Differential Operator For Cachazo-He-Yuan Formalism

The study of scattering amplitudes in quantum field theory has been a highly active field and gone through a series of milestone developments over the past two decades.This dissertation focuses on two obj ects that both have attracted wide attention from the community of scattering amplitudes:the Cachazo-He-Yuan(CHY)formalism and the Bern-Carrasco-Johansson(BCJ)numerators.In this dissertation we develop a systematic and theory-independent method that effectively computes the CHY forms to obtain the corresponding scattering amplitudes or loop integrands of scattering amplitudes.This method is then employed in the construction of the BCJ numerators from the CHY forms.We first propose a differential operator for computing multidimensional residues associated with meromorphic forms at isolated poles.The differential operator is conjectured to be completely deter-mined by the local duality theorem and the intersection number requirement.We apply this differential operator to the evaluation of the CHY forms,making use of the polynomial forms of the scattering equa-tions.We upgrade this method by further studying the polynomial scattering equations.The combinatoric properties of these equations lead to an algorithm that systematically computes the parameters in the aforementioned differential operator associated with a particular type of CHY forms,which we call the prepared form.This algorithm can be conveniently depicted using a tableau representation we design for these parameters.At tree level,we observe that the number of independent parameters,after solving the local duality constraints associated with the polynomial scattering equations,depends only on the number of exter-nal lines.Based on this observation,we propose the reduction matrix method that transforms a given CHY form with a factorized integrand to the aforementioned prepared form.The factorized form of the integrand can be guaranteed.We construct the BCJ numerators from the CHY form of a color-ordered partial amplitude,using the differential operator and the reduction matrix.The BCJ numerators are given by decomposing the CHY form into a set of CHY-like integrals,which can be identified as the minimal basis.Such decomposi-tions are translated to decompositions of the corresponding differential operator and the reduction matrix method renders the latter straightforward.The method developed in this dissertation is theory-independent and we expect a natural generaliza-tion of the method to higher loop levels.