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Take the following bounded analytic example for a dimensionless positive coupling $g$, with fixed $ϵ>0$ and $g(0)=g_0>0$:Increasing $\ell$ means increasing observation length $L$, hence motion toward the infrared fixed point. This convention matters: if the inverse-length scale is $\mu ={\mu }_0{e}^{-\ell }$, the same trajectory has renormalization-group beta function ${\beta }_{\mu }(g)=\mu dg/d\mu =-ϵg+g^2$. The two derivatives have opposite signs.

To solve the equation, set $v=1/g$. It obeys $dv/d\ell =-ϵv+1$, a linear ordinary differential equation. ThereforeFor $\ell \ge 0$ its denominator is positive, and $g$ stays between $g_0$ and $ϵ$. Thus the trajectory is bounded on this domain. The renormalization-group fixed points are $g=0$ and $g_{\ast }=ϵ$. The derivative of the length-flow right-hand side is $ϵ-2g$: zero is repulsive from positive $g$, whereas $g_{\ast }$ has eigenvalue $-ϵ$ and attracts positive trajectories. Linearization gives $g-g_{\ast }\propto e^{-ϵ\ell }=b^{-ϵ}$ near $g_{\ast }$. The inverse-length beta function has positive derivative at the same infrared-attractive point.

For the chosen $ϵ=1/5$, take two starting models with $g_0=1/20$ and $g_0=3/5$. At $b=1024$, $e^{-ϵ\ell }=1/4$, so their couplings are respectively $4/35$ and $6/25$. Both move toward $1/5$, from opposite sides. They are still visibly different: slow decay of an irrelevant operator can leave substantial corrections at accessible lengths. The case $g_0=0$ stays exactly zero and is outside this positive-coupling basin.

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 $a=g^2/(16{\pi }^2)$ and evolution in $\log {\mu }^2$, $\beta (a)=-{\beta }_0{a}^2-{\beta }_1{a}^3$ has the displayed zero and slope $-{\beta }_0^2/{\beta }_1>0$. Fundamental-fermion flavor number just below the asymptotic-freedom boundary produces this regime.

For the magnetic theory,It ceases to be asymptotically free whenThis is precisely the lower edge of the electric conformal window. Inside the window both descriptions flow to the same interacting infrared fixed point; below it the magnetic variables provide a weakly coupled infrared description.