= Solution
In units $\hbar=c=1$, a dimensionless <action> fixes $[\mathcal L]=4$. Canonical <kinetic terms> then give $[\phi]=[A_\mu]=1$, $[\psi]=3/2$ and $[\partial_\mu]=1$. If an interaction operator has dimension $d_v$, its coefficient has <mass dimension> $[g_v]=4-d_v$. Thus scalar quartic, Yukawa and <gauge couplings> are dimensionless; a scalar cubic coupling has dimension one, a four-fermion coupling has dimension minus two, and the Weinberg coefficient has dimension minus one. These are engineering dimensions; anomalous scaling is an additional quantum effect.
For a connected graph with $L$ independent loop momenta, $I_B$ <boson> lines, $I_F$ <fermion> lines and $r_v$ <derivatives> at vertex $v$, ordinary covariant-gauge ultraviolet counting gives the <superficial degree of divergence>
$$
D=4L-2I_B-I_F+\sum_vr_v
=4-E_B-\frac32E_F-\sum_v[g_v].
$$
The second form follows from $L=I_B+I_F-V+1$ and counting field ends at vertices. It presumes the canonical large-momentum propagators; unitary-gauge vector propagators can obscure the cancellations of a renormalizable gauge theory. A nonnegative $D$ signals a possible overall <ultraviolet divergence>, not proof of one. Symmetries can cancel it, while a graph with $D<0$ can still contain divergent subgraphs.
With only operators of dimension at most four, increasing the vertex count does not require <counterterms> of unbounded dimension. Subject to the necessary symmetry identities and absence of gauge anomalies, scalar quartic theory, Yukawa theories and the <Standard Model> are perturbatively renormalizable when all required interactions are included. In particular, coupling a scalar to other fields can require a scalar quartic <counterterm> even if its coefficient was initially set to zero. The <Higgs mechanism> does not destroy renormalizability of the underlying anomaly-free gauge theory.
By contrast, repeated insertions of the dimension-six <Fermi interaction> or a dimension-five operator allow increasingly high-dimensional <counterterms>. Such terms are useful in an <effective field theory> with a stated expansion in <energy> over a high scale; nonrenormalizable does not mean unpredictive at a fixed order. A scalar $\phi^6$ interaction and perturbative Einstein gravity in four dimensions provide other examples. Renormalizability also does not assert small coupling at every scale or a complete ultraviolet theory.
Back to article page
