#feynmangraph — Public Fediverse posts

The #paperOfTheDay is "Exponents for the excluded volume problem as derived by the Wilson Method" from 1972. This one-page letter makes one elementary, but far-reaching observation: The O(N)-invariant scalar #quantumFieldTheory , which had a few months before been treated by Wilson in terms of dimensional regularization, has a special interpretation when N=0. Namely, one assigns to every #FeynmanGraph a "symmetry factor", which is a polynomial in N. The coefficient of N^k in this polynomial counts how many ways there are to decompose the 4-valent vertices of the graph such that one obtains exactly k cycles. If one sets N=0, all that remains is the constant term: It counts the ways of decomposing the graph without forming any cycle.
One is interested in the statistical behaviour of non-crossing paths on a lattice, called a self-excluding walk. This can be studied with the methods of statistical #physics . One introduces a Boltzmann-type weight exp(-n*p), where p is a parameter (analogous to the inverse temperature or the Planck constant), and n is the length of a self-excluding walk. Let N_n be the number of different such walks (for a fixed size of the lattice, or counted relative to the lattice size), then the sum of N_n*exp(-n*p) is analogous to a partition function, or path integral. Hence, it can be analyzed perturbatively with Feynman integrals, namely those mentioned above of the O(N) theory at N=0. This way, one obtains, for example, the critical exponents.
https://www.sciencedirect.com/science/article/pii/0375960172901491

My #paperOfTheDay is "Event generation with exponential scaling in multiplicity using AmpliCol" https://arxiv.org/abs/2601.19483 . Scattering experiments of elementary particles, e.g. at CERN, are very complicated even beyond the "core" physical process, for example the detectors can only measure in certain angles and with certain thresholds. To calibrate them, one needs to simulate the whole processes numerically, which is done with Monte Carlo methods. Among other things, the behaviour of particles in quantum chromodynamics depends on their "color", which is a type of charge somewhat analogous to electrical charge. Since that dependence is very complicated, the idea of the present paper is to use a simplified "leading" color-dependence as a proxy for importance sampling in the Monte Carlo simulation. This is analogous, but for a totally different physical question, to the weighting of individual random #FeynmanGraph s that I did in my "Predicting Feynman Periods" paper a while ago. https://link.springer.com/article/10.1007/JHEP11(2024)038 #dailyPaperChallenge

New theoretical #physics preprint https://arxiv.org/abs/2412.08617
We looked at the asymptotic growth rate of the beta function in #quantumFieldTheory , and the relative importance of subdivergence-free #Feynmangraph s. These graphs correspond to integrals, and the size of the graph is measured by its loop number, which also indicates how hard it is to solve the integral. State of the art computations in realistic theories are anywhere between 1 and 6 loops. The asymptotics of the perturbation series is known from instanton calculations. We now showed (in a model theory), that the leading asymptotics describes the true growth rate only for more than 25 loops, way beyond anything that can realistically be computed.

This is good news: It tells us that asymptotic instanton calculations provide non-trivial additional information that can not be trivially inferred from low-order perturbation theory.
In the plot, the red dots are numerical data points for the subdivergence-free graphs in phi^4 theory up to 18 loops, the green lines are the leading instanton asymptotics.

I recently discovered an excellent #math article by Krajewski and Martinetti that I had overlooked so far: https://arxiv.org/abs/0806.4309
Basically, renormalization of #Feynmangraph s in #qft is organized in terms of rooted trees, and so are solutions to differential equations, concatenation of differential operators, numerical integration schemes, and the Hopf algebra of power series. It is intuitively clear that all these things must be closely related, but I wasn't aware that there is this article where the relations are actually spelled out in detail. They also include a derivation of Wigner's semicircle law for Gaussian random matrices in the rooted-tree formalism, which is something I didn't think about at all. Learned another unexpected connection 😀 .

Here is a curious finding from our statistical analysis https://arxiv.org/abs/2403.16217 :
A #Feynmangraph is a graphical short hand notation for a complicated integral that computes the probability for scattering processes in #quantum field theory.
An electrical circuit can also be described as a graph. What happens if we interpret the Feynman graph as an #electrical network, where each edge is a 1 Ohm resistor? We can then compute the resistance between any pair of vertices and collect all these values in a "resistance matrix", as shown below. The average of all these resistances is called "Kirchhoff index". Now it turns out that this average resistance is correlated fairly strongly with the Feynman integral of that graph: A graph with large contribution to quantum scattering amplitudes on average also has a large electrical resistance. Isn't that a nice connection between two seemingly distinct branches of theoretical #physics ?

The #Feynmangraph s that contribute to scattering amplitudes in #quantum field theory come in all shapes and sizes. Can one guess how much their Feynman integral will contribute just from looking at them 🤔? It turns out one can! Let's consider subdivergence-free graphs at 12 loops. The pictures show the two largest and the two smallest contributors. All graphs have the same number of edges and vertices. The graphs that contribute strongly look "larger", and the small contributors look "more dense", but drawings are of course arbitrary. A closer examination shows that "symmetry", as measured by graph automorphisms, is not clearly related to the value of the Feynman integral.