Abstract
Recently, Bern et al. observed that a certain class of next-to-planar Feynman integrals possess a bonus symmetry that is closely related to dual conformal symmetry. It corresponds to a projection of the latter along a certain lightlike direction. Previous studies were performed at the level of the loop integrand, and a Ward identity for the integral was formulated. We investigate the implications of the symmetry at the level of the integrated quantities. In particular, we focus on the phenomenologically important case of five-particle scattering. The symmetry simplifies the four-variable problem to a three-variable one. In the context of the recently proposed space of pentagon functions, the symmetry is much stronger. We find that it drastically reduces the allowed function space, leading to a well-known space of three-variable functions. Furthermore, we show how to use the symmetry in the presence of infrared divergences, where one obtains an anomalous Ward identity. We verify that the Ward identity is satisfied by the leading and subleading poles of several nontrivial five-particle integrals. Finally, we present examples of integrals that possess both ordinary and dual conformal symmetry.
Highlights
An important feature of planar scattering amplitudes in N = 4 sYM is that they have a hidden dualconformal and Yangian symmetry [4,5,6,7,8,9,10,11,12]
Recently, Bern et al observed that a certain class of next-to-planar Feynman integrals possess a bonus symmetry that is closely related to dual conformal symmetry
We show how to use the symmetry in the presence of infrared divergences, where one obtains an anomalous Ward identity
Summary
The notion of dual conformal invariance (DCI) for a Feynman integral relies on its dual space description. [13, 14] is that for this and similar nonplanar integrals one can preserve part of the dual conformal symmetry It corresponds to projecting the boost generator with the shift parameter, in our case p3 · K. The shifted and unshifted propagators transform in exactly the same way, so that We call this property directional dual conformal invariance (DDCI), meaning that the boost is projected on the lightlike direction of the external momentum p3. One should bear in mind that the trick of preserving part of the DCI can only work for nonplanar graphs that can be reduced to planar ones by removing a single external leg The reason for this is the key property (2.5): it will not work for more than one projection of the boost generator Kμ. This is very similar in spirit to the recent applications of conformal and superconformal symmetry to Feynman integrals [20, 21]
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