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For $\beta =-b{g}^3$, ${\gamma }_{\varphi }=c{g}^2$, and $b>0$, one has ${\bar{g}}^2=g^2/(1+b{g}^2\log z)$. Integrating the field-renormalization anomalous dimension gives the displayed logarithmic factor. At large $z$, a nonzero free matching factor yields $d\sim C(\log z{)}^{c/b}$ in the printed RG sign convention.

For $A,B>0$, the nonzero zero of the renormalization-group beta function has slope ${\beta }^{\prime }(g_{\ast })=-2B^2/A<0$ and is ultraviolet attractive. The origin is infrared attractive. A fixed-point field-renormalization anomalous dimension gives a power-law quantum field theory propagator factor $d(z)\sim C{z}^{{\gamma }_{\varphi }(g_{\ast })}$. A weak fixed point is needed for control by a perturbative truncation.

Loop integrals require a subtraction prescription. Even when the classical coupling constant is dimensionless and there is no mass, specifying its renormalized value means specifying a momentum or length scale at which it is measured. Dimensionless logarithms then involve that renormalization scale $\mu$. Changing $\mu$ changes the renormalized coupling constant and wave-function renormalization while leaving bare quantities fixed. In a theory whose renormalization-group beta function and anomalous dimension both vanish, this scale dependence may disappear; the generic statement concerns the interacting renormalized theory.

Write ${\varphi }_B=Z_{\varphi }^{1/2}{\varphi }_R$ and $G_{n,R}=Z_{\varphi }^{-n/2}{G}_{n,B}$. DefineBare correlation functions do not depend on the arbitrary subtraction scale. Differentiating their relation to renormalized correlation functions gives the Callan-Symanzik equationIt expresses the compensation between explicit scale dependence, running coupling and wave-function renormalization. In this convention the field scaling dimension is its engineering dimension plus ${\gamma }_{\varphi }$ at a fixed point; the critical-exponent notation $\eta$ often used for the anomalous dimension equals $2{\gamma }_{\varphi }$.

A renormalization-group fixed point satisfies $\beta (g_{\ast })=0$. It is ultraviolet attractive if the flow approaches it as $\mu \to \infty$, and infrared attractive if the flow approaches it as $\mu \to 0$. For a simple isolated zero, $\delta g\propto {\mu }^{{\beta }^{\prime }(g_{\ast })}$: a negative derivative gives ultraviolet attraction, a positive derivative infrared attraction. Marginal zeros, such as a cubic renormalization-group beta function at the origin, require looking at the first nonzero nonlinear term.

Put $z=p^2/{\mu }^2>0$, $t=\frac{1}{2}\log z$, and write the dimensionless quantum field theory propagator factor as $d(z,g)$. The two-point Callan-Symanzik equation becomesLet $\bar{g}(0)=g$ and $d\bar{g}/dt=\beta (\bar{g})$. The renormalization-group characteristic solution for a two-point function isWhere $\beta \ne 0$, the exponent is $2{\int }_g^{\bar{g}(t)}{\gamma }_{\varphi }(u)/\beta (u)du$. Thus the ultraviolet behavior depends on where the running coupling goes and on the anomalous dimension accumulated along the flow. A weak ultraviolet fixed point or asymptotically free limit permits perturbative evaluation; a flow to large values of the coupling constant does not.

For $\beta (g)=-b{g}^3$ and ${\gamma }_{\varphi }=c{g}^2$, $b>0$,ConsequentlyFor large positive $z$, the coupling constant tends to zero. If the normalization function $d(1,g)$ has a finite nonzero free-field limit $d(1,0)$, then $d(z,g)\sim d(1,0)[b{g}^2\log z{]}^{c/b}$. Equivalently, with $\Lambda =\mu \exp [-1/(2b{g}^2)]$,The positive exponent follows from the sign of $+n{\gamma }_{\varphi }$ in the printed RG equation; changing the field-renormalization anomalous dimension convention would change both signs together.

For the quintic renormalization-group beta function, set $B=-b>0$ and rename its positive quintic coefficient $A$ to avoid confusion with the later gauge-theory coupling constant. Then $\beta (g)=g^3(B-A{g}^2)$ has a nonzero fixed pointEvery nonzero positive running coupling in the perturbative basin approaches this ultraviolet fixed point, rather than approaching zero. The zero coupling constant is infrared attractive. If the same ${\gamma }_{\varphi }=c{g}^2$ is retained, the large-momentum factor is $d(z,g)\sim C{z}^{cB/A}$, assuming a regular nonzero matching factor at the fixed point. More generally the exponent is ${\gamma }_{\varphi }(g_{\ast })$. A fixed point at a large coupling constant inferred from a truncated renormalization-group beta function is only a formal extrapolation; perturbative control requires $B/A$ small.

For the gauge theory use the supplied convention $a=g^2/(16{\pi }^2)$ and evolution variable $L=\log {\mu }^2$. Asymptotic freedom requires ${\beta }_0>0$, or $n<11N/2$. To one-loop accuracy,The integration constant $\Lambda$ is an example of dimensional transmutation. With two loops this is the leading asymptotic expression, not an exact equality: if $\mathcal{L}=\log ({\mu }^2/{\Lambda }^2)$ is large,The expansion presumes fixed coefficients and a scale high enough to enter the ultraviolet regime.

Choose $n=11N/2-\delta$ with $0<\delta \ll N$, respecting integer flavor counts. The supplied coefficients becomeThus ${\beta }_1<0$ for sufficiently small $\delta /N$, and the two-loop flow has the Banks-Zaks fixed pointBoth $a_{\ast }$ and the loop-counting combination $N{a}_{\ast }$ are small. Its slope is ${\beta }^{\prime }(a_{\ast })=-{\beta }_0^2/{\beta }_1>0$, so it is infrared attractive. For even large $N$, taking $n=11N/2-1$ provides an explicit integer sequence with the desired parametric hierarchy. As a finite example, $N=3,n=16$ gives ${\beta }_0=1/3$, ${\beta }_1=-302/3$ and $a_{\ast }=1/302$. The fermion-count convention is the one encoded by the given coefficients.

For $t=\log (|p|/\mu )$ and $d\bar{g}/dt=\beta (\bar{g})$, the displayed characteristic solution follows from $(-{\partial }_t+\beta {\partial }_g+2{\gamma }_{\varphi })d=0$. It separates the running coupling from the accumulated field-renormalization anomalous dimension. The exponent sign is determined by the precise Callan-Symanzik equation convention.