Abstract
In previous works, we devised a differential operator for evaluating typical integrals appearing in the Cachazo–He–Yuan (CHY) forms and in this paper we further streamline this method. We observe that at tree level, the number of free parameters controlling the differential operator depends solely on the number of external lines, after solving the constraints arising from the scattering equations. This allows us to construct a reduction matrix that relates the parameters of a higher-order differential operator to those of a lower-order one. The reduction matrix is theory-independent and can be obtained by solving a set of explicitly given linear conditions. The repeated application of such reduction matrices eventually transforms a given tree-level CHY-like integral to a prepared form. We also provide analytic expressions for the parameters associated with any such prepared form at tree level. We finally give a compact expression for the multidimensional residue for any CHY-like integral in terms of the reduction matrices. We adopt a dual basis projector which leads to the CHY-like representation for the non-local Bern–Carrasco–Johansson (BCJ) numerators at tree level in Yang–Mills theory. These BCJ numerators are efficiently computed by the improved method involving the reduction matrix.
Highlights
In [23,24] Cheung, Xu and the current authors proposed a method for evaluating the CHY forms using a differential operator and studied the combinatoric properties of the scattering equations
We further streamline the method at tree level by relating a generic CHY-like expression to a prepared form
We have developed a systematic method to construct the BCJ numerators, starting from the CHY forms of scattering amplitudes and using the differential operator proposed in our previous work [23]
Summary
In [23,24] Cheung, Xu and the current authors proposed a method for evaluating the CHY forms using a differential operator and studied the combinatoric properties of the scattering equations. This method bypasses the need for solving the scattering equations and leads to the analytic evaluation of a particular class of CHY forms, called the prepared forms. In [38,42], the twistor string theory have been
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