Landau Discriminants

Abstract

Scattering amplitudes in quantum field theories have intricate analytic properties as functions of the energies and momenta of the scattered particles. In perturbation theory, their singularities are governed by a set of nonlinear polynomial equations, known as Landau equations, for each individual Feynman diagram. The singularity locus of the associated Feynman integral is made precise with the notion of the Landau discriminant, which characterizes when the Landau equations admit a solution. In order to compute this discriminant, we present approaches from classical elimination theory, as well as a numerical algorithm based on homotopy continuation. These methods allow us to compute Landau discriminants of various Feynman diagrams up to 3 loops, which were previously out of reach. For instance, the Landau discriminant of the envelope diagram is a reducible surface of degree 45 in the three-dimensional space of kinematic invariants. We investigate geometric properties of the Landau discriminant, such as irreducibility, dimension and degree. In particular, we find simple examples in which the Landau discriminant has codimension greater than one. Furthermore, we describe a numerical procedure for determining which parts of the Landau discriminant lie in the physical regions. In order to study degenerate limits of Landau equations and bounds on the degree of the Landau discriminant, we introduce Landau polytopes and study their facet structure. Finally, we provide an efficient numerical algorithm for the computation of the number of master integrals based on the connection to algebraic statistics. The algorithms used in this work are implemented in the open-source Julia package Landau.jl available at https://mathrepo.mis.mpg.de/Landau/.

67 pages, 8 figures, journal version

References in corpus (38)

• Causality, Analyticity and an IR Obstruction to UV Completion
• Motivic Amplitudes and Cluster Coordinates
• Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
• Positive Moments for Scattering Amplitudes
• New positivity bounds from full crossing symmetry
• Feynman Integrals and Intersection Theory
• Extremal Effective Field Theories
• Vector Space of Feynman Integrals and Multivariate Intersection Numbers
• On Motives Associated to Graph Polynomials
• Scattering Amplitudes from Intersection Theory
• Geometric approach to asymptotic expansion of Feynman integrals
• Critical points and number of master integrals
• Six-Gluon Amplitudes in Planar $\mathcal{N}=4$ Super-Yang-Mills Theory at Six and Seven Loops
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• Decomposition of Feynman Integrals by Multivariate Intersection Numbers
• The maximum likelihood degree of a very affine variety
• S-matrix Bootstrap for Effective Field Theories: Massless Pions
• Cuts from residues: the one-loop case
• Cluster algebras for Feynman integrals
• Feynman integral relations from parametric annihilators
• The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods
• Duals of Feynman Integrals, I: Differential Equations
• Feynman integrals as A-hypergeometric functions
• From Infinity to Four Dimensions: Higher Residue Pairings and Feynman Integrals
• On the computation of intersection numbers for twisted cocycles
• How tropical are seven- and eight-particle amplitudes?
• Crossing Symmetry in the Planar Limit
• Implications of the Landau Equations for Iterated Integrals
• Feynman Polytopes and the Tropical Geometry of UV and IR Divergences
• Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems
• On the status of expansion by regions
• All-loop singularities of scattering amplitudes in massless planar theories
• Unveiling Regions in multi-scale Feynman Integrals using Singularities and Power Geometry
• Rigorous bounds on the Analytic $S$-matrix
• GKZ-hypergeometric systems for Feynman integrals
• Counting the number of master integrals for sunrise-type Feynman diagrams via Mellin-Barnes representation
• Symbol Alphabets from Plabic Graphs
• Bounds on Crossing Symmetry

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• Loop-by-loop Differential Equations for Dual (Elliptic) Feynman Integrals
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• Exposing the threshold structure of loop integrals
• Towards Analytic Structure of Feynman Parameter Integrals with Rational Curves
• Natural Boundaries for Scattering Amplitudes
• Feynman Integrals of Grassmannians
• Hierarchies in relative Picard-Lefschetz theory
• Probing multi-particle unitarity with the Landau equations
• Cutkosky's Theorem for Massive One-Loop Feynman Integrals -- Part 1