CHY-Graphs on a Torus

Abstract

Recently, we proposed a new approach using a punctured Elliptic curve in the CHY framework in order to compute one-loop scattering amplitudes. In this note, we further develop this approach by introducing a set of connectors, which become the main ingredient to build integrands on ${\mathfrak{M}}_{1,n}$, the moduli space of n-punctured Elliptic curves. As a particular application, we study the ${\Phi }^3$ bi-adjoint scalar theory. We propose a set of rules to construct integrands on ${\mathfrak{M}}_{1,n}$ from ${\Phi }^3$ integrands on ${\mathfrak{M}}_{0,n}$, the moduli space of n-punctured spheres. We illustrate these rules by computing a variety of ${\Phi }^3$ one-loop Feynman diagrams. Conversely, we also provide another set of rules to compute the corresponding CHY-integrand on ${\mathfrak{M}}_{1,n}$ by starting instead from a given ${\Phi }^3$ one-loop Feynman diagram. In addition, our results can easily be extended to higher loops.

34 pages, 32 figures

References in corpus (37)

• Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory
• New Relations for Gauge-Theory Amplitudes
• Perturbative Gauge Theory As A String Theory In Twistor Space
• 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
• Scattering Equations and KLT Orthogonality
• Ambitwistor strings and the scattering equations
• Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations
• 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
• 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
• Cross-ratio Identities and Higher-order Poles of CHY-integrand
• Computation of Contour Integrals on ${\mathcal{M}}_{0,n}$
• An Algebraic Approach to the Scattering Equations
• Massless scattering at special kinematics as Jacobi polynomials
• Elliptic scattering equations
• Elimination and recursions in the scattering equations
• On the solutions of the scattering equations
• $\Lambda$ Scattering Equations
• Scattering equations, generating functions and all massless five point tree amplitudes
• Comments on the evaluation of massless scattering
• The Polynomial Form of the Scattering Equations is an H-Basis
• Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes
• Permutation Symmetry of the Scattering Equations
• Feynman Rules of Higher-order Poles in CHY Construction

Cited by in corpus (23)

• 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
• Gluons and gravitons at one loop from ambitwistor strings
• Two-Loop Scattering Amplitudes from Ambitwistor Strings: from Genus Two to the Nodal Riemann Sphere
• Relations for Einstein-Yang-Mills amplitudes from the CHY representation
• One-loop Correlators and BCJ Numerators from Forward Limits
• CHY Loop Integrands from Holomorphic Forms
• Quadratic Feynman Loop Integrands From Massless Scattering Equations
• 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
• One-loop diagrams with quadratic propagators from the worldsheet
• One-loop CHY-Integrand of Bi-adjoint Scalar Theory
• A String Deformation of the Parke-Taylor Factor
• 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
• New Factorization Relations for Yang Mills Amplitudes
• Scattering Equations and Factorization of Amplitudes II: Effective Field Theories
• Note on solutions of scattering equations
• Characterizing the solutions to scattering equations that support tree-level ${\text{N}}^k\text{MHV}$ gauge/gravity amplitudes
• One-loop matrix elements of effective superstring interactions: ${\alpha }^{\prime }$-expanding loop integrands