Determination of the QCD Coupling $α_s$

TL;DR

This comprehensive review consolidates experimental and theoretical determinations of the QCD coupling α_s across DIS, e+e− annihilation, hadron colliders, heavy quarkonia, and lattice studies, emphasizing NNLO perturbative QCD and renormalization-group running. By combining diverse measurements with careful treatment of correlations and scheme dependencies, it yields a world average α_s(M_Z) = 0.1184 ± 0.0031 and demonstrates the predicted logarithmic running of α_s, consistent with Nc ≈ 3. The work highlights both the remarkable agreement with QCD predictions and the remaining theoretical and experimental challenges, including higher-order calculations, improved parton densities, and nonperturbative effects such as power corrections. Overall, it solidifies α_s as a precisely constrained fundamental parameter and a stringent test of the QCD framework.

Abstract

Theoretical basics and experimental determinations of the coupling parameter of the Strong Interaction, $α_s$, are reviewed. The world average value of $α_s$, expressed at the energy scale of the rest mass of the $Z^0$ boson, is determined from analyses which are based on complete NNLO perturbative QCD. The result is $α_s (M_Z) = 0.1184 \pm 0.0031$. No significant deviations or systematic biases of subsamples of experimental results are found. From the observed energy dependence of $α_s$, which is in excellent agreement with the expectations of QCD, the number of colour degrees of freedom can be constrained to $N_c = 3.03 \pm 0.12$.

Figures (12)

• Figure 1: Examples of Feynman diagrams describing hadronic final states in processes which are used to measure $\alpha_{\rm s}$.
• Figure 2: (a) The running of $\alpha_{\rm s} (Q)$, according to equation \ref{['eq-as4loop']}, in 1-, 2- and 3-loop approximation, for $N_f = 5$ and with an identical value of $\Lambda_{\overline{\hbox{\scriptsize MS}}} = 0.22$ GeV. The 4-loop prediction is indistinguishable from the 3-loop curve. (b) Fractional difference between the 4-loop and the 1-, 2- and 3-loop presentations of $\alpha_{\rm s} (Q)$, for $N_f = 5$ and $\Lambda_{\overline{\hbox{\scriptsize MS}}}$ chosen such that, in each order, $\alpha_{\rm s}(M_{\rm Z^0}) = 0.119$.
• Figure 3: (a) 4-loop running of $\alpha_{\rm s} (Q)$ with 3-loop quark threshold matching according to equations \ref{['eq-as4loop']} and \ref{['Mq-matching']}, with $\Lambda_{\overline{\hbox{\scriptsize MS}}}^{(N_f = 5)}$ = 220 MeV and charm- and bottom-quark thresholds at the pole masses, $\mu^{(N_f = 4)}_c \equiv M_c = 1.5$ GeV and $\mu^{(N_f = 5)}_b \equiv M_b = 4.7$ GeV (full line), compared with the unmatched 4-loop result (dashed line). (b) The fractional difference between the two curves in (a).
• Figure 4: (a) $\alpha_{\rm s}(M_{\rm Z^0})$ determined from the scaled hadronic width of the $\rm Z^0$, $R_{\rm Z} = 20.768$, in leading, next-to-leading and in next-to-next-to leading order QCD, as a function of the renormalization scale factor $x_{\mu} = \mu / M_{\rm Z^0}$. NLO solutions according to the PMS ($\bullet$), the EC ($\circ$) and the BLM ($\diamond$) scale optimization methods are marked. (b) Scale dependence of the QCD coefficients $R_n$ of $R_{\rm Z}$.
• Figure 5: Comparison of CCFR data ccfr with NNLO QCD fits xf3-recent.