Lattice QCD determination of the $\gamma Z$ box contribution to the proton weak charge

Abstract

We present the first lattice QCD determination of the $\gamma Z$ box contribution to parity-violating electron-proton scattering, ${\square }_{\gamma Z}$ , a key ingredient for the precise tests of the Standard Model via the proton weak charge. Our calculation covers the electron beam energies up to $E=155MeV$. For the axial-vector component, we achieve reduced uncertainties across the entire energy range compared with phenomenological estimates. For the vector component, the uncertainties remain slightly larger after continuum extrapolation. At $E=0$, where the vector part vanishes, we obtain ${\square }_{\gamma Z}=0.00412(9)$ , reducing the uncertainty by a factor of $2$ relative to the most precise previous determination. Incorporating this result yields an updated weak charge of $Q_W^p=0.06987(50)$ . The calculated energy dependence of ${\square }_{\gamma Z}$ further provides a first-principles input for the upcoming P2 experiment at Mainz, which will operate at the optimized beam energy of $155MeV$ to extract $Q_W^p$.

15 pages, v1