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Measurements which probe the energy dependence of $alpha_s$, the coupling strength of the strong interaction, are reviewed. Jet counting in $e^+ e^-$ annihilation, combining results obtained in the centre of mass energy range from 22 to 133 GeV, provides direct evidence for an asymptotically free coupling, without the need to determine explicit values of $alpha_s$. Recent results from jet production in $e p$ and in $p overline{p}$ collisions, obtained in single experiments spanning large ranges of momentum transfer, $Q^2$, are in good agreement with the running of $alpha_s$ as predicted by QCD. Mass spectra of hadronic decays of $tau$-leptons are analysed to probe the running $alpha_s$ in the very low energy domain, $0.7 GeV^2 < Q^2 < M_tau^2$. An update of the world summary of measurements of $alpha_s(Q^2)$ consistently proves the energy dependence of $alpha_s$ and results in a combined average of $alpha_s(M_Z) = 0.118 pm 0.006$.
We provide strong evidence that the asymptotically free (1+1)-dimensional non-linear O(3) sigma model can be regularized using a quantum lattice Hamiltonian, referred to as the Heisenberg-comb, that acts on a Hilbert space with only two qubits per sp
Based on the spontaneous breaking of chiral symmetry, chiral perturbation theory (ChPT) is believed to approximate confinement scale QCD. Dedicated and increasingly accurate experiments and improving lattice calculations are confirming this belief, a
We study four-dimensional gauge theories coupled to fermions in the fundamental and meson-like scalars. All requisite beta functions are provided for general gauge group and fermion representation. In the regime where asymptotic freedom is absent, we
An effective field theory model of the massive Yang-Mills theory is considered. Assuming that the renormalized coupling constants of non-renormalizable interactions are suppressed by a large scale parameter it is shown that in analogy to the non-abel
${Z}_2$-Yukawa-QCD models are a minimalistic model class with a Yukawa and a QCD-like gauge sector that exhibits a regime with asymptotic freedom in all its marginal couplings in standard perturbation theory. We discover the existence of further asym