All-loop singularities of scattering amplitudes in massless planar theories

Abstract

In massless quantum field theories the Landau equations are invariant under graph operations familiar from the theory of electrical circuits. Using a theorem on the $Y$-$\Delta$ reducibility of planar circuits we prove that the set of first-type Landau singularities of an $n$-particle scattering amplitude in any massless planar theory, in any spacetime dimension $D$, at any finite loop order in perturbation theory, is a subset of those of a certain $n$-particle $\lfloor (n-2{)}^2/4\rfloor$-loop "ziggurat" graph. We determine this singularity locus explicitly for $D=4$ and $n=6$ and find that it corresponds precisely to the vanishing of the symbol letters familiar from the hexagon bootstrap in SYM theory. Further implications for SYM theory are discussed.

arXiv comment6 pages, 4 figures; v2: minor improvements and clarifications, including an expanded section VIII

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Cited by in corpus (23)

• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• Landau Discriminants
• Implications of the Landau Equations for Iterated Integrals
• Crossing Symmetry in the Planar Limit
• What is the $i\epsilon$ for the S-matrix?
• Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry
• Principal Landau Determinants
• Symbol Alphabets from Plabic Graphs
• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
• Symbol Alphabets from the Landau Singular Locus
• The Sklyanin Bracket and Cluster Adjacency at All Multiplicity
• Symbol Alphabets from Tensor Diagrams
• Constraints on Sequential Discontinuities from the Geometry of On-shell Spaces
• Algebraic branch points at all loop orders from positive kinematics and wall crossing
• A Note on Letters of Yangian Invariants
• The Two-Loop Remainder Function for Eight and Nine Particles
• Bounds on Crossing Symmetry
• Symbol Alphabets from Plabic Graphs II: Rational Letters
• Mirror channel eigenvectors of the $d$-dimensional fishnets
• Canonical differential equations for the elliptic two-loop five-point integral family relevant to $t\bar{t}+$jet production at leading colour
• Emergent unitarity from the amplituhedron
• SymBuild: a package for the computation of integrable symbols in scattering amplitudes
• Peaks and cusps: anomalous thresholds and LHC physics