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

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Zeitschrift für Physik C Particles and Fields

Abstract

Counterterms that are not reducible to ζn are generated by 3F2 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.

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Authors and Affiliations

  1. Physics Department, Open University, Milton Keynes, MK7 6AA, UK

    D.J. Broadhurst

  2. Department of Mathematical Sciences, University of Liverpool, PO Box 147, Liverpool, L69 3BX, UK

    J.A. Gracey

  3. Institut für Physik der Universität Mainz, D-55099, Mainz, Germany

    D. Kreimer

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  1. D.J. Broadhurst
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  2. J.A. Gracey
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  3. D. Kreimer
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Correspondence to D.J. Broadhurst.

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Work supported in part by grant CHRX-CT94-0579, from HUCAM.

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Broadhurst, D., Gracey, J. & Kreimer, D. Beyond the triangle and uniqueness relations: non-zeta counterterms at large N from positive knots. Z Phys C - Particles and Fields 75, 559–574 (1997). https://doi.org/10.1007/s002880050500

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