Predicting Feynman periods in ${\varphi }^4$-theory

Abstract

We present efficient data-driven approaches to predict Feynman periods in ${\varphi }^4$-theory from properties of the underlying Feynman graphs. We find that the numbers of cuts and cycles determines the period to approximately 2% accuracy. Hepp bound and Martin invariant allow to predict the period with accuracy much better than 1%. In most cases, the period is a multi-linear function of the parameters in question. Besides classical correlation analysis, we also 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 ${\varphi }^4$-theory at $L\in \left\{13,14,15,16\right\}$ 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 reaches 35-fold higher accuracy in fixed runtime, or requires 1000-fold less runtime to reach a given accuracy. The data set of all periods computed for this work, combined with a previous data set, is made publicly available. Besides the physical application, it could serve as a benchmark for graph-based machine learning algorithms.

arXiv comment45 pages

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Cited by in corpus (2)

• Feynman integrals at large loop order and the $\log$-$\Gamma$ distribution
• Tropicalized quantum field theory and global tropical sampling