The Large Quark Mass Expansion of Gamma (Z^0 -> hadrons) and Gamma(tau^- -> nu_tau + hadrons) in the Order alpha_s^3

TL;DR

This work delivers a complete $O(α_s^3)$ treatment of heavy-quark mass effects in hadronic Z decays and related tau decays, using a systematic large-mass expansion to include terms up to $(m_Z^2/m_t^2)^3$ for both vector and axial contributions (including singlet/non-singlet), and performing a careful decoupling analysis between 6- and 5-flavor QCD. It provides decoupled, five-flavor results at μ = m_Z and confirms RG invariance and gauge-independence, with independent verification of the decoupling relations. The paper also extends the approach to the large-$m_c$ expansion for $R_τ$ at $O(α_s^3)$, with explicit coefficients showing rapid convergence. Together, these results enable precise determinations of α_s from Z decays and tau decays and illustrate the non-decoupling behavior of the axial part due to anomalies.

Abstract

We present the analytical $α_{s}^3$ correction to the $Z^{0}$ decay rate into hadrons. We calculate this correction up to (and including) terms of the order $(m_Z^2/m_{top}^2)^3$ in the large top quark mass expansion. We rely on the technique of the large mass expansion of individual Feynman diagrams and treat its application in detail. We convert the obtained results of six flavour QCD to the results in the effective theory with five active flavours, checking the decoupling relation of the QCD coupling constant. We also derive the large charm quark mass expansion of the semihadronic $τ$ lepton decay rate in the $α_{s}^3$ approximation.