 Quartic complex-scalar mass renormalization

For $\mathcal{L}=|\partial \varphi {|}^2-m^2|\varphi {|}^2-{\mu }^{ϵ}g|\varphi {|}^4/4$ in $d=4-ϵ$, the directed tadpole diagram has symmetry factor one. Its divergent Feynman contribution to $i{\Gamma }^{(2)}$ is $ig{m}^2/(8{\pi }^2ϵ)$. A mass insertion $-i{\delta }_{m^2}$ therefore gives the displayed minimal subtraction scheme counterterm. Differentiating at fixed bare mass and coupling gives the displayed ${\gamma }_m=\frac{1}{2}\mu d\log m^2/d\mu$. The equivalent two-real-field formulation has vertex coefficients $3g/2$ for a pure species and $g/2$ for two of each, giving total tadpole weight $(3g/2+g/2)/2=g$.