 Solution

Use the notation ${\Delta }_{12}=\Delta (y_1-y_2)$, ${\Delta }_{xz}=\Delta (x-z)$ and ${\int }_{xz}=\int d^{D}x{d}^{D}z$. The order-$\lambda$ term vanishes because it has five fields. At order ${\lambda }^2$, enumerate the eight-field Wick contractions, whose expansion denominator is $2!(3!{)}^2=72$.

If the two external fields contract with each other, the remaining vacuum diagrams are a pair of cubic vertices joined by three lines, with six Wick contractions, and a pair joined by one line with a tadpole diagram at each end, with nine Wick contractions. Define their combined integrated weight byThus $Z/Z_0=1+(-i\lambda {)}^2{V}_2+O({\lambda }^4)$. The numerator isThe three latter terms have respectively 36, 36 and 18 Wick contractions. Together with the six and nine Wick contractions, they exhaust all $7!!=105$ pairings.

Dividing by $Z/Z_0$ cancels the two vacuum diagrams, givingThe first correction is the two-line bubble diagram, with symmetry factor $S=2$. The second is the connected tadpole contribution to a cubic-theory two-point function: both external lines meet one vertex, joined by a single line to a second vertex bearing a tadpole diagram; its symmetry factor is also $S=2$. Interchanging the two integrated vertices is already included in the 36 contractions and gives no extra factor.

The final term is the disconnected Feynman diagram consisting of two separate one-point tadpole diagrams, with coefficient $1/4$. It survives cancellation of vacuum bubbles, since each component touches an external point. Indeed it is precisely the leading $⟨\varphi (y_1)⟩⟨\varphi (y_2)⟩$. It would disappear from the connected two-point correlation function, but the question defines the full correlation function. No normal ordering or tadpole subtraction of the interaction was specified. All terms are depicted in solution, and all spacetime integrals remain unevaluated.