Hadronic weak charges and parity-violating forward Compton scattering

Background: 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, $Q_W^{p,\mathrm{tree}}=1-4{sin}^2{\theta }_W\approx 0.07$. Modern experiments aim at extracting $Q_W^p$ at $\sim 1\%$ accuracy. Similarly, parity nonconservation in atoms allows to access the weak charge of atomic nuclei.

Purpose: We consider a novel class of radiative corrections due to the exchange of two photons, with parity violation in the hadronic/nuclear system. These corrections are prone to long-range interactions and may affect the extraction of ${sin}^2{\theta }_W$ from the experimental data at the relevant level of precision.

Methods: The two-photon exchange contribution to the parity-violating electron-proton scattering amplitude is studied in the framework of forward dispersion relations. We address the general properties of the parity-violating forward Compton scattering amplitude and use relativistic chiral perturbation theory to provide the first field-theoretical proof that it obeys a superconvergence relation.

Results: We show that the significance of this new correction increases with the beam energy in parity-violating electron scattering, but the superconvergence relation protects the formal definition of the weak charge as a limit at zero-momentum transfer and zero energy. We evaluate the new correction in a hadronic model with pion loops and the $\mathrm{\Delta }\left(1232\right)$ resonance, supplemented with a high-energy contribution. For the kinematic conditions of existing and upcoming experiments we show that two-photon exchange corrections with hadronic or nuclear parity violation do not pose a problem for the interpretation of the data in terms of the weak mixing angle at the present level of accuracy.

Conclusions: Two-photon exchange in presence of hadronic or nuclear parity violation gives rise to long-range parity-violating interactions. Depending on the kinematic conditions and precision goal, this novel correction may affect the extraction of weak charges from experiments with atoms and electron scattering.

Figure 1

The two-boson exchange diagram ($\gamma \gamma$ or $\gamma Z$) with the relevant kinematic variables.

Figure 2

A schematic representation of the elastic contribution to the imaginary part of the two-boson exchange correction to the elastic PVES amplitude. The left and right parts show the $\gamma Z$ and PV $\gamma \gamma$ contributions, respectively. The vertical dashed line cutting through the diagrams indicates that the intermediate $ep$ state is on-shell.

Figure 3

Correction ${\square }_{\gamma Z+\gamma \gamma }^{\mathrm{el}}={\square }_{\gamma Z}^{A,\mathrm{el}}+{\square }_{\gamma \gamma }^{\mathrm{PV},\mathrm{el}}$ to the effective weak charge of the proton in units of ${10}^{-4}$ in the exact forward limit and as a function of the electron energy in GeV. The black curve shows the result for ${\square }_{\gamma Z}^{A,\mathrm{el}}$ with $G_A=-1.2701$. The red curve is our new central value obtained from the sum ${\square }_{\gamma Z}^{A,\mathrm{el}}+{\square }_{\gamma \gamma }^{PV,\mathrm{el}}$. The shaded region corresponds to the uncertainty from the proton's anapole moment.

Figure 4

Tree-level Feynman diagrams in BChPT needed to calculate the integrand of Eq. (35). Small black dots indicate the photon coupling to the nucleon charge, solid black squares denote couplings to the anomalous magnetic moment, and large blue circles describe the PV $\pi NN$ coupling.

Figure 5

The contributions of the terms proportional to the anomalous magnetic moments of the proton ${\mu }^p=1+{\kappa }^p$ (left) and of the neutron ${\mu }^n={\kappa }^n$ (right) to the integrand of Eq. (41) in arbitrary units.