Mass Corrections to the Tau Decay Rate

TL;DR

Chetyrkin and Kwiatkowski examine mass corrections to the tau hadronic decay rate within perturbative QCD and the operator product expansion. They compute quadratic (D=2) mass corrections to $R_\tau$ up to ${\cal O}(\alpha_s^2 m^2)$ and analyze $D=4$ nonperturbative corrections using renormalization-scale–specific condensates, arguing for RG-invariant inputs at $\hat{\mu}=M_\tau$. Their results reveal significant strange-quark-mass effects in Cabibbo-suppressed channels and show that second-order mass corrections propagate into non-strange decays via strange loops, while $D=4$ contributions to $R_\tau$ are small. Overall, the massless perturbative terms dominate $R_\tau$, but the mass corrections remain important for precision tests of QCD with tau decays.

Abstract

In this note radiative corrections to the total hadronic decay rate of the $τ$-lepton are studied employing perturbative QCD and the operator product expansion. We calculate quadratic quark mass corrections to the decay rate ration $R_τ$ to the order ${\cal O}(α_s^2 m^2)$ and find that they contribute appreciably to the Cabbibo supressed decay modes of the $τ$-lepton. We also discuss corrections of mass dimension D=4, where we emphasize the need of a suitable choice of the renormalization scale of the quark and gluon condensates.