# A Combinatoric Shortcut to Evaluate CHY-forms

## TL;DR

This paper introduces a combinatorial method using tableau representations and differential operators to efficiently evaluate CHY forms of scattering amplitudes, simplifying calculations for complex loop amplitudes.

## Contribution

It presents a novel combinatorial approach with tableau representation to evaluate generalized CHY forms, including one-loop amplitudes, streamlining previous residue calculations.

## Key findings

- Simplified evaluation of n-gon amplitudes at one-loop level.
- Complete calculation of one-loop five-point amplitude in SYM theory.
- Method yields results identical to established Q-Cut approach.

## 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.