Periodic Reporting for period 1 - MPAmplitudes (Multi-particle scattering amplitudes for precision collider physics)
Okres sprawozdawczy: 2018-09-01 do 2020-08-31
The context of the project is the study of the fundamental particles and interactions of nature. This is done in the framework of Quantum Field Theory (QFT), which treats elementary particles as excited quantum states of fields. The study of elementary particles and their interactions is mainly driven by high-energy collision experiments, where particles are collided at very high energies. The Large Hadron Collider (LHC) at CERN is the largest and most powerful particle collider ever built and it is collecting an unprecedented amount of data of particle collisions, probing previously unexplored energy scales and interactions.
All interactions and particles observed so far in collision experiments are compatible with the Standard Model (SM), namely a QFT which describes the strong interactions of quantum chromodynamics (QCD), weak and electromagnetic interactions. The SM is currently the most accurate and successful theory of the fundamental interactions of nature. The SM has however several important shortcomings and scientists are putting great effort in the search of new physics Beyond the Standard Model (BSM). Our ability to test the SM as accurately as possible is crucial for detecting possible deviations representing BSM signals and therefore the discovery of new particles and interactions.
Scattering amplitudes are essential ingredients for theoretical predictions for high-energy experiments. They are related to the probability of interactions between fundamental particles. They are computed in perturbation theory and next-to-next-to-leading-order (NNLO) accuracy is typically needed to match the one of the experimental data. This involves the calculation of complicated integrals, called Feynman integrals. The calculation of amplitudes is especially difficult for processes with many external particles (or legs), which are however very common at the LHC, due to the high centre-of-mass energy of the collisions. Multi-particle scattering amplitudes are therefore a crucial objective of current high-energy phenomenology.
The main goal of this project is a breakthrough in our capability of computing complicated multi-loop quantities, focusing on two-loop corrections to five-point scattering amplitudes in QCD. This involves theoretical work for better understanding multi-loop integrals, the development of new computational tools and techniques capable of tackling these complex calculations, as well as the explicit computation of important multi-loop results relevant for modern phenomenology, such as two-loop five-point amplitudes.
The usage of finite fields is an important innovative aspect of this project. Finite fields are numerical fields with a finite number of elements that can be represented by machine-size integers. Operations over finite fields are therefore relatively fast but also exact and free of numerical errors. Multivariate reconstruction algorithms allow to infer full numerical or analytic results out of several numerical evaluations over finite fields. This allows to sidestep the appearance of large analytic expressions in intermediate stages, which are often the main bottleneck of amplitude calculations, since only the final result is reconstructed analytically. These methods are very general and have a huge range of possible applications in science, even outside physics.
All interactions and particles observed so far in collision experiments are compatible with the Standard Model (SM), namely a QFT which describes the strong interactions of quantum chromodynamics (QCD), weak and electromagnetic interactions. The SM is currently the most accurate and successful theory of the fundamental interactions of nature. The SM has however several important shortcomings and scientists are putting great effort in the search of new physics Beyond the Standard Model (BSM). Our ability to test the SM as accurately as possible is crucial for detecting possible deviations representing BSM signals and therefore the discovery of new particles and interactions.
Scattering amplitudes are essential ingredients for theoretical predictions for high-energy experiments. They are related to the probability of interactions between fundamental particles. They are computed in perturbation theory and next-to-next-to-leading-order (NNLO) accuracy is typically needed to match the one of the experimental data. This involves the calculation of complicated integrals, called Feynman integrals. The calculation of amplitudes is especially difficult for processes with many external particles (or legs), which are however very common at the LHC, due to the high centre-of-mass energy of the collisions. Multi-particle scattering amplitudes are therefore a crucial objective of current high-energy phenomenology.
The main goal of this project is a breakthrough in our capability of computing complicated multi-loop quantities, focusing on two-loop corrections to five-point scattering amplitudes in QCD. This involves theoretical work for better understanding multi-loop integrals, the development of new computational tools and techniques capable of tackling these complex calculations, as well as the explicit computation of important multi-loop results relevant for modern phenomenology, such as two-loop five-point amplitudes.
The usage of finite fields is an important innovative aspect of this project. Finite fields are numerical fields with a finite number of elements that can be represented by machine-size integers. Operations over finite fields are therefore relatively fast but also exact and free of numerical errors. Multivariate reconstruction algorithms allow to infer full numerical or analytic results out of several numerical evaluations over finite fields. This allows to sidestep the appearance of large analytic expressions in intermediate stages, which are often the main bottleneck of amplitude calculations, since only the final result is reconstructed analytically. These methods are very general and have a huge range of possible applications in science, even outside physics.
