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 paper introduces a generalized forest technique for renormalization of Feynman integrals, simplifying divergence subtraction and deriving asymptotic expansions at large momenta and masses.

Contribution

It develops a generalized $R^{-1}$-operation, simplifies the $R^*$-operation, and provides a straightforward algorithm for asymptotic expansions of Feynman integrals.

Findings

Unified framework for ultraviolet and infrared divergence subtraction.

Simplified algorithm for evaluating divergences in arbitrary dimensions.

Explicit proof of asymptotic expansion for Feynman integrals.

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.