Building Bases of Loop Integrands

Abstract

We describe a systematic approach to the construction of loop-integrand bases at arbitrary loop-order, sufficient for the representation of general quantum field theories. We provide a graph-theoretic definition of `power-counting' for multi-loop integrands beyond the planar limit, and show how this can be used to organize bases according to ultraviolet behavior. This allows amplitude integrands to be constructed iteratively. We illustrate these ideas with concrete applications. In particular, we describe complete integrand bases at two loops sufficient to represent arbitrary-multiplicity amplitudes in four (or fewer) dimensions in any massless quantum field theory with the ultraviolet behavior of the Standard Model or better. We also comment on possible extensions of our framework to arbitrary (including regulated) numbers of dimensions, and to theories with arbitrary mass spectra and charges. At three loops, we describe a basis sufficient to capture all `leading-(transcendental-)weight' contributions of any four-dimensional quantum theory; for maximally supersymmetric Yang-Mills theory, this basis should be sufficient to represent all scattering amplitude integrands in the theory---for generic helicities and arbitrary multiplicity.

arXiv comment76 pages, 6 tables, hundreds of figures. Ancillary file includes our results for three loops

References in corpus (115)

• Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory
• One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits
• Review of AdS/CFT Integrability: An Overview
• Fusing Gauge Theory Tree Amplitudes Into Loop Amplitudes
• New Recursion Relations for Tree Amplitudes of Gluons
• Multiloop integrals in dimensional regularization made simple
• Gluon scattering amplitudes at strong coupling
• Differential Equations for Two-Loop Four-Point Functions
• Generalized Unitarity and One-Loop Amplitudes in N=4 Super-Yang-Mills
• Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond
• The Singular Behaviour of QCD Amplitudes at Two-loop Order
• Reducing full one-loop amplitudes to scalar integrals at the integrand level
• Dimensionally Regulated Pentagon Integrals
• Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory
• Reduction schemes for one-loop tensor integrals
• The Amplituhedron
• Magic identities for conformal four-point integrals
• Algorithm FIRE -- Feynman Integral REduction
• One-Loop Amplitudes for e^+ e^- to Four Partons
• An Automated Implementation of On-Shell Methods for One-Loop Amplitudes
• Infrared singularities of scattering amplitudes in perturbative QCD
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• Local Integrals for Planar Scattering Amplitudes
• Multi-loop Amplitudes and Resummation
• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• Projectors, Shadows, and Conformal Blocks
• Non-planar massless two-loop Feynman diagrams with four on-shell legs
• Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes
• Massive Loop Amplitudes from Unitarity
• Dimensionally Regulated One-Loop Integrals
• Reduction of one-loop tensor 5-point integrals
• Progress in One-Loop QCD Computations
• Two-Loop Four-Gluon Amplitudes in N=4 Super-Yang-Mills
• Three-Loop Superfiniteness of N=8 Supergravity
• Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
• Scattering amplitudes over finite fields and multivariate functional reconstruction
• The Two-loop Soft Anomalous Dimension Matrix and Resummation at Next-to-next-to Leading Pole
• One-Loop Self-Dual and N=4 Super Yang-Mills
• FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs
• One-loop calculations in quantum field theory: from Feynman diagrams to unitarity cuts
• Towards a Basis for Planar Two-Loop Integrals
• Factorization in One-Loop Gauge Theory
• Scattering AMplitudes from Unitarity-based Reduction Algorithm at the Integrand-level
• Five-Point Amplitudes in N=4 Super-Yang-Mills Theory and N=8 Supergravity
• Supersymmetric Regularization, Two-Loop QCD Amplitudes and Coupling Shifts
• Two-loop Integrand Decomposition into Master Integrals and Surface Terms
• Six-dimensional Methods for Four-dimensional Conformal Field Theories
• Universal structure of subleading infrared poles in gauge theory amplitudes
• Uniqueness of two-loop master contours
• Manifest Ultraviolet Behavior for the Three-Loop Four-Point Amplitude of N=8 Supergravity
• Three-loop corrections to the soft anomalous dimension in multi-leg scattering
• Is N = 8 Supergravity Ultraviolet Finite?
• A Two-Loop Four-Gluon Helicity Amplitude in QCD
• A Two-Loop Five-Gluon Helicity Amplitude in QCD
• Pentagon functions for massless planar scattering amplitudes
• Ultraviolet Properties of N = 8 Supergravity at Five Loops
