Executive Summary : | The LHC has entered an era of precision physics, but the precise theoretical calculation of multi-loop diagrams remains lagging due to various bottlenecks. The use of differential equation method for calculating multi-loop diagrams gained momentum in the last decade after the discovery of the canonical basis of master integrals for some processes. However, many processes still need to be calculated precisely. This research plan proposes to calculate a set of complicated topologies for t+tbar+jet production at two loops and H+jet production at three loops using the differential equation method. The first process will help determine top quark mass and measure the various couplings of the Higgs boson precisely. To calculate these processes, the complicated integrals of each topology are reduced to comparatively easier integrals, known as master integrals, using the Laporta Algorithm. However, solving these equations becomes extremely difficult due to their rapid growth. Improvements have been proposed, such as the use of finite field arithmetic to reconstruct rational coefficients. After reducing the diagrams to master integrals, differential equations are solved by taking derivatives of master integrals with respect to kinematic variables. |