A Combinatoric Shortcut to Evaluate CHY-forms

Abstract

In \cite{Chen:2016fgi} we proposed a differential operator for the evaluation of the multi-dimensional residues on isolated (zero-dimensional) poles.In this paper we discuss some new insight on evaluating the (generalized) Cachazo-He-Yuan (CHY) forms of the scattering amplitudes using this differential operator. We introduce a tableau representation for the coefficients appearing in the proposed differential operator. Combining the tableaux with the polynomial forms of the scattering equations, the evaluation of the generalized CHY form becomes a simple combinatoric problem. It is thus possible to obtain the coefficients arising in the differential operator in a straightforward way. We present the procedure for a complete solution of the $n$-gon amplitudes at one-loop level in a generalized CHY form. We also apply our method to fully evaluate the one-loop five-point amplitude in the maximally supersymmetric Yang-Mills theory; the final result is identical to the one obtained by Q-Cut.

34 pages, 10 figures

References in corpus (8)

• Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations
• Ambitwistor strings in 4-dimensions
• Two-Loop Scattering Amplitudes from the Riemann Sphere
• Fundamental BCJ Relation in N=4 SYM From The Connected Formulation
• On the solutions of the scattering equations
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• CHY Loop Integrands from Holomorphic Forms
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Cited by in corpus (11)

• Aspects of Scattering Amplitudes and Moduli Space Localization
• CHY Loop Integrands from Holomorphic Forms
• Quadratic Feynman Loop Integrands From Massless Scattering Equations
• Properties of scattering forms and their relation to associahedra
• One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations
• Understanding the Cancelation of Double Poles in the Pfaffian of CHY-formulism
• Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators
• BCJ Numerators from Differential Operator of Multidimensional Residue
• Note on solutions of scattering equations
• CHY Theory for Several Fields
• Properties of Yang-Mills scattering forms