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

Paul-Hermann Balduf, Kimia Shaban, Johannes Thürigen

TL;DR

The paper investigates the statistical behavior of subdivergence-free Feynman integrals (periods) in massless $\phi^4$ theory at large loop order. Using a dataset of over 2 million computed periods up to $\ell\le13$ (with partial samples to $\ell\le18$) and analyses of graph symmetries, the authors show that the mean period grows exponentially and that a few outliers skew the distribution, making uniform sampling inefficient. They demonstrate strong correlations between periods and graph invariants such as the Hepp bound and Martin invariant, enabling importance sampling that achieves roughly a $10^3$ speedup in summing periods at moderate loops. The study confirms the expected large-$\ell$ asymptotics only beyond $\ell\approx25$ and argues that statistical methods, rather than exhaustive enumeration, offer a practical path to computing high-loop amplitudes, with potential applicability to broader Feynman integrals.

Abstract

Recent algorithmic improvements have made it possible to evaluate subdivergence-free (=primitive=skeleton) Feynman integrals in $φ^4$ 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 $φ^4$ 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.

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

TL;DR

The paper investigates the statistical behavior of subdivergence-free Feynman integrals (periods) in massless theory at large loop order. Using a dataset of over 2 million computed periods up to (with partial samples to ) and analyses of graph symmetries, the authors show that the mean period grows exponentially and that a few outliers skew the distribution, making uniform sampling inefficient. They demonstrate strong correlations between periods and graph invariants such as the Hepp bound and Martin invariant, enabling importance sampling that achieves roughly a speedup in summing periods at moderate loops. The study confirms the expected large- asymptotics only beyond and argues that statistical methods, rather than exhaustive enumeration, offer a practical path to computing high-loop amplitudes, with potential applicability to broader Feynman integrals.

Abstract

Recent algorithmic improvements have made it possible to evaluate subdivergence-free (=primitive=skeleton) Feynman integrals in 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 -dependence of the -symmetric 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.

Paper Structure

This paper contains 6 sections, 3 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Graph count and period symmetries
  3. Numerical evaluation and Distribution
  4. Asymptotics of the period mean and the primitive $\beta$ function
  5. Correlations and Predictors
  6. Outlook

Figures (5)

  • Figure 1: (a) A completion on eight vertices. It has eight decompletions, but all of them are isomorphic to either one of the three graphs shown in (b). The decompletions have $\ell=6$ loops.
  • Figure 2: Convergence of graph–symmetry indicators to the asymptotic regime, double logarithmic plot. The quantities quickly approach zero upwards of 10 loops.
  • Figure 3: Distribution of periods at loop orders 12 (blue) and 13 (red), scaled to their respective mean. The histograms largely overlap, except for the outliers marked by arrows. Figure taken from balduf_statistics_2023.
  • Figure 4: (a) Growth ratio $f_\ell$ (\ref{['fL']}), plotted as function of $\frac{1}{\ell}$. (b) Growth ratio $r_\ell$ (\ref{['rL']}). In both cases, an extrapolation of the numerical data (red, dashed) does not reproduce the correct large-$\ell$ asymptotics (green lines). Still, the $N$-dependence of the numerical data has the expected pattern.
  • Figure 5: (a) Period as a function of the Hepp bound, for various loop orders. The correlation is very strong. (b) Period as a function of the average resistance. The correlation is weaker, but still clearly visible.