Periodic Reporting for period 1 - FFHiggsTop (High-precision multi-leg Higgs and top physics with finite fields)
Reporting period: 2022-09-01 to 2025-02-28
Experiments at particle colliders are the most effective way of probing the most fundamental constituents of nature - the elementary particles - and the way they interact with each other. The Large Hadron Collider (LHC) at CERN is the most powerful collider ever built. Data collected by the LHC, including future runs and upgrades, will have an unprecedented reach and accuracy. Theoretical predictions with a similar level of precision, often at the per-cent level, are thus needed to fully exploit its potential. The Standard Model (SM), which describes the strong interactions of quantum chromodynamics (QCD) and the electroweak (EW) interactions, currently provides the most accurate description of the observed fundamental particles and interactions. It has been validated by a huge number of experiments and observations but it is an incomplete theory, which lacks a high-energy description of gravity, dark energy and dark matter, and has other undesirable features when extrapolated at high energies. The search for new physics is thus one of the main objectives of modern high-energy experiments. High-precision predictions in the SM are crucial for detecting signals of new particles and interactions, which might be hidden by the large SM background. In particular, new data collected in Run 3 of LHC will allow to perform high-precision tests involving multi-particle final states as well as massive particles, which are of great importance e.g. for jet physics, top physics and Higgs physics. This data needs to be compared to similarly accurate theoretical predictions.
At the core of theoretical predictions in collision experiments are scattering amplitudes. They encode the probability of the interactions and are computed perturbatively in the coupling constants. In order to match the expected experimental precision, we often need to reach next-to-next-to-leading order (NNLO) in perturbation theory. This requires the calculation of complex integrals over the momenta of unresolved virtual particles. These integrals are known as loop integrals or Feynman integrals. NNLO predictions usually require two-loop amplitudes, which involve the integration over two independent loop momenta. Computing these integrals and combining them in scattering amplitudes is however beyond the state of the art for complex multi-leg processes involving internal and external masses.
The main objective of this project is a breakthrough in our ability to make high-precision predictions for collider experiments for processes involving massive internal and external particles, as well as multi- particle final states. This relies on new methods for scattering amplitudes and Feynman integrals, as well as finite fields and functional reconstruction techniques. The objectives include the development of a highly automated framework for the semi-numerical calculation of multi-loop amplitudes which can be systematically used for theoretical predictions for complex scattering processes at the LHC of high relevance for top physics and Higgs physics. These include top pair production in association with a Higgs boson and top pair production in association with a vector boson at NNLO. These will lead us to a deeper understanding of the top quark and the symmetry breaking mechanism of the SM.
The project is developing ground-breaking new methods for theoretical predictions in high-energy physics for the efficient reduction of loop integrals to independent master integrals, the analytic or semi-numerical integration of Feynman integrals, and will make NNLO phenomenological predictions involving top quark and massive bosons (Higgs, W and Z).
At the core of theoretical predictions in collision experiments are scattering amplitudes. They encode the probability of the interactions and are computed perturbatively in the coupling constants. In order to match the expected experimental precision, we often need to reach next-to-next-to-leading order (NNLO) in perturbation theory. This requires the calculation of complex integrals over the momenta of unresolved virtual particles. These integrals are known as loop integrals or Feynman integrals. NNLO predictions usually require two-loop amplitudes, which involve the integration over two independent loop momenta. Computing these integrals and combining them in scattering amplitudes is however beyond the state of the art for complex multi-leg processes involving internal and external masses.
The main objective of this project is a breakthrough in our ability to make high-precision predictions for collider experiments for processes involving massive internal and external particles, as well as multi- particle final states. This relies on new methods for scattering amplitudes and Feynman integrals, as well as finite fields and functional reconstruction techniques. The objectives include the development of a highly automated framework for the semi-numerical calculation of multi-loop amplitudes which can be systematically used for theoretical predictions for complex scattering processes at the LHC of high relevance for top physics and Higgs physics. These include top pair production in association with a Higgs boson and top pair production in association with a vector boson at NNLO. These will lead us to a deeper understanding of the top quark and the symmetry breaking mechanism of the SM.
The project is developing ground-breaking new methods for theoretical predictions in high-energy physics for the efficient reduction of loop integrals to independent master integrals, the analytic or semi-numerical integration of Feynman integrals, and will make NNLO phenomenological predictions involving top quark and massive bosons (Higgs, W and Z).
