One-loop proper vertices of massless phi-fourth theory (${\hat{\tau }}_4^{(1)}=\frac{{\lambda }^2}{2}(f_s+f_t+f_u)$)

The interaction vertex $-i\lambda$ and scalar propagator $-i/(p^2-iϵ)$ give one four-point bubble diagram for each of three momentum channels, each with Feynman-diagram symmetry factor $1/2$. The two-point tadpole diagram is zero as a scaleless integral in dimensional regularization. If full proper vertices include the free quadratic kernel, ${\hat{\tau }}_2^{(0)}=-p^2$; the interaction self-energy convention instead starts at zero.