Elliptic scattering equations

Abstract

Recently the CHY approach has been extended to one loop level using elliptic functions and modular forms over a Jacobian variety. Due to the difficulty in manipulating these kind of functions, we propose an alternative prescription that is totally algebraic. This new proposal is based on an elliptic algebraic curve embedded in a $\mathbb{C}{P}^2$ space. We show that for the simplest integrand, namely the $\mathrm{n}-\mathrm{gon}$, our proposal indeed reproduces the expected result. By using the recently formulated $\Lambda -$algorithm, we found a novel recurrence relation expansion in terms of tree level off-shell amplitudes. Our results connect nicely with recent results on the one-loop formulation of the scattering equations. In addition, this new proposal can be easily stretched out to hyperelliptic curves in order to compute higher genus.

arXiv commentv3: Typo

References in corpus (35)

• Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory
• Scattering of Massless Particles in Arbitrary Dimension
• Scattering of Massless Particles: Scalars, Gluons and Gravitons
• Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM
• Ambitwistor strings and the scattering equations
• Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations
• Ambitwistor strings and the scattering equations at one loop
• Loop Integrands for Scattering Amplitudes from the Riemann Sphere
• One-loop amplitudes on the Riemann sphere
• Infinite Tension Limit of the Pure Spinor Superstring
• Scattering in Three Dimensions from Rational Maps
• Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles
• Proof of the Formula of Cachazo, He and Yuan for Yang-Mills Tree Amplitudes in Arbitrary Dimension
• One-Loop Corrections from Higher Dimensional Tree Amplitudes
• Berends-Giele recursion for double-color-ordered amplitudes
• One-loop Scattering Equations and Amplitudes from Forward Limit
• The Polynomial Form of the Scattering Equations
• New Representations of the Perturbative S-Matrix
• N-Point Tree-Level Scattering Amplitude in the New Berkovits' String
• Integration Rules for Loop Scattering Equations
• Scattering Equations and Feynman Diagrams
• Integration Rules for Scattering Equations
• Computation of Contour Integrals on ${\mathcal{M}}_{0,n}$
• An Algebraic Approach to the Scattering Equations
• Massless scattering at special kinematics as Jacobi polynomials
• Scattering Equations and Global Duality of Residues
• Elimination and recursions in the scattering equations
• On the solutions of the scattering equations
• General Solution of the Scattering Equations
• Scattering equations, generating functions and all massless five point tree amplitudes
• Comments on the evaluation of massless scattering
• Permutation Symmetry of the Scattering Equations
• Feynman Rules of Higher-order Poles in CHY Construction
• The Q-cut Representation of One-loop Integrands and Unitarity Cut Method
• Minimal Basis in Four Dimensions and Scalar Blocks

Cited by in corpus (35)

• Two-Loop Scattering Amplitudes from the Riemann Sphere
• New BCJ representations for one-loop amplitudes in gauge theories and gravity
• Einstein-Yang-Mills from pure Yang-Mills amplitudes
• Manifesting Color-Kinematics Duality in the Scattering Equation Formalism
• Gluons and gravitons at one loop from ambitwistor strings
• Two-Loop Scattering Amplitudes from Ambitwistor Strings: from Genus Two to the Nodal Riemann Sphere
• Cross-ratio Identities and Higher-order Poles of CHY-integrand
• One-loop Correlators and BCJ Numerators from Forward Limits
• Relations for Einstein-Yang-Mills amplitudes from the CHY representation
• CHY Loop Integrands from Holomorphic Forms
• The Polynomial Form of the Scattering Equations is an H-Basis
• CHY-Graphs on a Torus
• Quadratic Feynman Loop Integrands From Massless Scattering Equations
• Scalar-Graviton Amplitudes
• 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
• Four dimensional ambitwistor strings and form factors of local and Wilson line operators
• A Combinatoric Shortcut to Evaluate CHY-forms
• One-loop diagrams with quadratic propagators from the worldsheet
• A String Deformation of the Parke-Taylor Factor
• 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
• New Factorization Relations for Non-Linear Sigma Model Amplitudes
• Derivation of Feynman Rules for Higher Order Poles Using Cross-ratio Identities in CHY Construction
• New Factorization Relations for Yang Mills Amplitudes
• Scattering Equations and Factorization of Amplitudes II: Effective Field Theories
• Note on recursion relations for the $\mathcal{Q}$-cut representation
• Transmuting off-shell CHY integrals in the double-cover framework
• Note on scalar-graviton and scalar-photon-graviton amplitudes
• Factorizations for tree amplitudes in the double-cover framework: from gravity to other theories
• Analytic expressions of amplitudes by the cross-ratio identity method
• One-loop matrix elements of effective superstring interactions: ${\alpha }^{\prime }$-expanding loop integrands