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

TL;DR

This paper delivers the first ab initio lattice QCD determination of the gammaZ box contribution to the proton’s weak charge, including both vector and axial-vector hadronic parts over the PVES-relevant energy range and up to 155 MeV beam energy. The authors develop a Wick–residue decomposition and an infinite-volume reconstruction framework to control short- and long-distance dynamics, explicitly treating Nπ intermediate states that crucially affect the vector channel. Their results yield a refined gammaZ box at zero energy, square_gammaZ(0) = 5.16(7)_(stat)(5)_(ES)×10^-3, and provide a nonperturbative energy dependence up to 155 MeV, essential for the upcoming P2 experiment. The updated proton weak charge is Q_W^p = 0.06987(50), with dominant residual uncertainties arising from s_W^2(0) input and from the P-odd piece of the gamma gamma box; the work demonstrates a viable path toward high-precision, first-principles determination of the gammaZ box in PVES. The methodology and results have significant implications for SM tests and precision electroweak program at low Q^2, including the Mainz P2 program.

Abstract

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 $Q_{W}^p$.

Figures (6)

• Figure 1: Numerical results for the Wick and residue contributions to $\square_{\gamma Z}^{V}(E)$ at a beam energy of $E=155$ MeV. The short-distance part, the long-distance nucleon contribution, the long-distance $N\pi$ contribution, and their total are indicated by circles, upward triangles, downward triangles, and squares, respectively.
• Figure 2: The beam energy dependence of vector box $\square_{\gamma Z}^{V}(E)$ and axial-vector box $\square_{\gamma Z}^{A}(E)$, with 24D, 32Dfine and continuum-extrapolated result.
• Figure S 1: Verification of Eq. (\ref{['eq:SD_relation_Npi']}). For all three isospin $\{I,I'\}$ channels, the relations in (\ref{['eq:SD_relation_Npi']}) are satisfied for $t_s>1.15$ fm on both 24D and 32Dfine ensembles.
• Figure S 2: Numerical results for the Wick and residue contributions to $\square_{\gamma Z}^{A,Q^2<2~\mathrm{GeV}^2}(E)$ at a beam energy of $E=155$ MeV. The short-distance, the long-distance and the total contributions are indicated by upward triangles, downward triangles and squares, respectively.
• Figure S 3: Numerical results for the $I=0$ and $I=1$ Sach form factors defined in Eq. (\ref{['eq:Sach_def']}) together with a dipole fit.