A Novel Approach to Understand the Color Structure of Multiparton Scattering Amplitudes

Soft function deals with the IR singularity structure of scattering amplitudes. It exponentiates in terms of Soft anomalous dimension \( \Gamma ^S \) . The Feynman diagrams that contribute to \( \Gamma ^S \) are called webs. The colour and kinematic factors contribute to webs mix via the web mixing matrices. Mixing matrix is a central object in non-abelian exponentiation and calculated using the replica trick. An essential and long-awaited aim is to construct these matrices, bypassing the replica trick. A novel approach for this is initiated in our recent work published in JHEP 06(2022)020. In the talk, I will present the ideas introduced in this work that pave a way of directly constructing these web mixing matrices. Several new concepts: (a) Normal ordering of the diagrams of a Cweb, (b) Fused-Webs (c) Basis and Family of Cwebs, along with a Uniqueness theorem provide an understanding of the diagonal blocks and several null matrices that appear in the mixing matrices. We demonstrate using this formalism that, once the basis Cwebs present upto order \( \alpha _s^n \) are determined, the number of exponentiated colour factors for several classes of Cwebs starting at order \( \alpha _s^{n+1} \) can be predicted. Further complete result for the mixing matrices, to all orders in perturbation theory, for two special classes of Cwebs is provided.