$α_s(m_Z)$ from tau decays with matching conditions at three loops

Abstract

Using the recent four-loop calculations of the QCD beta-function and the three-loop matching coefficients we study the induced error in $α_s(m_Z)$ obtained from $α_s(m_tau)$ due to the evolution procedure. We show that, when consistent matching and running is used at this order, these errors are pushed below 0.0005 in $α_s(m_Z)$.

Figures (3)

• Figure 1: Values one obtains for $\alpha^{(5)}_s(m_Z)$ starting from $\alpha^{(5)}_s(m_\tau)=0.336$ at 2,3 and 4 loops by using the different approximations discussed in the text: i) the usual $1/\log(\Lambda_{QCD})$ expansion chetyrkin.kniehl.ea:97 (white), ii) numerical integration of the renormalization group equation (soft hatching), iii) our expansion in eq. (\ref{['alfarun']}) (hard hatching).
• Figure 2: $\alpha_s(m_Z)$ obtained by running the coupling from $\alpha_s(m_\tau)=0.35$, as a function of the matching point taken to cross the $b$-quark threshold. Dotted line: two-loop beta functions and one-loop matching conditions. Dashed line: three-loop beta functions and two-loop matching conditions. The hatched band is obtained with four-loop beta functions and three-loop matching conditions when the $b$-quark mass is varied within its error interval.
• Figure 3: $\alpha_s(m_Z)$ as a function of the matching point scales $\mu^c_{th}$ and $\mu^b_{th}$. The contours are for $\alpha_s(m_Z)= 0.12214$ to $\alpha_s(m_Z)= 0.12182$ in steps of $0.00004$. The area within the strait lines is obtained by taking $\bar{m}_c < \mu^c_{th} < \mu^b_{th} < \bar{m}_b^2/\mu^c_{th}$.