In #physics, #quantum field theory is used to describe the behavior of elementary particles. #Feynman diagrams are used to visualize, and compute, the "elementary" processes that can happen. However, the processes that really occur in nature are a sum of infinitely many Feynman diagrams. Of course, in an actual computation, one can only include finitely many processes, and all the other ones need to absorbed into some "effective" parameters, such as effective charges. This is called #renormalization, and it involves a freedom regarding how exactly one defines the effective parameters. Two renormalization schemes are common in high energy physics: In "kinematic renormalization", one defines the effective parameters as the actually measured values of a certain scattering process. In "minimal subtraction", one chooses the effective parameters such that the computation is as easy as possible, regardless of what the parameters mean concretely.
Certain infinite sums of Feynman diagrams, called "rainbows" (see picture), had been computed in minimal subtraction 30 years ago. In a recent preprint https://arxiv.org/abs/2503.02079 I computed the analogous sums in the minimal subtraction scheme. The solution is structurally similar to the known one, but they involve slightly more complicated functions.
The sum of rainbows by itself is not a physically relevant observable. But since it is one of the few infinite classes of Feynman diagrams that can be solved exactly, it is often used as a model to describe qualitative features, such as how quickly these sums grow if one includes more and more terms.
