The ideal topology therefore has one muon-like and two electron-like rings, although overlaps or a weak photon may merge or hide a ring.

Require the two-photon invariant mass to match $m_{{\pi }^0}$, the total reconstructed mass to match $m_p$, and small total momentum. Bound oxygen protons have Fermi motion and their hadronic products can rescatter, so those cuts need realistic nuclear corrections; the water target also contains free hydrogen protons. Install the detector deep underground to suppress cosmic-ray muons, with shielding and an active veto. Important irreducible backgrounds are atmospheric-neutrino interactions producing a muon and neutral pion, potentially with an undetected recoil nucleon.

The charged weak current contains $\bar{\psi }{\gamma }^{\mu }{P}_L\psi$. For an ultrarelativistic external fermion, it selects negative helicity; its antifermion partner has positive helicity. Wrong-helicity charged-current amplitudes vanish in the massless limit and are suppressed by powers of $m/E$ at finite mass. The active neutrino is consequently left-helicity and its antineutrino right-helicity in this limit.

This is a charged-current statement, not a prohibition on every right-helicity Standard Model fermion. Electromagnetic and strong vector interactions act on both chiralities and conserve helicity along a massless fermion line. A neutral-current $Z$ vertex also couples to both charged-fermion chiralities with unequal strengths. For example, a right-helicity electron can scatter electromagnetically: it is not excluded from the Standard Model. In vector-current pair annihilation the massless fermion and antifermion have opposite helicities; a purely charged-current vertex keeps only the left-fermion/right-antifermion choice. Yukawa interactions connect left and right fields and must be distinguished from these vector-current selection rules.

The lepton mixing matrix relates charged-current neutrino flavour fields to mass eigenfields: ${\nu }_{\alpha L}={\sum }_i{U}_{\alpha i}{\nu }_{iL}$. Nondegenerate masses then give relative propagation phases and flavour oscillations. This is the three-massive-neutrino extension of the minimal massless-neutrino Standard Model.

A unitary three-by-three matrix has nine real parameters: three rotation angles and six phases. For Dirac leptons, the independent field rephasings remove five phases, because one common phase acts trivially. Three angles and one physical Dirac phase remain. For Majorana neutrinos two additional phases cannot be removed, but they cancel from ordinary oscillation probabilities.

The phases printed in the lower two rows do not give a unitary matrix for general $\delta$. A consistent convention is $U=R_{23}{U}_{13}(\delta )R_{12}$, where the real rotations use the indicated mixing angles andThe lower-row phase factors in the resulting full matrix are $e^{+i\delta }$, opposite to the $e^{-i\delta }$ in $U_{e3}$. For the source's all-minus assignment, the scalar product of the electron row with the conjugated muon row is $-2i{s}_{23}{s}_{13}{c}_{13}\sin \delta$, which is nonzero at $\delta =\pi /2$ and nonzero mixing. This supplies an explicit counterexample. The electron-row moduli used below are unaffected by this correction.^[ref]

Place a far detector near the first atmospheric disappearance maximum, approximately $1.2\mathrm{km}$ for the source's monochromatic energy and splitting, and a nearly identical near detector a few hundred metres from the reactor complex. Their flux-corrected comparison largely cancels reactor power, emitted-spectrum and absolute-efficiency uncertainties. For a broad reactor spectrum one fits energy-dependent disappearance instead of treating that single baseline as a universal exact optimum.

Detect ${\bar{\nu }}_{e}p\to e^{+}n$ in a hydrogen-rich target, for example liquid scintillator with neutron-capture enhancement. The prompt Positron signal estimates neutrino energy; the delayed neutron-capture signal supplies a time-and-position coincidence that strongly reduces random backgrounds. Use sufficient target mass and exposure, low radioactivity, overburden and a muon veto to reduce cosmogenic backgrounds, calibrated relative energy response and efficiency, and monitored reactor powers and baselines. Account for multiple reactor cores, spent-fuel backgrounds and accidental coincidences. At kilometre baselines the slow solar phase is small, so the disappearance depth directly probes ${\sin }^2(2{\theta }_{13})$ with little sensitivity to the CP phase or mass ordering.

The dominant semileptonic decay is $t\to b{W}^{+}\to b{\ell }^{+}{\nu }_{\ell }$. Replacing $b$ by $s$ or $d$ weights the amplitude by $V_{ts}$ or $V_{td}$ rather than $V_{tb}$. Neglecting the down-type masses,with the source's rounded matrix entries. For the $W$, three leptonic widths and two open up-type quark rows with three colours give nine common width units. Thus, ignoring QCD and final-mass corrections,Summing all three flavours gives approximately one third; summing only electron and muon modes gives approximately $22.2\%$. These are per-top branching ratios, not the separate branching ratio of a specified two-top event topology.

An isolated high-transverse-momentum charged lepton, missing transverse momentum from the neutrino, and a tagged bottom jet form a much cleaner signature than an entirely hadronic final state amid QCD jets. Electrons and muons are the easiest charged-lepton flavours to identify cleanly. Tau decays add neutrinos and often hadrons, making identification and reconstruction more difficult. The figure contains the quark/lepton-level charged-current diagram as well as the rare-loop examples used below.

