Periodic Reporting for period 4 - MathAm (Mathematical Structures in Scattering Amplitudes) Reporting period: 2020-03-01 to 2020-08-31 Summary of the context and overall objectives of the project One of the main ingredients that allow physicists to derive predictions from quantum field theory are the so-called scattering amplitudes, a set of mathematical quantities that encode the probabilities of how quantum particles interact. While the basic definition of a scattering amplitude dates back to the early days of quantum field theory, explicit computations of the so-called quantum loop corrections are still one of the major bottlenecks in theoretical particle physics. Since scattering amplitudes play such a central role in quantum physics, understanding their properties and mathematical structures is crucial in order to improve our knowledge of the fundamental laws of nature. Moreover, scattering amplitudes are not only interesting objects to study in their own right, but they are the main theoretical tool to make predictions for collider experiments like the LHC.Despite their importance and ubiquity in modern theoretical physics, the explicit computation of scattering amplitudes is often still a bottleneck. One of the main issues when computing scattering amplitudes is the necessity to evaluate certain classes of integrals, known as Feynman integrals, which allow one to compute quantum corrections to a scattering process due to the exchange of virtual quanta. These integrals often evaluate to special functions that are poorly understood even from the mathematical side. In recent years it was realised that there are deep and far-reaching connections between scattering amplitudes and certain areas of modern mathematics, like algebraic geometry and number theory. The goal of the project MathAm is to apply cutting-edge techniques from modern mathematics to the computation of multi-loop multi-leg scattering amplitudes in quantum field theory. Guided by advances in modern mathematics in the last decade, the goal of the project is to derive a radically new viewpoint on scattering amplitudes in quantum field theory from which the properties and the simplicity of amplitudes will be manifest at every step of the computation. The results obtained by MathAm do not only shed light on the mathematical underpinnings of quantum field theory and scattering amplitudes in general, but they also enable us to make predictions for collider experiments like the LHC at an unprecedented level of accuracy. For example, MathAm has provided the most precise predictions for the production of a Higgs boson or a pair of leptons at the LHC. After 5 years, MathAm has been able to break new ground in a wide range of areas of physics. On the purely mathematical ground, the project has provided deep insight into the mathematical structure of Feynman integrals, and we have proposed a new algebraic structure underlying perturbative quantum field theory. The conjectures made by MathAm have received a lot of attention also from the mathematical community, and several papers have been published by mathematicians where the conjectures by MathAm have been rigorously proven. On the application side, the results of MathAm for the production of a Higgs boson are the main reference of the LHC experiments, where they play a crucial role in measuring and studying the properties of the Higgs boson, for example its couplings to matter. The results of MathAm have therefore had a very big impact on a large range of topics, and they will continue to play a crucial role in understanding of quantum field theory and the Higgs sector, and therefore fundamental laws of nature and the origin of mass. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far "The work of MathAm is organised into 3 different work packages, each focusing on different aspects of scattering amplitudes and their mathematical properties.Work Package 1 focuses on scattering amplitudes from a purely mathematical point of view. MathAm has uncovered a novel algebraic structure underlying Feynman integrals. This structure connects Feynman integrals to recent developments in number theory and algebraic geometry. Our results have sparked novel research in pure mathematics: 2 peer-reviewed papers in pure mathematics have been published which develop further the ideas put forward by MathAm and put them into a rigorous mathematical framework. Furthermore, MathAm has pioneered the study and the use of ""elliptic multiple polylogarithms"" for multi-loop Feynman integrals. It has applied them to obtain various novel results. MathAm has been able to provide for the first time fully analytic results in terms of these functions for some integrals which had been thought out of reach before.While multi-loop scattering amplitudes are in general difficult to evaluate analytically, it is possible to obtain high-loop results in certain special supersymmetric quantum field theories. Work Package 2 investigates scattering amplitudes at high loop orders in these theories, with the aim of elucidating their mathematical properties. MathAm has focused on two particular theories. In the context of the so-called (planar) N=4 Super Yang-Mills theory, MathAm has shown that in a certain kinematical limit the amplitudes are described by certain geometric structures which are well studied in pure mathematics. Pairing this insight with input from integrability, MathAm has conjectured an all-order formula for all scattering amplitudes in this kinematic limit in this theory. The conjecture has passed a large number of non-trivial checks. MathAm has used the conjecture to provide for the first time a proof of the so-called ""principle of maximal and uniform transcendentality"" in this limit. MathAm has also initiated the first study of a scattering amplitude in a non-maximally supersymmetric gauge theory. We have shown that there are intriguing connections between infrared divergencies and violations of the principle of maximal and uniform transcendentality in this theory.Scattering amplitudes are not only of formal interest, but they also allow us to make predictions for collider experiments like the Large Hadron Collider (LHC) at CERN. Work Package 3 aims at applying the novel techniques developed by MathAm to obtain the most precise predictions for collider observables which would be impossible to obtain by standard methods. MathAm has pioneered the computation of LHC cross sections through third order in the strong coupling constant (N3LO). Our results are by now the official reference for the LHC experiments for the production probability of a Higgs boson. In a second phase, MathAm has extended these results to other LHC processes, including the important case of the Drell-Yan process, which plays an important role in understanding the internal structure of the proton and to calibrate the LHC detectors. These phenomenological results rely heavily on the mathematical tools developed by MathAm in Work Package 1, and without MathAm's insight into the relevance of abstract mathematics for Feynman integrals, these important concrete results could not have been obtained.To summarise, MathAm has obtained results ranging over a large range of topics, from pure mathematics to LHC phenomenology. In total, members of the MathAm team have published more than 40 papers in peer-reviewed journals over the last 5 years. These papers have been cited more than 1.800 times by other scientific works. In addition, the results have been presented and discussed at many conferences. These data show that MathAm has had a lasting impact on research in fundamental and physics." 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 computation of the Higgs cross section through third order is not only the most precise determination of the Higgs cross section to date, but it is at the same time the first computation ever performed at this order. As a consequence, the results of MathAm go beyond the current state of the art and have broken new ground in precision computations for hadron colliders. Moreover, this result has a lasting impact on the physics program carried out at the Large Hadron Collider (LHC) at CERN. Indeed, the results obtained by the MathAm group are by now the official reference for the LHC experiments for the production probability of a Higgs boson. They play a crucial role in the study of the properties of the Higgs boson, and hence in improving our understanding of one of the fundamental laws of nature. The MathAm group has recently extended this result to other physics processes, including the important case of the production of a virtual photon or W boson. In this way MathAm has broken new ground in providing precise theoretical predictions for collider physics.More generally, if no signal of new physics is discovered at the LHC, the precision program of the LHC will dominate research in high-energy physics for the decade to come. On the the theoretical side, this program will require the evaluation of complicated multi-loop Feynman integrals, many of which are out of reach by state-of-the-art techniques. Through an interdisciplinary approach combining physics and modern mathematics, the MathAm team has developing novel techniques to evaluate these integrals, and it has already applied them to evaluate analytically some of the most complicated two- and three-loop integrals. In particular, MathAm has pioneered the use of the theory of elliptic multiple polylogarithms, introduced in pure mathematics only decade ago, to obtain concrete new results for Feynman integrals. Currently, the team is applying these new mathematical tools to compute two-loop corrections to LHC processes, including the production of a charmonuim state at the LHC.