Periodic Reporting for period 1 - LoopAnsatz (Analytic Loop Amplitudes from Numerics and Ansatz)
Okres sprawozdawczy: 2021-04-01 do 2023-03-31
The quest to understand the fundamental constituents of Nature can be traced at least as far back as the ancient Greeks. Today, experiments at the smallest length scales take place at the Large Hadron Collider (LHC). Here, pairs of protons collide at speeds close to that of light and, for the briefest of moments, generate other, short-lived, particles. These decay, converting into high-energy sprays of particles that are deposited in detectors. Researchers then study the patterns they draw in the experiment, comparing them to those predicted by theory. In the summer of 2012, this resulted in the ground-breaking discovery of the Higgs boson, which further validated the ability of the Standard Model of Particle Physics (SM) to describe Nature at short distances. Nevertheless, open questions in particle physics remain unanswered by the SM. Given the evidence for dark matter in astronomical measurements, how does it manifest at the microscopic level? What is the resolution of the hierarchy problem, that is, what causes the large divide between the Higgs mass and the Planck scale? As a scalar particle, the Higgs boson is inherently sensitive to scales of new physics. It is therefore essential to ask if its properties and interactions with heavy particles match the predictions of the SM. To answer these questions and identify signals of new physics, theorists must make precise predictions of the SM. In the coming years, with the projected increase in collision rates, experiments at the LHC will hugely increase the amount of recorded data. We will enter into an era of precision, where uncertainties will reduce to the percent level for a broad class of collision observables. To understand this data theorists must make correspondingly precise predictions. These precise predictions pose an immense theoretical challenge.
A critical step in making these predictions is the calculation of a "scattering amplitude". Scattering amplitudes are mathematically beautiful objects that encode transition probabilities of particle collisions. In a textbook approach, a scattering amplitude is thought of as a sum of Feynman diagrams. As the precision of the associated prediction rises, so does the number of closed loops of particles in a diagram. The number of diagrams in the sum grows factorially and in cases relevant for LHC physics they can number in the tens of thousands. Each one is interpreted as an integral over the unobserved internal states of the diagram. These multi-dimensional integrals are known as “Feynman integrals” and their computation is a demanding undertaking.
In recent years, a new framework for the calculation of amplitudes has emerged. Building on a deep understanding of the physical and mathematical properties of the amplitudes, one uses numerical evaluations to constrain an Ansatz for the analytic form of the amplitude. These Ansaetze are constructed from knowledge of the geometric and mathematical structures which underpin the amplitudes. The project LoopAnsatz employs and develops these methods to handle increasingly complex loop amplitude computations.
Over its duration, LoopAnsatz has produced a number of state of the art results. Firstly, it has developed geometric techniques for both the computation of amplitudes as well as real radiation contributions. Secondly, it has produced cutting-edge computations of Feynman integrals for five-particle scattering. Finally, it has both produced scattering amplitudes for W+2-jet production at the LHC as well as amplitudes for the production of three photons.
A critical step in making these predictions is the calculation of a "scattering amplitude". Scattering amplitudes are mathematically beautiful objects that encode transition probabilities of particle collisions. In a textbook approach, a scattering amplitude is thought of as a sum of Feynman diagrams. As the precision of the associated prediction rises, so does the number of closed loops of particles in a diagram. The number of diagrams in the sum grows factorially and in cases relevant for LHC physics they can number in the tens of thousands. Each one is interpreted as an integral over the unobserved internal states of the diagram. These multi-dimensional integrals are known as “Feynman integrals” and their computation is a demanding undertaking.
In recent years, a new framework for the calculation of amplitudes has emerged. Building on a deep understanding of the physical and mathematical properties of the amplitudes, one uses numerical evaluations to constrain an Ansatz for the analytic form of the amplitude. These Ansaetze are constructed from knowledge of the geometric and mathematical structures which underpin the amplitudes. The project LoopAnsatz employs and develops these methods to handle increasingly complex loop amplitude computations.
Over its duration, LoopAnsatz has produced a number of state of the art results. Firstly, it has developed geometric techniques for both the computation of amplitudes as well as real radiation contributions. Secondly, it has produced cutting-edge computations of Feynman integrals for five-particle scattering. Finally, it has both produced scattering amplitudes for W+2-jet production at the LHC as well as amplitudes for the production of three photons.
