Past exam of the particle physics course of the University of Oxford / 2015 (A11047W1, C4)

Source paper: www.whatdotheyknow.com/request/oxford_physics_past_papers_2/response/2340696/attach/3/Physics%20Papers%202014.pdf?cookie_passthrough=1

Solutions use natural units unless a conversion is displayed. The surviving paper has damaged source illustrations in c and c; the corresponding numerical readings remain explicit TODOs. All eight questions, including all five alternatives in 8, are treated.

Treat the electron and electron antineutrino as effectively massless particles in the centre-of-momentum frame, retain only the charged current, and use the Fermi interaction: $m_e^2\ll s\ll M_W^2$. Take ${\theta }^{\ast }$ between incoming and outgoing electron antineutrino directions. The V-A interaction selects a right-helicity electron antineutrino and a left-helicity electron. Their opposite incoming directions give total angular momentum projection one. The outgoing pair has the same projection along its own axis. The Wigner D-matrix element is $d_{11}^1({\theta }^{\ast })=(1+\cos {\theta }^{\ast })/2$, so the squared helicity amplitude is proportional to $(1+\cos {\theta }^{\ast }{)}^2$. The inactive initial electron helicity is included in the unpolarized normalization supplied by the paper.

The boost from the centre-of-momentum frame to the laboratory frame has $\beta =E/(E+m_e)$, while the incoming electron antineutrino has $E_{\ast }=(s-m_e^2)/(2\sqrt{s})=m_{e}E/\sqrt{s}$ and $s=m_e^2+2m_{e}E$. ThusThe requested identity therefore uses $E\gg m_e$; it is not exact at finite target mass. With $c_{\ast }=\cos {\theta }^{\ast }$, the absolute Jacobian determinant is $|dy/d{c}_{\ast }|=1/2$, givingThe suppression at backward scattering follows from angular momentum conservation: rotating projection $+1$ through $\pi$ cannot leave projection $+1$ along the original axis. Replacing $1+c_{\ast }$ by $2(1-y)$ gives the supplied $(1-y{)}^2$ dependence.

Use the parton model, neglect antiquarks, binding and target mass corrections, quark masses, muon mass, QCD scaling violation and the W boson propagator correction. The struck quark is effectively free during the short interaction and carries parton momentum fraction $x$, giving $\hat{s}\simeq xs$. The charged current reaction is ${\bar{\nu }}_{\mu }{u}_i\to {\mu }^{+}{d}_j$. Its helicity amplitude has the same $(1-y)$ factor as in a. Define the effective parton distribution function by $q(x)={\sum }_{i,j}|V_{ij}{|}^2{u}_i(x)$, summing only kinematically accessible final quarks; for valence-only protons and accessible $d,s$ this is approximately $u(x)$. It is not the unweighted sum $u+d$.Here $y=p\cdot (k-k^{\prime })/(p\cdot k)$ is the inclusive deep inelastic scattering variable, and $x=Q^2/(2p\cdot (k-k^{\prime }))$ in the scaling limit. Integrating $0<y<1$ gives $1/3$, henceThe moment measures the momentum fraction in the active quarks. Approximate linearity is evidence for scattering from pointlike spin-one-half constituents with approximately scale-independent parton distribution functions, rather than coherent scattering from an extended proton. It is an intermediate-energy prediction: thresholds, experimental deep inelastic scattering cuts, QCD scaling violation and eventually the finite W boson propagator modify it.

Strike a target with high-energy protons, collect negative pions and kaons with magnetic horns, and let them decay in a long evacuated decay tunnel: ${\pi }^{-}\to {\mu }^{-}{\bar{\nu }}_{\mu }$ and $K^{-}\to {\mu }^{-}{\bar{\nu }}_{\mu }$. Absorb residual hadrons and range out muons in shielding. Focusing selects the negative parent sign and therefore enriches muon antineutrinos; other parent decays leave a finite contamination. Monitor the parent flux and use a near detector to constrain the unoscillated spectrum.

