$\gamma Z$-Exchange Contributions in Low-Energy Parity-Violating $ep$ Scattering

In this work, the $\gamma Z$-exchange contributions in the low-energy elastic parity-violating $ep$ scattering are discussed with the approximation $m_e=0$. By expanding the $\gamma pp$ and $Zpp$ interactions on the momentum of photon and considering both the leading-order and the next-to-leading order interactions, we calculate the amplitudes of the $\gamma Z$-exchange diagrams. After performing the loop integral, we expand the results in the low energy limit, and obtain the analytic expressions for the amplitudes. Numerical comparisons show that the analytic expressions are very close to the full results over a large region. We investigate the power behaviors of these contributions and find that some are enhanced by a kinematic factor in the low energy limit. Additionally, in some cases, the imaginary parts of the contributions from the next-to-leading-order interactions are at the same order as those from the leading-order interactions. Furthermore, the corresponding contributions to the physical observable quantity $A_{\textrm{PV}}$ are also discussed. Combining all the properties together, we conclude that these analytic expressions describe the leading-order contributions of all $\gamma Z$-exchange helicity amplitudes in the region with $\alpha_e\ll Q/M_N\sim \delta/M_N^2\ll 1$, where $\alpha_{e}$ is the fine structure constant, $M_N$ is the mass of proton, $Q$ and $\delta$ are the small quantities related to the momentum transfer and the center-of-mass energy.