Differential equations on unitarity cut surfaces

Abstract

We reformulate differential equations (DEs) for Feynman integrals to avoid doubled propagators in intermediate steps. External momentum derivatives are dressed with loop momentum derivatives to form tangent vectors to unitarity cut surfaces, in a way inspired by unitarity-compatible IBP reduction. For the one-loop box, our method directly produces the final DEs without any integration-by-parts reduction. We further illustrate the method by deriving maximal-cut level differential equations for two-loop nonplanar five-point integrals, whose exact expressions are yet unknown. We speed up the computation using finite field techniques and rational function reconstruction.

arXiv comment17 pages, 3 figures; v2: added more results and references, final journal version

References in corpus (48)

• Fusing Gauge Theory Tree Amplitudes Into Loop Amplitudes
• Multiloop integrals in dimensional regularization made simple
• Differential Equations for Two-Loop Four-Point Functions
• Generalized Unitarity and One-Loop Amplitudes in N=4 Super-Yang-Mills
• Reducing full one-loop amplitudes to scalar integrals at the integrand level
• Differential Equations for Feynman Graph Amplitudes
• Dimensionally Regulated Pentagon Integrals
• Algorithm FIRE -- Feynman Integral REduction
• One-Loop Amplitudes for e^+ e^- to Four Partons
• The analytical value of the electron (g-2) at order alpha^3 in QED
• FIRE5: a C++ implementation of Feynman Integral REduction
• Local Integrals for Planar Scattering Amplitudes
• Massive Loop Amplitudes from Unitarity
• Lectures on differential equations for Feynman integrals
• Direct extraction of one-loop integral coefficients
• A novel approach to integration by parts reduction
• Scattering amplitudes over finite fields and multivariate functional reconstruction
• One-Loop Self-Dual and N=4 Super Yang-Mills
• One-loop calculations in quantum field theory: from Feynman diagrams to unitarity cuts
• Towards a Basis for Planar Two-Loop Integrals
• Feynman Diagrams and Differential Equations
• Two-loop Integrand Decomposition into Master Integrals and Surface Terms
• Integration-by-parts reductions from unitarity cuts and algebraic geometry
• Uniqueness of two-loop master contours
• Explicit solutions of the multi-loop integral recurrence relations and its application
• Explicit solutions of the 3--loop vacuum integral recurrence relations
• Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD
• On the maximal cut of Feynman integrals and the solution of their differential equations
• Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA
• A Two-Loop Five-Gluon Helicity Amplitude in QCD
• Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form
• Analytical Results for Dimensionally Regularized Massless On-shell Double Boxes with Arbitrary Indices and Numerators
• Cuts of Feynman Integrals in Baikov representation
• epsilon: A tool to find a canonical basis of master integrals
• A New Algorithm For The Generation Of Unitarity-Compatible Integration By Parts Relations
• Evaluating single-scale and/or non-planar diagrams by differential equations
• Transforming differential equations of multi-loop Feynman integrals into canonical form
• Evidence for a Nonplanar Amplituhedron
• Azurite: An algebraic geometry based package for finding bases of loop integrals
• Quark and gluon form factors to four loop order in QCD: the $N_f^3$ contributions
• Integration by parts: An introduction
• Elliptic Functions and Maximal Unitarity
• The Dimensional Recurrence and Analyticity Method for Multicomponent Master Integrals: Using Unitarity Cuts to Construct Homogeneous Solutions
• Susy Theories and QCD: Numerical Approaches
• Integration by parts identities in integer numbers of dimensions. A criterion for decoupling systems of differential equations
• Lecture Notes on Multi-loop Integral Reduction and Applied Algebraic Geometry
• Unitarity Cuts of Integrals with Doubled Propagators
• Syzygies Probing Scattering Amplitudes

Cited by in corpus (22)

• Feynman Integrals and Intersection Theory
• Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA
• Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• NNLO QCD corrections to three-photon production at the LHC
• Analytic helicity amplitudes for two-loop five-gluon scattering: the single-minus case
• Two-Loop Integrals for Planar Five-Point One-Mass Processes
• Diagrammatic Hopf algebra of cut Feynman integrals: the one-loop case
• Differential equations from unitarity cuts: nonplanar hexa-box integrals
• Maximal Cuts in Arbitrary Dimension
• Direct Solution of Integration-by-Parts Systems
• Dual Conformal Symmetry, Integration-by-Parts Reduction, Differential Equations and the Nonplanar Sector
• Mathematical properties of nested residues and their application to multi-loop scattering amplitudes
• Dual Conformal Structure Beyond the Planar Limit
• Differential equations for loop integrals in Baikov representation
• Extracting analytical one-loop amplitudes from numerical evaluations
• Maximal transcendental weight contribution of scattering amplitudes
• Generalized unitarity method for unstable particles
• Integrated Unitarity for Scattering Amplitudes
• Two-loop D-dimensional unitarity and dual conformal symmetry
• A stroll through the loop-tree duality
• Leading singularities in higher-derivative Yang-Mills theory and quadratic gravity