Over the course of LoopAnsatz, work has been performed in three major directions. Firstly, new techniques were developed for the computation of two-loop Feynman integrals involving four massless and a single massive particle. Using advanced Ansatz technologies, LoopAnsatz constructed and solved differential equations for the Feynman integrals, producing state of the art building blocks for precise predictions for the production of a Higgs boson alongside two energetic jets. Secondly, new Ansatz methodologies were constructed in order to handle the complex computation of two-loop amplitudes with a massive external leg. This allowed LoopAnsatz to compute the most complicated two-loop amplitudes to date -- those for the production of a W boson in association with two jets. Thirdly, LoopAnsatz developed a new, geometric approach to the construction of Ansaetze for scattering amplitudes. This novel approach allows one to compute scattering amplitudes in a manner that is orders of magnitude more efficient than competing approaches. In a first application of these technologies, LoopAnsatz has computed the full two-loop amplitudes for the production of three photons at hadron colliders.
The results of LoopAnsatz have been written up in a collection of papers. Each of these papers is publicly and freely available on the pre-print server, arXiv and were further published in open access journals such as the Journal for High Energy Physics or Physical Review D. Analytic results for amplitudes and master integrals have all been made publicly available as ancillary files. Public, C++ codes "FivePointAmplitudes++" and "PentagonFunctions++" were produced for the evaluation of both master integrals and scattering amplitudes. The work of LoopAnsatz has been presented in many seminars in European and American institutions as well as talks at internationally recognized conferences.
The results of LoopAnsatz have been written up in a collection of papers. Each of these papers is publicly and freely available on the pre-print server, arXiv and were further published in open access journals such as the Journal for High Energy Physics or Physical Review D. Analytic results for amplitudes and master integrals have all been made publicly available as ancillary files. Public, C++ codes "FivePointAmplitudes++" and "PentagonFunctions++" were produced for the evaluation of both master integrals and scattering amplitudes. The work of LoopAnsatz has been presented in many seminars in European and American institutions as well as talks at internationally recognized conferences.
LoopAnsatz has resulted in a number of novel theoretical developments. Firstly, LoopAnsatz has developed a novel geometric approach for the construction of Ansaetze for two-loop amplitudes. This approach opens a path to orders of magnitudes of improvement in Ansatz methodologies for loop amplitude computation. Moreoever, LoopAnsatz has developed a novel geometric approach for real radiation contributions, providing a new direction for the computation of cross sections.
Beyond this, LoopAnsatz has made several cutting-edge computations. LoopAnsatz has provided the first computations of amplitudes for the production of a W-boson alongside two energetic jets at the LHC. These have already been used in the first percent-level-precise prediction of a cross-section for the production of a W-boson in association with a massive pair of bottom quarks and will make similarly precise predictions possible for the production of a W-boson in association with two jets. Moreover, the Feynman integrals computed by LoopAnsatz have already been applied to deepen our understanding of the so-called "anti-podal" duality of maximally supersymmetric gauge theory. Furthermore, they will soon lead to the computation of scattering amplitudes for the production of a Higgs boson alongside two energetic jets at the LHC. Finally, LoopAnsatz's computation of the scattering amplitudes for production of three photons at a hadron collider will soon lead to the associated percent-level precise prediction.
Beyond this, LoopAnsatz has made several cutting-edge computations. LoopAnsatz has provided the first computations of amplitudes for the production of a W-boson alongside two energetic jets at the LHC. These have already been used in the first percent-level-precise prediction of a cross-section for the production of a W-boson in association with a massive pair of bottom quarks and will make similarly precise predictions possible for the production of a W-boson in association with two jets. Moreover, the Feynman integrals computed by LoopAnsatz have already been applied to deepen our understanding of the so-called "anti-podal" duality of maximally supersymmetric gauge theory. Furthermore, they will soon lead to the computation of scattering amplitudes for the production of a Higgs boson alongside two energetic jets at the LHC. Finally, LoopAnsatz's computation of the scattering amplitudes for production of three photons at a hadron collider will soon lead to the associated percent-level precise prediction.