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We evaluate all the primitive divergences contributing to the 7--loop$\beta$\/--function of $\phi^4$ theory, i.e.\ all 59 diagrams that are free ofsubdivergences and hence give scheme--independent contributions. Guided by theassociation of diagrams with knots, we obtain analytical results for 56diagrams. The remaining three diagrams, associated with the knots $10_{124}$,$10_{139}$, and $10_{152}$, are evaluated numerically, to 10 sf. Only onesatellite knot with 11 crossings is encountered and the transcendental numberassociated with it is found. Thus we achieve an analytical result for the6--loop contributions, and a numerical result at 7 loops that is accurate toone part in $10^{11}$. The series of `zig--zag' counterterms,$\{6\zeta_3,\,20\zeta_5,\, \frac{441}{8}\zeta_7,\,168\zeta_9,\,\ldots\}$,previously known for $n=3,4,5,6$ loops, is evaluated to 10 loops, correspondingto 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 $C_n=\frac{1}{n+1}{2n \choosen}$ are the Catalan numbers, familiar in knot theory. The investigationsreported here entailed intensive use of REDUCE, to generate ${\rm O}(10^4)$lines of code for multiple precision FORTRAN computations, enabled by Bailey'sMPFUN routines, running for ${\rm O}(10^3)$ CPUhours on DecAlpha machines.Comment: 6 pages plain LaTe

Knots and Numbers in ϕ4 Theory to 7 Loops and Beyond

Knots and Numbers in ϕ⁴ Theory to 7 Loops and Beyond