Although the early termination of the grant prevented us from reaching the final objective - the calculation of two-loop amplitudes for top-pair production in association with a jet at the LHC - the project nonetheless accomplished several significant objectives.
1) My collaborators and I proposed a novel method for the numerical evaluation of Feynman integrals. The key idea is to apply the recently-introduced framework of physics-informed deep learning to train neural networks that approximate the solutions to the differential equations governing the Feynman integrals. We developed and publicly released a proof-of-concept implementation, which we successfully tested on a range of advanced examples. With further development, this method could prove invaluable in cases where an analytic solution to these differential equations is beyond reach.
2) We also constructed the necessary bases of Feynman integrals required for calculating the targeted amplitudes in the leading-color approximation, and computed the differential equations they satisfy. This progress put us in the position to numerically evaluate these crucial building blocks, and to initiate the computation of the actual amplitudes and the development of a novel method for their expansion in a basis of special functions, along with their efficient numerical evaluation.
3) In addition to the Feynman integrals mentioned in point 2, we derived bases and differential equations for another set of Feynman integrals. This result paves the way for calculating the two-loop QCD amplitudes for another crucial LHC process: the production of two heavy vector bosons in association with a jet.
4) We completed the computation of the two-loop QCD amplitudes for yet another interesting scattering process within the LHC's physics program: the production of a W-boson in association with two photons. These amplitudes, which exhibit algebraic complexity similar to that of top-pair production in association with a photon, allowed us to stress-test our methodology and develop new techniques that we are now applying to the project's main objective.
In addition to these achievements, I also established the foundation for several further objectives, which will be completed in the months following the conclusion of the grant.
