= Solution
Take integer $n\geq3$ and positive integer $d$, so the interaction is genuinely non-Gaussian. For a connected two-point <Feynman diagram> with $V$ <interaction vertices> and $I$ internal lines, counting half-edges and using the <loop order> gives
$$
nV=2I+2,\qquad L=I-V+1.
$$
At one loop $I=V$, hence $(n-2)V=2$. There are only two possibilities: a two-<interaction vertex> <bubble diagram> for $n=3$, or a one-<interaction vertex> <tadpole diagram> for $n=4$. The latter is independent of external momentum and changes the mass, not the <kinetic term>.
For <cubic scalar field theory>, introduce a positive auxiliary mass $\mu$ if needed to separate <infrared divergences> from <ultraviolet divergences>. The one-loop <self-energy> in the <quantum effective action> is $-\lambda^2 B_d(p)/2$, with
$$
B_d(p)=\int_0^1du\int\frac{d^dq}{(2\pi)^d}\frac{1}{[q^2+\mu^2+u(1-u)p^2]^2}.
$$
This <Feynman parameter> representation follows by shifting the <loop momentum>; it is directly valid for a translation-invariant regulator, or for the convergent subtracted integral. Differentiating at $p^2=0$ gives
$$
\left.\frac{d B_d}{dp^2}\right|_0=-\frac13\int\frac{d^dq}{(2\pi)^d}\frac{1}{(q^2+\mu^2)^3}.
$$
Thus the coefficient of the <kinetic term> changes by $\lambda^2\int d^dq/[(2\pi)^d(q^2+\mu^2)^3]/6$.
\b[Only the cubic interaction has a momentum-dependent one-loop two-point correction.]
$$
\boxed{n=3\quad\text{for every }d\geq1.}
$$
There is a terminology qualification: if “<wave-function renormalization>” means a required ultraviolet <counterterm>, rather than a finite correction to the field normalization, the large-$q$ radial integral behaves as $\int dq\,q^{d-7}$, and the answer is
$$
\boxed{n=3,\ d\geq6\quad\text{for an ultraviolet-divergent kinetic correction}.}
$$
It is logarithmically divergent at $d=6$ and power divergent above $6$ with a momentum cutoff. Among interactions with a <relevant coupling> or <marginal coupling>, the divergent case is uniquely $(n,d)=(3,6)$. The massless low-dimensional amplitude needs an <infrared divergence> prescription; one cannot take a local expansion at zero momentum without one. The $n=1,2$ cases, if admitted, are a source and a Gaussian mass term and have no interaction-induced <wave-function renormalization>.
Back to article page