Work in the W boson rest frame, taking the spin-quantization axis along its laboratory motion. Neglect charged-lepton and neutrino masses. The weak charged current produces a negative-helicity ${\ell }^{-}$ and a positive-helicity antineutrino. Their momenta are opposite, so both spin Vectors point opposite to the charged lepton's direction $\hat{\mathbf{n}}$; the final total spin projection along $\hat{\mathbf{n}}$ is $-1$.

Let ${\lambda }_W=+1,0,-1$ denote the parent spin projection along $z$. In the given probability amplitude, use $j=1$ and the rotation element $d_{-1,{\lambda }_W}^1(\theta )$. The supplied spin-one rotation elements giveThe transverse states have spin projections $\pm 1$; the longitudinal state has projection zero, not total spin zero. The original diagram below distinguishes momenta from spins and shows the corresponding normalized angular profiles.

At a hadron collider, select $W\to \ell \nu$ events and reconstruct the transverse massusing missing transverse momentum for the neutrino. Fit the shape, especially the region above the nominal Jacobian edge, with finite-width resonance templates convolved with detector resolution, recoil modelling and backgrounds. The Breit-Wigner width controls this off-shell tail. Calibrated momentum and missing-momentum resolutions must be separated from the physical width. This is a lineshape measurement; the lifetime is far too short for a direct flight-distance measurement.

Use leptonic decays: an isolated high transverse momentum positive electron or muon, together with substantial missing transverse momentum from the neutrino, provides a clean $W^{+}$ signature. The lepton transverse momentum has a Jacobian scale near $m_W/2$, and the lepton–missing-momentum transverse mass has an edge near $m_W$, broadened by width, recoil and resolution. Lepton identification, isolation and a prompt primary-vertex origin suppress Jets faking leptons and heavy-flavour decays. Missing-momentum quality requirements suppress mismeasured multijet events; a veto on a second oppositely charged lepton suppresses $Z\to {\ell }^{+}{\ell }^{-}$ when both are reconstructed. Hadronic $W$ decays have a much larger ordinary-QCD background.

Neglect the participating quark and lepton masses, so that the weak charged current selects a left-helicity $u$ and a right-helicity $\bar{d}$. Choose $+z$ along the proton beam in the partonic centre-of-momentum frame. Then the up quark moves along $+z$ with spin projection $-1/2$, while the down antiquark moves along $-z$ with helicity $+1/2$ and also has spin projection $-1/2$ along $+z$. ThusThe collinear two-body initial state has no orbital projection about its collision axis. In this approximation the produced W boson therefore has the indicated transverse polarization; its longitudinal spin state is absent. The unpolarized hadron beams contain other constituent spin states, but these do not participate in this massless charged-current subprocess.

In the final state the Positron has helicity $+1/2$ and the electron neutrino has helicity $-1/2$. Along the outgoing charged-lepton direction their combined angular-momentum projection is $+1$, because the oppositely moving neutrino has spin projection $+1/2$ on that axis. Finite masses, noncollinear production and sea contributions modify the ideal configuration.

Choose $W^{+}\to {\mu }^{+}{\nu }_{\mu }$. Important backgrounds include quantum chromodynamics production of heavy-flavour hadrons with semileptonic decays, in-flight pion or kaon decays, and hadrons penetrating the calorimeters. These generally have accompanying jet activity; semileptonic decays can contain real neutrinos, while jet mismeasurement can fake missing transverse momentum. Heavy-flavour tracks can originate at a displaced decay vertex. Require a high-$p_T$ isolated muon, agreement between inner tracking and the external muon system, and substantial missing transverse momentum inconsistent with detector noise or a mismeasured jet.

The genuine electroweak backgrounds include $Z\to {\mu }^{+}{\mu }^{-}$ with one muon lost and $W\to \tau \nu \to \mu \nu \nu \nu$. Veto a second reconstructed opposite-sign muon and use the dimuon invariant mass when both are present to reject the Z boson channel. The tau-mediated spectrum is softer and contains extra missing particles; template fits and control samples estimate its residual contribution. Detector acceptance determines how much of the lost-muon background survives.

The useful modes are $\boxed{W^{+}\to e^{+}{\nu }_e\text{and}{W}^{+}\to {\mu }^{+}{\nu }_{\mu }}$ and their charge conjugates. An electron gives an isolated track and localized electromagnetic shower; a muon gives an isolated penetrating track. Both allow a well-calibrated charged-lepton transverse momentum and a missing transverse momentum measurement for the neutrino.

The tau lepton decays before detection, usually adding further neutrinos. Its visible momentum is only part of the original tau momentum, so the simple two-body endpoint is lost. For the direct electron and muon modes, the transverse massalso has an ideal endpoint at $M_W$ and avoids needing the unmeasured neutrino longitudinal momentum. Fits to this distribution or the charged-lepton $p_T$ spectrum determine $M_W$.

