Statistics of Feynman amplitudes in $\phi^4$-theory

Abstract

The amplitude of subdivergence-free logarithmically divergent Feynman graphs in $\phi^4$-theory in 4 spacetime dimensions is given by a single number, the Feynman period. We numerically compute the periods of 1.3 million completed graphs, this represents more than 33 million graphs contributing to the beta function. Our data set includes all primitive graphs up to 13 loops, and non-complete samples up to 18 loops, with an accuracy of ca. 4 significant digits. We implement all known symmetries of the period in a new computer program and count them up to 14 loops. Combining the symmetries, we discover relations between periods that had been overlooked earlier. All expected symmetries are respected by the numerical values of periods. We examine the distribution of the numerically computed Feynman periods. We confirm the leading asymptotic growth of the average period with growing loop order, up to a factor of 2. At high loop order, a limiting distribution is reached for the amplitudes near the mean. A small class of graphs, most notably the zigzags, grows significantly faster than the mean and causes the limiting distribution to have divergent moments even when normalized to unit mean. We examine the relation between the period and various properties of the underlying graphs. We confirm the strong correlation with the Hepp bound, the Martin invariant, and the number of 6-edge cuts. We find that, on average, the amplitude of planar graphs is significantly larger than that of non-planar graphs, irrespective of $O(N)$ symmetry. We estimate the primitive contribution to the 18-loop beta function of the $O(N)$-symmetric theory. We find that primitive graphs constitute a large part of the beta function in MS for $L\rightarrow \infty$ loops. The relative contribution of planar graphs increases with growing $N$ and decreases with growing loop order $L$.