On trace invariance and energy-dependent entanglement in muon decay

Abstract

We revisit a proposed correction to the muon magnetic anomaly measurement and compare it with energy-binned scans reported by the Fermilab $g-2$ collaboration. The central point is that, for a relativistic three-body decay inferred from an ensemble of detected positrons, the relevant neutrino partial trace is defined on the asymptotic state at the detector, not on an idealised state at the decay vertex. Because the fitted positron sample mixes events with different decay times, flight times, energies and acceptances, tracing over the unobserved neutrino helicities leaves a time-dependent mixed state in the observed sector. Using angular-momentum balance, we obtain an energy-dependent correction in the positron energy interval $E_p \in (1.5,2.9)$ GeV. The corrected experimental interval for the muon anomaly moves towards the BaBar- and $\tau$-based determinations, while remaining compatible at the $1\sigma$ level with both data-driven and lattice-based Standard Model evaluations. We then fit digitised public $R(E_p)$ scans for Fermilab's runs $1$ to $6$, with two one-parameter hypotheses: a constant shift and the entanglement-motivated energy-dependent shift. Within digitisation accuracy, the differences in $\chi^2$ remain modest and the current binned data do not provide decisive discrimination between them.