Numerical approaches for two-loop amplitudes

• Applicant: Dr. Markus Christian Schulze
• Subject Area: Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields
• Term: from 2017 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 329845933

• Final Report Year: 2021

Final Report Abstract

We explored the prospects of evaluating Quaantum Field Thieory amplitudes with virtual loops using new numerical algorithms. Thiese amplitudes are needed for the precise description of the fundamental dynamics of elementary particles as observed in collisions at the Large Hadron Collider. Our purely numerical approach is interesting because it is orthogonal to conventional analytic solutions. Large expressions due to many diffeerent scales and complicated elliptic functions due to internal masses are typical show stoppers of these conventional approaches. In contrast, our numerical approach does not suffeer from these problems at all. Thiere are, however, new challenges related to numerical stability because the integrations are performed in multidimensional complex spaces with a very complicated singular surface structure. We developed a Monte-Carlo simulation that addresses these problems through a direct contour deformation in a multi-dimensional complex space. One-loop integrals as well as full amplitudes including tensor integrals can be solved with this algorithm at high efficciency. Higher-loop integrals constitute a challenge for this approach because of large numerical cancellations. Here, more research is required to understand these cancellations betteer. However, related research indicates that it is more efficcient if the integrals are analytically integrated over a one-dimensional subspace, which is then followed by a numerical integration similar to the one explored in this project.

Publications

• Phys. Rev. B 98 (2018) no. 16, 165125
  Zerf, Nikolai; Marquard, Peter; Boyack, Rufus & Maciejko, Joseph
  
• Phys. Rev. B 98 (2018) no.3, 035137
  Boyack, Rufus; Lin, Chien-Hung; Zerf, Nikolai; Rayyan, Ahmed & Maciejko, Joseph
  
• Phys. Rev. B 100 (2019) no. 13
  Ihrig, Bernhard; Zerf, Nikolai; Marquard, Peter; Herbut, Igor F. & Scherer, Michael M.
  
• Phys. Rev. B 100 (2019) no.23, 235130
  Zerf, Nikolai; Boyack, Rufus; Marquard, Peter; Gracey, John A. & Maciejko, Joseph