# Is it possible that a series of Feynman diagrams converge?

These series are power series like $\sum_{n=0}^{\infty}a_n g^n$ in some coupling $g$. Power series are special, versus all seriesm in that one has a priori a good understanding of what can possibly happen. Namely, the radius of convergence $R$ defined by $$ R=\frac{1}{\limsup\limits_{n\rightarrow \infty} |a_n|^{\frac{1}{n}}} $$ with the convention that $1/0=\infty$ and $1/\infty=0$, is such that the following holds.

- If $|g|>R$ the series is trivially divergent, i.e., the general term $a_n g^n$ does not converge to zero.
- If $|g|<R$ the series converges and does so absolutely, i.e., $\sum_{n=0}^{\infty}|a_n g^n|<\infty$.

Now for typical Feynman diagram power series, $a_n=\sum_D b_D$ is a finite sum over diagrams $D$. Figuring out the size $|a_n|$ is more complicated than just counting how many $D$'s there are at a certain order $n$ in perturbation theory. This number is typically factorial, but there can be cancellations so that $|\sum_D b_D|$ ends up being much smaller than $\sum_D |b_D|$. Moreover, the contributions of the diagrams do not have the same size.

Now for a Bosonic theory like Euclidean $\phi^4$ with cutoffs, at fixed $n$, the $b_D$ are real numbers and they all have the *same sign*, so there cannot be any cancellations. If one neglects the variation of size of diagram contributions, one gets a rough estimate
$$
|a_n|\sim \frac{1}{n!}\times \frac{(4n)!}{2^{2n}(2n)!}\sim n!
$$
ignoring anything of the form $C^n$.
This results in $R=0$ and divergence of the series, no matter how small the coupling $g$ is.

For Fermionic theories, there are cancellations. In the fact, in the presence of cutoffs (UV and IR), the series is convergent for $g$ small, i.e., $R>0$.

Another notable model where the perturbation series converges in a very subtle way is the dipole gas. See

- K. Gawedzski and A. Kupiainen, "Block spin renormalization group for dipole gas and $(\nabla\varphi)^4$" in Ann. Phys.
- K. Gawedzski and A. Kupiainen, "Lattice dipole gas and $(\nabla\varphi)^4$ models at long distances: decay of correlations and scaling limit" in CMP.

Also note that Dyson's argument is nowhere near being a proof, it is just some handwavy heuristic. Finally, to get a better feel for these convergence issues, and resummation techniques like Borel's method, it is good to consider the pedagogical example of QFT in zero dimension as explained in: Rivasseau, "Constructive Field Theory in Zero Dimension".