Landau and cluster structures of one-loop amplitudes in ${\mathcal N}=4$ SYM in dimensional regularization

Abstract

Generalized unitarity, Landau analysis, and cluster adjacency encode complementary aspects of scattering amplitudes. We use one-loop planar $\mathcal N=4$ 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 $\mathrm{Fl}_{2,4;n}$ for all multiplicities and helicities. Within the sector depending only on momentum-twistor four-brackets, we conjecture a stronger cluster property in $\operatorname{Gr}(4,n)$ 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.