= Solution
Loop integrals require a subtraction prescription. Even when the classical <coupling constant (physics)> 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 (physics)> 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 $\phi_B=Z_\phi^{1/2}\phi_R$ and $G_{n,R}=Z_\phi^{-n/2}G_{n,B}$. Define
$$
\beta(g)=\left.\mu\frac{dg}{d\mu}\right|_{\mathrm{bare}},\qquad
\gamma_\phi(g)=\left.\frac12\mu\frac{d\log Z_\phi}{d\mu}\right|_{\mathrm{bare}}.
$$
Bare <correlation functions> do not depend on the arbitrary subtraction scale. Differentiating their relation to renormalized <correlation functions> gives the <Callan-Symanzik equation>
$$
\boxed{\left(\mu\partial_\mu+\beta(g)\partial_g+n\gamma_\phi(g)\right)G_{n,R}=0.}
$$
It 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 $\gamma_\phi$ at a fixed point; the critical-exponent notation $\eta$ often used for the <anomalous dimension> equals $2\gamma_\phi$.
A <renormalization-group fixed point> satisfies $\beta(g_*)=0$. It is ultraviolet attractive if the flow approaches it as $\mu\to\infty$, and infrared attractive if the flow approaches it as $\mu\to0$. For a simple isolated zero, $\delta g\propto\mu^{\beta'(g_*)}$: 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 $z=p^2/\mu^2>0$, $t=\tfrac12\log z$, and write the dimensionless <quantum field theory propagator> factor as $d(z,g)$. The two-point <Callan-Symanzik equation> becomes
$$
\left(-\partial_t+\beta(g)\partial_g+2\gamma_\phi(g)\right)d(e^{2t},g)=0.
$$
Let $\bar g(0)=g$ and $d\bar g/dt=\beta(\bar g)$. The <renormalization-group characteristic solution for a two-point function> is
$$
\boxed{d(e^{2t},g)=d(1,\bar g(t))
\exp\left[2\int_0^t\gamma_\phi(\bar g(s))\,ds\right].}
$$
Where $\beta\ne0$, the exponent is $2\int_g^{\bar g(t)}\gamma_\phi(u)/\beta(u)\,du$. 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 (physics)> does not.
For $\beta(g)=-bg^3$ and $\gamma_\phi=cg^2$, $b>0$,
$$
\bar g(t)^2=\frac{g^2}{1+2bg^2t},\qquad
2\int_0^t c\bar g(s)^2\,ds=\frac cb\log(1+2bg^2t).
$$
Consequently
$$
\boxed{d(z,g)=d(1,\bar g(\tfrac12\log z))\,[1+bg^2\log z]^{c/b}.}
$$
For large positive $z$, the <coupling constant (physics)> tends to zero. If the normalization function $d(1,g)$ has a finite nonzero free-field limit $d(1,0)$, then $d(z,g)\sim d(1,0)[bg^2\log z]^{c/b}$. Equivalently, with $\Lambda=\mu\exp[-1/(2bg^2)]$,
$$
\bar g^2=\frac1{b\log(p^2/\Lambda^2)},\qquad
d(z,g)\sim d(1,0)\left[\frac{\log(p^2/\Lambda^2)}{\log(\mu^2/\Lambda^2)}\right]^{c/b}.
$$
The positive exponent follows from the sign of $+n\gamma_\phi$ 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 (physics)>. Then $\beta(g)=g^3(B-Ag^2)$ has a nonzero fixed point
$$
g_*^2=\frac BA,\qquad \beta'(g_*)=-\frac{2B^2}{A}<0.
$$
Every nonzero positive <running coupling> in the perturbative basin approaches this \b[<ultraviolet fixed point>], rather than approaching zero. The zero <coupling constant (physics)> is infrared attractive. If the same $\gamma_\phi=cg^2$ 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 $\gamma_\phi(g_*)$. A fixed point at a large <coupling constant (physics)> 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,
$$
\frac{da}{dL}=-\beta_0a^2,\qquad
\boxed{a(\mu^2)=\frac1{\beta_0\log(\mu^2/\Lambda^2)}.}
$$
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,
$$
a=\frac1{\beta_0\mathcal L}
\left[1-\frac{\beta_1}{\beta_0^2}\frac{\log\mathcal L}{\mathcal L}+O(\mathcal L^{-1})\right].
$$
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 become
$$
\beta_0=\frac23\delta,\qquad
\beta_1=-\frac{25}{2}N^2+\frac{11}{2}
+\left(\frac{13}{3}N-\frac1N\right)\delta.
$$
Thus $\beta_1<0$ for sufficiently small $\delta/N$, and the two-loop flow has the <Banks-Zaks fixed point>
$$
\boxed{a_*=-\frac{\beta_0}{\beta_1}\simeq\frac{4\delta}{75N^2}.}
$$
Both $a_*$ and the loop-counting combination $Na_*$ are small. Its slope is $\beta'(a_*)=-\beta_0^2/\beta_1>0$, so it is \b[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 $a_*=1/302$. The fermion-count convention is the one encoded by the given coefficients.
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