The dimension-shift conjecture for one-loop amplitudes
Résumé
A conjecture made by Bern, Dixon, Dunbar, and Kosower asserts a simple dimension shifting relationship between the one-loop structure of $ \mathcal{N} $ = 4 MHV amplitudes and all-plus helicity amplitudes in pure Yang-Mills theory. We prove this conjecture to all orders in dimensional regularisation using unitarity cuts, and evaluate the form of these simplest one-loop amplitudes using a generalised D-dimensional unitarity technique which captures the full amplitude to all multiplicities.