Knots and Numbers in $φ^4$ Theory to 7 Loops and Beyond

TL;DR

The paper advances the understanding of perturbative renormalization in $\phi^4$ theory by computing all primitive divergences up to 7 loops, revealing a deep link between Feynman diagrams and knot theory. By combining knot-theoretic guidance with high-precision numerical and symbolic methods, the authors obtain analytical results for 56 diagrams and accurate numerical results for the remaining three, extending the zig-zag counterterm series to 10 loops via a closed-form expression involving Catalan numbers. They identify and catalog both zeta-valued contributions and a small set of non-$\zeta$ transcendental constants tied to specific knots, such as $K_{353}$, $K_{53}$, and $K_{73}$, illustrating the knot-number correspondence in primitive counterterms. This work demonstrates a robust, cross-disciplinary approach that sharpens predictions for the $\beta$-function and provides a framework for exploring perturbative structure through knot theory.

Abstract

We evaluate all the primitive divergences contributing to the 7--loop $β$\/--function of $φ^4$ theory, i.e.\ all 59 diagrams that are free of subdivergences and hence give scheme--independent contributions. Guided by the association of diagrams with knots, we obtain analytical results for 56 diagrams. The remaining three diagrams, associated with the knots $10_{124}$, $10_{139}$, and $10_{152}$, are evaluated numerically, to 10 sf. Only one satellite knot with 11 crossings is encountered and the transcendental number associated with it is found. Thus we achieve an analytical result for the 6--loop contributions, and a numerical result at 7 loops that is accurate to one part in $10^{11}$. The series of `zig--zag' counterterms, $\{6ζ_3,\,20ζ_5,\, \frac{441}{8}ζ_7,\,168ζ_9,\,\ldots\}$, previously known for $n=3,4,5,6$ loops, is evaluated to 10 loops, corresponding to 17 crossings, revealing that the $n$\/--loop zig--zag term is $4C_{n-1} \sum_{p>0}\frac{(-1)^{p n - n}}{p^{2n-3}}$, where $C_n=\frac{1}{n+1}{2n \choose n}$ are the Catalan numbers, familiar in knot theory. The investigations reported here entailed intensive use of REDUCE, to generate ${\rm O}(10^4)$ lines of code for multiple precision FORTRAN computations, enabled by Bailey's MPFUN routines, running for ${\rm O}(10^3)$ CPUhours on DecAlpha machines.