Parity Violation on Longitudinal Single-Spin Asymmetries at the EicC

We explore two longitudinal single-spin asymmetries induced from parity violation in neutral-current deep inelastic scattering at the proposed Electron-ion collider in China (EicC): $A_{PV}^{e\,(p)}$ from longitudinally polarized (unpolarized) electrons scattering off unpolarized (longitudinally polarized) protons. We find $A_{PV}^e$, of $\mathcal{O}(10^{-4})$, is generically one to three orders of magnitude larger than $A_{PV}^p$. We further estimate different uncertainty sources including statistics, parton distribution functions, and beam polarization, for both asymmetries, and then identify individually their dominance in different regimes of the Bjorken-$x$. Based on these results, we then advocate utilizing $A_{PV}^p$ for the extraction of the weak mixing angle at two representative momentum transfer scales unexplored before, and we find a relative precision below 10% can be achieved at the EicC with an effective one-year operation time.