A massive segmented calorimeter supplies interaction probability and measures the hadronic shower. Alternate dense absorber plates with scintillation counters or tracking planes, and magnetize the iron or add a downstream magnetic muon detector. The signal is a positive muon plus a hadronic shower; the sign rejects much of the muon neutrino contamination. For a contained muon, its range gives energy; for a penetrating track, curvature gives $p_{\perp }[\mathrm{GeV}]=0.2998B[\mathrm{T}]R[\mathrm{m}]$. Multiple measurements and a field map determine the fitted momentum and sign. Summed calibrated shower response estimates the struck quark's visible hadronization energy, not the energy of a free observable quark. Correct for invisible nuclear losses, noncompensating response, leakage and acceptance. Neglecting target recoil corrections, $E_{\nu }\simeq E_{\mu }+E_{\mathrm{had}}$ and $y\simeq E_{\mathrm{had}}/E_{\nu }$.

At energies large compared with fermion masses, the gauge interactions preserve chirality on each fermion line. A photon couples to charged fermions with equal left and right coefficients $e{Q}_f$. A gluon couples to either quark chirality with coefficient $g_s{T}^a$. A Z boson couples to both, with coefficients $(g/\cos {\theta }_W)(T_3-Q_f{\sin }^2{\theta }_W)$ on left-chiral fields and $(g/\cos {\theta }_W)(-Q_f{\sin }^2{\theta }_W)$ on right-chiral fields. A W boson couples only to left-chiral fermions and the associated right-helicity antifermions: $W^{-}\to {\ell }_L^{-}{\bar{\nu }}_R$ or $d_L{\bar{u}}_R$, and the charge-conjugate $W^{+}$ channels. For quarks the coefficient includes the CKM matrix. There is no Standard Model right-chiral neutrino gauge vertex.

For outgoing pairs at a neutral vertex, the allowed helicities are $f_L^{-}{\bar{f}}_R^{+}$ and $f_R^{-}{\bar{f}}_L^{+}$; the latter labels helicity, not the chirality of the corresponding antifield. Only the $s$-channel photon and Z boson diagrams contribute to $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$ at tree level among these gauge exchanges. A W boson changes charged leptons into neutrinos, and gluons do not couple to leptons.

Assume unpolarized beams, $s\gg m_{\mu }^2$, a single exchanged photon, full azimuthal coverage and no radiative corrections. Take $c_{\ast }=\cos {\theta }^{\ast }$ between incoming $e^{-}$ and outgoing ${\mu }^{-}$. The four nonzero helicity amplitudes have magnitudes $e^2(1+c_{\ast })$ for equal incoming/outgoing negative-particle helicities, and $e^2(1-c_{\ast })$ for opposite ones. Averaging four initial spin states and summing the final ones givesThe massless two-body phase space givesThe even distribution reflects the equal left/right photon couplings and is symmetric under interchange of forward and backward directions.

Include the Z boson amplitude and its quantum interference with the photon amplitude. Define $s_W=\sin {\theta }_W$, $c_W=\cos {\theta }_W$, $e=g{s}_W$, $g_L^f=T_3^f-Q_f{s}_W^2$, $g_R^f=-Q_f{s}_W^2$, andFor the same unpolarized and massless approximations as b, the complete tree-level angular distribution isThe absolute squares include the pure terms and their interference. Put $A=|C_{LL}{|}^2+|C_{RR}{|}^2$ and $B=|C_{LR}{|}^2+|C_{RL}{|}^2$. The even part is $(A+B)(1+c_{\ast }^2)$ and the odd part is $2(A-B)c_{\ast }$. Thus the forward-backward asymmetry is $A_{\mathrm{FB}}=3(A-B)/(4(A+B))$.

At the Z boson pole, neglect the small photon contribution and use real tree-level couplings. Thenwhere $g_V=g_L+g_R$, $g_A=g_L-g_R$. For charged leptons, $g_V/g_A=1-4{\sin }^2{\theta }_W$. Fitting the forward/backward counts, after detector and radiation corrections, therefore measures the weak mixing angle, conventionally an effective leptonic angle once electroweak corrections are included. For illustration ${\sin }^2{\theta }_W=0.231$ gives ${\mathcal{A}}_{\ell }\simeq 0.151$ and $A_{\mathrm{FB}}^0\simeq 0.0171$. This illustrative parameter choice is not extracted from this paper's data. The real tree-level photon-Z boson interference vanishes exactly at the pole but matters away from it.

