Beam polarization precision requirements for future $e^+e^-$ Higgs factories

We present work on quantifying the minimum requirements for beam polarization precision at future $e^+e^-$ Higgs factories. We find that, under the assumption of a high electron beam polarization ($P_-$) that the positron polarization ($P_+$) is of key importance but for reasons both known and newly discovered. We have discovered that improved positron polarization leads to a less strict requirement on the beam polarization precision for measurements that scale only with the effective polarization, $P_\mathrm{eff}$. Conversely, measurements that scale with the product of beam polarizations, $P_-P_+$, such as those that contain the $eeZ$ or $ee\gamma$ vertex, have their polarization precision demands get more strict as positron polarization increases. We check the polarization precision demands for $10^{-3}$ on the Higgsstrahlung cross-section ($\sigma_{ZH}$) at 250~GeV, $10^{-4}$ on the electron left-right asymmetry ($A_{\rm e}$) at the Z pole, and $10^{-4}$ on the di-photon cross-section ($\sigma_{\gamma \gamma}$) from the Z pole to 3~TeV. We find that, for measurements away from the Z pole, the goals can plausibly be attained if one can achieve precision on beam polarization better than $0.1\%$. For measurements of $A_{\rm e}$ at $10^{-4}$, colliders must do better than $0.02\%$ on beam polarization precision, or determine ways to upgrade their beam polarization values towards unity, where the requirements decrease to better than $0.15\%$.