Feynman integrals at very many loops

by Paul-Hermann Balduf (Oxford)

Monday 16 Feb 2026, 14:00 → 15:00 Europe/Zurich

4/3-006 - TH Conference Room (CERN)

Modern computations in perturbative QFT easily involve thousands of Feynman diagrams, and their number grows factorially with the loop order. On the one hand, this poses a computational challenge. On the other hand, it is an opportunity to treat the diagrams statistically rather than solving each one individually. In my talk, I consider the O(N) symmetric scalar phi^4 theory, where more than 2 million subdivergence-free diagrams have been computed numerically by now, up to 18 loops. This allows to study their statistical properties, growth rates, and correlations on a robust empirical basis. I demonstrate in a simple example that such data can be very useful in weighted Monte Carlo integration. Moreover, it allows to partially answer some old conjectures regarding the role of instantons, renormalons, and subdivergence-free graphs in the beta function of phi^4 theory. In the last part of the talk, I present recent joint work with Erik Panzer on tropical field theory. A special case of tropical field theory gives rise to a combinatorial model of phi^4 theory, in which the large-order behaviour of the perturbation series can be determined to high accuracy. Although the concrete numerical values are different, the results from tropical field theory confirms the qualitative effects we had observed for non-tropical phi^4 theory to 18 loops.

Based on
2305.13506 (statistics of diagrams),
2403.16217 (weighted diagram sampling),
2412.08617 (graph counts and asymptotics),
2512.06898 (summary proceedings), 
2512.21091 (tropical field theory).

2026_02_Balduf_statistics.pdf