Direct formation in $e^{+}{e}^{-}$ through one virtual photon selects the vector meson quantum numbers $J^{PC}=1^{--}$. The nonzero $K_S^0{K}_L^0$ mode and absence of $K_S^0{K}_S^0$ are consistent with this assignment: the two spin-zero kaons require odd orbital angular momentum for negative parity, whereas identical $K_S^0$ bosons cannot have an odd relative wave. This argument neglects the tiny CP violation in neutral kaons.

Assuming the $\rho \pi$ decays conserve isospin, their equal rates indicate $I=0$. The Clebsch-Gordan coefficients giveAn $I=1,I_3=0$ state lacks ${\rho }^0{\pi }^0$, while $I=2,I_3=0$ has squared coefficients $1:4:1$ rather than $1:1:1$. The branching fractions alone determine neither $J$ nor these relative phases; the single-photon formation and approximate strong interaction isospin conservation are essential additional inputs.

For the resonant contribution only, subtract nonresonant Bhabha scattering and its interference before interpreting a fitted area. The integrated narrow vector resonance area follows from the Breit-Wigner resonance cross section, with $J=1$, $s_{e^{-}}=s_{e^{+}}=1/2$, and $p\simeq M/2$, becomesAcross a narrow scattering resonance, treat the prefactor and partial decay widths as constant and extend the integral over $E-M$ to the real line. Since ${\int }_{-\infty }^{\infty }du/(u^2+a^2)=\pi /a$,A normalized beam-energy response convolves the line shape without changing this full area. It can completely change the observed peak height and apparent width. If a uniform acceptance $ϵ$ is not already unfolded, use ${\mathcal{I}}_{ee}={\mathcal{I}}_{ee}^{\mathrm{obs}}/ϵ$. The distinction between $B_{ee}^2$ for this exclusive final state and $B_{ee}$ for an inclusive final state is essential.

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The recoverable numerical conversion is nevertheless definite. With $M\simeq 3.1\mathrm{GeV}$, $B_{ee}=0.060$, $ϵ=0.50$, and $1{\mathrm{GeV}}^{-2}=389400\mathrm{nb}$, b givesThis coefficient assumes the plotted exclusive electronic sample has not already been corrected for acceptance. A restored area, with continuum subtraction and acceptance convention checked, would complete the estimate. A tabulated modern J/psi meson width would be external information rather than a reading of these data.

The narrow scattering resonance indicates suppressed strong interaction decay. In the quark model, $c\bar{c}$ is below the open-charm quark $D\bar{D}$ threshold, so its constituents cannot simply emerge in separate charmed mesons. The OZI rule suppresses annihilation into light hadrons; for a $1^{--}$ color-singlet state the leading purely gluonic channel needs three gluons. The observed leptonic channels remain competitive. Its mass and $I=0$ fit charmonium; the small width alone would not uniquely identify the new quark flavour without the wider spectrum and threshold evidence.

For a neutral $c\bar{c}$ bound state, meson parity and charge conjugation give $P=(-1{)}^{L+1}$ and $C=(-1{)}^{L+S}$. In spectroscopic notation $n{}^{2S+1}{L}_J$, the integer $n$ labels radial excitation, $L$ is orbital angular momentum, $S$ the total constituent spin, and $J$ their total angular momentum. The surviving level diagram contains eight states:

The spin-dependent splittings distinguish the singlet and triplet levels and the three triplet P levels. The radial excitation changes neither parity nor charge conjugation.

The four allowed two-real-photon states are ${\eta }_c$, ${\eta }_c(2S)$, ${\chi }_{c0}$ and ${\chi }_{c2}$. Two photons have $C=+1$, excluding $J/\psi$, $\psi (2S)$ and $h_c$; the Landau-Yang theorem excludes ${\chi }_{c1}$ because $J=1$. Both scalar and pseudoscalar two-photon amplitudes exist, so $0^{++}$ and $0^{-+}$ are allowed. The J/psi meson was discovered first because its $1^{--}$ assignment permits direct, resonantly enhanced single-photon formation in $e^{+}{e}^{-}$ and a clean dilepton peak in hadron production. The lighter ${\eta }_c$ cannot be formed by one photon; it needs less favorable production channels and has a broader two-gluon decay.

