Cuts from residues: the one-loop case

Abstract

Using the multivariate residue calculus of Leray, we give a precise definition of the notion of a cut Feynman integral in dimensional regularization, as a residue evaluated on the variety where some of the propagators are put on shell. These are naturally associated to Landau singularities of the first type. Focusing on the one-loop case, we give an explicit parametrization to compute such cut integrals, with which we study some of their properties and list explicit results for maximal and next-to-maximal cuts. By analyzing homology groups, we show that cut integrals associated to Landau singularities of the second type are specific combinations of the usual cut integrals, and we obtain linear relations among different cuts of the same integral. We also show that all one-loop Feynman integrals and their cuts belong to the same class of functions, which can be written as parametric integrals.

The diagrammatic coaction and the algebraic structure of cut Feynman integrals

“…The question of whether all these contours are independent can be answered in the context of homology theory. This was addressed in the 60s [14] for integer space-time dimensions, and we have checked that the conclusions still hold in dimensional regularisation [8]. The so-called Decomposition Theorem establishes that the contours satisfy the following relations:…”

“…(2.1) applies to a large class of integrals. Here, we will show how it applies to one-loop Feynman integrals in dimensional regularisation and explore some of the consequences [6,7,8]. This is a very convenient choice for testing the applicability of eq.…”

“…Indeed, we must evaluate integrals of the form 5) which correspond to setting the propagators in C on-shell. In the one-loop case, these integrals have been extensively discussed in [8]. We note that the relations in eqs.…”

“…(2.1), and we must show that it gives consistent results. We now briefly review the several checks discussed in [7].…”

“…[5]. Here, we discuss the coaction introduced in [6] and its application to one-loop Feynman integrals where it has a simple diagrammatic representation [7]. We illustrate how it constrains the algebraic structure of these functions by discussing its implications in the study of their discontinuities and the differential equations they admit.…”