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Parametric representations of Feynman amplitudes
Analysis of convergence
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1PI subgraphs a-representation absolutely convergent analytic continuation analytic regularization applied arbitrary asymptotic expansion auxiliary mass based on subtractions basic multilinear functions c-component calculations Chapter coefficient functions composite operators consider contribution coordinate space counterterm operation counterterm technique define denote derived diagrammatic dimensional renormalization dimensionally regularized Feynman elements equation Euclidean explicitly external lines external momenta external vertices factor Feynman amplitudes finite forest F forest formula Fourier transform Furthermore graph of Fig Green's functions inserting integrand irreducible Lagrangian loop momenta mass renormalization massless maximal elements minimal mixed representation momentum space monomials non-exceptional momenta obtain one-loop parametric integral poles polynomial procedure products of composite proof propagators properties prove R-operation regularization parameter regularized Feynman integral relations renormalization schemes result S-matrix scalar Section sector variables self-energy singularities spinney subsets subtraction operator Taylor expansion tempered distribution theorem theory tion UV divergences UV forest vertex wave front set Wilson expansion zero