Building bases of loop integrands
Abstract
We describe a systematic approach to the construction of loopintegrand bases at arbitrary looporder, sufficient for the representation of general quantum field theories. We provide a graphtheoretic definition of `powercounting' for multiloop integrands beyond the planar limit, and show how this can be used to organize bases according to ultraviolet behavior. This allows amplitude integrands to be constructed iteratively. We illustrate these ideas with concrete applications. In particular, we describe complete integrand bases at two loops sufficient to represent arbitrarymultiplicity amplitudes in four (or fewer) dimensions in any massless quantum field theory with the ultraviolet behavior of the Standard Model or better. We also comment on possible extensions of our framework to arbitrary (including regulated) numbers of dimensions, and to theories with arbitrary mass spectra and charges. At three loops, we describe a basis sufficient to capture all `leading(transcendental)weight' contributions of any fourdimensional quantum theory; for maximally supersymmetric YangMills theory, this basis should be sufficient to represent all scattering amplitude integrands in the theory — for generic helicities and arbitrary multiplicity.
 Publication:

Journal of High Energy Physics
 Pub Date:
 November 2020
 DOI:
 10.1007/JHEP11(2020)116
 arXiv:
 arXiv:2007.13905
 Bibcode:
 2020JHEP...11..116B
 Keywords:

 Scattering Amplitudes;
 1/N Expansion;
 Gauge Symmetry;
 High Energy Physics  Theory
 EPrint:
 76 pages, 6 tables, hundreds of figures. Ancillary file includes our results for three loops