Momentum-cutoff two-point function in six-dimensional cubic scalar theory (${\hat{F}}_2^{(1)}(p)=\frac{g^2}{2}\int \frac{d^{6}k}{(2\pi {)}^6(k^2+m^2)((k+p{)}^2+m^2)}$)

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 $|k|\le \Lambda$ and $m^2>0$, its value at $p=0$ isThe 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.