Integration Rules for Loop Scattering Equations

Abstract

We formulate new integration rules for one-loop scattering equations analogous to those at tree-level, and test them in a number of non-trivial cases for amplitudes in scalar ${\varphi }^3$-theory. This formalism greatly facilitates the evaluation of amplitudes in the CHY representation at one-loop order, without the need to explicitly sum over the solutions to the loop-level scattering equations.

22 pages, 17 figures

References in corpus (8)

• Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM
• The Momentum Kernel of Gauge and Gravity Theories
• Loop Integrands for Scattering Amplitudes from the Riemann Sphere
• Gravity and Yang-Mills Amplitude Relations
• Scattering Equations and Feynman Diagrams
• Integration Rules for Scattering Equations
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Cited by in corpus (58)

• One-loop amplitudes on the Riemann sphere
• Aspects of Scattering Amplitudes and Moduli Space Localization
• One-Loop Corrections from Higher Dimensional Tree Amplitudes
• New BCJ representations for one-loop amplitudes in gauge theories and gravity
• Prescriptive Unitarity
• Expansion of Einstein-Yang-Mills Amplitude
• New Representations of the Perturbative S-Matrix
• Manifesting Color-Kinematics Duality in the Scattering Equation Formalism
• Cross-ratio Identities and Higher-order Poles of CHY-integrand
• An Algebraic Approach to the Scattering Equations
• Elliptic scattering equations
• Scattering Equations and Global Duality of Residues
• Relations for Einstein-Yang-Mills amplitudes from the CHY representation
• $\Lambda$ Scattering Equations
• General Solution of the Scattering Equations
• Analytic Representations of Yang-Mills Amplitudes
• Labelled tree graphs, Feynman diagrams and disk integrals
• CHY-Graphs on a Torus
• The Polynomial Form of the Scattering Equations is an H-Basis
• Quadratic Feynman Loop Integrands From Massless Scattering Equations
• Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes
• Understanding the Cancelation of Double Poles in the Pfaffian of CHY-formulism
• One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations
• Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators
• Ambitwistor Integrands from Tensionless Chiral Superstring Integrands
• Feynman Rules of Higher-order Poles in CHY Construction
• The Q-cut Representation of One-loop Integrands and Unitarity Cut Method
• One-Loop Yang-Mills Integrands from Scattering Equations
• The Role of Möbius Constants and Scattering Functions in CHY Scalar Amplitudes
• BCJ relations from a new symmetry of gauge-theory amplitudes
• Direct Evaluation of $n$-point single-trace MHV amplitudes in 4d Einstein-Yang-Mills theory using the CHY Formalism
• A Combinatoric Shortcut to Evaluate CHY-forms
• String-Like Dual Models for Scalar Theories
• One-loop CHY-Integrand of Bi-adjoint Scalar Theory
• One-loop diagrams with quadratic propagators from the worldsheet
• CHY formula and MHV amplitudes
• A differential operator for integrating one-loop scattering equations
• One-loop Bern-Carrasco-Johansson Numerators on Quadratic Propagators from the Worldsheet
• Scattering Equations and a new Factorization for Amplitudes I: Gauge Theories
• One loop amplitude from null string
• A Monte Carlo Approach to the 4D Scattering Equations
• On Polytopes and Generalizations of the KLT Relations
• Derivation of Feynman Rules for Higher Order Poles Using Cross-ratio Identities in CHY Construction
• New Factorization Relations for Non-Linear Sigma Model Amplitudes
• New Factorization Relations for Yang Mills Amplitudes
• BCJ Numerators from Differential Operator of Multidimensional Residue
• Scattering Equations and Factorization of Amplitudes II: Effective Field Theories
• Note on recursion relations for the $\mathcal{Q}$-cut representation
• Off-Shell Yang-Mills Amplitude in the CHY Formalism
• Geometric Recursion from Polytope Triangulations and Twisted Homology
• Infrared and Holographic Aspects of the $S$-Matrix in Gauge Theory and Gravity
• Note on the Labelled tree graphs
• Permutation in the CHY-Formulation
• Note on solutions of scattering equations
• Characterizing the solutions to scattering equations that support tree-level ${\text{N}}^k\text{MHV}$ gauge/gravity amplitudes
• CHY-construction of Planar Loop Integrands of Cubic Scalar Theory
• CHY Theory for Several Fields
• One-loop matrix elements of effective superstring interactions: ${\alpha }^{\prime }$-expanding loop integrands