| Summary: | <p>In this work, we consider scattering amplitudes relevant for high-precision Large Hadron Collider (LHC) phenomenology. They provide the most fundamental representation of the quantum probability amplitude for the scattering of elementary particles. As such, they are a necessary ingredient of theoretical predictions for hadronic differential cross section distributions for any process. Our predictions rely on the Quantum Field Theory of the Standard Model (SM) of Elementary Particles. Any confirmed deviation found between theoretical predictions and experimental measurements would yield the discovery of New Physics.</p>
<p>We analyse the general structure of amplitudes, and we review state-of-the-art methods for computing them. We discuss advantages and shortcomings of these methods, and we point out the bottlenecks in modern amplitude computations. As a practical illustration, we present frontier applications relevant for multi-loop multi-scale processes.</p>
<p>We compute the helicity amplitudes for diphoton production in gluon fusion and photon+jet production in proton scattering in three-loop massless Quantum Chromodynamics (QCD). We have adopted a new projector-based prescription to compute helicity amplitudes in the 't Hooft-Veltman scheme. We also rederived the minimal set of independent Feynman integrals for this problem using the differential equations method, and we confirmed their intricate analytic properties. By employing modern methods for integral reduction, we provide the final results in a compact form, which is appropriate for efficient numerical evaluation.</p>
<p>Beyond QCD, we have computed the two-loop mixed QCD-electroweak amplitudes for Z+jet production in proton scattering in light-quark-initiated channels, without closed fermion loops. This process provides important insight into the high-precision studies of the SM, as well as into Dark Matter searches at the LHC. We have employed a numerical approach based on high-precision evaluation of Feynman integrals with the modern Auxiliary Mass Flow method. The obtained numerical results in all relevant partonic channels are evaluated on a two-dimensional grid appropriate for further phenomenological applications.</p>
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