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arXiv · hep-th/9912093

Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pade-Borel resummation

D. J. Broadhurst·D. Kreimer
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Abstract

It is easy to sum chain-free self-energy rainbows, to obtain contributions to anomalous dimensions. It is also easy to resum rainbow-free self-energy chains. Taming the combinatoric explosion of all possible nestings and chainings of a primitive self-energy divergence is a much more demanding problem. We solve it in terms of the coproduct $Δ$, antipode S, and grading operator Y of the Hopf algebra of undecorated rooted trees. The vital operator is $S\star Y$, with a star product effected by $Δ$. We perform 30-loop Padé-Borel resummation of 463 020 146 037 416 130 934 BPHZ subtractions in Yukawa theory, at spacetime dimension d=4, and in a trivalent scalar theory, at d=6, encountering residues of $S\star Y$ that involve primes with up to 60 digits. Even with a very large Yukawa coupling, g=30, the precision of resummation is remarkable; a 31-loop calculation suggests that it is of order $10^{-8}$.

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D. J. Broadhurst, D. Kreimer. 2000-01-07. Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pade-Borel resummation. https://doi.org/10.1016/s0370-2693(00)00051-4

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