= Solution
A <renormalizable quantum field theory> requires only finitely many independent field and parameter redefinitions to remove regulator dependence from its perturbative <correlation functions>, order by order. The allowed local <counterterms> must close within that finite set; it is not necessary that the unrenormalized integrals be finite.
In four spacetime dimensions the <kinetic term> fixes the <mass dimension> of $\phi$ to one. A coefficient multiplying $\phi^k$ therefore has dimension $4-k$. <Power counting> excludes couplings of negative <mass dimension> for perturbative renormalizability. For a single scalar with a conventional nonderivative <polynomial> potential, the allowed form is
$$
\boxed{V(\phi)=v_0+h\phi+\frac12m^2\phi^2+\frac{\lambda_3}{3!}\phi^3+\frac{\lambda_4}{4!}\phi^4.}
$$
The constant controls vacuum energy and the linear term a possible tadpole. A field-reflection symmetry can remove the odd terms but is not necessary for renormalizability. Terms of degree above four generally require an infinite tower of <counterterms> when treated as fundamental interactions; they can instead be used in an <effective field theory>. Stability of the potential is a separate requirement, for example a positive quartic leading term, and should not be confused with the power-counting criterion.
For a graph with $E$ external scalar lines, $I$ internal lines, $L$ loops and $V_k$ vertices of degree $k$, the superficial <ultraviolet divergence> degree is $\omega=4L-2I$. The identities $L=I-\sum_kV_k+1$ and $2I+E=\sum_k kV_k$ give
$$
\omega=4-E+\sum_k(k-4)V_k.
$$
When every $k\le4$, only the finitely many low-point local divergences, including the <kinetic term>, can require subtraction. Subdivergences are removed recursively by the same local <counterterms>. A vertex with $k>4$ instead permits arbitrarily high divergence degrees as more such vertices are added. This explains the restriction rather than merely asserting it.
The bare <Lagrangian density> is written in terms of regulator-dependent bare quantities, before taking the regulator away:
$$
\mathcal L_B=-\frac12(\partial\phi_B)^2-V_B(\phi_B),\qquad\phi_B=Z_\phi^{1/2}\phi,\qquad\mathcal L_B=\mathcal L_R+\mathcal L_{\mathrm{ct}}.
$$
The coefficients in $V_B$ are related to the <renormalized> parameters by regulator-dependent redefinitions. The <wave-function renormalization> supplies a kinetic <counterterm>. In <dimensional regularization> $d=4-\epsilon$, a quartic coupling has a bare relation of the form $\lambda_B=\mu^\epsilon[\lambda+\delta\lambda(\lambda,\epsilon)]$, with field factors incorporated according to the chosen definition. The bare parameters and bare <correlation functions> are held fixed when changing the <renormalization scale>.
Even without a physical <mass>, interacting loop subtractions require a reference scale $\mu$ to define the <renormalized> dimensionless coupling and field normalization. Finite logarithms then involve dimensionless combinations such as $\mu|x_r-x_s|$ or $p^2/\mu^2$. This scale is a subtraction convention, not a new physical <mass> parameter. Its explicit dependence is compensated by the <running coupling> and field normalization.
Let $G_n=\langle\phi(x_1)\cdots\phi(x_n)\rangle$ at separated points. Its bare counterpart satisfies $G_{B,n}=Z_\phi^{n/2}G_n$. Define
$$
\beta(g)=\left.\mu\frac{dg}{d\mu}\right|_{\mathrm{bare}},\qquad\gamma(g)=\left.\frac12\mu\frac{d\log Z_\phi}{d\mu}\right|_{\mathrm{bare}}.
$$
Differentiating $G_{B,n}$ at fixed bare theory gives
$$
\boxed{\left(\mu\partial_\mu+\beta(g)\partial_g+n\gamma(g)\right)G_n=0.}
$$
This <Callan-Symanzik equation> expresses independence of the arbitrary subtraction scale. The <renormalization-group beta function> describes the compensating coupling change; the <field-renormalization anomalous dimension> describes the compensating rescaling of each insertion. Coincident composite insertions would require their own additional <renormalization>, rather than automatically obeying this separated-point equation.
For the two-point function, set $z=p^2/\mu^2$ and $t=\tfrac12\log z$. We consider large positive spacelike $p^2$, with continuation to other momenta as required. The dimensionless <propagator> factor obeys
$$
(-\partial_t+\beta\partial_g+2\gamma)d(e^{2t},g)=0.
$$
Define $\bar g(0)=g$ and $d\bar g/dt=\beta(\bar g)$. The <renormalization-group characteristic solution for a two-point function> is
$$
\boxed{d(e^{2t},g)=d(1,\bar g(t))\exp\left\{2\int_0^t\gamma(\bar g(s))\,ds\right\}.}
$$
To verify it, the running-coupling flow has $\beta(g)\partial_g\bar g(t)=\beta(\bar g(t))$. Also, $\beta(g)\partial_g\int_0^t\gamma(\bar g(s))ds=\gamma(\bar g(t))-\gamma(g)$. Applying $\partial_t-\beta(g)\partial_g$ to the displayed expression therefore gives $2\gamma(g)d$, exactly the required differential equation. The boundary value $d(1,g)$ records the <renormalization> condition. This shows how the ultraviolet behaviour is determined by the coupling trajectory and its accumulated <anomalous dimension>: it can approach zero coupling, a nonzero fixed point, or a finite-scale singularity, depending on $\beta$.
For $\beta=-bg^3$ with $b>0$, direct integration gives
$$
\bar g(t)^2=\frac{g^2}{1+2bg^2t},\qquad2\int_0^t c\bar g(s)^2\,ds=\frac cb\log(1+2bg^2t).
$$
Consequently
$$
\boxed{d(z,g)=d\left(1,\frac{g}{\sqrt{1+bg^2\log z}}\right)\left(1+bg^2\log z\right)^{c/b}.}
$$
For a regular free-theory normalization $d(1,0)=1$, the leading large-$z$ form is $(1+bg^2\log z)^{c/b}$, times corrections from the small <running coupling>. This is <asymptotic freedom> with a logarithmic, rather than a fixed-power, <propagator> correction. The hypothetical beta and gamma functions are being used as given; this calculation does not assert that they are the actual beta and gamma functions of an arbitrary quartic scalar theory.
For the second beta function put $B=-b>0$, so $\beta(g)=Bg^3-ag^5$. Its positive nonzero fixed point and linearized slope are
$$
\boxed{g_*^2=\frac{B}{a}=-\frac ba,\qquad\beta^{\prime}(g_*)=-\frac{2B^2}{a}<0.}
$$
For $0<g<g_*$ the coupling increases with momentum, while for $g>g_*$ it decreases; each positive trajectory approaches $g_*$ in the ultraviolet. Negative initial couplings approach $-g_*$ by the odd symmetry of the beta function, with the same limiting squared coupling. The exactly zero trajectory remains zero. Thus the <nonzero ultraviolet fixed point of a cubic-quintic beta function> replaces asymptotic freedom by a finite-coupling ultraviolet limit, with $\bar g(t)-g_*=O(e^{-2B^2t/a})$. If the <anomalous dimension> is regular there, the <propagator> factor has fixed-point behaviour $d(z,g)\sim A(g)z^{\gamma(g_*)}$ when its fixed-point boundary value is nonzero. If $\gamma(g)=cg^2$ is retained, the exponent is $\gamma(g_*)=-cb/a$. Trusting this perturbative truncation as a statement about the underlying theory requires $g_*$ to be sufficiently small; the flow result itself follows from the specified beta function.
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