 Solution

Set the external source to zero in solution and apply Wick's theorem. At a single quartic vertex, the four position insertions have three pairings; at a single sextic vertex, six insertions have $5!!=15$ pairings. ThusThermal time-translation invariance makes this $1-\lambda \beta g_0^2/8-\mu \beta g_0^3/48+\cdots$, with $g_0$ from solution. If desired the absolute corrections follow by multiplying by the free partition function $Z_0=[2\sinh (\beta \Omega /2){]}^{-1}$.

The first Vacuum Feynman diagram is one quartic vertex carrying two self-loops, with weight $-\lambda /8$ before time integration. The second is one sextic vertex carrying three self-loops, with weight $-\mu /48$. No disconnected subtraction is appropriate for $Z$ itself. The top row of the following figure shows these first-order terms and the conventions for solution.