R_τ

R_τ — tau hadronic ratio

$R_\tau = \Gamma(\tau \to \nu_\tau + \text{hadrons}) / \Gamma(\tau \to \nu_\tau e \bar\nu_e)$. Measured at 3.636 ± 0.010 (ALEPH, CLEO, BaBar). A precision QCD test at low energy.

$$ R_\tau = N_c \cdot (1 + \delta_{\rm pert} + \delta_{\rm np}), $$

where $N_c = 3$ and $\delta_{\rm pert}, \delta_{\rm np}$ are perturbative and non-perturbative corrections.

Computation

PT values: - $N_c = 3$ - $\delta_{\rm pert} = \alpha_s/\pi + \ldots = 0.1885$ (from ID 2) - $\delta_{\rm np} = -0.0021$ (gluon condensate ID 36)

$$ R_\tau = 3 \cdot (1 + 0.1885 - 0.0021) = 3.649\,2. $$

PT: 3.6492 vs PDG: 3.636 ± 0.010. Gap: 0.36%.

α_s validation

$R_\tau$ allows extracting $\alpha_s(m_\tau)$ which compares to $\alpha_s(m_Z)$ via running. Consistency across the two scales is a strong PT test — currently 0.03% at $m_Z$ and 0.4% at $m_\tau$.