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Renormalization in quantum field theory and the Riemann-Hilbert problem. 2. The beta function, diffeomorphisms and the renormalization group

Mar, 2000
35 pages
Published in:
  • Commun.Math.Phys. 216 (2001) 215-241
e-Print:
DOI:
Report number:
  • IHES-M-00-22,
  • MZ-TH-00-10
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Abstract:
We showed in part I (hep-th/9912092) that the Hopf algebra H{\cal H} of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group GG and that the renormalized theory is obtained from the unrenormalized one by evaluating at \ve=0 the holomorphic part \gamma_+(\ve) of the Riemann-Hilbert decomposition \gamma_-(\ve)^{-1}\gamma_+(\ve) of the loop \gamma(\ve)\in G provided by dimensional regularization. We show in this paper that the group GG acts naturally on the complex space XX of dimensionless coupling constants of the theory. More precisely, the formula g0=gZ1Z3−3/2g_0=gZ_1Z_3^{-3/2} for the effective coupling constant, when viewed as a formal power series, does define a Hopf algebra homomorphism between the Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra H{\cal H}. This allows first of all to read off directly, without using the group GG, the bare coupling constant and the renormalized one from the Riemann-Hilbert decomposition of the unrenormalized effective coupling constant viewed as a loop of formal diffeomorphisms. This shows that renormalization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter \ve. It also allows to lift both the renormalization group and the β\beta-function as the asymptotic scaling in the group GG. This exploits the full power of the Riemann-Hilbert decomposition together with the invariance of \gamma_-(\ve) under a change of unit of mass. This not only gives a conceptual proof of the existence of the renormalization group but also delivers a scattering formula in the group GG for the full higher pole structure of minimal subtracted counterterms in terms of the residue.
Note:
  • 35 pages, eps figures Report-no: IHES/M/00/22, MZ-TH/00-10
  • Feynman graph: higher-order
  • renormalization group: transformation
  • renormalization group: beta function
  • loop space
  • algebra: Hopf
  • transformation: diffeomorphism
  • Riemann surface
  • fibre bundle
  • regularization: dimensional