Lam-Tung relation breaking effects and weak dipole moments at lepton colliders

The breaking of the Lam-Tung relation in the Drell-Yan process at the LHC exhibits a long-standing tension with the Standard Model (SM) prediction at $O\left({\alpha }_s^3\right)$ accuracy. This tension could be explained by weak dipole interactions of leptons and quarks, associated with the $Z$-boson within the framework of the Standard Model Effective Field Theory (SMEFT). In this paper, we propose to cross-check these weak dipole interactions by measuring the violation effects of the Lam-Tung relation at future lepton colliders through the processes $e^{+}{e}^{-}\to Z\gamma \to \ell \overline{\ell }\gamma$ and $e^{+}{e}^{-}\to Z\gamma \to q\overline{q}\gamma$. By considering different decay modes of the $Z$-boson, these channels exhibit distinct sensitivities to various dipole operators, providing a way to disentangle their individual effects. Additionally, the high flavor-tagging efficiencies at lepton colliders could provide strong constraints on the dipole interactions of heavy quarks, such as $b$ and $c$ quarks, which are challenging to probe in the Drell-Yan process at the LHC due to the suppression of parton distribution functions.