= Solution
For positive $g$ in the perturbative regime, the sign of the <renormalization-group beta function> fixes the direction of flow. If $\beta_0>0$, then $g$ decreases when the <renormalization scale> increases: \b[the theory exhibits <asymptotic freedom>]. Toward low scales the interaction grows stronger, and its one-loop approximation reaches an infrared <strong-coupling scale>. This perturbative divergence does not by itself prove <confinement>.
If $\beta_0<0$, $g$ increases toward the ultraviolet and decreases toward the infrared: \b[the one-loop flow is infrared free and has an ultraviolet <Landau pole>]. Indeed
$$
\frac1{g^2(\mu)}=\frac1{g^2(\mu_0)}+\frac{\beta_0}{8\pi^2}\ln\frac{\mu}{\mu_0},
$$
whose right side reaches zero at a finite ultraviolet scale for negative $\beta_0$. Beyond strong coupling this formula is not controlled. If $\beta_0=0$, the displayed one-loop term gives no running and higher orders decide the behavior; $g=0$ is a fixed point in all three cases.
Back to article page
