Hadronic weak charges and parity-violating forward Compton scattering

Parity-violating elastic electron-nucleon scattering at low momentum transfer allows one to access the nucleon's weak charge, the vector coupling of the $Z$-boson to the nucleon. In the Standard Model and at tree level, the weak charge of the proton is related to the weak mixing angle and accidentally suppressed, $Q_W^{p,\,{\rm tree}}=1-4\sin^2\theta_W\approx0.07$. Modern experiments aim at extracting $Q_W^p$ at $\sim1\%$ accuracy. Similarly, parity non-conservation in atoms allows to access the weak charge of atomic nuclei. We consider a novel class of radiative corrections, an exchange of two photons with parity violation in the hadronic/nuclear system. These corrections may affect the extraction of $\sin^2\theta_W$ from the experimental data at the relevant level of precision because they are affected by long-range interactions similar to other parity-violating radiative corrections, such as, e.g., the $\gamma Z$-exchange, which has obtained much attention recently. We show that the significance of this new correction increases with the beam energy in parity-violating electron scattering, but the general properties of the parity-violating forward Compton amplitude protect the formal definition of the weak charge as a limit at zero-momentum transfer and zero-energy. We also discuss the relevance of the new correction for upcoming experiments.