Abstract
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.
Highlights
In 1996, Bern, Dixon, Dunbar and Kosower (BDDK) [23] conjectured a relation between the simplest one-loop gluon amplitudes in these two theories
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
The AP amplitude has been explicitly shown to be equivalent to the one-loop amplitude in self-dual Yang-Mills theory: the latter was built from the Lagrangian by Cangemi [24], as well as by Chalmers and Siegel [25] which built on previous work by Bardeen [26], who computed four and five-point amplitudes directly from all-plus tree level amplitudes with two off-shell legs, which is very close to the analysis we carry out in section 2 to all multiplicity
Summary
The two sides of the conjecture (1.1) were computed in different ways by BDDK: the all-epsilon structure of MHV amplitude was computed using the string-based formalism whereas AP amplitude, was computed with a prototypical version of D-dimensional unitarity This is effectively equivalent to taking unitarity cuts with massive on-shell states, applying the techniques of Bern and Morgan [49] to compute the expressions in equations (1.10), (1.11) and (1.12); a technique made especially simple thanks to the equivalence in equation (1.9). This method is directly analogous to the one first introduced for four-dimensional cuts in [7]; it produces a compact result thanks to the simple form of a solution to the on-shell conditions for a given pentagon, I5[i1,i2,i3,i4,i5].
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