From multiple unitarity cuts to the coproduct of Feynman integrals

Abstract

We develop techniques for computing and analyzing multiple unitarity cuts of Feynman integrals, and reconstructing the integral from these cuts. We study the relations among unitarity cuts of a Feynman integral computed via diagrammatic cutting rules, the discontinuity across the corresponding branch cut, and the coproduct of the integral. For single unitarity cuts, these relations are familiar. Here we show that they can be generalized to sequences of unitarity cuts in different channels. Using concrete one-and two-loop scalar integral examples we demonstrate that it is possible to reconstruct a Feynman integral from either single or double unitarity cuts. Our results offer insight into the analytic structure of Feynman integrals as well as a new approach to computing them.

Cuts from residues: the one-loop case

“…Combining the two cases, we see that 25) where θ(x) denotes the Heaviside step function. We see that the two notions of discontinuities carry equivalent information, at least in the case where the branch point lies on the real axis.…”

“…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.…”

“…Such precise definitions exist for certain types of cuts. 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].…”

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

“…Although the latter involve double pentagon integrals which are superficially more complicated than the examples studied in refs. [8,9], our analysis benefits from the fact that the kinematic domain for scattering amplitudes in SYM theory is much simpler than in general field theories.…”

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