Cuts and coproducts of massive triangle diagrams

Abstract

Relations between multiple unitarity cuts and coproducts of Feynman integrals are extended to allow for internal masses. These masses introduce new branch cuts, whose discontinuities can be derived by placing single propagators on shell and identified as particular entries of the coproduct. First entries of the coproduct are then seen to include mass invariants alone, as well as threshold corrections for external momentum channels. As in the massless case, the original integral can possibly be recovered from its cuts by starting with the known part of the coproduct and imposing integrability contraints. We formulate precise rules for cuts of diagrams, and we gather evidence for the relations to coproducts through a detailed study of one-loop triangle integrals with various combinations of external and internal masses.

Cuts from residues: the one-loop case

“…(4.15) can be done in closed form, and we obtain 26) which shows that [f n 1 ] i is well behaved as m 2 i → 0. This completes our proof of the vanishing of the massless single cut.…”

“…At one loop, every subset of two edges disconnects the graph, and so we obtain the well-known result that the discontinuity in a channel Q 2 is computed by the integral where two propagators are 'cut', i.e., replaced by δ functions. Similarly, it is known that one-propagator cuts compute the discontinuity in the mass of the cut propagator [26].…”

“…The motivation to study these objects is mostly driven by a desire to improve our understanding of the analytic structure of loop integrals, in particular in the light of novel mathematical developments and an improved understanding of the functions that appear in loop computations. For example, it was shown in concrete examples [25,26] that the coproduct of loop integrals can be cast in a form such that the rightmost entries are cut integrals. A complete understanding of this observation, however, requires a rigorous definition of the relevant integration contours and of how to evaluate the cut integrals, including for non-integer dimensions in order to work in dimensional regularization.…”

“…For instance, one can consider so-called unitarity cuts [3,21,22], which select a particular external channel, or iterated unitarity cuts [23][24][25] which select different channels. When propagators are massive, one can also consider single-propagator cuts [26]. These precise definitions are tailored to compute discontinuities in the variables identified with external channels or internal masses.…”

Landau singularities and symbology: one- and two-loop MHV amplitudes in SYM theory

“…Aspects of this connection have played a role in refs. [8,9], where various one-and two-loop examples were studied.…”