Author(s): D. J. Broadhurst, D. Kreimer

  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 $10^{-8}$.

Paper: hep-th/9912093
Dated: Sat, 11 Dec 1999 18:34:07 GMT   (11kb)
Comments: 10 pages, LaTeX
Report-no: OUT-4102-84, MZ-TH/99-57
Subj-class: High Energy Physics - Theory; Quantum Algebra

URL: http://xxx.lanl.gov/abs/hep-th/9912093