I'm currently reading A. Zee "Quantum Field Theory in Nutshell". There is a problem called baby problem which is to evaluate following partition function. $$ Z(\lambda ,J) := \int_{-\infty}^{\infty}dq \exp\left(-{\frac{1}{2}m^2q^2+Jq-\frac{\lambda}{4!}q^4}\right)$$ first expanding by $\lambda$ and then differentiating repeatedly by $J$ one gets following expression. $$Z(\lambda ,J) = \Bigl(\frac{2\pi}{m^2} \Bigr)^\frac12 \exp\left[{-\frac{\lambda}{4!}\Bigl(\frac{\partial}{\partial{J}} \Bigr)^4} \right]\exp\left(\frac{1}{2m^2}J^2\right)$$ after getting this we are interested in term of given $\lambda$ and $J$.
For example if we want to check the term with power of $\lambda=1$ and power of $J=4$ one gets, $$Z(\lambda=1 ,J=4) = \frac{8!(-\lambda{J^4})}{{4!}^3(2m^2)^4}$$ where I mean $Z(\lambda=1,J=4)$ as term with 1th power of $\lambda$ and 4th power of $J$. after that there are bunch of "baby Feynman diagrams" associated with each given $\lambda$ and $J$.
I want to understand relation of these diagrams and combinatoric term (e.g. $8!/(4!)^3$).
First thing I tried was, to evaluate general term for given $\lambda=n,J=2k$, and got result, $$Z(\lambda=n ,J=2k) = (-1)^n\frac{(4n+2k)!}{(4!)^n(2n+k)!(2k)!}\frac{\lambda^nJ^{2k}}{(2m^2)^{2n+k}}$$ here combinatorial term is $\dfrac{(4n+2k)!}{(4!)^n(2n+k)!(2k)!}$.
What I understood is, to get this fraction one needs,
- $n$ identical elements each constructed by $4$ identical elements, giving $(4!)^n$
- $2n+k$ identical elements giving $(2n+k)!$ term.
- $2k$ identical elements giving $(2k)!$ term.
but I don't understand the term $(4n+2k)!$ term, which one may think as permutation of $4n+2k$ identical elements, but what is this element associated with? (e.g. by looking at above general expression $2n+k$ term can be associated to $2m^{2}$)
Also, what kind of combination this fraction all together represents?
Also, how can one interpret this fraction by "Feynman diagram"?
Sorry for somewhat ambiguous question. Hope this makes sense.
Thank you.