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For Phi-six theory, let $D_0=D_F(0)$ and $D_{xy}=D_F(x-y)$. The Dyson series givesAt one vertex, Wick theorem supplies $5!!=15$ pairings. At two vertices let $r$ be the number of propagators joining them. It must be $0,2,4,$ or $6$, and the number of contractions isThereforeThe four Vacuum Feynman diagram types are shown below. The $r=0$ term is two disconnected copies of the order-$\lambda$ three-tadpole graph; the other three are connected.

Defineand let $B_2$ be the sum of the $r=2,4,6$ terms in the second line. The disconnected $r=0$ contribution is exactly $B_1^2/2$. Hencewhich is the linked-cluster theorem: the logarithm of the vacuum amplitude is the sum of connected vacuum bubbles.

The additional six-leg interaction vertex has Feynman rule $-i{g}_6$. Since both interactions lack derivatives, the graph identity still gives $\omega =2(1-V)$, with $V=V_4+V_6$. The difference is that a one-vertex contraction of the phi-six theory can leave four external legs: choosing the contracted pair gives $\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$. Its quartic interaction correction isConsequently a one-loop quartic interaction counterterm is now necessary:The one-loop mass counterterm from c is unchanged. Contracting four legs of a single six-leg interaction vertex gives $\left(\genfrac{}{}{0ex}{}{6}{2}\right)3=45$ double contractions, hence a two-loop raw self-energy $g_6{T}_{\Lambda }^2/8$. Its subtraction must include the quartic interaction counterterm insertion, rather than subtracting this raw expression in isolation.

All divergent field-dependent coefficients are most simply organized by the normal-ordering subtraction scheme in two dimensions. With $T=T_{\Lambda }$, express a potential normal ordered at mass $m$ in ordinary fields:Thus its divergent subtraction coefficients, in this scheme, areIndeed the two-loop self-energy terms are $g_6{T}^2/8+\delta g_{4}T/2=-g_6{T}^2/8$, which the displayed mass counterterm cancels. The constant terms only affect vacuum energy. After removing all one-vertex self-contractions, every remaining subgraph is ultraviolet finite. Additional finite on-shell renormalization scheme adjustments are subject to the same qualification as d.

Phi-six theory has scalar interaction $-\lambda {\varphi }^6/6!$. In four spacetime dimensions its coupling has mass dimension $-2$, and a six-valent vertex contributes the momentum-space factor $-i\lambda$.