Numerische Rechnungen in höherer Ordnung
Zusammenfassung der Projektergebnisse
The central theme of this project was centred around the question if higher-order calculation can be performed numerically by Monte Carlo methods. Quantum field theory tells us, that we should integrated over each unconstrained loop momentum. In a numerical approach we would like to this in four space-time dimensions, e.g. without dimensional regularisation. This requires that the integrand is locally integrable and singularities have been subtracted out. While this is straightforward for ultraviolet singularities, the situation for infrared singularities is more complicated. Part of this complication is related to causality: We cannot simply integrated over all real values of the loop momenta, but have to implement correctly Feynman’s iδ-prescription by escaping into the complex domain. In this project we investigated the implementation of causality within the loop-tree duality approach, which reduces the numerical integration from four space-time dimensions to three space dimensions. One trivial integration is performed analytically with the help of the residue theorem. It was known, that the correct implementation of causality within the loop-tree duality requires a modification of Feynman’s iδ-prescription. A correct, but rather complicated and clumsy formula for the correct prescription was known for two-loops and beyond. Through the results of this research project we could show, that the modified iδprescription can be given in a much simpler formula, directly related to the graph and the cuts under consideration. This is an important result and has been published in Physical Review Letters. There were more results from this research project: The appearance of higher poles within the loop-tree duality approach complicates the extraction of the residues. This requires that we carry out a differentiation. We could show that higher poles are absent, if ultraviolet subtraction terms are chosen in the on-shell scheme. This is also a significant simplification. The numerical approach is intended for processes with a high number of external particles. It offers a good scaling behaviour (required CPU time versus number of external particles), if all ingredients can be computed in polynomial time. We showed that this can be arranged by demonstrating that the integrand of a loop amplitude (i.e. the sum of all individual graphs) can be computed efficiently with the help of recurrence relations. We worked out non-trivial combinatorial factors explicitly up to three loops.
Projektbezogene Publikationen (Auswahl)
- Causality and loop-tree duality at higher loops, Phys. Rev. Lett. 122, (2019), Erratum: 111603, Phys. Rev. Lett. 123, (2019), 059902
R. Runkel, Z. Ször, J. P. Vesga and S. Weinzierl
(Siehe online unter https://doi.org/10.1103/PhysRevLett.123.059902) - On the vanishing of certain cuts or residues of loop integrals with higher powers of the propagators, Phys. Rev. D99, (2019), 096023
R. Baumeister, D. Mediger, J. Pecovnik and S. Weinzierl
(Siehe online unter https://doi.org/10.1103/PhysRevD.99.096023) - Integrands of loop amplitudes, Phys. Rev. D101, (2020), 116014
R. Runkel, Z. Ször, J. P. Vesga and S. Weinzierl
(Siehe online unter https://doi.org/10.1103/PhysRevD.101.116014)
