Integrated negative geometries in ABJM

We study, in the context of the three-dimensional N = 6 Chern-Simonsmatter (ABJM) theory, the infrared-finite functions that result from performing L - 1 loop integrations over the L-loop integrand of the logarithm of the four-particle scattering amplitude. Our starting point are the integrands obtained from the recently proposed all-loop projected amplituhedron for the ABJM theory. Organizing them in terms of negative geometries ensures that no divergences occur upon integration if at least one loop variable is left unintegrated. We explicitly perform the integrations up to L = 3, finding both parity-even and-odd terms. Moreover, we discuss a prescription to compute the cusp anomalous dimension Gcusp of ABJM in terms of the integrated negative geometries, and we use it to reproduce the first non-trivial order of Gcusp. Finally, we show that the leading singularities that characterize the integrated results are conformally invariant.

Automated summary

In this study, we investigate the infrared-finite functions obtained from integrating the logarithm of the four-particle scattering amplitude over L - 1 loops in the three-dimensional N = 6 Chern-Simonsmatter (ABJM) theory. By organizing the integrands using negative geometries, we ensure that the integrals do not diverge if at least one loop variable is left unintegrated. We explicitly perform the integrations up to L = 3 and find both parity-even and-odd terms. Additionally, we propose a prescription to compute the cusp anomalous dimension Gcusp of ABJM using the integrated negative geometries and successfully reproduce its first non-trivial order. Finally, we demonstrate that the leading singularities in the integrated results are conformally invariant.