= 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 $q\bar q\to g^*\to c\bar c$ 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>.
$$
D_s^+(c\bar s)\to W^{+*}\to\tau^+\nu_\tau,\qquad
\tau^+\to\bar\nu_\tau W^{+*}\to\bar\nu_\tau(\ell^+\nu_\ell\ \text{or}\ u\bar d,u\bar s).
$$
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 <2/a>.
For the <CKM matrix>, diagonalizing the up/down <quark mass> matrices by $U_{uL},U_{dL}$ leaves $V=U_{uL}^\dagger U_{dL}$ in the <charged current>:
$$
\mathcal L_{\rm cc}=-\frac g{\sqrt2}\bar u_{Li}\gamma^\mu V_{ij}d_{Lj}W^+_\mu+\mathrm{h.c.},\qquad V^\dagger V=I.
$$
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> gives
$$
\Gamma(P^+\to\tau^+\nu_\tau)=\frac{G_F^2f_P^2|V_{cq}|^2m_Pm_\tau^2}{8\pi}\left(1-\frac{m_\tau^2}{m_P^2}\right)^2.
$$
The <Ds meson> channel is Cabibbo-favoured, $|V_{cs}|\simeq0.974$, whereas the charged <D meson> channel uses $|V_{cd}|\simeq0.224$. Taking $m_{D_s}\simeq1.968$, $m_D\simeq1.870$ and $m_\tau\simeq1.777$ in GeV gives
$$
\frac{\Gamma(D_s^+\to\tau^+\nu)}{\Gamma(D^+\to\tau^+\nu)}\simeq75\left(\frac{f_{D_s}}{f_D}\right)^2.
$$
Including the approximate lifetime ratio $\tau_{D_s}/\tau_D\simeq0.48$ 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. <Final tau-neutrino results from the DONuT experiment by DONUT Collaboration>[DONuT final analysis: prompt beam production and tau-neutrino flux model]{ref}
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