The researcher has implemented several algorithms over finite fields, which are used as the building blocks of the calculation. These include the solution of dense and sparse linear systems, evaluation of polynomials and rational functions, linear fits, matrix multiplication, generalized unitarity cuts, and many more. He also implemented a framework allowing to combine these building blocks into a complete calculation, and functional reconstruction techniques for obtaining analytic results out of numerical calculations. This framework, called FiniteFlow, has then been published and open-sourced.
FiniteFlow has been used to implement two-loop multi-gluon amplitudes using integrand reduction and generalized unitarity. This implementation produced the first planar five-gluon results beyond the all-plus configuration, namely the single-minus case. This is the first analytic calculation of a two-loop five-point amplitude beyond the simplest case where all external particles have the same helicity. We obtained compact analytic results.
The researcher contributed to the calculation of the matter dependent contributions to the light-like cusp anomalous dimension, an important universal quantity in QCD, at four loops, making extensive usage of the tools implemented for the project.
The researcher and his collaborators then computed the non-planar contributions to the all-plus five-gluon amplitude. This is the first non-planar two-loop five-point amplitude ever computed in a non-supersymmetric theory.
The researcher and its collaborators then applied the integrand reduction method to helicity amplitudes computed via Feynman diagrams. This allowed to set up a framework capable of tackling a wider range of processes. We used this for producing numerical results for the planar two-loop helicity amplitudes for a W-boson plus four partons. This is the first calculation of two-loop five-point amplitudes with a massive external particle.
The researcher also contributed to the development of an alternative and competitive method for generating multi-loop helicity amplitudes. It is based on the existing method of decomposition in form factors and projectors, but it is drastically simplified taking into account information about the helicity of external states. The method is simple, general. It can be applied to any process in any QFT and it is suitable for multi-loop multi-scale processes.
FiniteFlow has been used to implement two-loop multi-gluon amplitudes using integrand reduction and generalized unitarity. This implementation produced the first planar five-gluon results beyond the all-plus configuration, namely the single-minus case. This is the first analytic calculation of a two-loop five-point amplitude beyond the simplest case where all external particles have the same helicity. We obtained compact analytic results.
The researcher contributed to the calculation of the matter dependent contributions to the light-like cusp anomalous dimension, an important universal quantity in QCD, at four loops, making extensive usage of the tools implemented for the project.
The researcher and his collaborators then computed the non-planar contributions to the all-plus five-gluon amplitude. This is the first non-planar two-loop five-point amplitude ever computed in a non-supersymmetric theory.
The researcher and its collaborators then applied the integrand reduction method to helicity amplitudes computed via Feynman diagrams. This allowed to set up a framework capable of tackling a wider range of processes. We used this for producing numerical results for the planar two-loop helicity amplitudes for a W-boson plus four partons. This is the first calculation of two-loop five-point amplitudes with a massive external particle.
The researcher also contributed to the development of an alternative and competitive method for generating multi-loop helicity amplitudes. It is based on the existing method of decomposition in form factors and projectors, but it is drastically simplified taking into account information about the helicity of external states. The method is simple, general. It can be applied to any process in any QFT and it is suitable for multi-loop multi-scale processes.
The project pushed the state of the art of what is possible in multi-loop calculations. In particular, the researcher and its collaborators published results for planar two-loop five-parton amplitudes, the first analytic results for non-planar five-point amplitudes, amplitudes involving a W boson and 4 partons and the matter-dependent contributions to the cusp anomalous dimension. These also had an important impact on the scientific community of the field, which is gradually adopting the methods developed and published by the author. The researcher has published and open-sourced, as part of the project, the computer program FiniteFlow. This program implements very general methods over finite fields and has a wide range of possible applications. Since its publication, it has been very well received and employed by several groups for obtaining cutting-edge results, including the very first phenomenological studies involving two-loop five-point amplitudes.
No website has been developed for the whole project. The now public code FiniteFlow, developed as part of the project, is available online at the URL https://github.com/peraro/finiteflow. More examples and tools using FiniteFlow are available at https://github.com/peraro.
No website has been developed for the whole project. The now public code FiniteFlow, developed as part of the project, is available online at the URL https://github.com/peraro/finiteflow. More examples and tools using FiniteFlow are available at https://github.com/peraro.