Higher-order calculations in quantum chromodynamics

In this thesis, several techniques and advances in higher-order Quantum Chromodynamics (QCD) calculations are presented. There is a particular focus on 2-loop 5-point massless QCD amplitudes, which are currently at the frontier of higher-order QCD calculations. Firstly, we study the Brodsky-Lepage-Mackenzie/Principle of Maximum Conformality (BLM/PMC) method for setting the renormalisation scale, μ_R, in higher-order QCD calculations. We identify three ambiguities in the BLM/PMC procedure and study their numerical impact using the example of the total cross-section for top-pair production at Next-to-Next-to-Leading Order (NNLO) in QCD. The numerical impact of these ambiguities on the BLM/PMC prediction for the cross-section is found to be comparable to the impact of the choice of μ_R in the conventional scale-setting approach. Secondly, we introduce a novel strategy for solving integration-by-parts (IBP) identities, which are widely used in the computation of multi-loop QCD amplitudes. We implement the strategy in an efficient C++ program and hence solve the IBP identities needed for the computation of any planar 2-loop 5-point massless amplitude in QCD. We also derive representative results for the most complicated non-planar family of integrals. Thirdly, we present an automated computational framework to reduce 2-loop 5-point massless amplitudes to a basis of pentagon functions. It uses finite-field evaluation and interpolation techniques, as well as the aforementioned analytical IBP results. We use this to calculate the leading-colour 2-loop QCD amplitude for qq̄→γγγ and then compute the NNLO QCD corrections to 3-photon production at the LHC. This is the first NNLO QCD calculation for a 2→3 process. We compare our predictions with the available 8 TeV measurements from the ATLAS collaboration and we find that the inclusion of the NNLO corrections eliminates the existing significant discrepancy with respect to NLO QCD predictions, paving the way for precision phenomenology in this process.