Measurement of the $τ$ electric dipole moment using longitudinal polarization of $e^+e^-$ beams

Abstract

Certain CP-odd momentum correlations in the production and subsequent decay of $τ$ pairs in $e^+ e^-$ collisions are enhanced significantly when the $e^+$ and $e^-$ beams are longitudinally polarized. These may be used to probe the real and imaginary parts of $d_τ^γ$, the electric dipole moment of the $τ$. Closed-form expressions for these ``vector correlations'' and the standard deviation of the operators defining them due to standard model interactions are presented for the two-body final states of $τ$ decays. If 42\% average polarization of each beam is achieved, as proposed for the tau-charm factories, with equal integrated luminosities for each sign of polarization and a total yield of $2\cdot 10^7$ $τ^+ τ^-$ pairs, it is possible to attain sensitivities for $\vertδ{\rm Re} d_τ^γ\vert$ of $8\cdot 10^{-19}$, $1\cdot 10^{-19}$, $1\cdot 10^{-19}$ $e$ cm respectively and for $\vertδ{\rm Im} d_τ^γ\vert$ of $4\cdot 10^{-14}$, $6\cdot 10^{-15}$, $5\cdot 10^{-16}$ $e$ cm respectively at the three operating center-of-mass energies of 3.67, 4.25 and 10.58 GeV. These bounds emerge when the effects of a posible weak dipole form factor $d_τ^Z$ are negligible as is the case when it is of the same order of magnitude as $d_τ^γ$. Furthermore, in such a polarization experiment where different polarizations are possible, a model-independent disentangling of their individual effects is possible, and a technique to achieve this is described. A strong longitudinal polarization physics programme at the tau-charm factory appears warranted.