 A solvable infrared renormalization flow

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.