An important part of our research is the development of novel techniques for the study and decomposition of loop integrals into a linear combination of an independent set of them, called master integrals. This is an essential step of amplitude calculations, but also a major bottleneck. We developed and implemented a new reduction algorithm that combines traditional identities with transverse integration identities, which map some of the integral sectors to simpler integral families, yielding substantial improvements in performance, especially on cutting edge two-loop examples with many external particles and massive states.
We elaborated on the novel approach of intersection theory, which defines scalar products between Feynman integrals (and other functions with suitable integral representations) called intersection numbers. This turns the reduction to master integrals into a projection. We developed a method that, for the first time, is purely rational operations and doe not need integral transformations or changes of bases and implemented it over finite fields. For this purpose, we introduced the method of polynomial expansions. This was later combined with the relative twisted cohomology and companion matrices, reformulating complex polynomial operations into simple matrix multiplication and proving major performance improvements to this technique.
We computed several two- and three-loop Feynman integrals as well as two-loop amplitudes with massive internal and external states, pushing the state of the art of theoretical predictions.
This work has been presented in a number of seminars, talks and conferences and workshops and scientific publications in high-impact journals.
We also had important breakthroughs in the study of differential equations and the evaluation of master integrals contributing to high-precision predictions of processes involving the production of top quarks, Higgs, W and Z bosons. We also developed new computer codes for efficient reduction of Feynman integrals and reconstruction of their differential equations. This progress is expected to be published in the near future.
We elaborated on the novel approach of intersection theory, which defines scalar products between Feynman integrals (and other functions with suitable integral representations) called intersection numbers. This turns the reduction to master integrals into a projection. We developed a method that, for the first time, is purely rational operations and doe not need integral transformations or changes of bases and implemented it over finite fields. For this purpose, we introduced the method of polynomial expansions. This was later combined with the relative twisted cohomology and companion matrices, reformulating complex polynomial operations into simple matrix multiplication and proving major performance improvements to this technique.
We computed several two- and three-loop Feynman integrals as well as two-loop amplitudes with massive internal and external states, pushing the state of the art of theoretical predictions.
This work has been presented in a number of seminars, talks and conferences and workshops and scientific publications in high-impact journals.
We also had important breakthroughs in the study of differential equations and the evaluation of master integrals contributing to high-precision predictions of processes involving the production of top quarks, Higgs, W and Z bosons. We also developed new computer codes for efficient reduction of Feynman integrals and reconstruction of their differential equations. This progress is expected to be published in the near future.
Among the results beyond the state-of-the-art that we produced and are already published or under peer review (all also accessible on the arXiv) are:
- a new approach to integral reduction that uses transverse integration identities and improves the efficiency of existing methods
- new results for two-loop integrals with six external particles
- the development of new rational algorithms for computing intersection numbers more efficiently, which involve polynomial expansions, relative cohomology and companion matrices
- two-loop QCD correction to a vector boson plus a jet including axial contributions
- phenomenological predictions for isolated photon production in association with a jet pair at NNLO in QCD
- new results for loop integrals for top pair production and a jet
- new results for one-loop QCD Corrections to W boson + top-pair production at higher orders in the dimensional regulator
- new Feynman integrals and amplitudes for NNLO QCD corrections to di-photon and di-jets production at hadronic colliders
- new three-loop four-point integrals with two off-shell external particles, relevant for di-boson production
Further cutting-edge advancements have been made in the development of integral reduction techniques, as well as in the study of differential equations and the evaluation of Feynman integrals, specifically for processes involving the top quark, Higgs boson, and other massive bosons in the Standard Model. These will soon yield major publications.
- a new approach to integral reduction that uses transverse integration identities and improves the efficiency of existing methods
- new results for two-loop integrals with six external particles
- the development of new rational algorithms for computing intersection numbers more efficiently, which involve polynomial expansions, relative cohomology and companion matrices
- two-loop QCD correction to a vector boson plus a jet including axial contributions
- phenomenological predictions for isolated photon production in association with a jet pair at NNLO in QCD
- new results for loop integrals for top pair production and a jet
- new results for one-loop QCD Corrections to W boson + top-pair production at higher orders in the dimensional regulator
- new Feynman integrals and amplitudes for NNLO QCD corrections to di-photon and di-jets production at hadronic colliders
- new three-loop four-point integrals with two off-shell external particles, relevant for di-boson production
Further cutting-edge advancements have been made in the development of integral reduction techniques, as well as in the study of differential equations and the evaluation of Feynman integrals, specifically for processes involving the top quark, Higgs boson, and other massive bosons in the Standard Model. These will soon yield major publications.