Periodic Reporting for period 2 - hipQCD (Highest Precision QCD predictions for a new era in Higgs boson phenomenology) Reporting period: 2020-12-01 to 2022-05-31 Summary of the context and overall objectives of the project High-energy colliders enable us to study the fundamental particles and interactions that make up our universe. To extract information from data, we require very sophisticated theoretical calculations that reliably model the complex environment of high-energy collisions. hipQCD develops innovative techniques that allow for high-precision studies at the Large Hadron Collider (LHC) and it applies them to in-depth phenomenological investigations, with a particular emphasis on processes related to the Higgs boson.On the theoretical side, we focus on computing higher-order perturbative corrections within the Standard Model, starting from first principles. The group deals with all the many aspects involved in such computations. These range from devising new ``subtraction frameworks'', which are required to properly organise the degrees of freedom at play and obtain physically meaningful results, to computing complex ``scattering amplitudes'', the theoretical objects that encode the quantum information on particle scattering probabilities. The goal is to develop robust and efficient theoretical predictions for key collider processes that allow for a wide range of phenomenological studies in different kinematical regimes, thus maximally profiting from LHC data. On the phenomenological side, we apply our results to stress-test the structure of the Standard Model at very high accuracy. Particular emphasis is devoted to the electroweak sector of the Standard Model and the Higgs boson, whose properties are yet unknown to a satisfactory level of precision. Our hope is that the techniques we are developing will help answering questions like ''what is the structure of Higgs interactions?'' or, more broadly, ''do fundamental particles and interactions behave as predicted by the Standard Model, or do we see hints for new physics?''. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far In this first part of the project, we have completed the formulation of the so-called ``nested soft-collinear subtraction'' (NSC) scheme, which provides a robust and efficient framework for computing next-to-next-to-leading order (NNLO) corrections in perturbative quantum chromodynamics (QCD) to a wide variety of important processes.As an unforeseen but welcome development, we have found that our framework is flexible enough to also cope with the case of mixed QCD-Electroweak corrections. We have followed this promising new line of investigations and we computedfor the first time mixed QCD-Electroweak corrections to gauge bosons production at the LHC for realistic experimental setups, and their implication for the extraction of fundamental parameters of the Standard Model. We have also applied our NSC framework to Higgs studies in important channels, namely associated Higgs production (VH) and Higgs production in vector boson fusion. In both cases, we have also properly modeled Higgs decays. In the VH case, we clearly emphasised the short-comings of the standard theoretical approach to higher order perturbative calculations for processes involving b-quarks in the final state, and developed a calculation that partially addresses them. We have also studied the interplay of higher-order corrections and effects coming from potential new physics contributions, to make sure that the former are not confused for the latter. The group obtained important results for complex scattering amplitudes. Highlights include the computation of two-loop amplitudes for di-photon production in association with one extra photon or a QCD parton. The latter was the first calculation of a two-loop scattering amplitude for a 2->3 process that takes into account the exact colour structure of QCD. The group also made progress towards predictions at even higher orders, and obtained for the very first time results for 2->2 scattering amplitudes at three loops in QCD. Group members have also devised techniques for computing scattering amplitudes fully numerically, to tackle cases which may be to complex to be treated with purely analytics methods.Finally, given the high level of accuracy of perturbative calculations, we have started investigating the limitations of the perturbative framework. These may become critical for high-precision analysis. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The main achievements reported above go beyond the current state-of-the art. Looking forward, we plan to apply our NSC framwork and our understanding of QCD scattering amplitude to perform further accurate phenomenological studies in the Higgs sector and beyond, both in ``standard'' and extreme kinematic regions. We also envision to further continue our investigation of Electroweak corrections, with a particular emphasis on extreme kinematic configurations. Furthermore, we hope to extend our framework towards even higher orders in perturbative QCD, and further investigate the issues that one may encounter in perturbative calculations beyond NNLO.