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Lattice QCD determination of the $γZ$ box contribution to the proton weak charge

We present the first lattice QCD determination of the $γZ$ box contribution to parity-violating electron-proton scattering, $\square_{γZ}$ , 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 $E =155 MeV$. 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 $\square_{γZ}= 0.00412(9)$ , 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 $\square_{γZ}$ further provides a first-principles input for the upcoming P2 experiment at Mainz, which will operate at the optimized beam energy of $155 MeV$ to extract $Q_{W}^p$.