Past exam of the particle physics course of the University of Oxford / 2015 / 5

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.