Primitive asymptotics in $\phi^4$ vector theory

TL;DR

This paper investigates the asymptotic behavior of primitive graphs in $\,phi^4_4$ theory, revealing that dominant contributions emerge only after about 25 loops and providing detailed combinatorial and numerical analyses.

Contribution

It extends the understanding of primitive graph asymptotics in $\,phi^4$ theory by analyzing $O(N)$ symmetric vector models, calculating generating functions, and providing bounds and explicit constructions for primitive graphs.

Findings

Leading asymptotics appear only above ~25 loops.

Numerical calculations up to 17 loops show qualitative similarity to 0D case.

Zeros of the beta function approach large-loop asymptotics at negative integer N.

Abstract

A longstanding conjecture in $\phi^4_4$ theory is that primitive graphs dominate the beta function asymptotically at large loop order in the minimal-subtraction scheme. Here we investigate this issue by exploiting additional combinatorial structure coming from an extension to vectors with $O(N)$ symmetry. For the 0-dimensional case, we calculate the $N$-dependent generating function of primitive graphs and its asymptotics, including arbitrarily many subleading corrections. We find that the leading asymptotic growth rate becomes visible only above $\approx 25$ loops, while data at lower order is suggestive of a wrong asymptotics. Our results also yield the symmetry-factor weighted sum of 3-connected cubic graphs, and the exact asymptotics of Martin invariants. For individual Feynman graphs, we give bounds on their degree in $N$ depending on their coradical degree, and construct the primitive graphs of highest degree explicitly. We calculate the 4D primitive beta function numerically up to 17 loops, and find its behaviour to be qualitatively similar to the 0D case. The locations of zeros quickly approach their large-loop asymptotics at negative integer $N$, while the growth rate of the beta function differs from the asymptotic prediction even at 17 loops.