# 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.

1. If $$|g|>R$$ the series is trivially divergent, i.e., the general term $$a_n g^n$$ does not converge to zero.
2. If $$|g| the series converges and does so absolutely, i.e., $$\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".