boils down to expanding the interaction term of the action in a Taylor series $$\label{Taylor}
\mathrm{e}^{- (g/4!) a^d \sum_x  \varphi^4_x} = 1 - \frac{g}{4!} 
a^d \sum_x  \varphi^4_x 
+ \frac{1}{2!} \left(\frac{g}{4!}\right)^2 a^{2d} \sum_{x,x'}  
\varphi^4_x \varphi^4_{x'} + \cdots .$$ The terms in this expansion can all be represented by Feynman diagrams.[@Amit; @KlSF] Figure \[fig:23loop\] shows the 2- and 3-loop diagrams contributing to $\ln Z$ as an example. Each loop, which are considered to have no orientation, carries a factor of $N$. The contact interaction is represented by a dashed line for convenience, each carrying a weight $\propto g$. In the equivalent spacetime approach, Feynman diagrams are understood as spacetime diagrams. Each loop of a Feynman diagram is expressed in a hopping expansion as in the noninteracting theory, with the interaction standing for an intersection between different worldlines or between different parts of a single worldline (self-intersection). In addition to the ones indicated, a worldline contributing to a given Feynman diagram may have further intersections, but these are treated as in the noninteracting case, i.e., as nonexisting. Put differently, apart from the intersections explicitly indicated, the particle trajectories still behave as phantom paths.

![Two- and 3-loop Feynman diagrams contributing to $\ln Z$ of 