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

TL;DR

This paper presents the first lattice QCD calculation of the $oxgammaZ$ contribution to the proton's weak charge, improving precision for tests of the Standard Model and providing essential input for upcoming experiments.

Contribution

It provides the first lattice QCD determination of the $oxgammaZ$ contribution across relevant energies, reducing uncertainties especially for the axial-vector component.

Findings

Reduced uncertainty in the axial-vector component across energies.

Precise value of $oxgammaZ$ at zero energy: 0.00412(9).

Updated proton weak charge: 0.06987(50).

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 $E =155 MeV$. 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 $\square_{\gamma Z}= 0.00412(9)$ , 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 $155 MeV$ to extract $Q_{W}^p$.