Parity violation in atoms and electron scattering revisited

Exchange by two neutrinos (as well as by other fermions) generates a long-range parity-violating (PV) potential of the form $\sim G^2/r^5$, with characteristic range $\hbar/(2m_νc)$. This interaction produces a $-0.8%$ correction to the effective weak charge of the Cs atom, thereby resolving the $2σ$ discrepancy between the Standard Model prediction and the Cs PV measurement, and yielding $\sin^2θ_W=0.2375(19)$ at $q^2\approx0$ (to be compared with the Standard Model value $\sin^2θ_W=0.2387$). We derive updated limits on an extra $Z’$ boson and obtain a constraint on isospin-conserving oblique radiative corrections characterized by the Peskin–Takeuchi parameter, $S=-0.32(53)$ at $q^2\approx0$. The corresponding relative correction to the proton weak charge is $3%$, and a similar correction is expected for the electron weak charge.