A spin-zero ${\eta }_c$ at rest has no preferred axis, so $d\Gamma /d{\Omega }_p$ is isotropic after summing final spins. A J/psi meson formed in unpolarized high-energy $e^{+}{e}^{-}$ has transverse $m=\pm 1$ alignment. Its $p\bar{p}$ distribution can be writtenwhere $a,b$ contain the two independent helicity amplitudes. It is generally anisotropic; the approximate massless, helicity-conserving limit has $b\to 0$. The tabulated branching fraction does not fix $a/b$. An ensemble equally populated in all three spin projections would instead be isotropic: alignment of the production ensemble matters.

Assume coherent production and detection, relativistic neutrinos, two active neutrino flavours, propagation in vacuum, negligible packet separation and no absorption. Invert the given real lepton mixing matrix to obtain $|{\nu }_{\mu }⟩=c|{\nu }_2⟩-s|{\nu }_3⟩$ and $|{\nu }_{\tau }⟩=s|{\nu }_2⟩+c|{\nu }_3⟩$, where $c=\cos \theta$, $s=\sin \theta$. With phases ${\varphi }_a=E_{a}t-{\mathbf{p}}_a\cdot \mathbf{x}$ in natural units,Using the paper's opposite sign convention conjugates this probability amplitude and leaves the probability unchanged. For equal momentum, $E_a\simeq p+m_a^2/(2p)$ and $t\simeq L$, so $\Delta \varphi =\Delta m^{2}L/(2E)$. The equal-energy convention gives the same leading result from the momentum difference; evaluating each component at a different arrival time would incorrectly double the phase.Since $\hslash c=1.97327\times 10^{-7}\mathrm{eV}\mathrm{m}$,This gives the stated factor $1.27$. Neutrino oscillation in this approximation measures $|\Delta m^2|$, not the absolute neutrino mass or its ordering.

The Fermilab primary proton beam strikes a target, producing secondary pions and kaons. Horn focusing selects positive parents; their decays ${\pi }^{+}\to {\mu }^{+}{\nu }_{\mu }$ and $K^{+}\to {\mu }^{+}{\nu }_{\mu }$ produce a predominantly muon neutrino beam. A decay tunnel permits decay before the remaining charged secondaries encounter an absorber. A near detector constrains the flux and interaction spectrum, while the distant MINOS detector measures its change.

The alternating scintillation counters and magnetized iron give a sampling calorimeter and a magnetic tracking volume. A charged current ${\nu }_{\mu }N\to {\mu }^{-}X$ event has a long negative muon track and a localized hadronic shower. Estimate $E_{\mu }$ by range if contained or magnetic curvature if exiting, and $E_X$ by the calibrated scintillator pulse-height sum. Reconstruct $E_{\nu }\simeq E_{\mu }+E_X$, with corrections for nuclear binding, invisible energy and detector response. The muon charge separates most wrong-sign muon antineutrino events; short showers without the track are primarily neutral current candidates. The source's $730\mathrm{km}$ baseline is used in the calculations below.

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.

The tau-neutrino charged-current threshold requires specifying the stationary target and the lightest allowed final state. For a free neutron, the lowest charged current channel is ${\nu }_{\tau }n\to {\tau }^{-}p$. At threshold both final particles are at rest relative to one another in the centre-of-momentum frame, soThe equal-nucleon-mass approximation gives $m_{\tau }+m_{\tau }^2/(2m_N)\simeq 3.46\mathrm{GeV}$. More generally $E_{\nu }^{\mathrm{thr}}=[(m_{\tau }+M_X{)}^2-M_T^2]/(2M_T)$ for a target of mass $M_T$ and a recoil system with minimum allowed invariant mass $M_X$. A stationary electron in ${\nu }_{\tau }{e}^{-}\to {\tau }^{-}{\nu }_e$ instead gives $(m_{\tau }^2-m_e^2)/(2m_e)\simeq 3.09\mathrm{TeV}$. The customary $3.5\mathrm{GeV}$ answer therefore presumes a nucleon target; nuclear thresholds and practical deep inelastic scattering thresholds differ.

