The parton-level structure of $e^{+}{e}^{-}$ to 2 jets at N${}^3$LO

We investigate the quantum chromodynamics (QCD) corrections to hadronic final states in electron-positron collisions at $\mathcal{O}({\alpha }_s^3)$ in the strong coupling constant ${\alpha }_s$. Namely, we analytically compute the total cross section for this process by separately integrating the tree-level five-parton, the one-loop four-parton, the two-loop three-parton, and the three-loop two-parton matrix elements over the respective phase space. All the contributions to the calculation are treated in a common framework whereby phase space integrals are expressed as physical cuts of the four-loop two-point function. We check the cancellation of infrared poles at all colour levels and we reproduce the known result for the $R$-ratio at order ${\alpha }_s^3$.