Corrections to the Forward Limit Dispersion Relations for $\gamma Z$-Exchange Contributions

The weak charge of the proton $Q_{\textrm{w}}$ is one of the most fundamental quantities in physics. It can be determined by measuring the parity asymmetry $A_{\textrm{PV}}$ in elastic $ep$ scattering, where the $\gamma Z$-exchange contributions are crucial. For the past fifteen years, dispersion relations (DRs) in the forward limit have been widely used as a model-independent method to estimate these contributions. In this work, we study corrections to these forward-limit DRs. We first estimate these corrections using pointlike interactions as an illustrative example. We then estimate the $\gamma Z$-exchange contributions for the upcoming P2 experiment through both direct calculation and the forward-limit DRs, within the framework of low-energy effective interactions. The results indicate that the correction to the forward-limit DR for $\Box_{\gamma Z}^{V}$ is around 47\% for the upcoming P2 experiment, which will significantly modify the extracted value of $Q_{\textrm{w}}$.