Predicting Feynman periods in ϕ4-theory

We present efficient data-driven approaches to predict the value of subdivergence-free Feynman integrals (Feynman periods) in ϕ⁴-theory from properties of the underlying Feynman graphs, based on a statistical examination of almost 2 million graphs. We find that the numbers of cuts and cycles determines the period to better than 2% relative accuracy. Hepp bound and Martin invariant allow for even more accurate predictions. In most cases, the period is a multi-linear function of the properties in question. Furthermore, we investigate the usefulness of machine-learning algorithms to predict the period. When sufficiently many properties of the graph are used, the period can be predicted with better than 0.05% relative accuracy.

We use one of the constructed prediction models for weighted Monte-Carlo sampling of Feynman graphs, and compute the primitive contribution to the beta function of ϕ⁴-theory at L ∈ {13, … , 17} loops. Our results confirm the previously known numerical estimates of the primitive beta function and improve their accuracy. Compared to uniform random sampling of graphs, our new algorithm is 1000-times faster to reach a desired accuracy, or reaches 32-fold higher accuracy in fixed runtime.

The dataset of all periods computed for this work, combined with a previous dataset, is made publicly available. Besides the physical application, it could serve as a benchmark for graph-based machine learning algorithms.