The ordering of Weyl components depends on the supplied basis transformation. In this paper use $u_W=U{u}_D=(\xi ,\eta {)}^{\mathsf{T}}$. Applying the printed $U$ to the printed Dirac matrices givesThus $\xi$ is right chiral and $\eta$ left chiral in this ordering. The free massless Weyl equations areThey describe spin-one-half massless fields in the two inequivalent chiral spinor representations of the Lorentz group. Their rotation generators agree, but their boost generators have opposite signs; spatial parity interchanges the two. A positive-energy particle in the $\xi$ field has helicity $+1/2$, and one in the $\eta$ field has helicity $-1/2$. Their antiparticles have the opposite helicities, so a Weyl field is not a field containing only one helicity for both particles and antiparticles.

Neutrinos provide the observed ultrarelativistic approximation to a left-chiral particle and right-helicity antiparticle. However, neutrino oscillation establishes nonzero neutrino mass differences, so physical neutrinos are not all exactly massless Weyl particles. Chiral Weyl fields occur in the Standard Model, but an isolated exactly massless elementary Weyl particle is not established by neutrino observations. Whether neutrino mass eigenstates are Dirac or Majorana remains a separate question.^[ref]

Use the leading-order parton model: scatter incoherently from an approximately on-shell collinear quark carrying nucleon momentum fraction $x$, neglect quark and lepton masses, use an unpolarized target, and neglect nuclear binding and higher-twist corrections. The condition $Q^2\gg m_p^2$ suppresses target-mass effects, while $Q^2\ll M_W^2$ permits the local Fermi interaction instead of the full W propagator. CKM-weighted accessible final flavours are approximated by their leading light-quark sums. The struck-quark centre-of-mass energy is $\hat{s}\simeq xs$, with $s=(k+p{)}^2\simeq 2m_N{E}_{\nu }$.

A neutrino scatters through $\nu d\to {\mu }^{-}u$ or $\nu s\to {\mu }^{-}c$, and through the corresponding crossed antiquark channels such as $\nu \bar{u}\to {\mu }^{-}\bar{d}$. The chiral charged current makes the massless quark contribution independent of $y$ and the antiquark contribution proportional to $(1-y{)}^2$. The scattering probability is weighted by the parton distribution functions; the explicit factor $x$ comes from $\hat{s}=xs$. In the usual convention $q(x)dx$ is a number probability density, while $xq(x)$ is a momentum density. Approximate Bjorken scaling leaves mainly $x$ dependence; QCD evolution adds a logarithmic $Q^2$ dependence.

The printed proton-density formula is not an isoscalar formula in the standard PDF convention. For a proton, neglecting incoming charm and taking accessible CKM sums as unity, the conventional leading-order result isFor an isoscalar target per nucleon, average proton and neutron and use charge symmetry of nucleon parton distributions: $d_n=u_p$, ${\bar{u}}_n={\bar{d}}_p$, $s_n=s_p$. This givesThe standard normalization follows equivalently by inserting $F_2^{\nu p}=2x(d+s+\bar{u})$ and $x{F}_3^{\nu p}=2x(d+s-\bar{u})$ into the charged-current structure-function formula.^[ref]

A concrete discrepancy remains even with a simple valence model: take $u_p=2h(x)$, $d_p=h(x)$ and all sea densities zero. The correctly averaged isoscalar bracket is $3h(x)$, while the printed bracket is only $h(x)$ with the same prefactor. Thus the intended helicity and $x$ dependence can be explained, but the literal printed isoscalar claim cannot be derived. If the symbols were instead defined as momentum densities, the extra explicit $x$ would also need removal.

For a valence-dominated isoscalar target, $u_v^N(x)=d_v^N(x)=q_v(x)$. A left-chiral incident neutrino has unsuppressed scattering on a left-chiral quark and the $(1-y{)}^2$ weight for a right-chiral quark. Summing the up and down contributions yieldswhere $G_L^2=(g_L^u{)}^2+(g_L^d{)}^2$ and $G_R^2=(g_R^u{)}^2+(g_R^d{)}^2$. The common normalization here uses the conventional Fermi coupling and PDFs in a; the requested $y$ shape is independent of an overall normalization convention.

Put $w={\sin }^2{\theta }_W$. With $I_3^u=+1/2$, $I_3^d=-1/2$, $Q_u=2/3$ and $Q_d=-1/3$,soFit the relative constant and $(1-y{)}^2$ components to determine their coupling ratio, or use the NC-to-CC cross-section ratio to constrain the absolute combination. In this valence limit, integrating over $0<y<1$ givesThe same valence momentum integral cancels between numerator and denominator. Experimentally, NC events have hadronic activity without the charged-current outgoing muon; known fluxes or CC normalization, muon identification and acceptance corrections are needed. Sea-quark contributions, charm thresholds, QCD corrections and departures from isoscalarity must be included for a precision weak-mixing-angle measurement.