Implications of the Landau Equations for Iterated Integrals

Abstract

We introduce a method for deriving constraints on the symbol of Feynman integrals from the form of their asymptotic expansions in the neighborhood of Landau loci. In particular, we show that the behavior of these integrals near singular points is directly related to the position in the symbol where one of the letters vanishes or becomes infinite. We illustrate this method on integrals with generic masses, and as a corollary prove the conjectured bound of $\lfloor \frac{D\ell }{2}\rfloor$ on the transcendental weight of polylogarithmic $\ell$-loop integrals of this type in integer numbers of dimensions $D$. We also derive new constraints on the kinematic dependence of certain products of symbol letters that remain finite near singular points.

arXiv comment5+2 pages; v2: minor clarifications and typos, updating to published version

References in corpus (45)

• Classical Polylogarithms for Amplitudes and Wilson Loops
• Reduction schemes for one-loop tensor integrals
• An Automated Implementation of On-Shell Methods for One-Loop Amplitudes
• Bootstrapping a Five-Loop Amplitude Using Steinmann Relations
• Superconformal symmetry and two-loop amplitudes in planar N=4 super Yang-Mills
• Heptagons from the Steinmann Cluster Bootstrap
• Jumpstarting the all-loop S-matrix of planar N=4 super Yang-Mills
• Elliptic polylogarithms and iterated integrals on elliptic curves I: general formalism
• The super-correlator/super-amplitude duality: Part I
• A Feynman integral via higher normal functions
• Pulling the straps of polygons
• Six-Gluon Amplitudes in Planar $\mathcal{N}=4$ Super-Yang-Mills Theory at Six and Seven Loops
• Elliptic Feynman integrals and pure functions
• A K3 in phi4
• Traintracks Through Calabi-Yaus: Amplitudes Beyond Elliptic Polylogarithms
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• Cluster adjacency properties of scattering amplitudes
• The Cosmic Galois Group and Extended Steinmann Relations for Planar $\mathcal{N}=4$ SYM Amplitudes
• Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series
• Bootstrapping pentagon functions
• Prescriptive Unitarity
• Cuts from residues: the one-loop case
• One-Loop Tensor Integrals in Dimensional Regularisation
• Cluster adjacency and the four-loop NMHV heptagon
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• From multiple unitarity cuts to the coproduct of Feynman integrals
• The Double Pentaladder Integral to All Orders
• A Three-Point Form Factor Through Five Loops
• Sequential Discontinuities of Feynman Integrals and the Monodromy Group
• Landau Singularities from the Amplituhedron
• Landau Singularities and Symbology: One- and Two-loop MHV Amplitudes in SYM Theory
• Two-loop Octagons, Algebraic Letters and $\bar{Q}$ Equations
• Bootstrapping two-loop Feynman integrals for planar N=4 sYM
• All-loop singularities of scattering amplitudes in massless planar theories
• All-Multiplicity Non-Planar MHV Amplitudes in sYM at Two Loops
• Boundaries of Amplituhedra and NMHV Symbol Alphabets at Two Loops
• Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops
• The symbol and alphabet of two-loop NMHV amplitudes from $\bar{Q}$ equations
• Notes on cluster algebras and some all-loop Feynman integrals
• Truncated cluster algebras and Feynman integrals with algebraic letters
• All-Mass $n$-gon Integrals in $n$ Dimensions
• All-Helicity Symbol Alphabets from Unwound Amplituhedra
• Building Bases of Loop Integrands
• Feynman Integrals and Scattering Amplitudes from Wilson Loops
• Comments on all-loop constraints for scattering amplitudes and Feynman integrals

Cited by in corpus (6)

• Landau Discriminants
• Antipodal Self-Duality for a Four-Particle Form Factor
• Two-loop master integrals for a planar topology contributing to $pp\to t\bar{t}j$
• Constraints on Sequential Discontinuities from the Geometry of On-shell Spaces
• One-loop hexagon integral to higher orders in the dimensional regulator
• Probing multi-particle unitarity with the Landau equations