• The Complete Four-Loop Four-Point Amplitude in N=4 Super-Yang-Mills Theory
• Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD
• Basics of Generalized Unitarity
• The Elliptic Double-Box Integral: Massless Amplitudes Beyond Polylogarithms
• Analytic Form of the Planar Two-Loop Five-Gluon Scattering Amplitudes in QCD
• A quasi-finite basis for multi-loop Feynman integrals
• Scattering Amplitudes from Multivariate Polynomial Division
• Traintracks Through Calabi-Yaus: Amplitudes Beyond Elliptic Polylogarithms
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• Maximal Unitarity at Two Loops
• The Five-Loop Four-Point Integrand of N=8 Supergravity as a Generalized Double Copy
• Constructing d-log integrands and computing master integrals for three-loop four-particle scattering
• On the Singularity Structure of Maximally Supersymmetric Scattering Amplitudes
• Two-Loop Four-Gluon Amplitudes with the Numerical Unitarity Method
• Sharpening The Leading Singularity
• The No-Triangle Hypothesis for N=8 Supergravity
• One-Loop Gluon Scattering Amplitudes in Theories with $N<4$ Supersymmetries
• Amplitudes and Correlators to Ten Loops Using Simple, Graphical Bootstraps
• Evidence for a Nonplanar Amplituhedron
• New differential equations for on-shell loop integrals
• Prescriptive Unitarity
• Absence of Triangles in Maximal Supergravity Amplitudes
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• The Soft-Collinear Bootstrap: N=4 Yang-Mills Amplitudes at Six and Seven Loops
• One-Loop n-Point Helicity Amplitudes in (Self-Dual) Gravity
• Adaptive Integrand Decomposition in parallel and orthogonal space
• Perturbation Theory at Eight Loops: Novel Structures and the Breakdown of Manifest Conformality
• Local Integrand Representations of All Two-Loop Amplitudes in Planar SYM
• Rationalizing Loop Integration
• Mellin Amplitudes for Dual Conformal Integrals
• The Five-Loop Four-Point Amplitude of N=4 super-Yang-Mills Theory
• Scattering Amplitudes -- Wilson Loops Duality for the First Non-planar Correction
• Multiloop Integrand Reduction for Dimensionally Regulated Amplitudes
• Gravity On-shell Diagrams
• Star Integrals, Convolutions and Simplices
• Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams
• Positroids, Plabic Graphs, and Scattering Amplitudes in Mathematica
• An Integrand Reconstruction Method for Three-Loop Amplitudes
• The classification of two-loop integrand basis in pure four-dimension
• Unitarity-based Techniques for One-Loop Calculations in QCD
• The Two-Loop Four-Graviton Scattering Amplitudes
• Efficient Tree-Amplitudes in N=4: Automatic BCFW Recursion in Mathematica
• Dual Conformal Structure Beyond the Planar Limit
• All-Multiplicity Non-Planar MHV Amplitudes in sYM at Two Loops
• Implications of nonplanar dual conformal symmetry
• Prescriptive Unitarity for Non-Planar Six-Particle Amplitudes at Two Loops
• Amplitudes at Infinity
• Manifestly Dual-Conformal Loop Integration
• Traintrack Calabi-Yaus from Twistor Geometry
• Maximal Unitarity for the Four-Mass Double Box
• Counting to one: reducibility of one- and two-loop amplitudes at the integrand level
• Lecture Notes on Multi-loop Integral Reduction and Applied Algebraic Geometry
• On integrands and loop momentum in string and field theory
• Unitarity Cuts of Integrals with Doubled Propagators
• UV cancelations in gravity loop integrands
• Rooting Out Letters: Octagonal Symbol Alphabets and Algebraic Number Theory
• On the vanishing of certain cuts or residues of loop integrals with higher powers of the propagators
• Multi-loop Integrand Reduction with Computational Algebraic Geometry
• Conformally Regulated Direct Integration of the Two-Loop Heptagon Remainder

Cited by in corpus (16)

• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• Implications of the Landau Equations for Iterated Integrals
• All-Multiplicity Non-Planar MHV Amplitudes in sYM at Two Loops
• Prescriptive Unitarity with Elliptic Leading Singularities
• An Elliptic Yangian-Invariant, `Leading Singularity'
• Radiative Classical Gravitational Observables at $\mathcal{O}(G^3)$ from Scattering Amplitudes
• An Infinite Family of Elliptic Ladder Integrals
• Gravity loop integrands from the ultraviolet
• Maximal transcendental weight contribution of scattering amplitudes
• Illustrations of Integrand-Basis Building at Two Loops
• On the finite basis topologies for multi-loop high-multiplicity Feynman integrals
• All Two-Loop, Color-Dressed, Six-Point Amplitude Integrands in sYM
• Locally Finite Observables in sYM
• Loop Amplitudes in the Coulomb Branch of $\mathcal{N}=4$ Super-Yang-Mills Theory
• Integrands of Less-Supersymmetric Yang-Mills at One Loop
• The Stratification of Rigidity