= Momentum-cutoff two-point function in six-dimensional cubic scalar theory
{title2=$\widehat F_2^{(1)}(p)=\frac{g^2}{2}\int\frac{d^6k}{(2\pi)^6(k^2+m^2)((k+p)^2+m^2)}$}
The <self-energy> <bubble diagram> at <loop order> one in <six-dimensional cubic scalar field theory> has <Feynman-diagram symmetry factor> two. With positive Euclidean <scalar propagators>, its <amputated Green's function> is the displayed integral. Its <superficial degree of divergence> is two. Expanding at large internal momentum and taking the angular average shows that its divergent part is a constant plus a multiple of $p^2$, precisely the <mass counterterm> and <wave-function renormalization> structures.
For a spherical <ultraviolet cutoff> $|k|\leq\Lambda$ and $m^2>0$, its value at $p=0$ is
$$
B_\Lambda=\frac{g^2}{256\pi^3}\left[\Lambda^2-2m^2\log\left(1+\frac{\Lambda^2}{m^2}\right)+\frac{m^2\Lambda^2}{\Lambda^2+m^2}\right].
$$
The divergent $p^2$ coefficient is $-g^2\log(\Lambda^2/m^2)/(768\pi^3)$. Finite terms depend on the <regularization> convention. <Tadpole diagram> insertions are excluded from the <one-particle-irreducible Feynman diagram>, and are removed from the connected <two-point function> by imposing a vanishing <one-point function>.
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