In a dense beam dump, long-lived pions and kaons usually interact and lose energy before decaying. Their residual stopped decays yield low-energy neutrinos, below the useful high-energy tau-interaction sample. Prompt charmed mesons have much shorter lifetimes and can decay before reinteraction. This suppresses the conventional muon neutrino background relative to the prompt component. The production sequence is Quantum chromodynamics creation of $c\bar{c}$, principally $gg\to c\bar{c}$ with $q\bar{q}\to g^{\ast }\to c\bar{c}$ also allowed, followed by hadronization $c\bar{s}\to D_s^{+}$ and its charge conjugate. The $gg$ process includes both quark-exchange diagrams and the three-gluon diagram. Fragmentation also produces other hadrons; the proton does not turn directly into one isolated Ds meson.The charge-conjugate chain gives a prompt tau antineutrino and a secondary tau neutrino. Thus each leptonic Ds meson chain produces both tau-flavour signs. The corresponding Feynman diagrams are shown in a.

For the CKM matrix, diagonalizing the up/down quark mass matrices by $U_{uL},U_{dL}$ leaves $V=U_{uL}^{†}{U}_{dL}$ in the charged current:The three-generation unitary matrix has three physical mixing angles and one CP violation phase after field rephasings. At fixed kinematics and hadronic matrix element, rates are proportional to $|V_{ij}{|}^2$; phase space, parent lifetimes, form factors and helicity suppression must also be included when comparing branching fractions.

For a charged pseudoscalar meson $P^{+}=(c\bar{q})$, leptonic pseudoscalar decay width givesThe Ds meson channel is Cabibbo-favoured, $|V_{cs}|\simeq 0.974$, whereas the charged D meson channel uses $|V_{cd}|\simeq 0.224$. Taking $m_{D_s}\simeq 1.968$, $m_D\simeq 1.870$ and $m_{\tau }\simeq 1.777$ in GeV givesIncluding the approximate lifetime ratio ${\tau }_{D_s}/{\tau }_D\simeq 0.48$ and the given production ratio $5.2/11$ gives a tau yield ratio about $17(f_{D_s}/f_D{)}^2$, already much greater than one even with equal decay constants. This is an illustrative estimate using approximate additional mass/lifetime inputs, not a determination from the production cross sections alone. A neutral D meson cannot annihilate to the charged $\tau \nu$ pair by electric charge conservation; its semileptonic tau modes have insufficient phase space once the required strange or nonstrange recoil hadron is included. Light mesons are below threshold; bottom production is much smaller at this fixed-target energy. Consequently neither the larger $D^0$ production rate nor ordinary light-hadron production outweighs the Ds meson tau source. ^[ref]

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.

Use $g^{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ and restore $\hslash$ temporarily. The free Dirac equation is $(i\hslash {\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$, with $c=1$. Its Hermitian conjugate is $-i\hslash ({\partial }_{\mu }{\psi }^{†})({\gamma }^{\mu }{)}^{†}-m{\psi }^{†}=0$. Multiplying on the right by ${\gamma }^0$ and using $({\gamma }^{\mu }{)}^{†}{\gamma }^0={\gamma }^0{\gamma }^{\mu }$ gives the adjoint equationThe Dirac adjoint and original equations give $({\partial }_{\mu }\bar{\psi }){\gamma }^{\mu }\psi =im\bar{\psi }\psi /\hslash$ and $\bar{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi =-im\bar{\psi }\psi /\hslash$. Adding gives the continuity equationThe Dirac current transforms as a four-vector; the positive density is the time component, not a Lorentz scalar. With ordinary units the number current is $c\bar{\psi }{\gamma }^{\mu }\psi$ and the electron charge current is $-e$ times it.

