= Solution
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>.
\Image[/past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2016-charged-current-partons.png]
{title=Leading charged-current electron-quark and positron-quark diagrams}
For unpolarized beams, the leading <charged current> <cross sections>, suppressing <Cabibbo-Kobayashi-Maskawa matrix> factors, have the form
$$
\begin{aligned}
\frac{d^2\sigma(e^-p)}{dx\,dQ^2}&=\frac{G_F^2}{2\pi}\left(\frac{M_W^2}{M_W^2+Q^2}\right)^2\left[u+c+(1-y)^2(\bar d+\bar s)\right],\\
\frac{d^2\sigma(e^+p)}{dx\,dQ^2}&=\frac{G_F^2}{2\pi}\left(\frac{M_W^2}{M_W^2+Q^2}\right)^2\left[\bar u+\bar c+(1-y)^2(d+s)\right].
\end{aligned}
$$
Here 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 form
$$
\delta_h=\sum_h(E_h-p_{zh}),\quad y_h=\frac{\delta_h}{2E_e},\quad Q_h^2=\frac{|\sum_h\mathbf p_{Th}|^2}{1-y_h},\quad \boxed{x_h=\frac{Q_h^2}{sy_h}}.
$$
These 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.
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