Statistics and asymptotics of subdivergence-free Feynman integrals in $\phi^4$ theory

TL;DR

This paper analyzes the statistical behavior of primitive Feynman integrals in $4$ theory at high loop orders, revealing exponential growth, distribution characteristics, and proposing importance sampling for efficient computation.

Contribution

It provides the first large-scale statistical analysis of high-loop Feynman integrals, introducing importance sampling to improve numerical evaluation efficiency.

Findings

Average integral value grows exponentially with loop order.

Distribution has many typical values and few outliers, affecting sampling.

Importance sampling achieves about 1000-fold speedup over uniform sampling.

Abstract

Recent algorithmic improvements have made it possible to evaluate subdivergence-free (=primitive=skeleton) Feynman integrals in $\phi^4$ theory numerically up to 18 loops. By now, all such integrals up to 13 loops and several hundred thousand at higher loop order have been computed. This data enables a statistical analysis of the typical behaviour of Feynman integrals at large loop order. We find that the average value grows exponentially, but the observed growth rate is accurately described by its leading asymptotics only upwards of 25 loops. This is in contrast with the $N$-dependence of the $ON(N)$-symmetric $\phi^4$ theory, which is close to its large-order asymptotics already around 10 loops. Secondly, the distribution of integrals has a largely continuous inner part but a few extreme outliers. This makes uniform random sampling inefficient. We find that the value of the integral is correlated with many features of the graph, which can be used for importance sampling. With a naive test implementation we obtained an approximately 1000-fold speedup compared with uniform sampling. This suggests that in future work, Feynman amplitudes at large loop order might be computed numerically with statistical methods, rather than through enumerating and evaluating every individual integral.