Analytical Methods for Phase-Space Integrals - With Applications to Semi-Inclusive Deep-Inelastic Scattering

This thesis develops reusable analytic infrastructure for the real-radiation contributions to perturbative calculations in quantum field theory. The central objects are the process-independent angular phase-space integrals that enter the next-to-next-to-leading-order (NNLO) subtraction and slicing frameworks of processes such as Drell--Yan, Higgs production, jet production, and semi-inclusive deep-inelastic scattering. After reviewing the reverse-unitarity formalism that places real and virtual corrections on a common algebraic footing, we use the radial--angular ecomposition to isolate these angular building blocks in their rotationally invariant (Somogyi) and frame-dependent (van Neerven) forms. We then construct a systematic Mellin--Barnes-to-polylogarithm method: after analytic continuation in the dimensional regulator and expansion, the balanced Mellin--Barnes integrals are converted into beta functions, then into real parametric integrals, and finally into Goncharov polylogarithms. With this method we compute, for the first time, a complete basis of the three- and four-denominator angular families, both massless and massive, to the orders in the regulator required for NNLO applications, together with partial-fraction and recursion relations for the massive cases and for higher denominator powers. Semi-inclusive deep-inelastic scattering at NNLO in QCD provides the demonstrator application: we combine the angular method with a one-fold radial integration and resolve the soft and collinear singularities, including soft singularities nested inside the angular part that must be extracted to all orders in the regulator. We then extend the same analytic-function technology from real-radiation phase space to the virtual sector, linearizing the generalized harmonic polylogarithms with square-root alphabets that the two-loop electroweak master integrals require. All results are validated against independent calculations, direct numerical Mellin--Barnes integration, permutation symmetries, and an independent differential-equation evaluation. The outcome is thus a durable, validated library of analytic real-radiation building blocks rather than a single process-specific calculation.