Resolving the Two-Fold Ambiguity in Tau Momentum Reconstruction via a Self-Consistent Weighting Method

Abstract

The reconstruction of $τ$ lepton momenta in $e^{+}e^{-} \to τ^{+}τ^{-}$ processes suffers from a two-fold kinematic ambiguity due to undetected neutrinos. We show that the common practice of retaining both solutions with equal weight systematically biases the extracted angular coefficients: the bias grows with the size of the $\cos^{2}θ$ modulation of the production distribution $f(\cosθ)$, but remains resolvable even in the nearly uniform QED limit at the $ψ(2S)$ threshold. We propose a self-consistent weighting method based on a fixed-point theorem: assigning normalized weights $w_{i} = f(\cosθ_{i})/[f(\cosθ_{1})+f(\cosθ_{2})]$ yields a weighted distribution $h(x) = f(x)[1+\varepsilon(x)]$ with $\varepsilon(x) \ll 1$ under a detailed balance condition. Monte Carlo simulations at $\sqrt{s}=3.686$~GeV and $4.8$~GeV show that the fixed-point iteration converges to the true angular coefficient $α$ within statistical uncertainty, providing a robust framework for angular analysis without event-by-event ambiguity resolution.