= Five-line banana integral
{title2=$B_5(p)$}
The <two-point function> <Feynman integral> with two <interaction vertices> joined by five parallel <scalar propagators> has four independent <loop momenta>:
$$
B_5(p)=\int\prod_{j=1}^4\frac{d^dk_j}{(2\pi)^d}
\frac1{\prod_{j=1}^4(k_j^2+m^2)\,[(p-\sum_jk_j)^2+m^2]}.
$$
This formula uses Euclidean <momenta>. In three dimensions the <convolution theorem> and $G(x)=e^{-m|x|}/(4\pi|x|)$ give
$$
B_5(p)=\frac1{256\pi^4}\int_0^\infty\rho^{-3}e^{-5m\rho}\frac{\sin(|p|\rho)}{|p|\rho}\,d\rho,
$$
with <cutoff regularization> required at $\rho=0$. With a lower radial cutoff $a$, the coefficient of $p^2$ has <ultraviolet divergence> $-\log(1/(ma))/(1536\pi^4)$. It therefore generates a divergent <wave-function renormalization> in three-dimensional sextic <scalar field theory>. By contrast the three-line <sunset diagram> has only a momentum-independent logarithmic <ultraviolet divergence> in three dimensions. This illustrates why a <superficial degree of divergence> of two allows both a <mass counterterm> and a kinetic <counterterm>, while degree zero allows only the former in a <two-point function>.
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