= Solution
Renormalization introduces an arbitrary <renormalization scale> $\mu$ even though the classical massless theory has no dimensionful parameter. Independence of the bare correlation function from that auxiliary scale gives the <Callan-Symanzik equation>
$$
\left(\mu\frac{\partial}{\partial\mu}+\beta(g)\frac{\partial}{\partial g}+n\gamma(g)\right)G^{(n)}=0.
$$
Here
$$
\beta(g)=\mu\frac{dg}{d\mu}\bigg|_{g_0}
$$
is the <beta function (physics)> and $\gamma(g)$ is the field <anomalous dimension>, with its sign fixed by the displayed equation. The beta function determines the <running coupling>. Its zeros are <renormalization-group fixed points>, where the theory can become scale invariant. A positive beta function makes a positive coupling increase toward larger $\mu$, while a negative one makes it decrease.
For the propagator coefficient $C$, follow a characteristic with $t=\log(\mu/\mu_0)$ and $dg/dt=\beta(g)$. The $n=2$ equation becomes
$$
\frac{d\log C}{dt}=-2\gamma(g(t)).
$$
Integration gives
$$
\boxed{C\left(\frac{p^2}{\mu^2},g(\mu)\right)
=f(\mu)C\left(\frac{p^2}{\mu_0^2},g(\mu_0)\right)},
$$
with
$$
\boxed{f(\mu)=e^{h(\mu)},
\qquad h(\mu)=-2\int_{g(\mu_0)}^{g(\mu)}\frac{\gamma(g)}{\beta(g)}\,dg}.
$$
For $\beta(g)=-bg^3$ with $b>0$, the only real fixed point is $g^*=0$. It is ultraviolet-attractive: the theory is <asymptotically free>. Integrating the running equation gives
$$
\frac1{g^2(\mu)}=\frac1{g^2(\mu_0)}+2b\log\frac\mu{\mu_0},
$$
or
$$
\boxed{g(\mu)=\frac{g(\mu_0)}
{\sqrt{1+2bg^2(\mu_0)\log(\mu/\mu_0)}}}
$$
for the branch with the same sign as $g(\mu_0)$. The perturbative expression has an infrared <Landau pole> at
$$
\Lambda=\mu_0\exp\left[-\frac1{2bg^2(\mu_0)}\right],
$$
and therefore
$$
\boxed{g^2(\mu)=\frac1{2b\log(\mu/\Lambda)}}.
$$
Finally set $\mu_0=p$ and use $\gamma(g)=cg^2$. The characteristic factor becomes
$$
\frac{C(p^2/\mu^2,g(\mu))}{C(1,g(p))}
=\exp\left[-2\int_{g(p)}^{g(\mu)}\frac{cg^2}{-bg^3}\,dg\right]
=\left(\frac{g(\mu)}{g(p)}\right)^{2c/b}.
$$
In terms of the dynamically generated scale,
$$
\boxed{\frac{C(p^2/\mu^2,g(\mu))}{C(1,g(p))}
=\left[\frac{\log(p/\Lambda)}{\log(\mu/\Lambda)}\right]^{c/b}}.
$$
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