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

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 $R^{\ast }$-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.