Past exam of the particle physics course of the University of Oxford / 2019 / 6 / c / Solution

For a valence-dominated isoscalar target, $u_v^N(x)=d_v^N(x)=q_v(x)$. A left-chiral incident neutrino has unsuppressed scattering on a left-chiral quark and the $(1-y{)}^2$ weight for a right-chiral quark. Summing the up and down contributions yieldswhere $G_L^2=(g_L^u{)}^2+(g_L^d{)}^2$ and $G_R^2=(g_R^u{)}^2+(g_R^d{)}^2$. The common normalization here uses the conventional Fermi coupling and PDFs in a; the requested $y$ shape is independent of an overall normalization convention.

Put $w={\sin }^2{\theta }_W$. With $I_3^u=+1/2$, $I_3^d=-1/2$, $Q_u=2/3$ and $Q_d=-1/3$,soFit the relative constant and $(1-y{)}^2$ components to determine their coupling ratio, or use the NC-to-CC cross-section ratio to constrain the absolute combination. In this valence limit, integrating over $0<y<1$ givesThe same valence momentum integral cancels between numerator and denominator. Experimentally, NC events have hadronic activity without the charged-current outgoing muon; known fluxes or CC normalization, muon identification and acceptance corrections are needed. Sea-quark contributions, charm thresholds, QCD corrections and departures from isoscalarity must be included for a precision weak-mixing-angle measurement.