Master thesis defense by Yuchan Miao

Kinematic Hopf Algebra in Scattering Amplitudes

Recent research has revealed that the kinematic algebra underlying the colour-kinematics duality for tree-level scattering amplitudes in heavy-mass effective field theory and Yang-Mills theory is a quasi-shuffle Hopf algebra. This thesis studies the generalization of this kinematic algebra to the interaction of spin-one-half matter fields with gluons/gravitons. It specifically concentrates on the Compton amplitude involving a pair of fermions interacting with an arbitrary number of gluons/gravitons.

Upon investigating the corresponding evaluation map, the kinematic algebra generators are realized as on-shell QCD currents and we have discovered the universality of the quasi-shuffle Hopf algebra in all theories. We present the closed form expression of the corresponding QCD BCJ numerator, which in our approach is manifestly gauge invariant and comprises only physical massive poles. In addition, we propose two novel recursive relations for the corresponding BCJ numerator. These relations are then proved from the analysis of factorisation behaviours in unitarity cuts on massive propagators.