Landau and cluster structures of one-loop amplitudes in 𝒩=4 SYM in dimensional regularization
Abstract
Generalized unitarity, Landau analysis, and cluster adjacency encode complementary aspects of scattering amplitudes. We use one-loop planar 𝒩=4 SYM amplitudes in dimensional regularization, at arbitrary multiplicity and helicity, to make their interface explicit. The weight-two symbol decomposes into an LS part, in which maximal-cut leading singularities furnish the coefficients and Landau loci associated with nested cuts organize the ordered symbol entries, an algebraic four-mass sector, and residual terms. Cancellations of certain letters contributed by individual box integrals, as well as further simplifications, are explained by the two-mass triangle relations among box coefficients. We prove these relations using a BCFW-like application of the global residue theorem and show that they can be understood geometrically as different dissections of the same region in the tree amplituhedron obtained by projecting the loop geometry of a triple cut. We then prove that the full rational symbol, including its infrared-divergent part, obeys cluster adjacency in the flag cluster algebra Fl2,4;n for all multiplicities and helicities. Within the sector depending only on momentum-twistor four-brackets, we conjecture a stronger cluster property in Gr(4,n) for all helicities and prove it for NMHV amplitudes: the amplitude admits a representation in which every pole of each coefficient is compatible with both symbol entries. Finally, we observe that the algebraic four-mass letters, although non-rational, exhibit a suggestive Sklyanin-bracket pattern.
Related
- Cusp regularization with spin structures and gravitational anomalies
- Topological defect trains excited by a boundary discontinuity in the two-dimensional complex Ginzburg-Landau equation
- Five-dimensional Geometry from Spinning Amplitudes
- Five-dimensional geometry from spinning amplitudes
- Functional Dimensional Regularization for the Lee-Yang Model
BibTeX
@misc{angelopoulou2026landau,
title = {{Landau and cluster structures of one-loop amplitudes in $\mathcal N=4$ SYM in dimensional regularization}},
author = {Athanasia-Konstantina Angelopoulou and Ruth Britto and Matteo Parisi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05716},
eprint = {2609.05716},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05716},
}
RIS
TY - UNPB
TI - Landau and cluster structures of one-loop amplitudes in 𝒩=4 SYM in dimensional regularization
AU - Athanasia-Konstantina Angelopoulou
AU - Ruth Britto
AU - Matteo Parisi
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05716
UR - https://arxiv.org/abs/2609.05716
AB - Generalized unitarity, Landau analysis, and cluster adjacency encode complementary aspects of scattering amplitudes. We use one-loop planar 𝒩=4 SYM amplitudes in dimensional regularization, at arbitrary multiplicity and helicity, to make their interface explicit. The weight-two symbol decomposes into an LS part, in which maximal-cut leading singularities furnish the coefficients and Landau loci associated with nested cuts organize the ordered symbol entries, an algebraic four-mass sector, and residual terms. Cancellations of certain letters contributed by individual box integrals, as well as further simplifications, are explained by the two-mass triangle relations among box coefficients. We prove these relations using a BCFW-like application of the global residue theorem and show that they can be understood geometrically as different dissections of the same region in the tree amplituhedron obtained by projecting the loop geometry of a triple cut. We then prove that the full rational symbol, including its infrared-divergent part, obeys cluster adjacency in the flag cluster algebra Fl_2,4;n for all multiplicities and helicities. Within the sector depending only on momentum-twistor four-brackets, we conjecture a stronger cluster property in Gr(4,n) for all helicities and prove it for NMHV amplitudes: the amplitude admits a representation in which every pole of each coefficient is compatible with both symbol entries. Finally, we observe that the algebraic four-mass letters, although non-rational, exhibit a suggestive Sklyanin-bracket pattern.
ER -