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Renormalization introduces an arbitrary renormalization scale $\mu$ even though the classical massless theory has no dimensionful parameter. Independence of the bare correlation function from that auxiliary scale gives the Callan-Symanzik equationHereis the beta function and $\gamma (g)$ is the field anomalous dimension, with its sign fixed by the displayed equation. The beta function determines the running coupling. Its zeros are renormalization-group fixed points, where the theory can become scale invariant. A positive beta function makes a positive coupling increase toward larger $\mu$, while a negative one makes it decrease.

For the propagator coefficient $C$, follow a characteristic with $t=\log (\mu /{\mu }_0)$ and $dg/dt=\beta (g)$. The $n=2$ equation becomesIntegration giveswith

For $\beta (g)=-b{g}^3$ with $b>0$, the only real fixed point is $g^{\ast }=0$. It is ultraviolet-attractive: the theory is asymptotically free. Integrating the running equation givesorfor the branch with the same sign as $g({\mu }_0)$. The perturbative expression has an infrared Landau pole atand therefore

Finally set ${\mu }_0=p$ and use $\gamma (g)=c{g}^2$. The characteristic factor becomesIn terms of the dynamically generated scale,

For $b>0$, the theory is asymptotically free: it is weakly coupled at high energies and strongly coupled in the infrared. For $b<0$, the coupling decreases toward the infrared but grows toward a finite ultraviolet Landau pole; such a theory is infrared free and requires an ultraviolet completion before that pole.