For either normalized two-spinor ${\chi }^s$, $(\mathbf{\sigma }\cdot \mathbf{p}{)}^2=p^{2}I$ givesThe Dirac equation alone does not fix $N$: a normalization convention is needed. In the usual covariant external-spinor convention ${\bar{u}}_r{u}_s=2m{\delta }_{rs}$, choose real positive $N=\sqrt{E+m}$. Then $u_r^{†}{u}_s=2E{\delta }_{rs}$ and ${\bar{u}}_s{\gamma }^{\mu }{u}_s=2p^{\mu }$. A second common convention is unit rest density, $\bar{u}u=1$, giving $N=\sqrt{(E+m)/(2m)}$ and density $E/m$. Both explain why the boosted density is not one particle per unit coordinate volume: the density is a four-current component and changes under a Lorentz transformation.

One may instead choose unit laboratory density, $N=\sqrt{(E+m)/(2E)}$, or box normalization $N=\sqrt{(E+m)/(2EV)}$ for one particle in volume $V$. An infinite plane wave is not normalizable to one particle over all space; a localized wave packet is required for that. Thus a unique $N$ cannot be inferred from the prompt without specifying which of these conventions is intended.

Let ${\chi }_{\pm }=(1,\pm 1{)}^T/\sqrt{2}$. The eigenvalues of ${\Sigma }_1$ are $\pm 1/2$, each with multiplicity two, and orthonormal bases for their eigenspaces areA general linear combination of the two positive-energy solutions has upper spinor $\xi =a{\chi }^1+b{\chi }^2$ and lower spinor $(\mathbf{\sigma }\cdot \mathbf{p})\xi /(E+m)$. An x-spin eigenvector needs $\xi \propto {\chi }_{\pm }$ and the lower spinor to have the same ${\sigma }_1$ eigenvalue. The Fixed-axis spin obstruction for positive-energy Dirac spinors isIts norm is $2\sqrt{p_y^2+p_z^2}$, so it vanishes exactly when $p_y=p_z=0$. For example with $\mathbf{p}=p\hat{z}$, the upper ${\chi }_{+}$ produces lower ${\chi }_{-}$, excluding a common eigenvector for every $p\ne 0$. In the nonrelativistic limit, the lower components are $O(p/m)$, and $({\psi }_1\pm {\psi }_2)/\sqrt{2}$ approach x-spin eigenvectors. At rest they are exact.

The prompt's unrestricted claim of impossibility at relativistic energies needs qualification: for momentum parallel to x these combinations remain exact eigenvectors at any energy. For a beam transverse to x, such as the conventional z-directed beam, the intended impossibility holds, and the lower components are no longer negligible. A transported relativistic spin observable differs from this fixed matrix; helicity is the conserved projection along momentum for a free particle.

An electron gun releases electrons by thermionic emission, field emission or photoelectric effect, accelerates them through an electric potential, and uses electrostatic or magnetic lenses to form a beam in vacuum. A hot unpolarized cathode does not by itself select spin.

For a polarized source, illuminate a suitable semiconductor photocathode, commonly strained GaAs, with circularly polarized light near its band-gap threshold. Optical angular-momentum selection rules preferentially populate one conduction-electron spin; emission from a negative-electron-affinity surface extracts the selected population. Reversing the light's circular polarization reverses the beam polarization. Strain and a suitable excitation wavelength suppress transitions that dilute the selection. Solenoids and spin rotators orient the spin after acceleration, and a scattering polarimeter measures the achieved polarization; it need not be $100\%$.

The basic Feynman diagrams have ${\tau }^{-}\to {\nu }_{\tau }{W}^{-\ast }$ followed by $W^{-\ast }\to e^{-}{\bar{\nu }}_e$, ${\mu }^{-}{\bar{\nu }}_{\mu }$, or $d\bar{u},s\bar{u}$, with the latter pairs undergoing hadronization. See the charged-current diagrams in a. Neglect all daughter masses, QCD corrections and inaccessible charm final states, and assume lepton universality. Each leptonic mode then has one unit of decay width; the two accessible quark combinations have three color charge states and total CKM weight $|V_{ud}{|}^2+|V_{us}{|}^2\simeq 1$. ThereforeThe result is an inclusive parton model estimate. Perturbative QCD corrections increase the hadronic contribution, while muon mass, nonperturbative dynamics and radiative corrections also matter; the physical fraction is roughly $0.65$.

