Counting master integrals: integration-by-parts procedure with effective mass

We show that the new relation between master integrals recently obtained in Ref. [1] can be reproduced using the integration-by-parts technique implemented with an effective mass. In fact, this relation is recovered as a special case of a whole family of new relations between master integrals.

Recently, one of us in collaboration with Mikhail Kalmykov found [1] a new relation between some specific Feynman integrals, which is actually absent in modern computer programs based on the integration-by-parts (IBP) technique [2] (for a recent review, see Ref. [3]). The new relation arises in the framework of the socalled differential reduction (see Refs. [1,4] and references cited therein) developed by these authors during last several years. This decreases the number of master integrals and, thus, leads to a simplification of calculations.
In this short note, we recover this relation directly in the framework of the IBP technique by introducing an effective mass originating from the reduction of oneloop integrals to simple propagators (see Refs. [5,6] and Eq. (2) below). In fact, this relation is found to be a special case of a whole family of new relations between master integrals. Following Ref. [1], let us consider the two-loop self-energy sunset-type diagram J 012 with on-shell kinematics, defined as where n = 4 − 2ε is the dimensionality of space time. It is depicted in Fig. 1.
Considering the standard Feynman representation of the following one-loop diagram as a one-fold integral with s = 1 − s, we can interpret this as an integral over a new propagator with the effective mass M 2 1 /s + M 2 2 /s. In previous papers [5,6], this procedure was used to decrease the numbers of loops in analyses of different types of master integrals and, thus, to simplify calculations.
Using Eq. (2) with M 1 = M and M 2 = 0, we can represent the considered two-loop diagram J 012 (σ, β, α) as the one-fold integral where is a one-loop on-shell diagram.
In conclusion, applying the IBP procedure to a one-loop integral with an effective mass in one of its propagators, we produced the new relation (8) between ordinary IBP master integrals. This relation coincides for σ = β = 1 with the one recently discovered in Ref. [1], but it is more general and obtained in a more straightforward way. We intend to extend this analysis to the case of the off-shell sunset diagrams in a future work. The effective-mass procedure applied here to reduce the number of master integrals with respect to the one achieved by the ordinary IBP procedure may in principle be applied whenever the considered topology contains a bubble subdiagram. We expect that relations between ordinary IBP master integrals thus obtained may be usefully implemented in modern computer packages based on the IBP procedure.