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

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]