Effects of order $\alpha^3$ on the determination of the Pauli form factor $F_2$ of the $\tau$-lepton

Abstract

We introduce optimal observables to measure the Pauli form factor $F_2$ of the $\tau$-lepton in the pair-production process $e^-e^+ \to \tau^- \tau^+$ from the intensity distribution of the decay products. The spin-density matrix for the production process is calculated in QED up to order $\alpha^3$ including virtual photon-loops and soft bremsstrahlung, as well the $\gamma Z^0$ interference. We find that the decay channel $(\rho^- \nu_\tau)\times(\rho^+ \bar\nu_\tau)$ yields the best resolution for Re$F_2(s)$ and Im$F_2(s)$ due to its high branching fraction. We also study the bias that is introduced in the determination of $F_2$ , if the production spin-density matrix is taken in tree-level (one-photon exchange) approximation.