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Renormalization in quantum field theory and the Riemann-Hilbert problem. II: The \(\beta\)-function, diffeomorphisms and the renormalization group. (English) Zbl 1042.81059
Summary: We showed in Part I [ibid. 210, No.1, 249-273 (2000; Zbl 1032.81026)] that the Hopf algebra \(H\) of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group \(G\) and that the renormalized theory is obtained from the unrenormalized one by evaluating at \(\varepsilon=0\) the holomorphic part \(\gamma_+\)(\(\varepsilon\)) of the Riemann-Hilbert decomposition \(\gamma_-(\varepsilon)^{-1}\gamma_+(\varepsilon\)) of the loop \(\gamma(\varepsilon)\in G\) provided by dimensional regularization. We show in this paper that the group \(G\) acts naturally on the complex space \(X\) of dimensionless coupling constants of the theory. More precisely, the formula \(g_0=gZ_1Z_3^{-3/2}\) 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 \(H\). This allows first of all to read off directly, without using the group \(G\), 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 nonlinear complex bundles on the Riemann sphere of the dimensional regularization parameter \(\varepsilon\). It also allows to lift both the renormalization group and the \(\beta\)-function as the asymptotic scaling in the group \(G\). This exploits the full power of the Riemann-Hilbert decomposition together with the invariance of \(\gamma_-(\varepsilon\)) 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 \(G\) for the full higher pole structure of minimal subtracted counterterms in terms of the residue.

81T15 Perturbative methods of renormalization applied to problems in quantum field theory
34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
81T18 Feynman diagrams
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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