Every ordinary hadronic mode already contains a tau neutrino, so it contains a lepton as well as hadrons if invisible particles are counted. For an additional visible charged lepton, a virtual photon can internally convert in a hadronic mode, for example ${\tau }^{-}\to {\nu }_{\tau }{\pi }^{-}{\gamma }^{\ast }\to {\nu }_{\tau }{\pi }^{-}{e}^{+}{e}^{-}$. The extra electromagnetic vertices suppress such modes; they are not a large fraction of the inclusive width. Alternatively a daughter pion or kaon can decay semileptonically or leptonically in flight, for example ${\tau }^{-}\to {\nu }_{\tau }{\pi }^{-}{K}^0$ followed by a semileptonic neutral-kaon decay. Those are secondary decay chains, not independent primary tau lepton branching fractions. Lepton-flavour-violating neutrinoless modes such as ${\tau }^{-}\to {\mu }^{-}{\pi }^{+}{\pi }^{-}$ do not contribute appreciably in the Standard Model.

At the Z boson pole, the accessible primary fermion pairs are $e^{+}{e}^{-}$, ${\mu }^{+}{\mu }^{-}$, ${\tau }^{+}{\tau }^{-}$, the three ${\nu }_i{\bar{\nu }}_i$ channels, and $u\bar{u},d\bar{d},s\bar{s},c\bar{c},b\bar{b}$. The top quark is too heavy. The charged channels also have photon exchange, and the electronic channel has a substantial nonresonant Bhabha scattering contribution. On-shell W boson pairs are above threshold; they are not primary $Z$ decays.

A tau lepton pair gives two approximately back-to-back narrow low-multiplicity cones, usually one or three charged tracks per cone, with overall charge zero and missing energy and momentum from neutrinos. Opposite hemispheres can show electron, muon or hadronic tags. A decay vertex or track impact-parameter information helps establish the short tau lepton flight. Reject Bhabha scattering and dimuon events by their nearly full visible beam energy, particle identification and little missing momentum; reject ordinary quark multijets by their broader high-multiplicity activity and larger visible invariant masses. Control two-photon backgrounds with visible transverse momentum, forward activity and event geometry. Selection efficiencies depend on decay mode and must be calibrated.

In ${\tau }^{-}\to {\ell }^{-}{\bar{\nu }}_{\ell }{\nu }_{\tau }$, charge and missing four-momentum show an invisible neutral component beyond the visible charged lepton. The distributions are consistent with three-body V-A interaction decay; hadronic modes such as ${\tau }^{-}\to {\pi }^{-}{\nu }_{\tau }$ instead have two-body missing-mass kinematics. Production and decay accounting assigns $L_{\tau }({\tau }^{-})=L_{\tau }({\nu }_{\tau })=1$ and their antiparticles minus one. Searches for wrong-flavour and neutrinoless channels constrain violations. These observations establish a useful approximately conserved tau-family number in ordinary interactions, rather than proving an exact law from missing energy alone.

To establish that the invisible particle is distinct from ${\nu }_e$ and ${\nu }_{\mu }$, combine decay evidence with the absence of unsuppressed tau production by conventional electron- and muon-flavour beams, and direct tau neutrino charged current interactions. DONUT observed the short displaced tau lepton decay in nuclear emulsion after a neutrino interaction. ^[ref] Neutrino oscillation means individual family numbers are not exact conserved quantities during propagation, even though they are an excellent approximation for these local decay assignments.

Write the leptonic partial decay width asHere $G_F$ is defined from muon decay, so the coupling denominator fixes the convention. Measure ${\Gamma }_{\ell }=B_{\ell }/{\tau }_{\tau }$. The ratio $B_{\mu }/B_e$ tests $|g_{\mu }/g_e{|}^2$ after the known phase space and radiative corrections. Comparing $B_e/{\tau }_{\tau }$ with the muon lifetime and the fifth power of the mass ratio tests $|g_{\tau }/g_{\mu }{|}^2$. Hadronic comparisons such as tau-to-pion versus pion-to-muon decay supply additional universality tests with their own hadronic and electromagnetic corrections.