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

Dec, 1999
9 pages
Published in:
  • Phys.Lett.B 475 (2000) 63-70
e-Print:
Report number:
  • OUT-4102-84,
  • MZ-TH-99-57

Citations per year

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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\Delta, antipode S, and grading operator Y of the Hopf algebra of undecorated rooted trees. The vital operator is SYS\star Y, with a star product effected by Δ\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 SYS\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 10810^{-8}.
  • renormalization
  • algebra: Hopf
  • Pade approximation
  • model: Yukawa
  • dimension: 4
  • field theory: scalar
  • dimension: 6
  • Borel transformation
  • Feynman graph: higher-order
  • higher-order: 30