An asymptotically free theory has a running coupling that tends to zero at arbitrarily high energy. Its negative weak-coupling beta function makes short-distance perturbation theory increasingly accurate.
A non-Abelian gauge theory can have a weak infrared fixed point when the one-loop coefficient of its renormalization-group beta function is small and positive while the two-loop coefficient is negative. For and evolution in , has the displayed zero and slope . Fundamental-fermion flavor number just below the asymptotic-freedom boundary produces this regime.
For and , integrate the running coupling equation and then the anomalous-dimension multiplier. The correlator correction grows as a power of the logarithm of energy, rather than as a fixed power of energy. Higher perturbative orders change subleading terms while retaining this leading logarithmic exponent.
The strong-coupling scale of an asymptotically free gauge theory is the renormalization-group invariant scale at which the one-loop running coupling formally diverges. It replaces a dimensionless ultraviolet coupling through dimensional transmutation.
Dimensional transmutation replaces a dimensionless coupling by a dimensionful renormalization-group invariant scale. In an asymptotically free theory, specifying the coupling at one scale is equivalent to specifying its strong-coupling scale.
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Asymptotic freedom is a property of some gauge theories, particularly quantum chromodynamics (QCD), which is the theory describing the strong interaction—the force that binds quarks and gluons into protons, neutrons, and other hadrons. The concept refers to the behavior of the coupling constant (which measures the strength of the interaction) as the energy scale of the interaction changes.