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arXiv · hep-th/9607174

Beyond the triangle and uniqueness relations: non-zeta counterterms at large N from positive knots

D. J. Broadhurst·J. A. Gracey·D. Kreimer
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Abstract

Counterterms that are not reducible to $ζ_{n}$ are generated by ${}_3F_2$ hypergeometric series arising from diagrams for which triangle and uniqueness relations furnish insufficient data. Irreducible double sums, corresponding to the torus knots $(4,3)=8_{19}$ and $(5,3)=10_{124}$, are found in anomalous dimensions at ${\rm O}(1/N^3)$ in the large-$N$ limit, which we compute analytically up to terms of level 11, corresponding to 11 loops for 4-dimensional field theories and 12 loops for 2-dimensional theories. High-precision numerical results are obtained up to 24 loops and used in Padé resummations of $\varepsilon$-expansions, which are compared with analytical results in 3 dimensions. The ${\rm O}(1/N^3)$ results entail knots generated by three dressed propagators in the master two-loop two-point diagram. At higher orders in $1/N$ one encounters the uniquely positive hyperbolic 11-crossing knot, associated with an irreducible triple sum. At 12 crossings, a pair of 3-braid knots is generated, corresponding to a pair of irreducible double sums with alternating signs. The hyperbolic positive knots $10_{139}$ and $10_{152}$ are not generated by such self-energy insertions.

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D. J. Broadhurst, J. A. Gracey, D. Kreimer. 1996-07-23. Beyond the triangle and uniqueness relations: non-zeta counterterms at large N from positive knots. https://arxiv.org/abs/hep-th/9607174

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