Put $I_r=\int_{\Lambda<|q|<\Lambda_0}d^6q\,(2\pi)^{-6}(q^2+m^2)^{-r}$, the <scalar shell integral in six dimensions>. Each of the three boxes is $g^4I_4$ as a connected insertion. To fix the effective-action sign and avoid a factorial ambiguity, use the <one-loop scalar effective action>:
$$
\frac12\operatorname{Tr}\log(D+g\phi)=\frac12\operatorname{Tr}\log D+\frac12\sum_{r\ge1}\frac{(-1)^{r+1}g^r}{r}\operatorname{Tr}(D^{-1}\phi)^r,\qquad D=-\partial^2+m^2.
$$
For a constant background the fourth-order density is $-g^4I_4\phi^4/8$. Thus the <cubic scalar box contribution to a quartic coupling> is $\delta g_4=-3g^4I_4$, whose vertex insertion is $-\delta g_4=+3g^4I_4$.

Using the supplied area of the unit five-sphere,
$$
I_4=\frac1{64\pi^3}\int_\Lambda^{\Lambda_0}\frac{q^5\,dq}{(q^2+m^2)^4}=\frac1{128\pi^3}\left[G(\Lambda^2+m^2)-G(\Lambda_0^2+m^2)\right],
$$
where $G(u)=u^{-1}-m^2u^{-2}+m^4/(3u^3)$. This follows by substituting $u=q^2+m^2$ and integrating $(u-m^2)^2/(2u^4)$. Hence
$$
\boxed{\delta g_4=-\frac{3g^4}{128\pi^3}\left[G(\Lambda^2+m^2)-G(\Lambda_0^2+m^2)\right].}
$$
As a check, the massless limit at positive $\Lambda$ is $-3g^4(\Lambda^{-2}-\Lambda_0^{-2})/(128\pi^3)$. Both versions have <mass dimension> minus two, as a six-dimensional local quartic coupling must.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 304 / 3 / iii / Solution

{title2=$\delta g_4=-3g^4I_4$}

In a <six-dimensional cubic scalar field theory>, three labelled <box Feynman diagrams> give the one-loop local quartic vertex at zero external momentum. With an effective-action term $g_4\phi^4/4!$, their contribution is $\delta g_4=-3g^4I_4$. To check its sign and multiplicity, expand the <one-loop scalar effective action> as $\tfrac12\operatorname{Tr}\log(D+g\phi)$, where $D=-\partial^2+m^2$. Its fourth-order term is $-g^4\operatorname{Tr}(D^{-1}\phi)^4/8$, giving the stated coefficient. The associated amputated connected diagram insertion has the opposite sign.

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 Incoming links: Cubic scalar box contribution to a quartic coupling