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

TL;DR

This paper develops a Hopf algebra-based method to tame the combinatoric explosion in high-loop renormalization calculations, enabling precise Padé-Borel resummation of over 463 quintillion subtractions in quantum field theories.

Contribution

It introduces a novel algebraic framework using Hopf algebra operators to efficiently handle complex nested divergences in high-order perturbative expansions.

Findings

Performed 30-loop resummation in Yukawa theory with high precision.

Successfully managed over 463 quintillion subtractions using the Hopf algebra approach.

Achieved residual errors around 10^{-8} even at large coupling constants.

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 $\Delta$, 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 $\Delta$. We perform 30-loop Pad\'e-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 $10^{-8}$.