The goal of SAMPL-QCD (Scattering Amplitudes for Multi-hadrons Processes in Lattice-QCD) is to compute, with a novel method, exclusive scattering amplitudes involving multiple hadrons in the final state, such as J to ππ or J to πππ, with J=B,D,e+e-,...from Euclidean correlation functions computed in lattice QCD. The motivation is to provide new theoretical predictions to compare with experimental data that already exists, and that will keep being produced, with the ultimate goal of finding evidence for physics beyond the Standard Model. The state-of-the-art for theory predictions of multi-hadron processes is the K→ ππ study from the RBC/UKQCD collaboration. The methodology employed there is based on the Lellouch Lüscher formalism, which appears hard to generalise to processes where many inelastic thresholds can open simultaneously. A different strategy that does not suffer the same complications is now possible, because of two factors: on one side, a mathematically-robust relation between Euclidean correlators (available from lattice QCD) and scattering amplitudes was given by A. Patella & N. Tantalo in terms of an inverse problem. On the technical side, progress made by my collaborators and I in the last few years in solving the inverse problem with reliable estimates of its systematics, paves the way for first QCD calculations of this kind, potentially opening the way to a new generation of lattice calculations. In this project, we intend to explore the level of precision that can be reached, within this formalism, for scattering amplitudes, given the current state-of-the-art lattice simulations.

The data taken at the Large Hadron Collider (LHC) at CERN and its future High-Luminosity upgrade enables the study of the fundamental interactions of matter at unprecedented precision and energy. After the discovery of the Higgs boson one of the major goals of the LHC is the precise measurement of its properties. In order to achieve the needed theoretical precision, two-loop corrections to many scattering amplitudes have to be computed. A crucial process to precisely determine the coupling of the Higgs boson to the bottom quark is the production of a Higgs, a Z-boson, and a jet. This process receives large contributions and accompanying uncertainties from the sub-process of gluon fusion, which only starts contributing at the one-loop level but is enhanced by the large gluon luminosity at hadron colliders. To improve the theoretical description of this sub-process and enable the most precise determination of the Higgs-bottom coupling, the two-loop five-point amplitudes of this sub-process will be computed in this project. The results of this project can also be applied to double Higgs production, which the crucial experimental signature to determine the Higgs self-interaction at hadron colliders in the future. The computation of five-point amplitudes with massive internal and external particles is a tremendously difficult task, due to the algebraic complexity of the amplitudes and the appearance of new classes of special functions. To tackle and circumvent some of the difficulties I will explore the systematic approximation of these complicated amplitudes in various kinematic limits, like small Mandelstam variables or the high-energy limit. This methodology has already proven successful for two-loop four-point amplitudes previously. The results of this project will, therefore, significantly enhance our ability to obtain precise theory predictions for high-multiplicity processes and simplify their calculation and numerical evaluation.