Abstract: In this paper, we take a major step towards the construction and applications of an all-loop, all-multiplicity amplituhedron for three-dimensional planar N = 6 Chern-Simons matter theory, or the ABJM amplituhedron. We show that by simply changing the overall sign of the positive region of the original amplituhedron for four-dimensional planar N = 4 super- Yang-Mills (sYM) and performing a symplectic reduction, only three-dimensional kinematics in the middle sector of even-multiplicity survive. The resulting form of the geometry, combined with its parity images, gives the full loop integrand. This simple modification geometrically enforces the vanishing of odd-multiplicity cuts, and manifests the correct soft cuts as well as two-particle unitarity cuts. Furthermore, the so-called “bipartite structures” of four-point all-loop negative geometries also directly generalize to all multiplicities. We introduce a novel approach for triangulating loop amplituhedra based on the kinematics of the tree region, resulting in local integrands tailored to “prescriptive unitarity”. This construction sheds fascinating new light on the interplay between loop and tree amplituhedra for both ABJM and N = 4 sYM: the loop geometry demands that the tree region must be dissected into chambers, defined by the simultaneous positivity of maximal cuts. The loop geometry is then the “fibration” of the tree region. Using the new construction, we give explicit results of one-loop integrands up to ten points and two-loop integrands up to eight points by computing the canonical form of ABJM loop amplituhedron. Keywords: Chern-Simons Theories, Scattering Amplitudes, Supersymmetric Gauge Theory ArXiv ePrint: 2306.00951 Open Access, © The Authors. Article funded by SCOAP3. https://doi.org/10.1007/JHEP04(2024)064 JHEP04(2024)064 In the Acknowledgments section of the original paper the information related to author SH needs to be modified from “SH thanks IAS (Princeton) for hospitality during the completion of the work; his research is supported in part by National Natural Science Foundation of China under Grant No. 11935013, 12047502, 12047503, 12247103, 12225510.” to the following: “SH thanks IAS (Princeton) for hospitality during the completion of the work; his research is supported in part by the National Natural Science Foundation of China under Grant No. 12225510, 11935013, 12047503, 12247103, and by the New Cornerstone Science Foundation through the XPLORER PRIZE.” The original article has been updated.

We describe a general formalism based on the partial-wave decomposition to compute the iterative s-channel discontinuity of four-point amplitudes at any loop order. As an application, we focus on the low-energy expansions of type I and II superstring amplitudes. Besides providing new results for their leading and sub-leading logarithmic contributions beyond genus one, our approach elucidates the general structure of non-analytic threshold terms. In the case of open strings, the use of orthogonal colour projectors allows us to efficiently compute all contributions from different worldsheet topologies at a given loop order.

In this letter, we consider a positive geometry conjectured to encode the loop integrand of four-point stress-energy correlators in planar N=4 super Yang-Mills. Beginning with four lines in twistor space, we characterize a positive subspace to which an ℓ-loop geometry is attached. The loop geometry then consists of ℓ lines in twistor space satisfying positivity conditions among themselves and with respect to the base. Consequently, the loop geometry can be viewed as fibration over a tree geometry. The fibration naturally dissects the base into chambers, in which the degree-4ℓ loop form is unique and distinct for each chamber. Interestingly, up to three loops, the chambers are simply organized by the six ordering of x1,22x3,42, x1,42x2,32, and x1,32x2,42. We explicitly verify our conjecture by computing the loop-forms in terms of a basis of planar conformal integrals up to ℓ=3, which indeed yield correct loop integrands for the four-point correlator.

Pei-Lin Lee serves as Clinical Associate Professor, School of Medicine, National Taiwan University; Consultant, Center of Sleep Disorder, National Taiwan University Hospital. Her current academic positions at international sleep societies include American Academy of Sleep Medicine Fellow and Co-Chair International Assembly; Asian Society of Sleep Medicine, Sleep Medicine Education Task Force committee member. Her current research focuses on the era of new technology and big data in sleep medicine; and intervention on sleep and metabolism in sleep disordered breathing.