= Solution
Use <metric signature> $(+,-,\ldots,-)$ and initially set $\hbar=1$. For an internal scalar line with <momentum> $p$, the <Feynman propagator> is $i/(p^2-m^2+i0)$. An $r$-leg <Feynman vertex>, for $r=3,4,6$, contributes $-i\lambda_r$, with all incident momenta incoming. Its $1/r!$ in the <Lagrangian density> is canceled by the permutations of its identical fields. Each vertex conserves <momentum> and supplies $(2\pi)^d\delta^{(d)}(\sum p)$. Integrate each independent <loop momentum> with $d^dk/(2\pi)^d$, multiply the line and vertex factors, and divide by the <Feynman-diagram symmetry factor> for labeled external legs. Sum the distinct graphs and external-momentum assignments without counting the same assignment twice.
An unamputated momentum-space <correlation function> also contains the external <propagators>. Dividing the <generating functional> by its zero-source value removes disconnected vacuum bubbles; taking its logarithm selects <connected correlation functions>. <Amputation of external propagators> removes the external line factors when computing vertex amplitudes. Translational invariance leaves one overall momentum-conservation delta function for each connected graph.
The <loop order> is the number of independent graph cycles, equivalently the number of independent momentum integrations left after vertex conservation. A spanning tree through $V$ interaction vertices contains $V-1$ internal edges; hence, with $I$ internal lines,
$$
\boxed{L=I-V+1.}
$$
Restoring $\hbar$ in $\exp(iS/\hbar)$ gives $\hbar$ for each internal <propagator> and $\hbar^{-1}$ for each interaction vertex. Thus an amputated connected amplitude has
$$
\boxed{\hbar^{I-V}=\hbar^{L-1}.}
$$
The same counting holds for a diagram in $\log Z[J]$ when the <external source> is coupled as $J\phi/\hbar$: every external propagator brings $\hbar$, canceled by the source insertion's $\hbar^{-1}$. This is <Planck-constant loop counting with source normalization>. An ordinary connected expectation of $n$ fields, with no such source factors, instead scales as $\hbar^{L+n-1}$; multiplying $\log Z$ by $-i\hbar$ to form $W$ shifts its source-kernel counting to $\hbar^L$. The normalization must be specified when applying the printed loop rule.
Let $V_r$ count $r$-valent interaction vertices. Counting their incident half-edges gives $2I+n=3V_3+4V_4+6V_6$. Substituting the loop identity yields
$$
L-1+\frac n2=\frac12V_3+V_4+2V_6,
\qquad V=V_3+V_4+V_6.
$$
Each coefficient on the right lies between $1/2$ and $2$, proving the <vertex-count bounds for bounded scalar valence>
$$
\boxed{\frac V2\le L-1+\frac n2\le2V.}
$$
The free two-point term, which has no interaction vertex, separately satisfies the bounds with $V=L=0$ and $n=2$.
A <one-particle-irreducible Feynman diagram> is connected and remains connected after any single internal line is cut. The left graph below contains two quartic vertices and two parallel internal lines: cutting either line leaves the vertices connected by the other. The right graph contains two cubic vertices joined by one internal line, whose removal disconnects it, so it is a <one-particle-reducible Feynman diagram>. Both have four external legs.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-65-four-point-diagrams.png]
{title=One-particle-irreducible quartic loop and one-particle-reducible cubic exchange with four external legs}
The <quantum effective action> generates the <one-particle-irreducible vertices>. Their ultraviolet divergences and those of their proper divergent subgraphs are removed by local <counterterms>. Reducible connected graphs are reconstructed by joining these vertices with full <propagators>; they do not introduce independent local interaction counterterms merely because they contain bridges. This makes the <one-particle-irreducible Feynman diagram> decomposition useful for organizing <renormalization>. Overall <power counting> still has to be supplemented by subtraction of divergent subgraphs.
For a scalar with a two-derivative kinetic term, <mass dimensions> obey
$$
[\phi]=\frac{d-2}{2},\qquad
[\lambda_r]=d-r\frac{d-2}{2}.
$$
The general power-counting criterion used here is that interactions of nonnegative coupling dimension can be included in a perturbatively <renormalizable quantum field theory>, with all symmetry-allowed required local counterterms included; negative-dimensional couplings generally require an unbounded tower of higher-dimensional counterterms. Indeed, the <scalar graph power-counting identity> gives
$$
D=d-\frac{d-2}{2}n-\sum_{r=3,4,6}[\lambda_r]V_r
$$
for the <superficial degree of divergence>. Nonnegative coupling dimensions prevent this degree from increasing indefinitely with the vertex count.
The coupling dimensions are
$$
\begin{array}{c|ccc|c}
d&[\lambda_3]&[\lambda_4]&[\lambda_6]&\text{allowed displayed interactions}\\\hline
3&3/2&1&0&\phi^3,\phi^4,\phi^6\\
4&1&0&-2&\phi^3,\phi^4\\
6&0&-2&-6&\phi^3
\end{array}
$$
Thus \b[all three are power-counting renormalizable in three dimensions; cubic and quartic in four; cubic only in six]. Positive-dimensional interactions are super-renormalizable, while the zero-dimensional ones are marginal by classical <power counting>. Usual kinetic, mass, vacuum and, when no symmetry forbids it, linear counterterms must also be admitted. This conclusion concerns perturbative ultraviolet <renormalizability>; it does not assert global stability of a cubic scalar potential.
Back to article page
