Manifestly dual-conformal loop integration

Abstract

Local, manifestly dual-conformally invariant loop integrands are now known for all finite quantities associated with observables in planar, maximally supersymmetric Yang-Mills theory through three loops. These representations, however, are not infrared-finite term by term and therefore require regularization; and even using a regulator consistent with dual-conformal invariance, ordinary methods of loop integration would naïvely obscure this symmetry. In this work, we show how any planar loop integral through at least two loops can be systematically regulated and evaluated directly in terms of strictly finite, manifestly dual-conformal Feynmanparameter integrals. We apply these methods to the case of the two-loop ratio and remainder functions for six particles, reproducing the known results in terms of individually regulated local loop integrals, and we comment on some of the novelties that arise for this regularization scheme not previously seen at one loop.

Yangian bootstrap for conformal Feynman integrals

“…In fact, this can also be observed directly by inspecting the recurrence relations (45,46). Note that if we move x or y slightly away from zero, the sum in equation (48) will extend over all of Z and diverge.…”

“…For a more systematic understanding, note that all 48 possible shifts of this type can be generated from the following three relations: The first two of these shifts correspond to the generators σ 1 , σ 2 of the permutation group, respectively. In order to derive relations such as (55) for all 12 series representations of the solutions of the Appell PDEs, we note the following shifts: In this appendix we list (dual conformal) Feynman parameter representations [43][44][45] for the integrals discussed in this paper. In their most general form, the integrals read…”

Conformally-regulated direct integration of the two-loop heptagon remainder

“…First, we should clarify why we expected I 5 to be pure despite its representation. Although the conformal regulator is known to spoil an integrand's purity (see the discussion in [70]), we strongly expect the logarithm of the amplitude (the cyclic sum of all seeds) to be pure; as {I 1 , . .…”

“…, z 6 }: its evaluation will be the same for seven particles as it was for six. More specifically, I 1 is essentially identical to what was computed (as part of what was called 'I 15 ') in [70] Notice that we are reserving calligraphic symbols to denote integrands and italic symbols to indicate integrals.…”

“…Among the integration techniques required are those that allow us to extract the leading terms in the limit of δ → 0 + (for the integrals which require regularization). We were able to effectively use the methods discussed in [70]; we refer the reader to appendix B.1 and the ancillary files of that work for a more thorough explanation and illustrative examples.…”