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

Counterterms that are not reducible to ζ n are generated by 3 F 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) = 819 and (5, 3) = 10124, are found in anomalous dimensions at 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 Pade resummations of e-expansions, which are compared with analytical results in 3 dimensions. The 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 10139 and 10152 are not generated by such self-energy insertions.