Probing the $Zb\bar{b}$ coupling at the $Z$-pole of future lepton colliders

The determination of the $Zb\bar{b}$ coupling in experiments has been a long-standing challenge, as the limited precision of off $Z$-pole measurements at the LEP has resulted in two degenerate solutions remained to be resolved. In this paper, we propose a novel method to probe the $Zb\bar{b}$ coupling by measuring the forward-backward asymmetry of the bottom quark, $A^{b}_\text{{FB}}$, in the $b\bar{b}$ system of the $e^+ e^- \to b \bar{b} \gamma$ and/or $e^+ e^- \to b \bar{b} g$ processes at the $Z$-pole of future lepton colliders. The additional hard photon or light jet radiation can mimic the energy scanning of the $e^+e^-\to b\bar{b}$ process, and the $A_{\rm FB}^b$ distribution from the $\gamma$-$Z$ interference process is linearly sensitive to the $Zb\bar{b}$ coupling. By combining the expected measurements of $R^0_b$ and $A^{0,b}_\text{{FB}}$ at the $Z$-pole at the CEPC, the $A^{b}_\text{{FB}}$ distributions can break the degeneracy observed at the LEP, leading to a unique determination of the $Zb\bar{b}$ coupling through $Z$-pole running alone.