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We study the renormalized perturbation theory of the single-impurity Andersonmodel, particularly the high-order terms in the expansion of the self-energy inpowers of the renormalized coupling $\tilde{U}$. Though the presence ofcounter-terms in the renormalized theory may appear to complicate thediagrammatics, we show how these can be seamlessly accommodated by carrying outthe calculation order-by-order in terms of skeleton diagrams. We describe howthe diagrams pertinent to the renormalized self-energy and four-vertex can beautomatically generated, translated into integrals and numerically integrated.To maximize the efficiency of our approach we introduce a generalized$k$-particle/hole propagator, which is used to analytically simplify theresultant integrals and reduce the dimensionality of the integration. Wepresent results for the self-energy and spectral density to fifth order in$\tilde{U}$, for various values of the model asymmetry, and compare them to aNumerical Renormalization Group calculation.

High-order terms in the renormalized perturbation theory for the Anderson impurity model