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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.

##### MSC:
 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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