  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.
