Integrated negative geometries in ABJM

In this paper, we take a major step towards the construction and applications of an all-loop, all-multiplicity amplituhedron for three-dimensional planar $$ \mathcal{N} $$ 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 $$ \mathcal{N} $$ 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 $$ \mathcal{N} $$ 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.

Section: Jhep09(2023)165mentioning

confidence: 99%

“…and similarly for I b , I c ; note that we have 1 I a integral, 2 × 3 = 6 I b integrals, and 3 + 6 = 9 I c integrals (6 cyclic images of I c (13, 24) and 3 of I c (13,46), and in the end we symmetrize…”

Section: Let Us Verify the Above Description By Computing ωαmentioning

confidence: 99%

The ABJM Amplituhedron

He,

Huang,

Kuo

2023

“…We are grateful to Johaness Note added. While this work was in progress, we were made aware of an upcoming work [44], where the finite function up to L = 3 has been computed and the two results agree.…”

Section: Acknowledgmentsmentioning

confidence: 94%

Emergent unitarity, all-loop cuts and integrations from the ABJM amplituhedron

Kuo

et al. 2023

We elaborate on aspects of a new positive geometry proposed recently, which was conjectured to be the four-point amplituhedron for ABJM theory. We study generalized unitarity cuts from the geometry, and in particular we prove that (1) the four-point integrand satisfies perturbative unitarity (or optical theorem) to all loops, which follows directly from the geometry, and (2) vanishing cuts involving odd-point amplitudes follow from the “bipartite” nature of the associated “negative geometries”, which justifies their appearance in ABJM theory. We also take a first step in integrating the forms of these negative geometries and obtain an infrared-finite quantity up to two loops, from which we extract the cusp anomalous dimension at leading order.

“…The result of performing L − 1 of the loop integrations over the L-loop integrand for the logarithm of the amplitude has been previously studied at four points both in N = 4 sYM [19][20][21][22][23][24][25][26] and in ABJM [27,28], up to L = 4 and L = 3 respectively. Based on the Wilson loops/scattering amplitudes duality [29][30][31][32][33][34][35][36], prescriptions have been proposed to compute the cusp anomalous dimension Γ cusp of each theory by applying certain functionals on the integrated results [19,24,25,27,28]. This has allowed to obtain the full four-loop value of Γ cusp in N = 4 sYM, including the first non-planar corrections [25].…”

Section: Introductionmentioning

confidence: 99%

“…This has allowed to obtain the full four-loop value of Γ cusp in N = 4 sYM, including the first non-planar corrections [25]. In the ABJM case, these prescriptions were used to compute the three-loop contribution to Γ cusp [27,28], in agreement with the all-loop integrability-based proposals [37]. Moreover, the integrated results have been shown to have uniform transcendental weight in their coupling expansion [19,20,[23][24][25][26][27][28], and the corresponding leading singularities have been proven to be conformally invariant [26,27].…”

Section: Introductionmentioning

confidence: 99%

Higher-loop integrated negative geometries in ABJM

Lagares,

Zhang

2024

Self Cite

In the three-dimensional $$\mathcal{N}$$ = 6 Chern-Simons matter (ABJM) theory, the integrand for the logarithm of the scattering amplitude admits a decomposition in terms of negative geometries, which implies that all the infrared divergences concentrate in the last loop integration. We compute the infrared-finite functions that arise from performing a three-loop integration over the four-loop integrand for the logarithm of the four-point amplitude, for which we use the method of differential equations. Our results provide a direct computation of the four-loop cusp anomalous dimension of the theory, in agreement with the current all-loop integrability-based proposal. We find an apparent simplicity in the leading singularities of the integrated results, provided one works in the frame in which the unintegrated loop variable goes to infinity. Finally, our results suggest an alternating sign pattern for the integrated negative geometries in the Euclidean region.