Combinatorics of $\mathbf{R}$-, $\mathbf{R^{-1}}$-, and $\mathbf{R^*}$-operations and asymptotic expansions of feynman integrals in the limit of large momenta and masses

TL;DR

This work develops a unified combinatorial framework for the $R$-, $R^{-1}$-, and $R^*$-operations in the context of dimensional regularization and the MS scheme, clarifying their interrelations and forest representations. It provides explicit constructions and proofs linking UV and IR subtraction, Zimmermann identities, and scheme conversions, and presents practical algorithms to compute UV counterterms and to obtain finite large-momentum and large-mass asymptotic expansions of Feynman integrals. The results extend the applicability of the forest technique to both UV and IR divergences, enable massless-propagator reductions for counterterm evaluation, and give a rigorous basis for the asymptotic expansion theorems (including $R^*$-normalized integrals) that are broadly useful in multi-loop perturbative QFT. The framework yields concrete, implementable procedures for high-loop calculations in gauge theories and two-dimensional models, with implications for RG functions and short-distance expansions.

Abstract

A generalization of the forest technique procedure --- the $R^{-1}$-operation---is elaborated and then employed to treat a variety of problems. First, it is employed to reveal the underlying simple structure of the Bogoliubov-Parasiuk renormalization prescription based on momentum subtractions. Second, we use this structure to derive a generalized Zimmermann identity connecting two different renormalized versions of a given Feynman integral. Third, the recursive procedure to minimally subtract the ultraviolet and infrared divergences from euclidean, dimensionally regularized Feynman integrals---the $R^*$-operation--- is simplified by reformulating it in terms of the R-operation alone. The new formulation is shown to lead immediately to a simple and regular algorithm for evaluating the overall ultraviolet divergences of arbitrary dimensionally regularized Feynman integrals, (including the ones appearing in two-dimensional field-theoretical models), the algorithm neatly reducing the problem to computing some massless propagator-type integrals. Finally, we construct a brief and concise proof of a general theorem which gives an explicitly finite large momenta and/or masses asymptotic expansion of an arbitrary (minimally subtracted) euclidean Feynman integral.