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

TL;DR

This paper develops optimal observables to measure the $ au$-lepton's Pauli form factor $F_2$ in $e^+e^-$ collisions, incorporating QED corrections up to order $oldsymbol{ extit{ extbf{ extalpha}}^3}$, and assesses the impact of higher-order effects on the measurement accuracy.

Contribution

It introduces a method to measure $F_2$ using optimal observables that include QED corrections up to order $oldsymbol{ extit{ extbf{ extalpha}}^3}$, improving precision over tree-level approaches.

Findings

The decay channel $(\rho^- \nu_\tau)\times(\rho^+ \bar\nu_\tau)$ provides optimal resolution for $F_2$.

QED corrections up to order $\alpha^3$ significantly affect the spin-density matrix.

Neglecting higher-order corrections biases the $F_2$ measurement.

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.