Probing the imaginary parts and their $q^2$ dependences for the tau $g-2$ and EDM
The $τ$ anomalous magnetic dipole moment (MDM) $a_τ= (g-2)_τ/2$ and electric dipole moment (EDM) $d_τ$, are precision probes of electroweak dynamics and possible new physics sources, yet both remain weakly constrained experimentally. Treated as generalized form factors, these quantities exhibit a generic $q^2$ dependence for an off-shell interacting photon. For timelike momentum transfer above the $τ^+τ^-$ threshold, $q^2 = s > 4m_τ^2$, the form factors can acquire absorptive imaginary parts. We investigate how such a $q^2$ dependence and the associated imaginary parts are generated from two complementary perspectives: the model-independent Standard Model Effective Field Theory (SMEFT) and a UV-complete Two-Higgs-Doublet Model (2HDM). The effective framework reveals the intimate correlation between $a_τ$ and $d_τ$. New CP-violating interactions which generate a non-zero $d_τ$, can also generically have non-zero contributions to $a_τ$, thereby deeply linking their phenomenological studies. Within the 2HDM, we demonstrate that sizable imaginary parts and significant $q^2$ running can be generated at levels accessible by $e^+e^-$ colliders. Motivated by these features, we propose experimental methods to extract both the real and imaginary components of the dipole form factors. Utilizing these techniques, we show that Belle II and the Super Tau-Charm Facility (STCF) can improve current bounds on $a_τ$ by more than one order of magnitude. Finally, we highlight that combining measurements across the distinct center-of-mass energies of Belle II and STCF provides a unique, previously unexplored avenue to explicitly obtain information about the $q^2$ evolution of these dipole form factors.