Manifestation of the electric dipole moment in the decays of $\tau$ leptons produced in $e^+e^-$ annihilation

TL;DR

This paper investigates CP-odd asymmetries in various $e^+e^-$ annihilation processes involving tau leptons, aiming to measure the electric dipole form factor $F_3^ au$ and its real and imaginary parts, with and without beam polarization.

Contribution

It introduces methods to measure the electric dipole form factor of tau leptons through asymmetries in specific processes, highlighting the role of beam polarization in these measurements.

Findings

Polarization allows measurement of the real part of $b$.

Unpolarized beams enable measurement of the imaginary part of $b$.

Certain processes can determine $b$ using differential cross sections.

Abstract

$\mbox{CP}$-odd asymmetries in the processes $e^+e^-\rightarrow \tau^+\pi^-\nu_\tau$, $e^+e^-\rightarrow \pi^+\tau^-\bar\nu_ \tau$, $e^+e^-\rightarrow \tau^+\rho^-\nu_\tau$, $e^+e^-\rightarrow \rho^+\tau^-\bar\nu_\tau $, $e^+e^-\to \tau^+e^-\nu_\tau{\bar\nu}_e\,,\,$ and $\,e^+e^-\to \tau^-e^+\nu_e{\bar\nu}_\tau$ are investigated with account for longitudinal polarization of electron (or positron) beam. These asymmetries is a manifestation of electric dipole form factor $F_3^\tau\equiv b$ in the $\gamma\tau^+\tau^-$ vertex. It is shown that, to measure $\mbox{Im}\,b$ in the specified processes, polarization is not needed, while to measure $\mbox{Re}\,b$ it is required. The processes $e^+e^-\to \pi^+\pi^-\nu_\tau{\bar\nu}_\tau$, $e^+e^-\to e^+e^-\nu_\tau{\bar\nu}_\tau\nu_e\bar\nu_e$, $e^+e^-\to \mu^+\mu^-\nu_\tau{\bar\nu}_\tau\nu_\mu\bar\nu_\mu$, $e^+e^-\to \mu^+e^-\nu_\tau{\bar\nu}_\tau\nu_\mu\bar\nu_e$, and $e^+e^-\to \mu^-e^+\nu_\tau{\bar\nu}_\tau\nu_e\bar\nu_\mu$ are also discussed for the case of unpolarized electron and positron beams. In the latter cases it is possible to measure $\mbox{Re}\,b$ using the differential cross section over momenta of both registered particles.