Constraining Neutrino--Nucleon Form Factors with Charged-Current Scattering at the Electron-Ion Collider

Next-generation neutrino oscillation experiments such as DUNE require percent-level knowledge of neutrino--nucleon interaction cross sections. The nucleon axial form factor $F_A(Q^2)$, parameterized by the axial mass $\MA$, is the dominant source of uncertainty in the quasi-elastic channel, and the parity-violating structure function $xF_3$ is poorly constrained on free nucleons. We propose using charged-current (CC) electron--proton scattering at the Electron-Ion Collider (EIC) to address both problems simultaneously. The measurement exploits three key features of the EIC: (1)~helicity-selective electron bunches provide \emph{in situ} electromagnetic background rejection; (2)~a longitudinally polarized proton target enables extraction of $F_A(Q^2)$ through the target-spin asymmetry $A_{UL}$; and (3)~the $y$-distribution leverage in CC deep inelastic scattering separates $F_2$ and $xF_3$ on a \emph{free proton}, without nuclear corrections. Using a Fisher-information analysis at $\sqrt{s} = 141\GeV$ with $500\fb^{-1}$ of integrated luminosity, we project the Cram\'{e}r--Rao statistical floor of $\delta\MA \approx 0.03\GeV$ (3\%). Incorporating first-order realistic detector effects: ZDC acceptance, $Q^2$ smearing (5\%), and background noise from helicity subtraction, the projected sensitivity is severely background-limited due to the small signal-to-background ratio ($S/B \approx 3 \times 10^{-4}$) in the elastic channel. Achieving competitive sensitivity ($\delta\MA \approx 0.14\GeV$) would require $\sim\!10^{-7}$ background suppression, three orders of magnitude beyond current projections. The CC DIS $y$-distribution provides sub-percent extraction of $xF_3^{\Wminus}$ over $0.05 < x < 0.5$, representing the most robust electroweak measurement in the near term.