Abstract:
In quantum field theory, the probability of outcomes of a scattering process can be calcu lated through scattering amplitudes. The scattering amplitudes can be calculated using the Feynman diagrams approach. However, the Feynman diagrams approach can become too difficult for beyond five particles. Focusing on the amplitude of Yang-Mills theory, the spinor-helicity formalism was used to help calculate the amplitude, resulting in a simple formula for an n particle scattering event with two negative helicity particles. For general helicity configuration, we focus on the work of Witten and Roiban, Spradlin, and Volovich (RSVW) that incorporated twistor theory to their formulation. This re- sults in a formula on twistor space for general helicity configuration of tree level gluon scattering, localized on constraint equations known as the refined scattering equations. The number of solutions to these constraints are observed to be the Eulerian numbers E(n−3, k−2), where k is the number of negative helicity particles. In this project, we will review the formulation of the tree level amplitude including the spinor-helicity for malism, twistor theory, and RSVW formula. Then, we proved that the number of solutions to the constraints to RSVW formula are the Eulerian numbers by establishing a recursion relation for the number of solutions, using the method of dominance balance. Recently, a twistor formula for one loop amplitude was published, and the number of solutions to these constraints – the loop polarized scattering equations – are still unknown. In order to be prepared for finding the number of solutions for the loop polarized scattering equations, were produce the proof for the number of solutions to the constraints of another one-loop formula, which is an extension of a formula proposed by Cachazo, He, and Yuan (CHY). The methods that were used to establish a recursion relation for the number of solutions to the one loop CHY’s constraints were then applied to the loop polarized scattering equations. Up to this point, we found that this recursion relation possesses some similar features to those of one loop CHY constraints.
