Bogolubov–Parasiuk theorem in the α-parametric representation

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Abstract
A renormalized Feynman amplitude expressed in the α‐parameters is defined by introducing a subtraction operator acting directly upon the α‐integrand. Different forms of this subtraction operator are discussed. We define the isotropic and nonisotropic normal products and we give a more general oversubtraction rule which ensures both the absolute convergence of the amplitude and the Bogolubov, Parasiuk and Hepp recurrence. The proof of absolute convergence of the amplitude is performed using Hepp’s sectors and equivalence classes of nests.